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stringlengths 59
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\begin{definition}[Definition:Interval/Notation/Unbounded Intervals]
In Wirth interval notation, unbounded intervals of an ordered set $\struct {S, \preccurlyeq}$ are written as follows:
{{begin-eqn}}
{{eqn | l = \hointr a \to
| o = :=
| r = \set {x \in S: a \preccurlyeq x}
}}
{{eqn | l = \hointl \gets a
| o = :=
| r = \set {x \in S: x \preccurlyeq a}
}}
{{eqn | l = \openint a \to
| o = :=
| r = \set {x \in S: a \prec x}
}}
{{eqn | l = \openint \gets a
| o = :=
| r = \set {x \in S: x \prec a}
}}
{{eqn | l = \openint \gets \to
| o = :=
| r = \set {x \in S} = S
}}
{{end-eqn}}
Category:Definitions/Intervals
\end{definition}
|
ProofWiki
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ESE 224 – Signal and Information Processing
Lab 5: Sampling
Signals exist in continuous time but it is not unusual for us to process them in discrete time. When we work in discrete time we say that we are doing discrete signal processing, something that is convenient due to the relative ease and lower cost of using computers to manipulate signals. When we use discrete time representations of continuous time signals we need to implement processes to move back and forth between continuous and discrete time. The process of obtaining a discrete time signal from a continuous time signal is called sampling. The process of recovering a continuous time signal from its discrete time samples is called signal reconstruction or interpolation.
• Click here to download the assignment.
• Due on Feb 21 by 5pm.
Sampling without aliasing
In the first problem of the lab $\textbf{(1.1)}$, you are asked to answer why sampling entails no loss of information for bandlimited signals with bandwith $W$; you are also asked to explain how the signal $x(t)$ can be recovered from the sampled signal $x_s(t)$.
For any sampling frequency $f_s \geq W$, since $X(f) = 0$ for all $f \notin [-W/2,W/2]$, $X[f-kf_s] = 0$ for all $f \notin [kf_s-f_s/2,kf_s+f_s/2]$. This implies $X[f-kf_s]$ and $X[f-lf_s]$ will not overlap for any $k \neq l$. For this reason, upon sampling the spectrum $\sum_{k=-\infty}^\infty X[f-kf_s]$ is periodized but not aliased and sampling entails no loss of information.
In order to recover $x(t)$ perfectly, we low pass filter the the Dirac train representation $x_\delta(t)$ with frequency threshold $\pm f_s/2$, that is,
\tilde{x} = x_\delta * [f_s \mbox{sinc}(\pi f_s t)]\ .
When signals are not bandlimited, loss of information is inevitable. Indeed, when non-bandlimited signals are sampled the spectral content in frequencies above $f_s/2$ and below $-f_s/2$ becomes aliased as exemplified by the red curve in Figure 1. In problem $\textbf{(1.2)}$, you are asked to explain how it is possible to avoid aliasing through judicious use of a low-pass filter.
Figure 1: Aliasing.
To avoid distortion we preprocess $x$ with a low pass filter, i.e., we transform $x$ into a signal $x_{f_s}$ with spectrum $X_{f_s} = \ccalF(x_{f_s})$ where $X_{f_s}(f) = X(f)\sqcap_{f_s}(f)$. This can be implemented as a convolution in the time domain,
x_{f_s} = x * f_s \mbox{sinc}(\pi f_s t)\ .
The signal $x_{f_s}$ has bandwith $f_s$ and can be sampled without aliasing. In other words, frequency components between $-f_s/2$ and $f_s/2$ are retained without distortion.
Reconstruction with arbitrary pulse trains
While it is mathematically possible to reconstruct $x(t)$ from $x_\delta(t)$, it is physically implausible to generate a Dirac train. However, we can approximate $\delta(t)$ with narrow pulses $p(t)$. In problem $\textbf{(1.3)}$, you are asked to derive the conditions on $p(t)$ that minimize signal distortion.
To start, the spectra of the actual modulated train pulse using $\delta(t)$ and the approximated modulated train pulse using $p(t)$ can be denoted
\begin{align}
X_\delta(f) &= \sum_{k=-\infty}^{\infty} X(f-kf_s), \\
X_p(f) &= P(f)X_\delta(f) = P(f)\sum_{k=-\infty}^{\infty}X(f-kf_s),
\end{align}
where $P(f)$ is the spectrum of $p(t)$. The reconstruction method described in $\textbf{(1.1)}$ recovers the signal by low pass filtering the sampled signal in the frequency domain. Using this approach, the spectrum $\tilde{X}_\delta (f)$ of the reconstructed signal using $X_\delta(f)$ and the spectrum $\tilde{X}_p(f)$ of the reconstructed signal using $X_p(f)$ can be computed as
\tilde{X}_\delta(f) &= \sqcap_{f_s}(f)\sum_{k=-\infty}^{\infty} X(f-kf_s) = \sqcap_{f_s}(f)X(f), \\
\tilde{X}_p(f) &= \sqcap_{f_s}(f)P(f)\sum_{k=-\infty}^{\infty} X(f-kf_s) = P(f)\sqcap_{f_s}(f)X(f),
where we use the fact that the low pass filter eliminates all frequencies outside of $[-f_s/2,f_s/2]$. Notice that $\tilde{X}_\delta(f)$ coincides with $\tilde{X}_p(f)$, i.e., no distortion results from using $p(t)$ to approximate $\delta(t)$, when the spectrum of $p(t)$ satisfies
P(f) = 1,\mbox{ for all } f \in [-f_s/2,f_s/2].
In particular, a pulse satisfying this property is $p(t) = f_s\mbox{sinc}(\pi f_s t)$ with $P(f) = \sqcap_{f_s}(f)$. Interestingly, this pulse is not that narrow.
Subsampling
In problem $\textbf{(2.1)}$, you are asked to derive a theorem relating the spectrum of the discrete time signal $x$ and its subsampled version $x_\delta$. Starting from equation (11) of the handout, note that it is equivalent to write
x_\delta(n) = x(n) \sum_{m=-\infty}^{\infty} \delta\left(n – m \frac{\tau}{T_s}\right)
which is $x(n)$ multiplied with a train of discrete deltas with spacing time $\tau$ and sampling time $T_s$. Suppose the sampling time is $\tau$; the train of discrete deltas becomes $\sum_{m=-\infty}^{\infty} \delta(n-m)$ which is a discrete time constant, and its DTFT is a Dirac train with spacing $\nu = 1/\tau$ (see slides 30-32 in the sampling lecture). The following equation in the DTFT derivation will be used later,
\tau \sum_{n=-\infty}^{\infty} e^{-j2\pi f n \tau} = \sum_{k=-\infty}^{\infty} \delta(f-k\nu)\ .
Back to our setting where the sampling time is $T_s$, the train of discrete deltas is a discrete signal with values either $0$ or $1$, depending on whether $n$ is a multiple of $\tau/T_s$. Its DTFT can be computed as
X_c(f) = T_s \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} \delta(n – m\tau/T_s) e^{-j 2 \pi f n T_s}\ .
We may define $n' = n T_s/\tau$ and utilize the result from (9) in evaluating (10),
X_c(f) = T_s \sum_{n'=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} \delta(n' – m) e^{-j2 \pi f n' \tau} &= T_s \sum_{n'=-\infty}^{\infty} e^{-j2 \pi f n' \tau} \\
&= \frac{T_s}{\tau} \sum_{k=-\infty}^{\infty} \delta(f-k\nu)\ .
The multiplication in the time domain in (8) implies that the spectrum $X_\delta(f)$ of $x_\delta(n)$ can be written as the convolution of $X(f)$ and the spectrum of the train of discrete deltas. Therefore, we may write
X_\delta(f) = X(f) * \frac{T_s}{\tau} \sum_{k=-\infty}^{\infty} \delta(f-k\nu) = \frac{T_s}{\tau} \sum_{k=0}^{\tau/T_s – 1} X\left(f-\frac{k}{\tau}\right)\ .
In the above derivation, we encountered two entities that appear to be similar — the train of discrete deltas $\sum_{m=-\infty}^{\infty} \delta(n – m \tau/T_s)$ and the Dirac train $\sum_{k=-\infty}^{\infty} \delta(f – k \nu)$ — and here we emphasize their difference. The train of discrete deltas $x(n) = \sum_{m=-\infty}^{\infty} \delta(n – m \tau/T_s)$ is a discrete signal and $x(n)=1$ whenever $n$ is a multiple of $\tau/T_s$ and zero otherwise. The Dirac train $\sum_{k=-\infty}^{\infty} \delta(f – k \nu)$, on the other hand, is a continuous signal and for each $k \in \mbZ$ there is a Dirac delta function (which is a continuous signal itself) centered at $k \nu$. When $\tau = T_s$, $x(n) = \sum_{m=-\infty}^{\infty} \delta(n – m \tau/T_s) = \sum_{m=-\infty}^{\infty} \delta(n-m)$ is also called a discrete time constant.
This spectrum periodization result is verified in problems $\textbf{(2.2)}$ and $\textbf{(2.3)}$. The gaussian pulse sampled at $f_s = 40$kHz with duration $T=2$ and parameters $\mu=1$ and $\sigma=0.1$ is shown on the top of Figure 2. Its subsampled version, with subsampling frequency $\nu = 4$kHz, is shown on the bottom of the same figure.
Figure 2: Original and subsampled gaussian pulse with $\sigma=0.1$.
There is not much of a difference between these two signals in the time domain, but if we looks at their DFTs as shown in Figure 3, we see that the DFT of $x_\delta$ is equal to the DFT of $x$ modulated by a Dirac train with spacing $\nu$.
Figure 3: DFTs of original and subsampled gaussian pulse with $\sigma=0.1$.
Because the DFT of $x$ is so narrow, there is no aliasing in Figure 3. But if we decrease $\sigma$ to $10^{-4}$, the DFT of the pulse becomes wider and we can see aliasing of consecutive periods of the DFT of $x_\delta$ as in Figure 4.
Figure 4: DFTs of original and subsampled gaussian pulse with $\sigma=10^{-4}$.
To avoid aliasing, in problems $\textbf{(2.4)}$ and $\textbf{(2.5)}$ we add a prefiltering step before subsampling. The DFT of the gaussian pulse with $\sigma=10^{-4}$, prefiltered to eliminate frequency components outside of the range $[-f_s/2,f_s/2]$, and subsequently subsampled with $\nu=4$kHz, is shown in Figure 5.
Figure 5: DFTs of original and subsampled gaussian pulse with $\sigma=10^{-4}$ (after prefiltering).
In problem $\textbf{(2.6)}$, you are asked to reconstruct the pulses you subsampled. We start by reconstructing the subsampled gaussian pulse with $\sigma=0.1$. The original and the reconstructed signal are shown in Figure 6. Notice that for this pulse subsampling is lossless as the original signal and its reconstruction have the same shape and energy.
Figure 6: Original and reconstructed gaussian pulse with $\sigma=0.1$.
Next, we reconstruct the pulse with $\sigma=10^{-4}$ that was subsampled without prefiltering. The original and reconstructed signals are shown in Figure 7.
Figure 7: Original and reconstructed gaussian pulse with $\sigma=10^{-4}$ (without prefiltering).
Notice that while the reconstructed pulse seems to have the same energy as the original pulse, it is not a faithful reconstruction because of aliasing. If we now reconstruct the subsampled version of this pulse that was previously prefiltered to avoid aliasing, the reconstructed signal is as in Figure 8.
Figure 8: Original and reconstructed gaussian pulse with $\sigma=10^{-4}$ (with prefiltering).
The reconstructed signal clearly has less energy than the original signal, but because there was no aliasing its shape is slightly more faithful to the shape of the original gaussian pulse.
Code links
The classes provided for Lab 5 can be found in the following folder: ESE224_Lab5_provided.zip. This folder contains the following 3 files:
$\p{dft.py}$: The class $\p{dft}$ implements the discrete Fourier transform in $3$ different ways.
$\p{idft.py}$: The class $\p{idft}$ implements the inverse discrete Fourier transform in $2$ different ways.
$\p{gaussian\_pulse.py}$: The class $\p{gaussian\_pulse}$ generates a gaussian pulse with a specified duration, sampling frequency, mean and standard deviation.
The code used to generate the figures above can be downloaded from ESE224_LAB5.
Copyright © 2023 ESE 224 – Signal and Information Processing. All rights reserved. Theme Spacious by ThemeGrill. Powered by: WordPress.
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CommonCrawl
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# 1. Understanding Hash Functions
Hash functions are an essential component of hash tables. They take a key as input and return an index into the table. The purpose of a hash function is to distribute keys as uniformly as possible in the hash table to avoid collisions. A good hash function should also be fast to compute and consistent with the equality testing function.
One example of a hash function is the modulo operator. For integer keys, a common hash function is `H(K) = K mod M`, where `M` is the size of the hash table. This function returns the remainder when the key `K` is divided by `M`. This function is fast to compute and can be used for general integer keys when `M` is prime.
Other contexts where hash functions are used include password systems, message digest systems, and digital signature systems. In these applications, the probability of collisions must be very low, and hash functions with a large set of possible values are required.
Using a hash function is straightforward. Given a key, you apply the hash function to determine the index in the hash table where the key should be stored. For example, if we have a hash table with 10 slots and a key of 701, we can use the hash function `H(K) = K mod 10` to find the index. In this case, `701 mod 10 = 1`, so the key would be stored at index 1.
# 1.1. Definition and Purpose of Hash Functions
A hash function is a function that takes an input (usually a key) and returns an index into a hash table. The purpose of a hash function is to distribute keys as uniformly as possible in the hash table to minimize collisions. A good hash function should also be fast to compute and consistent with the equality testing function.
Hash functions are commonly used in data structures like hash tables to provide efficient access to data. By mapping keys to indices in the hash table, hash functions enable constant-time average-case insert and find operations. This makes hash tables a very useful and commonly used data structure.
# 1.2. Types of Hash Functions
There are various types of hash functions that can be used depending on the specific requirements of the application. Some common types include:
1. Modulo hash function: This type of hash function uses the modulo operator to compute the hash value. It is often used for integer keys and is simple to implement.
2. Cryptographic hash function: These hash functions are designed to be secure and resistant to various types of attacks. They are commonly used in password systems, message digest systems, and digital signature systems.
3. Custom hash function: In some cases, a custom hash function may be developed to meet specific requirements of the application. This can involve complex algorithms and techniques tailored to the specific data being hashed.
The choice of hash function depends on factors such as the type of keys, the distribution of key insert requests, and the size of the hash table. Designing a good hash function can be challenging as it requires balancing factors such as computation speed, uniform distribution, and collision resistance.
# 1.3. Properties of a Good Hash Function
A good hash function possesses several important properties:
1. Fast computation: The hash function should be computationally efficient to minimize the time required for hashing operations.
2. Uniform distribution: The hash function should distribute keys as uniformly as possible in the hash table to avoid collisions. Ideally, each key should have an equal chance of being mapped to any index in the table.
3. Consistency with equality testing: If two keys are equal, the hash function should map them to the same table location. This ensures that the fundamental hash table operations, such as find and delete, work correctly.
4. Large set of possible values: In certain applications, such as password systems and message digest systems, the hash function should have a large set of possible values to minimize the probability of collisions.
Designing a hash function that satisfies all these properties can be challenging and depends on factors such as the type of keys and the size of the hash table. It often requires careful consideration and testing to ensure optimal performance.
# 2. Introduction to Hash Tables
Hash tables are a widely used data structure that allows for efficient storage and retrieval of key-value pairs. They are particularly useful when quick access to data is required, such as in search algorithms or database systems.
A hash table consists of two main components: an array and a hash function. The array is used to store the key-value pairs, while the hash function is responsible for mapping keys to indices in the array.
When a key-value pair is inserted into a hash table, the hash function is applied to the key to determine the index where the pair should be stored. This process is known as hashing. The value is then stored at the computed index in the array.
To retrieve a value from a hash table, the hash function is again applied to the key to determine the index where the value is stored. The value can then be accessed directly from the array at the computed index.
The key advantage of hash tables is their constant-time complexity for both insertion and retrieval operations on average. This means that the time required to perform these operations does not depend on the size of the hash table.
However, hash tables can also have some disadvantages. One potential issue is collisions, which occur when two different keys are mapped to the same index in the array. Collisions can be resolved using various techniques, such as linear probing or separate chaining.
# 2.1. Definition and Components of a Hash Table
A hash table is a data structure that allows for efficient storage and retrieval of key-value pairs. It consists of an array and a hash function.
The array is used to store the key-value pairs. Each element of the array is called a "bucket" or a "slot". The number of buckets in the array is typically determined by the size of the hash table.
The hash function is responsible for mapping keys to indices in the array. It takes a key as input and computes a hash code, which is an integer value. The hash code is then used to determine the index where the key-value pair should be stored in the array.
The goal of the hash function is to distribute the keys as uniformly as possible across the array. This helps to minimize collisions and ensure efficient access to the values.
When a key-value pair is inserted into the hash table, the hash function is applied to the key to compute the index. The value is then stored at the computed index in the array.
To retrieve a value from the hash table, the hash function is again applied to the key to compute the index. The value can then be accessed directly from the array at the computed index.
In summary, a hash table consists of an array and a hash function. The array is used to store the key-value pairs, and the hash function is responsible for mapping keys to indices in the array. This allows for efficient storage and retrieval of values based on their keys.
# 2.2. Implementing a Hash Table
Implementing a hash table involves several key steps. Let's walk through the process.
1. Choose the size of the hash table: The size of the hash table is typically determined by the expected number of key-value pairs that will be stored. It's important to choose a size that allows for efficient storage and retrieval of values. A common approach is to use a prime number as the size of the hash table, as this helps to distribute the keys more evenly.
2. Define the hash function: The hash function is responsible for mapping keys to indices in the array. It takes a key as input and computes a hash code, which is an integer value. The hash code is then used to determine the index where the key-value pair should be stored in the array. The hash function should be designed to distribute the keys as uniformly as possible across the array to minimize collisions.
3. Initialize the array: Create an array with the chosen size. Each element of the array represents a bucket or a slot where a key-value pair can be stored.
4. Insert key-value pairs: To insert a key-value pair into the hash table, apply the hash function to the key to compute the index. If the index is empty, store the key-value pair at that index. If the index is not empty, handle the collision according to the chosen collision resolution strategy (which we will cover in the next section).
5. Retrieve values: To retrieve a value from the hash table, apply the hash function to the key to compute the index. Access the value directly from the array at the computed index.
6. Handle collisions: Collisions occur when multiple keys map to the same index in the array. There are several strategies for handling collisions, including linear probing, double hashing, random hashing, and separate chaining. Each strategy has its own advantages and disadvantages, and the choice of strategy depends on the specific requirements of the application.
By following these steps, you can implement a hash table that allows for efficient storage and retrieval of key-value pairs. The choice of hash function and collision resolution strategy is crucial in ensuring the performance and effectiveness of the hash table.
# 2.3. Advantages and Disadvantages of Hash Tables
Hash tables offer several advantages and disadvantages that make them suitable for certain applications.
Advantages:
- Fast access: Hash tables provide constant-time access to values based on their keys. This makes them ideal for applications that require quick retrieval of data.
- Efficient storage: Hash tables only require a small amount of memory to store key-value pairs. This makes them space-efficient compared to other data structures.
- Flexible key types: Hash tables can handle a wide range of key types, including strings, integers, and objects. This flexibility allows for versatile use in various applications.
Disadvantages:
- Collision resolution: Collisions can occur when two or more keys map to the same index in the array. This can lead to performance issues and slower access times. Choosing an appropriate collision resolution strategy is crucial to mitigate this problem.
- Limited ordering: Hash tables do not maintain the order of insertion or the order of keys. If the order of elements is important, a different data structure may be more suitable.
- Memory usage: Hash tables require additional memory to store the array and handle collisions. In some cases, this can lead to increased memory usage compared to other data structures.
Despite these disadvantages, hash tables are widely used and offer efficient storage and retrieval of key-value pairs. By carefully choosing the hash function and collision resolution strategy, the performance of a hash table can be optimized for specific applications.
# 3. Collision Resolution
Collision resolution is an important concept in hash tables. It refers to the process of handling collisions that occur when two or more keys map to the same index in the array.
There are several collision resolution methods, each with its own advantages and disadvantages. In this section, we will explore different collision resolution methods and compare them to understand their strengths and weaknesses.
# 3.1. Definition and Types of Collision Resolution
Collision resolution is the process of resolving collisions that occur when two or more keys hash to the same index in a hash table.
There are several types of collision resolution methods, including:
- Linear probing
- Double hashing
- Random hashing
- Separate chaining
Each method has its own approach to resolving collisions and offers different trade-offs in terms of performance and space efficiency.
# 3.2. Linear Probing as a Collision Resolution Method
Linear probing is a collision resolution method that handles collisions by sequentially searching for the next available slot in the hash table.
When a collision occurs, linear probing checks the next slot in the array. If it is empty, the key-value pair is inserted into that slot. If the next slot is also occupied, linear probing continues to search for the next available slot until an empty slot is found.
Linear probing has the advantage of simplicity and efficiency in terms of space usage. It allows for a compact representation of the hash table, as there is no need for additional data structures to store collided keys. However, linear probing can lead to clustering, where keys that hash to the same index tend to cluster together. This can result in degraded performance as the search time increases when a cluster forms.
# 3.3. Comparison with Other Methods
Linear probing is just one of several collision resolution methods available. Let's compare linear probing with other methods to understand their differences.
- Linear probing vs. double hashing: Double hashing uses a second hash function to determine the next slot to probe in case of a collision. This can help reduce clustering and improve performance compared to linear probing. However, double hashing requires additional computation for the second hash function, which can impact performance.
- Linear probing vs. random hashing: Random hashing uses a pseudo-random number generator to determine the next slot to probe in case of a collision. This can help distribute keys more evenly and reduce clustering compared to linear probing. However, random hashing requires additional computation for the random number generation, which can impact performance.
- Linear probing vs. separate chaining: Separate chaining handles collisions by storing collided keys in separate linked lists. This avoids clustering and provides constant-time access to collided keys. However, separate chaining requires additional memory to store the linked lists, which can impact space efficiency compared to linear probing.
Each collision resolution method has its own advantages and disadvantages, and the choice depends on the specific requirements of the application. Linear probing offers simplicity and efficiency in terms of space usage, but it can suffer from clustering.
# 4. Understanding Linear Probing
Linear probing is a collision resolution method used in hash tables. When a collision occurs, linear probing searches for the next available slot in a linear sequence until an empty slot is found. This means that if a key cannot be inserted in its original hash position, it will be placed in the next available slot in the sequence.
The main idea behind linear probing is to keep the load factor low and minimize clustering. Clustering occurs when keys are inserted into consecutive slots, causing longer probe sequences and slower performance. By spreading out the keys in the table, linear probing aims to reduce clustering and improve lookup efficiency.
To understand linear probing, let's look at the steps involved in the process:
1. Calculate the hash value of the key using a hash function.
2. Check if the slot corresponding to the hash value is empty. If it is, insert the key into that slot and the process is complete.
3. If the slot is not empty, probe the next slot in the sequence.
4. Repeat step 3 until an empty slot is found or the entire table has been probed.
5. If an empty slot is found, insert the key into that slot and the process is complete.
6. If the entire table has been probed and no empty slot is found, the table is considered full and resizing may be necessary.
The linear probing process continues until an empty slot is found or the entire table has been probed. This ensures that every key will eventually be inserted into the hash table, even if a collision occurs.
One important consideration when using linear probing is the order in which keys are probed. Linear probing uses a simple linear sequence, where the next slot to probe is determined by incrementing the current slot by one. This means that if a collision occurs at slot `i`, the next slot to probe will be `i+1`, then `i+2`, and so on.
The order of probing can have an impact on the performance of linear probing. If keys are inserted in a specific order that causes many collisions, clustering can occur and the probe sequence can become longer. To mitigate this, it is important to choose a good hash function that distributes keys evenly and minimizes collisions.
Let's say we have a hash table with 10 slots and the following keys: 5, 15, 25, 35, 45. We'll use a simple hash function that takes the key modulo the table size.
1. The hash value of 5 is 5 % 10 = 5. Slot 5 is empty, so we insert 5 into that slot.
2. The hash value of 15 is 15 % 10 = 5. Slot 5 is already occupied, so we probe the next slot, which is 6. Slot 6 is empty, so we insert 15 into that slot.
3. The hash value of 25 is 25 % 10 = 5. Slot 5 and slot 6 are already occupied, so we probe the next slot, which is 7. Slot 7 is empty, so we insert 25 into that slot.
4. The hash value of 35 is 35 % 10 = 5. Slot 5, slot 6, and slot 7 are already occupied, so we probe the next slot, which is 8. Slot 8 is empty, so we insert 35 into that slot.
5. The hash value of 45 is 45 % 10 = 5. Slot 5, slot 6, slot 7, and slot 8 are already occupied, so we probe the next slot, which is 9. Slot 9 is empty, so we insert 45 into that slot.
In this example, all keys are successfully inserted into the hash table using linear probing. The order of probing is determined by the linear sequence, which ensures that every key will eventually be inserted into an empty slot.
## Exercise
Consider a hash table with 10 slots and the following keys: 12, 22, 32, 42, 52. Use linear probing with a simple hash function that takes the key modulo the table size.
Insert the keys into the hash table using linear probing. Write down the final state of the hash table after all keys have been inserted.
### Solution
```
0:
1:
2:
3:
4:
5: 12
6: 22
7: 32
8: 42
9: 52
```
# 4.1. Definition and Purpose of Linear Probing
Linear probing is a collision resolution method used in hash tables. When a collision occurs, linear probing searches for the next available slot in a linear sequence until an empty slot is found. This means that if a key cannot be inserted in its original hash position, it will be placed in the next available slot in the sequence.
The purpose of linear probing is to minimize clustering and improve lookup efficiency. Clustering occurs when keys are inserted into consecutive slots, causing longer probe sequences and slower performance. By spreading out the keys in the table, linear probing aims to reduce clustering and improve the average case time complexity of lookups.
Linear probing offers simplicity and efficiency in terms of space usage. Unlike separate chaining, which requires additional memory to store linked lists, linear probing stores collided keys directly in the hash table. This can result in better space efficiency, especially when the load factor is low and the table is not very full.
However, linear probing can suffer from clustering when the load factor is high. Clustering occurs when keys are inserted into consecutive slots, causing longer probe sequences and slower performance. To mitigate clustering, it is important to choose a good hash function that distributes keys evenly and minimizes collisions.
Let's say we have a hash table with 10 slots and the following keys: 5, 15, 25, 35, 45. We'll use a simple hash function that takes the key modulo the table size.
1. The hash value of 5 is 5 % 10 = 5. Slot 5 is empty, so we insert 5 into that slot.
2. The hash value of 15 is 15 % 10 = 5. Slot 5 is already occupied, so we probe the next slot, which is 6. Slot 6 is empty, so we insert 15 into that slot.
3. The hash value of 25 is 25 % 10 = 5. Slot 5 and slot 6 are already occupied, so we probe the next slot, which is 7. Slot 7 is empty, so we insert 25 into that slot.
4. The hash value of 35 is 35 % 10 = 5. Slot 5, slot 6, and slot 7 are already occupied, so we probe the next slot, which is 8. Slot 8 is empty, so we insert 35 into that slot.
5. The hash value of 45 is 45 % 10 = 5. Slot 5, slot 6, slot 7, and slot 8 are already occupied, so we probe the next slot, which is 9. Slot 9 is empty, so we insert 45 into that slot.
In this example, all keys are successfully inserted into the hash table using linear probing. The order of probing is determined by the linear sequence, which ensures that every key will eventually be inserted into an empty slot.
## Exercise
Consider a hash table with 10 slots and the following keys: 12, 22, 32, 42, 52. Use linear probing with a simple hash function that takes the key modulo the table size.
Insert the keys into the hash table using linear probing. Write down the final state of the hash table after all keys have been inserted.
### Solution
```
0:
1:
2:
3:
4:
5: 12
6: 22
7: 32
8: 42
9: 52
```
# 4.2. Steps Involved in Linear Probing
Linear probing involves a series of steps to insert a key into a hash table when a collision occurs. Here are the steps involved in linear probing:
1. Compute the hash value of the key using a hash function. The hash function should distribute the keys evenly across the hash table.
2. Check if the slot corresponding to the hash value is empty. If it is empty, insert the key into that slot and the insertion is complete.
3. If the slot is already occupied, probe the next slot in the linear sequence. The next slot can be calculated by adding a fixed offset to the current slot index. The offset is usually 1, but it can be any positive integer.
4. Continue probing the next slot until an empty slot is found. If the end of the hash table is reached, wrap around to the beginning of the table and continue probing.
5. Once an empty slot is found, insert the key into that slot and the insertion is complete.
6. If the entire hash table is full and no empty slot is found, the insertion cannot be completed. In this case, the hash table needs to be resized or rehashed to accommodate more keys.
These steps ensure that every key is eventually inserted into an empty slot in the hash table using linear probing. The order of probing is determined by the linear sequence, which spreads out the keys and minimizes clustering.
# 4.3. Analyzing Time and Space Complexity
Linear probing has some advantages and disadvantages in terms of time and space complexity.
In terms of time complexity, the average case for successful find and insert operations is O(1). This means that on average, it takes constant time to find or insert a key into the hash table. However, in the worst case, the time complexity can be O(n), where n is the number of keys in the hash table. This occurs when there are many collisions and the linear probing sequence becomes long.
The space complexity of linear probing is O(n), where n is the number of keys in the hash table. This is because the hash table needs to allocate space for each key-value pair. However, the actual space used may be less than the allocated space if there are many empty slots in the hash table.
It is important to note that the performance of linear probing can be affected by the load factor, which is the ratio of the number of keys to the number of slots in the hash table. A high load factor can lead to more collisions and longer linear probing sequences, resulting in increased time complexity.
Overall, linear probing provides a simple and efficient way to resolve collisions in hash tables. However, it is important to carefully choose the hash function and handle the load factor to ensure optimal performance.
# 5. Load Factor and Its Impact on Linear Probing
The load factor is an important concept in hash tables, including those that use linear probing for collision resolution. It is defined as the ratio of the number of keys in the hash table to the number of slots available.
A high load factor means that the hash table is almost full, while a low load factor means that there are many empty slots. The load factor has a significant impact on the performance of linear probing.
When the load factor is low, there are fewer collisions and the linear probing sequence is short. This results in faster find and insert operations, as the desired slot is likely to be empty or close to empty.
On the other hand, when the load factor is high, there are more collisions and the linear probing sequence becomes longer. This increases the time complexity of find and insert operations, as more slots need to be checked before finding an empty or matching slot.
To ensure optimal performance, it is important to choose an appropriate load factor for a hash table that uses linear probing. A load factor of around 0.7 to 0.8 is often recommended, as it provides a good balance between space usage and performance.
# 5.1. Definition and Importance of Load Factor
The load factor of a hash table is defined as the ratio of the number of keys in the table to the total number of slots available. It is calculated using the formula:
$$\text{Load Factor} = \frac{\text{Number of keys}}{\text{Total number of slots}}$$
The load factor is an important metric because it determines how full the hash table is. A high load factor means that the hash table is almost full, while a low load factor means that there are many empty slots.
The load factor is important because it affects the performance of operations such as find and insert. When the load factor is low, there are fewer collisions and the linear probing sequence is short. This results in faster find and insert operations, as the desired slot is likely to be empty or close to empty.
On the other hand, when the load factor is high, there are more collisions and the linear probing sequence becomes longer. This increases the time complexity of find and insert operations, as more slots need to be checked before finding an empty or matching slot.
Choosing an appropriate load factor is crucial for ensuring optimal performance of a hash table that uses linear probing. A load factor of around 0.7 to 0.8 is often recommended, as it provides a good balance between space usage and performance.
# 5.2. How Load Factor Affects Linear Probing
The load factor has a significant impact on the performance of linear probing. As the load factor increases, the number of collisions also increases. This leads to longer linear probing sequences and slower find and insert operations.
When the load factor is low, there are fewer keys in the hash table compared to the number of slots available. This means that there are many empty slots, and the linear probing sequence is short. Finding an empty slot or a matching key is relatively fast in this case.
However, as the load factor increases, the number of keys in the hash table approaches the number of slots available. This increases the likelihood of collisions, where two or more keys hash to the same slot. When a collision occurs, linear probing searches for the next available slot by incrementing the index and wrapping around to the beginning of the table if necessary.
With a high load factor, the linear probing sequence becomes longer, as more slots need to be checked before finding an empty or matching slot. This increases the time complexity of find and insert operations, as the search becomes more like an exhaustive search of the entire table.
Therefore, it is important to choose an appropriate load factor to balance space usage and performance. A load factor of around 0.7 to 0.8 is often recommended, as it provides a good trade-off between the number of keys and the number of slots in the hash table.
# 5.3. Choosing an Appropriate Load Factor
Choosing an appropriate load factor is crucial for the performance of a hash table that uses linear probing. The load factor determines the ratio of the number of keys in the hash table to the number of slots available.
A load factor that is too low can result in wasted space, as there are many empty slots in the hash table. This can lead to inefficient memory usage and slower find and insert operations, as the linear probing sequence is short.
On the other hand, a load factor that is too high can result in a high number of collisions. This increases the length of the linear probing sequence, making find and insert operations slower. In extreme cases, the hash table can become completely full, resulting in failed insertions.
A commonly recommended load factor for hash tables that use linear probing is around 0.7 to 0.8. This means that the number of keys in the hash table should be around 70% to 80% of the number of slots available. This load factor provides a good balance between space usage and performance.
However, the optimal load factor can vary depending on the specific use case and the characteristics of the data being stored. It is important to consider factors such as the expected number of keys, the expected distribution of the keys, and the memory constraints when choosing an appropriate load factor.
# 6. Practical Examples of Linear Probing
6.1. Implementing Linear Probing in Java
Java provides built-in support for hash tables through the `HashMap` class. By default, `HashMap` uses chaining as the collision resolution method. However, we can implement linear probing by extending the `HashMap` class and overriding the necessary methods.
Here is an example implementation of a linear probing hash table in Java:
```java
import java.util.Arrays;
public class LinearProbingHashMap<K, V> extends HashMap<K, V> {
private int size;
private K[] keys;
private V[] values;
public LinearProbingHashMap() {
size = 0;
keys = (K[]) new Object[16];
values = (V[]) new Object[16];
}
public void put(K key, V value) {
int index = getIndex(key);
while (keys[index] != null) {
if (keys[index].equals(key)) {
values[index] = value;
return;
}
index = (index + 1) % keys.length;
}
keys[index] = key;
values[index] = value;
size++;
}
public V get(K key) {
int index = getIndex(key);
while (keys[index] != null) {
if (keys[index].equals(key)) {
return values[index];
}
index = (index + 1) % keys.length;
}
return null;
}
private int getIndex(K key) {
int hashCode = key.hashCode();
return (hashCode & 0x7fffffff) % keys.length;
}
// Other methods such as remove, size, etc.
@Override
public String toString() {
return "LinearProbingHashMap{" +
"size=" + size +
", keys=" + Arrays.toString(keys) +
", values=" + Arrays.toString(values) +
'}';
}
}
```
In this example, we use arrays to store the keys and values. When a collision occurs, we increment the index by 1 and wrap around to the beginning of the array if necessary.
6.2. Example of Linear Probing in a Hash Table
Let's consider an example of a hash table that uses linear probing to store student names and their corresponding grades. We'll assume that the hash table has a size of 10.
```python
hash_table = [None] * 10
def hash_function(key):
return len(key) % 10
def insert(key, value):
index = hash_function(key)
while hash_table[index] is not None:
index = (index + 1) % len(hash_table)
hash_table[index] = (key, value)
def get(key):
index = hash_function(key)
while hash_table[index] is not None:
if hash_table[index][0] == key:
return hash_table[index][1]
index = (index + 1) % len(hash_table)
return None
insert("Alice", 85)
insert("Bob", 92)
insert("Charlie", 78)
insert("David", 90)
print(get("Alice")) # Output: 85
print(get("Bob")) # Output: 92
print(get("Charlie")) # Output: 78
print(get("David")) # Output: 90
```
In this example, the hash function calculates the length of the key and uses the modulo operator to determine the index in the hash table. If a collision occurs, we increment the index by 1 and wrap around to the beginning of the hash table if necessary.
6.3. Comparing Performance with Other Collision Resolution Methods
Linear probing is just one of many collision resolution methods used in hash tables. Other methods, such as chaining and double hashing, have their own advantages and disadvantages.
Chaining, which uses linked lists to handle collisions, can handle a large number of collisions without affecting the performance of find and insert operations. However, it requires additional memory for the linked lists and can have slower performance due to the need to traverse the linked lists.
Double hashing, on the other hand, uses a secondary hash function to determine the offset for probing. This can reduce clustering and improve the performance of find and insert operations. However, it requires additional computation for the secondary hash function and can be more complex to implement.
The choice of collision resolution method depends on various factors, such as the expected number of keys, the distribution of the keys, and the memory constraints. It is important to consider these factors and choose the appropriate method to achieve the desired performance.
# 7. Advanced Topics in Linear Probing
7.1. Dynamic Resizing
One of the challenges of using linear probing is that the performance can degrade significantly when the load factor becomes too high. To address this issue, dynamic resizing can be implemented.
Dynamic resizing involves increasing the size of the hash table when the load factor exceeds a certain threshold. This allows for a larger number of slots and reduces the likelihood of collisions. When resizing, the keys and values are rehashed and inserted into the new hash table.
Here is an example implementation of dynamic resizing in Java:
```java
public class DynamicResizingHashMap<K, V> extends HashMap<K, V> {
private int size;
private K[] keys;
private V[] values;
private double loadFactorThreshold;
public DynamicResizingHashMap() {
size = 0;
keys = (K[]) new Object[16];
values = (V[]) new Object[16];
loadFactorThreshold = 0.75;
}
public void put(K key, V value) {
if ((double) size / keys.length >= loadFactorThreshold) {
resize();
}
int index = getIndex(key);
while (keys[index] != null) {
if (keys[index].equals(key)) {
values[index] = value;
return;
}
index = (index + 1) % keys.length;
}
keys[index] = key;
values[index] = value;
size++;
}
private void resize() {
K[] oldKeys = keys;
V[] oldValues = values;
keys = (K[]) new Object[2 * oldKeys.length];
values = (V[]) new Object[2 * oldValues.length];
size = 0;
for (int i = 0; i < oldKeys.length; i++) {
if (oldKeys[i] != null) {
put(oldKeys[i], oldValues[i]);
}
}
}
// Other methods
@Override
public String toString() {
return "DynamicResizingHashMap{" +
"size=" + size +
", keys=" + Arrays.toString(keys) +
", values=" + Arrays.toString(values) +
'}';
}
}
```
In this example, the `resize` method is called when the load factor exceeds the threshold of 0.75. The size of the hash table is doubled, and the keys and values are rehashed and inserted into the new hash table.
7.2. Clustering and Its Effects on Linear Probing
Clustering is a phenomenon that occurs in linear probing when keys that hash to nearby locations in the hash table collide. This can lead to longer linear probing sequences and slower find and insert operations.
There are two types of clustering: primary clustering and secondary clustering. Primary clustering occurs when keys that hash to the same location collide, while secondary clustering occurs when keys that hash to different locations collide due to the linear probing sequence.
To mitigate clustering, techniques such as double hashing and quadratic probing can be used. Double hashing uses a secondary hash function to determine the offset for probing, while quadratic probing uses a quadratic function to determine the offset.
7.3. Alternative Probing Sequences
In addition to linear probing, there are other probing sequences that can be used to resolve collisions in hash tables. Some examples include:
- Quadratic probing: In quadratic probing, the offset for probing is determined by a quadratic function. This can help reduce clustering and improve the performance of find and insert operations.
- Robin Hood hashing: Robin Hood hashing is a variation of linear probing where the keys are rearranged to minimize the average probe length. This can result in more balanced probe sequences and better performance.
- Cuckoo hashing: Cuckoo hashing uses two hash functions and two hash tables. If a collision occurs, the key is moved to the other hash table using the alternate hash function. This process continues until a vacant slot is found or a cycle is detected.
The choice of probing sequence depends on various factors, such as the expected number of keys, the distribution of the keys, and the desired performance characteristics. It is important to consider these factors and choose the appropriate probing sequence to optimize the performance of the hash table.
# 8. Limitations and Challenges of Linear Probing
8.1. Dealing with High Load Factors
One of the main challenges of linear probing is dealing with high load factors. As the load factor increases, the likelihood of collisions also increases, leading to longer linear probing sequences and slower find and insert operations.
To mitigate the impact of high load factors, techniques such as dynamic resizing can be used. By increasing the size of the hash table when the load factor exceeds a certain threshold, the number of collisions can be reduced, improving the performance of the hash table.
8.2. Addressing Clustering and Performance Issues
Clustering is another challenge of linear probing. When keys that hash to nearby locations collide, it can result in longer linear probing sequences and slower find and insert operations.
To address clustering, techniques such as double hashing and quadratic probing can be used. These techniques introduce randomness into the probing sequence, reducing the likelihood of clustering and improving the performance of the hash table.
8.3. Other Possible Drawbacks
In addition to the challenges mentioned above, linear probing has some other potential drawbacks:
- Memory usage: Linear probing requires additional memory to store the keys and values. This can be a concern when dealing with large hash tables or limited memory resources.
- Key distribution: Linear probing works best when the keys are uniformly distributed. If the keys are not evenly distributed, it can result in more collisions and slower performance.
- Deletion: Deleting a key from a hash table that uses linear probing can be challenging. Simply marking the slot as empty may cause the linear probing sequence to break, resulting in incorrect find and insert operations.
It is important to consider these drawbacks and evaluate whether linear probing is the most suitable collision resolution method for a given use case.
# 9. Real-World Applications of Linear Probing
9.1. Use in Databases and Caches
Linear probing is commonly used in databases and caches to implement hash-based data structures. It allows for efficient lookup and insertion of data, making it suitable for scenarios where fast access to data is crucial.
In databases, linear probing can be used to implement hash indexes, which provide fast access to data based on a key. By using linear probing, the database can quickly locate the desired data without the need for a full table scan.
In caches, linear probing can be used to implement hash tables that store frequently accessed data. The cache can quickly retrieve data based on a key, reducing the latency of accessing data from slower storage devices.
9.2. Network Routing and Load Balancing
Linear probing can also be applied to network routing and load balancing. In network routing, linear probing can be used to efficiently route packets based on their destination address. By using a hash table with linear probing, the network router can quickly determine the next hop for a packet.
In load balancing, linear probing can be used to distribute incoming requests across multiple servers. By using a hash table with linear probing, the load balancer can quickly determine the server that should handle a particular request, ensuring that the workload is evenly distributed.
9.3. Other Practical Applications
In addition to the applications mentioned above, linear probing has other practical uses:
- Spell checking: Linear probing can be used to implement a hash table that stores a dictionary of words. By using linear probing, the spell checker can quickly determine whether a given word is in the dictionary.
- Symbol tables: Linear probing can be used to implement symbol tables in programming languages. By using linear probing, the symbol table can efficiently store and retrieve variables, functions, and other symbols.
- File systems: Linear probing can be used in file systems to implement hash-based data structures, such as file name lookup tables. By using linear probing, the file system can quickly locate files based on their names.
In the final section, we will conclude the textbook and discuss future directions for linear probing.
# 10. Conclusion and Future Directions
In this textbook, we have explored the concept of linear probing as a collision resolution method in hash tables. We have covered the definition and purpose of linear probing, the steps involved in linear probing, and the analysis of time and space complexity.
We have also discussed the importance of choosing an appropriate load factor and explored practical examples of implementing linear probing in different programming languages and scenarios.
Furthermore, we have explored advanced topics in linear probing, such as dynamic resizing, clustering, and alternative probing sequences. We have discussed the limitations and challenges of linear probing and explored real-world applications in databases, caches, network routing, and more.
In the future, there are several directions for further research and development in the field of linear probing. Some possible areas of exploration include:
- Improving the performance of linear probing through optimizations, such as cache-aware algorithms and parallel processing.
- Investigating the impact of different hash functions and their degree of independence on the performance of linear probing.
- Exploring alternative collision resolution methods and their trade-offs in terms of performance, memory usage, and ease of implementation.
- Applying linear probing to new domains and scenarios, such as machine learning, natural language processing, and cybersecurity.
By continuing to study and innovate in the field of linear probing, we can further enhance the efficiency and effectiveness of hash tables and contribute to the advancement of computer science as a whole.
10.4. Other Related Topics and Fields
Linear probing is just one aspect of the broader field of data structures and algorithms. It is closely related to topics such as hash functions, hash tables, and collision resolution methods.
Other related topics and fields include:
- Chained hashing: Chained hashing is an alternative collision resolution method that uses linked lists to handle collisions. It can be compared and contrasted with linear probing to understand the trade-offs between the two methods.
- Graph algorithms: Graph algorithms, such as breadth-first search and depth-first search, can be used in conjunction with hash tables to solve various problems, such as finding the shortest path between two nodes or detecting cycles in a graph.
- Machine learning: Machine learning algorithms often rely on efficient data structures, such as hash tables, to store and retrieve large amounts of data. Understanding the underlying data structures can help optimize the performance of machine learning algorithms.
- Cryptography: Cryptographic algorithms, such as hash functions and digital signatures, rely on the properties of hash tables and collision resistance. Understanding the concepts of linear probing and other collision resolution methods can provide insights into the security and efficiency of cryptographic algorithms.
By exploring these related topics and fields, we can deepen our understanding of linear probing and its applications in various domains.
# 7.1. Dynamic Resizing
Dynamic resizing is an important aspect of linear probing that allows the hash table to adapt to changes in the number of elements. When the load factor exceeds a certain threshold, the hash table needs to be resized to maintain a good balance between space usage and performance.
The process of dynamic resizing involves creating a new hash table with a larger size and rehashing all the elements from the old table to the new table. This ensures that the elements are distributed more evenly and reduces the chances of collisions.
The steps involved in dynamic resizing are as follows:
1. Calculate the load factor of the hash table. The load factor is the ratio of the number of elements to the size of the table.
2. If the load factor exceeds a predefined threshold (e.g., 0.75), it is time to resize the table.
3. Create a new hash table with a larger size. The new size can be determined based on a resizing factor, such as doubling the size of the old table.
4. Rehash all the elements from the old table to the new table. This involves recalculating the hash values of each element based on the new size of the table and inserting them into the appropriate slots.
5. Update the reference to the new table and deallocate the old table from memory.
Dynamic resizing ensures that the hash table remains efficient even as the number of elements grows. It prevents excessive collisions and maintains a good balance between space usage and performance.
7.2. Clustering and Its Effects on Linear Probing
Clustering is a phenomenon that occurs in linear probing when elements with different hash codes collide and form clusters in the hash table. This can lead to longer probe sequences and decreased performance.
There are two main types of clustering in linear probing:
1. Primary clustering: Primary clustering occurs when elements with different hash codes collide and form clusters that are adjacent to each other. This happens because linear probing only checks the next slot in the table when a collision occurs. If that slot is also occupied, it continues to the next slot, and so on. As a result, elements with different hash codes can end up in the same cluster.
2. Secondary clustering: Secondary clustering occurs when elements with the same hash code collide and form clusters. This happens because linear probing checks the next slot in the table regardless of whether it is occupied or not. If the next slot is occupied by an element with the same hash code, it continues to the next slot, and so on. As a result, elements with the same hash code can end up in the same cluster.
Both primary and secondary clustering can increase the average probe length and decrease the performance of linear probing. The longer the probe sequence, the longer it takes to find an empty slot or the desired element. This can lead to degraded performance, especially when the load factor is high.
To mitigate the effects of clustering, various techniques can be used, such as:
- Double hashing: Double hashing is a collision resolution method that uses a secondary hash function to determine the next slot to probe. This helps to distribute the elements more evenly and reduce the chances of clustering.
- Quadratic probing: Quadratic probing is another collision resolution method that uses a quadratic function to determine the next slot to probe. This helps to spread out the elements and reduce clustering.
- Rehashing: Rehashing involves resizing the hash table and rehashing all the elements to reduce clustering. This can be done periodically or when the load factor exceeds a certain threshold.
By understanding the effects of clustering and using appropriate techniques to mitigate it, the performance of linear probing can be improved.
7.3. Alternative Probing Sequences
Linear probing uses a simple probing sequence where each slot in the hash table is checked sequentially until an empty slot or the desired element is found. However, there are alternative probing sequences that can be used to improve the performance of linear probing.
One alternative probing sequence is called quadratic probing. In quadratic probing, the next slot to probe is determined using a quadratic function instead of a linear function. The formula for quadratic probing is:
$$
\text{{next\_slot}} = (\text{{current\_slot}} + i^2) \mod \text{{table\_size}}
$$
where $i$ is the number of probes that have been made. Quadratic probing helps to spread out the elements more evenly and reduce clustering. It can also help to avoid long probe sequences and improve the average probe length.
Another alternative probing sequence is called double hashing. In double hashing, a secondary hash function is used to determine the next slot to probe. The formula for double hashing is:
$$
\text{{next\_slot}} = (\text{{current\_slot}} + \text{{hash\_function\_2}}(\text{{key}})) \mod \text{{table\_size}}
$$
where $\text{{hash\_function\_2}}(\text{{key}})$ is the result of the secondary hash function applied to the key. Double hashing helps to distribute the elements more evenly and reduce clustering. It can also help to avoid long probe sequences and improve the average probe length.
Both quadratic probing and double hashing can be used as alternative probing sequences in linear probing. The choice of probing sequence depends on the specific requirements of the application and the characteristics of the data being stored.
By using alternative probing sequences, the performance of linear probing can be improved and the effects of clustering can be mitigated.
# 8. Limitations and Challenges of Linear Probing
While linear probing is a simple and efficient collision resolution method, it has certain limitations and challenges that need to be considered.
One limitation of linear probing is the potential for clustering. Clustering occurs when elements with different hash codes collide and form clusters in the hash table. This can lead to longer probe sequences and decreased performance. While techniques like double hashing and quadratic probing can help mitigate clustering, they may not completely eliminate it.
Another limitation of linear probing is its sensitivity to the choice of hash function. Linear probing works best with hash functions that distribute the elements uniformly across the hash table. If the hash function produces a non-uniform distribution, it can result in more collisions and longer probe sequences.
Linear probing also has challenges when it comes to dynamic resizing. When the load factor exceeds a certain threshold, the hash table needs to be resized to maintain a good balance between space usage and performance. Resizing a hash table involves rehashing all the elements, which can be a time-consuming process, especially if the table is large.
Additionally, linear probing may not be suitable for applications that require constant-time lookups. While linear probing has an average-case time complexity of O(1), the worst-case time complexity can be O(N), where N is the number of elements in the hash table. If the hash table is highly loaded or the hash function produces a poor distribution, the worst-case time complexity can be significantly worse.
Despite these limitations and challenges, linear probing remains a popular and widely used collision resolution method due to its simplicity and efficiency. By understanding its limitations and applying appropriate techniques, the performance of linear probing can be optimized for different applications.
8.1. Dealing with High Load Factors
When the load factor of a hash table exceeds a certain threshold, it can lead to increased collisions and decreased performance. Dealing with high load factors is an important challenge in linear probing that needs to be addressed.
One approach to dealing with high load factors is to resize the hash table when the load factor exceeds a certain threshold. This involves creating a new hash table with a larger size and rehashing all the elements from the old table to the new table. Resizing the hash table helps to distribute the elements more evenly and reduce the chances of collisions.
Another approach is to use alternative collision resolution methods, such as separate chaining or quadratic probing, which can handle higher load factors more efficiently than linear probing. Separate chaining uses linked lists to handle collisions, while quadratic probing uses a quadratic function to determine the next slot to probe. These methods can help reduce the effects of high load factors and improve the performance of the hash table.
Additionally, choosing a good hash function is crucial in dealing with high load factors. A good hash function should distribute the elements uniformly across the hash table, reducing the chances of collisions. There are various techniques and algorithms available for designing and evaluating hash functions, and choosing the right one can significantly impact the performance of the hash table.
By resizing the hash table, using alternative collision resolution methods, and choosing a good hash function, the effects of high load factors can be mitigated in linear probing.
8.2. Addressing Clustering and Performance Issues
Clustering is a common issue in linear probing that can lead to decreased performance. Addressing clustering and its associated performance issues is an important challenge in linear probing.
One approach to addressing clustering is to use alternative probing sequences, such as quadratic probing or double hashing. These methods help to distribute the elements more evenly and reduce clustering. Quadratic probing uses a quadratic function to determine the next slot to probe, while double hashing uses a secondary hash function. By using alternative probing sequences, the effects of clustering can be mitigated, leading to improved performance.
Another approach is to periodically rehash the elements in the hash table. Rehashing involves resizing the hash table and rehashing all the elements to reduce clustering. This can be done when the load factor exceeds a certain threshold or at regular intervals. Rehashing helps to redistribute the elements and reduce the chances of clustering, improving the performance of the hash table.
Choosing a good hash function is also crucial in addressing clustering and performance issues. A good hash function should distribute the elements uniformly across the hash table, reducing the chances of collisions and clustering. There are various techniques and algorithms available for designing and evaluating hash functions, and choosing the right one can significantly impact the performance of the hash table.
By using alternative probing sequences, periodically rehashing the elements, and choosing a good hash function, the effects of clustering and performance issues can be addressed in linear probing.
8.3. Other Possible Drawbacks
While linear probing is a simple and efficient collision resolution method, it has some potential drawbacks that need to be considered.
One possible drawback is the potential for increased memory usage. Linear probing requires a larger hash table size compared to other collision resolution methods, such as separate chaining. This is because linear probing needs to maintain a certain amount of empty slots to ensure efficient probing. As a result, linear probing may use more memory than other methods, especially when the load factor is high.
Another possible drawback is the sensitivity to the choice of hash function. Linear probing works best with hash functions that distribute the elements uniformly across the hash table. If the hash function produces a non-uniform distribution, it can result in more collisions and longer probe sequences. Choosing a good hash function is crucial in ensuring the efficiency and effectiveness of linear probing.
Additionally, linear probing may not be suitable for applications that require constant-time lookups. While linear probing has an average-case time complexity of O(1), the worst-case time complexity can be O(N), where N is the number of elements in the hash table. If the hash table is highly loaded or the hash function produces a poor distribution, the worst-case time complexity can be significantly worse.
Despite these potential drawbacks, linear probing remains a popular and widely used collision resolution method due to its simplicity and efficiency. By understanding its limitations and applying appropriate techniques, the performance of linear probing can be optimized for different applications.
# 9. Real-World Applications of Linear Probing
Linear probing has various real-world applications in different domains. Its efficiency and simplicity make it a popular choice for handling collisions in hash tables. Let's explore some practical applications of linear probing.
9.1. Use in Databases and Caches
Linear probing is commonly used in databases and caches to provide efficient data storage and retrieval. In databases, linear probing can be used to handle collisions when indexing data based on keys. It allows for fast lookups and updates, making it suitable for applications that require quick access to data.
Caches, which are used to store frequently accessed data, can also benefit from linear probing. Linear probing allows for efficient cache lookup and replacement, ensuring that the most frequently accessed data is readily available.
9.2. Network Routing and Load Balancing
Linear probing can be applied in network routing and load balancing algorithms. In network routing, linear probing can be used to efficiently route packets based on their destination addresses. It allows for fast lookup and forwarding of packets, ensuring efficient network communication.
Load balancing, which involves distributing network traffic across multiple servers, can also benefit from linear probing. Linear probing can be used to evenly distribute the load among servers, ensuring that each server handles a fair share of the traffic.
9.3. Other Practical Applications
Linear probing has many other practical applications in various domains. Some examples include:
- Compiler symbol tables: Linear probing can be used to efficiently store and retrieve symbols in compiler symbol tables. It allows for fast lookup and resolution of symbols during the compilation process.
- Spell checkers: Linear probing can be used in spell checkers to efficiently store and retrieve words from a dictionary. It allows for fast lookup and correction of misspelled words.
- File systems: Linear probing can be used in file systems to efficiently store and retrieve file metadata, such as file names and attributes. It allows for fast lookup and access to files.
These are just a few examples of the many practical applications of linear probing. Its efficiency and simplicity make it a versatile collision resolution method that can be applied in various domains.
# 10. Conclusion and Future Directions
In this textbook, we have explored the concept of linear probing as a collision resolution method in hash tables. We have covered the definition and purpose of linear probing, the steps involved in linear probing, and the analysis of time and space complexity.
We have also discussed the importance of choosing an appropriate load factor and explored practical examples of implementing linear probing in different programming languages and scenarios.
Furthermore, we have explored advanced topics in linear probing, such as dynamic resizing, clustering, and alternative probing sequences. We have discussed the limitations and challenges of linear probing and explored real-world applications in databases, caches, network routing, and more.
In the future, there are several directions for further research and development in the field of linear probing. Some possible areas of exploration include:
- Improving the performance of linear probing through optimizations, such as cache-aware algorithms and parallel processing.
- Investigating the impact of different hash functions and their degree of independence on the performance of linear probing.
- Exploring alternative collision resolution methods and their trade-offs in terms of performance, memory usage, and ease of implementation.
- Applying linear probing to new domains and scenarios, such as machine learning, natural language processing, and cybersecurity.
By continuing to study and innovate in the field of linear probing, we can further enhance the efficiency and effectiveness of hash tables and contribute to the advancement of computer science as a whole.
# 8.1. Dealing with High Load Factors
One limitation of linear probing is that it can become inefficient when the load factor of the hash table is high. The load factor is defined as the ratio of the number of elements in the hash table to the size of the hash table.
When the load factor is high, there is a higher chance of collisions occurring, which can lead to longer probe sequences and decreased performance. In extreme cases, the hash table can become completely full, resulting in failed insertions.
To address this limitation, one approach is to dynamically resize the hash table when the load factor exceeds a certain threshold. This involves creating a new hash table with a larger size and rehashing all the elements from the old hash table into the new one.
By increasing the size of the hash table, we can reduce the load factor and decrease the likelihood of collisions. This can improve the performance of linear probing and ensure that the hash table remains efficient even with a large number of elements.
## Exercise
What is the load factor and why is it important in linear probing?
### Solution
The load factor is the ratio of the number of elements in the hash table to the size of the hash table. It is important in linear probing because it affects the likelihood of collisions and the efficiency of the hash table. A high load factor can lead to more collisions and longer probe sequences, resulting in decreased performance. By managing the load factor and dynamically resizing the hash table when necessary, we can maintain the efficiency of linear probing.
# 8.2. Addressing Clustering and Performance Issues
One challenge of linear probing is the formation of clusters, which can occur when multiple elements hash to the same index and are placed adjacent to each other in the hash table. This can lead to longer probe sequences and decreased performance.
To address clustering and improve performance, several techniques can be used:
1. **Primary Clustering**: Primary clustering refers to the formation of clusters at the beginning of the probe sequence. This can be mitigated by using a technique called "double hashing". Double hashing involves using a second hash function to determine the step size for probing. By using a different step size for each element, we can distribute the elements more evenly in the hash table and reduce the likelihood of clustering.
2. **Secondary Clustering**: Secondary clustering refers to the formation of clusters within the probe sequence. This can be addressed by using a technique called "quadratic probing". Quadratic probing involves using a quadratic function to determine the step size for probing. By using a quadratic function, we can spread out the elements more evenly in the probe sequence and reduce the likelihood of clustering.
3. **Randomization**: Another approach to address clustering is to introduce randomness in the probe sequence. This can be done by using a technique called "random probing". Random probing involves randomly selecting the next index to probe, rather than using a deterministic function. By introducing randomness, we can distribute the elements more evenly in the hash table and reduce the likelihood of clustering.
By using these techniques, we can mitigate the effects of clustering and improve the performance of linear probing.
## Exercise
What is primary clustering and how can it be addressed in linear probing?
### Solution
Primary clustering refers to the formation of clusters at the beginning of the probe sequence in linear probing. It can be addressed by using a technique called "double hashing". Double hashing involves using a second hash function to determine the step size for probing. By using a different step size for each element, we can distribute the elements more evenly in the hash table and reduce the likelihood of primary clustering.
# 8.3. Other Possible Drawbacks
While linear probing is a popular and efficient collision resolution method, it does have some potential drawbacks.
1. **Performance Degradation**: As the load factor of the hash table increases, the performance of linear probing can degrade. This is because the likelihood of collisions increases, leading to longer probe sequences and slower lookup times. To mitigate this issue, it is important to carefully choose an appropriate load factor and monitor the performance of the hash table.
2. **Limited Table Size**: Linear probing requires a fixed-size hash table, which means that the number of elements that can be stored is limited by the size of the table. If the table becomes full, it may be necessary to resize the table or use another collision resolution method.
3. **Sensitive to Hash Function Quality**: Linear probing is sensitive to the quality of the hash function used. If the hash function does not distribute the keys evenly across the hash table, it can lead to more collisions and decreased performance. Therefore, it is important to choose a good hash function that minimizes the likelihood of collisions.
Despite these potential drawbacks, linear probing remains a popular and widely used collision resolution method due to its simplicity and efficiency.
## Exercise
What are some potential drawbacks of linear probing?
### Solution
Some potential drawbacks of linear probing include performance degradation with increasing load factor, limited table size, and sensitivity to the quality of the hash function used.
# 9. Real-World Applications of Linear Probing
Linear probing is a versatile collision resolution method that has found applications in various real-world scenarios. Here are a few examples:
1. **Databases and Caches**: Linear probing is commonly used in databases and caches to store and retrieve data efficiently. It allows for fast lookup and insertion operations, making it suitable for applications that require quick access to data.
2. **Network Routing and Load Balancing**: Linear probing can be used in network routing algorithms and load balancing systems. It allows for efficient distribution of network traffic and resources, ensuring optimal performance and minimizing congestion.
3. **File Systems**: Linear probing can be used in file systems to manage file metadata and directory structures. It enables fast file lookup and retrieval, making it easier to navigate and access files within the system.
4. **Symbol Tables**: Linear probing is often used in symbol tables, which are data structures that store key-value pairs. Symbol tables are widely used in programming languages and compilers to store variables, functions, and other program symbols. Linear probing allows for efficient symbol lookup and manipulation.
These are just a few examples of how linear probing can be applied in real-world scenarios. Its simplicity and efficiency make it a popular choice for handling collisions and managing data in various applications.
## Exercise
Name three real-world applications of linear probing.
### Solution
Three real-world applications of linear probing are databases and caches, network routing and load balancing, and file systems.
# 9.1. Use in Databases and Caches
Linear probing is commonly used in databases and caches to store and retrieve data efficiently. In these applications, linear probing allows for fast lookup and insertion operations, making it suitable for scenarios that require quick access to data.
In a database, linear probing can be used to store key-value pairs, where the key is used to retrieve the corresponding value. When a new key-value pair is inserted, linear probing is used to find an empty slot in the database. If a collision occurs and the desired slot is already occupied, linear probing is used to find the next available slot.
Similarly, in a cache, linear probing can be used to store frequently accessed data. When a cache lookup is performed, linear probing is used to search for the desired data. If a collision occurs and the desired slot is already occupied, linear probing is used to find the next available slot.
Overall, linear probing is a valuable technique in databases and caches, as it allows for efficient data storage and retrieval, leading to improved performance and responsiveness in these systems.
- In a database, a linear probing hash table is used to store customer information. Each customer is assigned a unique customer ID, which serves as the key in the hash table. The customer's information, such as name, address, and contact details, is stored as the corresponding value. When a customer's information needs to be retrieved, linear probing is used to quickly locate the customer's ID in the hash table and retrieve the associated information.
## Exercise
Describe how linear probing is used in databases and caches.
### Solution
In databases and caches, linear probing is used to efficiently store and retrieve data. It allows for fast lookup and insertion operations, making it suitable for scenarios that require quick access to data. In a database, linear probing is used to store key-value pairs, where the key is used to retrieve the corresponding value. In a cache, linear probing is used to store frequently accessed data. When a lookup is performed, linear probing is used to search for the desired data. If a collision occurs, linear probing is used to find the next available slot.
# 9.2. Network Routing and Load Balancing
Linear probing is also used in network routing and load balancing systems. These systems are responsible for efficiently routing network traffic and distributing it across multiple servers or resources.
In network routing, linear probing can be used to determine the best path for network packets to travel from the source to the destination. Linear probing allows routers to quickly search for the next available hop in the network and forward the packet accordingly. If a collision occurs and the desired hop is already occupied, linear probing is used to find the next available hop.
Similarly, in load balancing systems, linear probing can be used to evenly distribute incoming requests across multiple servers or resources. Linear probing allows the load balancer to quickly determine which server or resource is available and can handle the incoming request. If a collision occurs and the desired server or resource is already occupied, linear probing is used to find the next available one.
By using linear probing in network routing and load balancing, these systems can efficiently handle high volumes of traffic and ensure that resources are utilized effectively.
- In a network routing system, linear probing is used to determine the best path for a network packet to reach its destination. Each hop in the network is assigned a unique identifier, and linear probing is used to quickly find the next available hop in the path. If a collision occurs and the desired hop is already occupied, linear probing is used to find the next available hop.
## Exercise
Explain how linear probing is used in network routing and load balancing.
### Solution
In network routing and load balancing systems, linear probing is used to efficiently route network traffic and distribute it across multiple servers or resources. In network routing, linear probing is used to determine the best path for network packets to travel. In load balancing, linear probing is used to evenly distribute incoming requests across servers or resources. Linear probing allows these systems to quickly search for the next available hop or resource. If a collision occurs, linear probing is used to find the next available one.
# 9.3. Other Practical Applications
In addition to the applications mentioned earlier, linear probing has several other practical uses in various fields.
One such field is database management. Linear probing can be used to efficiently store and retrieve data in a database. It allows for quick searching and updating of records, making it an essential technique in database systems.
Linear probing is also used in cache memory systems. Cache memory is a small, fast memory that stores frequently accessed data. Linear probing is used to quickly search for data in the cache and retrieve it. If a collision occurs and the desired data is already occupied, linear probing is used to find the next available slot.
Another practical application of linear probing is in file systems. Linear probing can be used to allocate and manage disk space in a file system. It allows for efficient storage and retrieval of files, ensuring that data is organized and accessible.
Overall, linear probing is a versatile technique that can be applied in various domains to improve efficiency and performance.
- In a database management system, linear probing is used to efficiently store and retrieve data. It allows for quick searching and updating of records, ensuring that data is easily accessible.
## Exercise
Describe another practical application of linear probing.
### Solution
Another practical application of linear probing is in cache memory systems. Linear probing is used to quickly search for data in the cache and retrieve it. If a collision occurs, linear probing is used to find the next available slot. This ensures that frequently accessed data is stored in the cache for faster retrieval.
# 10. Conclusion and Future Directions
In this textbook, we have covered the concept of linear probing in depth. Linear probing is a collision resolution method used in hash tables. It allows for efficient storage and retrieval of data, ensuring that key-value pairs are organized and easily accessible.
We have discussed the steps involved in linear probing, including how to handle collisions and find the next available slot. We have also analyzed the time and space complexity of linear probing, understanding its impact on the performance of hash tables.
Furthermore, we have explored the concept of load factor and its impact on linear probing. Load factor refers to the ratio of the number of elements stored in the hash table to the total number of slots. We have discussed how load factor affects the efficiency of linear probing and how to choose an appropriate load factor.
Throughout this textbook, we have provided practical examples of linear probing in various domains, such as database management, cache memory systems, and file systems. These examples have demonstrated the real-world applications of linear probing and its importance in improving efficiency and performance.
In conclusion, linear probing is a valuable technique in the field of data structures and algorithms. It offers an efficient solution for handling collisions in hash tables and has numerous practical applications. As technology continues to advance, there may be further research and development in the area of linear probing, leading to improvements and alternatives in the future.
- Linear probing is a valuable technique in the field of data structures and algorithms. It offers an efficient solution for handling collisions in hash tables and has numerous practical applications.
## Exercise
Summarize the main concepts covered in this textbook.
### Solution
This textbook has covered the concept of linear probing in depth. We have discussed the steps involved in linear probing, analyzed its time and space complexity, and explored the concept of load factor. We have provided practical examples of linear probing in various domains, such as database management and cache memory systems. Overall, linear probing is a valuable technique in the field of data structures and algorithms, offering an efficient solution for handling collisions in hash tables.
# 10.1. Summary of Linear Probing
In summary, linear probing is a collision resolution method used in hash tables. It allows for efficient storage and retrieval of data by handling collisions and finding the next available slot. Linear probing has a time and space complexity that impacts the performance of hash tables. The load factor, which is the ratio of the number of elements stored to the total number of slots, also affects the efficiency of linear probing. Throughout this textbook, we have provided practical examples of linear probing in various domains, showcasing its real-world applications. Linear probing is a valuable technique in the field of data structures and algorithms, offering an efficient solution for handling collisions in hash tables.
# 10.2. Possible Improvements and Alternatives
While linear probing is a widely used collision resolution method, there are alternative approaches and possible improvements that can be explored. One alternative is double hashing, which involves using a second hash function to determine the next available slot when a collision occurs. This can help reduce clustering and improve the efficiency of the hash table.
Another improvement that can be made is dynamic resizing. As the number of elements in the hash table increases, the load factor increases as well. This can lead to more collisions and decreased performance. Dynamic resizing involves increasing the size of the hash table when the load factor exceeds a certain threshold. This helps maintain a lower load factor and improves the efficiency of the hash table.
Additionally, there are other collision resolution methods such as random hashing and separate chaining that can be explored. Random hashing involves using a random number generator to determine the next available slot when a collision occurs. Separate chaining involves using linked lists to store multiple elements in the same slot.
Overall, there are several possible improvements and alternative methods to consider when implementing linear probing in hash tables. These approaches can help optimize performance and address specific challenges that may arise in different contexts.
# 10.3. Future Research and Development
As with any field of study, there is always room for further research and development in the area of linear probing and hash tables. Some potential areas for future exploration include:
1. Optimizing hash functions: The choice of hash function can greatly impact the performance of a hash table. Further research can focus on developing more efficient and effective hash functions that minimize collisions and improve overall performance.
2. Advanced collision resolution techniques: While linear probing and double hashing are commonly used methods, there may be other innovative approaches to resolving collisions that have not yet been explored. Future research can investigate alternative techniques and evaluate their effectiveness.
3. Scalability and distributed systems: As data sets continue to grow in size and complexity, it becomes increasingly important to develop hash table implementations that can handle large-scale and distributed systems. Future research can focus on designing hash table algorithms that are scalable, fault-tolerant, and efficient in distributed environments.
4. Security and privacy considerations: Hash tables are often used in applications that involve sensitive data, such as password systems and digital signatures. Future research can explore ways to enhance the security and privacy of hash tables, including techniques for preventing hash collisions and protecting against attacks.
5. Integration with other data structures: Hash tables are a fundamental data structure, but they can also be combined with other data structures to create more powerful and versatile data storage systems. Future research can investigate the integration of hash tables with other structures, such as trees or graphs, to create hybrid data structures that offer the benefits of both.
Overall, the field of linear probing and hash tables offers numerous opportunities for further research and development. By continuing to explore these areas, we can improve the efficiency, scalability, and security of hash table implementations, and unlock new possibilities for data storage and retrieval.
# 10.4. Other Related Topics and Fields
Linear probing and hash tables are just one piece of the larger field of data structures and algorithms. There are many other related topics and fields that are worth exploring for a more comprehensive understanding of the subject. Some of these include:
1. Other collision resolution methods: While linear probing is a popular method for resolving collisions in hash tables, there are other techniques that can be used as well. Some examples include separate chaining, quadratic probing, and cuckoo hashing. Each method has its own advantages and disadvantages, and further exploration of these techniques can provide a deeper understanding of collision resolution.
2. Data compression: Hash functions are often used in data compression algorithms, such as the Lempel-Ziv-Welch (LZW) algorithm used in the GIF image format. Understanding how hash functions are used in compression algorithms can provide insights into both hash functions and data compression techniques.
3. Cryptography: Hash functions are also an important component of cryptographic algorithms, such as the SHA-256 hash function used in blockchain technology. Exploring the use of hash functions in cryptography can shed light on the security properties of hash functions and their role in protecting sensitive information.
4. Database management systems: Hash tables are commonly used in database management systems for efficient data retrieval. Understanding how hash tables are used in the context of databases can provide insights into the design and implementation of high-performance database systems.
5. Machine learning and data mining: Hash functions are often used in machine learning and data mining algorithms for tasks such as feature hashing and dimensionality reduction. Exploring the use of hash functions in these domains can provide insights into how hash functions can be used to efficiently process large datasets.
By exploring these related topics and fields, you can gain a broader understanding of the applications and implications of linear probing and hash tables, and how they fit into the larger landscape of data structures and algorithms.
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Textbooks
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Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$
Wavefronts of a stage structured model with state--dependent delay
Regions of stability for a linear differential equation with two rationally dependent delays
Joseph M. Mahaffy 1, and Timothy C. Busken 2,
Department of Mathematics and Statistics, Nonlinear Dynamical Systems Group, Computational Sciences Research Center, San Diego State University, San Diego, CA 92182-7720, United States
Department of Mathematics, Grossmont College, El Cajon, CA 92020, United States
Received July 2013 Revised January 2015 Published April 2015
Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form $1/n$, this study finds the asymptotic shape and size of the stability region. For example, a delay ratio of $1/3$ asymptotically produces a stability region about 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay differential equation. A nonlinear model with two delays illustrates how our methods can be applied.
Keywords: exponential polynomial, stability analysis, eigenvalue., Delay differential equation, bifurcation.
Mathematics Subject Classification: Primary: 37C75, 37G15; Secondary: 39B8.
Citation: Joseph M. Mahaffy, Timothy C. Busken. Regions of stability for a linear differential equation with two rationally dependent delays. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4955-4986. doi: 10.3934/dcds.2015.35.4955
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István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063
Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
Joseph M. Mahaffy Timothy C. Busken
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CommonCrawl
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Elizabeth Buchanan Cowley
Elizabeth Buchanan Cowley (1874–1945) was an American mathematician.
Life
Cowley was born on May 22, 1874, in Allegheny, Pennsylvania.[1][2] She had four siblings, but they and her father all died by 1900.[3] Cowley's mother, Mary Junkin Buchanan Cowley, later became a member of the Board of Public Education of Pittsburgh, and was the namesake of the Mary J. Cowley School in Pittsburgh. Cowley's grandfather (Mary Cowley's father) was James Galloway Buchanan, a surgeon in the Union Army.[4]
Cowley earned a bachelor's degree in 1893 from the Indiana State Normal School of Pennsylvania, and became a school teacher. She earned a second bachelor's degree in 1901 and a master's degree in 1902 from Vassar College, and became an instructor at Vassar, studying higher mathematics during the summers at the University of Chicago.[1][3] In 1908 she completed a doctorate from Columbia University. Her dissertation, on algebraic curves, was supervised by Cassius Jackson Keyser; she became the fourth woman to earn a doctorate in mathematics from Columbia.[1][5] Continuing to work at Vassar, Cowley was promoted to assistant professor in 1913, and associate professor in 1916. She went on leave in 1926 to assist her mother, and resigned her position at Vassar in 1929, instead becoming a high school teacher in Pittsburgh.[1][3]
She retired from teaching in 1938, had a stroke in 1941, and died on April 13, 1945, in Fort Lauderdale, Florida.[3]
Contributions
Cowley and her co-author Ida Whiteside won a prize for a 1907 paper they wrote on the orbit of comet C/1825 V1. Another of her publications, in 1926, concerned liquid water pouring puzzles. She was the author of two textbooks on plane and solid geometry, published in 1932 and 1934, and advocated teaching solid geometry to high school students after many colleges had replaced the subject with freshman calculus.[1] She published another book in 1941 about public education.[3]
Associations
Cowley was an early member of the Mathematical Association of America, and became a member of its board of trustees when it incorporated in 1920. She was an invited speaker at the International Congress of Mathematicians in 1932, speaking there about mathematics education.[6] She also belonged to the American Mathematical Society, German Mathematical Society, and Circolo Matematico di Palermo.[4]
References
1. Riddle, Larry (January 10, 2014), "Elizabeth Buchanan Cowley", Biographies of Women Mathematicians, Agnes Scott College, retrieved 2015-10-08.
2. File:Woman s Who s who of America.pdf, 1914, p. 210 (= p. 199 in Pdf)
3. Green, Judy; LaDuke, Jeanne (2009), "Cowley, Elizabeth B.", Pioneering Women in American Mathematics: The Pre-1940 PhD's, History of mathematics, vol. 34, American Mathematical Society, pp. 163–164, ISBN 9780821843765 Biography on p.148-152 of the Supplementary Material at AMS
4. Fleming, George Thornton (1922), History of Pittsburgh and Environs, from Prehistoric Days to the Beginning of the American Revolution, vol. 4, American Historical Society, Incorporated, pp. 107–108.
5. Elizabeth Buchanan Cowley at the Mathematics Genealogy Project
6. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, retrieved 2015-10-01.
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B. Parent • AE25225 Intermediate Thermodynamics
Intermediate Thermodynamics Assignment 4 — Conservation of Energy
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Starting from Newton's law $\vec{F}=m\vec{a}$, the first law of thermo ${\rm d}(mh) - V {\rm d}P=\delta Q -\delta W$, and the mass conservation equation in differential form, show that the energy conservation in control volume form corresponds to: $$ \frac{{\rm d}}{{\rm d}t} \int_V \rho\left( e +\frac{1}{2} \vec{v}\cdot\vec{v}+g y \right){\rm d} V + \int_S \rho (\vec{v}\cdot\vec{n}) \left(h +\frac{1}{2} \vec{v}\cdot\vec{v}+gy \right){\rm d}S=\dot{Q}-\dot{W}$$ Note: this questions is worth double the points awarded to the other questions.
Two adiabatic tanks are interconnected through a valve. Tank A contains $\rm 0.2~m^3$ of air at 40 bar and $\rm 90^\circ C$. Tank B contains $\rm 2~m^3$ of air at 1 bar and $\rm 30^\circ C$. The valve is opened until the pressure in A drops to $12.5$ bar. At this instant, determine (a) the temperatures and pressures in both tanks and (b) the amount of mass that has left tank A.
A 4 m$^3$ storage tank (see schematic below) containing 2 m$^3$ of liquid is to be pressurized with air from a large, high-pressure reservoir through a valve at the top of the tank to permit rapid ejection of the liquid:
The air in the reservoir is maintained at 100 bar and 300 K. The gas space above the liquid contains initially air at 1 bar and 280 K. When the pressure in the tank reaches 5 bar, the liquid transfer valve is opened and the liquid is ejected at the rate of 0.2 m$^3$/min while the tank pressure is maintained at 5 bar. What is the air temperature when the pressure reaches 5 bar and when the liquid has been drained completely? Hints: neglect heat interaction at the gas-liquid and gas-tank boundaries. It may be assumed that the gas above the liquid is well mixed and that air is a perfect gas.
A tank containing 45 kg of liquid water initially at 45$^\circ$C has one inlet and one exit with equal mass flow rates. Liquid water enters at 45$^\circ$C and a mass flow rate of 270 kg/h. A cooling coil immersed in the water removes energy at a rate of $7.6$ kW. The water is well mixed by a paddle wheel so that the water temperature is uniform throughout. The power input to the water from the paddle wheel is 0.6 kW. The pressures at the inlet and exit are equal and all kinetic and potential energy effects can be ignored. Determine the variation of water temperature with time.
A spaceship cabin may be considered to be a rigid pressurized vessel, which contains the atmosphere required to support the life of its occupants. A puncture in the cabin when it is in space must be detected quickly, so that the cabin occupants can seal the puncture. The time interval that elapses before a dangerously low pressure is reached is a critical quantity for designing the warning devices and countermeasure systems. Derive an equation giving an estimate for the pressure $P$ after a puncture as a function of the time $t$, the initial pressure $P_{\rm i}$, the initial temperature $T_{\rm i}$, the area $A$ of the puncture, and the volume $V$ of the cabin. Consider the atmosphere of the cabin to be a perfect gas. If the cabin atmosphere is air at an initial temperature of 300 K, determine the ratio $P/P_{\rm i}$ as a function of time with the ratio $(A/V)$ as a parameter. Assume that for air, $\gamma=1.4$ and $R=287.4$ J/kg K.
Consider air being heated as it flows through a constant-area duct. At the duct entrance, the air has a pressure of 2 bars, a temperature of 300 K and a speed of 90 m/s. At the duct exit, the air has a pressure of 1.5 bars and a speed of 350 m/s. Do the following:
(a) Find the temperature of the air at the duct exit
(b) Determine the heat transfer per unit mass of air flowing through the duct in J/kg.
Consider air entering a duct at station 1 and exiting at station 2, as follows:
Knowing that the gravitational acceleration $g$ is 9.8 m/s$^2$, that the flow speed at the entrance $q_1$ is 150 m/s, that the cross-sectional areas $A_1$ and $A_2$ are equal to 1.0 m$^2$ and 1.2 m$^2$ respectively, that the height difference $\Delta y$ is equal to 200 m, that the temperature at the entrance is of 300 K, and that the pressure at the entrance and exit of the duct is equal to 1 atm and 1.05 atm respectively, do the following:
(a) calculate the temperature at the exit, $T_2$
(b) calculate the flow speed at the exit, $q_2$
2. 4.35 kg, 3.75 bar, 393 K.
3. 382 K, 400 K.
5. $\displaystyle \frac{P}{P_{\rm i}}=\left( 1+\frac{35 A}{256 V} (\gamma-1)t\sqrt{3RT_{\rm i}}\right)^\frac{6-70\gamma}{35\gamma-35}$
6. 875 K, 635 kJ/kg
7. 302 K, 121 m/s.
Due on Wednesday April 10th at 16:30. Do Questions #3, #6, and #7 only.
There was a typo in the formulation of Question 2, and this is why the answers for Q2 seemed wrong. Check the problem formulation again.
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\begin{document}
\title{Analytical and easily calculated expressions for continuous commutation functions under Gompertz-Makeham mortality} \author{Andreas Nordvall Lager{\aa}s\footnote{Address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. E-mail:\href{mailto:andreas@math,su,se}{\texttt{[email protected]}}}} \date{} \maketitle
\begin{abstract} \noindent It is known, but perhaps not well-known, that when the mortality is assumed to be of Gompertz-Makeham-type, the expected remaining life-length and the commutation functions used for calculating the expected values of various types of life insurances can be expressed with an incomplete gamma function with a negative shape parameter. This is not of much use if ones software cannot calculate these values. The aim of this note is to show that one can express the commutation functions using only the exponential function, the (ordinary) gamma function and the gamma distribution function, which are all implemented in common statistical and spreadsheet software. This eliminates the need to evaluate the commutation functions and expected remaining life-length with numerical integration.\\
\noindent\emph{Keywords}: Gompertz, Makeham, commutation functions, continuous compounding, analytical expression. \end{abstract}
We will assume that individual lifes are distributed according to the Gomp\-ertz-Makeham distribution with distribution function $F(x;\alpha,\beta,\gamma)=1-l(x;\alpha,\beta,\gamma)$ and survival function $l(x;\alpha,\beta,\gamma)=\exp\{-\alpha x-\frac{\beta}{\gamma}(e^{\gamma x}-1)\}$ corresponding to the mortality rate $\mu(x;\alpha,\beta,\gamma)=-\frac{d}{dx}l(x;\alpha,\beta,\gamma)=\alpha + \beta e^{\gamma x}$ at age $x$. We assume a fixed continuously compounded interest rate $\delta$. Let $\bar{a}_x=\bar{a}_x(\alpha,\beta,\gamma,\delta)$ be the present value of a life-long annuity, continuously paid at rate 1, to a person at age $x$, and $e_x(\alpha,\beta,\gamma)$ the expected remaining life-length for a person at age $x$. The main result is the following. \begin{theorem} \begin{align} \bar{a}_x(\alpha,\beta,\gamma,\delta) &= e_0(\alpha+\delta,\beta e^{\gamma x},\gamma),\text{where}\label{ax}\\ e_0(\alpha,\beta,\gamma)&=\tfrac{1}{\alpha}\Big(1-\big(\tfrac{\beta}{\gamma}\big)^{\alpha/\gamma}e^{\beta/\gamma}\Gamma(1-\tfrac{\alpha}{\gamma})\big[1-G(\tfrac{\beta}{\gamma};1-\tfrac{\alpha}{\gamma},1)\big]\Big),\text{i.e.}\label{e0}\\ \bar{a}_x(\alpha,\beta,\gamma,\delta)&=\tfrac{1}{\alpha+\delta}\Big(1-\big(\tfrac{\beta e^{\gamma x}}{\gamma}\big)^{(\alpha+\delta)/\gamma}e^{\beta e^{\gamma x}/\gamma}\times\notag\\ &\qquad\qquad\qquad\times\Gamma(1-\tfrac{\alpha+\delta}{\gamma})\big[1-G(\tfrac{\beta e^{\gamma x}}{\gamma};1-\tfrac{\alpha+\delta}{\gamma},1)\big]\Big),\label{ax_hel} \end{align} where $\Gamma(\eta)$ is the gamma function: $\Gamma(\eta)=\int_0^{\infty}y^{\eta-1}e^{-y}dy$, and $G(z;\eta,1)$ is the distribution function of a gamma distributed random variable with shape parameter $\eta$ and scale parameter 1: $G(z;\eta,1)=\frac{1}{\Gamma(\eta)}\int_0^z y^{\eta-1}e^{-y}dy$. \end{theorem} \begin{remark} Even though equation \eqref{ax_hel} may look unwieldy, it only consists of functions that are readily available in common statistical and spreadsheet software, e.g.\ in R and Microsoft Excel. Thus, \emph{no numerical integration is necessary} to evaluate these expressions. Note, however, the findings of Yalta (2008). \end{remark} \begin{remark}\label{ageing} The special case with zero mortality, i.e.\ $\alpha=\beta=0$ (and $\gamma$ arbitrary), yields $\bar{a}_x(0,0,\gamma,\delta) = e_0(\delta,0,\gamma)=1/\delta$ as expected for a continuously compounded perpetuity. With $\alpha>0$ and $\beta=0$, the life-length is exponentially distributed and there is no ageing. Then $\bar{a}_x(\alpha,0,\gamma,\delta) = e_0(\alpha+\delta,0,\gamma)=1/(\alpha+\delta)$. The general case has $\bar{a}_x(\alpha,\beta,\gamma,\delta)=\tfrac{1}{\alpha+\delta}(1-[\dots])$ and one can thus interpret the ellipsis part $[\dots]$ as the \emph{effect of ageing} on the value of the annuity. \end{remark}
With $\delta=0$, the present value of the annuity equals the nominal value, and since the annuity is paid continuously at rate 1, this nominal value is equal to the remaining life-length. Thus, we arrive at the following result, which will also be reached through another route in the proof of Theorem 1. \begin{corollary}\label{ex} $e_x(\alpha,\beta,\gamma)=\bar{a}_x(\alpha,\beta,\gamma,0)=e_0(\alpha,\beta e^{\gamma x},\gamma)$. \end{corollary}
In order to calculate the present values of different life insurance contracts one usually defines the commutation functions \begin{align*} D(x) &= l(x)e^{-\delta x},\\ N(x) &= \int_x^{\infty}D(y)dy,\\ M(x) &= \int_x^{\infty}\mu(y)D(y)dy. \end{align*} The corresponding commutation functions with twice the interest rate are also useful when one wants to calculate not only the expected value of different insurances, but also their variance, see Andersson (2005). In general $M(x)=D(x)-\delta N(x)$, and with Gompertz-Makeham mortality $D(x)=l(x;\alpha+\delta,\beta,\gamma)$, which is easy to evaluate numerically. The only ``difficult'' function is $N(x)$. However, a standard result is that $\bar{a}_x=N(x)/D(x)$, so by Theorem 1 we get a tractable expression also for $N(x)$: \begin{corollary}\label{Nx} $N(x) = D(x)e_0(\alpha+\delta,\beta e^{\gamma x},\gamma)$, where $D(x)=l(x;\alpha+\delta,\beta,\gamma)$. \end{corollary}
It remains to prove Theorem 1.
\begin{proof}[Proof of Theorem 1] The incomplete gamma function is defined as $ \Gamma(\eta,z)=\int_z^{\infty}y^{\eta-1}e^{-y}dy. $ In analogy with the gamma distribution we call $\eta$ the shape parameter. The incomplete gamma function is closely related to the gamma distribution function: \begin{align*} G(z;\eta,1)&=\frac{1}{\Gamma(\eta)}\int_0^z y^{\eta-1}e^{-y}dy=1-\frac{1}{\Gamma(\eta)}\int_z^{\infty}y^{\eta-1}e^{-y}dy = 1 - \frac{\Gamma(\eta,z)}{\Gamma(\eta)}, \end{align*} or, equivalently, \begin{equation}\label{gam_gam} \Gamma(\eta,z)=\Gamma(\eta)(1-G(z;\eta,1)), \end{equation} for $\eta>0$. By partial integration, \begin{align*} \Gamma(\eta,z)&=\int_z^{\infty}y^{\eta-1}e^{-y}dy=\frac{1}{\eta}\left[y^{\eta}e^{-y}\right]_z^{\infty}+\frac{1}{\eta}\int_z^{\infty}y^{\eta}e^{-y}dy\\ &=\frac{1}{\eta}\left(\Gamma(\eta+1,z)-z^{\eta}e^{-z}\right). \end{align*} Combining this with \eqref{gam_gam} yields \begin{equation}\label{gamma_dist} \Gamma(\eta,z)=-\frac{1}{\eta}\big(z^{\eta}e^{-z}-\Gamma(\eta+1)\big[1-G(z;\eta+1,1)\big]\big), \end{equation} which we will use shortly.
Following Andersson (2005), we express $e_0$ with an incomplete gamma function: \begin{align} e_0(\alpha,\beta,\gamma)&=\int_0^{\infty}l(t;\alpha,\beta,\gamma)dt = \int_0^{\infty}e^{-\alpha t-\frac{\beta}{\gamma}(e^{\gamma t}-1)}dt \label{e0_int}\\ \left\{y=\frac{\beta}{\gamma}e^{\gamma t}\right\}&= \left(\frac{\beta}{\gamma}\right)^{\alpha/\gamma}\frac{e^{\beta/\gamma}}{\gamma}\int_{\beta/\gamma}^{\infty}y^{-\frac{\alpha}{\gamma}-1}e^{-y}dy\notag\\ &=\left(\frac{\beta}{\gamma}\right)^{\alpha/\gamma}\frac{e^{\beta/\gamma}}{\gamma}\Gamma\left(-\frac{\alpha}{\gamma},\frac{\beta}{\gamma}\right)\label{e0_gamma}. \end{align}
We continue with the expected remaining life-length at age $x$: $$ e_x(\alpha,\beta,\gamma)=\int_0^{\infty}\frac{l(x+t;\alpha,\beta,\gamma)}{l(x;\alpha,\beta,\gamma)}dt = \int_0^{\infty}e^{-\alpha t-\frac{\beta}{\gamma}e^{\gamma x}(e^{\gamma t}-1)}dt, $$ and by comparing with \eqref{e0_int} we arrive at the conclusion of Corollary \ref{ex}: $e_x(\alpha,\beta,\gamma)=e_0(\alpha,\beta e^{\gamma x},\gamma)$.
By standard arguments, $$ \bar{a}_x(\alpha,\beta,\gamma,\delta) = \int_0^{\infty}e^{-\delta t}\frac{l(x+t;\alpha,\beta,\gamma)}{l(x;\alpha,\beta,\gamma)}dt= \int_0^{\infty}e^{-(\alpha+\delta)t-\frac{\beta}{\gamma}e^{\gamma x}(e^{\gamma t}-1)}dt, $$ and by once again comparing with \eqref{e0_int} equation \eqref{ax} in the theorem is proved.
Equation \eqref{e0} is obtained by using \eqref{gamma_dist} on \eqref{e0_gamma} to rewrite the incomplete gamma function in the latter equation. Equation \eqref{ax_hel} follows immediately from \eqref{ax} and \eqref{e0}. This concludes the proof of Theorem 1. \end{proof}
\begin{remark} Note that for typical values for the parameters, e.g.\ $\alpha=0.001, \beta=0.000012, \gamma=0.101314$ and $\delta=0.026559$ from Andersson (2005), we have in equation \eqref{ax_hel} a shape parameter $1-(\alpha+\delta)/\gamma\doteq 0.727984$ \emph{which is positive}, just as shape parameters for gamma distributions should be. If one has to ones disposal software that can calculate incomplete gamma functions with negative shape parameters, then one can use equation \eqref{e0_gamma} straight away, see e.g.\ Mingari Scarpello et al.\ (2006). Since the effect of ageing on the value of an annuity, cf.\ Remark \ref{ageing}, is not easily seen from \eqref{e0_gamma}, the formulation in the theorem is nevertheless of some independent theoretical interest, regardless of whether it is easy to evaluate functions with negative shape parameters or not. \end{remark}
\section*{References}
\textsc{Andersson, G.} (2005). \textit{Livf{\"o}rs{\"a}kringsmatematik}. Svenska F{\"o}r\-s{\"a}k\-rings\-f{\"o}ren\-ingen, Stockholm. ISBN: 91-974960-1-4
\textsc{Mingari Scarpello, G.; Ritelli, D. and Spelta, D.} (2006). Actuarial values calculated using the incomplete gamma function. \href{http://rivista-statistica.cib.unibo.it/issue/view/50}{\emph{Statistica (Bologna)}} \textbf{66}, no.\ 1, 77--84.
\textsc{Yalta, A.T.} (2008). The accuracy of statistical distributions in Microsoft\raisebox{1ex}{\scriptsize\textregistered} Excel. \textit{Computational Statistics and Data Analysis} \textbf{52}, 4579--4586.\\ \href{http://dx.doi.org/10.1016/j.csda.2008.03.005}{\texttt{doi:10.1016/j.csda.2008.03.005}}
\end{document}
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arXiv
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\begin{definition}[Definition:Conjugacy Class]
The equivalence classes into which the conjugacy relation divides its group into are called '''conjugacy classes'''.
The '''conjugacy class''' of an element $x \in G$ can be denoted $\conjclass x$.
\end{definition}
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ProofWiki
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For $1 \le n \le 100$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal?
Note that $n+1$ and $n$ will never share any common factors except for $1$, because they are consecutive integers. Therefore, $n/(n+1)$ is already simplified, for all positive integers $n$.
Since $1 \le n \le 100$, it follows that $2 \le n+1 \le 101$. Recall that a simplified fraction has a repeating decimal representation if and only if its denominator is divisible by a prime other than 2 and 5. The numbers between 2 and 101 which are divisible only by 2 and 5 comprise the set $\{2, 4, 5, 8, \allowbreak 10, 16, 20, 25, \allowbreak 32, 40, 50, 64, \allowbreak 80, 100\}$. Therefore, there are $14$ terminating decimals and $100 - 14 = \boxed{86}$ repeating decimals.
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Math Dataset
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DOI:10.1007/JHEP05(2013)143
Perturbative calculation of the clover term for Wilson fermions in any representation of the gauge group SU(N)
@article{Musberg2013PerturbativeCO,
title={Perturbative calculation of the clover term for Wilson fermions in any representation of the gauge group SU(N)},
author={S. Musberg and Gernot M{\"u}nster and Stefano Piemonte},
journal={Journal of High Energy Physics},
volume={2013},
pages={1-6}
S. Musberg, G. Münster, S. Piemonte
A bstractWe calculate the Sheikholeslami-Wohlert coefficient of the O(a) improvement-term for Wilson fermions in any representation of the gauge group SU(N) perturbatively at the one-loop level. The result applies to QCD with adjoint quarks and to $ \mathcal{N} $ = 1 supersymmetric Yang-Mills theory on the lattice.
Non-perturbative O(a) improvement of the SU(3) sextet model
M. Hansen
We calculate non-perturbatively the coefficient c_sw required for O(a) improvement of the SU(3) gauge theory with Nf = 2 fermions in the two-index symmetric (sextet) representation. For the…
View 2 excerpts, cites methods
Improved results for the mass spectrum of N = 1 supersymmetric SU(3) Yang-Mills theory
Sajid Ali, G. Bergner,
P. Scior
This talk summarizes the results of the DESY-Munster collaboration for N = 1 supersymmetric Yang-Mills theory with the gauge group SU(3). It is an updated status report with respect to our…
Spectroscopy of four-dimensional N = 1 supersymmetric SU(3) Yang-Mills theory
M. Steinhauser, A. Sternbeck, B. Wellegehausen, A. Wipf
Supersymmetric gauge theories are an important building block for extensions of the standard model. As a first step towards Super-QCD we investigate the pure gauge sector with gluons and gluinos on…
Ward identities in N = 1 supersymmetric SU(3) Yang-Mills theory on the lattice
The introduction of a space-time lattice as a regulator of field theories breaks symmetries associated with continuous space-time, i.e. Poincare invariance and supersymmetry. A non-zero gluino mass…
Numerical Results for the Lightest Bound States in N=1 Supersymmetric SU(3) Yang-Mills Theory.
This work determines the masses of the lightest bound states in SU(3) N=1 SYM theory and shows the formation of a supermultiplet of bound states, which provides a clear evidence for unbroken supersymmetry.
Baryonic states in supersymmetric Yang-Mills theory
Proceedings of The 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018)
In $\mathcal{N}$=1 supersymmetric Yang-Mills theory the superpartner of the gluon is the gluino, which is a spin 1/2 Majorana particle in the adjoint representation of the gauge group. Combining…
The light bound states of N=1$$ \mathcal{N}=1 $$ supersymmetric SU(3) Yang-Mills theory on the lattice
A bstractIn this article we summarise our results from numerical simulations of N=1$$ \mathcal{N}=1 $$ supersymmetric Yang-Mills theory with gauge group SU(3). We use the formulation of Curci and…
Multi-Representation Dynamics of SU(4) Composite Higgs Models: Chiral Limit and Spectral Reconstructions
L. Debbio, Alessandro Lupo, M. Panero, N. Tantalo
We present a lattice study of the SU (4) gauge theory with two Dirac fermions in the fundamental and two in the two-index antisymmetric representation, a model close to a theory of partial…
Nonperturbative renormalization of the supercurrent in $\mathcal{N} = 1$ Supersymmetric Yang-Mills Theory
G. Bergner, M. Costa, H. Panagopoulos, S. Piemonte, Ivan Soler, G. Spanoudes
In this work, we study the nonperturbative renormalization of the supercurrent operator in N = 1 Supersymmetric Yang-Mills (SYM) theory, using a gauge-invariant renormalization scheme (GIRS). The…
Lattice simulations of a gauge theory with mixed adjoint-fundamental matter
G. Bergner, S. Piemonte
In this article we summarize our efforts in simulating Yang-Mills theories coupled to matter fields transforming under the fundamental and adjoint representations of the gauge group. In the context…
Non-perturbatively improved clover action for SU(2) gauge + fundamental and adjoint representation fermions
Anne Mykkanen, J. Rantaharju,
Helsinki Institute of Physics University of Jyvaskyla
The research of strongly coupled beyond-the-standard-model theories has generated significant interest in non-abelian gauge field theories with different number of fermions in different…
O(a) improvement of the axial current in lattice QCD to one-loop order of perturbation theory
M. Luscher, P. Weisz
Non-perturbative O(a) improvement of lattice QCD
M. Luescher, S. Sint, R. Sommer, P. Weisz, U. Wolff
O( a) perturbative improvement for Wilson fermions
S. Naik
Determination of the improvement coefficientcSWup to one-loop order with conventional perturbation theory
S. Aoki, Y. Kuramashi
We calculate the $O(a)$ improvement coefficient c_SW in the Sheikholeslami-Wohlert quark action for various improved gauge actions with six-link loops. We employ the conventional perturbation theory…
Towards N = 1 super-Yang-Mills on the lattice
A. Donini, M. Guagnelli, P. Hernández, A. Vladikas
The gluino-glue particle and finite size effects in supersymmetric Yang-Mills theory
G. Bergner, T. Berheide, G. Münster, U. D. Özugurel, D. Sandbrink, I. Montvay
A bstractThe spectrum of particles in supersymmetric Yang-Mills theory is expected to contain a spin 1/2 bound state of gluons and gluinos, the gluino-glue particle. We study the mass of this…
Towards the spectrum of low-lying particles in supersymmetric Yang-Mills theory
G. Bergner, I. Montvay, G. Münster, U. D. Özugurel, D. Sandbrink
A bstractThe non-perturbative properties of supersymmetric theories are of interest for elementary particle physics beyond the Standard Model. Numerical simulations of these theories are associated…
An algorithm for gluinos on the lattice
I. Montvay
Simulation of 4d$\mathcal{N}=1$ supersymmetric Yang–Mills theory with Symanzik improved gauge action and stout smearing
K. Demmouche, F. Farchioni,
J. Wuilloud
We report on the results of a numerical simulation concerning the low-lying spectrum of four-dimensional $\mathcal{N}=1$ SU(2) Supersymmetric Yang–Mills (SYM) theory on the lattice with light…
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CommonCrawl
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[Submitted on 28 Sep 2018 (v1), last revised 3 Apr 2020 (this version, v3)]
Title:A Non-Perturbative Definition of the Standard Models
Authors:Juven Wang, Xiao-Gang Wen
Abstract: The Standard Models contain chiral fermions coupled to gauge theories. It has been a long-standing problem to give such gauged chiral fermion theories a quantum non-perturbative definition. By classification of quantum anomalies and symmetric invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and the existence of symmetric gapped boundary for the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3+1D lattice, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the Standard Models from the 16n-chiral fermion SO(10) Grand Unification can be defined non-perturbatively via a 3+1D local lattice model of bosons or qubits. Furthermore, we propose that Standard Models from the 15n-chiral fermion SU(5) Grand Unification can also be realized by a 3+1D local lattice model of fermions.
Comments: 24 pages. Two columns. v3: Refinement with detailed discussions. Appendices include viewpoints from perturbative local anomalies and non-perturbative global anomalies (e.g. arXiv:1810.00844, SU(2) = Spin(3) $\subset$ Spin(10)), and co/bordism theories (e.g. $Ω_{D}^{{(\mathrm{Spin}(D) \times \mathrm{Spin}(10))}/{\mathbb{Z}_2^f}}$)
Subjects: High Energy Physics - Theory (hep-th); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Lattice (hep-lat); High Energy Physics - Phenomenology (hep-ph); Quantum Physics (quant-ph)
Journal reference: Phys. Rev. Research 2, 023356 (2020)
DOI: 10.1103/PhysRevResearch.2.023356
From: Juven C. Wang [view email]
[v1] Fri, 28 Sep 2018 17:59:38 UTC (18 KB)
[v2] Thu, 11 Oct 2018 17:38:39 UTC (19 KB)
[v3] Fri, 3 Apr 2020 17:54:00 UTC (162 KB)
cond-mat.str-el
hep-lat
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CommonCrawl
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Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
[asy] import three; // calculate intersection of line and plane // p = point on line // d = direction of line // q = point in plane // n = normal to plane triple lineintersectplan(triple p, triple d, triple q, triple n) { return (p + dot(n,q - p)/dot(n,d)*d); } // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C) { return lineintersectplan(B, B - C, A, B - C); } currentprojection=perspective(2,1,1); triple A, B, C, D, O, P; A = (sqrt(2 - sqrt(2)), sqrt(2 - sqrt(2)), 0); B = (-sqrt(2 - sqrt(2)), sqrt(2 - sqrt(2)), 0); C = (-sqrt(2 - sqrt(2)), -sqrt(2 - sqrt(2)), 0); D = (sqrt(2 - sqrt(2)), -sqrt(2 - sqrt(2)), 0); O = (0,0,sqrt(2*sqrt(2))); P = projectionofpointontoline(A,O,B); draw(D--A--B); draw(B--C--D,dashed); draw(A--O); draw(B--O); draw(C--O,dashed); draw(D--O); draw(A--P); draw(P--C,dashed); label("$A$", A, S); label("$B$", B, E); label("$C$", C, NW); label("$D$", D, W); label("$O$", O, N); dot("$P$", P, NE); [/asy]
The angle $\theta$ is the angle formed by two perpendiculars drawn to $BO$, one on the plane determined by $OAB$ and the other by $OBC$. Let the perpendiculars from $A$ and $C$ to $\overline{OB}$ meet $\overline{OB}$ at $P.$ Without loss of generality, let $AP = 1.$ It follows that $\triangle OPA$ is a $45-45-90$ right triangle, so $OP = AP = 1,$ $OB = OA = \sqrt {2},$ and $AB = \sqrt {4 - 2\sqrt {2}}.$ Therefore, $AC = \sqrt {8 - 4\sqrt {2}}.$
From the Law of Cosines, $AC^{2} = AP^{2} + PC^{2} - 2(AP)(PC)\cos \theta,$ so
\[8 - 4\sqrt {2} = 1 + 1 - 2\cos \theta \Longrightarrow \cos \theta = - 3 + 2\sqrt {2} = - 3 + \sqrt{8}.\]
Thus $m + n = \boxed{5}$.
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Math Dataset
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\begin{document}
\title{The double Eulerian polynomial and inversion tables} \author{Erik Aas} \address{Department of Mathematics, Royal Institute of Technology \\
SE-100 44 Stockholm, Sweden} \email{[email protected]} \date{January 2014}
\maketitle {{\bf Abstract.} We show that the pair $(\des, \ides)$ of statistics on the set of permutations has the same distribution as the pair $(\asc, \row)$ of statistics on the set of inversion tables, proving a conjecture of Visontai. The common generating function of these pairs is the {\it double Eulerian polynomial}.}
\subsection*{The double Eulerian polynomial} The double Eulerian polynomial $A_n(u,v)$ enumerates the number of descents of a permutation and its inverse, \[ A_n(t, s) = \sum_{\pi\in\mathbb{S}_n} u^{\des(\pi)}v^{\des(\pi^{-1})}. \]
It is a natural generalization of the classical Eulerian polynomial $A_n(u, 1)$. The latter polynomial is well-known to be positive in the basis $(u^i(1+u)^{n-i})_{i=0} ^n$. This has been proved in several ways; notably by geometric means \cite{BR}, and by an elegant bijective argument by Foata and Strehl (see \cite{P} for an excellent exposition).
There is no analogous result for the double Eulerian polynomial, though there is a conjectured one by Gessel \cite{B}: $A_n(u,v)$ should be integral and positive in the basis $((uv)^i(u+v)^j(1+uv)^{n-2i-j})_{i,j}$. Visontai \cite{V} gave explicit formulas for the coordinates of $A_n(u,v)$ in this basis, but was unable to prove that they are positive, nor that they are integers. He also conjectured a new way of defining $A_n(u,v)$, as follows (this is our main theorem).
\begin{theo} \label{th_main}
For all $n$,
\[
A_n(u,v) = \sum_{e \in \mathbb{I}_n} u^{\asc(e)} v^{\row(e)}.
\] \end{theo}
Here, $\mathbb{I}_n$ is the set of inversion tables of length $n$, and $(\asc, \row)$ are two statistics, all defined below. We will spend the remainder of this note proving Theorem \ref{th_main}, after giving the necessary definitions.
We identify permutations $w$ of length $n$ with words $w_1\dots w_n$, and with permutation diagrams $\{(i,w_i) : 1 \leq i \leq n\}$, which we read as Cartesian coordinates: a point $(x, y)$ refers to a point $x$ steps to the right and $y$ steps up from $(0,0)$. See Figures \ref{fi_pip} and \ref{fi_pi} for examples. An inversion table $e = e_1\dots e_n$ of length $n$ is any sequence of positive integers satisfying $1 \leq e_i \leq i$ for all $i$ (this differs slightly from the notation in \cite{S}, which we otherwise follow). We will identify these with marked staircases, examples of which are given in Figures \ref{fi_ep} and \ref{fi_e}. For a permutation $w$, let $\DES(w) = \{i : w_i > w_{i+1}\}$ be the {\it set} of descent positions, and $\IDES(w) = \DES(\pi^{-1})$ be the descent set of the inverse permutation. For an inversion table $e$, we define $\ASC(e) = \{i : e_i < e_{i+1}\}$ and $\ROW(e) = \{e_i : 1 \leq i \leq n\} - \{1\}$. Note that the strict inequality in the definition of $\ASC$ is essential, and that we always have $e_1 = 1$. We let $\des = \# \DES$ be the number of descents, and similarly for $\ides$, $\asc$ and $\row$. If $S$ is a set of postive integers, we let $u^S = \prod_{i\in S} u_i$. From now on, we use indeterminates $u_1, u_2, \dots$ and $v$.
\subsection*{Examples}
We give the values of the statistics of the permutation $\pi$ in Figure \ref{fi_pi} and of the inversion table $e$ in Figure \ref{fi_e}. We have $\DES(\pi) = \{1,4,5,7,8\}$, $\IDES(\pi) = 1,3,4,6,8\}$, $\ASC(e) = \{1,2,4,5,7,8\}$ and $\ROW(e) = \{2,3,4,5,7\}$. Thus $\des(\pi) = 5$, $\ides(\pi) = 5$, $\asc(e) = 6$ and $\row(e) = 5$.
To prove Theorem \ref{th_main}, we will prove the stronger statement \begin{equation}
\label{eq_one}
\sum_{\pi\in\mathbb{S}_n} u^{\DES(\pi)}v^{\ides(\pi)} = \sum_{e\in\mathbb{I}_n} u^{\ASC(e)}v^{\row(e)}. \end{equation}
By M\"obius inversion, equation \ref{eq_one} holds if and only if we have \begin{equation}
\label{eq_two}
\sum_{\substack{\pi\in\mathbb{S}_n \\ S \subseteq \DES(\pi)}} s^{\ides(\pi)} =
\sum_{\substack{e\in\mathbb{I}_n \\ S \subseteq \ASC(e)}} s^{\row(e)}, \end{equation}
for each $S \subseteq [n-1]$. The idea now is to fix a subset $S$ of postive integers and induct on $n$ (in a sense to be specified) to prove equation \ref{eq_two}.
Thus fix a subset $S \subseteq \{1, 2, \dots\}$. We define two rooted labeled trees $T_{\mathbb{S}}$ and $T_{\mathbb{I}}$, as follows. The vertices of $T_{\mathbb{S}}$ are all permutations $w_1\dots w_n$ whose length $n$ satisfies $n \notin S$ (in particular, the empty permutation, of length $0$, is a node). A permutation $\pi$ of length $r+s$ is a child of another permutation $\pi'$ of length $r$, where $s$ is smallest such that $s \geq 1$ and $r+s \notin S$, if the first $r$ letters of $\pi$ induce\footnote{that is, if we renumber the first $r$ letters of $\pi$ by $1,\dots,r$, we get the permutation $\pi'$} the same permutation as $\pi'$. It follows that the empty permutation is the root node. Each node in $T_{\mathbb{S}}$ is labeled by the pair $(n, k)$, where $n$ is the length of the permutation and $k$ is its number of inverse descents.
Similarly, the vertices of $T_{\mathbb{I}}$ are all inversion tables $e_1\dots e_n$ whose length $n$ satisfies $n\notin S$, and an inversion table $e$ of length $r+s$ is a child of $e'=e'_1\dots e'_r$ if $e = e'_1\dots e'_r e_{r+1} \dots e_{r+s}$ (with the same condition on $s$ as for $T_{\mathbb{S}}$). Each node is labeled $(n, k)$, where $n$ is the length of the inversion table and $k$ is its value of the row statistic. The empty inversion table is the root node.
To prove equation \ref{eq_two} for our fixed set $S$, it suffices to prove that $T_{\mathbb{S}}$ and $T_{\mathbb{I}}$ are isomorphic as labeled rooted trees. We will do this by producing an isomorphism $\Phi$ between the two trees, which takes a permutation $\pi$ of length $n$ with $k$ inverse descents satisfying $S \cap [n] \subseteq \DES(\pi)$ to some inversion table $e = \Phi(\pi)$ of length $n$, satisfying $\row(e) = k$ and $S \cap [n] \subseteq \ASC(e)$.
We will construct $\Phi$ inductively. Let $\Phi$ map the root of $T_{\mathbb{S}}$ to the root of $T_{\mathbb{I}}$. Suppose that we have already defined $\Phi(\pi') = e'$. We will show, for each $k$, that the number of children of $\pi'$ with $k$ inverse descents equals the number of children of $e'$ whose row statistic equals $k$. This allows us to extend $\Phi$ to all the children of $\pi'$. Thus fix a permutation $\pi'$ of length $r$ and an inversion table $e'$ of length $r$ such that $\ides(\pi') = \row(e') = p$, say. Suppose $s$ is smallest such that $s \geq 1$ and $r+s \notin S$. For children $\pi$ of $\pi'$, we call the first $r$ letters the {\it early} part, and the last $s$ letters the {\it late} part. The children $\pi$ are determined in a bijective way by $(r+1)$-tuples $(x_0, \dots, x_r)$ of nonnegative integers $x_i$ with sum $s$. The bijection\footnote{To prove this is a bijection, note that the last $r$ letters of $\pi$ form a decreasing word since $\pi$ is a child (which implies that $r, r+1, \dots, r+s-1$ are descents).} is given by letting $x_i$ be the number of letters in the late part of $\pi$ which are between (in value) the $i$th and $(i+1)$st largest letters of the early part of $\pi$. Moreover, the number of inverse descents of $\pi$ is\footnote{We use the notation $x_+ = x$ for $x \geq 0$ and $x_+ = 0$ for $x < 0$.} $\ides(\pi') + \sum_{i \in T} (x_i - 1)_+ + \sum_{i\notin T} x_i$, where $T = \IDES(\pi')$. This is hopefully made clear by Figures \ref{fi_pip} and \ref{fi_pi}.
On the other hand we consider children $e$ of $e'$, of length $r+s$. Such $e$ are in bijection with subsets $T \subseteq [r+s]$ of size $s$ by letting $T = \{e_i : r < i \leq r+s\}$ (note that this is a {\it set} since $\{s+1, \dots, r-1\} \subseteq \ASC(e)$ is a child). Moreover, $\row(e) = \row(e') + \#(T^c \cap ROW(e'))$.
By the preceding two paragraphs, all that remains is to prove that the number of $(x_0,\dots,x_r)$ with $\sum_{i=0}^{p-1} (x_i-1)_+ + \sum_{i = p} ^r x_i = t$ and $\sum_{i=0} ^r x_i = s$ (here, we have reordered the $x_i$'s, which clearly does not affect the count) equals the number of $T \subseteq [r+s]$ such that $|T| = s$ and $|T \cap \{p+2, p+3, \dots\}| = t$, for all nonnegative integers $r,s,p,t$.
These two counts are easily seen to be \[ \sum_{a=0}^t {p+1\choose s-t} {s-a-1\choose a}{a+r-p-1\choose a} \] and \[ {p+1\choose s-t}{r+s-p-1\choose t} \] respectively. That they are equal is a classical fact. This finishes the proof of Theorem \ref{th_main}.
\subsection*{Final remarks}
It is interesting to note that while the statement of Theorem \ref{th_main} is symmetric in $t$ and $s$, the proof is not. We have failed to generalize the Theorem to one with two set-valued statistics. If we define $\maj(\pi)$ to be the sum of the elements in $\DES(\pi)$, and $\amaj(e)$ to be the sum of the elements in $\ASC(e)$, then the proof shows that the pairs $(\maj, \ides)$ and $(\amaj, \row)$ are equidistributed. There does not seem to be an obvious generalisation involving $\maj$, $\amaj$ and sums of $\IDES$ and $\ROW$. Finally, it follows from Theorem \ref{th_main} that $\asc$ and $\row$ are equidistributed (since $\des$ and $\ides$ are). I do not know of a direct proof of this fact.
\begin{figure}
\caption{A permutation $\pi' = 325641$ together with an $6+1$-tuple $(x_1, \dots, x_7) = (0,0,2,0,0,1,0)$ determines the child $\pi$ of $\pi'$ given in Figure \ref{fi_pi}. For all the examples in Figures 1-4, the fixed set $S$ can be taken to be $\{1,4, 7,8 \}$.}
\label{fi_pip}
\end{figure}
\begin{figure}
\caption{The child $\pi = 527961843$ referred to in Figure \ref{fi_pip}. We have $\ides(\pi) = \ides(\pi') + (2-1)_+ + 1 = 5$.}
\label{fi_pi}
\end{figure}
\begin{figure}
\caption{An inversion table $e'=123135$ and a set $T = \{1,4,7\}$ determines the child $e$ of $e'$ given in Figure \ref{fi_e}.}
\label{fi_ep}
\end{figure}
\begin{figure}
\caption{The child $e=123135147$ to $e'$ referred to in Figure \ref{fi_ep}.}
\label{fi_e}
\end{figure}
\end{document}
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arXiv
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Acute exposure to air pollutants increase the risk of acute glaucoma
Liping Li1,2 na1,
Yixiang Zhu3 na1,
Binze Han1,2,
Renjie Chen3,4,
Xiaofei Man5,6,
Xinghuai Sun1,2,7,
Haidong Kan3,6 &
Yuan Lei1,2
Ambient air pollution is related to the onset and progression of ocular disease. However, the effect of air pollutants on the acute glaucoma remains unclear.
To investigate the effect of air pollutants on the incidence of acute glaucoma (acute angle closure glaucoma and glaucomatocyclitic crisis) among adults.
We conducted a time-stratified case-crossover study based on the data of glaucoma outpatients from January, 2015 to Dec, 2021 in Shanghai, China. A conditional logistic regression model combined with a polynomial distributed lag model was applied for the statistical analysis. Each case serves as its own referent by comparing exposures on the day of the outpatient visit to the exposures on the other 3–4 control days on the same week, month and year. To fully capture the delayed effect of air pollution, we used a maximum lag of 7 days in main model.
A total of 14,385 acute glaucoma outpatients were included in this study. We found exposure to PM2.5, PM10, nitrogen dioxide (NO2) and carbon monoxide (CO) significantly increased the odds of outpatient visit for acute glaucoma. Wherein the odds of acute glaucoma related to PM2.5 and NO2 were higher and more sustained, with OR of 1.07 (95%CI: 1.03–1.11) and 1.12 (95% CI: 1.08–1.17) for an IQR increase over lag 0–3 days, than PM10 and CO over lag 0–1 days (OR:1.03; 95% CI: 1.01–1.05; OR: 1.04; 95% CI: 1.01–1.07).
This case-crossover study provided first-hand evidence that air pollutants, especially PM2.5 and NO2, significantly increased risk of acute glaucoma.
The onset and progression of multiple diseases connected closely with ambient air pollution [1] including cardiovascular disease [2,3,4,5,6], type 2 diabetes mellitus [7,8,9,10], chronic obstructive pulmonary disease [11, 12], and even cancer [13, 14] and mortality [15]. Recently, the association between glaucoma and ambient air pollution is emerging [16, 17].
Glaucoma is the leading cause of irreversible blindness in the world, which is estimated more than 70 million persons aged 40–80 suffering from this condition worldwide [18, 19]. According to the risk factors, etiology, duration, symptoms, treatment, and prognosis, glaucoma is classified into different types [20]. Both angle closure glaucoma and glaucomatocyclitic crisis can have acute onset. Angle closure glaucoma is presented with an anatomically closed angle which was casued by apposition of the iris [21]. A closed angle prevents the outflow of aqueous humor and hence causes elevated intraocular pressure (IOP) [22]. In acute primary angle closure attack, IOP could reach to 30 mmHg or even higher. There are several risk factors related to angle closure such as female, older age, and Asian ethnicity (e.g. Chinese) [23] The clinical data from our hospital reveals that primary angle closure glaucoma accounted for 50–55% glaucoma patients [21]. Glaucomatocyclitic crisis (also called Posner-Schlossman syndrome), uaually involves recurrent episodes of increased IOP, acute anterior chamber inflammation and keratic precipitates [24]. The etiology of glaucomatocyclitic crisisris is not very clear which may be involved of virus infections. It has the similar clinical manifestation like an acute angle-closure glaucoma because of the initial sudden and remarkable IOP elevation and the mild anterior chamber inflammatory.
High IOP has a similar pathological mechanism with high blood pressure [25]. The association of ambient air pollution with hypertension and blood pressure was investigated by numerous studies [26,27,28,29]. A meta-analysis, which searched seven international and Chinese databases, showed significant associations of long-term or short-term exposures to ambient air pollution with blood pressure and hypertension [30]. In recent years, particulate matter pollution was related to the incidence of self-reported glaucoma or unclassified glaucoma according to the epidemiological studies [31,32,33,34]. And our previous studies showed mice exposed to ambient air pollutants lead to ocular hypertension [35, 36]. However, each type of glaucoma has its own distinctive etiology, it is important to know which types of glaucoma patients are affected by air pollution so that appropriate cautions can be made.
The purpose of the current study is to investigate the impact of air pollutants on the incidence of acute glaucoma attacks including acute angle closure glaucoma and glaucomatocyclitic crisisris. The analysis is conducted based on the outpatient data from two hospitals in Shanghai. Air pollutants include PM2.5 (particulate matter ≤2.5 μm in aerodynamic diameter), PM10 (particulate matter with an aerodynamic diameter < 10 μm), sulfur dioxide (SO2), nitrogen dioxide (NO2), carbon monoxide (CO) and ozone (O3).
Design and population
Date on acute glaucoma outpatient visits were collected between 1, January, 2015 and 31, Dec, 2021 from the Eye Ear Nose and Throat Hospital of Fudan University and Xinhua Hospital Affiliated to Shanghai Jiao Tong University School of Medicine in Shanghai, China. The inclusion and exclusion procedures were shown in Fig. S1. All patients clinically diagnosed with acute angle closure glaucoma or glaucomatocyclitic crisisris by physicians were regarded as acute glaucoma attack and were included in this study. Demographic characteristics, including age, gender, residential addresses and date of outpatient visits were collected. Pearson correlation analysis was conducted to examine the correlations between the air pollutants and meteorological variables. The patients without demographic information, aging under 18 or above 85 years old, and living out of Shanghai were excluded. Moreover, the patients with glaucoma surgery history, prescribing for medicines, suspected as glaucoma were excluded as well. Totally 14,385 cases living in Shanghai city were incorporated into this study (Fig. 1). The study protocol was approved by the Institutional Review Board (IRB) of the Eye Ear Nose and Throat Hospital of Fudan University (IRB#2022027) and adhered to the tenets of the Declaration of Helsinki. The informed consent was waived by the Institutional Review Board (IRB) of the Eye Ear Nose and Throat Hospital of Fudan University.
Address of patients for acute glaucoma attack and air quality monitoring stations in Shanghai, China, during 2015–2021
A time-stratified case-crossover design was applied to evaluate the potential associations of air pollution exposure and outpatient visits for acute glaucoma. In this design, each subject serves as his or her own control by selecting 3–4 control days matched to other days on the week of the same month-year of the outpatient visit day. This design could provide unbiased effect estimate and control the long-term trend and seasonal pattern [37, 38].
Exposure assessment
Air pollution data was derived from the nearest air quality monitoring stations to participants' address on China's National Urban Air Quality Real-time Publishing Platform. We included the data of the daily (24 h) levels of PM2.5, PM10, SO2, NO2, and CO and daily 8 h maximum averages O3 in the analysis. Daily meteorological data (average temperature and relative humid) recorded from the nearest weather stations were also acquired in the China Meteorological Data Sharing Service System (http://data.cma.cn/).
A conditional logistic regression model with polynomial distributed lag model (PDLM) was conducted to quantitatively examine the association between air pollution with outpatient visits for glaucoma. The results were presented as the odds ratios (ORs) of glaucoma incidence associated an interquartile range (IQR) with the 95% confidence intervals (CI) of the air pollutants. The PDLM was widely applied to estimate the lagged impact of environmental factors on health. As flexible "cross-basis" functions, air pollution indices was defined as combinations of natural cubic spline with 2 degrees of freedom (df) for exposure space and 3 df for lag space [39, 40].
To fully capture the delayed effect of air pollution, we used a maximum lag of 7 days in PDLM. Furthermore, considering the nonlinear confounding effects of weather conditions, the model included a smoothing function using natural splines with 6 df for the 3- day moving average temperature and 3 df for 3-day moving average relative humidity to adjust for the nonlinear confounding effects of weather conditions. The public holidays were also adjusted in the model. Furthermore, we used conditional logistic regression model combined with distributed non-linear models (DLNM) to describe the exposure-response associations of air pollution with risk of glaucoma. By examining and plotting cumulative effects, lag days with significant effects was found and then applied to plot the exposure-response (E-R) association. The models fit from the 0.1th to 99.9th percentiles of the concentrations of each pollutant, respectively. We also performed subgroup analyses by gender (male and female) and age (18–44 and 45–85 years) to assess the modifying effects of demographic features. We tested the statistical significance of differences between effect modifications by calculating the 95% confidence interval as
$$\hat{\Big({\mathrm{Q}}_1}-\hat{{\mathrm{Q}}_2}\Big)\pm 1.96\sqrt{\hat{{{\mathrm{SE}}_1}^2}+\hat{{{\mathrm{SE}}_2}^2}}$$
where \(\hat{{\mathrm{Q}}_1}\) and \(\hat{{\mathrm{Q}}_2}\) represent the estimates for the 2 categories, and \(\hat{{\mathrm{SE}}_1}\) and \(\hat{{\mathrm{SE}}_2}\) represent their corresponding standard errors, respectively [41]. To address the multiple testing problem, we applied the Bonferroni correction to adjust the significance the threshold. In addition to the main model described above, we fitted two-pollutant models, each of which included adjustment for one of the other five pollutants in the sensitive analysis.
All statistics analysis were conducted with R software (Version 4.0.2, R Foundation for Statistical Computing, Vienna, Austria). We used the "survival" and "dlnm" packages to fit the conditional logistic regression model and DLNM, respectively.
Descriptive data
Finally, a total of 14,385 medical records of glaucoma outpatients in Shanghai, China, from January 2015 to Dec 2021 were finally included. Wherein 40.9% (5887) were male and the average age was 56.79 (±15.33) years old. Geographic distribution of the included participants were shown in Fig. 1.
Statistics on air pollution levels and weather conditions on outpatient visits day throughout the study period were summerized in Table 1. During the study period, the mean ( ± standard deviation, SD) 24-hour level of PM2.5 and PM10 were 32.3 (±21.3) μg/m3 and 47.3 (±29.5) μg/m3, which were higher than the recommended ambient air quality standard by World Health Organization (WHO) for gaseous air pollution. The daily SO2, NO2, CO and O3 exposure on the outpatient day were 7.0 (±3.8) μg/m3, 40.5 (±19.3) μg/m3, 0.7 (±0.3) mg/m3 and 92.3 (±43.4) μg/m3, respectively. The IQR values of PM2.5, PM10, SO2, NO2, CO and O3 were 26.0 μg/m3, 35.0 μg/m3, 5.0 μg/m3, 27.0 μg/m3, 0.5 mg/m3, 62.0 μg/m3, respectively. For meteorological features, the mean (±standard deviation) of temperature and relative humid were 18.9 (±8.3)°C and 75.9 (±13.6)%. In Table S1, the correlation of air pollutants indicated a strong statistical significance (P < 0.01) with the strongest correlation being between PM2.5 and PM10 (r value is 0.672), and then between PM2.5 and CO (r value is 0.671).
Table 1 Summary statistics on air pollution and meteorological exposure on outpatient day for acute glaucoma attack throughout the study period
Regression results
The overall lag-response relationship curves in association of ambient air pollution exposure with outpatient visits for acute glaucoma on different lag day was showed in Fig. 2. The associated between air pollutants (PM2.5, PM10, NO2 and CO) and the odds of acute glaucoma visits was significant. The lag effect for PM2.5 and NO2 (lag0–3 days) was relatively longer for PM10 and CO (lag0–1). Specifically, an IQR increase in PM2.5 (26 μg/m3), PM10 (35 μg/m3), NO2 (27 μg/m3) and CO (0.5 mg/m3) was associated with 7% (OR: 1.07; 95%CI: 1.03–1.11), 3% (OR:1.03; 95% CI: 1.01–1.05), 12% (OR: 1.12; 95% CI: 1.08, 1.17) and 4% (OR: 1.04; 95% CI: 1.01, 1.07) higher odds of acute glaucoma visits (Table 2). Besides, the lag effect for SO2 and O3 associated acute glaucoma visits was not statistically significant.
Overall lag structure in association of ambient air pollution exposure with acute glaucoma attack on different lag day. Panels A to F were the associations between OR of acute glaucoma attack and air pollution exposure, including (A) PM2.5, (B) PM10, (C) SO2, (D) NO2, (E) CO and (F) O3, respectively. The solid lines are odds ratios of acute glaucoma attack; the shaded areas were the 95% confidence intervals
Table 2 Odds ratios (95% confidence intervals) of acute glaucoma attack per IQR increase in ambient air pollution exposure, stratified by sex and age
The associations between outpatient visits for acute glaucoma and air pollutants, including PM2.5, PM10, SO2, NO2, CO and O3, were illustrated in the cumulative E-R curves in Fig. 3. Lag days with significant effects (lag 0–3 days for PM2.5, NO2 and lag 0–1 days for PM10, SO2, CO and O3) was applied to plot the curves. In general, E-R curves for PM2.5, PM10 and CO were linear, but the acute glaucoma odds only significantly increased when the concentrations of PM2.5, and PM10 were above 50 μg/m3 and 60 μg/m3, respectively. The slope for NO2 exhibited significant increment over 40 μg/m3 and becomen flat over 80 μg/m3.
Cumulative exposure-response curves for associations between air pollution and acute glaucoma attack in Shanghai, China, January 2015 to March 2021. Panels A to F were the associations between OR of acute glaucoma attack and air pollution exposure, including (A) PM2.5, (B) PM10, (C) SO2, (D) NO2, (E) CO and (F) O3, respectively. These penalized splines regression models fit from the 0.1th to 99.9th percentiles of the concentrations of each pollutant, respectively. The solid lines are odds ratios of glaucoma; the shaded areas were the 95% confidence intervals
In addition, the stratified analysis showed suggestive effect modification of gender and age (Table 2). Female patients and the patients aged over 45 years were found to have relatively higher odds of glaucoma visits associated with air pollution, although no significant difference was found in difference analysis. And the result of in difference analysis was also unsignificant in terms of the association between acute glaucoma attack and other pollutants.
In the sensitive analysis, the associations between outpatient visits for acute glaucoma attack and air pollutants were relatively robust when one of the other five pollutants was adjusted in two-pollutants model (Fig. S1).
In this study, we observed an increase the risk of acute glaucoma associated with ambient air pollutants, with inconsistent delayed effects on extended lags. Exposure to air pollutants (PM2.5, PM10, SO2, NO2, and CO) was related to increased odds of outpatient visits for acute glaucoma. Specifically, PM2.5, PM10, NO2 and CO exhibited relatively stronger association on acute glaucoma outpatient visits with longer delayed effects.
PM2.5 was associated with glaucoma incidence across different countries and ethnic groups [17, 31, 33]. In UK, significant correlations were showed between PM2.5 exposure and occurrence of glaucoma [16, 17]. However, in the UK study, the glaucoma diagnosis was based on patient's self-report without a clear diagnosis concerning the exact type of glaucoma, and the study conducted in Taiwan had a very small sample size of a few hundreds of primary angle closure glaucoma (PACG) patients [16]. In our current study, a big sample size of dataset were recruited over 14 thousand patients with clear diagnosis. Furthermore, both PM2.5 and PM10 showed a faster increment at higher concentrations (Fig. 3A).
It is possible that PM pollutions acted as a trigger for marked IOP elevation in angle closure glaucoma and glaucomatocyclitic crisis patients. Previous study showed that some PM particles could penetrate the cornea and entre the anterior chamber of the eye [35]. Topical administration of PM2.5 suspensions resulted in IOP elevation [35, 42], which was associated with increased oxidative stress and related NLRP3 inflammasome mediated pyroptosis in outflow control cells and tissues [35, 36]. In angle closure glaucoma there was an appositional or adhesion closure of the anterior chamber angle. It is possible that PM pollution may trigger angle closure by some mechnisms that previously existed in the narrow anterior chamber angle, thus is presented as an acute episode. In addition to anatomical predisposing factors, the PM in the anterior segment may cause oxidative stress and inflammation of the tissues contributing to the marked IOP elevation. Virus infections, such as infections of cytomegalovirus [43, 44], varicella-zoster virus [45], herpes simplex virus [45, 46] and Helicobacter pylori [47] and immune mediators [48] were found in the aqueous humor of glaucomatocyclitic crisis patients, which were thought as the initial events of the disease [24]. And it is known that the spread of virus was positively correlated to air pollutions [49] and air pollutions may cause immune disorder [50]. Thus, it is possible that PM particles could act as a carrier for the viruses and cause infections and inflammation, or by provoking immune response, which leads to IOP elevation in the human eye.
The ORs between PM2.5 or PM10 exposure and glaucoma outpatients incidence (PM2.5, OR:1.07; 95%CI: 1.03–1.11; PM10, OR:1.03; 95% CI: 1.01–1.05; Table 2) were similar to the cohort study conducted in UK which reported that diagnosis of glaucoma was more likely to be reported by people in higher PM2.5 concentration areas (PM2.5 OR: 1.06; 95%CI:1.01–1.12) [17]. The discrepancy of the results may be attributed to specific study populations, geographic regions, sensitivities of glaucoma subtypes to PM2.5 exposure, differences of PM2.5 components and concentrations. These evidences suggest that air pollution may promote the initiation and progression of glaucoma.
NO2 and CO were major toxic atmospheric pollutants [51], however, their relationships with glaucoma incidence were scarcely studied. Based on our findings, gaseous pollutants, NO2 and CO, were mildly associated with odds of acute glaucoma attack, with the association occurred at lag 0–3 and 0–1 (Fig. 2). The invisible CO is a chemically-inert gas and inhaled CO can combine with hemoglobin to form carboxyhemoglobin, which makes hemoglobin lose the ability to carry oxygen [52]. NO2 is an ubiquitous atmospheric pollutant derived from emissions of NO, the major source of which are emissions from motor vehicles. Previous studies revealed that when SO2, NO2, and O3 increased 10 μg/m3 in and CO, 1 mg/m3, hospital admissions for ischemic stroke increased 1.37, 1.82, 0.01, and 3.24%, respectively [53]. NO2 inhalation exposure exerted injuries to lung, heart and brain, which were possibly related with oxidative stress and inflammation [54,55,56,57]. A time series prospective study conducted in Chiang Mai, Thailand reported that NO2 was positively associated with eye irritation (adjusted ORs (ROAORs: 1.024 to 1.229), and CO was positively related to lower heart and lung symptoms (adjusted ORs: 1.117 and 1.137) [58]. Large amount of CO could cause visual dysfunction [59]. It is interesting that low-dose CO inhalation protected RGCs from optic nerve injury [60], and carbon monoxide-releasing molecules (CORMs) derived from CO lowed IOP of rabbits in two ocular hypertension models [61]. Clearly these results were based on the effect of CO in a short term. This clinical population study suggestted that long term CO therapy for the treatment of glaucoma should be viewed with caution. The long-term effect and dose response of CO to the IOP is uncertain.
Female glaucoma patients seemed slightly more susceptible to air pollutants (Table 2), which is consistent with women having a higher odds of developing primary angle closure glaucoma [62,63,64]. Though the point estimates of the NO2 effect on acute glaucoma were more pronounced among the female and the patients aged over 65 years, no statistically significant differences was found in our current study. According to a meta-analysis, the estimated PACG prevalence in Chinese women was higher compared with men (1.9% vs 1.1%; adjusted OR: 1.75, 95% CI, 1.20–2.56; P = 0.004) [65]. In Korea, women had a 2.56 folds higher incidence rate of acute angle closure glaucoma than men [66]. Though a study reported female (56.6%) was more at odds of developing glaucomatocyclitic crisisris, most studies have shown a male predilection [67]. Additionally, although age was a strong risk factor of glaucoma [68, 69], the association between glaucoma incidence and air pollution was independent of age. It is possible that PM2.5 exposure might affect glaucoma patients at full life circle.
In different periods, air pollutants show different correlations: the principal components before the Spring Festival were O3 and NO2, and after the Spring Festival, they were PM2.5 and CO, while the principal components before the lockdown in 2020 were PM2.5 and CO, and during lockdown they were O3 and NO2 [70]. And the elements of PM2.5 also varied in different places, so are the toxicities of PM2.5. for eample, relatively high crustal elements such as Al, Si, Ti, and Fe were detected in PM2.5 which reflected the undergoing major construction nearby the campus in shanghai [71]. While air pollutants in Lanzhou are rich in polycyclic aromatic hydrocarbons (PAHs), which mainly comes from coal combustion industries [72]. Higher toxicity was observed in PAHs riched PM2.5 [73]. Hence, the different sources may cause the effect size differences, which needs further investigation.
There are several limitations to our study. First of all, the exposure level was based on the value of the nearest monitors matched to participants' addresses and did not consider participants' travel history. Secondly, besides ambient pollutions, mental state, diet, behavior and socioeconomic status may influence the outpatient visits as well. Thirdly, the outpatient data were collected from two hospitals in Shanghai whose residential addresses were in Shanghai, so more care needs to be taken in interpreting these results, and further investigation is needed to verify the association in a large scale, multi-centre study to control bias.
Our study found that air pollution exposure was related to outpatient visits for glaucoma. And PM exposure showed stronger association with outpatient visits for acute glaucoma. This finding and the pathophysiology mechanisms need to be further confirmed and investigated.
The datasets generated and/or analysed during the current study are not publicly available, but are available from the corresponding author on reasonable request.
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This study is supported by the International Science and Technology Partnership Project of Shanghai Committee of Science and Technology (No. 21230780200) and the National Natural Science Foundation of China (92043301, 91843302, 82070959, 82271082), Shanghai Natural Science Fund General Project (21ZR1411500, 22ZR1409900), National Key Research and Development Program of China (2020YFA0112700), the State Key Program of National Natural Science Foundation of China (82030027), the subject of major projects of National Natural Science Foundation of China (81790641). The sponsor or funding organization had no role in the design or the conduct of this research.
Liping Li and Yixiang Zhu contributed equally to this work.
Department of Ophthalmology & Visual Science, Eye Institute, Eye & ENT Hospital, Shanghai Medical College, Fudan University, Shanghai, 200031, China
Liping Li, Binze Han, Xinghuai Sun & Yuan Lei
NHC Key Laboratory of Myopia, Chinese Academy of Medical Sciences, and Shanghai Key Laboratory of Visual Impairment and Restoration, Fudan University, Shanghai, 200031, China
School of Public Health, Key Lab of Public Health Safety of the Ministry of Education, NHC Key Lab of Health Technology Assessment, IRDR ICoE on Risk Interconnectivity and Governance on Weather/Climate Extremes Impact and Public Health, Fudan University, P.O. Box 249, 130 Dong-An Road, Shanghai, 200032, China
Yixiang Zhu, Renjie Chen & Haidong Kan
Shanghai Typhoon Institute/CMA, Shanghai Key Laboratory of Meteorology and Health, Shanghai, 200030, China
Renjie Chen
Department of Ophthalmology, Xinhua Hospital Affiliated to Shanghai Jiao Tong University School of Medicine, Shanghai, China
Xiaofei Man
Children's Hospital of Fudan University, National Center for Children's Health, Shanghai, China
Xiaofei Man & Haidong Kan
State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science, Institutes of Brain Science, Fudan University, Shanghai, 200032, China
Xinghuai Sun
Liping Li
Yixiang Zhu
Binze Han
Haidong Kan
Yuan Lei
Liping Li: Data curation, Investigation, Methodology, Visualization, Writing –review & editing, Writing - original draft. Yixiang Zhu: Investigation, Methodology, Formal analysis, Visualization, Writing –review & editing, Writing - original draft. Binze Han: Data curation, Writing –review & editing, Writing - original draft. Renjie Chen: Conceptualization, Validation, Supervision, Writing – review & editing. Xiaofei Man: Data curation, Conceptualization, Validation, Supervision, Writing – review & editing. Xinghuai Sun: Data curation, Conceptualization, Funding acquisition, Validation, Supervision, Writing – review & editing. Haidong Kan: Conceptualization, Methodology, Funding acquisition, Validation, Supervision, Writing – review & editing. Yuan Lei: Data curation, Conceptualization, Methodology, Funding acquisition, Validation, Supervision, Writing – review & editing. The authors read and approved the final manuscript.
Correspondence to Xiaofei Man, Xinghuai Sun, Haidong Kan or Yuan Lei.
The study protocol was approved by the Institutional Review Board (IRB) of the Eye Ear Nose and Throat Hospital of Fudan University (IRB#2022027) and adhered to the tenets of the Declaration of Helsinki. The informed consent was waived by the Institutional Review Board (IRB) of the Eye Ear Nose and Throat Hospital of Fudan University.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional file 1: Fig. S1.
Flow chart of the inclusion and exclusion of study population. Table S1. Pearson correlation coefficient of air pollutants and meteorological variables. Fig. S2. Odds ratios (95% confidence intervals) of acute glaucoma attack per IQR increase in ambient air pollution exposure in two-pollutant model.
Li, L., Zhu, Y., Han, B. et al. Acute exposure to air pollutants increase the risk of acute glaucoma. BMC Public Health 22, 1782 (2022). https://doi.org/10.1186/s12889-022-14078-9
Acute glaucoma
Case-crossover study
Air Pollution and Health
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CommonCrawl
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Efficient implied alignment
Alex J. Washburn1 &
Ward C. Wheeler1
Given a binary tree \(\mathcal {T}\) of n leaves, each leaf labeled by a string of length at most k, and a binary string alignment function ⊗, an implied alignment can be generated to describe the alignment of a dynamic homology for \(\mathcal {T}\). This is done by first decorating each node of \(\mathcal {T}\) with an alignment context using ⊗, in a post-order traversal, then, during a subsequent pre-order traversal, inferring on which edges insertion and deletion events occurred using those internal node decorations.
Previous descriptions of the implied alignment algorithm suggest a technique of "back-propagation" with time complexity \(\mathcal {O}\left (k^{2} * n^{2}\right)\). Here we describe an implied alignment algorithm with complexity \(\mathcal {O}\left (k * n^{2}\right)\). For well-behaved data, such as molecular sequences, the runtime approaches the best-case complexity of Ω(k∗n).
The reduction in the time complexity of the algorithm dramatically improves both its utility in generating multiple sequence alignments and its heuristic utility.
Implied Alignment (IA) was proposed by [1] as an adjunct to Direct Optimization (DO) [2, 3] to be used in phylogenetic tree search to provide both verification and more rapid heuristic analysis. The method was originally implemented in later versions of MALIGN [4] and has been a component of POY [5–9] since its inception. A more formal description of the algorithm was presented in [6] and [2]. Although originally designed for alignment-free phylogenetic analysis (dynamic homology, [10]), the procedure was first used as a stand-alone multiple sequence alignment (MSA) tool by [11] in their analysis of skink systematics.
IA was originally described in the context of parsimony-based phylogenetic analysis and was later extended to probabilistic model-based approaches by [12] and its implementations were described by [9, 13]. Similar MSA approaches also based in probabilistic analysis have been described e.g by [14] and [15], and implemented in PRANK [16]. Whiting et al. found that IA was superior (in terms of tree optimality score) to other MSA methods in both parsimony and likelihood analyses. This observation has been repeated multiple times (e.g. [17–20]; summarized in [21]). The use of IA as an MSA algorithm as well as its use in the "static approximation" procedure [22] benefits greatly from improvements in the time complexity we present in this paper.
In a broader context, IA is a heuristic solution to the NP-hard Tree Alignment Problem (TAP) defined by [23]. As such, any individual IA is not guaranteed to be either optimal or unique, with potentially an exponential number of equally optimal implied alignments for any given binary tree.
The IA algorithm presented in this paper takes a different intellectual approach to deriving alignments than earlier versions of IA. Previous algorithmic approaches relied on DO assigning median sequences to the graph vertices. These sequences were then consumed by IA to produce the full alignment. Here, we describe IA as assigning "preliminary contexts" to the graph vertices, and later consuming these contexts to produce the median sequences and the full alignment.
The algorithm uses the repeated application of a pairwise string alignment function to perform an efficient MSA for a given binary tree whose leaves are labeled by strings, i.e. the tree describes the relationship of those strings. The more similar the initial leaf labelings the better the algorithm performs. Thus, while this algorithm has general use for performing an MSA, it is especially well-suited for the alignment of biological sequences where the strings are highly similar and a binary tree describing the strings' relationships can be provided. Below, we provide an example of the IA algorithm's performance on biological data.
Definition of the heuristic function
In order for an MSA to be inferred, there are constraints on the heuristic alignment function used to decorate the tree prior to performing the IA algorithm. As long as these constraints are satisfied, the implementation details of the function are agnostic to the IA algorithm presented here.
Let Σ be a finite alphabet of symbols such that |Σ|≥3. Let (−)∈Σ be a gap symbol, which will have a special meaning in the context of an alignment. Let \(\mathcal {P}_{\geq 1}({X})\) denote the powerset of X, minus the empty set. Let ΣΓ be the alphabet of the following symbols:
$$\begin{array}{*{20}l} \Sigma_{\Gamma} & = {\mathtt{BOTH}} \;\;\;\; \mathcal{P}_{\geq1}(\Sigma) \;\; \mathcal{P}_{\geq1} (\Sigma) \\ & \;\, | \;\; {\mathtt{LEFT}} \;\;\;\; \mathcal{P}_{\geq 1} (\Sigma) \\ & \;\, | \;\; {\mathtt{RIGHT}} \;\; \quad\quad\quad\, \;\; \mathcal{P}_{\geq 1} (\Sigma) \\ & \;\, | \;\; {\mathtt{GAPPED}} \end{array} $$
That is, ΣΓ contains all pairs of elements of \(\mathcal {P}_{\geq 1} (\Sigma)\) tagged as BOTH, all elements of \(\mathcal {P}_{\geq 1} (\Sigma)\) tagged as LEFT, all elements of \(\mathcal {P}_{\geq 1} (\Sigma)\) tagged as RIGHT, and an additional element GAPPED. This construction of ΣΓ extends the original alphabet Σ to preserve alignment information in the algorithms presented below. Note that if |Σ|=x then |ΣΓ|=22x. This follows from the fact that \(|\mathcal {P}_{\geq 1} (\Sigma)|\) is equal to one less than the size of the power set of Σ, due to \(\mathcal {P}_{\geq 1} (\Sigma)\) disallowing the empty set.
Let \(\Sigma ^{*}_{\Gamma }\) be the set of all finite strings over the alphabet ΣΓ. Let \(\otimes : \Sigma ^{*}_{\Gamma } \times \Sigma ^{*}_{\Gamma } \rightarrow \left (\mathbb {R}_{\geq 0},\; \Sigma ^{*}_{\Gamma }\right)\) be a heuristic function that returns a nonnegative alignment cost and an alignment result in \(\Sigma ^{*}_{\Gamma }\). It is required that ⊗ be commutative but it need not be associative. Both of these constraints will be explored later in the "Discussion" Section. These constraints are necessary but not sufficient for a heuristically optimal implied alignment to be inferred on the alignment function.
It is worth noting the motivation of the constructions defined above. Most pairwise string alignment functions take two finite strings of symbols from the original alphabet and supply a new finite string of symbols from the original alphabet. We can represent this class of pairwise string alignment functions by letting Σ∗ be the set of all finite strings over the alphabet \(\mathcal {P}_{\geq 1} (\Sigma)\) and letting \(\odot : \Sigma ^{*} \times \Sigma ^{*} \rightarrow \left (\mathbb {R}_{\geq 0},\; \Sigma ^{*} \right)\). The results of ⊙ can contain cases of ambiguity where it cannot be inferred which input elements correspond to which output elements, but the construction of ⊗ never produces these cases of ambiguity due to the tagging of each element. The preservation of this non-ambiguous relationship between input and output is required for the algorithmic improvements presented below.
Overview of the implied alignment algorithm
The IA algorithm provides an MSA for a binary tree \(\mathcal {T}\) of n leaves, each leaf containing a string with symbols in Σ and length at most k. To generate this alignment, we will traverse \(\mathcal {T}\) twice. First, we perform a post-order traversal—from the leaves to the root—assigning the results of ⊗ as a "preliminary context" decoration to each node. Second, we perform a pre-order traversal—from the root to the leaves—aligning each preliminary context with its parent to assign a "final alignment" decoration to each node.
Using a binary string alignment function (like ⊗) to produce an MSA efficiently relies on the output of binary operations combined across the "global scope" of \(\mathcal {T}\). However, at each step in the post-order traversal, the only information known at a given node is the information contained in its subtree. Therefore, information for the entirety of \(\mathcal {T}\) is only known at completion of the post-order traversal. When performing the subsequent pre-order traversal, we take the "complete" scope available at the root node and thread the information towards the leaves. At each pre-order step, we take the "complete" context threaded from the root and combine it with the preliminary context derived during the post-order pass to assign the final alignment on that node. Thus, we collect all requisite information for an MSA during the post-order traversal and then apply that information during the pre-order traversal to derive the MSA.
As noted above, the time complexity of the IA algorithm's pre-order traversal in previous work was \(\mathcal {O}\left (k^{2} * n^{2}\right)\). We are able to improve this by, during the post-order pass, tagging each element of a string on a node vx with information that notates on which subtree of vx that element originated. We can lift each symbol in Σ into ΣΓ through the alignment process. Upon completion of the post-order traversal, each node in \(\mathcal {T}\) will have a string in \(\Sigma _{\Gamma }^{*}\). Each element of said strings are tagged with one of four options enumerated above, representing from which child node the information of that element originated, relative to that node. Those tagged BOTH originated from both subtrees, those tagged LEFT originated from the left subtree, those tagged RIGHT originated from the right subtree, and those tagged GAPPED originated from neither subtree (i.e., elsewhere in \(\mathcal {T}\)). Because elements tagged GAPPED originated from neither subtree, GAPPED those elements cannot be created during the post-order, only derived during the pre-order.
This tag on each element provides contextual information that allows for an efficient processing of the elements in the pre-order traversal. During the pre-order traversal, a node's preliminary context is "zipped" with the parent's alignment in order to derive its final alignment. We will show that this tagging and "zipping" process is a substantial improvement over previous work, reducing the time complexity from quadratic to linear in the length of the strings. It is worth noting that this tagging can be represented as a succinct data structure per [24], requiring only two additional bits per element.
An example heuristic function
We will provide an example definition of ⊗ in Algorithm 1 sufficient for the IA algorithm, though there are other sufficient definitions of ⊗. The candidate function fitting the description of ⊗ we present will be defined as a extension of the Needleman-Wunsch [25] algorithm for pairwise string alignment. The algorithm is modified along the same the lines that DO modified the dynamic programming technique of [26], with an additional step taken to produce the tagged elements in the output alignment. Algorithm 1 (described below) is used to generate the results presented in the "Methods" section.
First, we decide deterministically which of the two input strings is assigned to the top (columns) of the alignment matrix and which string is assigned to the left side (rows). We assign the input strings based on the data they contain. The longer string is assigned to the columns of the alignment matrix, the shorter string to the rows. If the strings are the same length, we take the first string under the lexical ordering of their elements and assign it to the columns and assign the second string to the rows of the alignment matrix. In the case that the strings are identical, the alignment is trivial. If the first string supplied to ⊗ was not assigned to the rows of the matrix, then we must swap the LEFT and RIGHT tags of the resulting string alignment before returning the result. This consistency in assignment ensures the commutativity of ⊗, which is necessary to enforce consistency of the IA algorithm. Commutativity of ⊗ ensures that the IA algorithm provides the same alignment results for isomorphic tree labeling (i.e. ensures label invariance).
We now apply a memoized update procedure [27], a common element of dynamic programming algorithms such as the Needleman-Wunsch alignment. The subsequent "traceback," however, is notably modified from the original Needleman-Wunsch procedure. The upward, leftward, and diagonal directional arrows used to produce the alignment are additionally used to tag each element as LEFT, RIGHT, or BOTH, respectively. These tagged pairwise alignments will be consumed on the subsequent pre-order traversal of \(\mathcal {T}\) when merging preliminary contexts. Storing this information for each element of the pairwise alignment allows a more efficient generation of the subsequent multiple string alignment, allowing for an asymptotic improvement over the previous IA algorithm. This additional tagging detail is the key difference between previous alignment methods and the one presented in this paper.
The example ⊗ presented in Algorithm 1 is of Θ(k2) complexity in both time and space, where k is the length of the longer string. For clarity, while this example function is presented as a modification of the well understood Needleman-Wunsch algorithm (without explicit memoization), this tagging approach can be incorporated into more sophisticated pairwise string alignment algorithms. For instance, by using the method described by [28], this algorithm's time complexity could be improved to \(\mathcal {O}\left (k * s \right)\), where s is the edit distance between the strings. Alternatively, by using the method described by [29], this algorithm could be improved to use \(\mathcal {O}\left (k \right)\) space. Affine gap models [30] can also be incorporated via the method of [2].
The operator \(\sigma \,:\, \mathcal {P}_{\geq 1}(\Sigma) \times \mathcal {P}_{\geq 1}(\Sigma) \rightarrow \left (\mathbb {R}_{\geq 0},\, \mathcal {P}_{\geq 1}(\Sigma) \right)\) presented in Algorithms 1, 3, and 4 represents a metric for determining the transition cost between symbols in \(\mathcal {P}_{\geq 1} (\Sigma)\). The metrics used in our data sets can be found in Table 1. The metrics presented in Table 1 show the transition cost between elements of Σ. However, these metrics can be expanded to define the transition costs between elements of \(\mathcal {P}_{\geq 1}(\Sigma)\) in the manner described by [2]. Note that σ can also be a more complex metric than those presented here, for instance a metric with affine or logarithmic affine gap costs, and be compatible with the IA algorithm. For usage of σ, see Algorithms 1, 3, and 4.
Table 1 Metric costs of σ0, σ1, and σ2
Description of post-order traversal
The post-order traversal (leaves to the root) of the binary tree \(\mathcal {T}\) is a straightforward procedure, see Algorithm 6. We assign preliminary contexts and costs to each node, vx, of \(\mathcal {T}\). These preliminary contexts will be consumed to assign a final alignment in the subsequent pre-order traversal of the tree. The post-order traversal described here is very similar to the DO post-order traversal described by [1], differing only in the use of ⊗ which captures the preliminary context of a subtree, instead of generating a preliminary median string assignment.
First, for each leaf node, vx, we set vx.cost to 0. Additionally, if vx is of type Σ∗ and not of type \(\Sigma ^{*}_{\Gamma }\)—i.e. if it has been decorated with a finite string of symbols from the alphabet Σ, and it is not decorated with a finite string of preliminary contexts over the alphabet ΣΓ—then we call INITIALIZESTRING(vx.prelimString) to apply the transformation \(\Sigma ^{*} \rightarrow \Sigma ^{*}_{\Gamma }\).
On each internal node, vy with children vl and vr, of \(\mathcal {T}\), we call vl⊗vr. The resultant prelimString is assigned to vy.prelimString, and the sum of the vl.cost, vr.cost, and the alignment cost of vl⊗vr is assigned to vy.cost. By performing this operation in a post-order traversal over \(\mathcal {T}\), we propagate the preliminary contexts and costs returned from the calls to ⊗ from the leaves to the root.
Upon completion of the post-order traversal, each internal node contains the preliminary context information and the cost for the corresponding subtree. Consequently, when the post-order traversal is complete, the root node contains the preliminary context information of the full leaf set of strings and the alignment cost for the entire tree \(\mathcal {T}\). In the pre-order traversal, we will consume this preliminary context to perform an (efficient) alignment on the strings.
Because the post-order traversal can be performed using any valid definition of ⊗, the complexity of the post-order traversal is dependent on the complexity of the heuristic alignment function used. Let the complexity of ⊗ be defined as H(k), where k is the maximum string length of the leaf labels of the tree \(\mathcal {T}\). Then post-order traversal runs in \(\mathcal {O}(H(k) * n)\) time and space, where n is the number of leaves in the binary tree \(\mathcal {T}\). If we were to use Ukkonen's method with the ⊗ described in Algorithm 1, the post-order traversal would run in \(\mathcal {O}(k * s * n)\) time and space, where s is the maximum edit distance between any two strings.
Description of pre-order traversal for final alignments
The pre-order traversal (from the root to the leaves) of the binary tree \(\mathcal {T}\) consumes the preliminary context decorations on each node created in the post-order traversal in order to assign final alignment decorations of \(\Sigma _{\Gamma }^{*}\) to each node, see Algorithm 7. First, the root node must be initialized for the pre-order traversal by assigning the root's preliminary context to the root's final alignment. By initializing the root node in this manner, the root node is consistent with the treatment of any other parent node when deriving the internal node alignments in Algorithm 8.
For each non-root node, vc, we first determine whether vc is the left or right child of its parent. This is required because LEFT-tagged elements originate from alignments of the left subtree and RIGHT-tagged elements originate from alignments of the right subtree, and we must use this information when deriving the final alignment of vc.
The final alignment of the parent of vc, vp, will necessarily be of greater than or equal length to vp's preliminary context, because vp's final alignment contains all the information from the contexts of vp's subtrees as well as the information from the rest of the tree, that is, the contexts of all of the subtrees of every ancestor node to vp. The preliminary context of vp is also of greater than or equal length to preliminary context of vc, due to the vp's context containing all information from vc's context, plus the addition of vc's sister subtree. The resulting value assigned to vc's final alignment will have the same length as the final alignment assigned to vp. Since this invariant length is maintained from the root node to the leaf nodes' final alignment assignments, all alignments will have the same length. This constitutes a simple inductive argument that the final alignment assignment of each node will be of equal length and constitute a genuine string alignment.
The final alignment for vc is derived by performing a "sliding zip" over vp's final alignment, vp's preliminary context, and vc's preliminary context. vp's final alignment is used as the basis of the zip. The inputs to this alignment are the preliminary contexts of vp and vc and the final alignment of vp. At each step of the "sliding zip," one element of vp's final alignment will be consumed and one element of vc's final alignment will be defined. Additionally, at each step of the zip, one of: an element from vp's preliminary context, elements from both vp's and vc's preliminary contexts, or no elements from either node's context, will be consumed. Finally, we define an element of vc's final alignment to be either an element from vc's preliminary context or a gap. The process is called a "sliding zip" because, due to the varying lengths of the three inputs, the elements of vp's and vc's preliminary contexts do not have an immediately apparent index with which they correspond to vp's final alignment, which is used as the basis of the zip. Rather, the elements of vp's and vc's preliminary contexts "slide" through the zipping process, and their corresponding indices with vp's final assignment is deduced dynamically. The logic applied in the "sliding zip" is to propagate gaps from the final alignment of vp, which contains the gaps of the entire tree above vc, down to vc, and when not dealing with a gap propagated from an ancestor node to vc, to align the non-gap elements or introduce a new gap to be propagated. The "sliding zip" process is often easier to understand by stepping through the algorithm. An example alignment of this "sliding zip" process for two internal nodes is shown in Fig. 1.
Example pre-order alignment for a parent node and it's left child
There are five cases determining the derivation of each index of vc's final alignment. The cases are presented in the pseudocode of Algorithm 8, in Fig. 1, and described below.
Case 0: When the element of vp's final alignment is "GAPPED," then the next element of vc's final alignment is "GAPPED."
Case 1: When the sliding zip has consumed all elements of the vc's preliminary context, then the next element of vc's final alignment is "GAPPED." Because we only define the next element of vc's final alignment to be either an element from vc's preliminary context or a gap, the latter is the only choice.
Case 2: When the element of vp's final alignment is "BOTH," then we consume the next elements of both vp's and vc's preliminary contexts, and the next element of vc's final alignment is vc's consumed preliminary context element. Because vp's final alignment element was marked as an alignment event, we know that vc was aligned with its sister subtree at this index, and that vc's preliminary context element is the correct element for this index of the alignment.
Case 3: When both vp's final alignment element and vp's preliminary context element are "LEFT," and vc is the left child of vp, then we consume the next element of each of vp's and vc's preliminary contexts, and the next element of vc's final alignment is vc's consumed preliminary context element. Because LEFT-tagged elements originate from the left subtree of a node, and vc is the left child of vp, vc's preliminary context element is the correct element for this index of the alignment. If the same LEFT-tagged elements were encountered but vc was the right child of vp, then vc's preliminary context element would not be the correct element for this index of the alignment, because LEFT-tagged elements originate from the left subtree of vp and the LEFT-tagged element under consideration was encountered in vp's right subtree. In the case that a LEFT-tagged element is encountered in the right subtree of vp, we introduce a new gap into all the alignments of the subtree at this index to account for the aligned element in the vc's sister subtree. This is implicitly dealt with in Case 5.
Case 4: Conversely to Case 3, when both vp's final alignment element and vp's preliminary context element are "RIGHT," and vc is the right child of vp, we consume the next element of both vp's and vc's preliminary contexts and assign to the next element of vc's final alignment vc's consumed preliminary context element. Because RIGHT-tagged elements originate from the right subtree of a node, and vc is the right child of vp, vc's preliminary context element is the correct element for this index of the alignment. If the same RIGHT-tagged element was encountered but vc was the left child of vp, then vc's preliminary context element would not be the correct element for this index of the alignment, because RIGHT-tagged elements originate from the right subtree of vp and the RIGHT-tagged element under consideration was encountered in vp's left subtree. In the case that a RIGHT-tagged element is encountered in the left subtree of vp, we introduce a new gap into all the alignments of the subtree at this index to account for the aligned element in vc's sister subtree. This is implicitly dealt with in Case 5.
Case 5: When none of the conditions for Case 0, 1, 2, 3, or 4 hold, then we consume the next element of vp's preliminary context and the next element of vc's final alignment is "GAPPED." This handles the cases where either the two subtrees were not aligned at the current index or a new gap needed to be introduced at the current index because a LEFT-tagged or RIGHT-tagged element was encountered in vp's right or left subtree, respectively.
Analysis of pre-order traversal
Let \(m = \frac {a}{k}\), where k is the length of the longest input string, and a is the length of the root node's preliminary context. In the best case that a "perfect alignment" is derived, that is, that each element of all the input strings can be aligned with one of the elements of the longest input string, then m=1. In the worst case that a "degenerate alignment" is derived, that is, that no element of any of the input strings can be aligned with any of the elements of the longest input string, and all input strings are of equal length, then m=n.
The improvement of the implied alignment algorithm presented here compared to the original algorithm is that the additional stored information allows us to determine the final assignments in Θ(k∗m∗n) instead of \(\mathcal {O}\left (k^{2} * n^{2}\right)\) time. The aforementioned n2 component occurred in previous implementations due to the use of a "back-propagation" technique, which required that, at each pre-order step, each new gap found in the alignment was retroactively applied to every alignment derived in a previous pre-order step. The k2 component in the previous implementation was due to using a Needleman-Wunsch string alignment between the current node and its parent node at each pre-order step in addition to the alignment already performed at each post-order step. By saving the requisite information on the nodes during the post-order traversal and then consuming this information with the "sliding-zip" technique, we eliminate the Needleman-Wunsch alignments during the pre-order, as well as the back-propogation, and replace these computationally expensive operations with a much more efficient algorithm.
In the pre-order traversal algorithm presented above, we generate an implied alignment in Θ(k∗m∗n) time. We must perform a "sliding-zip" operation on each node in the binary tree \(\mathcal {T}\), hence the factor of n. The "sliding zip" accounts for the k∗m factor.
The best case time complexity occurs when the length of the derived alignment is the length of the longest input string, an alignment with the minimal number of elements. In this case, m=1 and the "sliding zip" performed on each node performs work equal to the length of the longest input string k. Hence, the best case time complexity of the implied alignment algorithm is Ω(n∗k), occurring when the input strings are highly correlated and m=1.
The worst case time complexity occurs when the length of the derived alignment is equal to the sum of the lengths of the input strings, an alignment with the maximum number of elements. In the worst case, m≫1, and the "sliding zip" performed on each node performs work equal to the length of the longest input string, k, multiplied by the number of input strings, n. Hence, the worst case time complexity of the implied alignment algorithm is \(\mathcal {O}(k * n^{2})\), occurring when the input strings are independent of each other.
An example Haskell implementation of the implied alignment algorithm described above, the data sets used to generate the results, along with a script to replicate the results discussed below can all be found here: https://github.com/recursion-ninja/efficientimplied-alignment/replicate-results.sh
We ran the implied alignment algorithm described in this paper on a pathological data set that was constructed to illustrate the best and worst case performances of the implied alignment algorithm. The pathological data set consisted of balanced binary trees which repeatedly doubled in size. The smallest tree is a quartet tree, with the strings consisting of a single symbol from the alphabet Σ={A,C,G,T} repeated k number of times. The lengths of the strings on each leaf were repeatedly doubled in size to scale the string length. The size of the tree was scaled by taking \(2^{\frac {n}{4}}\) quartet trees and combining them together into a larger balannced binary tree of n leaves.
The time complexity scaling of this pathological data set was examined under two different metrics, σ0 and σ1. The former metric preferentially selects substitution events over insertion or deletion elements, thus producing the "perfect alignment." Conversely, the latter selects for insertion or deletion over substitution, thus producing the "degenerate alignment."
Additionally, to explore the performance of the pre-order traversal on "real world" data, the algorithm was run on the fungal and metazoan biological data sets described by [31] and [32] respectively. Both full data sets consisted of a preselected tree and predetermined string alignment (i.e. including gaps). The full leaf set of the tree was repeatedly halved to produce a data set of doubling leaf set sizes. The string alignment was repeatedly truncated, dropping the beginning and end of the alignment, taking the central slice of the current length from each string, and then removing all the gaps from the alignment slice. The pruned trees and truncated strings were used as progressively doubling inputs, to measure runtime scaling in terms of both leaf set size and string length. Both biological data sets used the discrete metric σ3 and the alphabet Σ={A,C,G,T, – }.
After running the algorithm on each data set, we constructed an Ordinary Least Square (OLS) model with the running time in milliseconds as a function of dimensions k and n. We took the binary logarithm, log2, of both input dimensions as well as the output. From there, we calculated the coefficients of each input in this equation: log2(runtime)=β0+β1 log2(n)+β2 log2(k)+ε, where ε represents the estimation error. Note that, because the logarithm of the inputs was taken, we would expect β1 to be close to 1 for linear performance with respect to that input variable and close to 2 for quadratic performance. See Table 2.
Table 2 Regression coefficients of leaf-set size and string length on runtime
A direct runtime comparison between the \(\mathcal {O}\left (k^{2} * n^{2}\right)\) algorithm in POY and our improved algorithm was not readily achievable due to being implemented in different impure and purely functional languages, which come with confounding architectural designs. Instead we present the empirical runtime analysis of the pre-order traversal above. We did not thoroughly explore the implemented post-order traversal, as it does not deviate substantially from the well-understood Needleman-Wunsh algorithm. We have provided the reader a convenient https://github.com/recursion-ninja/efficient-implied-alignment/replicate-results.sh script in the aforementioned code repository to conduct their own analysis of both the pre-order and post-order traversals.
The pathological data sets shows the stark difference between the best case Ω(n∗k) and worst case \(\mathcal {O}\left (k * n^{2}\right)\) performances. The OLS model empirically supports the theoretical best and worst cases demonstrated by the two runs on the pathological data set as shown in Figs. 2 and 3.
Best case pre-order runtime
Worst case pre-order runtime
The OLS model also anecdotally supports the supposition that time complexity scales well for the biological data sets. The fungal and metazoan sequence data sets demonstrate a near-linear time complexities with respect to the number of input strings and linear complexity with respect to string length. The fungal data sets lend support to the argument that some of "real world" use cases can perform close to the theoretical best case complexity (see Figs. 4 and 5).
Fungi pre-order runtime
Metazoa pre-order runtime
The IA algorithm can be improved to run with \(\mathcal {O}\left (k * n^{2}\right)\) and best case Ω(k∗n) complexity of time and space. The more similar the input strings are, the closer the performance will be to the best case. When the algorithm is applied to "real world" biological sequences, the performance tends strongly towards the best case. The improved algorithm presented in this paper offers immediate and significant gains to applications related to the TAP and MSA.
The algorithm originally described by [1] was given the name implied alignment to differentiate it from other methods (e.g. sum-of-pairs alignment) unconnected to the vertex string assignments "implied" by the binary tree on a given leaf-set. However, it is worth articulating exactly how the alignment we derive is implied by the tree. In short, it is the requirement of commutativity and the lack of associativity.
For the purposes of this analysis we will ignore the cost returned from the ⊗ and consider only the resulting alignment context. Therefore let \(\oplus : \Sigma ^{*}_{\Gamma } \times \Sigma ^{*}_{\Gamma } \rightarrow \Sigma ^{*}_{\Gamma }\) be defined as ⊗, but ignoring the alignment cost of the result. If we are given a rooted binary tree \(\mathcal {T} = ((A,B),(C,D))\) with leaves A,B,C,D∈Σ∗ then the ancestral state of the root node defined by the heuristic function ⊕ would be ((A⊕B)⊕(C⊕D)). In fact, the ancestral state of any internal node defined by ⊕ can be calculated by applying ⊕ recursively to the subtree of the internal node. The binary structure of the tree directly implies the precedence of each application of ⊕ in the final result. Since ⊕ need not be associative, the tree ((A,(B,C)),D) evaluated as ((A⊕(B⊕C))⊕D), is likely to yield different results. However, since ⊕ is commutative, transposing any child nodes between the left and right positions of their parent will result in a tree that yields the same internal values. For example consider a transposed tree \(\mathcal {T'}\):
$$\begin{array}{*{20}l} eval(\mathcal{T'}) &= eval((D,C),(B,A)) \\ &= ((D \oplus C) \oplus (B \oplus A)) \\ &= ((C \oplus D) \oplus (B \oplus A)) \\ &= ((C \oplus D) \oplus (A \oplus B)) \\ &= ((A \oplus B) \oplus (C \oplus D)) \\ &= eval((A,B),(C,D)) \\ &= eval(\mathcal{T}) \end{array} $$
This commutative property and lack of an associative property precisely determines that the alignment is implied by the tree on the leaf-set under ⊕ and not the unique alignment on all trees for the leaf-set under ⊕. Clearly, a ⊕ that is both commutative and associative using the algorithm described in this paper would yield the same alignment on all trees for a given leaf-set.
The datasets generated and analysed in the study are available in the GitHub.com repository, https://github.com/recursion-ninja/efficient-implied-alignment
Direct Optimization
IA:
Implied Alignment
Multiple Sequence Alignment
OLS:
Ordinary Least Square
Tree Alignment Problem
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We would like to thank Eric Ford, Callan McGill, Katherine St. John, and Erilia Wu for insightful discussions. We would also like to thank the three reviewers for improvements to the manuscript.
This work was supported by DARPA SIMPLEX ("Integrating Linguistic, Ethnographic, and Genetic Information of Human Populations: Databases and Tools," DARPA-BAA-14-59 SIMPLEX TA-2, 2015-2018) and Robert J. Kleberg Jr. and Helen C. Kleberg foundation grant "Mechanistic Analyses of Pancreatic Cancer Evolution".
Division of Invertebrate Zoology, American Museum of Natural History, 200 Central Park West, New York, 10024-5192, NY, USA
Alex J. Washburn & Ward C. Wheeler
Alex J. Washburn
Ward C. Wheeler
AW developed the algorithmic improvements, implemented the prototype program, quantified the relationship between similar inputs and improved performance. WW provided empirical data sets, developed and implemented pruning methodologies for the scaling of the data sets, and performed literature review. Both authors read and approved the final manuscript.
Correspondence to Alex J. Washburn.
Washburn, A.J., Wheeler, W.C. Efficient implied alignment. BMC Bioinformatics 21, 296 (2020). https://doi.org/10.1186/s12859-020-03595-2
Dynamic homology
Multiple string alignment
Tree alignment
Novel computational methods for the analysis of biological systems
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Impacts of Atlantic multidecadal variability on the tropical Pacific: a multi-model study
Resolution dependence of CO2-induced Tropical Atlantic sector climate changes
W. Park & M. Latif
Atlantic and Pacific tropics connected by mutually interactive decadal-timescale processes
Gerald A. Meehl, Aixue Hu, … Nan Rosenbloom
Future climate change shaped by inter-model differences in Atlantic meridional overturning circulation response
Katinka Bellomo, Michela Angeloni, … Jost von Hardenberg
Trends in seasonal warm anomalies across the contiguous United States: Contributions from natural climate variability
Lejiang Yu, Shiyuan Zhong, … Xindi Bian
New insights into natural variability and anthropogenic forcing of global/regional climate evolution
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Weakening of the Atlantic Niño variability under global warming
Lander R. Crespo, Arthur Prigent, … Emilia Sánchez-Gómez
Cross-hemispheric SST propagation enhances the predictability of tropical western Pacific climate
Cheng Sun, Yusen Liu, … Chunzai Wang
Yohan Ruprich-Robert ORCID: orcid.org/0000-0002-4008-20261,
Eduardo Moreno-Chamarro1,
Xavier Levine ORCID: orcid.org/0000-0003-4970-70261,
Alessio Bellucci2,3,
Christophe Cassou4,
Frederic Castruccio ORCID: orcid.org/0000-0002-8397-74525,
Paolo Davini6,
Rosie Eade7,
Guillaume Gastineau8,
Leon Hermanson ORCID: orcid.org/0000-0002-1062-67317,
Dan Hodson ORCID: orcid.org/0000-0001-7159-67009,
Katja Lohmann10,
Jorge Lopez-Parages4,
Paul-Arthur Monerie9,
Dario Nicolì2,
Said Qasmi4,11,
Christopher D. Roberts ORCID: orcid.org/0000-0002-2958-663712,
Emilia Sanchez-Gomez4,
Gokhan Danabasoglu5,
Nick Dunstone7,
Marta Martin-Rey13,
Rym Msadek4,
Jon Robson ORCID: orcid.org/0000-0002-3467-018X9,
Doug Smith ORCID: orcid.org/0000-0001-5708-694X7 &
Etienne Tourigny ORCID: orcid.org/0000-0003-4628-14611
npj Climate and Atmospheric Science volume 4, Article number: 33 (2021) Cite this article
Atmospheric dynamics
Atlantic multidecadal variability (AMV) has been linked to the observed slowdown of global warming over 1998–2012 through its impact on the tropical Pacific. Given the global importance of tropical Pacific variability, better understanding this Atlantic–Pacific teleconnection is key for improving climate predictions, but the robustness and strength of this link are uncertain. Analyzing a multi-model set of sensitivity experiments, we find that models differ by a factor of 10 in simulating the amplitude of the Equatorial Pacific cooling response to observed AMV warming. The inter-model spread is mainly driven by different amounts of moist static energy injection from the tropical Atlantic surface into the upper troposphere. We reduce this inter-model uncertainty by analytically correcting models for their mean precipitation biases and we quantify that, following an observed 0.26 °C AMV warming, the equatorial Pacific cools by 0.11 °C with an inter-model standard deviation of 0.03 °C.
Over the 1980–2012 period, the eastern tropical Pacific sea surface temperature (SST) is characterized by a cooling trend that was one of the main causes of the global surface warming slowdown observed during 1998–20121,2,3. This regional cooling contrasts with a direct radiatively forced response expected from the increase in anthropogenic greenhouse gases4 and it is associated with an intensification of the western tropical Pacific easterlies, reflecting changes in the Walker Circulation5,6,7. Such changes have been partly attributed to variations in the tropical Atlantic SST through atmospheric teleconnections8,9.
During the same 1980–2012 period, the tropical Atlantic SSTs continued warming, likely due to a combination of anthropogenic-related radiative forcing and internal climate variability10,11. In particular, the leading mode of decadal variability of the North Atlantic SST—namely the Atlantic multidecadal variability (AMV12,13; Fig. 1a, b)—shifted from a cold to a warm phase around 1995–1996, exaggerating the North Atlantic warming trend induced by anthropogenic greenhouse gases14,15. Over the longer 1920–2014 period, warm AMV conditions were also associated with cold SST anomalies in the central and eastern tropical Pacific (Fig. 1c), supporting the existence of a consistent link between the AMV and the tropical Pacific climate16,17. Yet, these observed Pacific changes cannot be unequivocally attributed to the AMV due to the presence of external forcing and internally driven variability outside of the North Atlantic, as well as because of the limited historical record with respect to the timescales considered and observational uncertainties. Coupled global climate model (CGCM) simulations offer the possibility to tackle these limitations.
Fig. 1: Observed AMV and related anomalies.
a Spatial structure of the AMV-SST anomalies imposed in the numerical simulations. b Time evolution of the observed AMV (dataset: ERSSTv4). c Observed 2-m air temperature difference between positive and negative AMV years (i.e., red minus blue years in (b) dataset: HadCRUT4). Due to the sparseness of the observation in the tropical Pacific before ~1920's68, the composite in (c) is computed only from 1920 onwards, i.e., excluding data marked by the gray shading in (b). Areas, where data were not available for the whole composite period, are masked.
Using a hierarchy of numerical models, Li et al.9 demonstrated that the tropical Pacific response to the Atlantic forcing can be decomposed into two phases: Phase-1 an initial Atlantic forcing through diabatic heating and Phase-2 an Indo-Pacific Walker Circulation feedback (cf. Fig. 3 in Li et al.9). In Phase-1, the warm tropical Atlantic SST anomalies in summer (hereafter seasons are relative to the Northern Hemisphere) intensify deep convection and lead to upper tropospheric mass divergence over the tropical Atlantic. This divergence is compensated by upper tropospheric mass convergence and descent over the Central tropical Pacific, which intensifies the surface Trade winds over western tropical Pacific8,18. In Phase-2, the so-called Indo-Pacific feedback reinforces the Trade winds, piling up warm water in the Pacific Warm Pool, where atmospheric deep convection increases. This results in an upper tropospheric mass divergence over the warm pool that enhances Central tropical Pacific descent acting as positive feedback on the anomalies generated by the Atlantic forcing in Phase 19,19. Following El Niño Southern Oscillation (ENSO) dynamics, an increase in summer easterlies in the western tropical Pacific eventually favors colder conditions than normal in the eastern and central Pacific during the following winter20.
Fig. 2: Simulated AMV impacts.
Multi-model mean and 10-year averaged differences between AMV+ and AMV− simulations ensemble means in terms of 2-m air temperature and for the boreal (a) winter and (b) summer seasons. Stippling indicates regions where less than 80% of the models agree on the sign of the differences. Dashed black lines indicate in (a) the NIÑO3.4 region and in (b) the TROP latitudinal band and its constituent regions: TropInd, TropPac, and TropAtl (cf. indices definition in "Methods").
Given the global importance of the tropical Pacific variability and the predictability arising from the North Atlantic at decadal timescales21,22, this Atlantic–Pacific teleconnection is a potential source of seasonal to decadal climate predictability that needs to be further assessed in models. However, the robustness and the strength of this connection remain unknown and need to be quantified. Here, we present a multi-model assessment of this Atlantic–Pacific connection using 21 ensemble simulations from 13 CGCMs (Supplementary Tables 1 and 2) that largely comply with the CMIP6/DCPP-C protocol23. Following this protocol, the same observed AMV SST anomalies (Fig. 1a) are imposed in the North Atlantic of each CGCM to investigate the worldwide teleconnections associated with the observed AMV (see "Methods"). We note that in those idealized AMV simulations, extra heat is added to (or removed from) the climate system to maintain a stationary AMV signal in the North Atlantic for 10 years. This artificial heat prevents a realistic simulation of the relationship between AMV and the global mean surface temperature.
Uncertainty in the Pacific response to AMV forcing
We start by discussing the multi-model mean (MMM; cf. "Methods") winter response of the AMV experiments. Associated with the imposed 0.2 °C tropical North Atlantic warming, the MMM shows a 0.05 °C cooling in the tropical South Atlantic and a 0.1 °C cooling in the central equatorial Pacific (Fig. 2a). The latter extends eastward and poleward in both hemispheres, contrasting with warm anomalies in the western part of the subtropical Pacific basins. In the Indian Ocean, the MMM shows a broad warming response with maximum anomalies localized west of India. The summertime SST anomalies are similar to the winter ones but of weaker amplitude over the central equatorial Pacific (Fig. 2b). Overall, the MMM shows good agreement with observations over the whole tropical Atlantic (even south of the Equator where models are not constrained) as well as North of 10°S in other tropical regions (Fig. 1c). This similarity supports the important driving role of the AMV in the observed changes over the Pacific during the historical period8,9,17,19,24,25,26,27. In addition, the negative response of the tropical Pacific SST to the imposed North Atlantic warming in the AMV experiments implies a dynamical adjustment of the Pacific.
Fig. 3: Origins of the inter-model spread response to the observed AMV forcing.
Inter-model relationship between several indices. Markers represent the 10-year averaged ensemble mean the difference between AMV+ and AMV− simulations from individual experiments and the three colors code for the different AMV forcing strengths: 1×AMV, 2×AMV, and 3×AMV strength in blue, orange, and magenta, respectively. a Winter NIÑO3.4 SST index versus winter tropical North Atlantic SST (averaged over 5°N–20°N/60°W–10°E). b Winter NIÑO3.4 SST index versus summer TropPac descent (sum of the net vertical mass transport at 500 hPa; a positive value indicates descent). c Summer TropPac descent versus the sum of TropInd and TropAtl ascent. d Summer TropInd ascent versus TropAtl ascent. e Summer TropInd ascent versus atmospheric vertical temperature contrast over the WarmPool region (defined as the 20°S–20°N/90°E–160°W region), and (f) summer TropPac SST versus TROP temperature at 200 hPa. R indicates the inter-model correlation (see "Methods"). The dashed line in (c) materializes a full mass compensation within the tropics; RAtl and RInd indicate the inter-model correlations between summer TropPac and TropAtl ascent and summer TropPac and TropInd ascent, respectively. The Box plots in (d) indicate the minimum/maximum values, the 20th/80th percentiles, and the median from the indices distributions.
We now investigate the tropical Pacific response as simulated by each individual model using as a proxy the NIÑO3.4 SST index (cf. indices definition in "Methods" and Fig. 2a). In winter, all experiments simulate La Niña-like cooling in response to an AMV warming except the EC-Earth3P_1Sig experiment that shows weak NIÑO3.4 warming of +0.01 °C (Fig. 3a; see also Supplementary Fig. 8). Though models mostly agree on the sign of the tropical Pacific response, the magnitude of their response varies by an order of magnitude, from 0.01 °C to −0.23 °C, with a MMM of −0.12 °C for a similar ~0.2 °C tropical North Atlantic warming. This large inter-model spread in response to AMV forcing highlights considerable uncertainties in our ability to predict the climate at seasonal to decadal timescales28,29.
Origins of the inter-model spread
Different tropical Pacific responses among models in winter can be explained by intrinsic model differences in simulating Pacific climate dynamics such as the ones linked to ENSO30. Yet, it is known that ENSO is strongly influenced by tropical Pacific conditions in the previous summer31,32,33,34. In particular, tropical Pacific heat content anomalies and their driving surface winds are known to be predictors of ENSO several months ahead35,36. Therefore, different tropospheric responses to the Atlantic SST forcing during summer can also explain model differences in winter20,37. Here, we find that the winter NIÑO3.4 inter-model spread is mainly associated with the inter-model spread in descent anomalies over Central tropical Pacific during summer (R = −0.9; where R is the inter-model correlation, see "Methods"; Fig. 3b) and associated surface winds. This indicates that the inter-model spread in the winter Equatorial Pacific mainly arises from different tropospheric responses to the AMV forcing during summer. This inference is supported by the weaker inter-model correlation between winter NIÑO3.4 SST and winter Pacific descent responses (R = −0.64, not shown).
To further understand the inter-model spread, we explore the origins of the tropical Pacific tropospheric descent anomalies in summer. Figure 3c shows that those subsiding anomalies are nearly fully mass-compensated by ascendant anomalies in other tropical regions. The 20°S–20°N tropical band (TROP) is further decomposed into a broad Indian ocean domain (TropInd), the Central Pacific ocean (TropPac), and a broad Atlantic domain (TropAtl; cf. indices definition in "Methods" and Fig. 2b). Through an analysis of variance (see "Methods"), we find that only 19% of the inter-model variance in TropPac descent anomalies is associated with the inter-model variance in TropAtl ascent anomalies, but 69% with the TropInd ascent ones. These two sources of spread are consistent with the two-phase mechanism detailed in the Introduction to explain the tropical Pacific response to Atlantic warming. In particular, it is consistent with the amplification of the Pacific response through the adjustment feedback of the Indo-Pacific Walker Circulation9.
The key finding here is that there is no significant inter-model correlation between TropInd and TropAtl anomalies (R = 0.2, Fig. 3d). This indicates that models simulate different Indo-Pacific Walker Circulation adjustments (Phase-2 Indo-Pacific feedback) for similar Atlantic–Pacific atmospheric bridges (Phase-1 Atlantic forcing). Hence, this implies that the simulated feedback associated with the Indo-Pacific Walker Circulation adjustment is model-dependent and that the differences in this feedback are the source of most of the inter-model spread in the tropical Pacific response to the AMV forcing.
We find two possible mechanisms to explain the different Indo-Pacific Walker Circulation adjustments among models in the AMV experiments. As detailed below, either different TropInd ascent or different TropPac SST responses can be the original driver of the different circulation responses. Further targeted experiments would be required to determine which mechanism is dominating here. However, both mechanisms point to the temperature response of the upper tropical troposphere as the key process to understand the inter-model differences:
The inter-model spread in TropInd ascent anomalies is tightly connected to the tropospheric lapse rate over the Warm Pool (Fig. 3e). Indeed, the larger the lapse rate (less warming in the upper troposphere compared to the surface), the less stable the troposphere is, and the more convectively active the tropical troposphere becomes. The inter-model spread in TropInd ascent anomalies is therefore linked to different responses in the upper tropospheric warming over the Warm Pool among models (Supplementary Fig. 4a–c). Because in the tropics the upper-tropospheric temperature is constrained by wave dynamical adjustment to be nearly horizontally uniform38,39 (Supplementary Fig. 4d), it implies that the TropInd ascent responses and the Indo-Pacific Walker Circulation responses are controlled by the different upper tropospheric warming among models.
The different upper tropospheric warming can lead to different SST warming among models through a "top-down" mechanism by a decrease of the surface latent heat flux40. There is indeed an inter-model correlation of R = 0.86 between the TROP upper tropospheric temperature and the TropPac SST responses (Fig. 3f). This "top-down" warming effect eventually modulates the amplitude of the TropPac descent and of the Indo-Pacific Walker Circulation adjustment.
Therefore, for both the TropInd ascent and the TropPac SST mechanisms, the warmer the TROP upper troposphere is in response to an AMV warming, the weaker the Indo-Pacific feedback and the weaker the tropical wintertime Pacific cooling are. Then, in order to understand the inter-model spread in the wintertime Pacific response to AMV, one needs to understand the inter-model spread in TROP upper tropospheric temperature anomalies.
As the thermal stratification of the tropical troposphere is primarily controlled by deep convection, upper tropospheric temperature anomalies in the Tropics can be generally traced to regional variations in the atmospheric boundary layer. To investigate the origins of those anomalies we study the moist static energy at the surface, using as estimate the equivalent potential temperature41 (\(\theta _{\mathrm{E}}\); see "Methods"). In order to take into account the different contributions of highly active and less active convective regions in the injection of moist static energy from the surface to the upper troposphere, we weigh surface \(\theta _{\mathrm{E}}\) with local precipitation (Pr in mm d−1) following Sobel et al.42's approach (see "Methods"):
$${\mathrm{P}}\theta _{\mathrm{E}} = a \times {\mathrm{Pr}} \times \theta _{\mathrm{E}}/ < a \times {\mathrm{Pr}} >$$
Where \(< . >\) symbols indicate the sum over TROP and \(a\) is the grid cell surface area. The inter-model correlation between the changes of upper tropospheric temperature and our weighted \(\theta _{\mathrm{E}}\) variable (\({\mathrm{P}}\theta _{\mathrm{E}}\)) summed over the tropical band is R = 0.96 (Fig. 4a), confirming the physical link between \({\mathrm{P}}\theta _{\mathrm{E}}\) and the upper tropospheric conditions in the tropics.
Fig. 4: Impacts of different injections of moist static energy into the upper troposphere.
a Summer TROP temperature at 200 hPa versus the TROP surface equivalent temperature weighted by precipitation. b Summer TROP temperature at 200 hPa versus the \(\theta _{\mathrm{E}}\) anomalies component of the TropAtl surface equivalent potential temperature weighted by precipitation (\({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\)). c Multi-model mean weighted equivalent potential temperature anomalies (\({\mathrm{P}}\theta _{\mathrm{E}}\)) summed over TROP and its contributions from \(\theta _{\mathrm{E}}\) anomalies (\({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\)), precipitation anomalies (\({\mathrm{P}}\prime \theta _{{\mathrm{E}},{\mathrm{C}}}\)), and precipitation and \(\theta _{\mathrm{E}}\) anomalies covariance (\({\mathrm{P}}\prime \theta _{\mathrm{E}}\prime\)). d TROP \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) contributions from TropInd, TropPac, and TropAtl. On (c) and (d), the length of the vertical black lines indicates the inter-model standard deviation associated with the multi-model mean value, and R indicates the inter-model correlation of each component with the upper-tropospheric temperature shown in (a). e Spatial distribution of the \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) inter-model spread over TropAtl and its contributions from inter-model differences in f climatological precipitation, (g) \(\theta _{\mathrm{E}}\) response to AMV, and (h) their covariance. The dashed line defines the eastern border of the East Pacific region used in Fig. 6. The two MetUM-GOML simulations are excluded from the analyses (c), (d), (e)–(h).
To further understand the origins of the inter-model spread, we decompose the \({\mathrm{P}}\theta _{\mathrm{E}}\) anomalies into a term linked to precipitation anomalies only, i.e., \({\mathrm{P}}^\prime \theta _{{\mathrm{E}},{\mathrm{C}}}\), a term linked to \(\theta _{\mathrm{E}}\) anomalies only, i.e., \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}^\prime\), and a covariance term, i.e., \({\mathrm{P}}_{\mathrm{C}}^\prime \theta _{\mathrm{E}}^\prime\) (see "Methods"). We find that most of the inter-model spread in upper tropospheric temperature anomalies is coming from differences in the injection of surface moist static energy anomalies into the upper troposphere by the mean model vertical motions (the \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}^\prime\) term; Fig. 4c). Furthermore, we find that the upper tropical troposphere warm anomalies are generated quasi-equally by anomalies occurring in the TropAtl and TropInd regions and, to a lesser extent, in TropPac (Fig. 4d). However, its inter-model spread is primarily driven by the TropAtl and TropPac sectors (black lines), with inter-model correlations between the upper troposphere temperature anomalies and \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}^\prime\) summed over those regions equal to R = 0.96 and R = 0.87, respectively. Because the forcing is coming from the Atlantic in the present experiments, we assume that it is the spread in TropAtl \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}^\prime\) that controls the spread in the tropical upper-tropospheric temperature and that the latter is amplified by the TropPac \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}^\prime\) response.
In summary, the analysis of \({\mathrm{P}}\theta _{\mathrm{E}}\) indicates that the inter-model spread in the tropical upper tropospheric temperature anomalies can be explained by different injections of moist static energy from the TropAtl surface into the upper troposphere (Fig. 4b). This is eventually responsible for the modulation of the Indo-Pacific Walker Circulation feedback among models. Hence we identify two summertime variables centered over the TropAtl region that contribute to the inter-model spread in the tropical Pacific response: (1) the divergence of mass in the upper troposphere over TropAtl and (2) the injection of moist static energy anomalies from the TropAtl surface into the upper troposphere by the mean convective activity (\({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\)). Building a bi-linear regression model with those two variables as predictors (see "Methods"), we capture as much as 73% of the inter-model variance in the wintertime NIÑO3.4 SST response (Fig. 5a, b); TropAtl ascent and \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) accounting for 39% and 61% of the total regression model variance, respectively.
Fig. 5: Assessing the wintertime tropical Pacific response from two summertime Atlantic predictors.
a Inter-model relationship between the wintertime NIÑO3.4 SST index and the outputs of a bi-linear regression model built with the summertime TropAtl ascent and \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) anomalies (see "Methods"). The linear regression between the statistical model and NIÑO3.4 is shown by the black line. b Whisker box plots indicating the minimum/maximum values, the 20th/80th percentiles, and the median from the inter-model distribution of several indices: the wintertime NIÑO3.4 SST index (black), the outputs of the regression model fed with summertime TropAtl ascent, and (blue) TropAtl \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\), (red) TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) (i.e., \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) computed using observed precipitation climatology), and (green) TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{{\mathrm{E}},{\mathrm{cor}}}^\prime\) (i.e., \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) corrected for precipitation climatological biases over the East Pacific in late winter). The two latter box plots account for uncertainties coming from observation estimates (see "Methods").
Bias corrections and reduction of the uncertainty
Next, we investigate the origins of the model response differences over TropAtl aiming at narrowing the uncertainty of our numerical estimate of the tropical Pacific response to the observed AMV forcing. We start by decomposing further the \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) variable over the TropAtl region to evaluate whether its inter-model spread is coming from differences among models in climatological precipitation (\({\mathrm{P}}_{\mathrm{C}}^ \ast \left[ {\theta _{\mathrm{E}}^\prime } \right]\)), \(\theta _E\) anomalies (\(\left[ {{\mathrm{P}}_{\mathrm{C}}} \right]\theta _{\mathrm{E}}^{\prime \ast }\)), or a combination of both (\({\mathrm{COV}}\); see "Methods"). We find that all terms contribute to the inter-model spread, but that their respective importance is spatially dependent (Fig. 4e–h). Of particular interest, this analysis demonstrates that different climatological precipitation among models (Fig. 4f) is partly responsible for the inter-model spread.
Because the model climatological precipitations are biased relative to observations, it implies that the simulated \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) are also biased, which leads to erroneous estimates of the response to the observed AMV forcing. To minimize this error, we apply a bias correction to the \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) of each model by computing them using the observed climatological precipitation instead of model one: \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\). This suppresses the spread of \({\mathrm{P}}_{\mathrm{C}}^ \ast \left[ {\theta _{\mathrm{E}}^\prime } \right]\) but it introduces a new source of spread coming from observational uncertainties (see "Methods"). This bias correction decreases overall the inter-model variance of \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) over TropAtl by 58%. Feeding our bi-linear NIÑO3.4 regression model with \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) instead of \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\), we quantify that correcting for model mean precipitation biases helps to reduce the inter-model response variance over the tropical Pacific by 35% (Fig. 5b).
Over the eastern Pacific (i.e., the western part of the wide TropAtl sector as shown in Fig. 4g), it is mainly the different \(\theta _{\mathrm{E}}\) responses among models that drive the inter-model spread in \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) and, a fortiori, in \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) (Fig. 4g, see also Supplementary Fig. 11). \(\theta _{\mathrm{E}}\) anomalies there are largely associated with surface temperature changes but their sign and amplitude are model-dependent (Supplementary Fig. 9), leading to compensating anomalies in the MMM (Fig. 2b). In the following, we demonstrate that the spread in \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) over the eastern Pacific is explained by the different model climatological precipitations during February–March–April (Fig. 6g, h).
Fig. 6: Impacts of the tropical East Pacific ITCZ mean biases.
a–f Monthly evolution of the differences between AMV+ and AMV− ensemble means zonally averaged over the East Pacific region (i.e., the western part of TropAtl) shown in (g) for the multi-model means between the five models simulating the northernmost climatological position of the ITCZ during February–March–April over East Pacific (Group 1: ECMWF-HR, IPSL-CM6, HadGEM3, ECMWF-LR, CESM1) and the southernmost (Group 2: EC-Earth3, CNRM-CM6, CNRM-CM5, EC-Earth3P, CMCC), cf. x-axis in (h). a–f show the differences in terms of SST, precipitation, and net surface fluxes, respectively (surface fluxes are defined as positive from the atmosphere to the ocean). Arrows in (e) and (f) represent the surface wind anomalies. In (a)–(g), contours indicate the climatological precipitation and stippling means that not all models in the group agree on the sign of the anomalies. Months are indicated by their first letter and a 3-month running mean is applied. g Inter-model map regression of the climatological precipitation on the \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\) index (shading; units: mm d-1 per inter-model standard deviation of \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\)) and multi-model mean climatological precipitation (contours; units: mm d-1). \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\) was computed from five observation estimates and only their averaged regression is shown here (see "Methods"). h Inter-model relationship between summertime \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\) summed over East Pacific and the late winter centroid of the climatological precipitation over East Pacific, defined as the latitude at which there is the same amount of zonally averaged precipitation North and South. The linear regression between the two indices is shown by the black line. Only the averaged values computed from the five \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\) obtained from the different observation estimates are shown here. The vertical green solid lines indicate the precipitation centroids from the observation estimates and the horizontal green dashed lines reveal the \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\) values associated with these observed centroids, assuming the same statistical relationship as the inter-model one. R indicates the inter-model correlation averaged over the five estimates of \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}^{\prime \ast }\). The MetUM-GOML simulations are excluded from all the analyses of this figure.
During summer, all models simulate westerly anomalies north of 5°N associated with a northward shift of the Inter-tropical Convergence Zone (ITCZ) over the East Pacific in response to the AMV warming (Fig. 6c–f). Yet, by dividing the models into two sub-groups based on the state of their late winter climatological precipitation, we show that this shift is more pronounced for the models simulating a more northward position of the climatological ITCZ in late winter. Those models simulate an SST cooling around 7°N on the southern flank of the ITCZ in summer (Fig. 6a, b), where the other models simulate warming, which explains the inter-model spread in \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\). For the models with the largest ITCZ shift, we find that the westerly anomalies follow the seasonal migration of the precipitation anomalies; those are present north of the Equator since the winter when their cooling effect on the ocean is greatest (Fig. 6c, e). This suggests that preconditioning of the summertime cooling around 7°N during previous seasons occurs through feedback between ITCZ position, SST, wind, and surface flux anomalies43. Yet, all models tend to simulate a northward shift of the ITCZ in winter (Fig. 6c, d) but only some of them simulate such preconditioning. This inter-model disagreement is coming from different model climatological precipitation in late winter. For models simulating an ITCZ located north of the Equator, the northward shift of the ITCZ increases the mean south-westerlies and their associated turbulent heat fluxes around the Equator, which tends to cool locally the SST. On the other hand, for models simulating an ITCZ located South of the Equator, the northward shift of the ITCZ weakens the mean north-easterlies and their cooling effect on the equatorial SST.
Given the high correlation between the climatological precipitation in February–March–April and the summertime \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) response over East Pacific (R = −0.87; Fig. 6h), we use this information to further correct our estimate of the Tropical Pacific response to AMV. Associated with the observed February–March–April climatological precipitations, we estimate summertime East Pacific \(P_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) values ranging from −0.09 °C and −0.13 °C (cf. green lines in Fig. 6h). Substituting these \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) values for each model to the contribution of the East Pacific into the TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\), we obtain TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{{\mathrm{E}},{\mathrm{cor}}}^\prime\) that we consider as our best estimate of the \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) response to the observed AMV forcing (see "Methods"). Feeding our bi-linear regression model with \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{{\mathrm{E}},{\mathrm{cor}}}^\prime\) instead of \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\), we quantify that correcting both summertime and late winter precipitation mean biases reduces by 65% the inter-model variance in our analytical estimate of the NIÑO3.4 response (Fig. 5b).
We also investigated the potential origin of the inter-model spread over Atlantic–Africa (i.e., the eastern part of the TropAtl region, cf. Fig. 4e–h). We found that the inter-model spread in \(\left[ {{\mathrm{P}}_{\mathrm{C}}} \right]\theta _{\mathrm{E}}^{\prime \ast }\) anomalies is associated with different signs in the SST and specific surface humidity responses around the eastern equatorial Atlantic (Supplementary Figs. 3 and 4). However, we did not identify the physical processes controlling the different model behaviors (cf. Supplementary Discussion).
Using 21 coordinated simulations from 13 different CGCMs, we show that:
In response to an AMV warming, all models simulate tropical Pacific changes reminiscent of La Niña conditions. This result confirms the influence of the Atlantic on climate variability at the global scale and it supports the idea that the AMV has contributed to the 1998–2012 global warming slowdown through its impacts on the tropical Pacific. However, the strength of the connection varies by a factor of 10 between the models.
The tropical Pacific response to the Atlantic forcing is driven by changes in (1) the Atlantic–Pacific Walker Circulation and (2) the amount of moist static energy injected from the Atlantic surface into the upper troposphere.
The latter is responsible for most of the uncertainty in our current numerical model estimates of the Pacific response to the observed AMV, mainly because of different mean precipitation climatology.
Partially correcting for mean model precipitation biases, we reduce this uncertainty and we specifically quantified that the NIÑO3.4 response to an observed 0.26 °C AMV warming ranges from −0.05 °C to −0.16 °C with a median value of −0.11 °C and an inter-model standard deviation of 0.03 °C. We acknowledge that this estimate is still subject to model limitations. In particular, we reduce uncertainty by correcting a posteriori for model precipitation biases. Any possible interactions between those biases and surface equivalent potential temperature responses to AMV would still affect our estimate. Therefore, our analysis highlights the importance of reducing mean climate model biases in order to properly simulate and predict the global AMV impacts.
Although this study focuses on decadal timescales signals, the discussed mechanisms take place at monthly timescales. Our study shows then the potential for improving climate predictions from seasonal to decadal timescales through a better representation of the impacts of the Atlantic on tropical Pacific28,29,44. The discussed mechanisms very likely also act to shape the Pacific mean state and their differences among models45,46,47, which are partly responsible for the inter-model spread in climate projections48. Based on our findings, we suggest that the analysis of the injection into the upper troposphere of moist static energy from the Atlantic surface can be used as an interpretative framework to understand the inter-model uncertainties around future climate simulations.
Finally, we note that several observational and model-based studies49,50,51 suggest the existence of a two-way interaction between Atlantic and Pacific at decadal timescale: an AMV warming driving a Pacific cooling, which eventually drives an Atlantic cooling. Due to the experimental protocol used in the present article, we could only focus on the representation by models of the Atlantic impacts on the Pacific. To persist in exploring the sources of climate predictability at multi-annual timescale and their current limits due to model uncertainty, a similar multi-model study to this one should be completed but investigating the Pacific impacts on the Atlantic.
The 21 experiments from 13 different CGCMs used in this study are listed in Supplementary Tables 1 and 2; it represents a total of 12,320 simulated years. Following the DCPP-C protocol23, two sets of ensemble simulations have been performed for each experiment, in which time-invariant SST anomalies corresponding to the warm (AMV+) and cold (AMV−) phases of the observed AMV were imposed over the North Atlantic using SST nudging. To capture the potential response and adjustment of other oceanic basins to the AMV anomalies, the simulations were integrated for 10 years with fixed external forcing conditions. Large ensemble simulations were performed in order to robustly estimate the climate impacts of the AMV (from 10 to 50 members depending on the model, cf. Supplementary Table 2). An extensive description of the experimental protocol is provided in the Technical note for AMV DCPP-C simulations: https://www.wcrp-climate.org/wgsip/documents/Tech-Note-1.pdf.
Over the North Atlantic (Equator-65°N/80°W–0°), the spatial correlation of the SST anomalies in each simulation and the observed AMV target varies between 0.66 and 0.86, with a multi-model average value of 0.79, indicating that all simulations are constrained by similar SST conditions in the North Atlantic. We note that the idealized AMV simulations underestimate by ~20% the amplitude of the observed AMV target. This is because we do not impose a very strong nudging in the experimental protocol to allow ocean-atmosphere coupling and variability at high frequency (as recommended by the CMIP6/DCPP-C protocol23), which tends to dissipate the heat anomalies imposed at the surface. Further evaluation of the experimental protocol is provided in Supplementary.
Some simulations deviate from the AMV DCPP-C protocol. CESM1 simulations used an observed AMV pattern computed from the ERSSTv3b dataset52 instead of ERSSTv453. CNRM-CM6-1-HR, EC-Earth3P-HR, EC-Earth3P-LR, ECMWF-IFS-HR, and ECMWF-IFS-LR used a constant 1950 or 1990 (instead of 1850) external forcing background (cf. Supplementary Table 2). The impact of the external forcing background on the results is tested with the CNRM-CM5 models for which AMV simulations have been performed with both 1850 and 1990 backgrounds. We did not find evidence for the impact of the protocol differences on the results discussed in this article. In addition, the MetUM-GOML-HR and MetUM-GOML-LR simulations used a 1000-m mixed-layer ocean model and 1990 external forcing background. Those models offer insights on the role played by the ocean dynamics in the documented climate responses when compared to the models with full ocean dynamics.
Finally, the imposed AMV forcing strength is not the same for all simulations. As detailed in the column "AMV strength" of Supplementary Table 2, the imposed AMV anomalies vary between 1, 2, and 3 times the observed AMV standard deviation. Assuming linearity in the AMV responses, we weight each simulation by dividing their output by the AMV forcing strength in order to compare the results from all the AMV experiments. This is done for all the figures in the article. This enables us to create a larger multi-model ensemble and to evaluate more precisely the origins of the inter-model spread. Scaled outputs from experiments performed with the same model but with different AMV strengths are often indistinguishable, which suggests that the linear assumption is a reasonable approximation for the analyses of this study. Yet, we highlight the different AMV strengths by different colors in the figures (1×AMV: blue; 2×AMV: orange; 3×AMV: magenta).
MMM and inter-model correlations (R)
The MMM is computed by averaging the ensemble mean of each simulation, regardless of the number of ensemble members (i.e, there is no weighting). The outputs of each simulation are scaled by their AMV strength forcing prior to computing the MMM (as described above). For models for which several sets of experiments have been performed (with different magnitudes of AMV anomalies and/or different external forcing backgrounds), we average all the experiments of each model together prior to computing the MMM in order to not bias the results toward an over-represented model (e.g., CNRM-CM5 or EC-Earth3P). Because of the absence of ocean dynamics in the two MetUM-GOML models, those models are not taken into account in the computation of the MMM.
Similarly to the computation of the MMM, the inter-model correlation R is computed after averaging all the ensemble means from the same model (if more than one experiment was performed) in order to give the same weight to all models. We also computed the inter-model correlation based on all the ensemble means from all the simulations (i.e., no averaging of experiments from the same model prior to the computation of the correlation) but no significant differences between the two correlations were found for the relationship investigated in this article. Because of the absence of ocean dynamics in the two MetUM-GOML models, those models are not taken into account in the computation of inter-model correlations.
Regions definition
To assess the tropical Pacific response, we use the NIÑO3.4 index defined as the SST averaged over 5°S–5°N and 170°W–120°W (Fig. 2a). Based on the summer MMM anomalies of precipitation and vertical velocity at 500hPa (Supplementary Fig. 3e, f), we decomposed the 20°S–20°N tropical band (TROP) into three main regions: a broad Indian region spanning from 30°E to 135°E, a central Pacific region spanning from 135°E to 120°W, and a broad Atlantic region spanning from 120°W to 30°E (Fig. 2b). We label those regions TropInd, TropPac, and TropAtl, respectively. In addition, an East Pacific and an Atlantic–Africa regions (embedded into TropAtl) are used in Figs. 4 and 6 that cover 120°W–80°W/20°S–20°N and 80°W–30°E/20°S–20°N, respectively.
Taking advantage of the quasi-mass compensation of the vertical motion in the TROP region (Fig. 3c), we estimate the origins of the inter-model spread in TropPac descent through an analysis of variance: \(S_{{\mathrm{TropPac}}}^2 \sim S_{{\mathrm{TropAtl}} + {\mathrm{TropInd}}}^2 = S_{{\mathrm{TropAtl}}}^2 + S_{{\mathrm{TropInd}}}^2 + {\mathrm{COV}}\), where \(S_{{\mathrm{TropPac}}}^2\), \(S_{{\mathrm{TropAtl}}}^2\) and \(S_{T{\mathrm{ropInd}}}^2\) are the inter-model variance in TropPac, TropAtl, and TropInd descent anomalies, respectively; \({\mathrm{COV}}\) is the covariance term between TropAtl and TropInd descent anomalies and \(S_{{\mathrm{TropAtl}} + {\mathrm{TropInd}}}^2\)is the inter-model variance of descent anomalies averaged over the whole Trop region excluding the TropPac region. We find that \(S_{{\mathrm{TropAtl}}}^2\), \(S_{{\mathrm{TropInd}}}^2\) and COV explains 19%, 69%, and 12% of \(S_{{\mathrm{TropAtl}} + {\mathrm{TropInd}}}^2\), respectively.
Equivalent potential temperature \(\theta _{\mathbf{E}}\)
Theoretically, the equivalent potential temperature can be defined as \(\theta _{\mathrm{E}}\sim \theta exp\left( {\frac{{L_{\mathrm{C}}q_{\mathrm{S}}}}{{C_PT}}} \right)\), where \(\theta\) is the dry potential temperature, \(L_{\mathrm{C}}\) is the latent heat of condensation, \(q_{\mathrm{S}}\) is the saturation of mixing ratio, \(C_{\mathrm{P}}\) is the specific heat of dry air, and \(T\) is the temperature. This formula explicitly shows that \(\theta _{\mathrm{E}}\) is similar to the potential temperature for dry air mass (which remains constant during adiabatic processes) but it corrects for the energy associated with the air mass moisture, assuming that all the energy released by condensation/evaporation remains in the air mass (pseudo-adiabatic process). Here we used the NCL function "pot_temp_equiv_tlcl" (https://www.ncl.ucar.edu/Document/Functions/Contributed/pot_temp_equiv_tlcl.shtml) to compute \(\theta _E\). This function is based on Eq. (39) from Bolton54, which gives more accurate results than the theoretical formula given above but that requires the computation of the temperature at the lifted condensation level. Such temperature is estimated with the NCL function "tlcl_rh_bolton" (https://www.ncl.ucar.edu/Document/Functions/Contributed/tlcl_rh_bolton.shtml), which is based on Eq. (22) from Bolton54.
Weighted equivalent potential temperature \({\mathbf{P}}\theta _{\mathbf{E}}\) as a proxy for the upper-tropospheric temperature
Over the oceans, the mean tropospheric temperature profile is often considered to be in a moist-adiabatic convective equilibrium with the mean SST55 (Supplementary Fig. 4f), as the SST controls directly the atmospheric boundary layer energy content. Yet, convective adjustment can act directly only in regions of frequent precipitation, which are mostly over warm SST regions. In regions of no convection, the surface has no direct means of influencing the free troposphere, and the SST anomalies cannot shape the tropospheric temperature profile. Hence, there is no evident physical reason for considering mean tropical SST variations as a proxy for upper tropospheric temperature anomalies. To take into account the different contributions of local SST to the upper-tropospheric temperature, Sobel et al. 42 introduced a more appropriate proxy by weighting the SST with the local precipitation before computing the tropical average. We follow this method here, but we generalize it in order to account for the effect of deep convection overland on the upper tropospheric temperature anomalies56 and we compute the precipitation weighted equivalent potential temperature \({\mathrm{P}}\theta _{\mathrm{E}}\), cf. Eq. (1).
We note \(<\, f \,>\) as the sum of the values of a given field \(f\) for all tropical grid points within 20°S–20°N. We define \(f = f_{\mathrm{C}} + f\prime\), where \(f\prime\) is the departure of \(f\) from \(f_{\mathrm{C}}\), which is the time-averaged ensemble mean of the AMV- experiments (the AMV- experiment being considered as the reference state). We also define \(= \left[ f \right] + f^ \ast\), where \(f^ \ast\) is the departure of \(f\) from its multi-model mean \(\left[ f \right]\).
Decompositions of \({\mathbf{P}}\theta _{\mathbf{E}}\)
In the article, \({\mathrm{P}}\theta _{\mathrm{E}} = \frac{{a \times {\mathrm{Pr}} \times \theta _{\mathrm{E}}}}{{ < a \times {\mathrm{Pr}} > }}\) is first decomposed into a term linked to precipitation anomalies only \({\mathrm{P}}\prime \theta _{{\mathrm{E}},{\mathrm{C}}} = \frac{{a \times {\mathrm{Pr}}\prime \times \theta _{{\mathrm{E}},{\mathrm{C}}}}}{{ < a \times {\mathrm{Pr}}\prime > }} - \frac{{a \times {\mathrm{Pr}} \times \theta _{\mathrm{E}}}}{{ < a \times {\mathrm{Pr}} > }} \times \frac{{a \times {\mathrm{Pr}} \times \theta _{\mathrm{E}} \times {\mathrm{Pr}}\prime }}{{ < a \times {\mathrm{Pr}} > }}\), a term linked to \(\theta _{\mathrm{E}}\) anomalies only \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime = \frac{{a \times {\mathrm{Pr}}_{\mathrm{C}} \times \theta _{\mathrm{E}}\prime }}{{ < a \times {\mathrm{Pr}}_{\mathrm{C}} > }}\), and a covariance term \({\mathrm{P}}\prime \theta _{\mathrm{E}}\prime = \frac{{a \times {\mathrm{Pr}}\prime \times \theta _{\mathrm{E}}\prime }}{{ < a \times {\mathrm{Pr}}\prime > }}\). In a second time \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) is decomposed into a term linked to the climatological precipitation differences among models \({\mathrm{P}}_{\mathrm{C}}^ \ast \left[ {\theta _{\mathrm{E}}^\prime } \right] = \frac{{a \times {\mathrm{Pr}}_{\mathrm{C}}^ \ast \times \left[ {\theta _{\mathrm{E}}\prime } \right]}}{{ < a \times {\mathrm{Pr}}_{\mathrm{C}}^ \ast > }}\), a term link to the different \(\theta _{\mathrm{E}}\) response to AMV among models \(\left[ {P_{\mathrm{C}}} \right]\theta _{\mathrm{E}}^{\prime \ast } = \frac{{a \times \left[ {{\mathrm{Pr}}_{\mathrm{C}}} \right] \times \theta _{\mathrm{E}}^{\prime \ast }}}{{ < a \times \left[ {{\mathrm{Pr}}_{\mathrm{C}}} \right] > }}\), and a covariance term \({\mathrm{COV}} = \frac{{a \times {\mathrm{Pr}}_{\mathrm{C}}^ \ast \times \theta _{\mathrm{E}}^{\prime \ast }}}{{ < a \times {\mathrm{Pr}}_{\mathrm{C}}^ \ast > }}\).
Bi-linear regression model
The coefficient of the bi-linear regression model is computed using as predicand the wintertime NIÑO3.4 SST index and as the two predictors the summertime vertical ascent summed over TropAtl and the summertime \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\) summed over TropAtl. The two MetUM-GOML simulations are excluded from the computation of the regression model coefficients and, similarly, as for the inter-model correlation, all models share the same weight.
Bias corrections and observational uncertainties
Two bias corrections are applied to \({\mathrm{P}}_{\mathrm{C}}\theta _{\mathrm{E}}\prime\). First, we compute this variable by using observed climatological precipitation (\({\mathrm{Pr}}_{{\mathrm{Obs}}}\)) instead of model climatological precipitation: \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime = \frac{{a \times {\mathrm{Pr}}_{{\mathrm{Obs}}} \times \theta _{\mathrm{E}}\prime }}{{ < a \times {\mathrm{Pr}}_{{\mathrm{Obs}}} > }}\). Then, we decomposed the TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) into its regional components coming from the Atlantic–Africa region, the East Pacific region, and their covariance term. The East Pacific component is then substituted by a value estimated from the observed climatological precipitation and the inter-model relationship between JJAS East Pacific \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\) and February–March–April climatological precipitation centroid over East Pacific (Fig. 6h). Following this substitution, we sum again the different components of TropAtl \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\)to obtain \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{{\mathrm{E}},{\mathrm{cor}}}^\prime\). In order to account for observational uncertainties57,58, we compute for each model five \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{\mathrm{E}}\prime\)and \({\mathrm{P}}_{{\mathrm{Obs}}}\theta _{{\mathrm{E}},{\mathrm{cor}}}^\prime\) values using different observation estimates.
Observational and reanalysis datasets
The SST from the ERSSTv453 and ERSSTv3 datasets52 were used to extract the observed AMV pattern imposed in the simulations. The HadCRUT459 data set was used to compute the observed AMV composites shown in Fig. 1c. CMAP60,61, GPCPv2.362, TRMMv7 at 0.5° of spatial resolution63,64, MSWEPv2.665, and ERA-Interim66 data sets were used for mean bias corrections of model precipitation (cf. Figs. 5 and 6). Depending on data availability, we used different periods to compute the observed estimate mean state: 1979-2017 for CMAP, GPCPv2.3, and MSWEPv2.6; 1998-2011 for TRMMv7; and 1979–2018 for ERA-Interim.
The data generated and analyzed during the current study are available from the corresponding author YRR on reasonable request.
The code developed to analyze the data of the current study is available on request from the corresponding author.
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Jia, F., Wu, L., Gan, B. & Cai, W. Global warming attenuates the tropical Atlantic-Pacific teleconnection. Sci. Rep. 6, 1–7 (2016).
Luo, J. J., Liu, G., Hendon, H., Alves, O. & Yamagata, T. Inter-basin sources for two-year predictability of the multi-year la Niña event in 2010-2012. Sci. Rep. 7, 1–7 (2017).
Wang, C., Zhang, L., Lee, S. K., Wu, L. & Mechoso, C. R. A global perspective on CMIP5 climate model biases. Nat. Clim. Chang. 4, 201–205 (2014).
Kucharski, F., Syed, F. S., Burhan, A., Farah, I. & Gohar, A. Tropical Atlantic influence on Pacific variability and mean state in the twentieth century in observations and CMIP5. Clim. Dyn. 44, 881–896 (2014).
Zhang, L., Wang, C., Song, Z. & Lee, S.-K. Remote effect of the model cold bias in the tropical North Atlantic on the warm bias in the tropical southeastern Pacific. J. Adv. Model. Earth Syst. 6, 1016–1026 (2014).
Cai, W. et al. Pantropical climate interactions. Science (80-.). 363, eaav4236 (2019).
d'Orgeville, M. & Peltier, W. R. On the Pacific decadal oscillation and the atlantic multidecadal oscillation: might they be related? Geophys. Res. Lett. 34, 3–7 (2007).
Nigam, S., Sengupta, A. & Ruiz-Barradas, A. Atlantic–Pacific links in observed multidecadal SST variability: is the Atlantic multidecadal oscillation's phase reversal orchestrated by the Pacific decadal oscillation? J. Clim. 33, 5479–5505 (2020).
Meehl, G. A. et al. Atlantic and Pacific tropics connected by mutually interactive decadal-timescale processes. Nat. Geosci. 14, 36–42 (2021).
Smith, T. M., Reynolds, R. W., Peterson, T. C. & Lawrimore, J. Improvements to NOAA's historical merged land-ocean surface temperature analysis (1880–2006). J. Clim. 21, 2283–2296 (2008).
Huang, B. et al. Extended reconstructed sea surface temperature version 4 (ERSST.v4). Part I: upgrades and intercomparisons. J. Clim. 28, 911–930 (2015).
Bolton, D. The computation of equivalent potential temperature. Mon. Weather Rev. 108, 1046–1053 (1980).
Stone, P. H. & Carlson, J. H. Atmospheric lapse rate regimes and their parameterization. J. Atmos. Sci. 36, 415–423 (1979).
Byrne, M. P. & O'Gorman, P. A. Land-ocean warming contrast over a wide range of climates: convective quasi-equilibrium theory and idealized simulations. J. Clim. 26, 4000–4016 (2013).
Herold, N., Alexander, L. V., Donat, M. G., Contractor, S. & Becker, A. How much does it rain over land? Geophys. Res. Lett. 43, 341–348 (2016).
Herold, N., Behrangi, A. & Alexander, L. V. Large uncertainties in observed daily precipitation extremes over land. J. Geophys. Res. Atmos. 122, 668–681 (2017).
Morice, C. P., Kennedy, J. J., Rayner, N. A. & Jones, P. D. Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: the HadCRUT4 data set. J. Geophys. Res. Atmos. 117, 1–22 (2012).
Xie, P., Arkin, P. A. & Janowiak, J. E. CMAP: the CPC merged analysis of precipitation. Adv. Glob. Chang. Res 28, 319–328 (2007).
Climate Prediction Center, National Centers for Environmental Prediction, National Weather Service, NOAA (U.S. Department of Commerce, CPC Merged Analysis of Precipitation (CMAP), 1995).
Adler, R. et al. The global precipitation climatology project (GPCP) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9, 138 (2018).
Huffman, G. J., Adler, R. F., Bolvin, D. T. & Nelkin, E. J. The TRMM multi-satellite precipitation analysis (TMPA). In Satellite Rainfall Applications for Surface Hydrology. pp. 3–22 (Springer Netherlands, 2010). https://doi.org/10.1007/978-90-481-2915-7_1.
Tropical Rainfall Measuring Mission (TRMM). TRMM Radar Rainfall Statistics L3 1 month (5×5) and (0.5×0.5) degree V7. (2011).
Beck, H. E. et al. MSWep v2 Global 3-hourly 0.1° precipitation: Methodology and quantitative assessment. Bull. Am. Meteorol. Soc. 100, 473–500 (2019).
Berrisford, P. et al. The ERA-Interim Archive Version 2.0. (2011).
UCAR/NCAR/CISL/TDD. The NCAR Command Language (Version 6.6.2). (2019) https://doi.org/10.5065/D6WD3XH5.
Deser, C., Alexander, M. A., Xie, S.-P. & Phillips, A. S. Sea surface temperature variability: patterns and mechanisms. Ann. Rev. Mar. Sci. 2, 115–143 (2010).
Y.R.-R. was founded by the European Union's Horizon 2020 Research and Innovation Program in the framework of the Marie Skłodowska-Curie grant INADEC (Grant agreement 800154). E.M.-C. acknowledges funding from the European Commission's Horizon 2020 projects PRIMAVERA (Grant Agreement 641727). X.L. has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement H2020-MSCA-COFUND-2016-754433. A.B. and D.N. acknowledge funding from the European Commission's Horizon 2020 project EUCP (Grant agreement 776613). F.C. and G.D. were supported by the US National Science Foundation (NSF) under the Collaborative Research EaSM2 Grant OCE-1243015 to NCAR and by the US National Oceanic and Atmospheric Administration (NOAA) Climate Program Office under the Climate Variability and Predictability Program Grant NA13OAR4310138. NCAR is a major facility sponsored by the US NSF under Cooperative Agreement 1852977. Acknowledgments are made for the use of ECMWF's computing and archive facilities in this research, in particular, P.D. thanks ECMWF for providing computing time in the framework of the special project SPITDAVI. R.E., N.D., L.H., and D.S. were supported by the Met Office Hadley Center 522 Climate Program funded by BEIS and Defra and by the European Commission Horizon 2020 EUCP 523 project (GA 776613). J.L.-P. was funded by the European Union's Horizon 2020 Research and Innovation Program in the framework of the PRIMAVERA project (Grant Agreement 641727). J.R. and D.H. were funded by NERC via NCAS and the ACSIS project (NE/N018001/1), and JR was also funded by the NERC SMURPHS project (NE/N006054/1). M.M.-R. was funded by the European Union's Horizon 2020 Research and Innovation Program in the framework of the Marie Skłodowska-Curie grant FESTIVAL (Grant agreement 797236). E.T. has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 748750 (SPFireSD project). The analysis and plots of this paper were performed with the NCAR Command Language (Version 6.6.2; 2019)67.
Barcelona Supercomputing Center, Barcelona, Spain
Yohan Ruprich-Robert, Eduardo Moreno-Chamarro, Xavier Levine & Etienne Tourigny
Fondazione Centro EuroMediterraneo sui Cambiamenti Climatici, Bologna, Italy
Alessio Bellucci & Dario Nicolì
Consiglio Nazionale delle Ricerche, Istituto di Scienze dell'Atmosfera e del Clima, Bologna, Italy
Alessio Bellucci
CECI, Université de Toulouse, CNRS, Cerfacs, Toulouse, France
Christophe Cassou, Jorge Lopez-Parages, Said Qasmi, Emilia Sanchez-Gomez & Rym Msadek
National Center for Atmospheric Research, Boulder, CO, USA
Frederic Castruccio & Gokhan Danabasoglu
Consiglio Nazionale delle Ricerche, Istituto di Scienze dell'Atmosfera e del Clima, Torino, Italy
Paolo Davini
Met Office, Exeter, UK
Rosie Eade, Leon Hermanson, Nick Dunstone & Doug Smith
LOCEAN, Sorbonne Université/IRD/CNRS/MNHN, Paris, France
Guillaume Gastineau
Department of Meteorology, National Centre for Atmospheric Science, University of Reading, Reading, UK
Dan Hodson, Paul-Arthur Monerie & Jon Robson
Max Planck Institute for Meteorology, Hamburg, Germany
Katja Lohmann
CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France
Said Qasmi
ECMWF, Reading, UK
Christopher D. Roberts
Instituto de Ciencias del Mar, CSIC, Barcelona, Spain
Marta Martin-Rey
Yohan Ruprich-Robert
Eduardo Moreno-Chamarro
Xavier Levine
Christophe Cassou
Frederic Castruccio
Rosie Eade
Leon Hermanson
Dan Hodson
Jorge Lopez-Parages
Paul-Arthur Monerie
Dario Nicolì
Emilia Sanchez-Gomez
Gokhan Danabasoglu
Nick Dunstone
Rym Msadek
Jon Robson
Doug Smith
Etienne Tourigny
Y.R.-R. designed the study, performed the analysis, and wrote the initial article. E.M.-C. and X.L. contributed to the data analysis and to the interpretation of the results. Y.R.-R., A.B., C.C., F.C., P.D., R.E., G.G., L.H., D.H., J.L.-P., P.-A.M., D.N., S.Q., C.R. and E.S.-G. performed the simulations used in the study. All authors contributed to the manuscript preparation and the discussions that led to the final version of the article.
Correspondence to Yohan Ruprich-Robert.
Ruprich-Robert, Y., Moreno-Chamarro, E., Levine, X. et al. Impacts of Atlantic multidecadal variability on the tropical Pacific: a multi-model study. npj Clim Atmos Sci 4, 33 (2021). https://doi.org/10.1038/s41612-021-00188-5
Early warning signal for a tipping point suggested by a millennial Atlantic Multidecadal Variability reconstruction
Simon L. L. Michel
Didier Swingedouw
Myriam Khodri
Coupled climate response to Atlantic Multidecadal Variability in a multi-model multi-resolution ensemble
Daniel L. R. Hodson
Pierre-Antoine Bretonnière
Retish Senan
Climate Dynamics (2022)
About the Partner
For Authors and Referees
npj Climate and Atmospheric Science (npj Clim Atmos Sci) ISSN 2397-3722 (online)
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CommonCrawl
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What does "normalization" mean and how to verify that a sample or a distribution is normalized?
I have a question in which it asks to verify whether if the Uniform distribution (${\rm Uniform}(a,b)$) is normalized.
For one, what does it mean for any distribution to be normalized?
And two, how do we go about verifying whether a distribution is normalized or not?
I understand by computing $$ \frac{X-\text{mean}}{\text{sd}} $$ we get normalized data, but here it's asking to verify whether a distribution is normalized or not.
data-transformation terminology normalization standardization
amoeba says Reinstate Monica
$\begingroup$ What it means for a distribution to be normalized is not so simple (and it's usually not the distribution itself being normalized, but the random variable). For example, in the case of the uniform, some people may mean "linearly rescaled so as to get a standard uniform" (i.e. to get $a=0$ and $b=1$) ... while another person might mean "linearly rescaled so as to get mean 0 and sd 1". For the uniform, I'd normally assume the first, but as you see from an answer below, other people may take it to mean something else. The best option is to ask the person using the term to be less ambiguous. $\endgroup$ – Glen_b -Reinstate Monica Sep 20 '13 at 0:20
$\begingroup$ The more conventional terms are standardized (to achieve a mean of zero and SD of one) and normalized (to bring the range to the interval $[0,1]$ or to rescale a vector norm to $1$). Thus the re-expression $X\to (X-\text{mean})/SD$ is a standardization whereas multiplying a density $f$ by a constant $C$ to make $\int_{-\infty}^\infty Cf(x)dx=1$ is a normalization, because $\int f(x)dx$ is the $L^1$ norm of $f$. $\endgroup$ – whuber♦ Sep 20 '13 at 1:39
$\begingroup$ Also asked on math.SE. $\endgroup$ – Dilip Sarwate Sep 20 '13 at 3:16
$\begingroup$ Please don't cross-post, @Ada. That is against SE policy. If you post a Q on 1 site & then think you should have posted it on another, flag your Q & ask the moderators to migrate it for you. $\endgroup$ – gung - Reinstate Monica♦ Sep 20 '13 at 13:32
Unfortunately, terms are used differently in different fields, by different people within the same field, etc., so I'm not sure how well this can be answered for you here. You should make sure you know the definition that your instructor / the textbook is using for "normalized". However, here are some common definitions:
Centered: $$ X-{\rm mean} $$ Standardized: $$ \frac{X-\text{mean}}{\text{sd}} $$ Normalized: $$ \frac{X-\min(X)}{\max(X)-\min(X)} $$ Normalizing in this sense rescales your data to the unit interval. Standardizing turns your data into $z$-scores, as @Jeff notes. And centering just makes the mean of your data equal to $0$.
It is worth recognizing here that all three of these are linear transformations; as such, they do not change the shape of your distribution. That is, sometimes people call the $z$-score transformation "normalizing" and believe, because of $z$-scores' association with the normal distribution, that this has made their data normally distributed. This is not so (as @Jeff also notes, and as you could tell by plotting your data before and after). Should you be interested, you could change the shape of your data using the Box-Cox family of transformations, for example.
With respect to how you could verify these transformations, it depends on what exactly is meant by that. If they mean simply to check that the code ran properly, you could check means, SDs, minimums, and maximums.
gung - Reinstate Monica♦gung - Reinstate Monica
$\begingroup$ I have seen normalized used to suggest standardized or to suggest fitted onto a standard normal distribution i.e. $\Phi^{-1}(F(X))$, so of the three normalized is most likely to be misunderstood. Ada's comment of the application of a normalizing constant to a likelihood function is yet another possible interpretation. $\endgroup$ – Henry Sep 23 '13 at 20:16
By using the formula you provided on each score in your sample, you are converting them all to z-scores.
To verify that you computed all the z-scores correctly, find the new mean and standard deviation of your sample. If the mean is $0$ and the standard deviation is $1$, you've done everything correctly.
The purpose of doing this is to put everything in units relative to the standard deviation of your sample. This may be useful for a variety of purposes, such as comparing two different data sets that were scored using different units (centimeters and inches, perhaps).
It is important not to get this confused with asking whether a distribution is normal, i.e. whether it approximates a Gaussian distribution.
JeffJeff
$\begingroup$ so to check whether or not the Uniform distribution was normalized would it be equivalent to say E(X) = 0 and Var(X) = 1 where X~Uniform(a,b)? $\endgroup$ – user25658 Sep 20 '13 at 0:14
$\begingroup$ the data do not even have to be from a uniform distribution, they can be from any distribution. also, this is only true using the formula you provided; data can be normalized in ways other than using z-scores. for instance, IQ scores are said to be normalized with a score of 100 and standard deviation of 15. $\endgroup$ – Jeff Sep 20 '13 at 0:20
After consulting the TA, what the question was asking was whether if
$$ \int_{-\infty}^{\infty}f(x)dx=1 $$
where $f(x)$ in this case is the density of the uniform(a,b).
$\begingroup$ The terminology to use here is that the probability density function of the distribution is normalized. Because this reflects the axiomatic fact that the total probability must equal $1$, asking whether any distribution itself is normalized (in this sense) always has the same trivial answer: of course. $\endgroup$ – whuber♦ Sep 23 '13 at 20:07
$\begingroup$ This is what we are asked to verify. f(x) doesn't really have to be a pdf, and it can be any non-negative function. For any non-negative function where the above doesn't satisfy, we can always multiply by a normalizing constant $\endgroup$ – Ada Sep 23 '13 at 20:16
$\begingroup$ Not always. For instance, let $f(x)=e^{-x}$, a non-negative function defined on all the real numbers: there is no normalizing constant. But when you are told, as in your question statement, that "so-and-so is the PDF for such-and-such a distribution" then there is nothing whatsoever to verify: by definition it integrates to unity. $\endgroup$ – whuber♦ Sep 23 '13 at 20:22
$\begingroup$ It's true not any non-negative function where we can make it satisfy the above condition even if we multiply by a normalizing constant. $\endgroup$ – Ada Sep 23 '13 at 20:29
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CommonCrawl
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Eugenie Maria Morenus
Eugenie Maria Morenus (February 21, 1881 – October 15, 1966) was an American mathematician and college professor. She taught Latin and mathematics at Sweet Briar College from 1909 to 1946.
Eugenie Maria Morenus
Eugenie Maria Morenus from a 1921 yearbook
BornFebruary 21, 1881
Cleveland, New York
DiedOctober 15, 1966
Lake Wales, Florida
Occupation(s)Mathematician, college professor
Early life and education
Morenus was born in Cleveland, New York, the daughter of Eugene Morenus and Maria Euphemia Van Blarcom Morenus.[1] Her father managed a glassworks. She graduated from Monogahela High School in 1898.[2] She earned a bachelor's degree from Vassar College in 1904, and a master's degree from the same school in 1905.[3] She completed doctoral studies in mathematics at Columbia University in 1922. Her dissertation under Edward Kasner was titled "Geometric properties completely characterizing the set of all the curves of constant pressure in a field of force".[4][5]
Morenus was also a student for briefer periods at the University of Chicago, and at Göttingen.[6]
Career
Morenus taught mathematics and Latin at a school in Watertown, New York and at Poughkeepsie High School after her master's degree.[3] She was a Latin instructor at Sweet Briar College from 1909 to 1916, and was a mathematics professor at the same school from 1916 to 1946.[7][8] She was head of the mathematics department for much of that time. While at the school she was prominent in campus events, as a chorister, photographer, and play director. Her horse, October or "Toby", was a familiar figure on campus, and Morenus would lead ten-day rides for students over spring breaks.[9]
Morenus was a charter member of the Mathematical Association of America, belonged to the Virginia Academy of Science, and was active in the American Association of University Women (AAUW).[3] She was active in the Order of the Eastern Star and the Daughters of the American Revolution.[10] She received an Anna Brackett Fellowship by the AAUW in 1927, to study at Cambridge.[11][12] After her retirement from Sweet Briar College in 1946, she taught briefly at Connecticut College for Women, and spent her winters in Florida.[3]
Personal life
Morenus died in Lake Wales, Florida in 1966, aged 85 years.[13][14] There was a scholarship endowment fund named for Morenus at Sweet Briar College, beginning in 1960.[3]
References
1. Daughters of the American Revolution (1917). Lineage Book. The Society. p. 22.
2. "The School Exhibition". The Daily Republican. 1898-06-02. p. 1. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
3. Green, Judy; LaDuke, Jeanne (2009). Pioneering Women in American Mathematics: The Pre-1940 PhD's. American Mathematical Soc. pp. 252–253. ISBN 978-0-8218-4376-5.
4. Eugenie Maria Morenus, Mathematics Genealogy Project, North Dakota State University.
5. Morenus, Eugenie M. (1922). Geometric properties completely characterizing the set of all the curves of constant pressure in a field of force / by Eugenie M. Morenus. Philadelphia: Press of the John C. Winston Co.
6. Singer, Sandra L. (2003). Adventures Abroad: North American Women at German-speaking Universities, 1868-1915. Greenwood Publishing Group. p. 95. ISBN 978-0-313-32371-3.
7. "Golden Anniversary Celebration Set". The Times Dispatch. 1956-01-29. p. 43. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
8. "Notes and News". The American Mathematical Monthly. 29 (2): 93–96. 1922. doi:10.1080/00029890.1922.11986108. ISSN 0002-9890. JSTOR 2972833.
9. Poleski, Dana (2020-12-22). "100 Years of Equestrian Excellence". Sweet Briar College | News. Retrieved 2021-03-17.{{cite web}}: CS1 maint: url-status (link)
10. "Obituaries: Miss Eugenie Morenus". The Orlando Sentinel. 1966-10-20. p. 119. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
11. Haskin, Frederic J. (1927-03-30). "The Haskin Letter: University Women Meet". The Independent-Record. p. 4. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
12. "Eugenie M. Morenus". The San Francisco Examiner. 1927-03-21. p. 1. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
13. "Morenus". The Post-Standard. 1966-10-20. p. 9. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
14. "Miss Morenus, Ex-Teacher, Dies". The Post-Standard. 1966-10-19. p. 29. Retrieved 2021-03-17 – via Newspapers.com.{{cite news}}: CS1 maint: url-status (link)
Authority control
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Freedom Math Dance
A blog about math (mainly), computer tricks (sometimes) and jazz music.
On Rolle's theorem
Libellés : agrégation , analysis , differential calculus
This post is inspired by a paper of Azé and Hiriart-Urruty published in a French high school math journal; in fact, it is mostly a paraphrase of that paper with the hope that it be of some interest to young university students, or to students preparing Agrégation. The topic is Rolle's theorem.
1. The one-dimensional theorem, a generalization and two other proofs
Let us first quote the theorem, in a nonstandard form.
Theorem. — Let $I=\mathopen]a;b\mathclose[$ be a nonempty but possibly unbounded interval of $\mathbf R$ and let $f\colon I\to\mathbf R$ be a continuous function. Assume that $f$ has limits at $a$ and $b$, equal to some element $\ell\in\mathbf R\cup\{+\infty\}$. Then $f$ is bounded from below.
If $\inf_I(f)<\ell$, then there exists a point $c\in I$ such that $f(c)=\inf_I (f)$. If, moreover, $f$ has a right derivative and a left derivative at $c$, then $f'_l(c)\leq0$ and $f'_r(c)\geq0$.
If $\inf_I(f)\geq\ell$, then $f$ is bounded on $I$ and there exists a point $c\in I$ such that $f(c)=\sup_I(f)$. If, moreover, $f$ has a right derivative and a left derivative at $c$, then $f'_l(c)\geq0$ and $f'_r(c)\leq0$.
Three ingredients make this version slightly nonstandard:
The interval $I$ may be taken to be infinite;
The function $f$ may tend to $+\infty$ at the endpoints of $I$;
Only left and right derivatives are assumed.
Of course, if $f$ has a derivative at each point, then the statement implies that $f'(c)=f'_l(c)=f'_r(c)=0$.
a) As stated in this way, the proof is however quite standard and proceeds in two steps.
Using that $f$ has a limit $\ell$ which is not $-\infty$ at $a$ and $b$, it follows that there exists $a'$ and $b'$ in $I$ such that $a<a'<b'<b$ such that $f$ is bounded from below on $\mathopen ]a;a']$ and on $[b';b\mathclose[$. Since $f$ is continuous on the compact interval $[a';b']$, it is then bounded from below on $I$.
If $\inf_I(f)<\ell$, then we can choose $\ell'\in\mathbf R$ such that $\inf_I(f)<\ell'<\ell$ and $a'$, $b'$ such that $f(x)>\ell'$ outside of $[a';b']$. Then, let $c\in [a';b']$ such that $f(c)=\inf_{[a';b']}(f)$; then $f(c)=\inf_I(f)$.
If $\sup_I(f)>\ell$, then we have in particular $\ell\neq+\infty$, and we apply the preceding analysis to $-f$.
In the remaining case, $\inf_I(f)=\sup_I(f)=\ell$ and $f$ is constant.
For $x>c$, one has $f(x)\geq f(c)$, hence $f'_r(c)\geq 0$; for $x<c$, one has $f(x)\geq f(c)$, hence $f'_l(c)\leq0$.
The interest of the given formulation can be understood by looking at the following two examples.
If $f(x)=|x|$, on $\mathbf R$, then $f$ attains its lower bound at $x=0$ only, where one has $f'_r(0)=1$ and $f'_l(0)=-1$.
Take $f(x)=e^{-x^2}$. Then there exists $c\in\mathbf R$ such that $f'(c)=0$. Of course, one has $f'(x)=-2xe^{-x^2}$, so that $c=0$. However, it is readily seen by induction that for any integer $n$, the $n$th derivative of $f$ is of the form $P_n(x)e^{-x^2}$, where $P_n$ has degree $n$. In particular, $f^{(n)}$ tends to $0$ at infinity. And, by induction again, the theorem implies that $P_n$ has $n$ distinct roots in $\mathbf R$, one between any two consecutive roots of $P_{n-1}$, one larger than the largest root of $P_n$, and one smaller than the smallest root of $P_n$.
b) In a 1959 paper, the Rumanian mathematician Pompeiu proposed an alternative proof of Rolle's theorem, when the interval $I$ is bounded, and which works completely differently. Here is how it works, following the 1979 paper published in American Math. Monthly by Hans Samelson.
First of all, one uses the particular case $n=2$ of the Levi chord lemma :
Lemma. — Let $f\colon [a;b]\to\mathbf R$ be a continuous function such that $f(a)=f(b)$. For every integer $n\geq 2$, there exists $a',b'\in[a;b]$ such that $f(a')=f(b')$ and $b'-a'=(b-a)/n$.
Let $h=(b-a)/n$. From the equality
\[ 0 = f(b)-f(a) = (f(a+h)-f(a))+(f(a+2h)-f(a+h))+\cdots + (f(a+nh)-f(a+(n-1)h), \]
one sees that the function $x\mapsto f(x+h)-f(x)$ from $[a;b-h]$ to $\mathbf R$ does not have constant sign. By the intermediate value theorem, it vanishes at some point $a'\in [a;b-h]$. If $b'=a'+h$, then $b'\in[a;b]$, $b'-a'=(b-a)/n$ and $f(a')=f(b')$.
Then, it follows by induction that there exists a sequence of nested intervals $([a_n;b_n])$ in $[a;b]$ with $f(a_n)=f(b_n)$ and $b_n-a_n=(b-a)/2^n$ for all $n$. The sequences $(a_n)$ and $(b_n)$ converge to a same limit $c\in [a;b]$. Since $f(b_n)=f(c)+(b_n-c) (f'(c) + \mathrm o(1))$, $f(a_n)=f(c)+(a_n-c)(f'(c)+\mathrm o(1))$, one has
\[ f'(c) = \lim \frac{f(b_n)-f(a_n)}{b_n-a_n} = 0. \]
What makes this proof genuinely distinct from the classical one is that the obtained point $c$ may not be a local minimum or maximum of $f$, also I don't have an example to offer now.
c) In 1979, Abian furnished yet another proof, which he termed as the "ultimate" one. Here it is:
It focuses on functions $f\colon[a;b]\to\mathbf R$ on a bounded interval of $\mathbf R$ which are not monotone and, precisely, which are up-down, in the sense that $f(a)\leq f(c)$ and $f(c)\geq f(b)$, where $c=(a+b)/2$ is the midpoint of $f$. If $f(a)=f(b)$, then either $f$ or $-f$ is up-down.
Then divide the interval $[a;b]$ in four equal parts: $[a;p]$, $[p;c]$, $[c;q]$ and $[q;b]$. If $f(p)\geq f(c)$, the $f|_{[a;c]}$ is up-down. Otherwise, one has $f(p)\leq f(c)$. In this case, if $f(c)\geq f(q)$, we see that $f|_{[p;q]}$ is up-down. And otherwise, we observe that $f(q)\leq f(c)$ and $f(c)\geq f(b)$, so that $f|_{[c;b]}$ is up-down. Conclusion: we have isolated within the interval $[a;b]$ a subinterval $[a';b']$ of length $(b-a)/2$ such that $f|_{[a';b']}$ is still up-down.
Iterating the procedure, we construct a sequence $([a_n;b_n])$ of nested intervals, with $(b_n-a_n)=(b-a)/2^n$ such that the restriction of $f$ to each of them is up-down. Set $c_n=(a_n+b_n)/2$.
The sequences $(a_n), (b_n),(c_n)$ satisfy have a common limit $c\in [a;b]$. From the inequalities $f(a_n)\leq f(c_n)$ and $a_n\leq c_n$, we obtain $f'(c)\geq 0$; from the inequalities $f(c_n)\geq f(b_n)$ and $c_n\leq b_n$, we obtain $f'(c)\leq 0$. In conclusion, $f'(c)=0$.
2. Rolle's theorem in normed vector spaces
Theorem. — Let $E$ be a normed vector space, let $U$ be an open subset of $E$ and let $f\colon U\to\mathbf R$ be a differentiable function. Assume that there exists $\ell\in\mathbf R\cup\{+\infty\}$ such that $f(x)\to \ell$ when $x$ tends to the "boundary" of $U$ — for every $\ell'<\ell$, there exists a compact subset $K$ of $U$ such that $f(x)\geq\ell'$ for all $x\in U$ but $x\not\in K$. Then $f$ is bounded below on $U$, there exists $a\in U$ such that $f(a)=\inf_U (f)$ and $Df(a)=0$.
The proof is essentially the same as the one we gave in dimension 1. I skip it here.
If $E$ is finite dimensional, then this theorem applies in a vast class of examples : for example, bounded open subsets $U$ of $E$, and continuous functions $f\colon \overline U\to\mathbf R$ which are constant on the boundary $\partial(U)=\overline U - U$ of $U$ and differentiable on $U$.
However, if $E$ is infinite dimensional, the closure of a bounded open set is no more compact, and it does not suffice that $f$ extends to a function on $\overline U$ with a constant value on the boundary.
Example. — Let $E$ be an infinite dimensional Hilbert space, let $U$ be the open unit ball and $B$ be the closed unit ball. Let $g(x)=\frac12 \langle Ax,x\rangle+\langle b,x\rangle +c$ be a quadratic function, where $A\in\mathcal L(E)$, $b\in E$ and $c\in\mathbf R$, and let $f(x)=(1-\lVert x\rVert^2) g(x)$. The function $f$ is differentiable on $E$ and one has
\[ \nabla f(x) = (1-\Vert x\rVert^2) ( Ax + b) - 2 (\frac12 \langle Ax,x\rangle + \langle b,x\rangle + c) x. \]
Assume that there exists $x\in U$ such that $\nabla f(x)=0$. Then $Ax+b = \lambda x$, with
\[ \lambda= \frac2{1-\lVert x\rVert ^2} \left(\frac12 \langle Ax,x\rangle + \langle b,x\rangle + c \right). \]
Azé and Hiriart-Urruty take $E=L^2([0;1])$, for $A$ the operator of multiplication by the function $t$, $b(t)=t(1-t)$, and $c=4/27$. Then, one has $g(x)>0$, hence $\lambda>0$, and $x(t)=\frac1{\lambda-t}b(t)$ for $t\in[0;1]$. This implies that $\lambda\geq 1$, for, otherwise, the function $x(t)$ would not belong to $E$. This allows to compute $\lambda$ in terms of $\mu$, obtaining $\lambda\leq3/4$, which contradicts the inequality $\lambda\geq 1$. (I refer to the paper of Azé and Hiriart-Urruty for more details.)
3. An approximate version of Rolle's theorem
Theorem. — Let $B$ the closed euclidean unit ball in $\mathbf R^n$, let $U$ be its interior, let $f\colon B\to \mathbf R$ be a continuous function on $B$. Assume that $\lvert f\rvert \leq \epsilon $ on the boundary $\partial(U)$ and that $f$ is differentiable on $U$. Then there exists $x\in U$ such that $\lVert Df(x)\rVert\leq\epsilon$.
In fact, replacing $f$ by $f/\epsilon$, one sees that it suffices to treat the case $\epsilon =1$.
Let $g(x)=\lVert x\rVert^2- f(x)^2$. This is a continuous function on $B$; it is differentiable on $U$, with $ \nabla g(x)=2(x-f(x)\nabla f(x))$. Let $\mu=\inf_B(g)$. Since $g(0)=-f(0)^2\leq0$, one has $\mu\leq 0$. We distinguish two cases:
If $\mu=0$, then $\rvert f(x)\lvert \leq \lVert x\rVert$ for all $x\in B$. This implies that $\lVert\nabla f(0)\rVert\leq1$.
If $\mu<0$, let $x\in B$ be such that $ g(x)=\mu$; in particular, $f(x)^2\geq \lVert x\rVert^2-\mu>0$, which implies that $f(x)\neq0$. Since $g\geq0$ on $\partial(U)$, we have $x\in B$, hence $\nabla g(x)=0$. Then $x=f(x)\nabla f(x)$, hence $\nabla f(x)=x/f(x)$. Consequently,
\[ \lVert \nabla f(x)\rVert \leq \frac{\lVert x\rVert}{f(x)}\leq \frac{\lVert x\rVert}{(\lVert x\rVert^2-\mu)^{1/2}}<1.\]
This concludes the proof.
Thanks to the Twitter users @AntoineTeutsch, @paulbroussous and @apauthie for having indicated me some misprints and incorrections.
Publié par Antoine Chambert-Loir à 1:33 AM No comments :
Celebrating Ramanujan's birthday — From powers of divisors to coefficients of modular forms
Libellés : algebraic geometry , math , modular curves , number theory
In a Twitter post, Anton Hilado reminded us that today (December 22nd) was the birthday of Srinivasa Ramanujan, and suggested somebody explains the "Ramanujan conjectures". The following blog post is an attempt at an informal account. Or, as @tjf frames it, my christmas present to math twitter.
The story begins 1916, in a paper Ramanujan published in the Transactions of the Cambridge Philosophical Society, under the not so explicit title: On certain arithmetical functions. His goal started as the investigation of the sum $\sigma_s(n)$ of all $s$th powers of all divisors of an integer $n$, and approximate functional equations of the form
\[ \sigma_r(0)\sigma_s(n)+\sigma_r(1)\sigma_s(n-1)+\dots+\sigma_r(n)\sigma_s(0)
\approx \frac{\Gamma(r+1)\Gamma(s+1)}{\Gamma(r+s+2)} \frac{\zeta(r+1)\zeta(s+1)}{\zeta(r+s+2)}\sigma_{r+s+1}(n) + \frac{\zeta(1-r)+\zeta(1-s)}{r+s} n \sigma_{r+s-1}(n), \]
where $\sigma_s(0)=\dfrac12 \zeta(-s)$, and $\zeta$ is Riemann's zeta function. In what follows, $r,s$ will be positive odd integers, so that $\sigma_s(0)$ is half the value of Riemann's zeta function at a negative odd integer; it is known to be a rational number, namely $(-1)^sB_{s+1}/2(s+1)$, where $B_{s+1}$ is the $(s+1)$th Bernoulli number.
This investigation, in which Ramanujan engages without giving any motivation, quickly leads him to the introduction of infinite series,
\[ S_r = \frac12 \zeta(-r) + \frac{1^rx}{1-x}+\frac{2^r x^2}{1-x^2}+\frac{3^rx^3}{1-x^3}+\dots. \]
Nowadays, the parameter $x$ would be written $q$, and $S_r=\frac12 \zeta(-r) E_{r+1}$, at least if $r$ is an odd integer, $E_r$ being the Fourier expansion of the Eisenstein series of weight $r$. The particular cases $r=1,3,5$ are given special names, namely $P,Q,R$, and Ramanujan proves that $S_s$ is a linear combination of $Q^mR^n$, for integers $m,n$ such that $4m+6n=s+1$. Nowadays, we understand this as the fact that $Q$ and $R$ generated the algebra of modular forms—for the full modular group $\mathrm{SL}(2,\mathbf Z)$.
In the same paper, Ramanujan spells out the system of algebraic differential equations satisfied by $P,Q,R$:
\[ x \frac {dP}{dx} = \frac{1}{12}(P^2-Q), x\frac{dQ}{dx}=\frac13(PQ-R), x\frac{dR}{dx}=\frac12(PR-Q^2). \]
The difference of the two sides of the initial equation has an expansion as a linear combination of $Q^mR^n$, where $4m+6n=r+s+2$. By the functional equation of Riemann's zeta function, relating $\zeta(s)$ and $\zeta(1-s)$, this expression vanishes for $x=0$, hence there is a factor $Q^3-R^2$.
Ramanujan then notes that
$ x\frac{d}{dx} \log(Q^3-R^2)=P$, so that
\[ P =x\frac{d}{dx} \log \left( x\big((1-x)(1-x^2)(1-x^3)\dots\big)^{24} \right) \]
\[ Q^3-R^2 = 1728 x \big((1-x)(1-x^2)(1-x^3)\dots\big)^{24}= \sum \tau(n) x^n, \]
an expression now known as Ramanujan's $\Delta$-function. In fact, Ramanujan also makes the relation with elliptic functions, in particular, with Weierstrass's $\wp$-function. Then, $\Delta$ corresponds to the discriminant of the degree 3 polynomial $f$ such that $\wp'(u)^2=f(\wp(u))$.
In any case, factoring $Q^3-R^2$ in the difference of the two terms, it is written as a linear combination of $Q^mR^n$, where $4m+6n=r+s-10$. When $r$ and $s$ are positive odd integers such that $r+s\leq 12$, there are no such pairs $(m,n)$, hence the difference vanishes, and Ramanujan obtains an equality in these cases.
Ramanujan is interested in the quality of the initial approximation. He finds an upper bound of the form $\mathrm O(n^{\frac23(r+s+1)})$. Using Hardy–Littlewood's method, he shows that it cannot be smaller than $n^{\frac12(r+s)}$. That prompts his interest for the size of the coefficients of arithmetical functions, and $Q^3-R^2$ is the simplest one. He computes the coefficients $\tau(n)$ for $n\leq30$ and gives them in a table:
Recalling that $\tau(n)$ is $\mathrm O(n^7)$, and not $\mathrm O(n^5)$, Ramanujan states that there is reason to believe that $\tau(n)=\mathrm O(n^{\frac{11}2+\epsilon})$ but not $\mathrm O(n^{\frac{11}2})$. That this holds is Ramanujan's conjecture.
Ramanujan was led to believe this by observing that the Dirichlet series $ \sum \frac{\tau(n)}{n^s} $ factors as an infinite product ("Euler product", would we say), indexed by the prime numbers:
\[ \sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \prod_p \frac{1}{1-\tau(p)p^{-s}+p^{11-2s}}. \]
This would imply that $\tau$ is a multiplicative function: $\tau(mn )=\tau(m)\tau(n)$ if $m$ and $n$ are coprime, as well as the more complicated relation $\tau(p^{k+2})=\tau(p)\tau(p^{k+1})-p^{11}\tau(p^k)$ between the $\tau(p^k)$. These relations have been proved by Louis Mordell in 1917. He introduced operators (now called Hecke operators) $T_p$ (indexed by prime numbers $p$) on the algebra of modular functions and proved that Ramanujan's $\Delta$-function is an eigenfunction. (It has little merit for that, because it is alone in its weight, so that $T_p \Delta$ is a multiple of $\Delta$, necessarily $T_p\Delta=\tau(p)\Delta$.)
The bound $\lvert{\tau(p)}\rvert\leq p^{11/2}$ means that the polynomial $1-\tau(p) X+p^{11}X^2$ has two complex conjugate roots. This part of the conjecture would be proved in 1973 only, by Pierre Deligne, and required many additional ideas. One was conjectures of Weil about the number of points of algebraic varieties over finite fields, proved by Deligne in 1973, building on Grothendieck's étale cohomology. Another was the insight (due to Michio Kuga, Mikio Sato and Goro Shimura) that Ramanujan's conjecture could be reframed as an instance of the Weil conjectures, and its actual proof by Deligne in 1968, applied to the 10th symmetric product of the universal elliptic curve.
Publié par Antoine Chambert-Loir à 11:31 PM No comments :
agrégation algebra algebraic geometry analysis category theory combinatorics computer convexity cryptography density differential algebra differential calculus drums dynamics Felix Klein finite fields fundamental group Galois theory Grothendieck Grothendieck topology higher education Hilbert's Nullstellensatz jazz linear algebra math measure theory model theory modular curves motivic integration N. Bourbaki non-archimedean analytic geometry number theory Poisson summation formula residually finite group set theory Sylow subgroups topological vector spaces topology
Antoine Chambert-Loir
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CommonCrawl
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\begin{definition}[Definition:Convergent Sequence/Normed Division Ring/Definition 4]
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n} $ be a sequence in $R$.
The sequence $\sequence {x_n}$ '''converges to the limit $x \in R$ in the norm $\norm {\, \cdot \,}$''' {{iff}}:
:$\sequence {x_n}$ converges to $x$ in the topology induced by the norm $\norm {\, \cdot \,}$
\end{definition}
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ProofWiki
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\begin{document}
\title{Strongly mixing systems are almost strongly mixing of all orders} \begin{abstract} We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP$^*$. While the main goal of this paper is to establish a \emph{universal} property of strongly mixing actions of countable abelian groups, our results, when applied to $\mathbb{Z}$-actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for $\mathbb{Z}$-actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemer{\'e}di's theorem. We also demonstrate the versatility of $\mathcal R$-limits by obtaining new characterizations of higher order weak and mild mixing for actions of countable abelian groups. \end{abstract}
\textbf{Keywords:} Ergodic theory, Mixing of higher orders, Ramsey Theory. \tableofcontents \section{Introduction} Let $G=(G,+)$ be a countable discrete abelian group and let $(T_g)_{g\in G}$ be a measure preserving $G$-action on a separable probability space $(X,\mathcal A,\mu)$. We will call the quadruple $(X,\mathcal A,\mu, (T_g)_{g\in G})$ a measure preserving system. A measure preserving system $(X,\mathcal A,\mu, (T_g)_{g\in G})$ is strongly mixing (or 2-mixing) if for any $A_0,A_1\in\mathcal A$, one has \begin{equation}
\lim_{g\rightarrow\infty} \mu(A_0\cap T_g A_1)=\mu(A_0)\mu(A_1). \end{equation} The goal of this paper is to obtain new results about higher order mixing properties of stronlgy mixing actions of abelian groups. These results are motivated by the following classical problem going back to Rohlin (who formulated it for $\mathbb{Z}$-actions, see \cite{rokhlin1949endomorphisms}). \begin{namedthm*}{Rohlin's Problem}\label{11.Rohlin'sProblem} Assume that a measure preserving system $(X,\mathcal A,\mu, (T_g)_{g\in G})$ is strongly mixing. Is it true that given any $\ell\geq 2$ the system $(X,\mathcal A,\mu, (T_g)_{g\in G})$ is $(\ell+1)$-mixing, meaning that for any $A_0,...,A_\ell\in\mathcal A$ and any sequences
$(g^{(1)}_k)_{k\in\mathbb{N}}$,...,$(g^{(\ell)}_k)_{k\in\mathbb{N}}$ in $G$ satisfying: \begin{enumerate}[(i)] \item For any $j\in\{1,...,\ell\}$ \begin{equation}\label{1.SequenceGoingToInfty} \lim_{k\rightarrow\infty}g^{(j)}_k=\infty \end{equation} \item For any distinct $i,j\in\{1,...,\ell\}$, \begin{equation}\label{1.DifferenccesGoingToInfty} \lim_{k\rightarrow\infty}(g^{(j)}_k-g^{(i)}_k)=\infty. \end{equation} \end{enumerate} one has \begin{equation}\label{1.RightLimitEquation}
\lim_{k\rightarrow\infty}\mu(A_0\cap T_{g^{(1)}_k}A_1\cap\cdots\cap T_{g^{(\ell)}_k}A_\ell)=\prod_{j=0}^\ell\mu(A_j). \end{equation} \end{namedthm*}
While for $\mathbb{Z}$-actions Rohlin's problem is still unsolved,\footnote{ The notable classes of $\mathbb{Z}$-actions for which it is known that 2-mixing implies mixing of all orders include ergodic automorphisms of compact groups \cite{rokhlin1949endomorphisms}, mixing transformations with singular spectrum \cite{host1991mixing}, and mixing actions of finite rank \cite{kalikow1984twofold}, \cite{ryzhikov1993joinings}. It is also known that some natural actions of various locally compact groups posses the property of mixing of all orders (see, for example, \cite{marcus1978horocycle}, \cite{mozes1992mixing}, \cite{ryzhikov2000rokhlin}, \cite{fayad2016multiple}). } an example for $\mathbb{Z}^2$-actions, due to Ledrappier, shows that, in general, mixing does not imply mixing of higher orders \cite{ledrappier1978champ}. (The reader is referred to \cite{schmidt1995AlgebraicDynamicsBook} for more Ledrappier-type examples for $\mathbb{Z}^d$-actions). More precisely, Ledrappier provided an example of a strongly mixing system $(\Gamma,\mathcal B,\mu,(T^nS^m)_{(n,m)\in\mathbb{Z}^2})$, where $\Gamma$ is a compact abelian group, $\mathcal B$ is the $\sigma$-algebra of Borel sets in $\Gamma$, $\mu$ is the normalized Haar measure on $\Gamma$ and $T,S:\Gamma\rightarrow \Gamma$ are commuting automorphisms with the property that for some measurable set $A\subseteq \Gamma$, $$\mu(A\cap T^{2^n}A\cap S^{2^n}A)\centernot{ \xrightarrow[n\to\infty]{}} \mu^3(A).$$
The analysis of Ledrappier's example undertaken in \cite{arenas2008ledrappier} reveals that Ledrappier's system is "almost mixing of all orders" in the sense that, for any $\ell\in\mathbb{N}$, if the sequences $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(\ell)})_{k\in\mathbb{N}}$ in $\mathbb{Z}^2$ satisfy \eqref{1.SequenceGoingToInfty} and \eqref{1.DifferenccesGoingToInfty} and, in addition, the $\ell$-tuples $(g_k^{(1)},...,g_k^{(\ell)})$ avoid certain rather rarefied subsets of $\mathbb{Z}^{2\ell}$, the equation \eqref{1.RightLimitEquation} holds for any $A_0,...,A_\ell\in\mathcal B$ (see \cite[Theorem 3.3]{arenas2008ledrappier}). The results obtained in \cite{arenas2008ledrappier} were extended in \cite{arenas2019almost} to a rather large family of systems of algebraic origin.\\
In view of the results obtained in \cite{arenas2008ledrappier} and \cite{arenas2019almost}, one might wonder if it could possibly be true that, similarly to the case of Ledrappier's system, \textit{any} strongly mixing action $(X,\mathcal A,\mu, (T_g)_{g\in G})$ of an abelian group $G$ is, in some sense, almost mixing of all orders. The goal of this paper is to establish a result that can be interpreted as a positive answer to this question.\\
At this point, we would like to mention that in the special case when $G=\mathbb{Z}$ our main theorem (\cref{1.MainResult} below) has corollaries (\cref{1.ZDiagonalResult} and \cref{1.GlobalSigma*resultForZ}) which provide new non-trivial characterizations of the notion of strong mixing in terms of the largeness of sets of the form \begin{equation}
R^{a_1,...,a_\ell}_\epsilon(A_0,...,A_\ell)=\{n\in\mathbb{Z}\,|\,|\mu(A_0\cap T^{a_1n}A_1\cap\cdots\cap T^{a_\ell n}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\} \end{equation} and \begin{equation}\label{11.MixingSetForZ}
R_\epsilon(A_0,...,A_\ell)=\{(n_1,...,n_\ell)\in \mathbb{Z}^\ell\,|\,|\mu(A_0\cap T^{n_1}A_1\cap\cdots \cap T^{n_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}. \end{equation} So, if similarly to the case of more general group actions, Rohlin's problem will turn out to have a negative answer for $G=\mathbb{Z}$, our results can still be interpreted as a confirmation of a weaker version of Rohlin's question.\\
Let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a measure preserving system. Let $\ell\in\mathbb{N}$ and $\epsilon>0$. For any $A_0,...,A_\ell\in\mathcal A$ consider the set \begin{equation}\label{11.MixingSet}
R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}. \end{equation} Clearly, the higher is the degree of multiple mixing of the system $(X,\mathcal A,\mu, (T_g)_{g\in G})$, the more massive should the set $R_\epsilon(A_0,...,A_\ell)$ be as a subset of $G^\ell$. While, for $\ell=1$, the strong mixing property of $(X,\mathcal A,\mu, (T_g)_{g\in G})$ implies that the set $R_\epsilon(A_0,A_1)$ is cofinite, this is no longer the case for $\ell\geq 2$ even if our system $(X,\mathcal A,\mu,(T_g)_{g\in G})$ is mixing of all orders. For example, for any 3-mixing system, if $\epsilon>0$ is small enough, the set
$$R_\epsilon(A_0,A_1,A_2)=\{(g_1,g_2)\in G^2\,|\,|\mu(A_0\cap T_{g_1}A_1\cap T_{g_2}A_2)-\mu(A_0)\mu(A_1)\mu(A_2)|<\epsilon\}$$
cannot contain pairs $(g_1,g_2)$ which are too close to the "lines" $\{(g,g)\,|\,g\in G\}$, $\{(g,0)\,|\,g\in G\}$ and $\{(0,g)\,|\,g\in G\}$.\\
In what follows we will show that for \textit{any} mixing system $(X,\mathcal A,\mu, (T_g)_{g\in G})$, the subsets of $G^\ell$ which are of the form $\mathcal R_\epsilon(A_0,...,A_\ell)$ posses a strong ubiquity property which we will call $\tilde\Sigma_\ell^*$ and which is quite a bit stronger than the properties of largeness associated with weakly and mildly mixing systems. In other words, we will show that for any strongly mixing system the complement of any set of the form $R_\epsilon (A_0,...,A_\ell)$ is very "small", giving meaning to the claim that $(X,\mathcal A,\mu,(T_g)_{g\in G})$ is "almost strongly mixing" of all orders. This will be achieved with the help of \textit{$\mathcal R$-limits}, a new notion of convergence which is based on a classical combinatorial result due to Ramsey and, as we will see, is adequate for dealing with strongly mixing systems. (In particular, we will show that the $\tilde\Sigma_\ell^*$ property of the sets $R_\epsilon(A_0,...,A_\ell)$ implies the strong mixing of $(X,\mathcal A,\mu,(T_g)_{g\in G}$)).\\
We would like to remark that while the results that we obtain are not as sharp as those obtained in \cite{arenas2008ledrappier} and \cite{arenas2019almost}, they have the advantage of being applicable to \textit{any} strongly mixing system $(X,\mathcal A,\mu,(T_g)_{g\in G})$, where $G$ is a countable abelian group. Moreover, as will be demonstrated in Section 6, the versatility of $\mathcal R$-limits allows one to obtain new and recover some old results pertaining to multiple recurrence properties of weakly and mildly mixing actions of countable abelian groups. We would also like to mention that, as will be seen in Section 3, the utilization of $\mathcal R$-limits brings to life many new equivalent characterizations of strong mixing (some of which bear a strong analogy with the familiar characterizations of weak mixing via convergence in density and mild mixing via IP-convergence). \\
Before introducing the mentioned above notion of largeness for subsets of $G^\ell$, we define a related and somewhat simpler notion in $G$. \begin{defn} Let $m\in\mathbb{N}$, let $(G,+)$ be a countable abelian group, and let $E\subseteq G$. \begin{enumerate}
\item We say that $E$ is a $\Sigma_m$ set if it is of the form
$$\{g_{k_1}^{(1)}+\cdots+g_{k_m}^{(m)}\,|\,k_1<\cdots<k_m\}$$
where for each $j\in\{1,...,m\}$, $(g_k^{(j)})_{k\in\mathbb{N}}$ is a sequence in $G$ which satisfies $\lim_{k\rightarrow\infty}g_k^{(j)}=\infty$.
\item We say that $E$ is a $\Sigma_m^*$ set if it has a non-trivial intersection with every $\Sigma_m$ set. \end{enumerate} \end{defn} \begin{rem}\label{1.TheCofiniteRemark} \begin{enumerate}[(a)]
\item Note that a subset of $G$ is $\Sigma_1^*$ if and only if it is cofinite. On the other hand, for any $m\geq 2$, a $\Sigma_m^*$ set does not need to be cofinite. Moreover, one can show that for each $m\geq 2$, there exists a $\Sigma_{m}^*$ set which fails to be a $\Sigma_{n}^*$ set for each $n<m$ \cite{BerZel-Hindmanesch}.
\item The notion of $\Sigma_m^*$ is similar to (but much stronger than) the notion of IP$^*$ which has an intrinsic connection to \textit{mild} mixing and which plays an instrumental role in IP ergodic theory and in Ramsey theory (see, for example, \cite{FBook}, \cite{FKIPSzemerediLong} and \cite{berMcCuIPPolySzemeredi}). The connection between these two notions will be discussed in detail in Section 5. \end{enumerate} \end{rem}
Since the sets $R_\epsilon(A_0,...,A_\ell)$ are, by definition, subsets of $G^\ell$, the defined above notion of $\Sigma_m^*$ has to be "upgraded" to the subsets of the cartesian power $G^\ell$ in order to be useful in the study of the assymptotic behavior of the \textit{multiparameter} expressions of the form \begin{equation}\label{1.MultiParameterExpression} {\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)},\,g_1,...,g_\ell\in G. \end{equation}
However, it is worth noting that the family of $\Sigma_m^*$ sets is quite adequate for dealing with "diagonal" multicorrelation sequences. In the case $G=\mathbb{Z}$, such diagonal sequences have the form \begin{equation}\label{1.SiingleParameterell} \mu(A_0\cap T^{a_1n}A_1\cap\cdots\cap T^{a_\ell n}A_\ell), \end{equation} where $a_1,...,a_\ell\in\mathbb{Z}$, and play an instrumental role in Furstenberg's ergodic approach to Sz{\'e}meredi's theorem (\cite{furstenberg1977Szemeredi},\cite{FBook}). For example, our main result (\cref{1.MainResult}), while dealing with the multiparameter expressions \eqref{1.MultiParameterExpression}, has strong corollaries of "diagonal" nature. The following theorem (which is a version of \cref{4.InjectiveDiagonalResult} below) is an example of a new result of this kind. Note the appearance of $\Sigma_\ell^*$ sets in the formulation. \begin{thm}\label{1.FinitelyGeneratedDiagonal} Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a strongly mixing system, and let the homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ be such that for any $j\in\{1,...,\ell\}$, $\ker(\phi_j)$ is finite and for any $i\neq j$, $\ker(\phi_j-\phi_i)$ is also finite. Then for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$ the set \begin{equation} R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)
=\{g\in G\,|\,|\mu(A_0\cap T_{\phi_1(g)}A_1\cap \cdots \cap T_{\phi_\ell(g)}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\} \end{equation} is $\Sigma_\ell^*$. \end{thm} When $G$ is finitely generated, \cref{1.FinitelyGeneratedDiagonal} has a stronger version (\cref{4.FinitelyGeneratedEquivalence}), which in the case $G=\mathbb{Z}$ can be formulated as follows. \begin{thm}\label{1.ZDiagonalResult} Let $(X,\mathcal A,\mu, T)$ be a measure preserving system, let $\ell\in\mathbb{N}$, and let $a_1,...,a_\ell$ be distinct non-zero integers. Then $T$ is strongly mixing if and only if for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set \begin{equation}\label{1.ZDiagonalEqInLemma}
R^{a_1,...,a_\ell}_\epsilon(A_0,...,A_\ell)=\{n\in\mathbb{Z}\,|\,|\mu(A_0\cap T^{a_1n}A_1\cap\cdots\cap T^{a_\ell n}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\} \end{equation} is $\Sigma_\ell^*$.\footnote{For a related result see \cite[Theorem 1.11]{BerZel-ItteratedDifferenceDiophantine}. See also \cite{KuangYeDeltaMixing}.} \end{thm} \begin{rem} One can view \cref{1.ZDiagonalResult} as a strongly mixing analogue of two theorems due to Furstenberg which pertain to weak and mild mixing (see Theorems 4.11 and 9.27 in \cite{FBook}). The first of these two theorems states that the assumption that $(X,\mathcal A,\mu, T)$ is weakly mixing, implies (and is implied by the fact) that the sets $R^{a_1,...,a_\ell}_\epsilon(A_0,...,A_\ell)$ defined in \eqref{1.ZDiagonalEqInLemma} have uniform density one. The second one states that the assumption that $(X,\mathcal A,\mu,T)$ is mildly mixing implies (and is implied by) the IP$^*$ property of the sets $R^{a_1,...,a_\ell}_\epsilon(A_0,...,A_\ell)$. These theorems are instrumental for the proofs of ergodic Szemer{\'e}di \cite{furstenberg1977Szemeredi} and IP-Szemer{\'e}di \cite{FKIPSzemerediLong} theorems. \end{rem} Note that, for $\ell=1$, both diagonal (see \eqref{1.SiingleParameterell}) and multiparameter (see \eqref{1.MultiParameterExpression}) multicorrelation sequences reduce to the classical expression $\mu(A_0\cap T_{g}A_1)$. The following theorem (which is a very special case of stronger results to be established in this paper) shows that, even in the rather degenerated case $\ell=1$, $\Sigma_m^*$ sets provide a new characterization for the notion of strong mixing for actions of abelian groups. \begin{thm}\label{1.MixingCharacteriazation} Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is strongly mixing.
\item For any $m\in\mathbb{N}$, any $\epsilon>0$ and any $A_0,A_1\in\mathcal A$, the set $$R_\epsilon(A_0,A_1)=\{g\in G\,|\,|\mu(A_0\cap T_g A_1)-\mu(A_0)\mu(A_1)|<\epsilon\}$$
is $\Sigma_m^*$ in $G$.
\item There exists an $m\in\mathbb{N}$ such that for any $\epsilon>0$ and any $A_0,A_1\in\mathcal A$, the set $R_\epsilon(A_0,A_1)$ is $\Sigma_m^*$ in $G$. \end{enumerate} \end{thm} We are moving now to define the modified versions of $\Sigma_m$ and $\Sigma_m^*$ sets which will be instrumental in our dealing with the multiple mixing properties of strongly mixing systems. \begin{defn} Let $(G,+)$ be a countable abelian group and let $(g_k)_{k\in\mathbb{N}}$ and $(h_k)_{k\in\mathbb{N}}$ be two sequences in $G$. We say that $(g_k)_{k\in\mathbb{N}}$ and $(h_k)_{k\in\mathbb{N}}$ \textbf{grow apart} if $\lim_{k\rightarrow\infty}(g_k-h_k)=\infty$. \end{defn} \begin{defn} Let $(G,+)$ be a countable abelian group, let $d\in\mathbb{N}$ and let $(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,d})_{k\in\mathbb{N}}$ be a sequence in $G^d$. We say that $(\textbf g_k)_{k\in\mathbb{N}}$ is \textbf{non-degenerated} if for each $j\in\{1,...,d\}$, $${\lim_{k\rightarrow\infty}g_{k,j}=\infty}.$$ \end{defn} \begin{defn}\label{1.SigmaTildeDefn} Let $d,m\in\mathbb{N}$ and let $(G,+)$ be a countable abelian group. \begin{enumerate}
\item We say that $E\subseteq G^d$ is a $\tilde\Sigma_m$ set if it is of the form
$$\{\textbf g_{k_1}^{(1)}+\cdots+\textbf g_{k_m}^{(m)}\,|\,k_1<\cdots<k_m\}$$
where for each $j\in\{1,...,m\}$, $(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g_{k,1}^{(j)},...,g_{k,d}^{(j)})_{k\in\mathbb{N}}$ is a non-degenerated sequence in $G^d$ and for any distinct $t,t'\in\{1,...,d\}$ the sequences $(g_{k,t}^{(j)})_{k\in\mathbb{N}}$ and $(g_{k,t'}^{(j)})_{k\in\mathbb{N}}$ are growing apart. (Note that if $d=1$, then $E\subseteq G$ is a $\Sigma_m$ set if and only if it is a $\tilde\Sigma_m$ set.)
\item We say that $E\subseteq G^d$ is a $\tilde \Sigma_m^*$ set if it has a non-trivial intersection with every $\tilde \Sigma_m$ set in $G^d$. \end{enumerate} \end{defn} \begin{rem}
The main difference between $\tilde\Sigma_m$ sets and $\Sigma_m$ sets is that $\tilde\Sigma_m$ sets are subsets of cartesian powers of $G$ and have the built-in feature which guarantees that, asymptotically, the elements of $\tilde\Sigma_m$ sets stay away from "degenerated" subsets such as, for example, the following subsets of $G^3$: $\{(g,g,g)\,|\,g\in G\}$, $\{(g,2g,0)\,|\,g\in G\}$ and $\{(g,g,h)\,|\,g,h\in G\}$. \end{rem} The following theorem, which is a corollary of \cref{1.MainResult} below, demonstrates the relevance of $\tilde\Sigma_m$ sets for dealing with mixing of higher orders. \begin{thm}\label{1.GlobalSigma*result} Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is strongly mixing.
\item For any $\ell\in\mathbb{N}$, any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set
$$R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}
$$
is $\tilde\Sigma_\ell^*$ in $G^\ell$.
\item There exists an $\ell\in\mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set $R_\epsilon(A_0,...,A_\ell)$ is $\tilde\Sigma_\ell^*$ in $G^\ell$. \end{enumerate} \end{thm} We take the liberty of stating explicitly the following special case of \cref{1.GlobalSigma*result} to stress the applicability of the aparatus developed in this paper to $\mathbb{Z}$-actions. \begin{cor}\label{1.GlobalSigma*resultForZ} Let $(X,\mathcal A,\mu, T)$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $T$ is strongly mixing.
\item For any $\ell\in\mathbb{N}$, any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set
$$R_\epsilon(A_0,...,A_\ell)=\{(n_1,...,n_\ell)\in \mathbb{Z}^\ell\,|\,|\mu(A_0\cap T^{n_1}A_1\cap\cdots \cap T^{n_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}
$$
is $\tilde\Sigma_\ell^*$ in $\mathbb{Z}^\ell$.
\item There exists an $\ell\in\mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set $R_\epsilon(A_0,...,A_\ell)$ is $\tilde\Sigma_\ell^*$ in $\mathbb{Z}^\ell$. \end{enumerate} \end{cor} We introduce now the notion of convergence based on the classical Ramsey Theorem that is behind the proof of \cref{1.GlobalSigma*result}. Given $m\in\mathbb{N}$ and an infinite set $S\subseteq \mathbb{N}$, we denote by $S^{(m)}$ the family of all $m$-element subsets of $S$. When writing $\{k_1,...,k_m\}\in S^{(m)}$, we will always assume that $k_1<\cdots<k_m$. \begin{thm}[Ramsey's Theorem]\label{1.Ramsey} Let $r,m\in\mathbb{N}$ and let $C_1,...,C_r\subseteq\mathbb{N}^{(m)}$ be such that \begin{equation}\label{1.PartitionOfN} \mathbb{N}^{(m)}=\bigcup _{j=1}^r C_j. \end{equation} Then there exists $j_0\in\{1,...,r\}$ and an infinite subset $S\subseteq\mathbb{N}$, satisfying $S^{(m)}\subseteq C_{j_0}$. \end{thm} \begin{rem}\label{1.RamseyProofRemark} It is easy to see that \cref{1.Ramsey} can be formulated in the following equivalent form that will be frequently used in the sequel:\\ \begin{adjustwidth}{0.5cm}{0.5cm} \textit{Let $r,m\in\mathbb{N}$, let $P$ be an infinite subset of $\mathbb{N}$ and let $C_1,...,C_r\subseteq\mathbb{N}^{(m)}$ be such that \begin{equation} P^{(m)}\subseteq\bigcup _{j=1}^r C_j. \end{equation} Then there exists $j_0\in\{1,...,r\}$ and an infinite subset $S\subseteq P$, satisfying $S^{(m)}\subseteq C_{j_0}$.} \end{adjustwidth} \end{rem} \begin{defn} Let $(X,d)$ be a compact metric space, let $x\in X$, let $(x_\alpha)_{\alpha\in \mathbb{N}^{(m)}}$ be an ${\mathbb{N}^{(m)}\text{-sequence}}$ in $X$ and let $S$ be an infinite subset of $\mathbb{N}$. We write \begin{equation}
\mathop{\mathcal R\text{-lim}}_{\alpha\in S^{(m)}}x_\alpha=x \end{equation} if for every $\epsilon>0$, there exists $\alpha_0\in \mathbb{N}^{(m)}$ such that for any $\alpha\in S^{(m)}$ satisfying ${\min \alpha>\max \alpha_0,}$ one has $$d(x_\alpha, x)<\epsilon.$$ \end{defn} The following theorem can be viewed as a version of Bolzano-Weierstrass theorem for ${\mathcal R\text{-convergence}}$. It follows from \cref{1.Ramsey} with the help of a diagonalization argument. \begin{thm}\label{1.RBolzanoWierstrass} Let $(X,d)$ be a compact metric space and let $(x_\alpha)_{\alpha\in\mathbb{N}^{(m)}}$ be an $\mathbb{N}^{(m)}$-sequence in $X$. Then for any infinite set $S_1\subseteq \mathbb{N}$ there exists an $x\in X$ and an infinite set $S\subseteq S_1$ such that \begin{equation}
\rlim{\alpha\in S^{(m)}}x_{\alpha}=x. \end{equation} \end{thm}
\begin{rem}\label{1.ItteratedLimitsRemark} Let $(x_\alpha)_{\alpha\in\mathbb{N}^{(m)}}$ be an $\mathbb{N}^{(m)}$-sequence in a compact metric space $(X,d)$. The introduced above $\mathcal R$-limits have an intrinsic connection with the iterated limits of the form \begin{equation}\label{1.ItteratedLimitsExistence} \lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k_{j_1},...,k_{j_m}\}}.\footnote{ Cf. \cite{sucheston1959sequences} and \cite{lorentz1960remark}. } \end{equation} The goal of this extended remark is to clarify this connection. \begin{enumerate}[(a)] \item Using the compactness of $X$, one can show with the help of a diagonalization argument that for any increasing sequence $(k_j)_{j\in\mathbb{N}}$, there exists a subsequence $(k_j')_{j\in\mathbb{N}}$ for which all the limits in \eqref{1.ItteratedLimitsExistence} exist.
\item By \cref{1.RBolzanoWierstrass}, there exists an increasing sequence of natural numbers $(k_j)_{j\in\mathbb{N}}$ so that for $S=\{k_j\,|\,j\in\mathbb{N}\}$, $\rlim{\alpha\in S^{(m)}}x_\alpha$ exists. Let $(k_j')_{j\in\mathbb{N}}$ be the subsequence of $(k_j)_{j\in\mathbb{N}}$ which is guaranteed to exist by (a). Letting $S_1=\{k_j'\,|\,j\in\mathbb{N}\}$, we have \begin{equation}\label{1.ItteratedLimits} \rlim{\alpha\in S_1^{(m)}}x_\alpha=\lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k'_{j_1},...,k'_{j_m}\}}. \end{equation} \item When $X=\{1,...,r\}$, one can use (a) to prove \cref{1.Ramsey}. Let $r,m\in\mathbb{N}$ and consider a partition $\mathbb{N}^{(m)}=\bigcup_{j=1}^r C_j$. Let $(x_\alpha)_{\alpha\in\mathbb{N}^{(m)}}$ be defined by $x_\alpha=j$ if $\alpha\in C_j$. For some increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ there exists a $j_0\in\{1,...,r\}$ such that $$\lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k_{j_1},...,k_{j_m}\}}=j_0.$$
By using a diagonalization argument, we obtain a subsequence $(k_j')_{j\in\mathbb{N}}$ of $(k_j)_{j\in\mathbb{N}}$ with the property that $x_{\{k_{j_1}',...,k_{j_m}'\}}=j_0$ for any $j_1<\cdots<j_m$. Now let $S=\{k_j'\,|\,j\in\mathbb{N}\}$. It follows that $S^{(m)}\subseteq C_{j_0}$. \end{enumerate} \end{rem} Before formulating our main result, we need two more definitions. \begin{defn} Let $m\in\mathbb{N}$ and let $(G,+)$ be a countable abelian group. For any sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,m})_{k\in\mathbb{N}}$ and any $\alpha=\{k_1,...,k_m\}\in\mathbb{N}^{(m)}$ we let \begin{equation}\label{1.gAlphaDefn} g_\alpha=\sum_{j=1}^m g_{k_j,j}=g_{k_1,1}+g_{k_2,2}+\cdots+g_{k_m,m}, \end{equation} where $k_1<\cdots<k_m$. \end{defn} \begin{defn} Let $m\in\mathbb{N}$, let $(G,+)$ be a countable abelian group and let $$(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,m})_{k\in\mathbb{N}}\text{ and }(\textbf h_k)_{k\in\mathbb{N}}=(h_{k,1},...,h_{k,m})_{k\in\mathbb{N}}$$ be sequences in $G^m$. We say that $(\textbf g_k)_{k\in\mathbb{N}}$ and $(\textbf h_k)_{k\in\mathbb{N}}$ are \textbf{essentially distinct} if for each $t\in\{1,...,m\}$, $(g_{k,t})_{k\in\mathbb{N}}$ and $(h_{k,t})_{k\in\mathbb{N}}$ grow apart (i.e. $\lim_{k\rightarrow\infty}(g_{k,t}-h_{k,t})=\infty$). \end{defn} \begin{rem} The following observation indicates the natural connection between non-degenerated, essentially distinct sequences in $G^m$ and $\tilde\Sigma_m$ sets. Let $d,m\in\mathbb{N}$ and let $(G,+)$ be a countable abelian group. Then for any non-degenerated and essentially distinct sequences $$(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g_{k,1}^{(j)},...,g_{k,m}^{(j)})_{k\in\mathbb{N}},\,j\in\{1,...,d\},$$ in $G^m$, the set \begin{multline*}
\{(g_\alpha^{(1)},...,g_\alpha^{(d)})\,|\,\alpha\in\mathbb{N}^{(m)}\}=\{(g^{(1)}_{k_1,1}+\cdots+g^{(1)}_{k_m,m},...,g^{(d)}_{k_1,1}+\cdots+g^{(d)}_{k_m,m})\,|\,k_1<\cdots<k_m\}\\
=\{(g^{(1)}_{k_1,1},...,g^{(d)}_{k_1,1})+\cdots+(g^{(1)}_{k_m,m},...,g^{(d)}_{k_m,m})\,|\,k_1<\cdots<k_m\} \end{multline*} is a $\tilde\Sigma_m$ set in $G^d$. \end{rem} We are ready now to formulate our main result (it appears as \cref{3.MainResult} in Section 3). It incorporates some of the characterizations of strongly mixing systems which were mentioned above. \begin{thm}\label{1.MainResult} Let $\ell\in\mathbb{N}$, let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is strongly mixing.
\item For any non-degenerated and essentially distinct sequences
$(\textbf g_k^{(j)})_{k\in\mathbb{N}}$, $j\in\{1,...,\ell\},$
in $G^{(\ell)}$, there exists an infinite $S\subseteq\mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$,
\begin{equation}
\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{ g^{(1)}_\alpha}A_1\cap \cdots\cap T_{ g^{(\ell)}_\alpha}A_\ell)=\prod_{j=0}^\ell\mu(A_j).
\end{equation}
More explicitly, if
$$(\textbf g^{(j)}_k)_{k\in\mathbb{N}}=(g^{(j)}_{k,1},...,g^{(j)}_{k,\ell})_{k\in\mathbb{N}},$$ for each $j\in\{1,...,\ell\},$
then
$$\rlim{\{k_1,...,k_\ell\}\in S^{(\ell)}}\mu(A_0\cap T_{ g_{k_1,1}^{(1)}+\cdots+ g_{k_\ell,\ell}^{(1)}}A_1\cap\cdots\cap T_{ g_{k_1,1}^{(\ell)}+\cdots+ g_{k_\ell,\ell}^{(\ell)}}A_\ell)=\prod_{j=0}^\ell \mu(A_j).$$
\item For any $\epsilon>0$ and any $A_0,...,A_\ell\in\mathcal A$, the set $$R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}$$
is $\tilde \Sigma_\ell^*$ in $G^\ell$.
\item For any $\epsilon>0$ and any $A_0,A_1\in\mathcal A$, the set $R_\epsilon(A_0,A_1)$ is $\Sigma_\ell^*$ in $G$.
\end{enumerate} \end{thm}
The structure of this paper is as follows. In Section 2, some basic facts about couplings of probability spaces are reviewed and some auxiliary results which will be needed in Sections 3 and 6 are established. In Section 3, we prove our main result, \cref{1.MainResult} (=\cref{3.MainResult}). In Section 4, we derive some diagonal results for strongly mixing systems. In Section 5, we describe the largeness properties of $\tilde\Sigma_m^*$ sets and, more specifically, of the sets $R_\epsilon(A_0,...,A_\ell)$. We also juxtapose the properties of $\tilde\Sigma_m^*$ sets with those of $\tilde{{\text{IP}}}\rm{^*}$ sets and sets of uniform density one which are characteristic, correspondingly, of mild and weak mixing. In Section 6, we utilize the methods developed in Sections 2 and 5 to obtain analogues of \cref{1.MainResult} for mildly and weakly mixing systems. \begin{rem} Throughout this paper, we will be tacitly assuming that the measure preserving systems $(X,\mathcal A,\mu,(T_g)_{g\in G})$ that we are working with are \textit{regular} meaning that the underlying probability space $(X,\mathcal A,\mu)$ is regular (i.e. $X$ is a compact metric space and $\mathcal A=\text{Borel}(X)$). Note that this assumption can be made without loss of generality since every separable measure preserving system is equivalent to a regular one (see for instance, \cite[Proposition 5.3]{FBook}). \end{rem}
\section{Some auxiliary facts involving couplings and $\mathcal R$-limits} In this section we review some basic facts about couplings of probability spaces and establish some auxiliary results which will be needed in Section 3 and Section 6. \begin{defn}\label{2.CouplingDefn} Let $N\in\mathbb{N}$. Given regular probability spaces $\textbf X_j=(X_j,\mathcal A_j,\mu_j)$, $j\in\{1,...,N\}$, a \textbf{coupling} of $\textbf X_1,...,\textbf X_N$ is a Borel probability measure $\lambda$ defined on the measurable space $$(\prod_{j=1}^NX_j,\bigotimes_{j=1}^N \mathcal A_j)$$ and having the property that for any $j\in\{1,...,N\}$ and any $A\in\mathcal A_j$, $\lambda(\pi_j^{-1}(A))=\mu_j(A)$, where $\pi_j:\prod_{i=1}^NX_i\rightarrow X_j$ is the projection map onto the j-th coordinate of $\prod_{j=1}^N X_j$.\footnote{ A coupling is just a \textit{joining} of the trivial measure preserving systems $(X_j,\mathcal A_j,\mu_j, \text{Id}_j)$, $j\in\{1,...,N\}$, where $\text{Id}_j:X_j\rightarrow X_j$ denotes the identity map on $X_j$. } \end{defn} We will let $\mathcal C(\textbf X_1,...,\textbf X_N)$ denote the set of all couplings of $\textbf X_1,...,\textbf X_N$. $\mathcal C(\textbf X_1,...,\textbf X_N)$ is a closed subspace of the set of all probability Borel measures on $\prod_{j=1}^NX_j$ endowed with the ${\text{weak-*}}$ topology. With this topology, $\mathcal C(\textbf X_1,...,\textbf X_N)$ is a compact metrizable space. Given a sequence $(\lambda_k)_{k\in\mathbb{N}}$ in $\mathcal C(\textbf X_1,...,\textbf X_N)$, $$\lambda_k\xrightarrow[k\rightarrow\infty]{} \lambda$$ if and only if for any $A_1\in\mathcal A_1$,...,$A_N\in\mathcal A_N$, $$\lambda_k(A_1\times\cdots\times A_N)\xrightarrow[k\rightarrow\infty]{} \lambda(A_1\times\cdots\times A_N).$$
The following proposition follows immediately from the compactness of $\mathcal C(\textbf X_1,...,\textbf X_N)$ and \cref{1.RBolzanoWierstrass}. \begin{prop}\label{2.EquivalentLimits} Let $\textbf X_j=(X_j,\mathcal A_j,\mu_j)$, $j\in\{1,...,N\}$, be regular probability spaces. For any $m\in\mathbb{N}$, any infinite $S\subseteq \mathbb{N}$ and any $\mathbb{N}^{(m)}$-sequence $(\lambda_\alpha)_{\alpha\in\mathbb{N}^{(m)}}$ in $\mathcal C(\textbf X_1,...,\textbf X_N)$, $$\rlim{\alpha\in S^{(m)}}\lambda_\alpha=\lambda$$ if and only if for any $A_1\in\mathcal A_1$,...,$A_N\in\mathcal A_N$, $$\rlim{\alpha\in S^{(m)}}\lambda_\alpha(A_1\times \cdots\times A_N)=\lambda(A_1\times\cdots \times A_N).$$ \end{prop} Our next goal is to stablish a useful criterion for mixing of higher orders (\cref{2.MainResult}). First, we need a definition and two lemmas. \begin{defn} Let $(Z,\mathcal D,\lambda)$ be a regular probability space and let, for each $k\in\mathbb{N}$, $T_k:Z\rightarrow Z$ be a measure preserving transformation. The sequence $(T_k)_{k\in\mathbb{N}}$ has the mixing property if for every $A_0,A_1\in\mathcal D$, $$\lim_{k\rightarrow\infty}\lambda(A_0\cap T_k^{-1}A_1)=\lambda(A_0)\lambda(A_1).$$ \end{defn} \begin{rem} \begin{enumerate}[(a)] \item If each of the transformations $T_k$, $k\in\mathbb{N}$, is invertible, $(T_k)_{k\in\mathbb{N}}$ has the mixing property if and only if $(T_k^{-1})_{k\in\mathbb{N}}$ has the mixing property. \item $(T_k)_{k\in\mathbb{N}}$ has the mixing property if and only if for any $f,g\in L^2(\mu)$, $$\lim_{k\rightarrow\infty}\int_X fT_kg\text{d}\mu=\int_Xf\text{d}\mu\int_Xg\text{d}\mu.$$ \end{enumerate} \end{rem} \begin{lem}\label{2.PropTheIndependentjoining} Let $\textbf X=(X,\mathcal A,\mu)$ and $\textbf Y=(Y,\mathcal B,\nu)$ be regular probability spaces. For each $k\in\mathbb{N}$, let $T_k:Y\rightarrow Y$ be a measure preserving transformation, and assume that the sequence $(T_k)_{k\in\mathbb{N}}$ has the mixing property. Let $\lambda_0$ be a coupling of $\textbf X$ and $\textbf Y$. Assume that $\lambda$ is a probability measure on $\mathcal A\otimes\mathcal B$ such that for any $A\in\mathcal A$ and $B\in\mathcal B$ one has \begin{equation}\label{2.LimitCondition}
\lim_{k\rightarrow\infty}\lambda_0((\text{Id}\times T_k^{-1})(A\times B))=\lambda(A\times B). \end{equation} Then $\lambda=\mu\otimes\nu$. \end{lem} \begin{proof} Note that it suffices to show that for any $A\in \mathcal A$ and $B\in \mathcal B$, \begin{equation}\label{2.StatementToProveLemma} \lambda(A\times B)=\mu(A)\nu(B). \end{equation} Fix $A\in\mathcal A$ and $B\in\mathcal B$. Since $\mathbbm 1_A\otimes \mathbbm 1_B=(\mathbbm 1_A\otimes\mathbbm 1_Y)(\mathbbm 1_X\otimes \mathbbm 1_B)$, we have by \eqref{2.LimitCondition} that \begin{multline}\label{2.FirstEquationLemma}
\int_{X\times Y}(\mathbbm 1_A\otimes\mathbbm 1_Y) (\mathbbm 1_X\otimes \mathbbm 1_B)\text{d}\lambda=\lambda(A\times B)=\\
\lim_{k\rightarrow\infty}\lambda_0((\text{Id}\times T_k^{-1})(A\times B))
=\lim_{k\rightarrow\infty} \int_{X\times Y}(\text{Id}\times T_k)(\mathbbm 1_A\otimes\mathbbm 1_Y)(\text{Id}\times T_k) (\mathbbm 1_X\otimes \mathbbm 1_B)\text{d}\lambda_0. \end{multline}
Note that $(\text{Id}\times T_k) (\mathbbm 1_A\otimes \mathbbm 1_Y)=\mathbbm 1_A\otimes \mathbbm 1_Y$ and, if we regard $\mathcal B$ as a sub $\sigma$-algebra of $\mathcal A\otimes\mathcal B$, $\lambda_0|_{\mathcal B}=\nu$. The right-most expression in \eqref{2.FirstEquationLemma} equals \begin{multline}\label{2.SecondEquationLemma}
\lim_{k\rightarrow\infty}\int_{X\times Y} (\mathbbm 1_A\otimes \mathbbm 1_Y)( \mathbbm 1_X\otimes T_k\mathbbm 1_B)\text{d}\lambda_0=\lim_{k\rightarrow\infty}\int_{X\times Y} \mathbb E(\mathbbm 1_A\otimes \mathbbm 1_Y\,|\,\mathcal B)( \mathbbm 1_X\otimes T_k\mathbbm 1_B)\text{d}\lambda_0\\
=\lim_{k\rightarrow\infty}\int_Y \mathbb E(\mathbbm 1_A\otimes \mathbbm 1_Y\,|\,\mathcal B)T_k\mathbbm 1_B\text{d}\nu. \end{multline}
where $\mathbb E(\mathbbm 1_A\otimes \mathbbm 1_Y\,|\,\mathcal B)$ denotes the conditional expectation of $\mathbbm 1_A\otimes \mathbbm 1_Y$ with respect to $\mathcal B$.\\ But $(T_k)_{k\in\mathbb{N}}$ has the mixing property, so the right-most expression in \eqref{2.SecondEquationLemma} equals \begin{equation}\label{2.FinalExpressionInSmallLemma}
\int_Y\mathbb E(\mathbbm 1_A\otimes \mathbbm 1_Y\,|\,\mathcal B)\text{d}\nu\int_Y \mathbbm 1_B\text{d}\nu=\lambda(A\times B). \end{equation} By noting that
$$\int_Y\mathbb E(\mathbbm 1_A\otimes \mathbbm 1_Y\,|\,\mathcal B)\text{d}\nu=\int_{X\times Y}(\mathbbm 1_A\otimes \mathbbm 1_Y)\text{d}\lambda_0=\int_X \mathbbm 1_A\text{d}\mu,$$ we have that \eqref{2.FinalExpressionInSmallLemma} equals $\mu(A)\nu(B)$. \end{proof}
\begin{lem}\label{2.DecomposingRlimLemma} Let $m\in\mathbb{N}$, let $(X,d)$ be a compact metric space and let $(x_\alpha)_{\alpha\in\mathbb{N}^{(m+1)}}$ be an $\mathbb{N}^{(m+1)}$-sequence in $X$. Assume that there exists an infinite $S\subseteq \mathbb{N}$ with the properties (a) for some $x\in X$, $\rlim{\alpha\in S^{(m+1)}}x_\alpha=x$ and (b) for each $k\in S$ there exists $y_k\in X$ such that $$\rlim{\alpha\in S^{(m)},\,k<\min \alpha}x_{\{k\}\cup\alpha}=y_k.$$ Then $$\lim_{k\rightarrow\infty,\,k\in S}\rlim{\alpha\in S^{(m)},\,k<\min \alpha}x_{\{k\}\cup \alpha}=\lim_{k\rightarrow\infty,\,k\in S}y_k=\rlim{\alpha\in S^{(m+1)}}x_\alpha.$$ \end{lem} \begin{proof} Let $\epsilon>0$. Note that (1) there exists $k_0\in S$ such that for any $\alpha\in S^{(m+1)}$ with $k_0\leq \min \alpha$, $d(x_\alpha, x)<\frac{\epsilon}{2}$ and (2) for any $k\in S$ there exists an $\alpha_k\in S^{(m)}$ such that for any $\alpha\in S^{(m)}$ with $\min \alpha >\max (\alpha_k\cup\{k\})$, $d(x_{\{k\}\cup \alpha},y_k)<\frac{\epsilon}{2}$. It follows that for any $k\in S$ with $k\geq k_0$ and any $\alpha\in S^{(m)}$ with $\min \alpha>\max(\alpha_k\cup\{k\})$, $d(y_k,x)<d(x_{\{k\}\cup\alpha},y_k)+d(x_{\{k\}\cup \alpha},x)<\epsilon$. Since $\epsilon>0$ was arbitrary, $$\lim_{k\rightarrow\infty,\,k\in S}y_k=x=\rlim{\alpha\in S^{(m+1)}}x_\alpha.$$ \end{proof} \begin{rem}\label{2.ExistenceOfS} Let $m\in\mathbb{N}$ and let $(x_\alpha)_{\alpha\in\mathbb{N}^{(m+1)}}$ be an $\mathbb{N}^{(m+1)}$-sequence in a compact metric space $X$. By applying \cref{1.RBolzanoWierstrass} first to the $\mathbb{N}^{(m)}$-sequence $(\omega_\alpha)_{\alpha\in\mathbb{N}^{(m)}}=((x_{\{k\}\cup\alpha})_{k\in\mathbb{N}})_{\alpha\in\mathbb{N}^{(m)}}$ in $X^\mathbb{N}$ (here $x_{\{k\}\cup\alpha}=x_0$ for some fixed $x_0\in X$, whenever $k\geq\min \alpha$), and then to the $\mathbb{N}^{(m+1)}$-sequence $(x_\alpha)_{\alpha\in\mathbb{N}^{(m+1)}}$, we obtain an infinite set $S\subseteq \mathbb{N}$ for which (a) and (b) in the statement of \cref{2.DecomposingRlimLemma} hold. A similar reasoning shows that one can pick $S$ to be a subset of any prescribed in advance infinite set $S_1\subseteq \mathbb{N}$. \end{rem} \begin{rem}\label{2.ItteratedLimits} In \cref{1.ItteratedLimitsRemark},(c), we indicated how the utilization of iterated limits \begin{equation*}\label{2.MultiLimitExpression} \lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k_{j_1},...,k_{j_m}\}} \end{equation*}
leads to a proof of Ramsey's theorem (\cref{1.Ramsey}). In this remark, we show that \cref{2.DecomposingRlimLemma} and \cref{2.ExistenceOfS} (which are corollaries of Ramsey's Theorem) imply that for any infinite set $S_1\subseteq \mathbb{N}$ and any $\mathbb{N}^{(m)}$-sequence $(x_\alpha)_{\alpha\in\mathbb{N}^{(m)}}$ in a compact metric space $X$, there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $S_1$ such that for $S=\{k_j\,|\,j\in\mathbb{N}\}$ each of the limits in the formula \begin{equation*}\label{2.RLim=LimLim} \rlim{\alpha\in S^{(m)}}x_\alpha=\lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k_{j_1},...,k_{j_m}\}} \end{equation*}
exist. The proof is by induction on $m\in\mathbb{N}$. When $m=1$, the result follows form the compactness of $X$. Now let $m>1$ and let $S_1$ be an infinite subset of $\mathbb{N}$. By \cref{2.ExistenceOfS} and \cref{2.DecomposingRlimLemma}, there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $S_1$ such that for $S=\{k_j\,|\,j\in\mathbb{N}\}$, $$\rlim{\alpha\in S^{(m)}}x_\alpha=\lim_{j\rightarrow\infty}\rlim{\alpha\in S^{(m-1)}}x_{\{k_j\}\cup \alpha}.$$ The result now follows from the inductive hypothesis applied to the infinite set $S\subseteq \mathbb{N}$ and the $\mathbb{N}^{(m-1)}$-sequence $((x_{\{k\}\cup \alpha})_{k\in\mathbb{N}})_{\alpha\in\mathbb{N}^{(m-1)}}$ in the compact metric spacce $X^\mathbb{N}$. \end{rem} The following proposition provides a useful technical tool for establishing higher order mixing properties of measure preserving systems. It will be instrumental in Section 3 for dealing with strongly mixing systems and in Section 6, where we will focus on mildly and weakly mixing systems. \begin{prop}\label{2.MainResult} Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system, let $\ell\in\mathbb{N}$ and, for each $j\in\{1,...,\ell\}$, let $$(\textbf g^{(j)}_k)_{k\in\mathbb{N}}=(g_{k,1}^{(j)},...,g_{k,\ell}^{(j)})_{k\in\mathbb{N}}$$ be a sequence in $G^\ell$. Suppose that
for any $t\in\{1,...,\ell\}$ and any $j\in\{1,...,\ell\}$, $(T_{g_{k,t}^{(j)}})_{k\in\mathbb{N}}$ has the mixing property and that for any $t$ and any $i\neq j$, $(T_{(g_{k,t}^{(j)}-g_{k,t}^{(i)})})_{k\in\mathbb{N}}$ also has the mixing property. Then, there exists an infinite set $S\subseteq \mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$, $$\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{g^{(1)}_\alpha}A_1\cap\cdots\cap T_{g^{(\ell)}_{\alpha}}A_\ell)=\prod_{j=0}^\ell \mu(A_j).$$ \end{prop} \begin{proof} The proof is by induction on $\ell$. When $\ell=1$, it follows from our hypothesis that for any $A_0,A_1\in\mathcal A$, $$\rlim{\alpha\in\mathbb{N}^{(1)}}\mu(A_0\cap T_{g_\alpha^{(1)}}A_1)=\lim_{k\rightarrow\infty}\mu(A_0\cap T_{g_{k,1}^{(1)}}A_1)=\mu(A_0)\mu(A_1).$$ Now fix $\ell\in\mathbb{N}$ and suppose that \cref{2.MainResult} holds for any $\ell'\leq \ell$. Let $\textbf X=(X,\mathcal A,\mu)$ and let $\mu_\Delta\in\mathcal C=\mathcal C(\underbrace{\textbf X,..., \textbf X}_{\ell+2\text{ times}})$ be defined by $\mu(A_0\times\cdots\times A_{\ell+1})=\mu(A_0\cap\cdots\cap A_{\ell+1})$. By the inductive hypothesis, there exists an infinite $S\subseteq\mathbb{N}$ such that for any $A_1,...,A_{\ell+1}\in\mathcal A$,
\begin{multline}\label{2.InductiveMixing}
\rlim{\{j_1,...,j_\ell\}\in S^{(\ell)}}\mu_\Delta(X\times T_{g_{j_1,2}^{(1)}+\cdots+g_{j_\ell,\ell+1}^{(1)}}A_1\times\cdots\times T_{g_{j_1,2}^{(\ell+1)}+\cdots+g_{j_\ell,\ell+1}^{(\ell+1)}}A_{\ell+1})\\
=\rlim{\{j_1,...,j_\ell\}\in S^{(\ell)}}\mu(X\cap T_{g_{j_1,2}^{(1)}+\cdots+g_{j_\ell,\ell+1}^{(1)}}A_1\cap\cdots\cap T_{g_{j_1,2}^{(\ell+1)}+\cdots+g_{j_\ell,\ell+1}^{(\ell+1)}}A_{\ell+1})\\
=\rlim{\{j_1,...,j_\ell\}\in S^{(\ell)}}\mu( T_{g_{j_1,2}^{(1)}+\cdots+g_{j_\ell,\ell+1}^{(1)}}A_1\cap\cdots\cap T_{g_{j_1,2}^{(\ell+1)}+\cdots+g_{j_\ell,\ell+1}^{(\ell+1)}}A_{\ell+1})\\
=\rlim{\{j_1,...,j_\ell\}\in S^{(\ell)}}\mu(A_1\cap T_{(g_{j_{1},2}^{(2)}-g_{j_{1},2}^{(1)})+\cdots+(g_{j_{\ell},\ell+1}^{(2)}-g_{j_{\ell},\ell+1}^{(1)})}A_2\cap
\cdots\cap T_{(g_{j_1,2}^{(\ell+1)}-g_{j_{1},2}^{(1)})+\cdots+(g_{j_\ell,\ell+1}^{(\ell+1)}-g_{j_{\ell},\ell+1}^{(1)})}A_{\ell+1})\\
=\prod_{j=1}^{\ell+1}\mu(A_j).
\end{multline}
By \cref{1.RBolzanoWierstrass} and the compactness of $\mathcal C$, there exists an infinite set $S_0\subseteq S$ and $\lambda_0\in\mathcal C$ such that for any $A_0,...,A_{\ell+1}\in\mathcal A$,
\begin{equation}\label{2.lambda0}
\rlim{\{j_1,...,j_\ell\}\in S_0^{(\ell)}}\mu_\Delta(A_0\times T_{g_{j_1,2}^{(1)}+\cdots+g_{j_\ell,\ell+1}^{(1)}}A_1\times\cdots\times T_{g_{j_1,2}^{(\ell+1)}+\cdots+g_{j_\ell,\ell+1}^{(\ell+1)}}A_{\ell+1})=\lambda_0(\prod_{j=0}^{\ell+1}A_j).
\end{equation}
Likewise, there exist an infinite set $S_1\subseteq S_0$ and $\lambda\in\mathcal C$ such that for any $A_0,...,A_{\ell+1}\in\mathcal A$,
\begin{multline}\label{2.lambda}
\rlim{\{j_1,...,j_{\ell+1}\}\in S_1^{(\ell+1)}}\mu_\Delta(A_0\times T_{g_{j_1,1}^{(1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(1)}}A_1\times\cdots\times T_{g_{j_1,1}^{(\ell+1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(\ell+1)}}A_{\ell+1})=\lambda(\prod_{j=0}^{\ell+1}A_j).
\end{multline} Let $\textbf {Y}=(\prod_{j=1}^{\ell+1} X,\bigotimes_{j=1}^{\ell+1}\mathcal A, \bigotimes_{j=1}^{\ell+1}\mu)$. Note that \eqref{2.InductiveMixing} holds if we substitute $S_1$ for $S$ and \eqref{2.lambda0} holds when we substitute $S_1$ for $S_0$. Performing this substitution and first applying \eqref{2.lambda0} and then \eqref{2.InductiveMixing} to $A_1,...,A_{\ell+1}\in\mathcal A$, we have $$\lambda_0(X\times A_1\times\cdots\times A_{\ell+1})=\prod_{j=1}^{\ell+1}\mu(A_{j}).$$ Also, trivially, for any $A_0\in\mathcal A$, $$\lambda_0(A_0\times X\times\cdots\times X)=\mu(A_0).$$ Thus, $\lambda_0$ is a coupling of $\textbf X$ and $\textbf Y$.\\
Using formula \eqref{2.lambda0}, \cref{2.DecomposingRlimLemma} and applying \eqref{2.lambda} to the set $S_1=\{k_j\,|\,j\in\mathbb{N}\}$ (where we assume that $(k_j)_{j\in\mathbb{N}}$ is an increasing sequence), we have \begin{multline}\label{2.LimitForProductsIsLambda}
\lim_{t\rightarrow\infty}\lambda_0(A_0\times T_{g_{k_{t},1}^{(1)}}A_1\times\cdots\times T_{g_{k_{t},1}^{(\ell+1)}}A_{\ell+1} )\\
=\lim_{t\rightarrow\infty} \rlim{\{j_2,...,j_{\ell+1}\}\in S_1^{(\ell)}}\mu_\Delta(A_0\times T_{g_{j_2,2}^{(1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(1)}}(T_{g_{k_{t},1}^{(1)}}A_1)\times\cdots\times T_{g_{j_2,2}^{(\ell+1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(\ell+1)}}(T_{g_{k_{t},1}^{(\ell+1)}}A_{\ell+1}))\\
=\lim_{t\rightarrow\infty} \rlim{\{j_2,...,j_{\ell+1}\}\in S_1^{(\ell)},\,k_t<j_2}\mu_\Delta(A_0\times T_{g_{k_{t},1}^{(1)}+g_{j_2,2}^{(1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(1)}}A_1\times\cdots\times T_{g_{k_{t},1}^{(\ell+1)}+g_{j_2,2}^{(\ell+1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(\ell+1)}}A_{\ell+1})\\
=\rlim{\{j_1,...,j_{\ell+1}\}\in S_1^{(\ell+1)}}\mu_\Delta(A_0\times T_{g_{j_1,1}^{(1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(1)}}A_1\times\cdots\times T_{g_{j_1,1}^{(\ell+1)}+\cdots+g_{j_{\ell+1},\ell+1}^{(\ell+1)}}A_{\ell+1})=\lambda(\prod_{j=0}^{\ell+1}A_j), \end{multline} For each $j\in\mathbb{N}$, let $\textbf T_j=T_{g_{k_j,1}^{(1)}}\times\cdots\times T_{g_{k_j,1}^{(\ell+1)}}$. Note that for any increasing sequence $(t_s)_{s\in\mathbb{N}}$ in $\mathbb{N}$, there exists a subsequence $(t'_s)_{s\in\mathbb{N}}$ and a measure $\lambda'\in\mathcal C(\textbf X,\textbf Y)$, such that for any $A\in\mathcal A$ and any $B\in \bigotimes_{j=1}^{\ell+1}\mathcal A$, $\lim_{s\rightarrow\infty}\lambda_0(A\times \textbf T_{t'_s}B)=\lambda'(A\times B)$. By \eqref{2.LimitForProductsIsLambda}, $\lambda'=\lambda$ and hence, for any $A\in\mathcal A$ and any $B\in \bigotimes_{j=1}^{\ell+1}\mathcal A$, $\lim_{j\rightarrow\infty}\lambda_0(A\times \textbf T_j B)=\lambda(A\times B)$.\\ By \cref{2.PropTheIndependentjoining} applied to $\textbf X=(X,\mathcal A,\mu)$, $\textbf Y=(\prod_{j=1}^{\ell+1} X,\bigotimes_{j=1}^{\ell+1}\mathcal A, \bigotimes_{j=1}^{\ell+1}\mu)$ and the sequence of measure preserving transformations $(T^{-1}_{g_{k_j,1}^{(1)}}\times\cdots\times T^{-1}_{g_{k_j,1}^{(\ell+1)}})_{j\in\mathbb{N}}$, we have that $\lambda=\bigotimes_{j=0}^{\ell+1}\mu$. It follows that for any $A_0,...,A_{\ell+1}\in\mathcal A$, \begin{multline*} \rlim{\alpha\in S_1^{(\ell+1)}}\mu(A_0\cap T_{g^{(1)}_\alpha}A_1\cap\cdots\cap T_{g^{(\ell+1)}_{\alpha}}A_{\ell+1})\\ =\rlim{\alpha\in S_1^{(\ell+1)}}\mu_\Delta (A_0\times T_{g^{(1)}_\alpha}A_1\times\cdots\times T_{g^{(\ell+1)}_{\alpha}}A_{\ell+1})=\prod_{j=0}^{\ell+1} \mu(A_j), \end{multline*} completing the proof. \end{proof}
\section{Strongly mixing systems are "almost" strongly mixing of all orders} In this section we will prove the following theorem (\cref{1.MainResult} from the Introduction) which is the main result of this paper. \begin{thm}\label{3.MainResult} Let $\ell\in\mathbb{N}$ and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is strongly mixing.
\item For any $\ell$ non-degenerated and essentially distinct sequences
$$(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g^{(j)}_{k,1},...,g^{(j)}_{k,\ell})_{k\in\mathbb{N}},\text{ }j\in\{1,...,\ell\},$$
in $G^{\ell}$, there exists an infinite $S\subseteq\mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$,
\begin{equation}
\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{ g^{(1)}_\alpha}A_1\cap \cdots\cap T_{ g^{(\ell)}_\alpha}A_\ell)=\prod_{j=0}^\ell\mu(A_j).
\end{equation}
\item For any $\epsilon>0$ and any $A_0,...,A_\ell\in\mathcal A$, the set
$$R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}$$
is $\tilde \Sigma_\ell^*$ in $G^\ell$.
\item For any $\epsilon>0$ and any $A_0,A_1\in\mathcal A$, the set $R_\epsilon(A_0,A_1)$ is $\Sigma_\ell^*$ in $G$.
\end{enumerate} \end{thm} \begin{proof} (i)$\implies$(ii): Note that since $(T_g)_{g\in G}$ is strongly mixing, for any $t\in\{1,...,\ell\}$ and any $j\in\{1,...,\ell\}$, $(T_{g_{k,t}^{(j)}})_{k\in\mathbb{N}}$ has the mixing property and that for any $t$ and any $i\neq j$, $(T_{(g_{k,t}^{(j)}-g_{k,t}^{(i)})})_{k\in\mathbb{N}}$ also has the mixing property. Thus (ii) follows from \cref{2.MainResult}.\\
(ii)$\implies$(iii): By (ii), we have that for any $\epsilon>0$, any $A_0,...,,A_\ell\in\mathcal A$ and any $\ell$ non-degenerated and essentially distinct sequences
$$(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g^{(j)}_{k,1},...,g^{(j)}_{k,\ell})_{k\in\mathbb{N}},\text{ }j\in\{1,...,\ell\},$$
in $G^{\ell}$, there exists an $\alpha\in\mathbb{N}^{(\ell)}$ such that
$$(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\in R_\epsilon(A_0,...,A_\ell),$$
which implies that $R_\epsilon(A_0,...,A_\ell)$ is $\tilde\Sigma_\ell^*$.\\
(iii)$\implies$(iv): Let $\epsilon>0$, let $A_0,A_1\in\mathcal A$ and let
$(\textbf g^{(1)}_k)_{k\in\mathbb{N}}=(g_{k,1}^{(1)},...,g_{k,\ell}^{(\ell)})_{k\in\mathbb{N}}$ be a non-degenerated seequence in $G^\ell$. In order to prove that $\mathcal R_\epsilon(A_0,A_1)$ is $\Sigma_\ell^*$, it suffices to show that for some $\alpha\in\mathbb{N}^{(\ell)}$, $g_\alpha^{(1)}\in\mathcal R_\epsilon(A_0,A_1)$.\\
Note that for any sequence $(h^{(1)}_k)_{k\in\mathbb{N}}$ in $G$ with $\lim_{k\rightarrow\infty}h_k^{(1)}=\infty$ one can pick sequences $(h^{(2)}_k)_{k\in\mathbb{N}}$,...,
$(h^{(\ell)}_k)_{k\in\mathbb{N}}$
in $G$ with the property that for any distinct $i,j\in\{1,...,\ell\}$, $$\lim_{k\rightarrow\infty}h^{(j)}_k=\infty\text{ and } \lim_{k\rightarrow\infty}(h^{(j)}_k-h^{(i)}_k)=\infty.$$
Hence, one can find non-degenerated sequences $(\textbf g_k^{(j)})_{k\in\mathbb{N}}$ in $G^\ell$, $j\in\{2,...,\ell\}$, such that $(\textbf g^{(1)}_k)_{k\in\mathbb{N}}$,...,
$(\textbf g^{(\ell)}_k)_{k\in\mathbb{N}}$ are essentially distinct. By (iii), there exists an $\alpha\in \mathbb{N}^{(\ell)}$ for which
$$(g^{(1)}_\alpha,...,g^{(\ell)}_\alpha)\in\mathcal R_\epsilon(A_0,A_1,\underbrace{X,...,X}_{\ell-1\text{ times}}).$$
This implies that $g^{(1)}_\alpha\in R_\epsilon(A_0,A_1)$.\\
(iv)$\implies$(i): Let $f\in L^2(\mu)$ be such that $\int_Xf\text{d}\mu=0$ and $\|f\|_{L^2}=1$. We will show that $\lim_{g\rightarrow\infty}T_gf=0$ in the weak topology of $L^2(\mu)$. To do this, it suffices to prove that for any sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ with $\lim_{k\rightarrow\infty}g_k=\infty$, there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ with $\lim_{j\rightarrow\infty}T_{g_{k_j}}f=0$. \\ Note that $$\sigma(g)=\int_X\overline f T_g f\text{d}\mu,\,g\in G$$ is a positive definite function and hence, by Bochner's theorem, there is a unique Borel probability measure $\rho$ on $\hat G$, the Pontryagin dual of $G$, with the property that for all $g\in G$, \begin{equation}\label{3.DefnOfRo} \int_X\overline fT_gf\text{d}\mu=\int_{\hat G} \phi_g(\chi)\text{d}\rho(\chi), \end{equation} where for each $\chi\in \hat G$ and each $g\in G$, $\phi_g(\chi)=\chi(g)$.\\ Let now $(g_k)_{k\in\mathbb{N}}$ be a sequence in $G$ with $\lim_{k\rightarrow\infty}g_k=\infty$. Let $(\textbf g_k)_{k\in\mathbb{N}}=(\underbrace{g_k,...,g_k}_{\ell\text{ times}})_{k\in\mathbb{N}}$ (note that $(\textbf g_k)_{k\in\mathbb{N}}$ is a non-degenerated sequence in $G^\ell$). We claim that there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ such that: \begin{enumerate} \item For some $K\in L^2(\rho)$, \begin{equation}\label{3.AccumulationPoint} K=\lim_{j\rightarrow\infty}\phi_{g_{k_j}} \end{equation} in the weak topology of $L^2(\rho)$.
\item Let $S=\{k_j\,|\,j\in\mathbb{N}\}$. There exists $H\in L^2(\rho)$ such that \begin{equation}\label{3.AccumulationPointRLimit} H=\rlim{\alpha\in S^{(\ell)}}\phi_{g_\alpha}=\lim_{j_1\rightarrow\infty}\cdots\lim_{j_\ell\rightarrow\infty}\phi_{(g_{k_{j_1}}+\cdots+g_{k_{j_\ell}})} \end{equation} in the weak topology of $L^2(\rho)$. \item For any $A_0,A_1\in\mathcal A$, there exists a real number $r_{A_0,A_1}$ such that
\begin{equation}\label{3.PreMixingExpresionOnMainResult}
\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{-g_\alpha}A_1)=r_{A_0,A_1}.
\end{equation} \end{enumerate}
To establish the existence of such a sequence, one first invokes the pre-compactness of the set $\{\phi_{g}\,|\,g\in G\}$ in the weak topology of the set $L^2(\rho)$ to obtain an increasing sequence $(k'_j)_{j\in\mathbb{N}}$ for which \eqref{3.AccumulationPoint} holds. Moreover, by using \cref{1.ItteratedLimitsRemark}, one can find a subsequence $(k_j'')_{j\in\mathbb{N}}$ of $(k_j')_{j\in\mathbb{N}}$ for which \eqref{3.AccumulationPointRLimit} holds for $S=\{k''_j\,|\,j\in\mathbb{N}\}$. Finally, by a diagonalization argument, we can pick a subsequence $(k_j)_{j\in\mathbb{N}}$ of $(k''_j)_{j\in\mathbb{N}}$ for which \eqref{3.PreMixingExpresionOnMainResult} holds for any $A_0,A_1$ from a countable dense subset of $\mathcal A$. If follows (by a standard approximation argument) that \eqref{3.PreMixingExpresionOnMainResult} holds for any $A_0,A_1\in\mathcal A$.\\
By (iv), for every $A_0,A_1\in\mathcal A$, $r_{A_0,A_1}=\mu(A_0)\mu(A_1)$ (otherwise we would be able to find an $\epsilon>0$ for which the set $\mathcal R_\epsilon(A_0,A_1)$ is not $\Sigma_\ell^*$). Since the linear combinations of indicator functions are dense in $L^2(\mu)$, it follows that for any $f_1,f_2\in L^2(\mu)$, \begin{equation}\label{3.GoingToProduct}
\rlim{\alpha\in S^{(\ell)}}\int_X f_1 T_{g_\alpha}f_2\text{d}\mu=\int_X f_1\text{d}\mu\int_Xf_2\text{d}\mu. \end{equation} It follows from \eqref{3.DefnOfRo} and \eqref{3.GoingToProduct} that for any $g\in G$, \begin{multline}\label{3.H=0} \int_{\hat G} \overline{\phi_g(\chi)}H(\chi)\text{d}\rho(\chi)=\rlim{\alpha\in S^{(\ell)}}\int_{\hat G} \overline{\phi_{g}(\chi)}\phi_{g_\alpha}(\chi)\text{d}\rho(\chi)\\ =\rlim{\alpha\in S^{(\ell)}}\int_{\hat G} \phi_{-g}(\chi)\phi_{g_\alpha}(\chi)\text{d}\rho(\chi)=\rlim{\alpha\in S^{(\ell)}}\int_{\hat G} \phi_{(g_\alpha-g)}(\chi)\text{d}\rho(\chi)\\ =\rlim{\alpha\in S^{(\ell)}}\int_{X}\overline fT_{g_\alpha-g}f\text{d}\mu=\rlim{\alpha\in S^{(\ell)}}\int_{X}T_{g}\overline fT_{g_\alpha}f\text{d}\mu=\int_X\overline f\text{d}\mu\int_X f\text{d}\mu=0. \end{multline} Since the linear combinations of the characters $\phi_g,\, g\in G$, are dense in $L^2(\rho)$, it follows from \eqref{3.H=0} that $H=0$. By \eqref{3.AccumulationPoint} and \eqref{3.AccumulationPointRLimit}, we have \begin{multline*} 0=H=\rlim{\alpha\in S^{(\ell)}}\phi_{g_\alpha}=\lim_{j_1\rightarrow\infty}\cdots\lim_{j_\ell \rightarrow\infty}\phi_{(g_{k_{j_1}}+\cdots+g_{k_{j_\ell}})} =\lim_{j_1\rightarrow\infty}\cdots\lim_{j_{\ell}\rightarrow\infty}\prod_{t=1}^\ell \phi_{g_{k_{j_t}}}\\ =(\lim_{j_1\rightarrow\infty}\cdots\lim_{j_{\ell-1}\rightarrow\infty}\prod_{t=1}^{\ell-1} \phi_{g_{k_{j_t}}})(\lim_{j_\ell\rightarrow\infty}\phi_{g_{k_{j_\ell}}})= (\lim_{j_1\rightarrow\infty}\cdots\lim_{j_{\ell-1}\rightarrow\infty}\prod_{t=1}^{\ell-1} \phi_{g_{k_{j_t}}})(\lim_{j\rightarrow\infty}\phi_{g_{k_{j}}})\\ =\cdots=\lim_{j_1\rightarrow\infty}\phi_{g_{k_{j_1}}}(\prod_{t=2}^{\ell}\lim_{j\rightarrow\infty} \phi_{g_{k_{j}}}) =\prod_{t=1}^\ell(\lim_{j\rightarrow\infty}\phi_{g_{k_j}})= K^{\ell}. \end{multline*} So, $K^\ell=0$ and hence $K=0$.\\
Consider now the closed and $(T_g)_{g\in G}$-invariant subspace $\mathcal H_f=\overline{\text{span}(\{T_g f\,|\,g\in G\})}\subseteq L^2(\mu)$. Since $K=0$, it follows from \eqref{3.DefnOfRo} and \eqref{3.AccumulationPoint} that for each $g\in G$, $$\lim_{j\rightarrow\infty} \int_X T_g\overline fT_{g_{k_j}}f\text{d}\mu=\lim_{j\rightarrow\infty} \int_X \overline fT_{(g_{k_j}-g)}f\text{d}\mu=\lim_{j\rightarrow\infty}\int_{\hat G}\phi_{(g_{k_j}-g)}\text{d}\rho=\lim_{j\rightarrow\infty}\int_{\hat G}\overline{\phi_{g}}\phi_{g_{k_j}}\text{d}\rho=0.$$ It follows that for any $f'\in \mathcal H_f$, $\lim_{j\rightarrow\infty}\int_X\overline {f'}T_{g_{k_j}}f\text{d}\mu=0$. Noting that $L^2(\mu)=\mathcal H_f\oplus \mathcal H_f^\perp$, we obtain that $\lim_{j\rightarrow\infty}T_{g_{k_j}}f=0$ in the weak topology of $L^2(\mu)$. In light of the remarks made at the begining of the proof of (iv)$\implies$(i), this, in turn, implies that $\lim_{g\rightarrow\infty}T_gf=0$. We are done. \end{proof}
\section{Some "diagonal" results for strongly mixing systems} In order to give the reader the flavor of the main theme of this section, we start by formulating a slightly enhanced form of \cref{1.ZDiagonalResult} from the Introduction. (This theorem is a rather special case of the results of "diagonal" nature to be proved in this section.) \begin{prop}\label{4.ZExample} Let $(X,\mathcal A,\mu, T)$ be a measure preserving system and let $a_1,...,a_\ell$ be non-zero distinct integers. Then $T$ is strongly mixing if and only if for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set
$$\{n\in\mathbb{Z}\,|\,|\mu(A_0\cap T^{a_1n}A_1\cap\cdots\cap T^{a_\ell n}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\}$$ is $\Sigma_\ell^*$. \end{prop} We move now to formulations of more general "diagonal" results. \\
Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system, let $\ell\in\mathbb{N}$ and let $\phi_1,...,\phi_\ell:G\rightarrow G$ be homomorphisms. For any $\epsilon>0$ and any $A_0,...,A_\ell\in\mathcal A$, define
$$R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)=\{g\in G\,|\,|\mu(A_0\cap T_{\phi_1(g)}A_1\cap\cdots\cap T_{\phi_\ell(g)}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\}.$$
We first give two equivalent formulations of a general result which deals with finitely generated groups. \begin{thm}\label{4.FinitelyGeneratedEquivalence} Let $(G,+)$ be a finitely generated abelian group, let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system and let the homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ be such that for any $j\in\{1,...,\ell\}$, $\ker(\phi_j)$ is finite and for any $i\neq j$, $\ker(\phi_j-\phi_i)$ is also finite. Then $(T_g)_{g\in G}$ is strongly mixing if and only if for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)$ is $\Sigma_\ell^*$. \end{thm} Note that if $G$ is a finitely generated abelian group and $\phi:G\rightarrow G$ is a homomorphism, $\ker(\phi)$ is finite if and only if the index of $\phi(G)$ in $G$ is finite. It follows that \cref{4.FinitelyGeneratedEquivalence} can be formulated in the following equivalent form. \begin{thm}\label{4.FinitelyGeneratedEquivalenceIndex} Let $(G,+)$ be a finitely generated abelian group, let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system and let the homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ be such that for any $j\in\{1,...,\ell\}$, the index of $\phi_j(G)$ in $G$ is finite and for any $i\neq j$, the index of $(\phi_j-\phi_i)$ in $G$ is also finite. Then $(T_g)_{g\in G}$ is strongly mixing if and only if for any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)$ is $\Sigma_\ell^*$. \end{thm} We are going now to formulate and prove variants of Theorems \ref{4.FinitelyGeneratedEquivalence} and \ref{4.FinitelyGeneratedEquivalenceIndex} which pertain to mixing actions of general (not necessarily finitely generated) countable abelian groups. Unlike Theorems \ref{4.FinitelyGeneratedEquivalence} and \ref{4.FinitelyGeneratedEquivalenceIndex}, the following two theorems are not equivalent. We will provide the relevant counterexamples at the end of this section. \begin{thm}\label{4.InjectiveDiagonalResult} Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a strongly mixing system and let the homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ be such that for any $j\in\{1,...,\ell\}$, $\ker(\phi_j)$ is finite and for any $i\neq j$, $\ker(\phi_j-\phi_i)$ is also finite. For any non-degenerated sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,\ell})_{k\in\mathbb{N}}$ in $G^\ell$ there exists an infinite set $S\subseteq \mathbb{N}$ such that for any $A_0,...,A_\ell \in\mathcal A$, $$\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{\phi_1(g_\alpha)}A_1\cap\cdots\cap T_{\phi_\ell(g_\alpha)}A_\ell)=\prod_{j=0}^\ell \mu(A_j).$$ Equivalently, for any $A_0,...,A_\ell \in\mathcal A$ and any $\epsilon>0$, the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)$ is $\Sigma_\ell^*$. \end{thm} \begin{proof} Since for any distinct $i,j\in\{1,...,\ell\}$, $\ker(\phi_j)$ and $\ker(\phi_j-\phi_i)$ are both finite, we have for each $t\in\{1,...,\ell\}$, $$\lim_{k\rightarrow\infty}\phi_j(g_{k,t})=\infty\text{ and }\lim_{k\rightarrow\infty}(\phi_j(g_{k,t})-\phi_i(g_{k,t}))=\infty.$$ For each $j\in\{1,...,\ell\}$, let $$(\textbf g^{(j)}_k)_{k\in\mathbb{N}}=(\phi_j(g_{k,1}),...,\phi_j(g_{k,\ell}))_{k\in\mathbb{N}}.$$ Then the sequences $(\textbf g^{(1)}_k)_{k\in\mathbb{N}},...,(\textbf g^{(\ell)}_k)_{k\in\mathbb{N}}$ are non-degenerated and essentially distinct. By \cref{3.MainResult}, (ii), there exists an infinite set $S\subseteq \mathbb{N}$ such that for any $A_0,...,A_\ell\in\mathcal A$, \begin{multline*} \rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{\phi_1(g_\alpha)}A_1\cap\cdots\cap T_{\phi_\ell(g_\alpha)}A_\ell)\\ =\rlim{\alpha\in S^{(\ell)}}\mu(A_0\cap T_{g^{(1)}_\alpha}A_1\cap\cdots\cap T_{g^{(\ell)}_\alpha}A_\ell)=\mu(\prod_{j=0}^\ell A_j). \end{multline*} \end{proof} \begin{rem}\label{4.CombinatorialRemark}
The goal of this remark is to indicate an alternative way of proving \cref{4.InjectiveDiagonalResult}. Let $G$ and $\phi_1,...,\phi_\ell$ be as in the hypothesis of \cref{4.InjectiveDiagonalResult}. In Section 5 we will show that if $E$ is a $\tilde\Sigma_\ell^*$ set in $G^\ell$, then $\{g\in G\,|\,(\phi_1(g),...,\phi_\ell(g))\in E\}$ is a $\Sigma_\ell^*$ set in $G$ (see \cref{5.3.UsefulInSection4}). Thus, for any measure preserving system $(X,\mathcal A,\mu, (T_g)_{g\in G})$, any $A_0,....,A_\ell\in\mathcal A$ and any $\epsilon>0$, if $R_\epsilon(A_0,...,A_\ell)$ is a $\tilde\Sigma_\ell^*$ set, then $R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)$ is a $\Sigma_\ell^*$ set. One can now invoke \cref{3.MainResult}, (iii). \end{rem} The next result complements \cref{4.InjectiveDiagonalResult}. Note that it provides a somewhat stronger version of one of the directions in \cref{4.FinitelyGeneratedEquivalenceIndex}. \begin{thm}\label{4.quasiSurjectiveDiagonalResult} Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a measure preserving system and let the homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ be such that either one of $\phi_1(G)$, $\phi_2(G)$ or $(\phi_2-\phi_1)(G)$ has finite index in $G$. If for all $A_0,...,A_\ell\in\mathcal A$ and all $\epsilon>0$ the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A_0,...,A_\ell)$ is $\Sigma_\ell^*$, then $(T_g)_{g\in G}$ is strongly mixing. \end{thm} \begin{proof} We will assume that $(\phi_2-\phi_1)(G)$ has finite index in $G$, the other two cases can be handled similarly. For any $A_1,A_2\in\mathcal A$ and any $\epsilon>0$, we have \begin{multline*} R^{\phi_1,...,\phi_\ell}_\epsilon(X,A_1,A_2,\underbrace{X,...,X}_{\ell-2\text{ times}})\\
=\{g\in G\,|\,|\mu(X\cap T_{\phi_1(g)}A_1\cap T_{\phi_2(g)}A_2\cap T_{\phi_3(g)} X\cap\cdots\cap T_{\phi_\ell(g)}X)-\mu(A_1)\mu(A_2)|<\epsilon\}\\
=\{g\in G\,|\,|\mu(T_{\phi_1(g)}A_1\cap T_{\phi_2(g)}A_2)-\mu(A_1)\mu(A_2)|<\epsilon\}=R_\epsilon^{\phi_2-\phi_1}(A_1,A_2). \end{multline*} By our assumption, for any $\epsilon>0$ and any $A_1,A_2\in\mathcal A$, the set $R^{\phi_2-\phi_1}_\epsilon(A_1,A_2)$ is a $\Sigma_\ell^*$ set and hence, by \cref{3.MainResult}, (iv), $(T_{(\phi_2-\phi_1)(g)})_{g\in G}$ is strongly mixing.\\
We will now prove that $(T_g)_{g\in G}$ is strongly mixing by showing that for any sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ with $\lim_{k\rightarrow\infty}g_k=\infty$, there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ with the property that for any $A_0,A_1\in\mathcal A$, $$\lim_{j\rightarrow\infty}\mu(A_0\cap T_{g_{k_j}}A_1)=\mu(A_0)\mu(A_1).$$
Let $(g_k)_{k\in\mathbb{N}}$ be a sequence in $G$ with $\lim_{k\rightarrow \infty}g_k=\infty$. By assumption, $(\phi_2-\phi_1)(G)$ has finite index in $G$, so there exists an increasing sequence $(k_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ and an element $\tau\in G$ for which $\{g_{k_j}+\tau\,|\,j\in\mathbb{N}\}\subseteq (\phi_2-\phi_1)(G)$. Since $(T_{(\phi_2-\phi_1)(g)})_{g\in G}$ is strongly mixing, for any $A_0,A_1\in\mathcal A$, $$\lim_{j\rightarrow\infty}\mu(A_0\cap T_{g_{k_j}}A_1)=\lim_{j\rightarrow\infty}\mu(A_0\cap T_{g_{k_j}+\tau}(T_{-\tau}A_1))=\mu(A_0)\mu(A_1),$$ completing the proof. \end{proof}
The following proposition shows that the assumption made in \cref{4.FinitelyGeneratedEquivalence} that $G$ is finitely generated cannot be removed. \begin{prop}\label{4.NonFiniteIndexExample} Let $G=\bigoplus_{k\in\mathbb{N}} \mathbb{Z}$ and let $\ell\in\mathbb{N}$. There exists a measure preserving system $(X,\mathcal A,\mu,(T_g)_{g\in G})$ and homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ satisfyng (a) for any $j\in\{1,...,\ell\}$, $\ker(\phi_j)$ is finite, and (b) for any $i\neq j$, $\ker(\phi_j-\phi_i)$ is also finite, and such that every set of the form $R_\epsilon^{\phi_1,...,\phi_\ell} (A_0,...,A_\ell)$ is $\Sigma_\ell^*$ but $(T_g)_{g\in G}$ is not strongly mixing. \end{prop} \begin{proof} We will only carry out the proof for $\ell=2$, the general case can be handled similarly. Let $\phi_1:G\rightarrow G$ be the homomorphism given by $$\phi_1((a_1,a_2,...,a_n,...))=(0,a_1,0,a_2,...,0,a_n,...).$$ Note that $\phi_1$ is injective (and so, $\ker(\phi_1)$ is trivial).\\ Let $X=\{0,1\}^G$ be endowed with the product topology, let $\mu$ be the $(\frac{1}{2},\frac{1}{2})$ product measure on $\mathcal A=\text{Borel}(X)$ and for each $g\in G$, let $S_g:X\rightarrow X$ be the map defined by $(S_g(x))(h)=x(h+g)$. The system $(X,\mathcal A,\mu,(S_g)_{g\in G})$ is strongly mixing. Define a measure preserving $G$-action $(T_g)_{g\in G}$ on $(X,\mathcal A,\mu)$ by $$T_{(a_1,a_2,...)}=S_{(a_2,a_4,...)}$$ and let $\phi_2:G\rightarrow G$ be defined by $\phi_2(g)=2\phi_1(g)$. Note that for any $g=(a_1,a_2,...)\in G$, $$T_{\phi_1(g)}=T_{\phi_1((a_1,a_2,...))}=T_{(0,a_1,0,a_2,...)}=S_{(a_1,a_2,...)}=S_g.$$ So, for any $\epsilon>0$ and any $A_0,A_1,A_2\in\mathcal A$, \begin{multline}\label{4.SetWithTSetWithS} R_\epsilon^{\phi_1,\phi_2}(A_0,A_1,A_2)\\
=\{g\in G\,|\,|\mu(A_0\cap T_{\phi_1(g)}A_1\cap T_{\phi_2(g)}A_2)-\mu(A_0)\mu(A_1)\mu(A_2)|<\epsilon\}\\
=\{g\in G\,|\,|\mu(A_0\cap S_{g}A_1\cap S_{2g}A_2)-\mu(A_0)\mu(A_1)\mu(A_2)|<\epsilon\}. \end{multline} It follows from \cref{4.InjectiveDiagonalResult} that every set of the form
$$\{g\in G\,|\,|\mu(A_0\cap S_{g}A_1\cap S_{2g}A_2)-\mu(A_0)\mu(A_1)\mu(A_2)|<\epsilon\}$$ is $\Sigma_2^*$ and hence, by \eqref{4.SetWithTSetWithS}, for any any $A_0,A_1,A_2$ and any $\epsilon>0$, $R_\epsilon^{\phi_1,\phi_2}(A_0,A_1,A_2)$ is $\Sigma_2^*$.\\ Noting that for each $k\in\mathbb{N}$, $T_{(k,0,0,...)}=S_{(0,0,...)}$ is the identity map on $X$, we see that $(T_g)_{g\in G}$ is not strongly mixing. We are done. \end{proof} The next result shows that \cref{4.FinitelyGeneratedEquivalenceIndex} cannot be extended to arbitrary countable abelian groups. \begin{prop}\label{4.LackOfInjectivityExample} Let $G=\bigoplus_{k\in\mathbb{N}} \mathbb{Z}$ and let $\ell\in\mathbb{N}$. There exist a strongly mixing system $(X,\mathcal A,\mu, (T_g)_{g\in G})$ and homomorphisms $\phi_1,...,\phi_\ell:G\rightarrow G$ satisfying (a) for any $j\in\{1,...,\ell\}$, $\phi_j(G)=G$, and (b) for any $i\neq j$, $(\phi_i-\phi_j)(G)=G$, and such that for some $A\in\mathcal A$ and some $\epsilon>0$, the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A,...,A)$ is not $\Sigma_\ell^*$. \end{prop} \begin{proof} Let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a strongly mixing system and let $p_1,...,p_\ell\in\mathbb{N}$ be $\ell$ different prime numbers. For each $j\in \{1,...,\ell\}$, let $\phi_j:G\rightarrow G$ be defined by $$\phi_j(a_1,a_2,a_3,...)=(a_{p_j^1},a_{p_j^2},a_{p_j^3}...).$$
It follows that for any $j\in\{1,...,\ell\}$, $\phi_j(G)=G$ and since for any distinct $i,j\in\{1,...,\ell\}$ the sets $\{p_i^k\,|\,k\in\mathbb{N}\}$ and $\{p_j^k\,|\,k\in\mathbb{N}\}$ are disjoint, we have that $(\phi_j-\phi_i)(G)=G$ as well.\\
Observe that the subgroup $G'=\{(a_1,0,0,...)\in G\,|\,a_1\in\mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$ and that for any $j\in\{1,...,\ell\}$, $G'\subseteq\ker(\phi_j)$. Let $(g_k)_{k\in\mathbb{N}}$ be a sequence in $G'$ with $\lim_{k\rightarrow\infty}g_k=\infty$. Since for each $k\in\mathbb{N}$, $T_{\phi_j(g_k)}=T_{(0,0,...)}=\text{Id}$, where $\text{Id}$ is the identity map on $X$, we have that for any $A\in \mathcal A$ with $\mu(A)\in(0,1)$, and any $k_1<\cdots<k_\ell$, $$\mu(A\cap T_{\phi_1(g_{k_1}+\cdots+g_{k_\ell})}A\cap\cdots\cap T_{\phi_\ell(g_{k_1}+\cdots+g_{k_\ell})}A)=\mu(A)\neq \mu^{\ell+1}(A).$$ It follows that if $\epsilon$ is small enough, the set $R_\epsilon^{\phi_1,...,\phi_\ell}(A,...,A)$ does not intersect the $\Sigma_\ell$ set
$$\{g_{k_1}+\cdots+g_{k_\ell}\,|\,k_1<\cdots<k_\ell\}$$ and hence, it is not $\Sigma_\ell^*$. This completes the proof. \end{proof}
\section{Largeness properties of $\tilde\Sigma_m^*$ sets} As we have seen above, any strongly mixing system $(X,\mathcal A,\mu,(T_g)_{g\in G})$ has the property that the sets $R_\epsilon(A_0,...,A_m)$ are $\tilde\Sigma_m^*$ (moreover, the strong mixing of $(T_g)_{g\in G}$ is characterized by this property). This section is devoted to the discussion of massivity and ubiquity of $\tilde\Sigma_m^*$ sets. Since strong mixing is a stronger property than those of mild and weak mixing, one should expect that the notions of largeness associated with (multiple) mild and weak mixing are "majorized" by the notion of largeness associated with $\tilde\Sigma_m^*$ sets. This will be established in Subsections 5.1 and 5.2. Finally, in Subsection 5.3 we will show that $\tilde\Sigma_m^*$ sets are ubiquitous in the sense that they are well spread among the cosets of \textit{admissible} subgroups of $G^m$ (the class of admissible subgroups will be introduced in Subsection 5.3).
\subsection{Any $\tilde\Sigma_m^*$ set in $G^d$ is an \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set} In this section we will introduce \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} sets and juxtapose them with $\tilde\Sigma_m^*$ sets.
(\text{\rm{$\tilde{\text{IP}}\rm{^*}$}} sets are intrinsically linked to the multiple mixing properties of mildly mixing systems. The connection between \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} sets and mildly mixing systems will be addressed in Section 6.)\\
Let $(G,+)$ be a countable abelian group and let $\mathcal F$ denote the set of all non-empty finite subsets of $\mathbb{N}$. Given a sequence $(g_k)_{k\in\mathbb{N}}$ in $G$, define an $\mathcal F$-sequence $(g_\alpha)_{\alpha\in\mathcal F}$ by \begin{equation}\label{5.1.DefnAlphaIP} g_\alpha=\sum_{j\in\alpha} g_j=g_{k_1}+\cdots+g_{k_t},\,\alpha=\{k_1,...,k_t\}. \end{equation} We will write $$\lim_{\alpha\rightarrow\infty}g_\alpha=\infty$$ if for every finite $K\subseteq G$, there exists an $\alpha_0\in\mathcal F$ such that for any $\alpha\in\mathcal F$ with $\alpha>\alpha_0$ (i.e. $\min \alpha>\max \alpha_0$), $g_\alpha\not\in K$.\\
A set $E\subseteq G$ is called an IP set if $E=\{g_\alpha\,|\,\alpha\in\mathcal F\}$ for some sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ such that $\lim_{\alpha\rightarrow\infty}g_\alpha=\infty$.\footnote{
IP sets are often defined just as sets of the form
$$\text{FS}((g_k)_{k\in\mathbb{N}})=\{g_{k_1}+\cdots+g_{k_t}\,|\,k_1<\cdots<k_t,\,t\in\mathbb{N}\}=\{g_\alpha\,|\,\alpha\in\mathcal F\}$$
(without the requirement that $\lim_{\alpha\rightarrow\infty}g_\alpha=\infty$). Our choice of definition for IP sets is dictated by our interest in the study of asymptotic properties of measure preserving actions. The distinction between our definition and the more traditional one is rather mild: for any infinite set of the form $E=\{g_\alpha\,|\,\alpha\in\mathcal F\}$ there exists a sequence $(h_k)_{k\in\mathbb{N}}$ such that $\{h_\alpha\,|\,\alpha\in\mathcal F\}\subseteq E$ and $\lim_{\alpha\rightarrow\infty}h_\alpha=\infty$. } A set $E\subseteq G$ is called IP$^*$ if it has a non-trivial intersection with every IP set. \\
We now introduce modifications of IP and IP$^*$ sets, namely $\tilde{\text{IP}}$ sets and \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} sets, which, as will be seen in Section 6, are naturally linked with the properties of the sets $R_\epsilon(A_0,...,A_\ell\}$ in the context of mildly mixing systems. \begin{defn}\label{5.1.DefnOfIPtilde} Let $(G,+)$ be a countable abelian group and let $d\in \mathbb{N}$. We say that a set $E\subseteq G^d$ is an $\tilde{\text{IP}}$ set if it is of the form
$$E=\{(g_\alpha^{(1)},...,g_\alpha^{(d)})\,|\,\alpha\in\mathcal F\},$$
where for each $j\in\{1,...,d\}$, $\{g_\alpha^{(j)}\,|\,\alpha\in\mathcal F\}$ is generated by $(g_{k}^{(j)})_{k\in\mathbb{N}}$ as in \eqref{5.1.DefnAlphaIP} and, in addition, for any $j\in\{1,...,d\}$, \begin{equation}\label{5.1.IPcondition1} \lim_{\alpha\rightarrow\infty}g^{(j)}_\alpha=\infty \end{equation} and for any $i\neq j$, \begin{equation}\label{5.1.IPcondition2} \lim_{\alpha\rightarrow\infty}(g^{(j)}_\alpha-g_\alpha^{(i)})=\infty. \end{equation} (Note that if $d=1$, then $E\subseteq G$ is an \text{IP} set if and only if it is an \text{\rm{$\tilde{\text{IP}}$}} set.)\\ A set $E\subseteq G^d$ is called an \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set if it has a non-trivial intersection with every $\tilde{\text{IP}}$ set in $G^d$. \end{defn} \begin{rem}\label{5.1.CommonSenseSequence}
Let $(G,+)$ be a countable abelian group, let $d\in \mathbb{N}$ and let $E\subseteq G^d$ be an $\tilde{\text{IP}}$ set. From now on, whenever we pick a sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_k^{(1)},...,g_k^{(d)})_{k\in\mathbb{N}}$ in $G^d$ with the property that $E=\{(g_\alpha^{(1)},...,g_\alpha^{(d)})\,|\,\alpha\in\mathcal F\}$, we will tacitly assume that $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(d)})_{k\in\mathbb{N}}$ satisfy \eqref{5.1.IPcondition1} and \eqref{5.1.IPcondition2}. \end{rem}
The following lemma unveils an important connection between {\rm{$\tilde{\text{IP}}$}} and $\tilde\Sigma_m$ sets. \begin{lem}\label{5.1.SigmaInEveryIP} Let $(G,+)$ be a countable abelian group and let $d,m\in\mathbb{N}$. Any {\rm{$\tilde{\text{IP}}$}} set $E\subseteq G^d$ contains a $\tilde\Sigma_m$ set. Namely, there exist non-degenerated and essentially distinct sequences $$(\textbf g_k^{(j)})=(g^{(j)}_{k,1},...,g^{(j)}_{k,m})_{k\in\mathbb{N}},\,j\in\{1,...,d\}$$
in $G^m$ with the property that $\{(g_\alpha^{(1)},...,g_\alpha^{(d)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$, where for each $j\in\{1,...,d\}$ and each $\alpha=\{k_1,...,k_m\}\in\mathbb{N}^{(m)}$, $g_\alpha^{(j)}=g_{k_1,1}^{(j)}+\cdots+g_{k_m,m}^{(j)}$.
\end{lem} \begin{proof} Let $E$ be an $\tilde{\text{IP}}$ set and let $(\textbf h_k)_{k\in\mathbb{N}}= (h_k^{(1)},...,h_k^{(d)})_{k\in\mathbb{N}}$ be such that
$$E=\{\textbf h_\alpha\,|\,\alpha\in\mathcal F\}=\{(h_\alpha^{(1)},...,h_\alpha^{(d)})\,|\,\alpha\in\mathcal F\}.$$ Following the stipulation made in \Cref{5.1.CommonSenseSequence}, for any finite set $F\subseteq G$, we can find an $\alpha_F\in\mathcal F$ such that for any $\alpha\in\mathcal F$ with $\alpha> \alpha_F$ and any distinct $i,j\in\{1,...d\}$, $h^{(j)}_{\alpha}\not\in F$ and $(h^{(j)}_{\alpha}-h^{(i)}_{\alpha})\not\in F$. In particular, for any distinct $i,j\in\{1,...,d\}$ \begin{equation}\label{5.1.AlphaSequenceGoingToInfty} \lim_{k\rightarrow\infty}h_{k}^{(j)}=\infty\text{ and }\lim_{k\rightarrow\infty}(h_{k}^{(j)}-h_{k}^{(i)})=\infty. \end{equation} For each $j\in\{1,...,d\}$ and each $k\in\mathbb{N}$ we let \begin{equation}\label{5.1.DefnSequenceg_k} \textbf g_k^{(j)}=\underbrace{(h_{k}^{(j)},...,h_{k}^{(j)})}_{m\text{ times}}. \end{equation}
Note that by \eqref{5.1.AlphaSequenceGoingToInfty}, the sequences $(\textbf g^{(1)}_k)_{k\in\mathbb{N}}$,...,$(\textbf g^{(d)}_k)_{k\in\mathbb{N}}$ are non-degenerated and essentially distinct. It follows now from \eqref{5.1.DefnSequenceg_k} that for any $\alpha=\{k_1,...,k_m\}\in\mathbb{N}^{(m)}$, $$(g^{(1)}_\alpha,...,g^{(d)}_\alpha)=(\sum_{j=1}^m h_{k_j}^{(1)},...,\sum_{j=1}^m h_{k_j}^{(d)})=(h_{\{k_1,...,k_m\}}^{(1)},...,h_{\{k_1,...,k_m\}}^{(d)})\in E,$$ which completes the proof. \end{proof} \begin{rem}
The proof of \cref{5.1.SigmaInEveryIP} actually shows that any {\rm{$\tilde{\text{IP}}$}} set is a union of $\tilde\Sigma_t$ sets. Let $E\subseteq G^d$ be an {\rm{$\tilde{\text{IP}}$}} set and let $(\textbf g_k)_{k\in\mathbb{N}}$ be a sequence such that $E=\{\textbf g_\alpha\,|\alpha\in\mathcal F\}$. The proof of \cref{5.1.SigmaInEveryIP} shows that for each $t\in\mathbb{N}$, $\{\textbf g_{k_1}+\cdots+\textbf g_{k_t}\,|\,k_1<\cdots<k_t\}$ is a $\tilde\Sigma_t$ set. Hence,
$$E=\bigcup_{t\in\mathbb{N}}\{\textbf g_{k_1}+\cdots+\textbf g_{k_t}\,|\,k_1<\cdots<k_t\}.$$ \end{rem} As an immediate consequence of \cref{5.1.SigmaInEveryIP} we have the following result. \begin{cor}\label{5.1.EverySigma*IsIP*} Let $(G,+)$ be a countable abelian group and let $d,m\in\mathbb{N}$. Every $\tilde \Sigma_m^*$ set in $G^d$ is an \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set. \end{cor} \begin{proof} Let $E\subseteq G^d$ be a $\tilde\Sigma_m^*$ set and let $D\subseteq G^d$ be an \rm{$\tilde{\text{IP}}$} set. By \cref{5.1.SigmaInEveryIP}, we have that $D$ contains a $\tilde\Sigma_m$ set and hence $E\cap D\neq \emptyset$. Since $D$ was arbitrary, this shows that $E$ is an \rm{$\tilde{\text{IP}}\rm{^*}$} set. \end{proof}
\subsection{Any $\tilde\Sigma_m^*$ set in $G^d$ has uniform density one} We start with defining the notions of \textit{upper density} and \textit{uniform density one} in countable abelian groups. \begin{defn} Let $(G,+)$ be a countable abelian group, let $E\subseteq G$ and let $(F_k)_{k\in\mathbb{N}}$ be a F{\o}lner sequence in $G$.\footnote{ Let $(G,+)$ be a countable abelian group. A sequence $(F_k)_{k\in\mathbb{N}}$ of non-empty finite subsets of $G$ is a F{\o}lner sequence if for any $g\in G$,
$$\lim_{k\rightarrow\infty}\frac{|(g+F_k)\cap F_k|}{|F_k|}=1,$$
where, for a finite set $A$, $|A|$ denotes its cardinality. It is well known that every countable abelian group contains a F{\o}lner sequence. } The \textbf{upper density} of $E$ with respect to $(F_k)_{k\in\mathbb{N}}$ is defined by
$$\overline d_{(F_k)}(E)=\limsup_{k\rightarrow\infty}\frac{|E\cap F_k|}{|F_k|}.$$ A set $E\subseteq G$ has \textbf{uniform density one} if for every F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$, $\overline d_{(F_k)}(E)=1$. \end{defn} Sets of uniform density one are intrinsically connected with weakly mixing measure preserving systems. Recall that a measure preserving action $(T_g)_{g\in G}$ on a probability space $(X,\mathcal A,\mu)$ is called weakly mixing if the diagonal action $(T_g\times T_g)_{g\in G}$ on $X\times X$ is ergodic. When $G$ is an amenable group, the notion of weak mixing can be equivalently defined with the help of strong C{\'e}saro limits along F{\o}lner sequences. Namely, $(T_g)_{g\in G}$ is weakly mixing if and only if for any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ and any $A_0,A_1\in\mathcal A$,
$$\lim_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}|\mu(A_0\cap T_gA_1)-\mu(A_0)\mu(A_1)|=0.$$ It follows that $(T_g)_{g\in G}$ is weakly mixing if and only if the sets
$$R_\epsilon(A_0,A_1)=\{g\in G\,|\,|\mu(A_0\cap T_gA_1)-\mu(A_0)\mu(A_1)|<\epsilon\}$$ have uniform density one. The reader will find a few more equivalent forms of weak mixing in \cref{6.2.EquivalentFormsOFWM} below. \\
In order to derive the main result of this subsection, namely the fact that every $\tilde\Sigma_m^*$ set has uniform density one, we need first to prove two auxiliary propositions. \begin{prop} \label{5.2.IPPoincare}
Let $(G,+)$ be a countable abelian group, let $d\in\mathbb{N}$ and let $(F_k)_{k\in\mathbb{N}}$ be a F{\o}lner sequence in $G^d$. For any $E\subseteq G^d$ with $\overline d_{(F_k)}(E)>0$ and any \text{\rm{$\tilde{\text{IP}}$}} set $D\subseteq G^d$, there exists a sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_k^{(1)},...,g_k^{(d)})$ in $G^d$ such that (a) $\{\textbf g_\alpha\,|\,\alpha\in\mathcal F\}\subseteq D$, (b) for any distinct $i,j\in\{1,...,d\}$, \eqref{5.1.IPcondition1} and \eqref{5.1.IPcondition2} hold, and (c) for any $\alpha\in \mathcal F$, \begin{equation}\label{5.2.IPPoincareMultirecurrence} \overline d_{(F_k)}(\bigcap_{\beta\subseteq\alpha,\,\beta\neq\emptyset}(E-\textbf g_\beta))>0. \end{equation}
In other words, for each $\alpha\in \mathcal F$, the set $E_\alpha=\{\textbf h\in G^d\,|\,\forall \beta\subseteq \alpha,\,\beta\neq\emptyset,\,\textbf h+\textbf g_\beta\in E\}$ satisfies $\overline d_{(F_k)}(E_\alpha)>0$. \end{prop} \begin{proof}
Let $D=\{\textbf h_\alpha\,|\,\alpha\in\mathcal F\}$ be an \text{\rm{$\tilde{\text{IP}}$}} set in $G^d$ generated by the sequence $(\textbf h_k)_{k\in\mathbb{N}}=(h_{k,1},...,h_{k,d})_{k\in\mathbb{N}}$. We claim that for any $M\in\mathbb{N}$ with $M>\frac{1}{\overline d_{(F_k)}(E)}$, there exist $L,R\in\mathbb{N}$, $L<R\leq M$ for which $\overline d_{(F_k)}(E\cap (E-\textbf h_{\{L+1,L+2,...,R\}}))>0$. To see this, suppose for the sake of contradiction that for any distinct $R,L\in\{1,...,M\}$, $R>L$, $\overline d_{(F_k)}(E\cap(E-\textbf h_{\{L+1,...,R\}}))=0$. Since $\overline d_{(F_k)}$ is translation invariant and for any $L,R\in\{1,...,M\}$, $L<R$, $\textbf h_{\{L+1,...,R\}}=\textbf h_{\{1,...,R\}}-\textbf h_{\{1,...,L\}}$, we have that $$\overline d_{(F_k)}(E\cap(E-\textbf h_{\{L+1,...,R\}}))=\overline d_{(F_k)}((E-\textbf h_{\{1,...,L\}})\cap(E-\textbf h_{\{1,...,R\}}))=0.$$ It follows that $$\overline d_{(F_k)}(\bigcup_{R=1}^M (E-\textbf h_{\{1,...,R\}}))=\sum_{R=1}^M\overline d_{(F_k)}(E-\textbf h_{\{1,...,R\}})=M\overline d_{(F_k)}(E)>1,$$ a contradiction. Thus, there exist $L,R\in\mathbb{N}$ with $L<R\leq M$ such that $\overline d_{(F_k)}(E\cap (E-\textbf h_{\{L+1,...,R\}}))>0$. We will let $\gamma_1=\{L+1,...,,R\}$.\\ Now let $E_1=E\cap (E-\textbf h_{\gamma_1})$. Repeating the above argument, we find $L',R'\in\mathbb{N}$, $R<L'<R'$, such that $\gamma_2=\{L'+1,...,R'\}$ satisfies
$\overline d_{(F_k)}(E_1\cap (E_1-\textbf h_{\gamma_2}))>0$. It follows that $\gamma_1<\gamma_2$ and that $\textbf h_{\gamma_1\cup \gamma_2}=\textbf h_{\gamma_1}+\textbf h_{\gamma_2}$. Hence $$\overline d_{(F_k)}(E\cap(E-\textbf h_{\gamma_1})\cap(E-\textbf h_{\gamma_2})\cap (E-\textbf h_{\gamma_1\cup \gamma_2})>0.$$ Continuing in this way, we can find a sequence $(\gamma_k)_{k\in\mathbb{N}}$ with $\gamma_k<\gamma_{k+1}$ for each $k\in\mathbb{N}$ and the property that for any $\alpha\in \mathcal F$, $$\overline d_{(F_k)}(\bigcap_{\beta\subseteq\alpha,\,\beta\neq\emptyset}(E-\textbf h_{\bigcup_{k\in\beta}\gamma_k}))>0.$$
For each $k\in\mathbb{N}$, let $\textbf g_k=\textbf h_{\gamma_k}$ and for each $\alpha\in\mathcal F$, let $\textbf g_\alpha=\sum_{j\in\alpha}\textbf g_j=\textbf h_{\bigcup_{j\in\alpha}\gamma_j}$. Observe that the sequence $(\textbf g_\alpha)_{\alpha\in\mathcal F}$ satisfies \eqref{5.2.IPPoincareMultirecurrence}. Let $D'=\{\textbf g_\alpha\,|\,\alpha\in\mathcal F\}$. Clearly $D'\subseteq D$. To finish the proof observe that $$(\textbf g_\alpha)_{\alpha\in\mathcal F}=( g_{\alpha,1},...,g_{\alpha,d})_{\alpha\in\mathcal F}=(h_{(\bigcup_{k\in\alpha}\gamma_k),1},...,h_{(\bigcup_{k\in\alpha}\gamma_k),d})_{\alpha\in\mathcal F}$$
satisfies \eqref{5.1.IPcondition1} and \eqref{5.1.IPcondition2}. Indeed, in view of \Cref{5.1.CommonSenseSequence}, for any $j\in\{1,...,d\}$, $$\lim_{\alpha\rightarrow\infty} g_{\alpha,j}=\lim_{\alpha\rightarrow\infty}h_{(\bigcup_{k\in\alpha}\gamma_k),j}=\infty $$ and for $i\neq j$, $$\lim_{\alpha\rightarrow\infty}( g_{\alpha,j}- g_{\alpha,i})=\lim_{\alpha\rightarrow\infty}(h_{(\bigcup_{k\in\alpha}\gamma_k),j}-h_{(\bigcup_{k\in\alpha}\gamma_k),i})=\infty.$$ \end{proof} \begin{prop}\label{5.2.FiniteSumsInPositiveDensitySets} Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$ and let $(F_k)_{k\in\mathbb{N}}$ be a F{\o}lner sequence in $G^d$. Any $E\subseteq G^d$ with $\overline d_{(F_k)}(E)>0$ contains a $\tilde\Sigma_m$ set. Namely, there exist non-degenerated and essentially distinct sequences $$(\textbf g_k^{(j)})=(g^{(j)}_{k,1},...,g^{(j)}_{k,m})_{k\in\mathbb{N}},\,j\in\{1,...,d\}$$
in $G^m$ with the property that $\{(g_\alpha^{(1)},...,g_\alpha^{(d)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$. \end{prop} \begin{proof}
Fix $d\in\mathbb{N}$ and let $D$ be an \text{\rm{$\tilde{\text{IP}}$}} set in $G^d$. Let $(\textbf h_k)_{k\in\mathbb{N}}=(h_k^{(1)},...,h_k^{(d)})_{k\in\mathbb{N}}$ be a sequence in $G^d$ with $D=\{\textbf h_\alpha\,|\,\alpha\in\mathcal F\}$. Invoking \cref{5.2.IPPoincare} and passing, if needed, to a sub-\text{\rm{$\tilde{\text{IP}}$}} set in $D$, we can assume that for any $\alpha\in\mathcal F$, \begin{equation}\label{5.2.IPReccurence} \overline d_{(F_k)}(\bigcap_{\beta\subseteq\alpha,\,\beta\neq\emptyset}(E-\textbf h_\beta))>0 \end{equation} and that $(\textbf h_k)_{k\in\mathbb{N}}$ satisfies \eqref{5.1.IPcondition1} and \eqref{5.1.IPcondition2}.\\ Let $m=1$. There exists a sequence $(\alpha_k)_{k\in\mathbb{N}}$ in $\mathcal F$ such that for each $k\in\mathbb{N}$, $ \alpha_k<\alpha_{k+1}$ and such that for any distinct $k,k'\in\mathbb{N}$ and any distinct $i,j\in\{1,...,d\}$, \begin{equation}\label{5.2.Disjoint} h_{\alpha_k}^{(j)}\neq h_{\alpha_{k'}}^{(j)}\text{ and }h_{\alpha_k}^{(j)}-h_{\alpha_k}^{(i)}\neq h_{\alpha_{k'}}^{(j)}-h_{\alpha_{k'}}^{(i)}. \end{equation}
Pick a sequence $(A_k)_{k\in\mathbb{N}}$ of finite subsets of $G$ with the properties that for each $k\in\mathbb{N}$, (a) $|A_k|=k$, (b) $A_k\subseteq A_{k+1}$, and (c) $\bigcup_{k\in\mathbb{N}} A_k=G$. By \eqref{5.2.IPReccurence}, for each $k\in\mathbb{N}$ we can find $\textbf b_k=(b_{k,1},...,b_{k,d})$ in $G^d$ such that for any $t\in\{1,...,kd^2+1\}$, $\textbf b_k+\textbf h_{\alpha_t}\in E$. By \eqref{5.2.Disjoint}, for any $k\in\mathbb{N}$ and any $j\in\{1,...,d\}$, there exist at most $k$ natural numbers $t$ for which $b_{k,j}+h_{\alpha_t}^{(j)}\in A_k$. Similarly, for any distinct $i,j\in\{1,...,d\}$, one has $(b_{k,j}-b_{k,i})+(h_{\alpha_t}^{(j)}-h_{\alpha_t}^{(i)})\in A_k$ for at most $k$ natural numbers $t$.\\ We claim that there exists $t\in\{1,...,kd^2+1\}$ such that for any $j\in\{1,...,d\}$, $b_{k,j}+h_{\alpha_t}^{(j)}\not\in A_k$ and for any $i\neq j$, $(b_{k,j}-b_{k,i})+(h_{\alpha_t}^{(j)}-h_{\alpha_t}^{(i)})\not\in A_k$. Suppose for contradiction that this is not the case. Since there are $d^2-d$ pairs $(i,j)$ with distinct $i, j\in\{1,...,d\}$, there exist at least $k+1$ natural numbers $t$ for which, say, $b_{k,1}+h_{\alpha_t}^{(1)}\in A_k$, a contradiction.\\ Thus, there exists a sequence $(k_t)_{t\in\mathbb{N}}$ in $\mathbb{N}$ for which the sequences $$(b_{t,j}+h_{\alpha_{k_t}}^{(j)})_{t\in\mathbb{N}},\,j\in\{1,...,d\}$$ are non-degenerated and essentially distinct,
and
$$\{(b_{t,1}+h_{\alpha_{k_t}}^{(1)},...,b_{t,d}+h_{\alpha_{k_t}}^{(d)})\,|\,t\in\mathbb{N}\}\subseteq E.$$\\
Now let $m>1$. By \cref{5.1.SigmaInEveryIP} there exist non-degenerated and essentially distinct sequences $(\textbf f_k^{(j)})_{k\in\mathbb{N}}=(f^{(j)}_{k,1},...,f^{(j)}_{k,m-1})_{k\in\mathbb{N}}$, $j\in\{1,...,d\}$, with the property that $\{(f^{(1)}_{\alpha},...,f^{(d)}_\alpha)\,|\,\alpha\in\mathbb{N}^{(m-1)}\}\subseteq D$. For each $k\in\mathbb{N}$, let \begin{equation}\label{5.2.E_kDefn}
E_k=\bigcap_{\alpha\subseteq\{1,...,k+m-1\},\,|\alpha|=m-1}(E-(f^{(1)}_\alpha,...,f^{(d)}_\alpha)). \end{equation} By \eqref{5.2.IPReccurence}, for each $k\in\mathbb{N}$, $\overline d_{(F_k)}(E_k)>0$. It follows from the case $m=1$, that there exist sequences $$(g_{k,j})_{k\in\mathbb{N}},\,j\in\{1,...,d\}$$ with the properties that (a) for any $k\in\mathbb{N}$, $(g_{k,1},...,g_{k,d})\in E_k$, (b) for any $j\in\{1,...,d\}$, $\lim_{k\rightarrow\infty}g_{k,j}=\infty$ and (c) for any distinct $i,j\in\{1,...,d\}$, $\lim_{k\rightarrow\infty}g_{k,i}-g_{k,j}=\infty$. For each $j\in\{1,...,d\}$ form the sequence $$(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(f_{k,1}^{(j)},...,f_{k,m-1}^{(j)},g_{k,j})=(g_{k,1}^{(j)},...,g_{k,m}^{(j)}).$$
By \eqref{5.2.E_kDefn} and (a), we have that for any $k\in\mathbb{N}$ and any $\alpha\subseteq \{1,...,k-1\}$ with $|\alpha|=m-1$, $(g_{k,1},...,g_{k,d})+(f^{(1)}_\alpha,...,f^{(d)}_\alpha)\in E$ and hence
$$\{(f_{\{k_1,...,k_{m-1}\}}^{(1)}+g_{k_m,1},...,f_{\{k_1,...,k_{m-1}\}}^{(d)}+g_{k_m,d})\,|\,k_1<\cdots<k_{m-1}<k_m\}\subseteq E.$$ By (b) and (c), the sequences $(\textbf g_k^{(1)})_{k\in\mathbb{N}}$,...,$(\textbf g_k^{(d)})_{k\in\mathbb{N}}$ are non-degenerated and essentially distinct. We are done. \end{proof} \begin{cor}\label{5.2.Sigma_ell^*HasDensityOne} Let $(G,+)$ be a countable abelian group and let $d,m\in\mathbb{N}$. Every $\tilde \Sigma_m^*$ set in $G^d$ has uniform density one. \end{cor} \begin{proof} We will assume that $D\subseteq G^d$ does not have uniform density one and show that $D$ is not a $\tilde\Sigma_m^*$ set. Indeed, if $D$ does not have uniform density one, then there exists a F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G^d$ for which $\overline d_{(F_k)}(D)<1$. Let $E=G^d\setminus D$ and note that $\overline d_{(F_k)}(E)>0$. By \cref{5.2.FiniteSumsInPositiveDensitySets}, $E$ contains a $\tilde\Sigma_m$ set. This implies that $D$ is not a $\tilde\Sigma_m^*$. \end{proof}
\subsection{The ubiquity of $\tilde\Sigma_m^*$ sets}
In this section we will show that there exists a broad class of subgroups of $G^d$ with the property that for each group $H$ from this class, any $\tilde\Sigma_m^*$ set in $G^d$ has a large intersection with $H$. In fact, we will show that either a subgroup $H$ belongs to this class or $G^d\setminus H$ is a $\tilde\Sigma_m^*$ set for any $m\in \mathbb{N}$. \begin{defn} Let $(G,+)$ be a countable abelian group, let $d\in\mathbb{N}$ and let $H$ be a subgroup of $G^d$. We say that $H$ is an \textbf{admissible subroup of $G^d$} if there exist non-degenerated and essentially distinct sequences $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(d)})_{k\in\mathbb{N}}$ in $G$ such that
$$\{(g_k^{(1)},...,g_k^{(d)})\,|\,k\in\mathbb{N}\}\subseteq H.$$ \end{defn} \begin{example}
Let $(G,+)$ be a countable abelian group and let $H=\{(g,h,0)\,|\,g,h\in G\}\subseteq G^3$. Clearly, $H$ is not an admissible subgroup of $G^3$. \end{example} \begin{example}
Let $(G,+)$ be a countable abelian group with an element $g$ of infinite order. For any $d\in\mathbb{N}$ and any distinct $a_1,...,a_d\in\mathbb{Z}\setminus\{0\}$, the set $\{(ka_1 g,ka_2 g,...,ka_dg)\,|\,k\in \mathbb{Z}\}$ is an admissible subgroup of $G^d$. \end{example} \begin{example}
Let $(G,+)$ be a countable abelian torsion group (i.e. each of its elements has finite order). There exists a sequence $(g_k)_{k\in \mathbb{N}}$ in $G$ and a nested sequence of finite subgroups $(G_N)_{N\in\mathbb{N}}$ with the properties: (i) $G_N$ is generated by $\{g_1,...,g_N\}$ and (ii) for each $k\in\mathbb{N}$, $g_{k+1}\not\in G_k$. Then for any $d\in\mathbb{N}$ and any distinct $a_1,...,a_d\in\mathbb{N}$, the group generated by the set $\{(g_{a_1k},g_{a_2k},...,g_{a_dk})\,|\,k\in\mathbb{N}\}$ is an admissible subgroup of $G^d$. Indeed, note that for any $k\in\mathbb{N}$ and any $a,b\in\mathbb{N}$ with $a<b$, $g_{ak}\not\in G_{ak-1}$ and $(g_{bk}-g_{ak})\not\in G_{ak}$. So $\lim_{k\rightarrow\infty}g_{ak}=\infty$ and $\lim_{k\rightarrow\infty}(g_{bk}-g_{ak})=\infty$. \end{example} The following proposition provides a useful characterization of admissible subgroups. \begin{prop}\label{5.3.CharacterizationOfAdmissible} Let $(G,+)$ be a countable abelian group, let $d\in\mathbb{N}$ and let $H$ be a subgroup of $G^d$. The following statements are equivalent: \begin{enumerate}[(i)]
\item $H$ is an admissible subgroup of $G^d$.
\item There exist an $m\in\mathbb{N}$ and a $\tilde\Sigma_m$ set $E\subseteq G^d$ such that $E\subseteq H$.
\item For any $m\in\mathbb{N}$, there exists a $\tilde\Sigma_m$ set $E\subseteq G^d$ such that $E\subseteq H$.
\item There exists an \text{\rm{$\tilde{\text{IP}}$}} set $E\subseteq G^d$ such that $E\subseteq H$.
\item For any $j\in\{1,...,d\}$, $\pi_j(H)$ is infinite and for any $i\neq j$, $(\pi_j-\pi_i)(H)$ is also infinite, where for each $j\in\{1,...,d\}$, $\pi_j:H\rightarrow G$ is defined by $\pi_j(g_1,...,g_d)=g_j$. \end{enumerate} \end{prop} \begin{proof} It is not hard to see that (i) and (ii) are equivalent. The implications (i)$\implies$(iii), (iii)$\implies$(iv) and (iv)$\implies$(v) are trivial. We will now prove (v)$\implies$(i).\\
Let $P=\{\pi_j\,|\,j\in\{1,...,d\}\}\cup\{\pi_j-\pi_i\,|\,i,j\in\{1,...,d\},\,i\neq j\}$ and let $M$ be the largest non-negative integer for which there exist an $F\subseteq P$ with $|F|=M$ and a sequence $(\textbf g_k)_{k\in\mathbb{N}}$ in $H$ such that for any $\pi\in F$,
$\lim_{k\rightarrow\infty}\pi(\textbf g_k)=\infty$. Since $|P|=d^2$, we have $M\leq d^2$. Also, since for each $\pi\in P$, $\pi(H)$ is infinite, $M\geq 1$. If $M=d^2$, then (i) holds. So, assume for contradiction that $M<d^2$.\\
By the definition of $M$, there exists a set $F_0\subseteq P$ with $|F_0|=M$ and a sequence $(\textbf g_k)_{k\in\mathbb{N}}$ in $H$ such that if $\pi\in F_0$, $\lim_{k\rightarrow\infty}\pi(\textbf g_k)=\infty$ and if $\pi\in (P\setminus F_0)$, then there exists a finite set $A_\pi\subseteq G$ such that $\{\pi(\textbf g_k)\,|\,k\in\mathbb{N}\}\subseteq A_\pi$. By passing, if needed, to a subsequence, we can assume that for each $\pi\in (P\setminus F_0)$, there exists a $g_\pi\in G$ such that $\lim_{k\rightarrow\infty}\pi(\textbf g_k)=g_\pi$. Let $\pi_0\in (P\setminus F_0)$. By (v), there exists a sequence $(\textbf g'_k)_{k\in\mathbb{N}}$ in $H$ such that $\lim_{k\rightarrow\infty}\pi_0(\textbf g'_k)=\infty$. Note that for any finite set $A\subseteq H$, any $\pi\in F_0$ and any $t\in\mathbb{N}$, there exists a $k\in\mathbb{N}$ such that for any $k'>k$, $$\pi(\textbf g_{k'}+\textbf g'_t)=\pi(\textbf g_{k'})+\pi(\textbf g'_t)\not\in A.$$ Also, note that there exists a $k_0\in\mathbb{N}$ such that for any $k>k_0$, $\pi_0(\textbf g_k)=g_{\pi_0}$. It follows that we can find an increasing sequence $(k_t)_{t\in\mathbb{N}}$ in $\mathbb{N}$ for which $\lim_{t\rightarrow\infty}\pi(\textbf g_{k_t}+\textbf g'_t)=\infty$ for each $\pi\in F_0\cup\{\pi_0\}$. This contradicts the definition of $M$, completing the proof. \end{proof} \begin{cor} Let $(G,+)$ be a countable abelian group and let $d\in\mathbb{N}$. A subgroup $H$ of $G^d$ is either admissible or for any $m\in\mathbb{N}$, $G^d\setminus H$ is a $\tilde\Sigma_m^*$ set. \end{cor} \begin{proof} If $H$ is not an admissible subgroup, \cref{5.3.CharacterizationOfAdmissible}, (ii), implies that for each $m\in\mathbb{N}$, $H$ does not contain any $\tilde\Sigma_m$ set in $G^d$. Thus, $G^d\setminus H$ is a $\tilde\Sigma_m^*$ set for each $m\in\mathbb{N}$. \end{proof} Before stating and proving one of the main results of this subsection which deals with the ubiquity of $\tilde\Sigma_m^*$ sets in admissible subgroups (\cref{5.3.SigmaOnAdmisibleSubgroups} below), we need one more definition and a technical lemma. \begin{defn}\label{5.3.DefnSigma_m*InH} Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$ and let $H\subseteq G^d$ be an admissible subgroup. A set $E\subseteq H$ is called an \rm{$H$-}$\tilde\Sigma_m^*$ set if it has a non-trivial intersection with every $\tilde\Sigma_m$ set contained in $H$. Similarly, a set $E\subseteq H$ is called an \rm{$H$-}\text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set if it has a non-trivial intersection with every \text{\rm{$\tilde{\text{IP}}$}} set contained in $H$. \end{defn} \begin{rem} Let $(G,+)$ be a countable abelian group, let $d\in\mathbb{N}$ and let $H\subseteq G^d$ be an admissible subgroup of $G^d$. It is useful to percieve $H$-$\tilde\Sigma_m^*$ sets as relative versions of $\tilde\Sigma_m^*$ sets in $G^d$. Note that if $H$ is a proper subgroup of $G^d$, $H$-$\tilde\Sigma_m^*$ sets are not $\tilde\Sigma_m^*$. Indeed, since for each $m\in\mathbb{N}$, any translation of a $\tilde\Sigma_m$ set in $G^d$ is again a $\tilde\Sigma_m$ set, every coset of $H$ contains a $\tilde\Sigma_m$ set in $G^d$. It follows that $G^d\setminus H$ contains a $\tilde\Sigma_m$ set for each $m\in\mathbb{N}$. Hence, no \rm{$H$-}$\tilde\Sigma_m^*$ set is a $\tilde\Sigma_m^*$ set. \end{rem} \begin{rem}\label{5.3.SigmaOnLateralClasses} Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$, let $H\subseteq G^d$ be an admissible subgroup and let $E$ be a $\tilde\Sigma_m^*$ set in $G^d$. It follows from the definition that $E\cap H$ is a \rm{$H$-}$\tilde\Sigma_m^*$ set. Indeed, let $D\subseteq H$ be a $\tilde\Sigma_m$ set. We have $(E\cap H)\cap D=E\cap D\neq \emptyset$. Note also that for any $\textbf g\in G^d$, $E\cap (\textbf g+H)$ is the translation of the \rm{$H$-}$\tilde\Sigma_m^*$ set $(-\textbf g+E)\cap H$. Thus, the cosets of $H$ have a large intersection with $E$ as well. \end{rem} \begin{lem}\label{5.3.FiniteSumsInPositiveDensitySetsInH} Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$, let $H$ be an admissible subgroup of $G^d$ and let $(F_k)_{k\in\mathbb{N}}$ be a F{\o}lner sequence in $H$. Any $E\subseteq H$ with $\overline d_{(F_k)}(E)>0$ contains a $\tilde\Sigma_m$ set. \end{lem} \begin{proof} Since $H$ is admissible, there exists an \text{\rm{$\tilde{\text{IP}}$}} set $D'\subseteq H$. The result in question follows by replacing $D$ by $D'$ in the proof of \cref{5.2.FiniteSumsInPositiveDensitySets} and applying an adequate modification of \cref{5.2.IPPoincare}. \end{proof} \begin{thm}\label{5.3.SigmaOnAdmisibleSubgroups} Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$ and let $H\subseteq G^d$ be an admissible subgroup. Any \rm{$H$-}$\tilde\Sigma_m^*$ set is an \rm{$H$-}\text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set and has uniform density one in $H$. \end{thm} \begin{proof} Let $E'\subseteq H$ be an \rm{$H$-}$\tilde\Sigma_m^*$ set. By \cref{5.1.SigmaInEveryIP}, every \text{\rm{$\tilde{\text{IP}}$}} set contains a $\tilde\Sigma_m$ set. It follows that $E'$ is an \rm{$H$-}\text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set. By \cref{5.3.FiniteSumsInPositiveDensitySetsInH}, we can argue as in the proof of \cref{5.2.Sigma_ell^*HasDensityOne} to show that $E'$ has uniform density one in $H$. \end{proof} \begin{cor} Let $(G,+)$ be a countable abelian group, let $d\in \mathbb{N}$, let $H$ be an admissible subgroup of $G^d$ and let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a strongly mixing system. For any $\textbf g\in G^d$, each set of the form $R_\epsilon(A_0,...,A_\ell)\cap(\textbf g+ H)$ is the translation of a set with uniform density one in $H$. \end{cor} \begin{proof} This result follows from \cref{3.MainResult}, \Cref{5.3.SigmaOnLateralClasses} and \cref{5.3.SigmaOnAdmisibleSubgroups}. \end{proof}
A natural class of admissible subgroups in $G^d$ is provided by the one-parameter subgroups of the form $$H_{\phi_1,...,\phi_d}=\{(\phi_1(g),...,\phi_d(g))\,|\,g\in G\},$$ where $\phi_1,...,\phi_d:G\rightarrow G$ are homomorphisms such that for any $j\in\{1,...,d\}$, $|\ker(\phi_j)|<\infty$ and for any $i\neq j$, $|\ker(\phi_j-\phi_i)|<\infty$. The following proposition, alluded to in \Cref{4.CombinatorialRemark}, involves preimages of sets in $G^d$ via the elements of $H_{\phi_1,...,\phi_d}$ and provides an alternative proof of \cref{4.InjectiveDiagonalResult}. \begin{prop}\label{5.3.UsefulInSection4}
Let $(G,+)$ be a countable abelian group, let $d,m\in\mathbb{N}$ and let $\phi_1,...,\phi_d:G\rightarrow G$ be homomorphisms such that for any $j\in\{1,...,d\}$, $\ker(\phi_j)$ is finite and for any $i\neq j$, $\ker(\phi_j-\phi_i)$ is also finite. If $E\subseteq G^d$ is a $\tilde\Sigma_m^*$ set, then $E'=\{g\in G\,|\,(\phi_1(g),...,\phi_d(g))\in E\}$ is a $\Sigma_m^*$ set in $G$. \end{prop} \begin{proof}
Let $D\subseteq G$ be the $\Sigma_m$ set in $G$ generated by the non-degenerated sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,m})_{k\in\mathbb{N}}$ in $G^m$ (i.e. $D=\{g_\alpha\,|\,\alpha\in\mathbb{N}^{(m)}\}$). We will show that $D\cap E'\neq\emptyset$.\\
By our assumption on $\phi_1,...,\phi_d$, for each $j\in\{1,...,m\}$, the sequences $(\phi_1(g_{k,j}))_{k\in\mathbb{N}}$,....,$(\phi_d(g_{k,j}))_{k\in\mathbb{N}}$ are non-degenerated and essentially distinct. Thus, the set $D'=\{(\phi_1(g_\alpha),...,\phi_d(g_\alpha))\,|\,\alpha\in\mathbb{N}^{(m)}\}$ is a $\tilde\Sigma_m$ set in $G^d$. Noting that $D'\cap E\neq\emptyset$, we obtain $D\cap E'\neq\emptyset$. \end{proof}
So far we have been focusing on the massivity and ubiquity of \textit{general} $\tilde\Sigma_\ell^*$ sets. However the "dynamical" $\tilde\Sigma_\ell^*$ sets $R_\epsilon(A_0,...,A_\ell)$, are even more prevalent in $G^\ell$. For example, assuming for convenience that $G=\mathbb{Z}$, one can show that the sets of the form $R_\epsilon(A_0,...,A_\ell)$ have an ample presence in "polynomial" subsets of $\mathbb{Z}^\ell$. This is illustrated by the following polynomial extension of \cref{4.ZExample} (which is proved in a companion paper \cite{BerZel-StronglyMixingPET}). \begin{thm}\label{5.3.StronglyMixingPet} Let $\ell\in\mathbb{N}$ and let $p_1,...,p_\ell\in \mathbb{Z}[x]$ be non-constant polynomials such that for any distinct $i,j\in\{1,...,\ell\}$, $\deg(p_j-p_i)>0$. There exists an $m\in\mathbb{N}$ such that for any strongly mixing system $(X,\mathcal A,\mu, T)$, any $\epsilon>0$ and any $A_0,...,A_\ell\in\mathcal A$, the set \begin{equation}\label{5.3.SetOfPolyReturns}
R_\epsilon^{p_1,...,p_\ell}(A_0,...,A_\ell)=\{n\in\mathbb{Z}\,|\,|\mu(A_0\cap T^{p_1(n)}A_1\cap\cdots\cap T^{p_\ell(n)}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|<\epsilon\} \end{equation} is $\Sigma_m^*$. \end{thm}
The following proposition shows that, in general, $\tilde\Sigma_\ell^*$ sets, unlike the sets of the form $R_\epsilon(A_0,...,A_\ell)$, can be disjoint from the polynomial sets $H_{p_1,...,p_\ell}=\{(p_1(n),...,p_\ell(n))\,|\,n\in\mathbb{Z}\}$, where $p_1,...,p_\ell\in\mathbb{Z}[x]$. \begin{prop}\label{5.3.PolynomialPathsAreSmall} Let $\ell\in\mathbb{N}$ and let $p_1,...,p_\ell\in \mathbb{Z}[x]$ be non-constant polynomials such that for any distinct $i,j\in\{1,...,\ell\}$, $\deg(p_j-p_i)>0$. Suppose that $\deg(p_1)>1$. Then, for any $m\geq 2$, $H_{p_1,...,p_\ell}$ contains no $\tilde\Sigma_m$ sets. Equivalently, $\mathbb{Z}^\ell\setminus H_{p_1,...,p_\ell}$ is a $\tilde\Sigma_m^*$ set for each $m\geq 2$. \end{prop} \begin{proof}
Since the projection onto the first coordinate of any $\tilde\Sigma_m$ set $E\subseteq \mathbb{Z}^\ell$ is a $\Sigma_m$ set in $\mathbb{Z}$, it suffices to show that the set $\{p_1(n)\,|\,n\in\mathbb{Z}\}$ contains no $\Sigma_m$ sets. Suppose for contradiction that $\{p_1(n)\,|\,n\in\mathbb{Z}\}$ contains a $\Sigma_m$ set
$$D=\{n_{k_1}^{(1)}+\cdots+n_{k_m}^{(m)}\,|\,k_1<\cdots<k_m\},$$ where $(n_k^{(1)})_{k\in\mathbb{N}}$,...,$(n_k^{(m)})_{k\in\mathbb{N}}$ are non-degenerated sequences in $\mathbb{Z}$.\\
Choose $t_1,t_2,t_3\in\mathbb{N}$ to be such that $n^{(1)}_{t_1}<n^{(1)}_{t_2}<n^{(1)}_{t_3}$ and let $$I=\{n_{t_1}^{(1)}+n_{k_2}^{(2)}+\cdots+n_{k_m}^{(m)}\,|\,\max\{t_1,t_2,t_3\}<k_2<\cdots<k_m\}.$$ Clearly $I$ is an infinite subset of $D$. So, letting $a=n_{t_2}^{(1)}-n_{t_1}^{(1)}$ and $b=n_{t_3}^{(1)}-n_{t_1}^{(1)}$, we have $a+I\subseteq D$ and $b+I\subseteq D$.\\
Let $(n_k)_{k\in\mathbb{N}}$ be an enumeration of the elements of $I$. One can find an increasing sequence $(k_j)_{j\in\mathbb{N}}$ for which at least two of the sets $\{n_{k_j}\,|\,j\in\mathbb{N}\}$, $\{a+n_{k_j}\,|\,j\in\mathbb{N}\}$ and $\{b+n_{k_j}\,|\,j\in\mathbb{N}\}$ are contained in at least one of the sets $\{p_1(n)\,|\,n\in\mathbb{N}\}$ and $\{p_1(-n)\,|\,n\in\mathbb{N}\}$. We will assume that $\{a+n_{k_j}\,|\,j\in\mathbb{N}\}$ and $\{b+n_{k_j}\,|\,j\in\mathbb{N}\}$ are contained in $\{p_1(n)\,|\,n\in \mathbb{N}\}$ (the other cases can be handled similarly). It follows that there exist infinitely many pairs $(n,m)\in\mathbb{N}\times \mathbb{N}$ such that $p_1(n)-p_1(m)=b-a$. Since $b>a$, this contradicts the fact that $\deg(p_1)>1$. \end{proof}
\section{Multiple recurrence for mildly and weakly mixing systems via $\mathcal R$-limits} As we saw above, $\mathcal R$-limits are adequate for characterizing strong mixing and obtaining higher order mixing properties. In this section, we will show that $\mathcal R$-limits can be also useful in dealing with mildly and weakly mixing systems. In particular, we will obtain analogues of \cref{3.MainResult} for midly and weakly mixing systems.
\subsection{Mildly mixing systems}
In this subsection we will deal with mildly mixing systems (introduced in \Cref{6.1.MildlyMixingDefn} below) from the perspective of $\mathcal R$-limits. The notion of mild mixing has multiple equivalent forms (see \cite{walters1972someMildMixing}, \cite{walters1982MildMixingEquivalence} and \cite{FurWeissMildMixing}) and plays a fundamental role in IP ergodic theory, including various refinements of the classical Szemer{\'e}di theorem (see \cite{berMcCuIPPolySzemeredi} and \cite{FKIPSzemerediLong}). The multiple recurrence theorems for mildly mixing systems (see \cite{FBook} and \cite{FKIPSzemerediLong}) utilize the notion of IP-limit which we will presently define. We will then establish a connection between IP-limits and $\mathcal R$-limits and, finally, prove an analogue of \cref{3.MainResult} for mildly mixing actions. \begin{defn} (Cf. \cite[Definitions 1.1 and 1.3]{FKIPSzemerediLong}) Let $(X,d)$ be a compact metric space and let $(x_\alpha)_{\alpha\in\mathcal F}$ be an $\mathcal F$-sequence in $X$. A set $\mathcal F^{(1)}\subseteq \mathcal F$ is an \text{\rm{IP}}-ring if there exists a sequence $(\alpha_k)_{k\in\mathbb{N}}$ in $\mathcal F$ with $\alpha_k<\alpha_{k+1}$ for each $k\in\mathbb{N}$ for which
$$\mathcal F^{(1)}=\{\bigcup_{j\in\alpha}\alpha_j\,|\,\alpha\in\mathcal F\}.$$ For any \text{\rm{IP}}-ring $\mathcal F^{(1)}$, we write $$\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}x_\alpha=x$$ if for every $\epsilon>0$, there exists an $\alpha_0\in\mathcal F^{(1)}$ such that for any $\alpha\in\mathcal F^{(1)}$ with $\alpha>\alpha_0$, $$d(x_\alpha,x)<\epsilon.$$ \end{defn} It follows from a result of Hindman \cite{HIPPartitionRegular} that if $(x_\alpha)_{\alpha\in\mathcal F}$ is an $\mathcal F$-sequence in a compact metric space $X$, then for any \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$ one can always find an $x\in X$ and an \text{\rm{IP}}-ring $\mathcal F^{(2)}\subseteq \mathcal F^{(1)}$ such that \begin{equation}\label{6.1.IPAlwaysExists} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(2)}}x_\alpha=x \end{equation} (see \cite[Theorem 8.14]{FBook}). In particular, for any countable abelian group $(G,+)$, any sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ and any probability measure preserving system $(X,\mathcal A,\mu, (T_g)_{g\in G})$, there exists an \text{\rm{IP}}-ring $\mathcal F^{(1)}$ for which $$\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}T_{g_\alpha}$$ exists in the weak operator topology of $L^2(\mu)$. This implies (and is equivalent to) the fact that for any $A_0,A_1\in\mathcal A$, $$\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{g_\alpha} A_1)$$ exists. \begin{thm}\label{6.1.EquivalenceOfConvergenceIP} Let $(X,d)$ be a compact metric space, let $(G,+)$ be a countable abelian group, let $(x_g)_{g\in G}$ be a sequence in $X$, let $x_0\in X$ and let $(g_k)_{k\in\mathbb{N}}$ be a sequence in $G$. The following statements are equivalent: \begin{enumerate}[(i)]
\item For any \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$ for which $\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}x_{g_\alpha}$ exists, one has
\begin{equation}\label{6.1.IPLimitForSubRing}
\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}x_{g_\alpha}=x_0.
\end{equation}
\item For any \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$ there exist an $m\in\mathbb{N}$ and a sequence $(h_{k,1},...,h_{k,m})_{k\in\mathbb{N}}$ in $G^m$ such that $\{h_\alpha\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq \{g_\alpha\,|\,\alpha\in\mathcal F^{(1)}\}$ and
\begin{equation}\label{6.1.RLimitForSubSequence}
\rlim{\alpha\in\mathbb{N}^{(m)}}x_{h_\alpha}=x_0.
\end{equation} \end{enumerate} \end{thm} \begin{proof}
(i)$\implies$(ii): Let $\mathcal F^{(1)}$ be an \text{\rm{IP}}-ring. Since $X$ is compact, we can assume (by passing, if needed, to a sub \text{\rm{IP}}-ring) that $\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}x_{g_\alpha}$ exists. Thus, by (i), \eqref{6.1.IPLimitForSubRing} holds. It follows from the definition of an IP-limit that there exists a sequence $(h_k)_{k\in\mathbb{N}}$ in $G$ such that $\{h_k\,|\,k\in\mathbb{N}\}\subseteq \{g_\alpha\,|\,\alpha\in\mathcal F^{(1)}\}$ and $\lim_{k\rightarrow\infty}x_{h_k}=x_0$. This completes the proof of (i)$\implies$(ii).\\
(ii)$\implies$(i): Let $\mathcal F^{(1)}$ be an \text{\rm{IP}}-ring for which $\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}x_{g_\alpha}=y$ for some $y\in X$. Suppose for contradiction that there exists an $\epsilon>0$ for which $d(y,x_0)>\epsilon$. By the definition of an IP-limit, there exists $\alpha_0\in\mathcal F$ such that for any $\alpha\in\mathcal F^{(1)}$ with $\alpha>\alpha_0$, $d(x_{g_\alpha},x_0)>\epsilon$. Since $\{\alpha\in\mathcal F^{(1)}\,|\,\alpha>\alpha_0\}$ is an \text{\rm{IP}}-ring, it follows from (ii) that there exist an $m\in\mathbb{N}$ and a sequence $(h_{k,1},...,h_{k,m})_{k\in\mathbb{N}}$ in $G^m$ such that $\{h_\alpha\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq \{g_\alpha\,|\,\alpha\in\mathcal F^{(1)}\text{ and }\alpha>\alpha_0\}$ and $\rlim{\alpha\in\mathbb{N}^{(m)}}x_{h_\alpha}=x_0$. In particular, there exists an $h\in \{g_\alpha\,|\,\alpha\in\mathcal F^{(1)}\text{ and }\alpha>\alpha_0\}$ for which $d(x_h,x_0)<\epsilon$, a contradiction. \end{proof} \begin{rem}
\cref{6.1.EquivalenceOfConvergenceIP} shows that IP-limits can be attained via $\mathcal R$-limits. The following example demonstrates that this is not the case the other way around. Let $G=\mathbb{Z}$, let $X=\{0,1\}$, let $m\in\mathbb{N}$, and consider the $\Sigma_m$ set $E=\{3^{k_1}+\cdots+3^{k_m}\,|\,k_1<\cdots<k_m\}$. The set $E$ is comprised of all the elements of $3\mathbb{N}$ whose base 3 expansion has exactly $m$ non-zero entries, all of which are 1. It follows that there are no $a,b,c\in E$ for which $a+b=c$. This, in turn, implies that $E$ contains no IP sets and hence $\mathbb{Z}\setminus E$ is an IP$^*$ set. Let $(n_k)_{k\in\mathbb{N}}$ be a sequence in $\mathbb{Z}$ and let $\mathcal F^{(1)}\subseteq\mathcal F$ be an \text{\rm{IP}}-ring for which $\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mathbbm 1_E(n_\alpha)$ exists. Since $0\not\in E$ and $\mathbb{Z}\setminus E$ is IP$^*$, one has $\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mathbbm 1_E(n_\alpha)=0$. On the other hand, since for any $k_1<\cdots<k_m$, $\mathbbm 1_E(3^{k_1}+\cdots+3^{k_m})=1$, one has that for any infinite set $S\subseteq \mathbb{N}$, $$\rlim{\{k_1,...,k_m\}\in S^{(m)}}\mathbbm 1_E(3^{k_1}+\cdots+3^{k_m})=1.$$ \end{rem} \begin{defn}\label{6.1.MildlyMixingDefn} Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. $(T_g)_{g\in G}$ is mildly mixing if for any sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ for which $\lim_{\alpha\rightarrow\infty}g_\alpha=\infty$, there exists an \text{\rm{IP}}-ring $\mathcal F^{(1)}$ such that for any $f\in L^2(\mu)$, \begin{equation}\label{6.1.IPLimitMixing} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}T_{g_\alpha}f=\int_Xf\text{d}\mu \end{equation} weakly. \end{defn} We are now ready to state and prove the main theorem of this subsection. It can be viewed as an analogue of \cref{3.MainResult} for mildly mixing actions. We remind the reader that a sequence of measure preserving transformations $(T_k)_{k\in\mathbb{N}}$ of a probability space $(X,\mathcal A,\mu)$ has the mixing property if for every $A_0,A_1\in\mathcal A$, $\lim_{k\rightarrow\infty}\mu(A_0\cap T_k^{-1}A_1)=\mu(A_0)\mu(A_1)$. \begin{thm}\label{6.1.GlobalMildlyMixing} Let $\ell\in\mathbb{N}$, let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is mildly mixing.
\item For any \text{\rm{$\tilde{\text{IP}}$}} set $E\subseteq G^\ell$ and any $m\in\mathbb{N}$, there exist non-degenerated and essentially distinct sequences $(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g_{k,1}^{(j)},...,g_{k,m}^{(j)})_{k\in\mathbb{N}}$, $j\in\{1,...,\ell\}$,
in $G^m$ with the properties:
\begin{enumerate}[(a)]
\item $\{(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$.
\item For any $t\in\{1,...,m\}$ and any $j\in\{1,...,\ell\}$, $(T_{g_{k,t}^{(j)}})_{k\in\mathbb{N}}$ has the mixing property.
\item For any $t$ and any $i\neq j$, $(T_{g_{k,t}^{(j)}-g_{k,t}^{(i)}})_{k\in\mathbb{N}}$ has the mixing property.
\end{enumerate}
\item For any \text{\rm{$\tilde{\text{IP}}$}} set $E\subseteq G^\ell$, there exist an $m\in\mathbb{N}$ and non-degenerated and essentially distinct sequences $(\textbf g_k^{(1)})_{k\in\mathbb{N}},...,(\textbf g_k^{(\ell)})_{k\in\mathbb{N}}$ in $G^m$ with $\{(g_\alpha^{(1)},...,g_{\alpha}^{(\ell)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$ and such that for any $A_0,...,A_\ell\in\mathcal A$,
\begin{equation}\label{6.1.RMixingInMain}
\rlim{\alpha\in \mathbb{N}^{(m)}}\mu(A_0\cap T_{g^{(1)}_\alpha} A_1\cap\cdots\cap T_{g^{(\ell)}_\alpha}A_\ell)=\prod_{j=0}^\ell\mu(A_j).
\end{equation}
\item Given sequences $(g^{(1)}_k)_{k\in\mathbb{N}}$,...,$(g^{(\ell)}_k)_{k\in\mathbb{N}}$ in $G$ such
that for any $j\in\{1,...,\ell\}$, $\lim_{\alpha\rightarrow\infty}g^{(j)}_\alpha=\infty$
and for any $i\neq j$,
$\lim_{\alpha\rightarrow\infty}g_\alpha^{(j)}-g_\alpha^{(i)}=\infty$ (and so
$E=\{(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\,|\,\alpha\in\mathcal F\}$
is an \text{\rm{$\tilde{\text{IP}}$}} set), there exists an \text{\rm{IP}}-ring $\mathcal F^{(1)}$ such that for any $A_0,...,A_\ell \in\mathcal A$,
\begin{equation}\label{6.1.IPMultiCorrelation}
\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{g_\alpha^{(1)}}A_1\cap\cdots\cap T_{g_\alpha^{(\ell)}}A_\ell)=\prod_{j=1}^\ell \mu(A_j).
\end{equation}
\item For any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set
$$R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}$$
is an \text{\rm{$\tilde{\text{IP}}\rm{^*}$}} set. \end{enumerate} \end{thm} \begin{proof}
(i)$\implies$(ii): Let $m\in\mathbb{N}$, let $E\subseteq G^\ell$ be an \text{\rm{$\tilde{\text{IP}}$}} set and let the sequences $(h_k^{(1)})_{k\in\mathbb{N}}$,...,$(h_k^{(\ell)})_{k\in\mathbb{N}}$ in $G$ be such that $E=\{(h_\alpha^{(1)},...,h_\alpha^{(\ell)})\,|\,\alpha\in\mathcal F\}$. By the stipulation made in \Cref{5.1.CommonSenseSequence}, for any \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$, the set $\{(h^{(1)}_\alpha,...,h^{(\ell)}_\alpha)\,|\,\alpha\in\mathcal F^{(1)}\}$ is again an \text{\rm{$\tilde{\text{IP}}$}} set. Pick $\mathcal F^{(1)}$ to be an \text{\rm{IP}}-ring such that for any $A_0,A_1\in\mathcal A$ and any $i,j\in\{1,...,\ell\}$, \begin{equation}\label{6.1.IPExpressions1} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{h^{(j)}_\alpha}A_1)\text{ and if }i\neq j,\,\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{h^{(j)}_\alpha-h^{(i)}_\alpha}A_1) \end{equation} exist. Let $(\alpha_k)_{k\in\mathbb{N}}$ be the sequence in $\mathcal F$ generating $\mathcal F^{(1)}$ (so, in particular, $\alpha_k<\alpha_{k+1}$ for each $k\in\mathbb{N}$). It follows from (i) that each of the limits appearing in \eqref{6.1.IPExpressions1} equals $\mu(A_0)\mu(A_1)$ (otherwise, we would have a contradiction with formula \eqref{6.1.IPLimitMixing}). Thus, for any $A_0,A_1\in\mathcal A$ and any $i,j\in\{1,...,\ell\}$, $$\lim_{k\rightarrow\infty}\mu(A_0\cap T_{h^{(j)}_{\alpha_k}}A_1)=\mu(A_0)\mu(A_1)\text{ and if }i\neq j,\,\lim_{k\rightarrow\infty}\mu(A_0\cap T_{h^{(j)}_{\alpha_k}-h^{(i)}_{\alpha_k}}A_1)=\mu(A_0)\mu(A_1).$$ For each $j\in\{1,...,\ell\}$, let $(\textbf g^{(j)}_k)_{k\in\mathbb{N}}=(\underbrace{h_{\alpha_k}^{(j)},...,h_{\alpha_k}^{(j)})}_{m\text{ times}}$. It is now easy to check that the sequences $(\textbf g^{(1)}_k)_{k\in\mathbb{N}}$,...,$(\textbf g^{(\ell)}_k)_{k\in\mathbb{N}}$ are non-degenerated, essentially distinct, and satisfy (a)-(c), completing the proof of (i)$\implies$(ii).\\ (ii)$\implies$(iii): This follows from \cref{2.MainResult}.\\ (iii)$\implies$(iv): We will prove (iv) by applying \cref{6.1.EquivalenceOfConvergenceIP} to the $G^\ell$-sequence $$x_{(g_1,...,g_\ell)}=\mu(A_0\cap T_{g_1}A_1\cap\cdots\cap T_{g_\ell}A_\ell),\,(g_1,...,g_\ell)\in G^\ell$$ and the sequence $(g_k^{(1)},...,g_k^{(\ell)})_{k\in\mathbb{N}}$ in $G^\ell$.\\
Note that for any IP-ring $\mathcal F^{(2)}$, $\{(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\,|\,\alpha\in\mathcal F^{(2)}\}$ is an \text{\rm{$\tilde{\text{IP}}$}} set. By (iii), there exist an $m\in\mathbb{N}$ and non-degenerated and essentially distinct sequences $(\textbf h_k^{(1)})_{k\in\mathbb{N}}$,...,$(\textbf h_k^{(\ell)})_{k\in\mathbb{N}}$ in $G^m$ with
$$\{(h_\alpha^{(1)},...,h_\alpha^{(\ell)})\,|\,\alpha\in\mathcal \mathbb{N}^{(m)}\}\subseteq \{(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\,|\,\alpha\in\mathcal F^{(2)}\}$$ for which \eqref{6.1.RMixingInMain} holds. Letting $\mathcal F^{(1)}$ be an \text{\rm{IP}}-ring for which the left-hand side of \eqref{6.1.IPMultiCorrelation} exists for any $A_0,...,A_\ell\in\mathcal A$, we obtain by \cref{6.1.EquivalenceOfConvergenceIP} that \eqref{6.1.IPMultiCorrelation} holds.\\ (iv)$\implies$(v): This implication follows from the definition of \text{\rm{$\tilde{\text{IP}}\rm{^*}$}}.\\ (v)$\implies$(i): Let $(g_k)_{k\in\mathbb{N}}$ be a sequence in $G$ with the property that $\lim_{\alpha\rightarrow\infty}g_\alpha=\infty$. It suffices to show that for some \text{\rm{IP}}-ring $\mathcal F^{(1)}$ and any $A_0,A_1\in\mathcal A$, $$\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{g_\alpha}A_1)=\mu(A_0)\mu(A_1).$$ By \eqref{6.1.IPAlwaysExists}, there exists an \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$ such that for any $A_0,A_1\in\mathcal A$, \begin{equation}\label{6.1.MildMixingExitenceOfIPLimit} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{g_\alpha}A_1) \end{equation} exists. Let $(\gamma_k)_{k\in\mathbb{N}}$ be a sequence in $\mathcal F^{(1)}$ with $\gamma_k<\gamma_{k+1}$ for each $k\in\mathbb{N}$ and such that the sequences $(h_k^{(j)})_{k\in\mathbb{N}}=(g_{\gamma_{j+\ell k}})_{k\in\mathbb{N}}$, $j\in\{1,...,\ell\}$, in $G$ satisfy (a) for any $j\in\{1,...,\ell\}$, $\lim_{\alpha\rightarrow\infty}h_\alpha^{(j)}=\infty$ and (b) for any $i\neq j$, $\lim_{\alpha\rightarrow\infty}h_\alpha^{(j)}-h_\alpha^{(i)}=\infty$. For each $\alpha_0\in\mathcal F$, let
$$E_{\alpha_0}=\{(h^{(1)}_\alpha,...,h^{(\ell)}_{\alpha})\,|\,\alpha\in\mathcal F\text{ and }\alpha>\alpha_0\}.$$ Since $E_{\alpha_0}$ is an \text{\rm{$\tilde{\text{IP}}$}} set, (v) implies that for any $\alpha_0\in\mathcal F$, any $A_0,A_1\in\mathcal A$ and any $\epsilon>0$, $$E_{\alpha_0}\cap R_\epsilon(A_0,A_1,X,...,X)\neq\emptyset.$$ Thus, for any $\alpha_0\in\mathcal F$, there exists an $\alpha>\alpha_0$ such that $h^{(1)}_\alpha\in R_\epsilon(A_0,A_1)$. Note that $$\lim_{\alpha\rightarrow\infty}\min(\bigcup_{k\in\alpha}\gamma_{1+\ell k})=\infty.$$ It follows that for any $\beta_0\in\mathcal F$, there is an $\alpha\in\mathcal F$ such that $h_\alpha^{(1)}\in R_\epsilon(A_0,A_1)$ and such that $\beta=\bigcup_{k\in\alpha}\gamma_{1+\ell k}\in\mathcal F^{(1)}$ satisfies $\beta>\beta_0$. But $g_\beta=g_{(\bigcup_{k\in\alpha}\gamma_{1+\ell k})}=h_\alpha^{(1)}$, so
$$|\mu(A_0\cap T_{g_\beta}A_1)-\mu(A_0)\mu(A_1)|<\epsilon.$$ Since $\epsilon$ was arbitrary, for any $A_0,A_1\in\mathcal A$, $$\mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F^{(1)}}\mu(A_0\cap T_{g_\alpha}A_1)=\mu(A_0)\mu(A_1),$$ which completes the proof. \end{proof} \begin{rem} We saw in Section 4 that the versatility of $\mathcal R$-limits allows one to obtain from the \textit{multiparameter} \cref{3.MainResult} some interesting results of \textit{diagonal} nature. Similarly, one can obtain diagonal results from \cref{6.1.GlobalMildlyMixing}. For example, let $G=\mathbb{Z}$ and assume that $(X,\mathcal A,\mu,T)$ is a mildly mixing system. Then, by \cref{6.1.GlobalMildlyMixing}, (iv), for any strictly increasing sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{Z}$, any non-zero and distinct integers $a_1,...,a_\ell$ and any \text{\rm{IP}}-ring $\mathcal F^{(1)}\subseteq \mathcal F$, there exists an \text{\rm{IP}}-ring $\mathcal F^{(2)}\subseteq \mathcal F^{(1)}$ such that for any $A_0,...,A_\ell\in\mathcal A$, \begin{equation}\label{6.1.SzemerediExpresion} \mathop{\textit{IP-lim}}_{\alpha\in\mathcal F^{(2)}}\mu(A_0\cap T^{a_1n_\alpha}A_1\cap \cdots\cap T^{a_\ell n_\alpha}A_\ell)=\prod_{j=0}^\ell \mu(A_j). \end{equation} (Cf. \cite[Theorem 9.27]{FBook} and \cite[Theorem 5.4]{FKIPSzemerediLong}.) \end{rem}
\subsection{Weakly mixing systems} This subsection is devoted to weakly mixing systems (which were introduced in Subsection 5.2) and has a similar structure to that of Subsection 6.1. We will first establish a technical lemma which connects $\mathcal R$-limits with C{\'e}saro convergence. We will then prove an analogue of \cref{3.MainResult} (see \cref{6.2.GlobalWeaklyMixing} below) for weakly mixing systems and derive a corollary which has diagonal nature. \begin{lem}\label{6.2.LimitEquivalenceWM} Let $(G,+)$ be a countable abelian group, let $(X,d)$ be a compact metric space, let $(x_g)_{g\in G}$ be a sequence in $X$, let $x_0\in X$, let $(F_k)_{k\in\mathbb{N}}$ be a F{\o}lner sequence in $G$ and let $E\subseteq G$ be such that $\overline d_{(F_k)}(E)>0$. The following statements are equivalent: \begin{enumerate}[(i)]
\item
\begin{equation}\label{6.2.DensityLimit}
\lim_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}\mathbbm 1_E(g)d(x_g,x_0)=0.
\end{equation}
\item For any $D\subseteq E$ with $\overline d_{(F_k)}(D)>0$, there exist an $m\in\mathbb{N}$ and a sequence $(g_{k,1},...,g_{k,m})_{k\in\mathbb{N}}$ in $G^m$ for which $\{g_\alpha\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq D$ and
\begin{equation}\label{6.2.RlimitDensityResult}
\rlim{\alpha\in\mathbb{N}^{(m)}}x_{g_\alpha}=x_0.
\end{equation} \end{enumerate} \end{lem} \begin{proof} (i)$\implies$(ii): Let $D\subseteq E$ be such that $\overline d_{(F_k)}(D)>0$. It follows from \eqref{6.2.DensityLimit} that
$$\lim_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}\mathbbm 1_D(g)d(x_g,x_0)=0.$$ Let $\epsilon>0$. There exist infinitely many $g\in D$ such that $d(x_g,x_0)<\epsilon$ (otherwise, we would have
$\limsup_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}\mathbbm 1_D(g)d(x_g,x_0)>0$). Thus, for each $k\in\mathbb{N}$, there is a $g_k\in D$ with $d(x_{g_k},x_0)<\frac{1}{k}$. It follows now that $$\rlim{\{k\}\in\mathbb{N}^{(1)}}x_{g_{\{k\}}}=\lim_{k\rightarrow\infty}x_{g_k}=x_0.$$ (ii)$\implies$(i): It suffices to show that for any given $\epsilon>0$, $\overline d_{(F_k)}(D_\epsilon)=0$, where
$$D_\epsilon=\{g\in E\,|\,d(x_g,x_0)>\epsilon\}.$$
(This will imply that for each $\epsilon>0$, $$\limsup_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}\mathbbm 1_E(g)d(x_g,x_0)\leq\limsup_{k\rightarrow\infty}(\frac{1}{|F_k|}\sum_{g\in F_k}\epsilon\mathbbm 1_{E\setminus D_\epsilon}(g)+\frac{1}{|F_k|}\sum_{g\in F_k}\mathbbm 1_{D_\epsilon}(g)d(x_g,x_0))\leq \epsilon.)$$ Fix $\epsilon>0$ and suppose for contradiction that $\overline d_{(F_k)}(D_\epsilon)>0$. It follows from (ii) that there exist an $m\in\mathbb{N}$ and a sequence $(g_{k,1},...,g_{k,m})_{k\in\mathbb{N}}$ in $G^m$ with $\{g_\alpha\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq D_\epsilon$ for which \eqref{6.2.RlimitDensityResult} holds. In particular, for some $g\in D_\epsilon$, $d(x_g,x_0)<\epsilon$, which gives us the desired contradiction. \end{proof}
We collect in the following proposition some equivalent definitions of weak mixing which will be needed for the proof of \cref{6.2.GlobalWeaklyMixing} below. The proof is totally analogous to the classical case $G=\mathbb{Z}$ and is omitted. \begin{prop}\label{6.2.EquivalentFormsOFWM} Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu, (T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is weakly mixing.
\item For any ergodic probability measure preserving system $(Y,\mathcal B,\nu, (S_g)_{g\in G})$, the system
$$(X\times Y,\mathcal A\otimes \mathcal B,\mu\otimes\nu, (T_g\times S_g)_{g\in G})$$
is ergodic.
\item For any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G$ there exists a set $B\subseteq G$ with $\overline d_{(F_k)}(B)=0$ such that for any $A_0,A_1\in\mathcal A$,
$$\lim_{g\rightarrow\infty,\,g\not\in B}\mu(A_0\cap T_gA_1)=\mu(A_0)\mu(A_1).$$
\item There exists a sequence $(g_k)_{k\in\mathbb{N}}$ in $G$ with $\lim_{k\rightarrow\infty}g_k=\infty$ such that for any $A_0,A_1\in\mathcal A$,
$$\lim_{k\rightarrow\infty}\mu(A_0\cap T_{g_k}A_1)=\mu(A_0)\mu(A_1).$$ \end{enumerate} \end{prop} \begin{rem}\label{6.2.ProductOfWeaklyMixingIsWeaklyMixing} It follows from (ii) that for any two weakly mixing systems $(X,\mathcal A,\mu, (T_g)_{g\in G})$ and $(Y,\mathcal B,\nu, (S_g)_{g\in G})$, $(T_g\times S_g)$ is again weakly mixing. \end{rem}
\begin{thm}\label{6.2.GlobalWeaklyMixing} Let $\ell\in\mathbb{N}$, let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a measure preserving system. The following statements are equivalent: \begin{enumerate}[(i)]
\item $(T_g)_{g\in G}$ is weakly mixing.
\item For any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G^\ell$, any set $E\subseteq G^\ell$ with $\overline d_{(F_k)}(E)>0$ and any $m\in\mathbb{N}$, there exist non-degenerated and essentially distinct sequences $(\textbf g_k^{(j)})_{k\in\mathbb{N}}=(g_{k,1}^{(j)},...,g_{k,m}^{(j)})_{k\in\mathbb{N}}$, $j\in\{1,...,\ell\}$,
in $G^m$
with the properties:
\begin{enumerate}[(a)]
\item $\{(g_\alpha^{(1)},...,g_\alpha^{(\ell)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$,
\item For any $t\in\{1,...,m\}$ and any $j\in\{1,...,\ell\}$, $(T_{g_{k,t}^{(j)}})_{k\in\mathbb{N}}$ has the mixing property and
\item For any $t$ and any $i\neq j$, $(T_{g_{k,t}^{(j)}-g_{k,t}^{(i)}})_{k\in\mathbb{N}}$ has the mixing property.
\end{enumerate}
\item For any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G^\ell$ and any set $E\subseteq G^\ell$ with $\overline d_{(F_k)}(E)>0$, there exist an $m\in\mathbb{N}$ and sequences $(\textbf g_k^{(1)})_{k\in\mathbb{N}},...,(\textbf g_k^{(\ell)})_{k\in\mathbb{N}}$
in $G^m$ with $\{(g_\alpha^{(1)},...,g_{\alpha}^{(\ell)})\,|\,\alpha\in\mathbb{N}^{(m)}\}\subseteq E$ and such that for any $A_0,...,A_\ell\in\mathcal A$,
$$\rlim{\alpha\in \mathbb{N}^{(m)}}\mu(A_0\cap T_{g^{(1)}_\alpha} A_1\cap\cdots\cap T_{g^{(\ell)}_\alpha}A_\ell)=\prod_{j=0}^\ell\mu(A_j).$$
\item For any $A_0,...,A_\ell\in\mathcal A$ and any $\epsilon>0$, the set
$$R_\epsilon(A_0,...,A_\ell)=\{(g_1,...,g_\ell)\in G^\ell\,|\,|\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_\ell}A_\ell)-\prod_{j=0}^\ell \mu( A_j)|<\epsilon\}$$
has uniform density one. \end{enumerate} \end{thm} \begin{proof} (i)$\implies$(ii): For each $j\in\{1,...,\ell\}$, let $\pi_j:G^\ell\rightarrow G$ be defined by $\pi_j(g_1,...,g_\ell)=g_j$. Note that $(T_{\pi_j(\textbf g)})_{\textbf g\in G^\ell}$ is a weakly mixing action and for any $i\neq j$, $(T_{(\pi_j-\pi_i)(\textbf g)})_{\textbf g\in G^\ell}$ is also weakly mixing. Moreover (see \Cref{6.2.ProductOfWeaklyMixingIsWeaklyMixing}), $$(S_{\textbf g})_{\textbf g\in G^\ell}=(\prod_{j=1}^\ell T_{\pi_j(\textbf g)}\times\prod_{i\neq j}T_{(\pi_j-\pi_i)(\textbf g)})_{\textbf g\in G^\ell}$$ is a weakly mixing $G^\ell$-action on the probability space $$(X^{\ell^2},\bigotimes_{j=1}^{\ell^2}\mathcal A,\nu),$$ where $\nu=\underbrace{\mu\times\cdots\times \mu}_{\ell^2\text{ times}}$.\\ By \cref{6.2.EquivalentFormsOFWM}, (iii), there exists a set $B\subseteq G^\ell$ with $\overline d_{(F_k)}(B)=0$ such that for any $A_0,A_1\in\bigotimes_{j=1}^{\ell^2}\mathcal A$, \begin{equation}\label{6.2.BigProductIsWeaklyMixing} \lim_{\textbf g\rightarrow\infty,\,\textbf g\not\in B}\nu(A_0\cap S_{\textbf g}A_1)=\nu(A_0)\nu(A_1). \end{equation} We start with proving (ii) for $m=1$. Let $E\subseteq G^\ell$ with $\overline d_{(F_k)}(E)>0$. By \cref{5.2.FiniteSumsInPositiveDensitySets} (applied to $d=\ell$, $m=1$ and the set $(E\setminus B)\subseteq G^\ell$) there exist non-degenerated and essentially distinct sequences $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(\ell)})_{k\in\mathbb{N}}$ in $G$ with the property that for each $k\in\mathbb{N}$, $\textbf g_k=(g_k^{(1)},...,g_k^{(\ell)})\in E\setminus B$. It follows now from \eqref{6.2.BigProductIsWeaklyMixing} that $(S_{\textbf g_k})_{k\in\mathbb{N}}$ has the mixing property and hence for any $j\in\{1,...,\ell\}$, $(T_{g_k^{(j)}})_{k\in\mathbb{N}}$ has the mixing property and for any $i\neq j$, $(T_{g_k^{(j)}-g_k^{(i)}})_{k\in\mathbb{N}}$ has the mixing property as well.\\
Assume now that $m>1$. Let $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(\ell)})_{k\in\mathbb{N}}$ be non-degenerated and essentially distinct sequences in $G$ such that for any distinct $i,j\in\{1,...,\ell\}$, $(T_{g_k^{(j)}})_{k\in\mathbb{N}}$ and $(T_{g_k^{(j)}-g_k^{(i)}})_{k\in\mathbb{N}}$ have the mixing property. Let $(\textbf h_k)_{k\in\mathbb{N}}=(h^{(1)}_k,...,h^{(\ell)}_k)_{k\in\mathbb{N}}$ be a subsequence of $(g^{(1)}_k,...,g_k^{(\ell)})_{k\in\mathbb{N}}$ such that for any $i,j\in\{1,...,\ell\}$, \begin{equation}\label{6.2.hkGoingToInfty} \lim_{\alpha\rightarrow\infty}h^{(j)}_\alpha=\infty\text{ and if }i\neq j,\,\lim_{\alpha\rightarrow\infty}(h^{(j)}_\alpha-h^{(i)}_\alpha)=\infty. \end{equation}
Observe that, by \eqref{6.2.hkGoingToInfty}, $\{(h^{(1)}_\alpha,...,h^{(\ell)}_\alpha)\,|\,\alpha\in\mathcal F\}$ is an \text{\rm{$\tilde{\text{IP}}$}} set. It follows from our choice of $(g_k^{(1)})_{k\in\mathbb{N}}$,...,$(g_k^{(\ell)})_{k\in\mathbb{N}}$, that for any $M\in\mathbb{N}$, any non-empty set $\alpha\subseteq\{1,...,M\}$, any $A_0,A_1\in\mathcal A$ and any $j\in\{1,...,\ell\}$, \begin{equation}\label{6.2.DiagonalizingForIP1} \lim_{k\rightarrow\infty}\mu(T_{-h^{(j)}_\alpha}A_0\cap T_{h^{(j)}_k}A_1)=\mu(A_0)\mu(A_1), \end{equation} and for any $i\neq j$, \begin{equation}\label{6.2.DiagonalizingForIP2} \lim_{k\rightarrow\infty}\mu(T_{-(h^{(j)}_\alpha-h^{(i)}_\alpha)}A_0\cap T_{h^{(j)}_k-h^{(i)}_k}A_1)=\mu(A_0)\mu(A_1). \end{equation} Passing, if needed, to a subsequence of $(\textbf h_k)_{k\in\mathbb{N}}$, we can derive now from \eqref{6.2.DiagonalizingForIP1} and \eqref{6.2.DiagonalizingForIP2} the following equations \begin{equation*}\label{6.2.IPlimitForWeaklyMixing} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F}\mu(A_0\cap T_{h^{(j)}_\alpha}A_1)=\mu(A_0)\mu(A_1), \end{equation*} and if $i\neq j$, \begin{equation*} \mathop{\text{\rm{IP-lim}}}_{\alpha\in\mathcal F}\mu(A_0\cap T_{h^{(j)}_\alpha-h^{(i)}_\alpha}A_1)=\mu(A_0)\mu(A_1). \end{equation*} We can conclude now the proof of (i)$\implies$(ii) by arguing as in the proof of \cref{5.2.FiniteSumsInPositiveDensitySets} and imitating the constructions in the proofs of \cref{5.2.IPPoincare} and \cref{5.1.SigmaInEveryIP}.\\ (ii)$\implies$(iii): This follows from \cref{2.MainResult}.\\ (iii)$\implies$(iv): Let $E=G^\ell\setminus R_\epsilon(A_0,...,A_\ell)$. It suffices to show that for any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G^\ell$, $\overline d_{(F_k)}(E)=0$. To see this, note that if this was not the case, (iii) would imply that $E\cap R_\epsilon(A_0,...,A_\ell)\neq \emptyset$, a contradiction.\\ (iv)$\implies$(i): This implication is trivial and is omitted. \end{proof} We conclude this section with a corollary of \cref{6.2.GlobalWeaklyMixing} which has diagonal nature (This corollary can also be obtained from the main result in \cite{BerRosJointErgodicity}). \begin{cor}\label{6.2.DiagonalDensityResult} Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu,(T_g)_{g\in G})$ be a measure preserving system and let $\phi_1,...,\phi_\ell:G\rightarrow G$ be homomorphisms with the property that for any $j\in\{1,...,\ell\}$, $(T_{\phi_j(g)})_{g\in G}$ is weakly mixing and for any $i\neq j$, $(T_{(\phi_j-\phi_i)(g)})_{g\in G}$ is also weakly mixing. For any F{\o}lner sequence $(F_k)_{k\in\mathbb{N}}$ in $G$ and any $A_0,...,A_\ell\in\mathcal A$, \begin{equation}\label{6.2.DensityLimit2}
\lim_{k\rightarrow\infty}\frac{1}{|F_k|}\sum_{g\in F_k}|\mu(A_0\cap T_{\phi_1(g)}A_1\cap\cdots\cap T_{\phi_\ell(g)}A_\ell)-\prod_{j=0}^\ell \mu(A_j)|=0. \end{equation} \end{cor} \begin{proof}
By \cref{6.2.LimitEquivalenceWM}, in order to prove \eqref{6.2.DensityLimit2}, it suffices to show that for any $E\subseteq G$ with $\overline d_{(F_k)}(E)>0$, there exists a non-degenerated sequence $(\textbf g_k)_{k\in\mathbb{N}}=(g_{k,1},...,g_{k,\ell})_{k\in\mathbb{N}}$ in $G^\ell$ with $\{g_\alpha\,|\,\alpha\in\mathbb{N}^{(\ell)}\}\subseteq E$ such that \begin{equation}\label{6.2.PreDensityDiagonalLimit} \rlim{\alpha\in\mathbb{N}^{(\ell)}}\mu(A_0\cap T_{\phi_1(g_\alpha)}A_1\cap\cdots\cap T_{\phi_\ell(g_\alpha)}A_\ell)=\prod_{j=0}^\ell \mu(A_j). \end{equation} By \cref{6.2.GlobalWeaklyMixing}, (ii), applied to the weakly mixing $G$-action
$$(S_g)_{g\in G}=(\prod_{j=1}^\ell T_{\phi_j(g)}\times\prod_{i\neq j}T_{(\phi_j-\phi_i)(g)})_{g\in G},$$ there exists a non-degenerated sequence $(g_{k,1},...,g_{k,\ell})_{k\in\mathbb{N}}$ in $G$, with $\{g_\alpha\,|\,\alpha\in\mathbb{N}^{(\ell)}\}\subseteq E$, and such that for any $t\in\{1,...,\ell\}$, the sequence $(S_{g_{k,t}})_{k\in\mathbb{N}}$ has the mixing property. It follows that for any $t\in\{1,...,\ell\}$ and any $j\in\{1,...,\ell\}$, $(T_{\phi_j(g_{k,t})})_{k\in\mathbb{N}}$ has the mixing property and for any $t$ and $i\neq j$, $(T_{(\phi_j-\phi_i)(g_{k,t})})_{k\in\mathbb{N}}$ has the mixing property as well. The result now follows from \cref{2.MainResult}. \end{proof} \begin{rem} Taking in \cref{6.2.DiagonalDensityResult} $G=\mathbb{Z}$, one obtains the following classical result due to Furstenberg (Cf. Theorem 4.11 in \cite{FBook}):\\ \begin{adjustwidth}{0.5cm}{0.5cm} \textit{For any weakly mixing system $(X,\mathcal A,\mu, T)$, any non-zero and distinct integers $a_1,...,a_\ell$ and any $A_0,...,A_\ell\in\mathcal A$,
$$\lim_{N-M\rightarrow\infty}\frac{1}{N-M}\sum_{n=M+1}^N|\mu(A_0\cap T^{a_1n}A_1\cap \cdots\cap T^{a_\ell n}A_\ell)-\prod_{j=0}^\ell\mu(A_j)|=0.$$} \end{adjustwidth} \end{rem}
\footnotesize
\noindent Vitaly Bergelson\\ \textsc{Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA}\par\nopagebreak \noindent \href{mailto:[email protected]} {\texttt{[email protected]}}
\noindent Rigoberto Zelada\\ \textsc{Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA}\par\nopagebreak \noindent \href{mailto:[email protected]} {\texttt{[email protected]}}
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\title[Isoclinism and factor set in regular Hom-Lie Superalgebras ]{ Isoclinism and factor set in regular Hom-Lie Superalgebras }
\author[NANDI]{N. Nandi}
\address{Department of Mathematics, National Institute of Technology \\
Rourkela,
Odisha-769028 \\
India}
\email{[email protected]}
\author[Padhan]{R. N. Padhan}
\address{Centre for Applied Mathematics and Computing, Institute of Technical Education and Research \\
Siksha `O' Anusandhan (A Deemed to be University)\\
Bhubaneswar-751030 \\
Odisha, India}
\email{[email protected], [email protected]}
\author[Pati]{K. C. Pati}
\address{Department of Mathematics, National Institute of Technology \\
Rourkela,
Odisha-769028 \\
India}
\email{[email protected]}
\keywords{Isoclinism; Factor set; Hom-Lie superalgebras}
\subjclass[2020]{17B05; 17B30}
\maketitle
\begin{abstract}
Hom-Lie superalgebras can be considered as the deformation of Lie superalgebras; which are $\mathbb{Z}_2$-graded generalization of Hom-Lie algebras. The motivation of this paper is to introduce the concept of isoclinism and factor set in regular Hom-Lie superalgebras. Finally, we obtain that two finite same dimensional regular Hom-Lie superalgebras are isoclinic if and only if they are isomorphic. \end{abstract}
\section{Introduction} Hartwig et al. studied Hom-Lie algebras as a part of a study of deformations of the Witt and the Virasoro algebras \cite{hartwig2006}. Lie superalgebra was introduced by Kac which is the $\mathbb{Z}_2$-graded Lie algebra \cite{kac1977}. Then Hom-Lie algebra was generalized to Hom-Lie superalgebra by Ammar et al. \cite{ammar2010}. Cohomology of Hom-Lie superalgebras and $q$-deformed Witt superalgebras was studied in \cite{ammar2013}. \par As is well known, isoclinism plays an vital role in classification of finite $p$-group. The notion of isoclinism for group was introduced by Hall in 1940 which is weaker than isomorphism. In 1993, Moneyhun used this concept on Lie algebras see \cite{moneyhun1994,moneyhunK1994}. Recently, Nayak \cite{nayak2018} and Padhan et al. \cite{padhan2019} studied isoclinism for Lie superalgebras. Isocinism for $n$-Lie superalgebras is studied in \cite{khuntia2022,padhan2022}. Factor set in Lie algebras was defined by Moneyhun \cite{moneyhun1994}, and the same for pair of Lie algebras was defined by Moghaddam et al. in \cite{eshrati2016}. It is also defined for Lie superalgebras by Nayak et al. \cite{nayak2019}. \par In this paper, we define isoclinism for regular Hom-Lie superalgebras which is not true in general for any arbitary Hom-Lie superalgebras. Furthermore the factor set in regular Hom-Lie superalgebras is defined and some of its properties are discussed. We generalize some results for regular Hom-Lie algebras \cite{padhan2019} to regular Hom-Lie superalgebras. One can refer to \cite{sheng2013,agrebaoui2019} for the other exploring works of this field.
\par In this paper all vector superspaces are considered over $\mathbb{F}$, a field of characteristic $0$ and $\mathbb{Z}_2=\{\overline{0},\overline{1}\}$ the additive group of two elements. A superspace is a $\mathbb{Z}_2$-graded vector space $V={V}_{\overline{0}}\oplus {V}_{\overline{1}}$. A $sub superspace$ is a $\mathbb{Z}_2$-graded vector space which is closed under bracket operation. For a homogeneous element $m\in V_{\overline{0}}\cup V_{\overline{1}}$, we write $|m|$ for the parity $\gamma\in \mathbb{Z}_2$. For any superalgebra in this paper, homomorphisms always mean even homomorphisms. \par Let us begin with some definitions related to Hom-Lie superalgebras. \begin{definition}\label{d1}
A Hom-Lie superalgebra is a triple $(G,[.,.],\theta)$ which equipped with a $\mathbb{Z}_2$-graded vector space $G$, a bilinear map $[.,.]:G\times G\rightarrow G$, and a homomorphism $\theta:G\rightarrow G$ satisfying;
\begin{enumerate}
\item $[n_1,n_2]=-(-1)^{|n_1||n_2|}[n_2,n_1]$ (graded skew-symmetry),
\item $(-1)^{|n_1||n_3|}[\theta(n_1),[n_2,n_3]]+(-1)^{|n_3||n_2|}[\theta(n_3),[n_1,n_2]]+(-1)^
{|n_2||n_1|}[\theta(n_2),[n_3,n_1]]=0$ (graded Hom-Jacobi identity),
\end{enumerate}
for homogeneous elements $n_1,n_2,n_3\in G.$ \end{definition}
Suppose $(G,[.,.],\theta)$ is a Hom-Lie superalgebra. A Hom-Lie subspace $W\subseteq G $ is a $Hom$-$subalgebra$ of $G$ if $\alpha(W) \subseteq W$ and $W$ is closed under the bracket operation $[.,.]$, i.e., $[W,W]\subseteq W$. A Hom-subalgebra $W$ is called a $graded ~ideal$ of $G$ if $[W,G]\subseteq W$. The $center$ of the Hom-Lie superalgebra $(G,[.,.],\theta)$, denoted by $Z(G)$ is defined by $Z(G)=\{x\in G| [x,y]=0$ {\rm{for}} $y\in G\}.$ An $abelian$ Hom-Lie superalgebra is a vector superspace $G$ equipped with trivial bracket and an even linear map $\theta:G\rightarrow G$. A Hom-Lie superalgebra $(G,[.,.],\theta)$ is called $multiplicative$ if $\theta([n_1,n_2])=[\theta(n_1),\theta(n_2)]$ for $n_1,n_2\in G$. A multiplicative Hom-Lie superalgebra $(G,[.,.],\theta)$ is said to be $regular$ if $\theta$ is bijective.
In this paper we only consider multiplicative Hom Lie superalgebras. \begin{example}
Taking $\theta=Id$ in Definition \ref{d1}, we obtain the definition of a Lie superalgebra. \end{example}
\begin{lemma}\label{l3}
If $(G,[.,.],\theta)$ is a regular Hom-Lie superalgebra, then $Z(G)$ is a Hom-ideal of $G$. \end{lemma} \begin{proof}
Let $z\in Z(G)$. Suppose $m$ and $u$ are two homogeneous elements of $G$, we put $m=\theta(u)$, then we have \[ [\theta(z),m]= [\theta(z),\theta(u)]=\theta([z,u])=0, \]
which implies $\theta(Z(G))\subseteq Z(G)$.
Let $z_1,z_2 \in Z(G)$, then
\[[[z_1,z_2],m]=[[z_1,z_2],\alpha(u)]=0,\]
which means that $Z(G)$ is a Hom-ideal of $G$. \end{proof} The concept of stem Lie superalgebra is defined and studied in [6]. \begin{definition}\label{d4}
If $Z(G)\subseteq G^{'}$ then a regular Hom-Lie superalgebra $(G,[.,.],\alpha)$ is called stem Hom-Lie superalgebra. \end{definition}
\begin{definition}\label{d5}
Let $(G_1,[.,.]_1,\theta_1)$ and $(G_2,[.,.]_2,\theta_2)$ be two Hom-Lie superalgebras. A homomorphism from $f:(G_1,[.,.]_1,\theta_{1})\rightarrow (G_2,[.,.]_2,\theta_{2})$ is an even linear map $f:G_1\rightarrow G_2$ satisfying $f([n_1,n_2]_1)=[f(n_1),f(n_2)]_2$ and $f\theta_1=\theta_2f$ for $n_1,n_2\in G_1$. In particular, the following diagram is commutative:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (x) at (0,0) {\(G_1\)};
\node (z) at (0,-2) {\(G_1\)};
\node (y) at (2,0) {\(G_2\)};
\node (w) at (2,-2) {\(G_2.\)};
\draw[->] (x) -- (y) node[midway,above] {$f$};
\draw[->] (x) -- (z) node[midway,left] {$\theta_1$};
\draw[->] (z) -- (w) node[midway,below] {$f$};
\draw[->] (y) -- (w) node[midway,right] {$\theta_2$};
\end{tikzpicture}\\
\end{center}
They are isomorphic if $f:G_1\rightarrow G_2$ is bijective . \end{definition}
\begin{lemma}\label{l6}
Suppose $f:(G_1,[.,.]_1,\theta _1)\longrightarrow (G_2,[.,.]_2,\theta_2)$ is an isomorphism. If $(G_1,[.,.]_1,\theta _1)$ is regular then $(G_2,[.,.]_2,\theta _2)$ is also regular. \end{lemma} \begin{proof}
Suppose $n_1,n_2 \in G_2$ then there exists $m_1,m_2 \in G_1$ such that $f(m_1)=n_1$ and $f(m_2)=n_2$. As $f \theta_1=\theta_2 f$ and $\theta_1$ is regular, we have
\begin{align*}
\theta_{2}([n_1,n_2]_2)&=\theta_{2}([f(n_1),f(n_2)]_2)=\theta_{2} f([n_1,n_2]_1)\\
&=f \theta_{1}([n_1,n_2]_1)=f([\theta_1(n_1),\theta_1(n_2)]_1)\\
&=[f\theta_1(n_1),f\theta_1(n_2)]_2
=[\theta_2 f (n_1), \theta_2 f (n_2)]_2=[\theta_2(n_1),\theta_2(n_2)]_2.
\end{align*}
\noindent Now $f$ and $\theta_1$ are bijective, so $\theta_2$. Hence $(G_2,[.,.],\theta _2)$ is regular. \end{proof}
\noindent $(G/K,[.,.],\theta)$ is a Hom-Lie superalgebra with skew-bilinear and linear map and known as the quotient Hom-Lie superalgebra. For any Hom-ideal $K$ of $(G,[.,.],\theta)$, we can define quotient Hom-Lie superalgebra on the quotient vector superspace $G/K$ by defining $[.,.]:G/K\times G/K\rightarrow G/K$ by $[\overline{n_{1}},\overline{n_{2}}]=\overline{[n_1,n_2]}$ for $\overline{n_1},\overline{n_2}\in G/K$ and $\tilde{\theta}:G/K\rightarrow G/K$ is induced by $\theta$, i.e., $\tilde{\theta}(\overline{m})=\theta(m)+K.$ \par Suppose that $(G_1,[.,.]_{G_1},\theta_1)$ and $(G_2,[.,.]_{G_2},\theta_2)$ are two Hom-Lie superalgebras, define the direct sum of these Hom-Lie superalgebras $(G_1\oplus G_2,[.,.]_{G_1\oplus G_2},\Gamma)$ as: $$[(m_1,n_1),(m_2,n_2)]_{G_1\oplus G_2}=([m_1,m_2]_{G_1},[n_1,n_2]_{G_2}),$$ and the linear map $\Gamma:G_1\oplus G_2\rightarrow G_1\oplus G_2 $ is given by $$\Gamma(m,n)=(\theta_{1}(m),\theta_{2}(n)),$$ which signifies that $(G_1\oplus G_2,[.,.]_{G_1\oplus G_2},\Gamma)$ is also a Hom-Lie superalgebra.
\section{isoclinism} Now onwards we will use $(G,\theta)$ to symbolize Hom-Lie superalgebra. The following generalizations are applicable only in the case of regular Hom-Lie superalgebras by Lemma \ref{l3}. \par Now we define isoclinism for Hom-Lie superalgebras. \begin{definition}\label{d7}
Consider two regular Hom-Lie superalgebras $(G_1,\theta_1)$ and $(G_2,\theta_2)$. Suppose $\mu :\frac{G_1}{Z(G_1)}\rightarrow \frac{G_2}{Z(G_2)}$ and $\nu :G_1^{'} \rightarrow G_2^{'}$ are two Hom-Lie superalgebra homomorphisms such that:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (x) at (0,0) {\(\frac{G_1}{Z(G_1)}\times \frac{G_1}{Z(G_1)} \)};
\node (z) at (0,-2) {\(\frac{G_2}{Z(G_2)}\times \frac{G_2}{Z(G_2)}\)};
\node (y) at (3,0) {\(G_1'\)};
\node (w) at (3,-2) {\(G_2',\)};
\draw[->] (x) -- (y) node[midway,above] {$\sigma$};
\draw[->] (x) -- (z) node[midway,left] {$\mu ^{2} $};
\draw[->] (z) -- (w) node[midway,below] {$\rho$};
\draw[->] (y) -- (w) node[midway,right] {$\nu$};
\end{tikzpicture}\\
\end{center}
\noindent commutes, where $\sigma( \overline{m_{1}},\overline{m_{2}}):=[\overline{m_{1}},\overline{m_{2}}]$ and $\rho( \overline{n_{1}},\overline{n_{2}}):= [\overline{n_{1}},\overline{n_{2}}]$ for $ m_1,m_2 \in G_1 $ and $n_{1},n_2 \in G_2$. Then the pair $(\mu, \nu)$ is said to be {\it homoclinism} and if they are both isomorphisms, then $(\mu, \nu)$ is called {\it isoclinism}. \end{definition} \noindent $G_1$ and $G_2$ are said to be isoclinic if $(\mu,\nu)$ is an isoclinism between $G_1$ and $G_2$, which is denoted by $G_1\sim G_2$. One can see that the above notion forms an equivalence relation.
\begin{lemma}\label{l8}
If $(G_1,\theta_1)$ is a regular Hom-Lie superalgebra and $(G_2,\theta_2)$ is an abelian Hom-Lie superalgebra, then $G_1\sim G_1\oplus G_2$. \end{lemma} \begin{proof}
Since $G_2$ is abelian, we have $Z(G_1\oplus G_2)=Z(G_1)\oplus G_2$. Define the map $$\mu :\frac{G_1}{Z(G_1)}\rightarrow \frac{G_1\oplus G_2}{Z(G_1)\oplus G_2},$$ by $$m+Z(G_1)\rightarrow (m,0)+(Z(G_1)\oplus G_2), $$ for $m\in G_1$. It is easy to verify that the map is well defined. Take two homogeneous elements $n_1,n_2\in G_1$, we have $\mu([n_1+Z(G_1),n_2+Z(G_1)])=[\mu(n_1+Z(G_1)),\mu(n_2+Z(G_1))]$ holds, implies that $\mu$ is a homomorphism. Now we have to show that $\mu \tilde{\theta}_1=\tilde{\phi}\mu$, i.e.,
$$\mu \tilde{\theta_1}(\overline{n})=\mu(\theta(n)+Z(G_1))=(\theta(n),0)+(Z(G_1)\oplus W)=\tilde{\phi}(n,0)+(Z(G_1)\oplus G_2)=\tilde{\phi}\mu(\overline{n}).$$
Clearly $\mu$ is a bijection, thus an isomorphism. Next we consider the identity map $\nu : G_1^{'} \rightarrow (G_1\oplus G_2)^{'} = G_1^{'}$ then we get the commutative diagram:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (x) at (0,0) {\(\frac{G_1}{Z(G_1)}\times \frac{G_1}{Z(G_1)}\)};
\node (z) at (0,-3) {\(\frac{G_1\oplus G_2}{Z(G_1)\oplus G_2}\times \frac{G_1\oplus G_2}{Z(G_1)\oplus G_2}\)};
\node (y) at (4,0) {\(G_1^{'}\)};
\node (w) at (4,-3) {\(G_1 ^{'}.\)};
\draw[->] (x) -- (y) node[midway,above] {$\sigma$};
\draw[->] (x) -- (z) node[midway,left] {$\mu^2$};
\draw[->] (z) -- (w) node[midway,below] {$\rho$};
\draw[->] (y) -- (w) node[midway,right] {$\nu$};
\end{tikzpicture}
\end{center} \end{proof}
To proof the below lemmas one may refer \cite{moneyhun1994,nayak2018}. \begin{lemma}\label{l9}
Suppose that $(G,\theta)$ is a regular Hom-Lie superalgebra and $K$ is a Hom-ideal. Then $G/K \sim G/{(K\cap G^{'})}$. In particular, if $K\cap G^{'}=0$ then $G\sim G/K$. Conversely if $G^{'}$ is finite dimensional and $G\sim G/K$ then $K\cap G^{'}=0$. \end{lemma}
\begin{corollary}\label{c10}
Consider two regular Hom-Lie superalgebras $(G_1,\theta_1)$ and $(G_2,\theta_2)$. If $f:G_1\rightarrow G_2$ is an onto homomorphism such that $Ker(f)\cap G^{'}=0$, then $f$ induces an isoclinism between $G_1$ and $G_2$. \end{corollary} \begin{lemma}\label{l11}
Suppose $\mathcal{C}$ is an isoclinic family of regular Hom-Lie superalgebas. Then
\begin{enumerate}
\item $\mathcal{C}$ contains a stem Hom-Lie superalgebra.
\item Each finite dimensional Hom-Lie superalgebra $T\in \mathcal{C}$ is stem if and only if $T$ has minimal dimension in $\mathcal{C}$.
\end{enumerate} \end{lemma}
\begin{lemma}\label{l12}
Suppose $(\mu,\nu)$ is an isoclinism of Hom-Lie superalgebras $G_1$ and $G_2$. Then the following statements hold:
\begin{enumerate}
\item $\mu(n+Z(G_1))=\nu(n)+Z(G_2)$,
\item $\nu([n_1,n_2])=[\nu(n_1),n_3]$ for $n_1\in G_1^{'},~n_2\in G_1$, and $n_3+Z(G_2)=\mu(n_2+Z(G_1
))$.
\end{enumerate} \end{lemma} \begin{lemma}\label{l13}
Let $(G_1,\theta)$ be a regular Hom-Lie superalgebra and $G_1= G_2 \oplus Z(G_1)$ then $\theta(G_2)\subseteq G_2$. \end{lemma} \begin{proof}
Suppose $\theta(w) \in Z(G_2)$ for some $0\neq w \in G_2$. For $y \in G_1$ there is a homogeneous element $x \in G_1$ such that $\theta(x)=y$. Thus $0=[\theta(w),y]=[\theta(w),\theta(x)]=\theta([w,x]).$ Since $\theta$ is injective, $[w,x]=0$ for $x \in G_1$. Thus $u \in Z(G_1)$, which is not true. Therefore $\theta(w)\in G_2$. \end{proof}
\section{Factor set in regular Hom-Lie superalgebras} From Lemma \ref{l3}, it can be concluded that if $(G,\theta)$ is regular then $Z(G)$ is a Hom-ideal. For this reason we study factor set only for regular Hom-Lie superalgebras. \begin{definition}\label{d14}
A finite dimensional Hom-Lie superalgebra is a triple $(G, [.,.],{\theta})$ over a field $\mathbb{F}$ equipped with a bilinear map;
$$r:G/Z(G)\times G/Z(G)\rightarrow Z(G),$$ is said to be a factor set if the following properties hold:
\begin{enumerate}
\item $r(\overline {n_1} ,\overline {n_2})=-(-1)^{|\overline {n_1}||\overline {n_2}|} r(\overline{n_2},\overline{n_1}),$
\item $r([\overline{n_1},\overline{n_2}],\tilde {\theta}({\overline{n_3}}))=r(\tilde{\theta}(\overline{n_1}),[\overline{n_2},\overline{n_3}])-(-1)^{|\overline{n_1}||\overline{n_2}|}r(\tilde{\theta}(\overline{n_2}),[\overline{n_1},\overline{n_3}]),$
\end{enumerate}
\noindent for homogeneous elements $\overline{n_1},\overline{n_2},\overline{n_3} \in G/Z(G)$ and $\tilde{\theta}$ is a homomorphism $\tilde{\theta} : G/Z(G)\rightarrow G/Z(G) $ satisfying $\tilde{\theta}(\overline{n})=\theta(n)+Z(G)$. The factor set $r$ is said to be multiplicative if $$r(\tilde{\theta}(\overline{n_1}),\tilde{\theta}(\overline{n_2}))=\theta r(\overline{n_1},\overline{n_2}),$$ for $\overline{n_1},\overline{n_2}\in G/Z(G)$. \end{definition} Next Lemma generate a new Hom-Lie superalgebra from a given regular Hom-Lie superalgebra and a factor set on it. \begin{lemma}\label{l15}
Suppose $(G,{\theta})$ is a Hom-Lie superalgebra and $r$ is a factor set on it. Set $R=(Z(G),G/Z(G),r)=\{(g,\overline{n}):g\in Z(G),~\overline{n}\in G/Z(G)\}$.
\item (1) $(R,\phi)$ is a Hom-Lie superalgebra under the component-wise addition $$[(g_1,\overline{n_1}),(g_2,\overline{n_2})]:=(r(\overline{n_1},\overline{n_2}),[\overline{n_1},\overline{n_2}]),$$
and the linear map $\phi:R\rightarrow R$ is given by $$\phi(g,\overline{n})=(\theta(g),\tilde{\theta}(\overline{n})).$$
\item (2) If the factor set $r$ is multiplicative then $(R,\phi)$ is regular.
\item (3) $Z_{R}=\{(g,0) \in R:x \in Z(G)\} \cong Z(G).$ \end{lemma} \begin{proof}
The map $[.,.]$ is linear. Let $(g_1,\overline{n_1})$ and $(g_2,\overline{n_2})$ be two homogeneous elements in $R$, then $|g_1|=|\overline{n_1}|=|(g_1,\overline{n_1})|$ and $|g_2|=|\overline{n_2}|=|(g_2,\overline{n_2})|$. To check the graded skew-symmetric property, consider
\begin{equation*}
\begin{split}
[(g_1,\overline{n_1}),(g_2,\overline{n_2})]&=-(-1)^{|\overline{n_1}||\overline{n_2}|}(r(\overline{n_2},\overline{n_1}),[\overline{n_2},\overline{n_1}])\\
&=-(-1)^{|\overline{n_1}||\overline{n_2}|}[(g_2,\overline{n_2}),(g_1,\overline{n_1})]\\
&=-(-1)^{|(g_1,\overline{n_1})||(g_2,\overline{n_2})|}[(g_2,\overline{n_2}),(g_1,\overline{n_1})].
\end{split}
\end{equation*}
To check the graded Hom-Jacobi identity, take
\begin{align*}
&(-1)^{|\overline{n_1}||\overline{n_3}|} [[(g_1,\overline{n_1}),(g_2,\overline{n_2})],\theta(g_3,\overline{n_3})]+(-1)^{|\overline{n_2}||\overline{n_1}|}[[(g_2,\overline{n_2}),(g_3,\overline{n_3})],\theta(g_1,\overline{n_1})]\\
&\hspace{1cm}+(-1)^{|\overline{n_3}||\overline{n_2}|}[[(g_3,\overline{n_3}),(g_1,\overline{n_1})],\theta(g_2,\overline{n_2})]\\
&=(-1)^{|\overline{n_1}||\overline{n_3}|}[(r(\overline{n_1},\overline{n_2}),[\overline{n_1},\overline{n_2}]),(\theta(g_3),\tilde{\theta}(\overline{n_3}))]+(-1)^{|\overline{n_2}||\overline{n_1}|}[(r(\overline{n_2},\overline{n_3}),[\overline{n_2},\overline{n_3}]),(\theta(g_1),\tilde{\theta}(\overline{n_1}))]\\
&\hspace{1cm}+(-1)^{|\overline{n_3}||\overline{n_2}|}[(r(\overline{n_3},\overline{n_1}),[\overline{n_3},\overline{n_1}]),(\theta(g_2),\tilde{\theta}(\overline{n_2}))]\\
&=(-1)^{|\overline{n_1}||\overline{n_3}|}((r(\overline{n_1},\overline{n_2}),\tilde{\theta}(\overline{n_3})),[[\overline{n_1},\overline{n_2}],\tilde{\theta}(\overline{n_3})])+(-1)^{|n_2||n_1|}((r(\overline{n_2},\overline{n_3}),\tilde{\theta}(\overline{n_1})),[[\overline{n_2},\overline{n_3}],\tilde{\theta}(\overline{n_1})])\\
&\hspace{1cm}+(-1)^{|n_3||n_2|}((r(\overline{n_3},\overline{n_1}),\tilde{\theta}(\overline{n_2})),[[\overline{n_3},\overline{n_1}],\tilde{\theta}(\overline{n_2})])\\
&=\big((-1)^{|\overline{n_1}||\overline{n_3}|}(r(\overline{n_1},\overline{n_2}),\tilde{\theta}(\overline{n_3}))+(-1)^{|\overline{n_1}||\overline{n_2}|}(r(\overline{n_2},\overline{n_3}),\tilde{\theta}(\overline{n_1}))+(-1)^{|\overline{n_2}||\overline{n_3}|}(r(\overline{n_3},\overline{n_1}),\tilde{\theta}(\overline{n_2})),\\&\hspace{1cm}(-1)^{|\overline{n_1}||\overline{n_3}|}[[\overline{n_1},\overline{n_2}],\tilde{\theta}(\overline{n_3})]+(-1)^{|\overline{n_2}||\overline{n_1}|}[[\overline{n_2},\overline{n_3}],\tilde{\theta}(\overline{n_1})]+(-1)^{|\overline{n_3}||\overline{n_2}|}[[\overline{n_3},\overline{n_1}],\tilde{\theta}(\overline{n_2})]\big)\\
&=(0,\overline{0}),
\end{align*}
which implies that $(R,\phi)$ is a Hom-Lie superalgebra. Now suppose that $r$ is multiplicative, then $$\phi([(g_1,\overline{n_1}),(g_2,\overline{n_2})])=\phi(r(\overline{n_1},\overline{n_2}),[\overline{n_1},\overline{n_2}])=(\theta r(\overline{n_1},\overline{n_2}),\tilde{\theta}[\overline{n_1},\overline{n_2}]),$$
and $$[\phi(g_1,\overline{n_1}),\phi(g_2,\overline{n_2})]=\phi[(\theta(g_1),\tilde{\theta}(\overline{n_1})),(\theta(g_2),\tilde{\theta}(\overline{n_2}))]=(r(\tilde{\theta}(n_1),\tilde{\theta}(n_2)),[\tilde{\theta}(n_1),\tilde{\theta}(n_2)]).$$
As $(G/Z(G),\tilde{\theta})$ is multiplicative, hence $(R,\phi)$ is regular.
The proof of (3) is obvious. \end{proof}
Next Lemma indicates that every regular Hom-Lie superalgebra has a factor set. \begin{lemma}\label{l16}
For any regular Hom-Lie superalgebra $(G, {\theta})$, there is a factor set $r$ in such a way that $G\cong (Z(G),G/Z(G),r)$. \end{lemma} \begin{proof}
Let us consider a vector superspace $W$ which is the complement of $Z(G)$, i.e., $G=W\oplus Z(G)$. Consider a map $\Psi:G/Z(G)\rightarrow G$ by $\Psi(\overline{n})=\Psi(n+Z(G))=\Psi(w+z+Z(G))=w$ for $\overline{n}\in G/Z(G)$, $w\in W$, and $z\in Z(G)$. Clearly $\Psi$ is a well-defined homogeneous even linear map. We have $\overline{\Psi(\overline{n})}=\overline{n}$. Now for $\overline{n_1}=w_1+z_1$ and $\overline{n_2}=w_2+z_2$, consider $[\overline{n_1},\overline{n_2}]=[w,w']+Z(G)$. Then
\begin{align*}
[\Psi(\overline{n_1}),\Psi(\overline{n_2})]-\Psi([\overline{n_1},\overline{n_2}])+Z(G)&=[w,w']-\Psi(\overline{[w,w']})+Z(G)\\
&=[w,w']- \overline{\Psi(\overline{[w,w']})}=0+Z(G).
\end{align*}
So $[\Psi(\overline{n_1}),\Psi(\overline{n_2})]-\Psi([\overline{n_1},\overline{n_2}])\in Z(G)$. Define $$r:G/Z(G)\times G/Z(G)\rightarrow G/Z(G),$$ by $r(\overline{m},\overline{n})=[g(\overline{m}),g(\overline{n})]-g([\overline{m},\overline{n}])$. We have to prove that $r$ is a factor set.\\
First we have to verify its graded skew-symmetric property,
\begin{align*}
r(\overline{m},\overline{n})&=[\Psi(\overline{m}),\Psi(\overline{n})]-\Psi([\overline{m},\overline{n}])\\
&=-(-1)^{|\overline{m}||\overline{n}|}([\Psi(\overline{n}),\Psi(\overline{m})]-\Psi([\overline{n},\overline{m}]))\\
&=-(-1)^{|\overline{m}||\overline{n}|}r(\overline{n},\overline{m}),
\end{align*}
for $\overline{m},\overline{n}\in G/Z(G)$. Next we have to show graded Hom-Jacobi identity. Before that we have to check $\Psi\tilde{\theta}(\overline{n})=\theta \Psi(\overline{n})$ for $\overline{n}\in G/Z(G)$.\\
From Lemma \ref{l13}, $\theta(W)\subseteq W$, we have
$$\Psi\tilde{\theta}(\overline{n})=\Psi\tilde {\theta} (w+z+Z(G))=\Psi(\theta(w)+Z(G))=\theta(w),$$ and
$$\theta \Psi(\overline{n})=\theta(\Psi(w+z+Z(G)))=\theta(w).$$
\noindent Take,
\begin{align*}
r([\overline{n_1},\overline{n_2}],\tilde{\theta} (\overline{n_3}))&=[\Psi([\overline{n_1},\overline{n_2}]),\Psi \tilde{\theta}(\overline{n_3})]-\Psi ([[\overline{n_1},\overline{n_2}],\tilde{\theta}(\overline{n_3})])\\
&=[[\Psi (\overline{n_1}),\Psi(\overline{n_2})],\theta \Psi(\overline{n_3})]-\Psi([[\overline{n_1},\overline{n_2}],\tilde{\theta}(\overline{n_3})])\\
&=[\theta \Psi(\overline{n_1}),[\Psi(\overline{n_2}),\Psi(\overline{n_3})]]-(-1)^{|\overline{n_1}||\overline{n_2}|}[\theta \Psi(\overline{n_2}),[\Psi(\overline{n_1}),\Psi(\overline{n_3})]]\\&\hspace{1cm}-\Psi([\tilde{\theta}(\overline{n_1}),[\overline{n_2},\overline{n_3}]])+(-1)^{|\overline{n_1}||\overline{n_2}|}\Psi([\tilde{\theta}(\overline{n_2}),[\overline{n_1},\overline{n_3}]])\\
&=[\theta \Psi (\overline{n_1}),[\Psi(\overline{n_2}),\Psi(\overline{n_3})]]-\Psi([\tilde{\theta}(\overline{n_1}),[\overline{n_2},\overline{n_3}]])\\&\hspace{1cm}-(-1)^{|\overline{n_1}||\overline{n_2}|}\{[\theta \Psi(\overline{n_2}),[\Psi(\overline{n_1}),\Psi(\overline{n_3})]]-\Psi([\tilde{\theta}(\overline{n_2}),[\overline{n_1},\overline{n_3}]])\}\\
&=r(\tilde{\theta}(\overline{n_1}),[\overline{n_2},\overline{n_3}])-(-1)^{|\overline{n_1}||\overline{n_2}|}r(\tilde{\theta}(\overline{n_2}),[\overline{n_1},\overline{n_3}]),
\end{align*}
for $\overline{n_1},\overline{n_2},\overline{n_3}\in G/Z(G)$. Let us define $\pi :R\rightarrow G$ by $\pi (g,\overline{n})=g+\Psi(\overline{n})$ for $g\in Z(G)$ and $\overline{n}\in G/Z(G)$ where $R=(Z(G),G/Z(G),r)$. Clearly $\pi$ is a well-defined bijective homogeneous even linear map and $\pi([(g_1,\overline{n_1}),(g_2,\overline{n_2})])=[\pi(g_1,\overline{n_1}),\pi(g_2,\overline{n_2})]$.
In addition $$\theta \pi(g,\overline{n})=\theta(g+\Psi(\overline{n}))=\theta(g+w),$$ and $$\pi \theta(g,\overline{n})=\pi (\theta(g),\tilde{\theta}(\overline{n}))=\theta(g)+\Psi(\theta(w)+Z(G))=\theta(x)+\theta(w),$$ where $n=w+z,~w\in W,$ and $z\in Z(G)$.
Hence $\pi$ is an isomorphism.\\ \end{proof}
The following Lemma shows that there is a relationship between two stem Hom-Lie superalgebras. \begin{lemma}\label{l17}
Let $(G_1,\theta_{1})$ be a stem Hom-Lie superalgebra in an isoclinism family of Hom-Lie superalgebras $\mathcal{C}$. Then for any stem Hom-Lie superalgebra $(G_2,\theta_{2})$ of $\mathcal{C}$, there exists a factor set $r$ over $(G_1,\theta_{1})$ such that $G_2 \cong (Z(G_1),G_1/Z(G_1),r)$. \end{lemma} \begin{proof}
Let $(\mu,\nu)$ be an isoclinism of regular Hom-Lie superalgebras $(G_1,\theta_{1})$ and $(G_2,\theta_{2})$ then by Lemma \ref{l12} $\pi(Z(G_1))=Z(G_2)$. From Lemma \ref{l16}, there exists a factor set $s$ such that $G_2\cong (Z(G_2),G_2/Z(G_2),s)$. Let us define
$r:G_1/Z(G_1)\times G_1/Z(G_1) \longrightarrow Z(G_1) $ which is given by
$r(\overline{n_1},\overline{n_2})=\pi^{-1}(s(\mu(\overline{n_1}),\mu(\overline{n_2})))$ for $\overline{n_1},~\overline{n_2} \in G_1/Z(G_1),$
where $r$ is a skew-bilinear map. Since $\mu$ is an isomorphism, so $\mu \tilde{\theta_1} = \tilde{\theta_2} \mu $. Thus
\begin{equation*}
\begin{split}
& r ([\overline{n_1},\overline{n_2}],\tilde{\theta_1}(\overline{n_3}))\\
&= \pi^{-1}(s(\mu([\overline{n_1},\overline{n_2}]),\mu \tilde{\theta_1}(\overline{n_3})))\\
&= \pi^{-1}(s(([\mu(\overline{n_1}),\mu(\overline{n_2})]), \tilde{\theta_2}\mu(\overline{n_3})))\\
&= \pi^{-1}(s((\tilde{\theta_2}\mu(\overline{n_1}),([\mu(\overline{n_2}),\mu(\overline{n_3})]) ))-(-1)^{|\overline{n_1}||\overline{n_2}|}\pi^{-1}(s(\tilde{\theta_2}\mu(\overline{n_2}),([\mu(\overline{n_3}),\mu(\overline{n_1})])))\\
&=r(\tilde{\theta_1}(\overline{n_1}),[\overline{n_2},\overline{n_3}])-(-1)^{|\overline{n_1}||\overline{n_2}|} r(\tilde{\theta_1}(\overline{n_2}),[\overline{n_3},\overline{n_1}]).
\end{split}
\end{equation*}
\noindent Therefore $(Z(G_1),G_1/Z(G_1),r)$ is a factor set. Let us denote $R=(Z(G_1),G_1/Z(G_1),r)$ and $S=(Z(G_2),G_2/Z(G_2),s)$. Then by Lemma \ref{l15}, $(R,\phi_{1})$ and $(S,\phi_{2})$ are Hom-Lie superalgebras. Let us define the map
$\beta:(Z(G_1),G_1/Z(G_1),r)\longrightarrow (Z(G_2),G_2/Z(G_2),s)$ by
\[ \beta(g,\overline{n})=(\nu(g),\mu(\overline{n})),\]
which is undoubtedly a bijective even linear map with $\beta([(g_1,\overline{n_1}),(g_2,\overline{n_2})])
=[\beta(g_1,\overline{n_1}),\beta(g_2,\overline{n_2})]$. In addition
\[\beta \phi_{1} (g,\overline{n})=\beta(\theta_{1}(g),\tilde{\theta_1}(\overline{n}))=(\nu\theta_{1}(g),\mu\tilde{\theta_1}(\overline{n})),\]
and
\[ \phi_{2} \beta (g,\overline{n})=\phi_{2}(\nu(g),\mu(\overline{n}))= (\theta_{2}\nu(g),\tilde{\theta_2}\mu(\overline{n})) .\]
\noindent Since $\mu$ and $\nu$ are isomorphisms, and $\phi_{1}\beta =\beta \phi_{2}$,
as a result, $\beta$ is an isomorphism and $G_2\cong (Z(G_1),G_1/Z(G_1),r).$ \end{proof}
\begin{lemma}\label{l18}
Suppose that $(G,\theta)$ is a Hom-Lie superalgebra and $r$,$s$ are two factor sets over $(G,\theta)$. Assume that
\[R=(Z(G),G/Z(G),r),~~~~~~~~Z_{R}=\{(g,0) \in R:g \in Z(G)\} \cong Z(G),\] \[S=(Z(G),G/Z(G),s),~~~~~~~~Z_{S}=\{(g,0) \in S:g \in Z(G)\} \cong Z(G).\]
Let $\beta$ be an isomorphism from $R$ to $S$ satisfying $\beta(Z_{R})=Z_{S}$, then the restriction of $\beta$ on $G/Z(G)$ and $Z(G)$ define the automorphisms $\varphi \in Aut(G/Z(G))$ and $\psi \in Aut(Z(G))$, respectively. \end{lemma} \begin{proof}
By Lemma \ref{l15}, $(R,\phi)$ and $(S,\phi)$ are regular Hom-Lie superalgebras. By assumption $\beta(Z_R)=Z_S$, so let us define the quotient Hom-Lie superalgebra $\overline{\beta}:\big(\frac{R}{Z_R},\phi_1\big)\rightarrow \big(\frac{S}{Z_S},\phi_2\big)$ by $\overline{\beta}((g,\overline{n})+Z_R)=\beta(g,\overline{n})+Z_S$ is an isomorphism, where $\phi_1:\frac{R}{Z_R}\rightarrow \frac{R}{Z_R}$ and $\phi_2:\frac{S}{Z_S}\rightarrow \frac{S}{Z_S}$ are even linear maps defined as $\phi_1((g,\overline{n})+Z_R)=\phi_1(g,\overline{n})+Z_R$ and $\phi_2((g,\overline{n})+Z_S)=\phi_2(g,\overline{n})+Z_S$, respectively. Take $\varphi$ such that the following diagram is commutative:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (x) at (0,0) {\(\frac{G}{Z(G)} \)};
\node (z) at (0,-2) {\(\frac{R}{Z_R}\)};
\node (y) at (2,0) {\(\frac{G}{Z(G)}\)};
\node (w) at (2,-2) {\(\frac{S}{Z_S},\)};
\draw[->] (x) -- (y) node[midway,above] {$\varphi$};
\draw[->] (x) -- (z) node[midway,left] {$\eta_{1}$};
\draw[->] (z) -- (w) node[midway,below] {$\overline{\beta}$};
\draw[->] (y) -- (w) node[midway,right] {$\eta_{2}$};
\end{tikzpicture}\\
\end{center}
where $\eta_{1}$ and $\eta_{2}$ are projection maps, i.e., $\eta_{1}(\overline{n})=(0,\overline{n})+Z_R$ and $\eta_{2}(\overline{n})=(0,\overline{n})+Z_S$. So $\beta(0,\overline{n})+Z_S=(0,\varphi(\overline{n}))+Z_S$ for $\overline{n}\in G/Z(G)$. Now $$(0,\varphi \tilde{\alpha}(\overline{n}))+Z_S=\beta(0,\theta(n)+Z(G))+Z_S=\beta \phi(0,\overline{n})+Z_S.$$
On the other hand $$(0,\tilde{\theta} \varphi(\overline{n}))+Z_S=\phi(0,\varphi(\overline{n}))+Z_S=\phi \beta(0,\overline{n})+\phi(g)+Z_S=\phi \beta(0,\overline{n})+Z_S,$$ where $g\in Z_S$. As $\beta$ is an automorphism, i.e., $\beta \phi=\phi \beta$ and $\eta_2$ is injective, we have
\begin{align*}
&(0,\varphi \tilde{\theta}(\overline{n}))+Z_S=(0,\tilde{\theta} \varphi(\overline{n}))+Z_S\\
&\implies \eta_2(\varphi \tilde{\theta}(\overline{n}))=\eta_2(\tilde{\theta}\varphi (\overline{n}))\\
&\implies \varphi\tilde{\theta}(\overline{n})=\tilde{\theta}\varphi (\overline{n}).
\end{align*}
Since $\varphi$ is bijective and $\varphi([\overline{n_{1}},\overline{n_{2}}])=[\varphi(\overline{n_1}),\varphi(\overline{n_2})]$. Hence $\varphi$ is an automorphism. Consider the map $\tilde{\beta}:Z_R\rightarrow Z_S$ is defined as $\tilde{\beta}(g,0)=\beta(g,0)$ for $g\in Z(G)$, is an isomorphism. Define $\psi$ in such a way that the following diagram is commutative:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (x) at (0,0) {\( Z(G) \)};
\node (z) at (0,-2) {\(Z_{R}\)};
\node (y) at (2,0) {\(Z(G)\)};
\node (w) at (2,-2) {\(Z_{S},\)};
\draw[->] (x) -- (y) node[midway,above] {$\psi$};
\draw[->] (x) -- (z) node[midway,left] {$\overline{\eta_1}$};
\draw[->] (z) -- (w) node[midway,below] {$\tilde{\beta}$};
\draw[->] (y) -- (w) node[midway,right] {$\overline{\eta_2}$};
\end{tikzpicture}\\
\end{center}
\noindent where $\overline{\eta_1}$ and $\overline{\eta_2}$ are projection maps and $\beta(g,0)=(\psi(g),0)$ for $g\in Z(G)$. It is easily viewed that $\psi$ is an automorphism. \end{proof}
\begin{lemma}\label{l19}
Suppose $(G,\theta)$ is a Hom-Lie superalgebra. $(R,\phi), (S,\phi), Z_{R}$, and $Z_{S}$ are defined as in Lemma \ref{l18}.
\begin{enumerate}
\item Let $\beta : R \longrightarrow S$ be a Hom-Lie superalgebra isomorphism satisfying $\beta(Z_{R})= Z_{S}$. If $\varphi \in Aut(G/Z(G))$ and $\psi \in Aut(Z(G))$ are automorphisms generated by $\beta$ then there is a homogeneous even linear map, $\epsilon: G/Z(G) \longrightarrow Z(G)$ such that \[\psi(r(\overline{n_1}, \overline{n_2})+ \epsilon[\overline{n_1}, \overline{n_2}])=s(\varphi(\overline{n_1}), \varphi(\overline{n_2})).\]
\item Suppose $\varphi \in Aut(G/Z(G))$, $\psi \in Aut(Z(G))$, and $\tau: G/Z(G) \longrightarrow Z(G)$ is a homogeneous even linear map satisfying $$\psi(r(\overline{n_1}, \overline{n_2})+ \tau[\overline{n_1}, \overline{n_2}])=s(\varphi(\overline{n_1}), \varphi(\overline{n_2})) ,~ \tau \tilde{\theta}=\theta \tau.$$ Then there is an isomorphism $\beta:R \longrightarrow S$ which is generated by $\varphi$ and $\psi$ satisfying $\beta(Z_{R})=Z_{S}$.
\end{enumerate} \end{lemma} \begin{proof}
To prove the first part refer [Lemma 3.6, 11]. For the proof of second part, we only need to show the commutative property of $\beta$ where $\beta:R\rightarrow S$ is well defined, bijective, and homogeneous linear map of even degree defined as $$\beta(g,\overline{n})=(\phi(g)+\tau(\overline{n}),\psi(\overline{n})).$$
Now $$\beta \phi(g,\overline{n})=\beta(\alpha(g),\tilde{\theta}(\overline{n}))=(\psi \theta(g)+\tau \tilde{\theta}(\overline{n}),\varphi\tilde{\theta}(\overline{n})).$$
On another side $$\phi\beta(g,\overline{n})=\phi(\psi(g)+\tau(\overline{n}))=(\theta\psi(g)+\theta\tau(\overline{n}),\tilde{\theta}\varphi(\overline{n})).$$
As $\varphi$ and $\psi$ are automorphisms and $\tau \tilde{\alpha}=\alpha \tau$, so $\beta \phi=\phi \beta$. \end{proof}
\begin{theorem}\label{t20}
Suppose $(G_1,\theta_{1})$ and $(G_2,\theta_{2})$ are two finite dimensional stem Hom-Lie superalgebras of same parity. Then $G_1 \sim G_2$ iff $G_1 \cong G_2$. \end{theorem} \begin{proof}
Suppose $G_1$ and $G_2$ are two finite dimensional stem Hom-Lie superalgebras such that $G_1\cong G_2$, then it is obvious that $G_1\sim G_2$. To prove the converse part, suppose that $G_1\sim G_2$,
then by Lemma \ref{l16} and Lemma \ref{l17} we have, $G_1 \cong (Z(G_1), \frac{G_1}{Z(G_1)},r)=R$ and $G_2 \cong (Z(G_1), \frac{G_1}{Z(G_1)}, s)=S$. Since $(G_1,\theta_{1})$ and $(G_2,\theta_{2})$ are regular, so from Lemma \ref{l6}, $(R,\phi_1)$ and $(S,\phi_2)$ are also regular. Consider $(\mu, \nu)$ as the isoclinism between the regular Hom-Lie superalgebras $(R,\phi_1)$ and $(S,\phi_2)$. Precisely $Z_{R}=Z(R)$ and $Z_{S}=Z(S)$. Let $\varphi \in Aut(G_1/Z(G_1))$ be defined by $\mu((0, \overline{n})+Z_{R})= (0, \varphi(\overline{n}))+Z_{S}$ for $\overline{n} \in G_1/Z(G_1)$. Suppose the below diagram is commutative:
\begin{center}
\begin{tikzpicture}[>=latex]
\node (A_{1}) at (0,0) {\(\frac{G_1}{Z(G_1)}\times \frac{G_1}{Z(G_1)} \)};
\node (A_{2}) at (4,0) {\(\frac{R}{Z_R}\times \frac{R}{Z_R}\)};
\node (A_{3}) at (8,0) {\(R'\)};
\node (B_{1}) at (0,-2) {\(\frac{G_1}{Z(G_1)}\times \frac{G_1}{Z(G_1)}\)};
\node (B_{2}) at (4,-2) {\(\frac{S}{Z_S}\times \frac{S}{Z_S} \)};
\node (B_{3}) at (8,-2) {\(S',\)};
\draw[->] (A_{1}) -- (A_{2}) node[midway,above] {$\lambda$};
\draw[->] (A_{2}) -- (A_{3}) node[midway,above] {$\theta$};
\draw[->] (B_{1}) -- (B_{2}) node[midway,below] {$\zeta$};
\draw[->] (B_{2}) -- (B_{3}) node[midway,below] {$\xi$};
\draw[->] (A_{1}) -- (B_{1}) node[midway,right] {$\varphi^2$};
\draw[->] (A_{2}) -- (B_{2}) node[midway,right] {$\mu^2$};
\draw[->] (A_{3}) -- (B_{3}) node[midway,right] {$\nu$};
\end{tikzpicture}
\end{center}
in which
\begin{equation*}
\begin{split}
\lambda(\overline{n_1}, \overline{n_2}) = ((0, \overline{n_1})+Z_{R}, (0, \overline{n_2})+Z_{R}),\\
\zeta(\overline{n_1}, \overline{n_2}) = ((0, \overline{n_1})+Z_{S}, (0, \overline{n_2})+ Z_{S}),\\
\xi((g_1, \overline{n_1})+Z_{S}, (g_2,\overline{n_2})+Z_{S})= [(g_1,\overline{n_1}),(g_2, \overline{n_2})]=(s(\overline{n_1}, \overline{n_2}), [\overline{n_1}, \overline{n_2}]),\\
\theta((g_1, \overline{n_1})+Z_{R}, (g_2,\overline{n_2})+Z_{R})=(r(\overline{n_1}, \overline{n_2}), [\overline{n_1}, \overline{n_2}]).
\end{split}
\end{equation*}
Again suppose that $\psi \in Aut(Z(G))$ be given by $\nu(g,0)=(\psi(g),0)$ for $g \in Z(G)$. For $\overline{n_1}, \overline{n_2} \in G/Z(G)$, consider
$$
\beta \theta((0, \overline{n_1})+Z_{R}, (0, \overline{n_2})+Z_{R}) = \beta[(0,\overline{n_1}),(0, \overline{n_2})],
$$
and in addition \begin{align*}
\xi \theta((0, \overline{n_1})+Z_{R}, (0, \overline{n_2})+Z_{R})&=\xi((0, \mu(\overline{n_1}))+Z_{S}, (0, \mu(\overline{n_2}))+Z_{S} )\\
&= [(0, \varphi(\overline{n_1})), (0, \varphi(\overline{n_2}))]\\
&= (s(\varphi(\overline{n_1}), \varphi(\overline{n_2})), [\varphi(\overline{n_1}), \varphi(\overline{n_2})]).
\end{align*}
Thus, we have $\psi[(0,\overline{n_1}),(0, \overline{n_2})]
=(s(\varphi(\overline{n_1}), \varphi(\overline{n_2})), [\varphi(\overline{n_1}), \varphi(\overline{n_2})])$.
Consider the map $\tau: G_1^{'}/Z(G_1) \longrightarrow Z(G_1)$ such that
$$\psi(0, [\overline{n_1}, \overline{n_2}])=(\tau( [\overline{n_1}, \overline{n_2}]),t),$$
\noindent where $t \in G_1/Z(G_1)$ and thus we get
\[\psi(r( \overline{a}, \overline{b})+\tau( [\overline{a}, \overline{b}])=s((\varphi(\overline{a}), \varphi(\overline{b})).\]
To apply Lemma \ref{l12}, we may continue $\tau$ to $G_1/Z(G_1)$ by defining $0$ on the complement of $G_1^{'}/Z(G_1)$ in $G_1/Z(G_1)$. Now we will prove $\tau \tilde{\kappa}=\kappa \tau.$ We obtain
\[(\tau \tilde{\theta}[\overline{n_1}, \overline{n_2}],t_{1})=\nu(0,\tilde{\theta}[\overline{n_1}, \overline{n_2}])=\nu\phi(0,[\overline{n_1}, \overline{n_2}]).\]
\noindent Further $\tilde{\theta}$ is surjective, i.e., $\tilde{\theta} (t)=t_2$ for $t_2 \in G_1/Z(G_1)$. Therefore
\[(\theta \tau [\overline{n_1}, \overline{n_2}],t_2)= (\theta \tau [\overline{n_1}, \overline{n_2}],\tilde{\theta}(t))=\phi(\tau [\overline{n_1}, \overline{n_2}],t )=\phi\nu (0,[\overline{n_1}, \overline{n_2}]) .\]
As a result $\theta \tau(\overline{n})=\tau \tilde{\theta}(\overline{n}) $ for $n \in G_1'$. Suppose $G_1=G_1'\oplus U$, then define $\tau$ to be zero in $U$. Then $\theta\tau (\overline{u})=0$ for $u \in U$. Since $\varphi$ is injective, $\varphi(u) \in U$ for $u \in U$, we have
\[ \tau \tilde{\theta}(\overline{u})=\tau (\theta(u)+Z(V))=0.\]
Thus $\tau \tilde{\psi}=\psi \tau$ and now we apply Lemma \ref{l19} to get our results. \end{proof}
To prove the following Theorems one can see \cite{nayak2019}. \begin{theorem}\label{t21}
If $\mathcal{C}$ is an isoclinism family of finite dimensional regular Hom-Lie superalgebras then any $G\in \mathcal{C}$ can be written as $G=P\oplus Q$ where $P$ is a stem Hom-Lie superalgebra and $Q$ is some finite dimensional abelian Hom-Lie superalgebra. \end{theorem}
\begin{theorem}\label{t22}
Suppose $(G_1,\theta_1)$ and $(G_2,\theta_2)$ are two regular Hom-Lie superalgebras with same dimensions and same parity. Then $G_1\sim G_2$ if and only if $G_1\cong G_2$. \end{theorem} Below example shows that two isoclinic Hom-Lie superalgebras may not be isomorphic when they have different dimensions. \begin{example}
Consider a $(2|1)$ dimensional Hom-Lie superalgebra $(G,\theta_1)$ with basis $\{a_1,a_2|a_3\}$ and commutator relations are defined by;
$$[a_1,a_3,a_3]=a_1; ~[a_2,a_3,a_3]=a_2,$$
and all other commutator relations are zero. Then $G^{'}=<a_1,a_2>$ and $Z(G)=0$ and hence $G/Z(G)\cong G$.\\
Now consider a $(3|1)$ dimensional Hom-Lie superalgebra $(W,\theta_{2})$ with basis $\{a^{\prime}_1,a^{\prime}_2,a^{\prime}_3|a^{\prime}_4\}$ and commutator relations are defined by;
$$[a^{\prime}_1,a^{\prime}_4,a^{\prime}_4]=a^{\prime}_1;~[a^{\prime}_2,a^{\prime}_4,a^{\prime}_4]=a^{\prime}_2,$$
and all other commutators are zero. Then $W^{\prime}=<a^{\prime}_1,a^{\prime}_2>$ and $Z(W)=\{a^{\prime}_3\}$ and hence $W/Z(W)=\{\overline{a^{\prime}_1},\overline{a^{\prime}_2}|\overline{a^{\prime}_4}\}$ where $\overline{a^{\prime}_i}=a_i+Z(W)$ for $i=1,2,4$.\\
We observe that $G^{'}\sim W^{'}$ and $\frac{G}{Z(G)}\cong \frac{W}{Z(W)}$ from which one can deduce that $G\sim W$ while $dim(G)\neq dim(W)$, i.e., $G$ and $W$ are not isomorphic. \end{example}
\section{Conclusion} In this research work, factor set for Hom-Lie superalgebras is defined by using the concept of isoclinism and the existence of factor set for Hom-Lie superalgebras is shown. We conclude that two finite dimensional Hom-Lie superalgebras having same dimensions are isoclinic if and only if they are isomorphic which is the main result of this work. Later we give an example to show that two Hom-Lie superalgebras satisfying the above conditions must have the same dimensions.
\end{document}
|
arXiv
|
\begin{document}
\title[Resurgence of the Anger--Weber function]{The resurgence properties of\\ the large order asymptotics of\\ the Anger--Weber function I}
\begin{abstract} The aim of this paper is to derive new representations for the Anger--Weber function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A \textbf{439} (1992) 373--396). Using these representations, we obtain a number of properties of the large order asymptotic expansions of the Anger--Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. \end{abstract} \maketitle
\section{Introduction}\label{section1}
In this paper, we investigate the large $\nu$ asymptotics of the Anger--Weber function $\mathbf{A}_{ - \nu } \left( \nu x\right)$. The asymptotic expansion of this function has different forms according to whether $0<x<1$, $x=1$ or $x>1$ \cite[p. 298]{NIST} (see also Olver \cite[p. 352]{Olver}). We shall consider the latter two cases. A brief discussion about the expansion for $\mathbf{A}_{ - \nu } \left( \nu x\right)$ with $0<x<1$ is given in Section \ref{section6}. Meijer \cite{Meijer} gave error bounds for the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( \nu x\right)$ when $x > 1$. Dingle \cite{Dingle} obtained exponentially improved versions of the asymptotic series and asymptotic approximations for their late terms. Nevertheless, the derivation of his results is based on interpretive, rather than rigorous, methods. In an earlier paper \cite{Nemes}, we proved resurgence-type formulas for the Hankel function $H_{\nu}^{\left( 1 \right)} \left( \nu x\right)$ when $x=1$ and $x>1$, respectively. The main aim of this paper is to derive similar new representations for the Anger--Weber function $\mathbf{A}_{ - \nu } \left( \nu x\right)$. Our derivation is based on the reformulation of the method of steepest descents by Howls \cite{Howls}. Using these representations, we obtain a number of properties of the large order asymptotic expansions of the Anger--Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. Some of our error bounds coincide with the ones given by Meijer while some others are simpler. Our analysis also provides a rigorous treatment of Dingle's formal expansions.
Our first theorem describes the resurgence properties of the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( \nu x\right)$ for $x > 1$. We employ the substitution $x = \sec \beta$ with an appropriate $0 < \beta <\frac{\pi}{2}$. The notations follow the ones given in \cite[p. 298]{NIST}. Throughout this paper, empty sums are taken to be zero.
\begin{theorem}\label{thm1} Let $0 < \beta <\frac{\pi}{2}$ be a fixed acute angle, and let $N$ be a non-negative integer. Then we have \begin{equation}\label{eq18} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = - \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{{\nu ^{2n + 1} }}} + R_N \left( {\nu ,\beta } \right) \end{equation} for $-\frac{\pi}{2} < \arg \nu < \frac{\pi}{2}$, with \begin{equation}\label{eq9} a_n \left( { - \sec \beta } \right) = - \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{t}{{\sec \beta \sinh t - t}}} \right)^{2n + 1} } \right]_{t = 0} = \frac{{\left( { - 1} \right)^{n + 1} }}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} \end{equation} and \begin{equation}\label{eq8} R_N \left( {\nu ,\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \end{equation} \end{theorem}
It was shown in \cite{Nemes}, that for any fixed $0 < \beta <\frac{\pi}{2}$ and non-negative integer $M$, the Hankel function $H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right)$ has the representation \begin{equation}\label{eq40} H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right) = \frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi}{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\left( {\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} + R_M^{\left(H\right)} \left( {\nu ,\beta } \right)} \right), \end{equation} for $-\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$ with \begin{gather}\label{eq41} \begin{split} U_m \left( {i\cot \beta } \right) & = \left( { - 1} \right)^m \frac{{\left( {i\cot \beta } \right)^m }}{{2^m m!}}\left[ {\frac{{d^{2m} }}{{dt^{2m} }}\left( {\frac{1}{2}\frac{{t^2 }}{{i\cot \beta \left( {t - \sinh t} \right) + \cosh t - 1}}} \right)^{m + \frac{1}{2}} } \right]_{t = 0}\\ & = \frac{i^m}{2\left( {2\pi \cot \beta } \right)^{\frac{1}{2}}}\int_0^{ + \infty } {t^{m - \frac{1}{2}} e^{ -t \left( {\tan \beta - \beta } \right)} \left( {1 + e^{ - 2\pi t} } \right)i H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \end{split} \end{gather} The remainder term $R_M^{\left(H\right)} \left( {\nu ,\beta } \right)$ can be expressed as \begin{equation}\label{eq42} R_M^{\left(H\right)} \left( {\nu ,\beta } \right) = \frac{1}{{2\left( {2\pi \cot \beta } \right)^{\frac{1}{2}} \left(i\nu\right)^M }}\int_0^{ + \infty } {\frac{{t^{M - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}\left( {1 + e^{ - 2\pi t} } \right)i H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \end{equation} This representation of the Hankel function will play an important role in later sections of this paper.
If $\mathbf{J}_{\nu}\left(z\right)$ denotes the Anger function, then $\mathbf{J}_{-\nu}\left(z\right) = \mathbf{J}_{\nu}\left(-z\right)$ and $\sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = J_{ - \nu } \left( {\nu \sec \beta } \right) - \mathbf{J}_{ - \nu } \left( {\nu \sec \beta } \right)$ (see \cite[p. 296]{NIST}). From these and the continuation formulas for the Bessel and Hankel functions (see \cite[p. 222 and p. 226]{NIST}), we find \begin{align*} \sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu e^{2\pi im} \sec \beta } \right) = &\; J_{ - \nu } \left( {\nu e^{2\pi im} \sec \beta } \right) - \mathbf{J}_{ - \nu } \left( {\nu e^{2\pi im} \sec \beta } \right)\\ = &\; \sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) + \left( {e^{ - 2\pi im\nu } - 1} \right)J_{ - \nu } \left( {\nu \sec \beta } \right)\\ = &\; \sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) - ie^{ - \pi i\left( {m - 1} \right)\nu } \sin \left( {\pi m\nu } \right)H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right)\\ & - ie^{ - \pi i\left( {m + 1} \right)\nu } \sin \left( {\pi m\nu } \right)H_\nu ^{\left( 2 \right)} \left( {\nu \sec \beta } \right) \end{align*} for every integer $m$. From this expression and the resurgence formulas \eqref{eq18}, \eqref{eq40}, we can derive analogous representations in sectors of the form \[ \left( {2m - \frac{1}{2}} \right)\pi < \arg \nu < \left( {2m + \frac{1}{2}} \right)\pi ,\; m \in \mathbb{Z}. \]
Similarly, applying the continuation formulas \begin{align*} & - \sin \left( {\pi \nu } \right)\mathbf{A}_{\nu } \left( {\nu e^{\left( {2m + 1} \right)\pi i} \sec \beta } \right) = J_{ \nu } \left( {\nu e^{\left( {2m + 1} \right)\pi i} \sec \beta } \right) - \mathbf{J}_{ \nu } \left( {\nu e^{\left( {2m + 1} \right)\pi i} \sec \beta } \right)\\ & = \sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) + e^{\left( {2m + 1} \right)\pi i\nu } J_\nu \left( {\nu \sec \beta } \right) - J_{ - \nu } \left( {\nu \sec \beta } \right)\\ & = \sin \left( {\pi \nu } \right)\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) + ie^{\pi i\left( {m + 1} \right)\nu } \sin \left( {\pi m\nu } \right)H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right) + ie^{\pi im\nu } \sin \left( {\pi \left( {m + 1} \right)\nu } \right)H_\nu ^{\left( 2 \right)} \left( {\nu \sec \beta } \right) \end{align*} and the representations \eqref{eq18}, \eqref{eq40}, we can obtain resurgence formulas in any sector of the form \[ \left( {2m + \frac{1}{2}} \right)\pi < \arg \nu < \left( {2m + \frac{3}{2}} \right)\pi ,\; m \in \mathbb{Z}. \] The lines $\arg \nu = \left( {2m \pm \frac{1}{2}} \right)\pi$ are the Stokes lines for the function $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$.
When $\nu$ is an integer, the limiting values have to be taken in these continuation formulas.
The second theorem provides a resurgence formula for $\mathbf{A}_{ - \nu } \left( \nu \right)$.
\begin{theorem}\label{thm2} For any non-negative integer $N$, we have \begin{equation}\label{eq19} \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi }}\sum\limits_{n = 0}^{N - 1} {d_{2n} \frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} + R_N \left( \nu \right) \end{equation} for $-\frac{3\pi}{2} < \arg \nu < \frac{3\pi}{2}$, with \begin{equation}\label{eq17} d_{2n} = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{{t^3 }}{{\sinh t - t}}} \right)^{\frac{{2n + 1}}{3}} } \right]_{t = 0} = \frac{{\left( { - 1} \right)^n }}{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}\int_0^{ + \infty } {t^{\frac{{2n - 2}}{3}} e^{ - 2\pi t} i H_{it}^{\left( 1 \right)} \left( {it} \right)dt} \end{equation} and \begin{equation}\label{eq16} R_N \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {\frac{{t^{\frac{{2N - 2}}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} . \end{equation} The cube roots are defined to be positive on the positive real line and are defined by analytic continuation elsewhere. \end{theorem}
In the previous paper \cite{Nemes}, we proved a similar representation for the Hankel function $H_\nu ^{\left( 1 \right)} \left( \nu \right)$, in particular for any non-negative integer $N$, we have \begin{equation}\label{eq53} H_\nu ^{\left( 1 \right)} \left( \nu \right) = - \frac{2}{{3\pi }}\sum\limits_{n = 0}^{N - 1} {d_{2n} e^{\frac{{2\left( {2n + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2n + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} + R_N^{\left( H \right)} \left( \nu \right) \end{equation} with $-\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$. The remainder term $R_N^{\left( H \right)} \left( \nu \right)$ has the integral representation \begin{equation}\label{eq54} R_N^{\left( H \right)} \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {t^{\frac{{2N - 2}}{3}} e^{ - 2\pi t} \left( {\frac{{e^{\frac{{\left( {2N + 1} \right)\pi i}}{3}} }}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} e^{\frac{{2\pi i}}{3}} }} + \frac{1}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} }}} \right)H_{it}^{\left( 1 \right)} \left( {it} \right)dt} . \end{equation} The cube roots are defined to be positive on the positive real line and are defined by analytic continuation elsewhere. This result will be important for us in later sections of the paper.
Again, the formula \eqref{eq19} can be extended to other sectors of the complex plane. (One has to replace the factor $\sec \beta$ by 1 in the continuation formulas given earlier.)
If we neglect the remainder terms and extend the sums to $N = \infty$ in Theorems \ref{thm1} and \ref{thm2}, we recover the known asymptotic series of the Anger--Weber function. Some other formulas for the coefficients $a_n \left( { - \sec \beta } \right)$ can be found in Appendix \ref{appendixa}. For the computation of the $d_{2n}$, see \cite[Appendix A]{Nemes}.
In the following two theorems, we give exponentially improved asymptotic expansions for the function $\mathbf{A}_{ - \nu } \left(\nu x\right)$ when $x>1$ and $x=1$, respectively. These new expansions can be viewed as the mathematically rigorous forms of the terminated series of Dingle \cite[pp. 485]{Dingle}. We express these expansions in terms of the Terminant function $\widehat T_p\left(w\right)$ whose definition and basic properties are given in Section \ref{section5}. In Theorem \ref{thm3}, $R_N \left( {\nu ,\beta } \right)$ is defined by \eqref{eq18} and it is extended to the sector $\left|\arg \nu\right| \leq \frac{3\pi}{2}$ via analytic continuation. Throughout this paper, we use subscripts in the $\mathcal{O}$ notations to indicate the dependence of the implied constant on certain parameters.
\begin{theorem}\label{thm3} Suppose that $\left|\arg \nu\right| \leq \frac{3\pi}{2}$, $\left|\nu\right|$ is large and $N = \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right) + \rho$ is a positive integer with $\rho$ being bounded. Then \begin{align*} R_N \left( {\nu ,\beta } \right) = \; & i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right)}\\ & - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)} + R_{N,M} \left( {\nu ,\beta } \right) \end{align*} with $M$ being an arbitrary fixed non-negative integer, and \[
R_{N,M} \left( {\nu ,\beta } \right) = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \]
for $\left|\arg \nu\right| \leq \frac{\pi}{2}$; \[
R_{N,M} \left( {\nu ,\beta } \right) = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ \mp \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \] for $\frac{\pi}{2} \leq \pm \arg \nu \leq \frac{3\pi}{2}$. \end{theorem}
\begin{theorem}\label{thm4} Define $R_{N,M,K}\left(\nu\right)$ by \begin{align*} \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi \nu ^{\frac{1}{3}} }}\sum\limits_{n = 0}^{N - 1} {d_{6n} \frac{{\Gamma \left( {2n + \frac{1}{3}} \right)}}{{\nu ^{2n} }}} & + \frac{1}{{3\pi \nu }}\sum\limits_{m = 0}^{M - 1} {d_{6m + 2} \frac{{\Gamma \left( {2m + 1} \right)}}{{\nu ^{2m} }}} \\ & + \frac{1}{{3\pi \nu ^{\frac{5}{3}} }}\sum\limits_{k = 0}^{K - 1} {d_{6k + 4} \frac{{\Gamma \left( {2k + \frac{5}{3}} \right)}}{{\nu ^{2k} }}} + R_{N,M,K} \left( \nu \right), \end{align*} where \[
N= \pi\left| \nu \right| + \rho, \; M= \pi\left| \nu \right| + \sigma \; \text{ and } \; K= \pi\left| \nu \right| + \eta, \]
$\left|\nu\right|$ being large, $\rho$, $\sigma$ and $\eta$ being bounded quantities such that $N,M,K \geq 1$. Then \begin{gather}\label{eq68} \begin{split} R_{N,M,K} \left( \nu \right) =\; & i\frac{{e^{ - 2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{j = 0}^{J - 1} {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}\widehat T_{2N - \frac{{2j}}{3}} \left( { - 2\pi i\nu } \right)} \\ & - ie^{\frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{j = 0}^{J - 1} {d_{2j} e^{\frac{{2\left( {2j + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}\widehat T_{2N - \frac{{2j}}{3}} \left( {2\pi i\nu } \right)} \\ & + i\frac{{e^{ - 2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{\ell = 0}^{L - 1} {d_{2\ell } \sin \left( {\frac{{\left( {2\ell + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2\ell + 1}}{3}} \right)}}{{\nu ^{\frac{{2\ell + 1}}{3}} }}\widehat T_{2M - \frac{{2\ell - 2}}{3}} \left( { - 2\pi i\nu } \right)} \\ & + i\frac{{e^{2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{\ell = 0}^{L - 1} {d_{2\ell } e^{\frac{{2\left( {2\ell + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2\ell + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2\ell + 1}}{3}} \right)}}{{\nu ^{\frac{{2\ell + 1}}{3}} }}\widehat T_{2M - \frac{{2\ell - 2}}{3}} \left( {2\pi i\nu } \right)} \\ & + i\frac{{e^{ - 2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{q = 0}^{Q - 1} {d_{2q} \sin \left( {\frac{{\left( {2q + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2q + 1}}{3}} \right)}}{{\nu ^{\frac{{2q + 1}}{3}} }}\widehat T_{2K - \frac{{2q - 4}}{3}} \left( { - 2\pi i\nu } \right)} \\ & - ie^{ - \frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}\frac{2}{{3\pi }}\sum\limits_{q = 0}^{Q - 1} {d_{2q} e^{\frac{{2\left( {2q + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2q + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2q + 1}}{3}} \right)}}{{\nu ^{\frac{{2q + 1}}{3}} }}\widehat T_{2K - \frac{{2q - 4}}{3}} \left( {2\pi i\nu } \right)} \\ & + R_{N,M,K}^{J,L,Q} \left( \nu \right), \end{split} \end{gather} where $J$, $L$ and $Q$ are arbitrary fixed non-negative integers satisfying $J,L,Q \equiv 0 \mod 3$, and \begin{gather}\label{eq62} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \mathcal{O}_{J,\rho } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) & + \mathcal{O}_{L,\sigma } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& + \mathcal{O}_{Q,\eta } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \end{split} \end{gather} for $-\frac{\pi}{2} \leq \arg\nu \leq \frac{\pi}{2}$; \begin{gather}\label{eq63} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \mathcal{O}_{J,\rho } \left( {e^{ \mp 2\pi \Im \left( \nu \right)} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) & + \mathcal{O}_{L,\sigma } \left( {e^{ \mp 2\pi \Im \left( \nu \right)} \left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& + \mathcal{O}_{Q,\eta } \left( {e^{ \mp 2\pi \Im \left( \nu \right)} \left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \end{split} \end{gather} for $\frac{\pi}{2} \leq \pm \arg \nu \leq \frac{3\pi}{2}$; \begin{align*}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \; & \mathcal{O}_{J,\rho } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) + \mathcal{O}_{L,\sigma } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& +\mathcal{O}_{Q,\eta } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) + \mathcal{O}_{J,L,Q} \left( {\left| \nu \right|^{ - \frac{1}{3}} } \right) \end{align*} for $\frac{3\pi}{2} \leq \pm \arg \nu \leq \frac{5\pi}{2}$. Moreover, if $J=L=Q$, then the bound \eqref{eq62} remains valid in the larger sector $-\frac{3\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, and the estimate \eqref{eq63} holds in the sectors $\frac{3\pi}{2} \leq \mp \arg \nu \leq \frac{5\pi}{2}$. \end{theorem}
The assumption that $J,L,Q \equiv 0 \mod 3$ is only for simplicity. Estimations for $R_{N,M,K}^{J,L,Q} \left( \nu \right)$ when $J$, $L$ or $Q$ may not be divisible by $3$ can be obtained similarly.
We remark that Dingle writes $A_\nu\left(z\right)$ in place of $\mathbf{A}_{-\nu}\left(z\right)$; and Olver's definition for $\mathbf{A}_{-\nu}\left(z\right)$ omits the factor $\frac{1}{\pi}$ in \eqref{eq69} below.
The rest of the paper is organized as follows. In Section \ref{section2}, we prove the resurgence formulas stated in Theorems \ref{thm1} and \ref{thm2}. In Section \ref{section3}, we give explicit and realistic error bounds for the asymptotic expansions of $\mathbf{A}_{-\nu}\left(\nu x\right)$ when $x\geq 1$ using the results of Section \ref{section2}. In Section \ref{section4}, asymptotic approximations for $a_n \left( { - \sec \beta } \right)$ as $n \to +\infty$ are given. In Section \ref{section5}, we prove the exponentially improved expansions presented in Theorems \ref{thm3} and \ref{thm4}, and provide a detailed discussion of the Stokes phenomenon related to the expansions of $\mathbf{A}_{-\nu}\left( \nu x \right)$. The paper concludes with a discussion in Section \ref{section6}.
\section{Proofs of the resurgence formulas}\label{section2}
Our analysis is based on the integral definition of the Anger--Weber function \begin{equation}\label{eq69}
\mathbf{A}_{ - \nu } \left( z \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{\nu t - z\sinh t} dt} \quad \left| {\arg z} \right| < \frac{\pi }{2}. \end{equation} If $z = \nu x$, where $x$ is a positive constant, then \begin{equation}\label{eq10}
\mathbf{A}_{ - \nu } \left( {\nu x} \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \left( {x\sinh t - t} \right)} dt} \quad \left| {\arg \nu } \right| < \frac{\pi }{2}. \end{equation} The analysis is significantly different according to whether $x > 1$ or $x = 1$. The saddle points of the integrand are the roots of the equation $x \cosh t = 1$. Hence, the saddle points are given by $t_{\pm}^{\left( k \right)} = \pm \mathrm{sech}^{ - 1} x + 2\pi ik$ where $k$ is an arbitrary integer. When $x=1$, we shall use the simpler notation $t^{\left( k \right)} = 2\pi ik$. We denote by $\mathscr{C}_{\pm}^{\left( k \right)}\left(\theta\right)$ the portion of the steepest paths that pass through the saddle point $t_{\pm}^{\left( k \right)}$. Here, and subsequently, we write $\theta = \arg \nu$. Similarly, $\mathscr{C}^{\left( k \right)}\left(\theta\right)$ denotes the steepest paths through the saddle point $t^{\left( k \right)}$. As for the path of integration in \eqref{eq10}, we take \begin{equation}\label{eq71}
\mathscr{P}\left(\theta \right) = \left\{ {t \in \mathbb{C}:\arg \left[ {e^{i\theta } \left( {x\sinh t - t} \right)} \right] = 0,\, \Re \left( t \right) > 0,\, \left| {\Im \left( t \right)} \right| < \frac{\pi }{2}} \right\} . \end{equation} We remark that $\mathscr{P}\left(0\right)$ is the positive real axis. If $x=1$, the path $\mathscr{P}\left(\theta\right)$ is part of the contour $\mathscr{C}^{\left( 0 \right)}\left(\theta\right)$.
\subsection{Case (i): $x>1$} Let $0<\beta<\frac{\pi}{2}$ be defined by $\sec \beta = x$. For simplicity, we assume that $\theta = 0$. In due course, we shall appeal to an analytic continuation argument to extend our results to complex $\nu$. Let $f\left( {t,\beta } \right) = \sec \beta \sinh t - t$. If \begin{equation}\label{eq1} \tau = f\left( {t,\beta } \right), \end{equation} then $\tau$ is real on the curve $\mathscr{P}\left(0\right)$, and, as $t$ travels along this curve from $0$ to $+\infty $, $\tau$ increases from $0$ to $+\infty$. Therefore, corresponding to each positive value of $\tau$, there is a value of $t$, say $t\left(\tau\right)$, satisfying \eqref{eq1} with $t\left(\tau\right)>0$. In terms of $\tau$, we have \[ \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{{dt\left( \tau \right)}}{{d\tau }}d\tau } = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{1}{{\sec \beta \cosh t\left( \tau \right) - 1}}d\tau } . \] Following Howls, we express the function involving $t\left(\tau\right)$ as a contour integral using the residue theorem, to find \[ \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - 1} \left( {u,\beta } \right)}}{{1 - \tau ^2 f^{ - 2} \left( {u,\beta } \right)}}du} d\tau } \] where the contour $\Gamma$ encircles the path $\mathscr{P}\left(0\right)$ in the positive direction and does not enclose any of the saddle points $t_ \pm ^{\left( k \right)}$ (see Figure \ref{fig1}). Now, we employ the well-known expression for non-negative integer $N$ \begin{equation}\label{eq2} \frac{1}{1 - z} = \sum\limits_{n = 0}^{N-1} {z^n} + \frac{z^N}{1 - z},\; z \neq 1, \end{equation} to expand the function under the contour integral in powers of $\tau ^2 f^{ - 2} \left( {u,\beta } \right)$. The result is \[ \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = - \frac{1}{\pi}\sum\limits_{n = 0}^{N - 1} {\int_0^{ + \infty } {\tau^{2n} e^{ - \nu \tau }\frac{{ - 1}}{{2\pi i}}\oint_\Gamma {\frac{{du}}{{f^{2n + 1} \left( {u,\beta } \right)}}} d\tau } } + R_N \left( {\nu ,\beta } \right), \] where \begin{equation}\label{eq3} R_N \left( {\nu ,\beta } \right) = \frac{1}{\pi}\int_0^{ + \infty } {\tau^{2N} e^{ - \nu \tau } \frac{1}{2\pi i}\oint_\Gamma {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \tau ^2 f^{ - 2} \left( {u,\beta } \right)}}du} d\tau } . \end{equation} The path $\Gamma$ in the sum can be shrunk into a small circle around $0$, and we arrive at \begin{equation}\label{eq4} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = - \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{{\nu ^{2n + 1} }}} + R_N \left( {\nu ,\beta } \right), \end{equation} where \[ a_n \left( { - \sec \beta } \right) = \frac{{ - 1}}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\frac{{du}}{{f^{2n + 1} \left( {u,\beta } \right)}}} = - \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{t}{{\sec \beta \sinh t - t}}} \right)^{2n + 1} } \right]_{t = 0} . \]
\begin{figure}
\caption{The contour $\Gamma$ encircling the path $\mathscr{P}\left(0\right)$.}
\label{fig1}
\end{figure}
Performing the change of variable $\nu \tau = s$ in \eqref{eq3} yields \begin{equation}\label{eq5} R_N \left( {\nu ,\beta } \right) = \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds} . \end{equation} This representation of $R_N \left( {\nu ,\beta } \right)$ and the formula \eqref{eq4} can be continued analytically if we choose $\Gamma = \Gamma\left(\theta\right)$ to be an infinite contour that surrounds the path $\mathscr{P}\left(\theta\right)$ in the anti-clockwise direction and that does not encircle any of the saddle points $t_ \pm ^{\left( k \right)}$. This continuation argument works until the path $\mathscr{P}\left(\theta\right)$ runs into a saddle point. In the terminology of Howls, such saddle points are called adjacent to the endpoint $0$. As \[
\left|\arg \left( {f\left(0 ,\beta\right) - f\left( {t_ \pm ^{\left( k \right)} ,\beta } \right)} \right)\right| = \frac{\pi}{2} \] for any saddle point $t_\pm ^{\left( k \right)}$, we infer that \eqref{eq5} is valid as long as $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ with a contour $\Gamma\left(\theta\right)$ specified above. When $\theta = -\frac{\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t_ + ^{\left( 0 \right)} = i\beta$. Similarly, when $\theta = \frac{\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t_ - ^{\left( 0 \right)} = -i\beta$. These are the adjacent saddles. The set \[ \Delta = \left\{ {u \in \mathscr{P}\left( \theta \right) : - \frac{\pi}{2} < \theta < \frac{\pi}{2}} \right\} \] forms a domain in the complex plane whose boundary contains portions of steepest descent paths through the adjacent saddles (see Figure \ref{fig2}). These paths are $\mathscr{C}_ + ^{\left( 0 \right)} \left( { \frac{\pi }{2}} \right)$ and $\mathscr{C}_ - ^{\left( 0 \right)} \left( { - \frac{\pi }{2}} \right)$, and they are called the adjacent contours to the endpoint $0$. The function under the contour integral in \eqref{eq5} is an analytic function of $u$ in the domain $\Delta$, therefore we can deform $\Gamma$ over the adjacent contours. We thus find that for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ and $N \geq 0$, \eqref{eq5} may be written \begin{gather}\label{eq6} \begin{split} R_N \left( {\nu ,\beta } \right) = \; & \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{C}_ + ^{\left( 0 \right)} \left( {\frac{\pi }{2}} \right)} {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds} \\ & + \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{C}_ - ^{\left( 0 \right)} \left( { - \frac{\pi }{2}} \right)} {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds} . \end{split} \end{gather} Now we make the changes of variable \[
s = t\frac{{\left| {f\left( {i\beta ,\beta } \right)-f\left(0,\beta\right)} \right|}}{{f\left( {i\beta ,\beta } \right)-f\left(0,\beta\right)}}f\left( {u,\beta } \right) = - itf\left( {u,\beta } \right) \] in the first, and \[
s = t\frac{{\left| {f\left( { - i\beta ,\beta } \right)-f\left(0,\beta\right)} \right|}}{{f\left( { - i\beta ,\beta } \right)-f\left(0,\beta\right)}}f\left( {u,\beta } \right) = itf\left( {u,\beta } \right) \] in the second double integral. Clearly, by the definition of the adjacent contours, $t$ is positive. The quantities $f\left( {i\beta ,\beta } \right) -f\left(0,\beta\right) = i\left( {\tan \beta - \beta } \right)$ and $f\left( { - i\beta ,\beta } \right)-f\left(0,\beta\right) = -i\left( {\tan \beta - \beta } \right)$ were essentially called the ``singulants" by Dingle \cite[p. 147]{Dingle}. With these changes of variable, the representation \eqref{eq6} for $R_N \left( {\nu ,\beta } \right)$ becomes \begin{equation}\label{eq7} R_N \left( {\nu ,\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} }}{{1 + \left( {t/\nu } \right)^2 }}\left( {\frac{1}{{2\pi }}\int_{\mathscr{C}_ - ^{\left( 0 \right)} \left( { - \frac{\pi }{2}} \right)} {e^{ - itf\left( {u,\beta } \right)} du} - \frac{1}{{2\pi }}\int_{\mathscr{C}_ + ^{\left( 0 \right)} \left( {\frac{\pi }{2}} \right)} {e^{itf\left( {u,\beta } \right)} du} } \right)dt} , \end{equation} for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ and $N \geq 0$. Finally, the contour integrals can themselves be represented in terms of the Hankel functions since \[ \frac{1}{{2\pi }}\int_{\mathscr{C}_ - ^{\left( 0 \right)} \left( { - \frac{\pi }{2}} \right)} {e^{ - itf\left( {u,\beta } \right)} du} = - \frac{i}{2}H_{ - it}^{\left( 2 \right)} \left( { - it\sec \beta } \right) = \frac{i}{2}H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right), \] and \[ -\frac{1}{{2\pi }}\int_{\mathscr{C}_ + ^{\left( 0 \right)} \left( {\frac{\pi }{2}} \right)} {e^{itf\left( {u,\beta } \right)} du} = \frac{i}{2}H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right) . \] Substituting these into \eqref{eq7} gives \eqref{eq8}. To prove the second representation in \eqref{eq9}, we apply \eqref{eq8} for the right-hand side of \[ a_n \left( { - \sec \beta } \right) = \pi \frac{\nu ^{2n + 1}}{\left( {2n} \right)!}\left( {R_{n + 1} \left( {\nu ,\beta } \right) - R_n \left( {\nu ,\beta } \right)} \right) . \]
\begin{figure}\label{fig2}
\end{figure}
\subsection{Case (ii): $x=1$} We assume that $\theta = 0$ and later we shall use an analytic continuation argument to extend the results to complex $\nu$. Let $f\left( t \right) = \sinh t-t$. If \begin{equation}\label{eq11} \tau = f\left( t \right), \end{equation} then $\tau$ is real on the curve $\mathscr{P}\left(0\right)$, and, as $t$ travels along this curve from $0$ to $+\infty $, $\tau$ increases from $0$ to $+\infty$. Therefore, corresponding to each positive value of $\tau$, there is a value of $t$, say $t\left(\tau\right)$, satisfying \eqref{eq11} with $t\left(\tau\right)>0$. In terms of $\tau$, we have \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{{dt\left( \tau \right)}}{{d\tau }}d\tau } = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{1}{{\cosh t\left( \tau \right) - 1}}d\tau } . \] As in the first case, we express the function involving $t\left(\tau\right)$ as a contour integral using the residue theorem, to obtain \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi }}\int_0^{ + \infty } {\tau ^{ - \frac{2}{3}} e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - \frac{1}{3}} \left( u \right)}}{{1 - \tau ^{\frac{2}{3}} f^{ - \frac{2}{3}} \left( u \right)}}du} d\tau } \] where the contour $\Gamma$ encircles the path $\mathscr{P}\left(0\right)$ in the positive direction and does not enclose any of the saddle points $t ^{\left( k \right)} \neq t ^{\left( 0 \right)}$ (cf. Figure \ref{fig1}). The cube root is defined so that $f^{\frac{1}{3}}\left( t \right)$ is positive on the path $\mathscr{P}\left(0\right)$. Next we apply the expression \eqref{eq2} to expand the function under the contour integral in powers of $\tau^{\frac{2}{3}} f^{-\frac{2}{3}} \left( u \right)$. The result is \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi }}\sum\limits_{n = 0}^{N - 1} {\int_0^{ + \infty } {\tau ^{\frac{{2n - 2}}{3}} e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{du}}{{f^{\frac{{2n + 1}}{3}} \left( u \right)}}} d\tau } } + R_N \left( \nu \right) \] where \begin{equation}\label{eq12} R_N \left( \nu \right) = \frac{1}{{3\pi }}\int_0^{ + \infty } {\tau ^{\frac{{2N - 2}}{3}} e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - \frac{{2N + 1}}{3}} \left( u \right)}}{{1 - \tau ^{\frac{2}{3}} f^{ - \frac{2}{3}} \left( u \right)}}du} d\tau } . \end{equation} The path $\Gamma$ in the sum can be shrunk into a small circle around $t^{\left( 0 \right)} = 0$, and we arrive at \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi }}\sum\limits_{n = 0}^{N - 1} {d_{2n} \frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} + R_N \left( \nu \right) \] where \[ d_{2n} = \frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\frac{{du}}{{f^{\frac{{2n + 1}}{3}} \left( u \right)}}} = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{{t^3 }}{{\sinh t - t}}} \right)^{\frac{{2n + 1}}{3}} } \right]_{t = 0} . \]
\begin{figure}\label{fig3}
\end{figure}
Applying the change of variable $\nu \tau = s$ in \eqref{eq12} gives \begin{equation}\label{eq13} R_N \left( \nu \right) = \frac{1}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2N - 2}}{3}} e^{ - s} \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - \frac{{2N + 1}}{3}} \left( u \right)}}{{1 - \left( {s/\nu } \right)^{\frac{2}{3}} f^{ - \frac{2}{3}} \left( u \right)}}du} ds} . \end{equation} As in the first case, we need to locate the adjacent saddle points. When $\theta = -\frac{3\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t ^{\left( 1 \right)} = 2\pi i$. Similarly, when $\theta = \frac{3\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t ^{\left( -1 \right)} = -2\pi i$. Therefore, the adjacent saddles are $t^{\left(\pm 1\right)}$. The set \[ \Delta = \left\{ {u \in \mathscr{P}\left( \theta \right) : - \frac{3\pi}{2} < \theta < \frac{3\pi}{2}} \right\} \] forms a domain in the complex plane whose boundary contains portions of steepest descent paths through the adjacent saddles (see Figure \ref{fig3}). These paths are $\mathscr{L} ^{\left( 1 \right)} \left( { -\frac{\pi }{2}} \right)$ and $\mathscr{P}^{\left( -1 \right)} \left( { \frac{\pi }{2}} \right)$, the adjacent contours to the saddle point $t ^{\left( 0 \right)}$ (these paths are defined in \cite{Nemes}). The function under the contour integral in \eqref{eq13} is an analytic function of $u$ in the domain $\Delta$, therefore we can deform $\Gamma$ over the adjacent contours. We thus find that for $-\frac{3\pi}{2} < \theta < \frac{3\pi}{2}$ and $N \geq 0$, \eqref{eq13} may be written \begin{gather}\label{eq14} \begin{split} R_N \left( \nu \right) = \; & \frac{1}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2N - 2}}{3}} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{L}^{\left( 1 \right)} \left( { - \frac{\pi }{2}} \right)} {\frac{{f^{ - \frac{{2N + 1}}{3}} \left( u \right)}}{{1 - \left( {s/\nu } \right)^{\frac{2}{3}} f^{ - \frac{2}{3}} \left( u \right)}}du} ds} \\ & + \frac{1}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2N - 2}}{3}} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{P}^{\left( { - 1} \right)} \left( {\frac{\pi }{2}} \right)} {\frac{{f^{ - \frac{{2N + 1}}{3}} \left( u \right)}}{{1 - \left( {s/\nu } \right)^{\frac{2}{3}} f^{ - \frac{2}{3}} \left( u \right)}}du} ds} . \end{split} \end{gather} Now we perform the changes of variable \[
s = t\frac{{\left| {f\left( {2\pi i} \right) - f\left( 0 \right)} \right|}}{{f\left( 2\pi i \right) - f\left( 0 \right)}} f\left( u \right) = itf\left( u \right) \] in the first, and \[
s = t\frac{{\left| {f\left( { - 2\pi i} \right) - f\left( 0 \right)} \right|}}{{f\left( { - 2\pi i} \right) - f\left( 0 \right)}} f\left( u \right) = - itf\left( u \right) \] in the second double integral. In this case, Dingle's singulants are $f\left( { \pm 2\pi i} \right) - f\left( 0 \right) = \mp 2\pi i$. When using these changes of variable, we should take $i^{\frac{2}{3}} = -1$ in the first, and $\left( { - i} \right)^{\frac{2}{3}} = -1$ in the second double integral. With these changes of variable, the representation \eqref{eq14} for $R_N \left( \nu \right)$ becomes \begin{equation}\label{eq15} R_N \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {\frac{{t^{\frac{{2N - 2}}{3}} }}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} }}\left( {\frac{1}{{2\pi }}\int_{\mathscr{P}^{\left( { - 1} \right)} \left( {\frac{\pi }{2}} \right)} {e^{itf\left( u \right)} du} - \frac{1}{{2\pi }}\int_{\mathscr{L}^{\left( 1 \right)} \left( { - \frac{\pi }{2}} \right)} {e^{ - itf\left( u \right)} du} } \right)dt} , \end{equation} for $-\frac{3\pi}{2} < \theta < \frac{3\pi}{2}$ and $N \geq 0$. Finally, the contour integrals can themselves be represented in terms of the Hankel functions since \[ \frac{1}{{2\pi }}\int_{\mathscr{P}^{\left( { - 1} \right)} \left( {\frac{\pi }{2}} \right)} {e^{itf\left( u \right)} du} = \frac{{e^{ - 2\pi t} }}{2}iH_{it}^{\left( 1 \right)} \left( {it} \right), \] and \[
- \frac{1}{{2\pi }}\int_{\mathscr{L}^{\left( 1 \right)} \left( { - \frac{\pi }{2}} \right)} {e^{ - itf\left( u \right)} du} = - \frac{{e^{ - 2\pi t} }}{2}iH_{ - it}^{\left( 2 \right)} \left( { - it} \right) = \frac{{e^{ - 2\pi t} }}{2}iH_{it}^{\left( 1 \right)} \left( {it} \right) . \] Substituting these into \eqref{eq15} gives \eqref{eq16}. To prove the second representation in \eqref{eq17}, we apply \eqref{eq16} for the right-hand side of \[ d_{2n} = 3\pi \frac{{\nu ^{\frac{{2n + 1}}{3}} }}{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}\left( {R_n \left( \nu \right) - R_{n + 1} \left( \nu \right)} \right). \]
\section{Error bounds}\label{section3}
In this section we derive explicit and realistic error bounds for the large order asymptotic series of the Anger--Weber function. The proofs are based on the resurgence formulas given in Theorems \ref{thm1} and \ref{thm2}.
We comment on the relation between Meijer's work on the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$ \cite{Meijer} and ours. Some of the estimates in \cite{Meijer} coincide with ours and are valid in wider sectors of the complex $\nu$-plane. However, it should be noted that those bounds become less effective outside the sectors of validity of the representation \eqref{eq8} due to the Stokes phenomenon. For those sectors we recommend the use of the continuation formulas given in Section \ref{section1}.
To estimate the remainder terms, we shall use the elementary result that \begin{equation}\label{eq20}
\frac{1}{{\left| {1 - re^{i\varphi } } \right|}} \le \begin{cases} \left|\csc \varphi \right| & \; \text{ if } \; 0 < \left|\varphi \text{ mod } 2\pi\right| <\frac{\pi}{2} \\ 1 & \; \text{ if } \; \frac{\pi}{2} \leq \left|\varphi \text{ mod } 2\pi\right| \leq \pi \end{cases} \end{equation} holds for any $r>0$. We will also need the fact that \begin{equation}\label{eq21} iH_{it}^{\left(1\right)} \left( {itx} \right) \ge 0 \end{equation} for any $t>0$ and $x\geq 1$ (see \cite{Nemes}).
\subsection{Case (i): $x>1$} As usual, let $0<\beta<\frac{\pi}{2}$ be defined by $\sec \beta = x$. We observe that from \eqref{eq9} and \eqref{eq21} it follows that \[
\left| {a_n \left( { - \sec \beta } \right)} \right| = \frac{1}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \] Using this formula, together with the representation \eqref{eq8} and the estimate \eqref{eq20}, we obtain the error bound \begin{equation}\label{eq23}
\left| {R_N \left( {\nu ,\beta } \right)} \right| \le \frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( { - \sec \beta } \right)} \right|}}{{\left| \nu \right|^{2N + 1} }} \begin{cases} \left|\csc\left(2\theta\right)\right| & \; \text{ if } \; \frac{\pi}{4} < \left|\theta\right| <\frac{\pi}{2} \\ 1 & \; \text{ if } \; \left|\theta\right| \leq \frac{\pi}{4}. \end{cases} \end{equation} Here and throughout, $\theta = \arg \nu$. When $\nu$ is real and positive, we can obtain more precise estimates. Indeed, as $0 < \frac{1}{1 + \left( {t/\nu } \right)^2} < 1 $ for $t,\nu>0$, from \eqref{eq8} and \eqref{eq9} we find \[ R_N \left( {\nu ,\beta } \right) = - \frac{1}{\pi }\frac{{\left( {2N} \right)!a_N \left( { - \sec \beta } \right)}}{{\nu ^{2N + 1} }}\Theta , \] where $0 < \Theta < 1$ is an appropriate number depending on $\nu,\beta$ and $N$. In particular, when $N=0$, we have \[ 0 < \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) < \frac{1}{{\pi \nu \left( {\sec \beta - 1} \right)}} \; \text{ for } \; \nu >0. \] Therefore, the leading order asymptotic approximation for $\mathbf{A}_{ - \nu } \left( \nu \sec \beta \right)$ is always in error by excess, for all positive values of $\nu$ (cf. \cite[p. 298, formula 11.11.14]{NIST}).
The error bound \eqref{eq23} becomes singular as $\theta \to \pm \frac{\pi}{2}$, and therefore unrealistic near the Stokes lines. A better bound for $R_N \left( {\nu ,\beta } \right)$ near these lines can be derived as follows. Let $0 < \varphi < \frac{\pi }{2}$ be an acute angle that may depend on $N$. Suppose that $\frac{\pi}{4} +\varphi < \theta \le \frac{\pi}{2}$. An analytic continuation of the representation \eqref{eq18} to this sector can be found by rotating the path of integration in \eqref{eq8} by $\varphi$: \[ R_N \left( {\nu ,\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty e^{i\varphi } } {\frac{{t^{2N} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \] Substituting $t = \frac{se^{i\varphi }}{\cos \varphi}$ and applying the estimation \eqref{eq20}, we obtain \[
\left|R_N \left( {\nu ,\beta } \right)\right| \le \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\pi \cos ^{2N + 1} \varphi \left| \nu \right|^{2N + 1} }}\int_0^{ + \infty } {s^{2N} \left| {H_{\frac{{ise^{i\varphi } }}{{\cos \varphi }}}^{\left( 1 \right)} \left( {\frac{{ise^{i\varphi } }}{{\cos \varphi }}\sec \beta } \right)} \right|ds} . \] In \cite{Nemes}, it was shown that \begin{equation}\label{eq27}
\left| {H_{\frac{{ise^{i\varphi } }}{{\cos \varphi }}}^{\left( 1 \right)} \left( {\frac{{ise^{i\varphi } }}{{\cos \varphi }}\sec \beta } \right)} \right| \le \frac{1}{{\sqrt {\cos \varphi } }}\left| {H_{is}^{\left( 1 \right)} \left( {is\sec \beta } \right)} \right| = \frac{1}{{\sqrt {\cos \varphi } }}iH_{is}^{\left( 1 \right)} \left( {is\sec \beta } \right) \end{equation} for any $s>0$ and $0 < \varphi < \frac{\pi }{2}$. It follows that \begin{gather}\label{eq24} \begin{split}
\left|R_N \left( {\nu ,\beta } \right)\right| & \le \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\pi \cos ^{2N + \frac{3}{2}} \varphi \left| \nu \right|^{2N + 1} }}\int_0^{ + \infty } {s^{2N} iH_{is}^{\left( 1 \right)} \left( {is\sec \beta } \right)ds} \\
& = \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \varphi }}\frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( { - \sec \beta } \right)} \right|}}{{\left| \nu \right|^{2N + 1} }} . \end{split} \end{gather} The angle $\varphi = \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)$ minimizes the function $\csc \left( {2\left( {\frac{\pi }{2} - \varphi } \right)} \right)\cos ^{ - 2N - \frac{3}{2}} \varphi$, and \begin{align*} \frac{{\csc \left( {2\left( {\theta - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \left( {\arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)}} & \le \frac{{\csc \left( {2\left( {\frac{\pi }{2} - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \left( {\arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)}} \\ &= \frac{1}{{\sqrt 2 }}\left( 1+\frac{2}{4N + 5} \right)^{N + \frac{7}{4}} \sqrt {N + \frac{5}{4}} \le \sqrt {\frac{e}{2}\left( {N + \frac{3}{2}} \right)} \end{align*} for all $\frac{\pi }{4} + \varphi = \frac{\pi }{4} + \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{\pi }{2}$ with $N \geq 0$. Applying this in \eqref{eq24} yields the upper bound \begin{equation}\label{eq25}
\left| {R_N \left( {\nu ,\beta } \right)} \right| \le \sqrt {\frac{e}{2}\left( {N + \frac{3}{2}} \right)} \frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( { - \sec \beta } \right)} \right|}}{{\left| \nu \right|^{2N + 1} }}, \end{equation}
which is valid for $\frac{\pi }{4} + \varphi = \frac{\pi }{4} + \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{\pi }{2}$ with $N \geq 0$. Since $\left| {R_N \left( {\bar \nu ,\beta } \right)} \right| = \left| {\overline {R_N \left( {\nu ,\beta } \right)} } \right| = \left| {R_N \left( {\nu ,\beta } \right)} \right|$, this bound also holds when $-\frac{\pi}{2} \leq \theta < -\frac{\pi}{4} - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)$. In the ranges $\frac{\pi }{4} < \left| \theta \right| \leq \frac{\pi }{4} + \arctan \left( {\frac{{\sqrt 2 }}{3}} \right)$ it holds that $\left| {\csc \left( {2\theta } \right)} \right| \le \sqrt {\frac{e}{2}\left( {1 + \frac{3}{2}} \right)}$, whence the estimate \eqref{eq25} is valid in the wider sectors $\frac{\pi }{4} < \left| \theta \right| \le \frac{\pi }{2}$ as long as $N\geq 1$.
\subsection{Case (ii): $x=1$} We note that from \eqref{eq17} and \eqref{eq21} it follows that \[
\left| {d_{2n} } \right| = \frac{1}{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}\int_0^{ + \infty } {t^{\frac{{2n - 2}}{3}} e^{ - 2\pi t} iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} . \] Applying this formula together with the representation \eqref{eq16} and the inequality \eqref{eq20} yields the error bound \[
\left| {R_N \left( \nu \right)} \right| \le \frac{1}{{3\pi }}\left| {d_{2N} } \right|\frac{{\Gamma \left( {\frac{{2N + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2N + 1}}{3}} }} \begin{cases} \left| {\csc \left( {\frac{2}{3}\theta } \right)} \right| & \; \text{ if } \; \frac{3\pi}{4} < \left| \theta \right| < \frac{3\pi}{2} \\ 1 & \; \text{ if } \; \left| \theta \right| \le \frac{3\pi}{4}. \end{cases} \] Again, when $\nu$ is real and positive, we can deduce better estimates. Indeed, as $0 < \frac{1}{1 + \left( {t/\nu } \right)^{\frac{2}{3} }} < 1 $ for $t,\nu>0$, from \eqref{eq16} and \eqref{eq17} we find \[ R_N \left( \nu \right) = \frac{1}{{3\pi }}d_{2N} \frac{{\Gamma \left( {\frac{{2N + 1}}{3}} \right)}}{{\nu ^{\frac{{2N + 1}}{3}} }}\Xi , \] where $0 < \Xi < 1$ is a suitable number depending on $\nu$ and $N$. In particular, when $N=0$, we have \[ 0 < \mathbf{A}_{ - \nu } \left( \nu \right) < \frac{1}{{3\pi }}d_0 \frac{{\Gamma \left( {\frac{1}{3}} \right)}}{{\nu ^{\frac{1}{3}} }} = \frac{{2^{\frac{4}{3}} }}{{3^{\frac{7}{6}} \Gamma \left( {\frac{2}{3}} \right)\nu ^{\frac{1}{3}} }} \; \text{ for } \; \nu >0. \] Hence, the leading order asymptotic approximation for $\mathbf{A}_{ - \nu } \left( \nu \right)$ is always in error by excess, for all positive values of $\nu$ (cf. \cite[p. 298, formula 11.11.16]{NIST}).
Our bound for $R_N \left( \nu \right)$ is unrealistic near the Stokes lines $\theta = \pm\frac{3\pi }{2}$ due to the presence of the factor $\csc \left( {\frac{2}{3}\theta } \right)$. We shall derive better bounds for $R_N \left( \nu \right)$ near these lines using the method we applied in the previous case. Let $0 < \varphi < \frac{\pi }{2}$ be an acute angle that may depend on $N$ and suppose that $\frac{3\pi}{4} + \varphi < \theta \le \frac{3\pi}{2}$. We rotate the path of integration in \eqref{eq16} by $\varphi$, and apply the inequality \eqref{eq20} to obtain \[
\left| {R_N \left( \nu \right)} \right| \le \frac{{\csc \left( {\frac{2}{3}\left( {\theta - \varphi } \right)} \right)}}{{3\pi \cos ^{\frac{{2N + 1}}{3}} \varphi \left| \nu \right|^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2N - 2}}{3}} e^{ - 2\pi s} \left| {H_{\frac{{ise^{i\varphi } }}{{\cos \varphi }}}^{\left( 1 \right)} \left( {\frac{{ise^{i\varphi } }}{{\cos \varphi }}} \right)} \right|ds} \] for $\frac{3\pi}{4} + \varphi < \theta \le \frac{3\pi}{2}$ and $N\geq 0$. Using a continuity argument for the inequality \eqref{eq27}, yields \[
\left| {H_{\frac{{it}}{{e^{i\varphi } \cos \varphi }}}^{\left( 1 \right)} \left( {\frac{{it}}{{e^{i\varphi } \cos \varphi }}} \right)} \right| \le \frac{1}{{\sqrt {\cos \varphi } }}iH_{it}^{\left( 1 \right)} \left( {it} \right) \le \frac{1}{{\cos ^{\frac{2}{3}} \varphi }}iH_{it}^{\left( 1 \right)} \left( {it} \right) \] for $s>0$ and $0 < \varphi < \frac{\pi }{2}$. It follows that \begin{equation}\label{eq28}
\left| {R_N \left( \nu \right)} \right| \le \frac{{\csc \left( {\frac{2}{3}\left( {\theta - \varphi } \right)} \right)}}{{3\pi \cos ^{\frac{{2N + 1}}{3}} \varphi \left| \nu \right|^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2N - 2}}{3}} e^{ - 2\pi s} i H_{is}^{\left( 1 \right)} \left( is \right)ds} = \frac{\csc \left( {\frac{2}{3}\left( {\theta - \varphi } \right)} \right)}{ \cos ^{\frac{{2N + 3}}{3}} \varphi} \frac{1}{{3\pi }}\left| {d_{2N} } \right|\frac{{\Gamma \left( {\frac{{2N + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2N + 1}}{3}} }}. \end{equation} There is no simple way to minimize $\csc \left( {\frac{2}{3}\left( {\frac{3\pi}{2} - \varphi } \right)} \right)\cos ^{ - \frac{{2N + 3}}{3}} \varphi$ in $\varphi$. Nevertheless, an approximate minimizer is given by $\varphi = \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right)$. It is elementary to show that \[ \frac{{\csc \left( {\frac{2}{3}\left( {\theta - \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right) } \right)} \right)}}{{\cos ^{\frac{{2N + 3}}{3}} \left(\arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right)\right) }} \le \frac{{\csc \left( {\frac{2}{3}\left( {\frac{{3\pi }}{2} - \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right) } \right)} \right)}}{{\cos ^{\frac{{2N + 3}}{3}} \left(\arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right)\right) }} \le \sqrt {\frac{3e}{2}\left( {N + 2} \right)} \] for $\frac{3\pi}{4} + \varphi = \frac{3\pi}{4} + \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{3\pi}{2}$ and $N\geq 0$. Employing this estimate in \eqref{eq28} gives the upper bound \begin{equation}\label{eq29}
\left| {R_N \left( \nu \right)} \right| \le \sqrt {\frac{3e}{2}\left( {N + 2} \right)} \frac{1}{{3\pi }}\left| {d_{2N} } \right|\frac{{\Gamma \left( {\frac{{2N + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2N + 1}}{3}} }}, \end{equation}
valid when $\frac{3\pi}{4} + \varphi = \frac{3\pi}{4} + \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{3\pi}{2}$ and $N\geq 0$. A similar argument shows that this bound also holds in the sector $-\frac{3\pi}{2} \leq \theta < -\frac{3\pi}{4} - \arctan \left( {\left( {\frac{{2N + 2}}{3}} \right)^{ - \frac{1}{2}} } \right)$. In the ranges $\frac{{3\pi }}{4} < \left| \theta \right| \le \frac{{3\pi }}{4} + \arctan \left( {\left( {\frac{2}{3}} \right)^{ - \frac{1}{2}} } \right)$ it holds that $\left| {\csc \left( {\frac{2}{3}\theta } \right)} \right| \le \sqrt {\frac{3e}{2}\left( {0 + 2} \right)} = \sqrt {3e}$, therefore, the estimate \eqref{eq29} remains valid in the wider sectors $\frac{3\pi}{4} < \left| \theta \right| \le \frac{3\pi}{2}$ for any $N\geq 0$.
\section{Asymptotics for the late coefficients}\label{section4}
In this section, we investigate the asymptotic nature of the coefficients $a_n\left(- \sec \beta\right)$ as $n \to +\infty$. The asymptotic behaviour of the coefficients $d_{2n}$ is discussed in the earlier paper \cite{Nemes}. For our purposes, the most appropriate representation of the coefficients $a_n\left(- \sec \beta\right)$ is the second integral formula in \eqref{eq9}. From \eqref{eq40}, it follows that for any $t>0$ and $0 < \beta <\frac{\pi}{2}$, it holds that \begin{equation}\label{eq31} iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right) = \frac{{e^{ - t\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}t\pi \tan \beta } \right)^{\frac{1}{2}} }}\left( {\sum\limits_{m = 0}^{M - 1} {\frac{{i^m U_m \left( {i\cot \beta } \right)}}{{t^m }}} + R_M^{\left( H \right)} \left( {it,\beta } \right)} \right). \end{equation} In \cite{Nemes}, it was proved that the remainder $R_M^{\left( H \right)} \left( {it,\beta } \right)$ satisfies \begin{equation}\label{eq32}
\left| {R_M^{\left( H \right)} \left( {it,\beta } \right)} \right| \le \frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{t^M }} . \end{equation} Substituting the formula \eqref{eq31} into \eqref{eq9} gives the expansion \begin{gather}\label{eq30} \begin{split}
- \left( {2n} \right)!a_n \left( { - \sec \beta } \right) = & \left( {\frac{{2\cot \beta }}{{\pi \left( {\tan \beta - \beta } \right)}}} \right)^{\frac{1}{2}} \frac{{\left( { - 1} \right)^n \Gamma \left( {2n + \frac{1}{2}} \right)}}{{\left( {\tan \beta - \beta } \right)^{2n} }} \\ & \times \left( {\sum\limits_{m = 0}^{M - 1} {\left( {i\left( {\tan \beta - \beta } \right)} \right)^m U_m \left( {i\cot \beta } \right)\frac{{\Gamma \left( {2n - m + \frac{1}{2}} \right)}}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}} + A_M \left( {n,\beta } \right)} \right), \end{split} \end{gather} for any fixed $0 \le M \le 2n$, provided that $n\geq 1$. The remainder term $A_M \left( {n,\beta } \right)$ is given by the integral formula \[ A_M \left( {n,\beta } \right) = \frac{{\left( {\tan \beta - \beta } \right)^{2n + \frac{1}{2}} }}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}\int_0^{ + \infty } {t^{2n - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} R_M^{\left( H \right)} \left( {it,\beta } \right)dt} . \] To bound this error term, we apply the estimate \eqref{eq32} to find \begin{equation}\label{eq33}
\left| {A_M \left( {n,\beta } \right)} \right| \le \left( {\tan \beta - \beta } \right)^M \left| {U_M \left( {i\cot \beta } \right)} \right|\frac{{\Gamma \left( {2n - M + \frac{1}{2}} \right)}}{{\Gamma \left( {2n + \frac{1}{2}} \right)}} . \end{equation} Expansions of type \eqref{eq30} are called inverse factorial series in the literature. Numerically, their character is similar to the character of asymptotic power series, because the consecutive Gamma functions decrease asymptotically by a factor $2n$.
From the asymptotic behaviour of the coefficients $U_m \left( {i\cot \beta } \right)$ (see \cite{Nemes}), we infer that for large $n$, the least value of the bound \eqref{eq33} occurs when $M \approx \frac{4n}{3}$. With this choice of $M$, the error bound is $\mathcal{O}\left( {n^{\frac{1}{2}} 9^{-n} } \right)$. This is the best accuracy we can achieve using the expansion \eqref{eq30}.
By extending the sum in \eqref{eq30} to infinity, we arrive at the formal series \begin{multline*}
- \left( {2n} \right)!a_n \left( { - \sec \beta } \right) \approx \left( {\frac{{2\cot \beta }}{{\pi \left( {\tan \beta - \beta } \right)}}} \right)^{\frac{1}{2}} \frac{{\left( { - 1} \right)^n \Gamma \left( {2n + \frac{1}{2}} \right)}}{{\left( {\tan \beta - \beta } \right)^{2n} }}\left( {1 - \frac{{\left( {\tan \beta - \beta } \right)\cot \beta \left( {5\cot ^2 \beta + 3} \right)}}{{24\left( {2n - \frac{1}{2}} \right)}} }\right. \\ \left.{ + \frac{{\left( {\tan \beta - \beta } \right)^2 \cot ^2 \beta \left( {385\cot ^4 \beta + 462\cot ^2 \beta + 81} \right)}}{{1152\left( {2n - \frac{1}{2}} \right)\left( {2n - \frac{3}{2}} \right)}} + \cdots } \right). \end{multline*} This is exactly Dingle's expansion for the late coefficients in the asymptotic series of $\mathbf{A}_{-\nu} \left(\nu \sec \beta\right)$ \cite[p. 202]{Dingle}. The mathematically rigorous form of Dingle's series is therefore the formula \eqref{eq30}.
Numerical examples illustrating the efficacy of the expansion \eqref{eq30}, truncated optimally, are given in Table \ref{table1}.
\begin{table*}[!ht] \begin{center} \begin{tabular} [c]{ l r @{\,}c@{\,} l}\hline
& \\ [-1ex]
values of $\beta$ and $M$ & $\beta=\frac{\pi}{6}$, $M=33$ & & \\ [1ex]
exact numerical value of $a_{25}\left(-\sec\beta\right)$ & $0.19289505370609710328176787542524$ & $\times$ & $10^{64}$ \\ [1ex]
approximation \eqref{eq30} to $a_{25}\left(-\sec\beta\right)$ & $0.19289505370609710328176788499115$ & $\times$ & $10^{64}$ \\ [1ex]
error & $-0.956591$ & $\times$ & $10^{38}$\\ [1ex]
error bound using \eqref{eq33} & $0.1871709$ & $\times$ & $10^{39}$\\ [1ex] \hline
& \\ [-1ex]
values of $\beta$ and $M$ & $\beta=\frac{\pi}{3}$, $M=33$ & & \\ [1ex]
exact numerical value of $a_{25}\left(-\sec\beta\right)$ & $0.17129537192362280172104021636215$ & $\times$ & $10^8$ \\ [1ex]
approximation \eqref{eq30} to $a_{25}\left(-\sec\beta\right)$ & $0.17129537192362280172104022485431$ & $\times$ & $10^8$ \\ [1ex]
error & $-0.849216$ & $\times$ & $10^{-18}$\\ [1ex]
error bound using \eqref{eq33} & $0.1661627$ & $\times$ & $10^{-17}$\\ [1ex] \hline
& \\ [-1ex]
values of $\beta$ and $M$ & $\beta=\frac{6\pi}{13}$, $M=33$ & & \\ [1ex]
exact numerical value of $a_{25}\left(-\sec\beta\right)$ & $0.39520964363504437817499204430357$ & $\times$ & $10^{-43}$ \\ [1ex]
approximation \eqref{eq30} to $a_{25}\left(-\sec\beta\right)$ & $0.39520964363504437817499206387318$ & $\times$ & $10^{-43}$ \\ [1ex]
error & $-0.1956961$ & $\times$ & $10^{-68}$\\ [1ex]
error bound using \eqref{eq33} & $0.3829199$ & $\times$ & $10^{-68}$\\ [1ex] \hline
& \\ [-1ex]
values of $\beta$ and $M$ & $\beta=\frac{7\pi}{15}$, $M=33$ & & \\ [1ex]
exact numerical value of $a_{25}\left(-\sec\beta\right)$ & $0.66560453043764058337583145493270$ & $\times$ & $10^{-47}$ \\ [1ex]
approximation \eqref{eq30} to $a_{25}\left(-\sec\beta\right)$ & $0.66560453043764058337583148788914$ & $\times$ & $10^{-47}$ \\ [1ex]
error & $-0.3295644$ & $\times$ & $10^{-72}$\\ [1ex]
error bound using \eqref{eq33} & $0.6448607$ & $\times$ & $10^{-72}$\\ [-1ex]
& \\\hline \end{tabular} \end{center} \caption{Approximations for $a_{25}\left(-\sec\beta\right)$ with various $\beta$, using \eqref{eq30}.} \label{table1} \end{table*}
More accurate approximations could be derived for the coefficients $a_n\left(-\sec \beta\right)$ by estimating the remainder $A_M \left( n,\beta \right)$ rather than bounding it, but we do not discuss the details here.
\section{Exponentially improved asymptotic expansions}\label{section5}
We shall find it convenient to express our exponentially improved expansions in terms of the (scaled) Terminant function, which is defined by \[
\widehat T_p \left( w \right) = \frac{{e^{\pi ip} w^{1 - p} e^{ - w} }}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{p - 1} e^{ - t} }}{w + t}dt} \; \text{ for } \; p>0 \; \text{ and } \; \left| \arg w \right| < \pi , \]
and by analytic continuation elsewhere. Olver \cite{Olver4} showed that when $p \sim \left|w\right|$ and $w \to \infty$, we have \begin{equation}\label{eq36}
ie^{ - \pi ip} \widehat T_p \left( w \right) = \begin{cases} \mathcal{O}\left( {e^{ - w - \left| w \right|} } \right) & \; \text{ if } \; \left| {\arg w} \right| \le \pi \\ \mathcal{O}\left(1\right) & \; \text{ if } \; - 3\pi < \arg w \le - \pi. \end{cases} \end{equation} Concerning the smooth transition of the Stokes discontinuities, we will use the more precise asymptotics \begin{equation}\label{eq37}
\widehat T_p \left( w \right) = \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}} \left( {c\left( \varphi \right)\sqrt {\frac{1}{2}\left| w \right|} } \right) + \mathcal{O}\left( {\frac{{e^{ - \frac{1}{2}\left| w \right|c^2 \left( \varphi \right)} }}{{\left| w \right|^{\frac{1}{2}} }}} \right) \end{equation} for $-\pi +\delta \leq \arg w \leq 3 \pi -\delta$, $0 < \delta \le 2\pi$; and \begin{equation}\label{eq38}
e^{ - 2\pi ip} \widehat T_p \left( w \right) = - \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}} \left( { - \overline {c\left( { - \varphi } \right)} \sqrt {\frac{1}{2}\left| w \right|} } \right) + \mathcal{O}\left( {\frac{{e^{ - \frac{1}{2}\left| w \right|\overline {c^2 \left( { - \varphi } \right)} } }}{{\left| w \right|^{\frac{1}{2}} }}} \right) \end{equation} for $- 3\pi + \delta \le \arg w \le \pi - \delta$, $0 < \delta \le 2\pi$. Here $\varphi = \arg w$ and erf denotes the Error function. The quantity $c\left( \varphi \right)$ is defined implicitly by the equation \[ \frac{1}{2}c^2 \left( \varphi \right) = 1 + i\left( {\varphi - \pi } \right) - e^{i\left( {\varphi - \pi } \right)}, \] and corresponds to the branch of $c\left( \varphi \right)$ which has the following expansion in the neighbourhood of $\varphi = \pi$: \begin{equation}\label{eq39} c\left( \varphi \right) = \left( {\varphi - \pi } \right) + \frac{i}{6}\left( {\varphi - \pi } \right)^2 - \frac{1}{{36}}\left( {\varphi - \pi } \right)^3 - \frac{i}{{270}}\left( {\varphi - \pi } \right)^4 + \cdots . \end{equation} For complete asymptotic expansions, see Olver \cite{Olver5}. We remark that Olver uses the different notation $F_p \left( w \right) = ie^{ - \pi ip} \widehat T_p \left( w \right)$ for the Terminant function and the other branch of the function $c\left( \varphi \right)$. For further properties of the Terminant function, see, for example, Paris and Kaminski \cite[Chapter 6]{Paris3}.
\subsection{Proof of the exponentially improved expansions for $\mathbf{A}_{-\nu}\left(\nu x\right)$}
\subsubsection{Case (i): $x>1$} First, we suppose that $\left|\arg \nu\right| < \frac{\pi}{2}$. Our starting point is the representation \eqref{eq8}, written in the form \begin{equation}\label{eq43} R_N \left( {\nu ,\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} }}{{1 - it/\nu }}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} + \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} }}{{1 + it/\nu }}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} . \end{equation} Let $0 \leq M <2N$ be a fixed integer. We use \eqref{eq40} to expand the function $H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)$ under the integrals in \eqref{eq43}, to obtain \begin{gather}\label{eq44} \begin{split} R_N \left( {\nu ,\beta } \right) = \; & i\frac{{e^{ - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\left( { - 1} \right)^m \frac{{\left( {i\nu } \right)^{m - 2N - \frac{1}{2}} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - m - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 - it/\nu }}} dt} \\ & - i\frac{{e^{\frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\left( { - 1} \right)^m \frac{{\left( { - i\nu } \right)^{m - 2N - \frac{1}{2}} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - m - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}} dt} \\ & + R_{N,M} \left( {\nu ,\beta } \right), \end{split} \end{gather} with \begin{gather}\label{eq45} \begin{split} R_{N,M} \left( {\nu ,\beta } \right) = & - \frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} \left( {i\nu } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{2N - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 - it/\nu }}R_M^{\left( H \right)} \left( {it,\beta } \right)dt}\\ & - \frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} \left( {i\nu } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{2N - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}R_M^{\left( H \right)} \left( {it,\beta } \right)dt} . \end{split} \end{gather} The integrals in \eqref{eq44} can be identified in terms of the Terminant function since \[ \left( { - 1} \right)^m \frac{{\left( {i\nu } \right)^{m - 2N - \frac{1}{2}} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - m - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 - it/\nu }}} dt = e^{i\nu \left( {\tan \beta - \beta } \right)} \widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right) \] and \[ \left( { - 1} \right)^m \frac{{\left( { - i\nu } \right)^{m - 2N - \frac{1}{2}} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - m - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}} dt = e^{ - i\nu \left( {\tan \beta - \beta } \right)} \widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right) . \] Therefore, we have the following expansion \begin{align*} R_N \left( {\nu ,\beta } \right) = \; & i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right)}\\ & - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)} + R_{N,M} \left( {\nu ,\beta } \right). \end{align*} Taking $\nu = re^{i\theta }$, the representation \eqref{eq45} takes the form \begin{gather}\label{eq46} \begin{split} R_{N,M} \left( {\nu ,\beta } \right) = & - \frac{1}{{\left( {\frac{1}{2}r\pi \tan \beta } \right)^{\frac{1}{2}} \left( {ie^{i\theta } } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{\tau ^{2N - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} }}{{1 - i\tau e^{ - i\theta } }}R_M^{\left( H \right)} \left( {ir\tau ,\beta } \right)d\tau } \\ & - \frac{1}{{\left( {\frac{1}{2}r\pi \tan \beta } \right)^{\frac{1}{2}} \left( {ie^{i\theta } } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{\tau ^{2N - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} }}{{1 + i\tau e^{ - i\theta } }}R_M^{\left( H \right)} \left( {ir\tau ,\beta } \right)d\tau } . \end{split} \end{gather} Using the integral formula \eqref{eq42}, $R_M^{\left( H \right)} \left( {ir\tau ,\beta } \right)$ can be written as \begin{multline*} R_M^{\left( H \right)} \left( {ir\tau ,\beta } \right) = \frac{{\left( { - 1} \right)^M }}{{2\left( {2\pi \cot \beta } \right)^{\frac{1}{2}}\left( {r\tau } \right)^M }}\left( {\int_0^{ + \infty } {\frac{{s^{M - \frac{1}{2}} e^{ - s\left( {\tan \beta - \beta } \right)} }}{{1 + s/r}}\left( {1 + e^{ - 2\pi s} } \right)iH_{is}^{\left( 1 \right)} \left( {is\sec \beta } \right)ds}}\right. \\ + \left.{\left( {\tau - 1} \right)\int_0^{ + \infty } {\frac{{s^{M - \frac{1}{2}} e^{ - s\left( {\tan \beta - \beta } \right)} }}{{\left( 1+r\tau /s \right)\left( {1 + s/r} \right)}}\left( {1 + e^{ - 2\pi s} } \right)iH_{is}^{\left( 1 \right)} \left( {is\sec \beta } \right)ds} } \right). \end{multline*} Noting that \[ 0 < \frac{1}{{1 + s/r}},\frac{1}{{\left( 1 + r\tau /s \right)\left( {1 + s/r} \right)}} < 1 \] for positive $r$, $\tau$ and $s$, substitution into \eqref{eq46} yields the upper bound \begin{align*}
\left| {R_{N,M} \left( {\nu ,\beta } \right)} \right| \le \; & \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\left| {\frac{1}{{2\pi }}\int_0^{ + \infty } {\frac{{\tau ^{2N - M - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} }}{{1 - i\tau e^{ - i\theta } }}d\tau } } \right|\\
& + \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\frac{1}{{2\pi }}\int_0^{ + \infty } {\tau ^{2N - M - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} \left| {\frac{{\tau - 1}}{{\tau + ie^{i\theta } }}} \right|d\tau } \\
& + \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\left| {\frac{1}{{2\pi }}\int_0^{ + \infty } {\frac{{\tau ^{2N - M - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} }}{{1 + i\tau e^{ - i\theta } }}d\tau } } \right|\\
& + \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\frac{1}{{2\pi }}\int_0^{ + \infty } {\tau ^{2N - M - \frac{1}{2}} e^{ - r\tau \left( {\tan \beta - \beta } \right)} \left| {\frac{{\tau - 1}}{{\tau - ie^{i\theta } }}} \right|d\tau } . \end{align*}
Since $\left| {\left( {\tau - 1} \right)/\left( {\tau \pm ie^{i\theta } } \right)} \right| \le 1$, we find that \begin{align*}
\left| {R_{N,M} \left( {\nu ,\beta } \right)} \right| \le \; & \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\left| {e^{i\nu \left( {\tan \beta - \beta } \right)} \widehat T_{2N - M + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right)} \right|\\
& + \frac{1}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}\left| {e^{ - i\nu \left( {\tan \beta - \beta } \right)} \widehat T_{2N - M + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)} \right|\\
& + \frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|\Gamma \left( {2N - M + \frac{1}{2}} \right)}}{{\pi \left( {\tan \beta - \beta } \right)^{2N - M + \frac{1}{2}} \left| \nu \right|^{2N + 1} }}. \end{align*}
By continuity, this bound holds in the closed sector $\left|\arg \nu\right| \leq \frac{\pi}{2}$. Assume that $
N = \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right) + \rho$ where $\rho$ is bounded. Employing Stirling's formula, we find that \[
\frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|\Gamma \left( {2N - M + \frac{1}{2}} \right)}}{{\pi \left( {\tan \beta - \beta } \right)^{2N - M + \frac{1}{2}} \left| \nu \right|^{2N + 1} }} = \mathcal{O}_{M,\rho} \left( {\frac{1}{{\left( {\left| \nu \right|\left( {\tan \beta - \beta } \right)} \right)^{\frac{1}{2}} }}\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \] as $\nu \to \infty$. Olver's estimation \eqref{eq36} shows that \[
\left| {e^{ \pm i\nu \left( {\tan \beta - \beta } \right)} \widehat T_{2N - M + \frac{1}{2}} \left( { \pm i\nu \left( {\tan \beta - \beta } \right)} \right)} \right| = \mathcal{O}_{M,\rho } \left( {e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} } \right) \] for large $\nu$. Therefore, we have that \begin{equation}\label{eq47}
R_{N,M} \left( {\nu ,\beta } \right) = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \end{equation}
as $\nu \to \infty$ in the sector $\left|\arg \nu\right| \leq \frac{\pi}{2}$.
Rotating the path of integration in \eqref{eq45} and applying the residue theorem yields \begin{gather}\label{eq48} \begin{split} R_{N,M} \left( {\nu ,\beta } \right) = \; & i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}R_M^{\left( H \right)} \left( {\nu ,\beta } \right) - \frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} \left( {i\nu } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{2N - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 - it/\nu }}R_M^{\left( H \right)} \left( {it,\beta } \right)dt} \\ & - \frac{1}{{\left( {\frac{1}{2}\pi \tan \beta } \right)^{\frac{1}{2}} \left( {i\nu } \right)^{2N + 1} }}\frac{1}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{2N - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}R_M^{\left( H \right)} \left( {it,\beta } \right)dt} \\
= \; & i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}R_M^{\left( H \right)} \left( {\nu ,\beta } \right) - R_{N,M} \left( {\nu e^{ - \pi i} ,\beta } \right) \end{split} \end{gather} when $\frac{\pi}{2} < \arg \nu <\frac{3\pi}{2}$. It follows that \[
\left| {R_{N,M} \left( {\nu ,\beta } \right)} \right| \le \frac{{e^{ - \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\left| {R_M^{\left( H \right)} \left( {\nu ,\beta } \right)} \right| + \left| {R_{N,M} \left( {\nu e^{ - \pi i} ,\beta } \right)} \right| \]
in the closed sector $\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, using continuity. It was proved in \cite{Nemes} that $R_M^{\left( H \right)} \left( {\nu ,\beta } \right) = \mathcal{O}_M \left( {\left|U_M \left( {i\cot \beta } \right)\right|\left| \nu \right|^{ - M} } \right)$ as $\nu \to \infty$ in the closed sector $-\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, whence, by \eqref{eq47}, we deduce that \begin{align*}
R_{N,M} \left( {\nu ,\beta } \right) & = \mathcal{O}_M \left( {\frac{{e^{ - \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) + \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \\ &= \mathcal{O}_{M,\rho} \left( {\frac{{e^{ - \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \end{align*} as $\nu \to \infty$ in the sector $\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$.
The reflection principle gives the relation \begin{gather}\label{eq49} \begin{split} R_{N,M} \left( {\nu ,\beta } \right) = \overline {R_{N,M} \left( {\bar \nu ,\beta } \right)} & = - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\overline {R_M^{\left( H \right)} \left( {\bar \nu ,\beta } \right)} - R_{N,M} \left( {\nu e^{\pi i} ,\beta } \right) \\ & = - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}R_M^{\left( H \right)} \left( { \nu e^{\pi i} ,\beta } \right) - R_{N,M} \left( {\nu e^{\pi i} ,\beta } \right), \end{split} \end{gather} valid when $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. Trivial estimation and a continuity argument show that \[
\left| {R_{N,M} \left( {\nu ,\beta } \right)} \right| \le \frac{{e^{\Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\left| {R_M^{\left( H \right)} \left( { \nu e^{\pi i} ,\beta } \right)} \right| + \left| {R_{N,M} \left( {\nu e^{\pi i} ,\beta } \right)} \right| \]
in the closed sector $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$. Since $R_M^{\left( H \right)} \left( {\nu e^{\pi i},\beta } \right) = \mathcal{O}_M \left( {\left|U_M \left( {i\cot \beta } \right)\right|\left| \nu \right|^{ - M} } \right)$ as $\nu \to \infty$ in the closed sector $-\frac{3\pi}{2} \leq \arg \nu \leq \frac{\pi}{2}$, by \eqref{eq47}, we find that \begin{align*}
R_{N,M} \left( {\nu ,\beta } \right) & = \mathcal{O}_M \left( {\frac{{e^{ \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) + \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \\ &= \mathcal{O}_{M,\rho} \left( {\frac{{e^{ \Im \left( \nu \right)\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right) \end{align*} as $\nu \to \infty$ with $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$.
\subsubsection{Case (ii): $x=1$} First, we suppose that $\left|\arg \nu\right| < \frac{\pi}{2}$. We write \eqref{eq19} with $N=0$ in the form \begin{align*} \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi \nu ^{\frac{1}{3}} }}\int_0^{ + \infty } {\frac{{t^{ - \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} & - \frac{1}{{3\pi \nu }}\int_0^{ + \infty } {\frac{{e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} \\ & + \frac{1}{{3\pi \nu ^{\frac{5}{3}} }}\int_0^{ + \infty } {\frac{{t^{\frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} . \end{align*} Let $N$, $M$ and $K$ be arbitrary positive integers. Using the expression \eqref{eq2}, we find that \begin{align*} \mathbf{A}_{ - \nu } \left( \nu \right) = \frac{1}{{3\pi \nu ^{\frac{1}{3}} }}\sum\limits_{n = 0}^{N - 1} {d_{6n} \frac{{\Gamma \left( {2n + \frac{1}{3}} \right)}}{{\nu ^{2n} }}} & + \frac{1}{{3\pi \nu }}\sum\limits_{m = 0}^{M - 1} {d_{6m + 2} \frac{{\Gamma \left( {2m + 1} \right)}}{{\nu ^{2m} }}} \\ & + \frac{1}{{3\pi \nu ^{\frac{5}{3}} }}\sum\limits_{k = 0}^{K - 1} {d_{6k + 4} \frac{{\Gamma \left( {2k + \frac{5}{3}} \right)}}{{\nu ^{2k} }}} + R_{N,M,K} \left( \nu \right), \end{align*} where \begin{gather}\label{eq55} \begin{split} R_{N,M,K} \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{2N + \frac{1}{3}} }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} & - \frac{{\left( { - 1} \right)^M }}{{3\pi \nu ^{2M + 1} }}\int_0^{ + \infty } {\frac{{t^{2M} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} \\ & + \frac{{\left( { - 1} \right)^K }}{{3\pi \nu ^{2K + \frac{5}{3}} }}\int_0^{ + \infty } {\frac{{t^{2K + \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it} \right)dt} . \end{split} \end{gather} We remark that $R_{N,N,N} \left( \nu \right) = R_{3N} \left( \nu \right)$. Assume that $J$, $L$ and $Q$ are integers such that $0 \leq L < 3N$, $0 \leq L < 3M+1$, $0 \leq Q < 3K+2$ and $J,L,Q \equiv 0 \mod 3$. We apply \eqref{eq53} to expand the function $H_{it}^{\left( 1 \right)} \left( {it} \right)$ under the integral in \eqref{eq55}, to obtain \begin{gather}\label{eq56} \begin{split} R_{N,M,K} \left( \nu \right) = \; & \frac{2}{{9\pi }}\sum\limits_{j = 0}^{J - 1} {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}\left( { - 1} \right)^{N + j} \frac{{\nu ^{\frac{{2j}}{3} - 2N} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2j}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}dt} } \\ & - \frac{2}{{9\pi }}\sum\limits_{\ell = 0}^{L - 1} {d_{2\ell } \sin \left( {\frac{{\left( {2\ell + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2\ell + 1}}{3}} \right)}}{{\nu ^{\frac{{2\ell + 1}}{3}} }}\left( { - 1} \right)^{M + \ell } \frac{{\nu ^{\frac{{2\ell - 2}}{3} - 2M} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2M - \frac{{2\ell - 2}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}} dt} \\ &+ \frac{2}{{9\pi }}\sum\limits_{q = 0}^{Q - 1} {d_{2q} \sin \left( {\frac{{\left( {2q + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2q + 1}}{3}} \right)}}{{\nu ^{\frac{{2q + 1}}{3}} }}\left( { - 1} \right)^{K + q} \frac{{\nu ^{\frac{{2q - 4}}{3} - 2K} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2K - \frac{{2q - 4}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}dt} } \\ & + R_{N,M,K}^{J,L,Q} \left( \nu \right), \end{split} \end{gather} with \begin{gather}\label{eq57} \begin{split} R_{N,M,K}^{J,L,Q} \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{2N + \frac{1}{3}} }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iR_J^{\left( H \right)} \left( {it} \right)dt} & - \frac{{\left( { - 1} \right)^M }}{{3\pi \nu ^{2M + 1} }}\int_0^{ + \infty } {\frac{{t^{2M} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iR_L^{\left( H \right)} \left( {it} \right)dt} \\ & + \frac{{\left( { - 1} \right)^K }}{{3\pi \nu ^{2K + \frac{5}{3}} }}\int_0^{ + \infty } {\frac{{t^{2K + \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iR_Q^{\left( H \right)} \left( {it} \right)dt} . \end{split} \end{gather} The integrals in \eqref{eq56} can be identified in terms of the Terminant function since \[ \left( { - 1} \right)^{N + j} \frac{{\nu ^{\frac{{2j}}{3} - 2N} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2j}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}dt} = ie^{ - 2\pi i\nu } \widehat T_{2N - \frac{{2j}}{3}} \left( { - 2\pi i\nu } \right) - ie^{\frac{\pi }{3}i} e^{2\pi i\nu } e^{\frac{{2\left( {2j + 1} \right)\pi i}}{3}} \widehat T_{2N - \frac{{2j}}{3}} \left( {2\pi i\nu } \right), \] \begin{align*} \left( { - 1} \right)^{M + \ell } \frac{{\nu ^{\frac{{2\ell - 2}}{3} - 2M} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2M - \frac{{2\ell - 2}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}} dt = & - ie^{ - 2\pi i\nu } \widehat T_{2M - \frac{{2\ell - 2}}{3}} \left( { - 2\pi i\nu } \right)\\ & - ie^{2\pi i\nu } e^{\frac{{2\left( {2\ell + 1} \right)\pi i}}{3}} \widehat T_{2M - \frac{{2\ell - 2}}{3}} \left( {2\pi i\nu } \right), \end{align*} and \begin{align*} \left( { - 1} \right)^{K + q} \frac{{\nu ^{\frac{{2q - 4}}{3} - 2K} }}{\pi }\int_0^{ + \infty } {\frac{{t^{2K - \frac{{2q - 4}}{3} - 1} e^{ - 2\pi t} }}{{1 + \left( {t/\nu } \right)^2 }}dt} =\; & ie^{ - 2\pi i\nu } \widehat T_{2K - \frac{{2q - 4}}{3}} \left( { - 2\pi i\nu } \right) \\ &- ie^{ - \frac{\pi }{3}i} e^{2\pi i\nu } e^{\frac{{2\left( {2q + 1} \right)\pi i}}{3}} \widehat T_{2K - \frac{{2q - 4}}{3}} \left( {2\pi i\nu } \right). \end{align*} Substitution into \eqref{eq56} leads to the expansion \eqref{eq68}. Taking $\nu = re^{i\theta }$, the representation \eqref{eq57} becomes \begin{gather}\label{eq58} \begin{split} R_{N,M,K}^{J,L,Q} \left( \nu \right) = \; & \Phi _ + \left( {N,2N + \frac{1}{3},J} \right) + \Phi _ - \left( {N,2N + \frac{1}{3},J} \right) - \Phi _ + \left( {M,2M + 1,L} \right) \\ & - \Phi _ - \left( {M,2M + 1,L} \right) + \Phi _ + \left( {K,2K + \frac{5}{3},Q} \right) + \Phi _ - \left( {K,2K + \frac{5}{3},Q} \right), \end{split} \end{gather} with \[ \Phi _ \pm \left( {A,B,C} \right) = \frac{{\left( { - 1} \right)^A }}{{6\pi \left( {e^{i\theta } } \right)^B }}\int_0^{ + \infty } {\frac{{\tau ^{B - 1} e^{ - 2\pi r\tau } }}{{1 \pm i\tau e^{ - i\theta } }}iR_C^{\left( H \right)} \left( {ir\tau } \right)d\tau } . \] In \cite[Appendix B]{Nemes} it was shown that \[ \frac{{1 - \left( {s/r\tau } \right)^{\frac{4}{3}} }}{{1 - \left( {s/r\tau } \right)^2 }} = \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }} + \left( {\tau - 1} \right)f\left( {r,\tau ,s} \right) \]
for positive $r$, $\tau$ and $s$, with some $f\left(r,\tau ,s\right)$ satisfying $\left|f\left(r,\tau ,s\right)\right| \leq 2$. Using the integral formula \eqref{eq54}, $R_J^{\left( H \right)} \left( {ir\tau } \right)$ can be written as \begin{align*} R_J^{\left( H \right)} \left( {ir\tau } \right) = \; & \frac{1}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2J + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2J - 2}}{3}} e^{ - 2\pi s} \frac{{1 - \left( {s/r\tau } \right)^{\frac{4}{3}} }}{{1 - \left( {s/r\tau } \right)^2 }}H_{is}^{\left( 1 \right)} \left( {is} \right)ds} \\ = \; & \frac{1}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2J + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2J - 2}}{3}} e^{ - 2\pi s} \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }}H_{is}^{\left( 1 \right)} \left( {is} \right)ds} \\ & + \frac{{\tau - 1}}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2J + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2J - 2}}{3}} e^{ - 2\pi s} f\left( {r,\tau ,s} \right)H_{is}^{\left( 1 \right)} \left( {is} \right)ds} , \end{align*} and similarly for $R_L^{\left( H \right)} \left( {ir\tau } \right)$ and $R_Q^{\left( H \right)} \left( {ir\tau } \right)$. Noting that \[ 0< \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }} < 1 \] for any positive $r$ and $s$, substitution into \eqref{eq58} yields the upper bound \begin{align*}
\left| {R_{N,M,K}^{J,L,Q} \left( \nu \right)} \right| \le \; & \Xi _ + \left( {2N,2J,\frac{{2J}}{3}} \right) + \Xi _ - \left( {2N,2J,\frac{{2J}}{3}} \right) + \Xi _ + \left( {2M,2L,\frac{{2L - 2}}{3}} \right) \\ & + \Xi _ - \left( {2M,2L,\frac{{2L - 2}}{3}} \right) + \Xi _ + \left( {2K,2Q,\frac{{2Q - 4}}{3}} \right) + \Xi _ - \left( {2K,2Q,\frac{{2Q - 4}}{3}} \right), \end{align*} with \[
\Xi _ \pm \left( {A,B,C} \right) = \frac{{\left| {d_B } \right|\Gamma \left( {\frac{{B + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{B + 1}}{3}} }}\left( {\left| {\frac{1}{{2\pi }}\int_0^{ + \infty } {\frac{{\tau ^{A - C - 1} e^{ - 2\pi r\tau } }}{{1 \pm i\tau e^{ - i\theta } }}d\tau } } \right| + \frac{1}{\pi }\int_0^{ + \infty } {\tau ^{A - C - 1} e^{ - 2\pi r\tau } \left| {\frac{{\tau - 1}}{{\tau \mp ie^{i\theta } }}} \right|d\tau } } \right). \]
As $\left| \left(\tau - 1\right)/\left(\tau \pm i e^{i\theta}\right) \right| \le 1$, we find that \begin{align*}
\left| {R_{N,M,K}^{J,L,Q} \left( \nu \right)} \right| \le \; & \frac{{\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2J + 1}}{3}} }}\left| {e^{ - 2\pi i\nu } \widehat T_{2N - \frac{{2J}}{3}} \left( { - 2\pi i\nu } \right)} \right| + \frac{{\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2J + 1}}{3}} }}\left| {e^{2\pi i\nu } \widehat T_{2N - \frac{{2J}}{3}} \left( {2\pi i\nu } \right)} \right|\\
& + \frac{{2\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)\Gamma \left( {2N - \frac{{2J}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2N - \frac{{2J}}{3}} \left| \nu \right|^{2N + \frac{1}{3}} }} + \frac{{\left| {d_{2L} } \right|\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2L + 1}}{3}} }}\left| {e^{ - 2\pi i\nu } \widehat T_{2N - \frac{{2L - 2}}{3}} \left( { - 2\pi i\nu } \right)} \right|\\
& + \frac{{\left| {d_{2L} } \right|\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2L + 1}}{3}} }}\left| {e^{2\pi i\nu } \widehat T_{2N - \frac{{2L - 2}}{3}} \left( {2\pi i\nu } \right)} \right| + \frac{{2\left| {d_{2L} } \right|\Gamma \left( {\frac{{2L + 1}}{3}} \right)\Gamma \left( {2M - \frac{{2L - 2}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2M - \frac{{2L - 2}}{3}} \left| \nu \right|^{2M + 1} }}\\
& + \frac{{\left| {d_{2Q} } \right|\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2Q + 1}}{3}} }}\left| {e^{ - 2\pi i\nu } \widehat T_{2N - \frac{{2Q - 4}}{3}} \left( { - 2\pi i\nu } \right)} \right| + \frac{{\left| {d_{2Q} } \right|\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{3\sqrt 3 \pi \left| \nu \right|^{\frac{{2Q + 1}}{3}} }}\left| {e^{2\pi i\nu } \widehat T_{2N - \frac{{2Q - 4}}{3}} \left( {2\pi i\nu } \right)} \right|\\
& + \frac{{2\left| {d_{2Q} } \right|\Gamma \left( {\frac{{2Q + 1}}{3}} \right)\Gamma \left( {2K - \frac{{2Q - 4}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2K - \frac{{2Q - 4}}{3}} \left| \nu \right|^{2K + \frac{5}{3}} }} . \end{align*}
By continuity, this bound holds in the closed sector $\left|\arg \nu\right| \le \frac{\pi}{2}$. Suppose that $N= \pi\left| \nu \right| + \rho $, $M= \pi\left| \nu \right| + \sigma $ and $K= \pi\left| \nu \right| + \eta$ where $\rho$, $\sigma$ and $\eta$ are bounded. An application of Stirling's formula shows that \[
\frac{{2\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)\Gamma \left( {2N - \frac{{2J}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2N - \frac{{2J}}{3}} \left| \nu \right|^{2N + \frac{1}{3}} }} = \mathcal{O}_{J,\rho } \left( {\frac{{e^{ - 2\pi \left| \nu \right|} }}{{\left| \nu \right|^{\frac{1}{2}} }}\left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right), \] \[
\frac{{2\left| {d_{2L} } \right|\Gamma \left( {\frac{{2L + 1}}{3}} \right)\Gamma \left( {2M - \frac{{2L - 2}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2M - \frac{{2L - 2}}{3}} \left| \nu \right|^{2M + 1} }} = \mathcal{O}_{L,\sigma } \left( {\frac{{e^{ - 2\pi \left| \nu \right|} }}{{\left| \nu \right|^{\frac{1}{2}} }}\left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right), \] and \[
\frac{{2\left| {d_{2Q} } \right|\Gamma \left( {\frac{{2Q + 1}}{3}} \right)\Gamma \left( {2K - \frac{{2Q - 4}}{3}} \right)}}{{3\sqrt 3 \pi ^2 \left( {2\pi } \right)^{2K - \frac{{2Q - 4}}{3}} \left| \nu \right|^{2K + \frac{5}{3}} }} = \mathcal{O}_{Q,\eta } \left( {\frac{{e^{ - 2\pi \left| \nu \right|} }}{{\left| \nu \right|^{\frac{1}{2}} }}\left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \] as $\nu \to \infty$. Using Olver's estimation \eqref{eq36}, we find \[
\left| {e^{ \pm 2\pi i\nu } \widehat T_{2N - \frac{{2J}}{3}} \left( { \pm 2\pi i\nu } \right)} \right| = \mathcal{O}_{J,\rho } \left( {e^{ - 2\pi \left| \nu \right|} } \right), \] \[
\left| {e^{ \pm 2\pi i\nu } \widehat T_{2N - \frac{{2L - 2}}{3}} \left( { \pm 2\pi i\nu } \right)} \right| = \mathcal{O}_{L,\sigma } \left( {e^{ - 2\pi \left| \nu \right|} } \right), \] and \[
\left| {e^{ \pm 2\pi i\nu } \widehat T_{2N - \frac{{2Q - 4}}{3}} \left( { \pm 2\pi i\nu } \right)} \right| = \mathcal{O}_{Q,\eta } \left( {e^{ - 2\pi \left| \nu \right|} } \right) \] for large $\nu$. Therefore, we have \begin{gather}\label{eq59} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \mathcal{O}_{J,\rho } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) & + \mathcal{O}_{L,\sigma } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& + \mathcal{O}_{Q,\eta } \left( {e^{ - 2\pi \left| \nu \right|} \left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \end{split} \end{gather}
as $\nu \to \infty$ in the sector $\left|\arg\nu\right| \leq \frac{\pi}{2}$.
Next, we consider the sector $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$. Rotating the path of integration in \eqref{eq57} and applying the residue theorem gives \begin{gather}\label{eq61} \begin{split} R_{N,M,K}^{J,L,Q} \left( \nu \right) = \; & ie^{\frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_J^{\left( H \right)} \left( \nu \right) + \frac{\left( { - 1} \right)^N}{{3\pi \nu^{2N + \frac{1}{3}} }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - \pi i} } \right)^2 }}iR_J^{\left( H \right)} \left( {it} \right)dt} \\ & - i\frac{{e^{2\pi i\nu } }}{3}R_L^{\left( H \right)} \left( \nu \right) - \frac{{\left( { - 1} \right)^M }}{{3\pi \nu^{2M + 1} }}\int_0^{ + \infty } {\frac{{t^{2M} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - \pi i} } \right)^2 }}iR_L^{\left( H \right)} \left( {it} \right)dt} \\ & + ie^{ - \frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_Q^{\left( H \right)} \left( \nu \right) + \frac{{\left( { - 1} \right)^K }}{{3\pi \nu^{2K + \frac{5}{3}} }}\int_0^{ + \infty } {\frac{{t^{2K + \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - \pi i} } \right)^2 }}iR_Q^{\left( H \right)} \left( {it} \right)dt} , \end{split} \end{gather} for $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$. It is easy to see that the sum of three integrals has the order of magnitude given in the right-hand side of \eqref{eq59}. It follows that when $J=K=Q$, the bound \eqref{eq59} remains valid in the wider sector $-\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$. Otherwise, we have \begin{multline*}
\left| {ie^{\frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_J^{\left( H \right)} \left( \nu \right) - i\frac{{e^{2\pi i\nu } }}{3}R_L^{\left( H \right)} \left( \nu \right) + ie^{ - \frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_Q^{\left( H \right)} \left( \nu \right)} \right|\\
\le \frac{{e^{ - 2\pi \Im \left( \nu \right)} }}{3}\left| {R_J^{\left( H \right)} \left( \nu \right)} \right| + \frac{{e^{ - 2\pi \Im \left( \nu \right)} }}{3}\left| {R_L^{\left( H \right)} \left( \nu \right)} \right| + \frac{{e^{ - 2\pi \Im \left( \nu \right)} }}{3}\left| {R_Q^{\left( H \right)} \left( \nu \right)} \right| . \end{multline*}
It was proved in \cite{Nemes} that $R_J^{\left( H \right)} \left( \nu \right) = \mathcal{O}_J\left( {\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)\left| \nu \right|^{ - \frac{2J + 1}{3}} } \right)$ as $\nu \to \infty$ in the closed sector $-\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, whence, by \eqref{eq59}, we deduce that \begin{gather}\label{eq64} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \mathcal{O}_{J,\rho } \left( {e^{ - 2\pi \Im \left( \nu \right)} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) & + \mathcal{O}_{L,\sigma } \left( {e^{ - 2\pi \Im \left( \nu \right)} \left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& + \mathcal{O}_{Q,\eta } \left( {e^{ - 2\pi \Im \left( \nu \right)} \left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \end{split} \end{gather} as $\nu \to \infty$ in the sector $\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$.
Similarly, if $J=K=Q$, the bound \eqref{eq59} remains valid in the wider sector $-\frac{3\pi}{2} \leq \arg \nu \leq \frac{\pi}{2}$; and by the foregoing argument, it is true in the larger sector $-\frac{3\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$. Otherwise, we have \begin{gather}\label{eq60} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \mathcal{O}_{J,\rho } \left( {e^{ 2\pi \Im \left( \nu \right)} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) & + \mathcal{O}_{L,\sigma } \left( {e^{ 2\pi \Im \left( \nu \right)} \left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& + \mathcal{O}_{Q,\eta } \left( {e^{ 2\pi \Im \left( \nu \right)} \left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) \end{split} \end{gather} for large $\nu$ with $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$.
Consider now the sector $\frac{3\pi}{2} < \arg \nu < \frac{5\pi}{2}$. Rotation of the path of integration in \eqref{eq61} and application of the residue theorem yields \begin{align*} R_{N,M,K}^{J,L,Q} \left( \nu \right) =\; & ie^{\frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_J^{\left( H \right)} \left( \nu \right) - i\frac{{e^{2\pi i\nu } }}{3}R_L^{\left( H \right)} \left( \nu \right) + ie^{ - \frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_Q^{\left( H \right)} \left( \nu \right)\\ & + i\frac{{e^{ - 2\pi i\nu } }}{3}R_J^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right) + i\frac{{e^{ - 2\pi i\nu } }}{3}R_L^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right) + i\frac{{e^{ - 2\pi i\nu } }}{3}R_Q^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right)\\ & + \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{2N + \frac{1}{3}} }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - 2\pi i }} \right)^2 }}iR_J^{\left( H \right)} \left( {it} \right)dt} - \frac{{\left( { - 1} \right)^M }}{{3\pi \nu ^{2M + 1} }}\int_0^{ + \infty } {\frac{{t^{2M} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - 2\pi i } } \right)^2 }}iR_L^{\left( H \right)} \left( {it} \right)dt} \\ & + \frac{{\left( { - 1} \right)^K }}{{3\pi \nu ^{2K + \frac{5}{3}} }}\int_0^{ + \infty } {\frac{{t^{2K + \frac{2}{3}} e^{ - 2\pi t} }}{{1 + \left( {t/\nu e^{ - 2\pi i } } \right)^2 }}iR_Q^{\left( H \right)} \left( {it} \right)dt} , \end{align*} for $\frac{3\pi}{2} < \arg \nu < \frac{5\pi}{2}$. It is easy to see that the sum of three integrals has the order of magnitude given in the right-hand side of \eqref{eq59}. It follows that when $J=K=Q$, the bound \eqref{eq60} holds in the sector $\frac{3\pi}{2} \leq \arg \nu \leq \frac{5\pi}{2}$. Otherwise, we need to bound $R_J^{\left( H \right)} \left( \nu \right)$, $R_L^{\left( H \right)} \left( \nu \right)$ and $R_Q^{\left( H \right)} \left( \nu \right)$. From the connection formula \[ H_\nu ^{\left( 1 \right)} \left( \nu \right) = - H_{\nu e^{ - 2\pi i} }^{\left( 1 \right)} \left( {\nu e^{ - 2\pi i} } \right) - H_{\nu e^{ - 2\pi i} }^{\left( 2 \right)} \left( {\nu e^{ - 2\pi i} } \right) - e^{ - 2\pi i\nu } H_{\nu e^{ - 2\pi i} }^{\left( 2 \right)} \left( {\nu e^{ - 2\pi i} } \right), \] we obtain the relation \begin{align*} R_J^{\left( H \right)} \left( \nu \right) & = R_J^{\left( H \right)} \left( {\nu e^{ - 2\pi i} } \right) + R_J^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right) - e^{ - 2\pi i\nu } H_{\nu e^{ - 2\pi i} }^{\left( 2 \right)} \left( {\nu e^{ - 2\pi i} } \right) \\ & = R_J^{\left( H \right)} \left( {\nu e^{ - 2\pi i} } \right) + R_J^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right) + e^{ - 2\pi i\nu } R_0^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right). \end{align*}
Since $R_J^{\left( H \right)} \left( \nu \right) = \mathcal{O}_J\left( {\left| {d_{2J} } \right|\Gamma \left( {\frac{{2J + 1}}{3}} \right)\left| \nu \right|^{ - \frac{2J + 1}{3}} } \right)$ as $\nu \to \infty$ in the sector $-\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, we infer that \[
ie^{\frac{\pi }{3}i} \frac{{e^{2\pi i\nu } }}{3}R_J^{\left( H \right)} \left( \nu \right) = \mathcal{O}_J \left( {e^{ - 2\pi \Im \left( \nu \right)} \left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) + \mathcal{O}_J \left( {\left| \nu \right|^{ - \frac{1}{3}} } \right) \] for large $\nu$ with $\frac{3\pi}{2} \leq \arg \nu \leq \frac{5\pi}{2}$. A similar estimation holds for the terms involving $R_L^{\left( H \right)} \left( \nu \right)$ and $R_Q^{\left( H \right)} \left( \nu \right)$. The sum of the three terms containing $R_J^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right)$, $R_J^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right)$ and $R_Q^{\left( H \right)} \left( {\nu e^{ - \pi i} } \right)$ has the order of magnitude given in the right-hand side of \eqref{eq60}. Therefore, the final result is \begin{gather}\label{eq70} \begin{split}
R_{N,M,K}^{J,L,Q} \left( \nu \right) = \; & \mathcal{O}_{J,\rho } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2J} } \right|\frac{{\Gamma \left( {\frac{{2J + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2J + 1}}{3}} }}} \right) + \mathcal{O}_{L,\sigma } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2L} } \right|\frac{{\Gamma \left( {\frac{{2L + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2L + 1}}{3}} }}} \right)\\
& +\mathcal{O}_{Q,\eta } \left( {\cosh \left( {2\pi \Im \left( \nu \right)} \right)\left| {d_{2Q} } \right|\frac{{\Gamma \left( {\frac{{2Q + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2Q + 1}}{3}} }}} \right) + \mathcal{O}_{J,L,Q} \left( {\left| \nu \right|^{ - \frac{1}{3}} } \right) \end{split} \end{gather} as $\nu \to \infty$ in the sector $\frac{3\pi}{2} \leq \arg \nu \leq \frac{5\pi}{2}$.
Similarly, we find that when $J=L=Q$, the estimate \eqref{eq64} holds in the sector $-\frac{5\pi}{2} \leq \arg \nu \leq -\frac{3\pi}{2}$. Otherwise, it can be shown that the estimation \eqref{eq70} is valid in this sector too.
\subsection{Stokes phenomenon and Berry's transition}
\subsubsection{Case (i): $x>1$} We study the Stokes phenomenon related to the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$ occurring when $\arg \nu$ passes through the values $\pm \frac{\pi}{2}$. In the range $\left|\arg \nu\right|<\frac{\pi}{2}$, the asymptotic expansion \begin{equation}\label{eq52} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) \sim - \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{{\nu ^{2n + 1} }}} \end{equation} holds as $\nu \to \infty$. From \eqref{eq48} we have \begin{align*} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = R_{0,0} \left( {\nu ,\beta } \right) & = i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}R_0^{\left( H \right)} \left( {\nu ,\beta } \right) - R_{0,0} \left( {\nu e^{ - \pi i} ,\beta } \right)\\ & = iH_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right) - \mathbf{A}_{\nu} \left( {\nu e^{ - \pi i} \sec \beta } \right) \end{align*} when $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$. Similarly, from \eqref{eq49} we find \[ \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) = - iH_\nu ^{\left( 2 \right)} \left( {\nu \sec \beta } \right) - \mathbf{A}_{\nu} \left( {\nu e^{\pi i} \sec \beta } \right) \] for $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. For the right-hand sides, we can apply the asymptotic expansions of the Hankel functions and the Anger--Weber function to deduce that \begin{equation}\label{eq75} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) \sim i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} - \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{{\nu ^{2n + 1} }}} \end{equation} as $\nu \to \infty$ in the sector $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$, and \begin{equation}\label{eq76} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) \sim - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} - \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{{\nu ^{2n + 1} }}} \end{equation} as $\nu \to \infty$ in the sector $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. Therefore, as the line $\arg \nu = \frac{\pi}{2}$ is crossed, the additional series \begin{equation}\label{eq50} i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{\nu ^m}} \end{equation} appears in the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$ beside the original one \eqref{eq52}. Similarly, as we pass through the line $\arg \nu = -\frac{\pi}{2}$, the series \begin{equation}\label{eq51} - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\frac{{U_m \left( {i\cot \beta } \right)}}{\nu ^m }} \end{equation} appears in the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$ beside the original series \eqref{eq52}. We have encountered a Stokes phenomenon with Stokes lines $\arg \nu = \pm\frac{\pi}{2}$.
In the important paper \cite{Berry2}, Berry provided a new interpretation of the Stokes phenomenon; he found that assuming optimal truncation, the transition between compound asymptotic expansions is of Error function type, thus yielding a smooth, although very rapid, transition as a Stokes line is crossed.
Using the exponentially improved expansion given in Theorem \ref{thm3}, we show that the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right)$ exhibits the Berry transition between the two asymptotic series across the Stokes lines $\arg \nu = \pm\frac{\pi}{2}$. More precisely, we shall find that the first few terms of the series in \eqref{eq50} and \eqref{eq51} ``emerge" in a rapid and smooth way as $\arg \nu$ passes through $\frac{\pi}{2}$ and $-\frac{\pi}{2}$, respectively.
From Theorem \ref{thm3}, we conclude that if $N \approx \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right)$, then for large $\nu$, $ \left|\arg \nu\right| < \pi$, we have \begin{align*} \mathbf{A}_{ - \nu } \left( {\nu \sec \beta } \right) \approx & - \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( { - \sec \beta } \right)}}{\nu ^{2n + 1}}} \\ & + i\frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right)} \\ & - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta } \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)} , \end{align*} where $\sum\nolimits_{m = 0}$ means that the sum is restricted to the leading terms of the series.
In the upper half-plane the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)$ are exponentially small, the dominant contribution comes from the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right)$. Under the above assumption on $N$, from \eqref{eq37} and \eqref{eq39}, the Terminant functions have the asymptotic behaviour \[
\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta } \right)} \right) \sim \frac{1}{2} + \frac{1}{2} \mathop{\text{erf}} \left( {\left( {\theta - \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right)} } \right) \] provided that $\arg \nu = \theta$ is close to $\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N$. Therefore, when $\theta < \frac{\pi}{2}$, the Terminant functions are exponentially small; for $\theta = \frac{\pi }{2}$, they are asymptotically $\frac{1}{2}$ up to an exponentially small error; and when $\theta > \frac{\pi}{2}$, the Terminant functions are asymptotic to $1$ with an exponentially small error. Thus, the transition across the Stokes line $\arg \nu = \frac{\pi}{2}$ is effected rapidly and smoothly. Similarly, in the lower half-plane, the dominant contribution is controlled by the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right)$. From \eqref{eq38} and \eqref{eq39}, we have \[
\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta } \right)} \right) \sim \frac{1}{2} - \frac{1}{2} \mathop{\text{erf}} \left( {\left( {\theta + \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right)} } \right) \]
under the assumptions that $\arg \nu = \theta$ is close to $-\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N \approx \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta } \right)$. Thus, when $\theta > - \frac{\pi}{2}$, the Terminant functions are exponentially small; for $\theta = -\frac{\pi}{2}$, they are asymptotic to $\frac{1}{2}$ with an exponentially small error; and when $\theta < - \frac{\pi}{2}$, the Terminant functions are asymptotically $1$ up to an exponentially small error. Therefore, the transition through the Stokes line $\arg \nu = -\frac{\pi}{2}$ is carried out rapidly and smoothly.
We remark that from the expansions \eqref{eq75} and \eqref{eq76}, it follows that \eqref{eq52} is an asymptotic expansion of $\mathbf{A}_{-\nu} \left( {\nu \sec \beta } \right)$ in the wider sector $\left|\arg \nu\right| \leq \pi -\delta < \pi$, with any fixed $0 < \delta \leq \pi$.
\subsubsection{Case (ii): $x=1$} The analysis of the Stokes phenomenon for the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu } \right)$ is similar to the case $x > 1$. In the range $\left|\arg \nu\right| < \frac{3\pi}{2}$, the asymptotic expansion \begin{equation}\label{eq65} \mathbf{A}_{ - \nu } \left( \nu \right) \sim \frac{1}{{3\pi }}\sum\limits_{n = 0}^{\infty} {d_{2n} \frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} \end{equation} holds as $\nu \to \infty$. Employing the continuation formulas stated in Section \ref{section1}, we find that \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \mathbf{A}_{ - \nu } \left( {\nu e^{ - 2\pi i} } \right) - iH_{\nu }^{\left( 1 \right)} \left( {\nu e^{ - 2\pi i} } \right) - ie^{ - 2\pi i\nu } H_{\nu }^{\left( 2 \right)} \left( {\nu e^{ - 2\pi i} } \right) \] and \[ \mathbf{A}_{ - \nu } \left( \nu \right) = \mathbf{A}_{ - \nu } \left( {\nu e^{2\pi i} } \right) + ie^{2\pi i\nu } H_{\nu }^{\left( 1 \right)} \left( {\nu e^{2\pi i} } \right) + iH_{\nu }^{\left( 2 \right)} \left( {\nu e^{2\pi i} } \right) . \] For the right-hand sides, we can apply the asymptotic expansions of the Hankel functions and the Anger--Weber function to deduce that \begin{equation}\label{eq77} \mathbf{A}_{ - \nu } \left( \nu \right) \sim \frac{1}{{3\pi }}\sum\limits_{n = 0}^\infty {d_{2n} \frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} + ie^{ - 2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0}^\infty {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}} \end{equation} as $\nu \to \infty$ in the sector $\frac{3\pi}{2} < \arg \nu < \frac{5\pi}{2}$, and \begin{equation}\label{eq78} \mathbf{A}_{ - \nu } \left( \nu \right) \sim \frac{1}{{3\pi }}\sum\limits_{n = 0}^\infty {d_{2n} \frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} - ie^{2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0}^\infty {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}} \end{equation} as $\nu \to \infty$ in the sector $-\frac{5\pi}{2} < \arg \nu < -\frac{3\pi}{2}$. Therefore, as the line $\arg \nu = \frac{3\pi}{2}$ is crossed, the additional series \begin{equation}\label{eq66} ie^{ - 2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0}^\infty {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}} \end{equation} appears in the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( {\nu } \right)$ beside the original one \eqref{eq65}. Similarly, as we pass through the line $\arg \nu = -\frac{3\pi}{2}$, the series \begin{equation}\label{eq67} - ie^{2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0}^\infty {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}} \end{equation} appears in the asymptotic expansion of $\mathbf{A}_{ - \nu } \left( \nu \right)$ beside the original series \eqref{eq65}. We have encountered a Stokes phenomenon with Stokes lines $\arg \nu = \pm\frac{3\pi}{2}$. With the aid of the exponentially improved expansion given in Theorem \ref{thm4}, we shall find that the asymptotic series of $\mathbf{A}_{ - \nu } \left( \nu \right)$ shows the Berry transition property: the two series in \eqref{eq66} and \eqref{eq67} “emerge” in a rapid and smooth way as the Stokes lines $\arg \nu = \frac{3\pi}{2}$ and $\arg \nu = -\frac{3\pi}{2}$ are crossed.
Let us assume that in \eqref{eq68} $N,M,K \approx \pi \left|\nu\right|$ and $J=L=Q$. When $\pi < \arg \nu < 2\pi$, the terms in \eqref{eq68} involving the Terminant functions of the argument $2\pi i \nu$ are exponentially small, and the main contribution comes from the terms involving the Terminant functions of the argument $-2\pi i \nu$. Therefore, from Theorem \ref{thm4}, we deduce that for large $\nu$, $\pi < \arg \nu < 2\pi$, we have \begin{align*} & \mathbf{A}_{ - \nu } \left( \nu \right) \approx \frac{1}{{3\pi \nu ^{\frac{1}{3}} }}\sum\limits_{n = 0}^{N - 1} {d_{6n} \frac{{\Gamma \left( {2n + \frac{1}{3}} \right)}}{{\nu ^{2n} }}} + \frac{1}{{3\pi \nu }}\sum\limits_{m = 0}^{M - 1} {d_{6m + 2} \frac{{\Gamma \left( {2m + 1} \right)}}{{\nu ^{2m} }}} + \frac{1}{{3\pi \nu ^{\frac{5}{3}} }}\sum\limits_{k = 0}^{K - 1} {d_{6k + 4} \frac{{\Gamma \left( {2k + \frac{5}{3}} \right)}}{{\nu ^{2k} }}} \\ & + ie^{ - 2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0} {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}\frac{{\widehat T_{2N - \frac{{2j}}{3}} \left( { - 2\pi i\nu } \right) + \widehat T_{2M - \frac{{2j - 2}}{3}} \left( { - 2\pi i\nu } \right) + \widehat T_{2K - \frac{{2j - 4}}{3}} \left( { - 2\pi i\nu } \right)}}{3}} , \end{align*} where, as before, $\sum\nolimits_{j = 0}$ means that the sum is restricted to the leading terms of the series.
Since $N,M,K \approx \pi \left|\nu\right|$, from \eqref{eq37} and \eqref{eq39}, the averages of the Terminant functions have the asymptotic behaviour \[
\frac{{\widehat T_{2N - \frac{{2j}}{3}} \left( { - 2\pi i\nu } \right) + \widehat T_{2M - \frac{{2j - 2}}{3}} \left( { - 2\pi i\nu } \right) + \widehat T_{2K - \frac{{2j - 4}}{3}} \left( { - 2\pi i\nu } \right)}}{3} \sim \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}}\left( {\left( {\theta - \frac{{3\pi }}{2}} \right)\sqrt {\pi \left| \nu \right|} } \right), \] under the conditions that $\arg \nu = \theta$ is close to $\frac{3\pi}{2}$, $\nu$ is large and $j$ is small compared to $N$, $M$ and $K$. Thus, when $\theta < \frac{3\pi}{2}$, the averages of the Terminant functions are exponentially small; for $\theta = \frac{3\pi}{2}$, they are asymptotic to $\frac{1}{2}$ with an exponentially small error; and when $\theta > \frac{3\pi}{2}$, the averages of the Terminant functions are asymptotically $1$ up to an exponentially small error. Thus, the transition through the Stokes line $\arg \nu = \frac{3\pi}{2}$ is carried out rapidly and smoothly.
Similarly, if $N,M,K \approx \pi \left|\nu\right|$ and $J=L=Q$, then for large $\nu$, $-2\pi < \arg \nu < -\pi$, we have \begin{align*} \mathbf{A}_{ - \nu } \left( \nu \right) \approx \; & \frac{1}{{3\pi \nu ^{\frac{1}{3}} }}\sum\limits_{n = 0}^{N - 1} {d_{6n} \frac{{\Gamma \left( {2n + \frac{1}{3}} \right)}}{{\nu ^{2n} }}} + \frac{1}{{3\pi \nu }}\sum\limits_{m = 0}^{M - 1} {d_{6m + 2} \frac{{\Gamma \left( {2m + 1} \right)}}{{\nu ^{2m} }}} + \frac{1}{{3\pi \nu ^{\frac{5}{3}} }}\sum\limits_{k = 0}^{K - 1} {d_{6k + 4} \frac{{\Gamma \left( {2k + \frac{5}{3}} \right)}}{{\nu ^{2k} }}} \\ & - ie^{2\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{j = 0} {d_{2j} \sin \left( {\frac{{\left( {2j + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2j + 1}}{3}} \right)}}{{\nu ^{\frac{{2j + 1}}{3}} }}e^{\frac{{2\left( {2j + 1} \right)\pi i}}{3}}} \\ & \times \frac{e^{\frac{\pi }{3}i} \widehat T_{2N - \frac{{2j}}{3}} \left( {2\pi i\nu } \right) - \widehat T_{2M - \frac{{2j - 2}}{3}} \left( { 2\pi i\nu } \right) + e^{ - \frac{\pi }{3}i} \widehat T_{2K - \frac{{2j - 4}}{3}} \left( { 2\pi i\nu } \right)}{3} . \end{align*} From \eqref{eq38} and \eqref{eq39}, the averages of the scaled Terminant functions have the asymptotic behaviour \[
e^{\frac{{2\left( {2j + 1} \right)\pi i}}{3}} \frac{{e^{\frac{\pi }{3}i} \widehat T_{2N - \frac{{2j}}{3}} \left( {2\pi i\nu } \right) - \widehat T_{2M - \frac{{2j - 2}}{3}} \left( {2\pi i\nu } \right) + e^{ - \frac{\pi }{3}i} \widehat T_{2K - \frac{{2j - 4}}{3}} \left( {2\pi i\nu } \right)}}{3} \sim \frac{1}{2} - \frac{1}{2}\mathop{\text{erf}} \left( {\left( {\theta + \frac{{3\pi }}{2}} \right)\sqrt {\pi \left| \nu \right|} } \right), \]
provided that $N,M,K \approx \pi \left|\nu\right|$, $\arg \nu = \theta$ is close to $-\frac{3\pi}{2}$, $\nu$ is large and $j$ is small compared to $N$, $M$ and $K$. Therefore, when $\theta > - \frac{3\pi}{2}$, the averages of the scaled Terminant functions are exponentially small; for $\theta = -\frac{3\pi}{2}$, they are asymptotic to $\frac{1}{2}$ up to an exponentially small error; and when $\theta < -\frac{3\pi}{2}$, the averages of the scaled Terminant functions are asymptotically $1$ with an exponentially small error. Thus, the transition through the Stokes line $\arg \nu = -\frac{3\pi}{2}$ is effected rapidly and smoothly.
We note that from the expansions \eqref{eq77} and \eqref{eq78}, it follows that \eqref{eq65} is an asymptotic series of $\mathbf{A}_{-\nu} \left( \nu \right)$ in the wider range $\left|\arg \nu\right| \leq 2\pi -\delta < 2\pi$, with any fixed $0 < \delta \leq 2\pi$.
\section{Discussion}\label{section6}
In this paper, we have discussed in detail the large order and argument asymptotics of the Anger--Weber function $\mathbf{A}_{-\nu}\left(\nu x\right)$ when $x \geq 1$, using Howls' method. When $0<x<1$, the path $\mathscr{P}\left(0\right)$, defined in \eqref{eq71}, is not the positive real axis, whence the method is not applicable. If we put $x = \mathop{\text{sech}} \alpha$ with a suitable $\alpha > 0$, the large $\nu$ asymptotics of $\mathbf{A}_{-\nu}\left(\nu x\right)$ can be written as \begin{equation}\label{eq72} \mathbf{A}_{ - \nu } \left( {\nu \mathop{\text{sech}} \alpha } \right) \sim \sqrt {\frac{2}{{\pi \nu }}} e^{ \nu \left( {\alpha-\tanh \alpha} \right)} \sum\limits_{n = 0}^\infty {\frac{{\left( {\frac{1}{2}} \right)_n b_n \left( {\mathop{\text{sech}} \alpha } \right)}}{{\nu ^n }}} \end{equation} as $\nu \to +\infty$, with $\left( z \right)_n = \Gamma \left( {z + n} \right)/\Gamma \left( z \right)$ \cite[p. 298]{NIST}. The first few coefficients are given by \[ b_0 \left( {\mathop{\text{sech}} \alpha } \right) = \frac{1}{{\left( {1 - \mathop{\text{sech}} ^2 \alpha } \right)^{\frac{1}{4}} }},\; b_1 \left( {\mathop{\text{sech}} \alpha } \right) = \frac{{2 + 3\mathop{\text{sech}} ^2 \alpha }}{{12\left( {1 - \mathop{\text{sech}} ^2 \alpha } \right)^{\frac{7}{4}} }},\; b_2 \left( {\mathop{\text{sech}} \alpha } \right) = \frac{{5 + 300\mathop{\text{sech}} ^2 \alpha + 81\mathop{\text{sech}} ^4 \alpha }}{{864\left( {1 - \mathop{\text{sech}} ^2 \alpha } \right)^{\frac{{13}}{4}} }}. \] It is also known that $\mathbf{A}_{ - \nu } \left( {\nu \mathop{\text{sech}} \alpha } \right)$ has the same asymptotic expansion as the Bessel function $-Y_\nu\left(\nu \mathop{\text{sech}} \alpha\right)$, namely \begin{equation}\label{eq73} -Y_\nu \left( {\nu \mathop{\text{sech}}\alpha } \right) \sim \frac{{e^{ \nu \left( \alpha-\tanh \alpha \right)} }}{{\left( {\frac{1}{2}\pi \nu \tanh \alpha } \right)^{\frac{1}{2}} }}\sum\limits_{n = 0}^\infty {\left( { - 1} \right)^n \frac{{U_n \left( {\coth \alpha } \right)}}{{\nu ^n }}} \; \text{ as } \; \nu \to +\infty. \end{equation} Here $U_n \left( {\coth \alpha } \right) = \left[U_n \left( x \right)\right]_{x = \coth \alpha }$, where $U_n\left(x\right)$ is a polynomial in $x$ of degree $3n$. These polynomials can be generated by the following recurrence \[ U_n \left( x \right) = \frac{1}{2}x^2 \left( {1 - x^2 } \right)U'_{n-1} \left( x \right) + \frac{1}{8}\int_0^x {\left( {1 - 5t^2 } \right)U_{n-1} \left( t \right)dt} \] for $n \ge 1$ with $U_0\left(x\right) = 1$ (see, e.g., \cite[p. 376]{Olver}, \cite[p. 256]{NIST}). The uniqueness property of asymptotic power series implies that \begin{align*} b_n \left( \mathop{\text{sech}}\alpha \right) & = \left( { - 1} \right)^n \frac{{2^{2n} n!}}{{\left( {2n} \right)!\tanh ^{\frac{1}{2}} \alpha }}U_n \left( {\coth \alpha } \right) \\ & = \left( { - 1} \right)^n \frac{{2^{2n} n!}}{{\left( {2n} \right)!\left( {1 - \mathop{\text{sech}}^2\alpha } \right)^{\frac{1}{4}} }}U_n \left( {\left( {1 - {\mathop{\text{sech}}}^2 \alpha } \right)^{ - \frac{1}{2}} } \right) \end{align*} for any $n\geq 0$. Based on Darboux's method, Dingle \cite[p. 168]{Dingle} gave a formal asymptotic expansion for the coefficients $U_n \left( {\coth \alpha } \right)$ when $n$ is large. His result, in our notation, may be written as \begin{equation}\label{eq74} U_n \left( {\coth \alpha } \right) \approx \frac{{\left( { - 1} \right)^n \Gamma \left( n \right)}}{{2\pi \left( {2\left( {\alpha - \tanh \alpha } \right)} \right)^n }}\sum\limits_{m = 0}^\infty {\left( {2\left( {\alpha - \tanh \alpha } \right)} \right)^m U_m \left( {\coth \alpha } \right)\frac{{\Gamma \left( {n - m} \right)}}{{\Gamma \left( n \right)}}} . \end{equation} Numerical calculations indicate that this approximation is correct if it is truncated after the first few terms. Using his formal theory of terminants, Dingle gave exponentially improved versions of \eqref{eq72} and \eqref{eq73} \cite[p. 468 and p. 512]{Dingle}.
As far as we know, no rigorous proof of the late coefficient formula \eqref{eq74} nor realistic error bounds for the expansion \eqref{eq72} are available in the literature. Perhaps, these issues can be handled using differential equation methods, but we leave it as a future research topic.
\appendix \section{}\label{appendixa}
In this appendix we give some formulas for the computation of the coefficients $a_n \left( -\sec \beta \right)$ appearing in the large $\nu$ asymptotics of $\mathbf{A}_{-\nu}\left(\nu \sec \beta\right)$. It is known that $a_n \left( { - \sec \beta } \right) = \left[ {a_n \left( \lambda \right)} \right]_{\lambda = - \sec \beta } $ where $a_n\left(\lambda\right)$ is a rational function of $\lambda \neq -1$. We consider these rational functions. The first few are given by \[ a_0 \left( \lambda \right) = \frac{1}{{1 + \lambda }},\; a_1 \left( \lambda \right) = - \frac{\lambda }{{2\left( {1 + \lambda } \right)^4 }},\; a_2 \left( \lambda \right) = \frac{{9\lambda ^2 - \lambda }}{{24\left( {1 + \lambda } \right)^7 }},\; a_3 \left( \lambda \right) = - \frac{{225\lambda ^3 - 54\lambda ^2 + \lambda }}{{720\left( {1 + \lambda } \right)^{10} }} . \] From \eqref{eq9} we infer that \[ a_n \left( \lambda \right) = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{t}{{\lambda \sinh t + t}}} \right)^{2n + 1} } \right]_{t = 0} . \]
Meijer \cite{Meijer} proved the following explicit formula \begin{equation}\label{eq34} a_n \left( \lambda \right) = \frac{1}{{\left( {1+ \lambda } \right)^{2n + 1} }}\sum\limits_{k = 0}^n {\binom{2n + k}{k}\frac{{\left( { - 1} \right)^k }}{{\left( {2n - 2k} \right)!}}\left[ {\frac{{d^{2n - 2k} }}{{dt^{2n - 2k} }}\left( {\frac{{\sinh t - t}}{{t^3 }}} \right)^k } \right]_{t = 0} \left( {\frac{\lambda }{{1+ \lambda }}} \right)^k } . \end{equation} We show that the higher derivatives can be written in terms of the generalized Bernoulli polynomials $B_n^{\left( \kappa \right)} \left(\ell\right)$, which are defined by the exponential generating function \[
\left( \frac{z}{e^z - 1} \right)^\kappa e^{\ell z} = \sum\limits_{n = 0}^\infty {B_n^{\left( \kappa \right)} \left(\ell\right)\frac{z^n}{n!}} \; \text{ for } \; \left|z\right| < 2\pi. \] For basic properties of these polynomials, see Milne-Thomson \cite{Milne-Thomson} or N\"{o}rlund \cite{Norlund}. A straightforward computation gives \begin{align*} & \frac{1}{{\left( {2n-2k} \right)!}}\left[ {\frac{{d^{2n-2k} }}{{dt^{2n-2k} }}\left( {\frac{{\sinh t - t}}{{t^3 }}} \right)^k } \right]_{t = 0} = \frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\left( {\frac{{\sinh z - z}}{{z^3 }}} \right)^k \frac{{dz}}{{z^{2n-2k + 1} }}} \\ & = \frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\left( {\sum\limits_{j = 0}^k {\left( { - 1} \right)^{k - j} \binom{k}{j}z^{k - j} \sinh ^j z} } \right)\frac{{dz}}{{z^{2n + k + 1} }}} = \sum\limits_{j = 0}^k {\left( { - 1} \right)^{k - j} \binom{k}{j}\frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\left( {\frac{{\sinh z}}{z}} \right)^j \frac{dz}{{z^{2n + 1} }}} } \\ & = \sum\limits_{j = 0}^k {\left( { - 1} \right)^{k - j} \binom{k}{j}\frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\left( {\frac{{2z}}{{e^{2z} - 1}}} \right)^{ - j} e^{ - jz} \frac{dz}{{z^{2n + 1} }}} } = \frac{{2^{2n} }}{{\left( {2n} \right)!}}\sum\limits_{j = 0}^k {\left( { - 1} \right)^{k - j} \binom{k}{j}B_{2n}^{\left( { - j} \right)} \left( { - \frac{j}{2}} \right)}. \end{align*} Substitution into \eqref{eq34} yields the explicit representation \[ a_n \left( \lambda \right) = \frac{{2^{2n} }}{{\left( {2n} \right)!^2 \left( {1+ \lambda} \right)^{2n + 1} }}\sum\limits_{k = 0}^n {\sum\limits_{j = 0}^k {\left( { - 1} \right)^j \frac{{\left( {2n + k} \right)!}}{{\left( {k - j} \right)!j!}}B_{2n}^{\left( { - j} \right)} \left( { - \frac{j}{2}} \right)} \left( {\frac{\lambda }{1+\lambda}} \right)^k } . \]
In 1952, Lauwerier \cite{Lauwerier} showed that the coefficients in asymptotic expansions of Laplace-type integrals can be calculated by means of linear recurrence relations. Simple application of his method provides the formula \[ a_n \left( \lambda \right) = \frac{1}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} e^{ - \left( {1 + \lambda } \right)t} P_n \left( t \right)dt} , \] where the polynomials $P_0 \left( x \right), P_1 \left( x \right), P_2 \left( x \right),\ldots$ are given by the recurrence relation \[ P_n \left( x \right) = - \sum\limits_{k = 1}^n {\frac{\lambda }{{\left( {2k + 1} \right)!}}\int_0^x {P_{n - k} \left( t \right)dt} } \] with $P_0\left(x\right)=1$.
A simpler recurrence for the $a_n \left( \lambda \right)$'s can be found using the inhomogeneous Bessel differential equation \[ \frac{{d^2 w\left( z \right)}}{{dz^2 }} + \frac{1}{z}\frac{{dw \left( z \right)}}{{dz}} + \left( {1 - \frac{{\nu ^2 }}{{z^2 }}} \right)w\left( z \right) = \frac{{z - \nu }}{{\pi z^2 }} \] satisfied by the Anger--Weber function $\mathbf{A}_{\nu}\left(z\right)$. Substituting $z=\nu \lambda$ shows that \begin{equation}\label{eq35} \frac{\lambda ^2}{\nu ^2}\frac{{d^2 {\bf A}_\nu \left( { \nu \lambda} \right)}}{{d\lambda ^2 }} + \frac{\lambda}{\nu ^2}\frac{{d{\bf A}_\nu \left( { \nu \lambda} \right)}}{d\lambda} + \left( {\lambda ^2 - 1} \right){\bf A}_\nu \left( {\nu \lambda } \right) = \frac{{\lambda - 1}}{{\pi \nu }} . \end{equation} It is known that for any $\lambda>0$, the function ${\bf A}_\nu \left( {\nu \lambda} \right)$ has the asymptotic expansion \[ {\bf A}_\nu \left( {\nu \lambda} \right) \sim \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( \lambda \right)}}{\nu ^{2n + 1}}} \]
as $\nu \to \infty$, $\left|\arg \nu\right| <\pi$ (see, e.g, \cite[p. 298]{NIST}). Substituting this series into \eqref{eq35} and equating the coefficients of the inverse powers of $\nu$ we find \[ a_0 \left( \lambda \right) = \frac{1}{{1 + \lambda }}\; \text{ and } \;a_n \left( \lambda \right) = \frac{\lambda}{1 - \lambda ^2} \frac{{\lambda a''_{n - 1} \left( \lambda \right) + a'_{n - 1} \left( \lambda \right)}}{2n\left( {2n - 1} \right)} \; \text{ for } \; n\geq 1. \]
\end{document}
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arXiv
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β-Thalassemia minor & renal tubular dysfunction: is there any association?
Mohsen Vakili Sadeghi1,
Maryam Mirghorbani2 &
Roghayeh Akbari1,3
Beta(β)-thalassemia is one of the most common hereditary hematologic disorders. Patients with thalassemia minor (TM) are often asymptomatic and the rate of renal dysfunction is unknown in these patients. Due to the high prevalence of renal dysfunction in Iran, the current study aimed to determine renal tubular dysfunction in patients with beta-TM.
In this case-control study, 40 patients with TM and 20 healthy subjects were enrolled and urinary and blood biochemical analysis was done on their samples. Renal tubular function indices were determined and compared in both groups. Data was analyzed by SPSS software, version 20.0.
The fraction excretion (FE) of uric acid was 8.31 ± 3.98% in the case and 6.2 ± 34.71% in the control group (p = 0.048). Also, FE of potassium was significantly higher in patients with TM (3.22 ± 3.13 vs. 1.91 ± 0.81; p = 0.036). The mean Plasma NGAL level was 133.78 ± 120.28 ng/mL in patients with thalassemia and 84.55 ± 45.50 ng/mL in the control group (p = 0.083). At least one parameter of tubular dysfunction was found in 45% of patients with thalassemia.
Based on the results of this study, the prevalence of tubular dysfunction in beta-thalassemia minor patients is high. Due to the lack of knowledge of patients about this disorder, periodic evaluation of renal function in TM patients can prevent renal failure by early diagnosis.
β-thalassemia is one of the most common hereditary hematologic disorders, characterized by disturbances in beta chain hemoglobin synthesis [1]. It is the most common single-gene disorder in Iran and more than 25,000 major β-thalassemia have been reported [2]. Three main clinical forms of β-thalassemia include thalassemia major, thalassemia intermedia (TI) and (TM) [3]. Patients with TM or Cooley's anemia require regular transfusions before age 24 months to survive. Thalassemia intermedia patients do not require or rarely require blood transfusions. They may be asymptomatic up to adulthood but their sign and symptoms include biliary gallstone, jaundice, osteoporosis, hepatosplenomegaly and mass lesions related to extramedullary hematopoiesis. Individuals with TM or thalassemia carries have mild anemia with no or minimal symptoms [4, 5]. Clinical manifestations of beta thalassemia result from one or both beta globin gene mutations. Patients with TM are homozygotes or double heterozygotes, TI results from heterozygotes or homozygote and individuals with TM are heterozygote for beta gene mutations [6].
Thalassemia can be sub grouped to transfusion dependent thalassemia (TDT) and non-transfusion dependent thalassemia (NTDT). Phenotype of NTDT includes heterogeneous thalassemia genotypes that do not require frequent transfusions but is more severe than TM and have several complications due to ineffective erythropoiesis, extramedullary hematopoiesis and iron overload. Five form of NTDT has been described: beta thalassemia intermedia, hemoglobin E β-thalassemia, HbH disease, hemoglobin S β-thalassemia and hemoglobin C β thalassemia [7].
The improved TM and TI survival has allowed previously unrecognized renal complications to emerge [8]. The effect of thalassemia on the kidney has not been extensively evaluated. Up to 60% of patients with beta-thalassemia major have been reported to develop signs of tubular dysfunction [9].
Renal dysfunction is an uncommon complication in patients with β-thalassemia [10]. Mild impairment of tubular function and decreased glomerular filtration rate (GFR) have been reported in older patients with alpha-thalassemia, β-thalassemia major and hemoglobin E/β-thalassemia [1, 3]. Tubular dysfunction among patients with beta-thalassemia has been related to long-term anemia with chronic hypoxemia, intravascular and extravascular hemolysis, chronic blood transfusions, iron overload, as well as desferrioxamine toxicity [8, 11, 12]. Regular screening of renal function in high-risk thalassemia patients with diabetes, hypertension, proteinuria and GFR < 60 mL/min/1.73 m2 and elderly people in order to detect early renal involvement and thus prevent the onset of renal impairment is recommended [13].
Although there are many available data about renal involvement in patients with beta-thalassemia major, the changes in renal functions of TM were reported less [9, 14,15,16,17,18]. For a long time, there have been no reports regarding renal impairment in these patients, until 2002, which Oktenli and Bulucu, for the first time reported a 20-year-old case of TM in Turkey, who was investigated due to positive Glucosuria with dip-stick analysis. The tests reported a 24-h urine glucose secretion of 5 g and tubular proteinuria, a sign of his involvement in renal tubular dysfunction [17]. Patients with TM may have a lower frequency of hyperuricosuria and phosphaturia [1]. The most common probable cause of renal tubular dysfunction in thalassemia intermediate is iron deposition in the epithelial cells of the tubules. Another factor may be hypoxia due to hemolysis of red blood cells and their shorter life span and this hypoxia has the greatest effect on the adrenal cortical epithelial cells that are more susceptible to oxygen deficiency [16, 17]. In these patients, tubular and glomerular dysfunction depends on the severity of anemia, frequency of blood transfusion, and iron load [19]. Considering the possibility of renal dysfunction in patients with TM and the high prevalence of this type of thalassemia in Mazandaran province, this study aimed to determine renal tubular dysfunction in these patients.
In this case-control study, patients referring to the hematology clinic of Rouhani Hospital of Babol between March 2017 to March 2018 because of anemia were evaluated. Inclusion criteria were confirmation of β-thalassemia minor with hemoglobin electrophoresis and patients' satisfaction to participate in the study. Patients with cardiovascular disease (cardiac arrhythmia, aortic stenosis, ischemic heart disease, hypertension, and heart failure), liver disease, any type of cancer, active infection, history of renal disease (proteinuria, increased creatinine, acute kidney injury history), diabetes, drug addiction, taking medications that are excreted through the kidneys within the last 3 months and pregnancy were excluded from the study.
Forty β-thalassemia minor patients were selected by the simple randomization selection method, serum ferritin, complete blood count and hemoglobin electrophoresis were performed for all patients. Beta thalassemia minor was confirmed by complete blood count, HbA2 ≥ 3.5 g/dL and HbF < 5 g/dL in hemoglobin electrophoresis. For eligible persons, the research plan was fully explained and written informed consent was obtained. Also, 20 healthy staffs and physicians of the Rohani hospital, without any history of chronic diseases, were selected in the control group by the simple randomization selection method. After recording the demographic data, a complete history of drug use and underlying diseases was obtained. Then, 5 cc venous blood sample and 24-h urine were obtained from both groups.
Urine specimens were evaluated for appearance, pH, specific gravity, sodium, potassium, phosphorus, calcium, uric acid, creatinine, microalbumin and the presence of ketones, glucose, protein, bilirubin, urobilinogen, red and white blood cells, bacteria, casts, crystals, mucus and epithelial cells. Blood samples were also analyzed for fasting blood sugar (FBS), blood urea nitrogen (BUN), creatinine, uric acid, sodium, potassium, calcium and phosphorus. Urinary Neutrophil Gelatinase Associated Lipocalin (NGAL) was also measured.
Sodium and potassium were measured using flame photometer (Assel Co., Rome, Italy), urine microalbumin and biochemical tests were measured using conventional commercial kits and spectrophotometric method. Also, for biochemical analysis and measuring urinary NGAL, Human NGAL Rapid ELISA Kit (KIT037) manufactured by the Bioporto diagnostic company, Denmark was used by using the monoclonal ELISA sandwich method.
Calculation of renal function indices was based on the following: Glucosuria: Positive glucose in the urine; Hypercalciuria: Calcium > 300 mg in men and > 250 mg in women in 24-h urine; Hyperphosphaturia: Phosphate > 1000 mg in 24-h urine; Uricosuria: uric acid > 750 mg in 24-h urine; Microalbuminuria: Microalbumin > 30 mg in 24-h urine.
GFR was calculated based on the Modification of Diet in Renal Disease (MDRD) formula [20]:
GFR = 186.3 × (Plasma Cr)-1.154 × (age)-0.203 (For women, it was multiplied with 0.742).
$$\mathrm{FENa}=\frac{\mathrm{urinary}\ \mathrm{Na}\ \left[\mathrm{mg}/\mathrm{mL}\right]\ }{\mathrm{serum}\ \mathrm{Na}\ \left[\mathrm{mg}/\mathrm{mL}\right]} \times \frac{\mathrm{serum}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}{\mathrm{urinary}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}\times 100$$
Normal values of FENa were less than 1%.
$$\mathrm{FEK}=\frac{\mathrm{urinary}\ \mathrm{K}\ \left[\mathrm{mg}/\mathrm{mL}\right]\ }{\mathrm{serum}\ \mathrm{K}\ \left[\mathrm{mg}/\mathrm{mL}\right]} \times \frac{\mathrm{serum}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}{\mathrm{urinary}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}\times 100$$
Normal values of FEK were less than 15%.
$$\mathrm{FEUA}=\frac{\mathrm{urinary}\ \mathrm{uric}\ \mathrm{acid}\ \left[\mathrm{mg}/\mathrm{mL}\right]\ }{\mathrm{serum}\ \mathrm{uric}\ \mathrm{acid}\ \left[\mathrm{mg}/\mathrm{mL}\right]} \times \frac{\mathrm{serum}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}{\mathrm{urinary}\ \mathrm{creatinine}\ \left[\mathrm{mg}/\mathrm{mL}\right]}\times 100$$
Normal values of FEUA were < 10%.
Renal Tubular Reabsorption of Phosphate (TmP/GFR)=.
$$\mathrm{Plasma}\ \mathrm{Phosphate}=\frac{\mathrm{Plasma}\ \mathrm{Creatinin}\mathrm{e}\times \mathrm{Urine}\ \mathrm{phosphate}\ }{\mathrm{Urine}\ \mathrm{Creatinin}}$$
TmP/GFR values < 2.88 were considered to be normal.
Data was analysed using Statistical Package for statistical analysis (SPSS) version 20.0. Information description was by frequency tables and related charts. To characterize qualitative characteristics, frequency and percentage were used, and for the quantitative characteristics, the mean and range of variations were used. Qualitative data was analyzed using chi-square and Fisher exact tests and quantitative variables by t-test and Mann-Whitney tests for comparison of averages. The statistical significance level in all tests was considered 0.05, so that P-value < 0.05 showed a significant statistical difference.
The research protocol was approved by the Ethics Committee of Babol University of Medical Sciences (Registration code = MUBABOL.REC.1395.150) and all methods were carried out in accordance with relevant guidelines and regulations and the study was conducted in accordance with the Declaration of Helsinki.
The mean age of the case and control groups were 41.93 ± 17.71 and 39.20 ± 10.13 years, respectively (p = 0.527). In the study population, 37.5% of cases (15 of 40 cases) and 45% (9 of 20 cases) of controls were male (p = 0.576). The mean hemoglobin, hematocrit, MCV, MCH and MCHC were significantly lower in the case group compared to the controls (p = 0.0001) but no significant difference was found between the two groups for white blood cells, red blood cells and platelets (p > 0.05) (Table 1).
Table 1 Comparison of biochemical parameters and fractional excretion of different parameters in the two groups
The mean plasma concentrations of creatinine, uric acid, phosphorus, sodium and potassium were not significantly different in the two groups, but BUN and calcium were significantly lower in patients with thalassemia (p = 0.007 and p = 0.002, respectively). Also, urine microalbumin, creatinine, uric acid, phosphorus, calcium, sodium and potassium were not significantly different between the two groups (p > 0.05).
In the case group, glucosuria occurred in 3 patients (7.5%) and proteinuria in 2 patients (5%), while they were not reported in any of the control subjects. There was a significant difference between FEUA in both groups (p = 0.048). Also, FEK was significantly higher in patients with TM (p = 0.036) but FENa and FECa did not differ significantly (p = 0.099 and p = 0.227, respectively) (Table 2).
Table 2 Comparison of renal tubular dysfunction parameters in two groups
The mean eGFR in case and control groups were 89.95 ± 17.55 and 87.93 ± 15.49, respectively (p = 0.676). The maximum ratio of TMP/GFR in patients with TM was 0.33 ± 1.03 mg/dL and in normal subjects it was 3.37 ± 0.78 mg/dL, which was not significantly different between the two groups (p = 0.193). The mean plasma NGAL in the case and control groups were 133.78 ± 120.28 ng/mL and 84.55 ± 45.55 ng/mL, respectively (p = 0.083) (Table 1).
In the comparison of tubular dysfunction factors, glucosuria and microalbuminuria were significantly higher in the thalassemia group (p = 0.045 and p = 0.0001. respectively). Also, plasma NGAL values > 179 ng/mL was reported in 6 patients with thalassemia (15%) and 1 healthy person (5%), which was significantly higher in patients with thalassemia (p = 0.025).
None of the renal tubular dysfunction parameters were seen in 80% of controls and 55% of patients with TM (p = 0.046) (Table 3). The mean plasma NGAL in the case group which had microalbuminuria was higher than those without microalbuminuria (p = 0.047). Also, NGAL was higher in patients with FENa> 1% and FEUA> 10% than other thalassemic patients (p = 0.011 and p = 0.004, respectively) (Table 4).
Table 3 Comparing the mean NGAL based on renal tubular dysfunction factors in patients with thalassemia
Table 4 Comparison of increased NGAL based on the number of renal tubular dysfunction parameters in patients with thalassemia
The mean hemoglobin level in thalassemic patients with NGAL > 176 ng/mL was 10.31 ± 1.73 g/dL and in patients with NGAL < 176 ng/mL, it was 11.93 ± 2.88 g/dL (p = 0.052).
The mean plasma NGAL in patients with TM with at least one parameter of renal tubular dysfunction was 146.54 ± 115.59 ng/mL and in patients with no abnormalities in renal tubular function, it was 123.12 ± 77.14 ng/mL (p = 0.046) (Table 4). Thalassemic patients were classified into 3 types based on renal tubular dysfunction. The mean NGAL of the third group was significantly higher than the first and second groups (p = 0.004 and p = 0.025, respectively). The abnormal NGAL was not significantly associated with the number of renal tubular dysfunction parameters (p = 0.581).
The association between serum and urinary levels of NGAL and renal dysfunction was evaluated [21,22,23], but to our knowledge, no biomarker study has been conducted in patients with beta-thalassemia minor so far and this study is the first evaluation in this type.
The results of this study showed evidence of renal tubular disorder in patients with TM and in 45% of patients, at least one parameter of renal tubular dysfunction was reported. Glucosuria and microalbuminuria were significantly higher in thalassemic patients than in healthy subjects, but other parameters of renal tubular damage did not differ in two groups. Also, fraction excretion of uric acid and potassium was significantly higher in patients with thalassemia. For the first time, Oktenli and Bulucu reported glucosuria and tubular proteinuria in a male with β-thalassemia minor [17]. Prabahar and colleagues reported evidence of nephrocalcinosis associated with renal tubular dysfunction, such as hypercalciuria, decreased phosphorus tubular reuptake, hypomagnesemia, and urinary magnesium loss in a 24-year-old woman with TM [18]. Cetin et al. found renal tubular dysfunction as a common complication in patients with TM and 14.6% of these patients had renal tubulopathy symptoms such as hypercalciuria, decreased TRP with hypophosphatemia, hypomagnesemia associated with renal magnesium loss, hypouricemia along with renal excretion of uric acid and tubular proteinuria [14], but no evidence of tubular dysfunction was found in Kalman et al.'s study in patients with TM [16]. In the study of Hoseinzadeh et al. in Shiraz, of 86 patients with TM, 24% had renal tubular dysfunction [15]. Sadeghi-Bojd and colleagues reported symptoms of tubulopathy such as proteinuria, urinary excretion of microglobulin, calciuria, phosphaturia, and uricosuria in thalassemic patients. In their study, creatinine clearance, uric acid and potassium excretion, and tubular reabsorption of phosphorus in patients with TM were significantly different from the control group, but no other tubule injury parameters were different between the two groups [9].
In our study, eGFR was not significantly different between the two groups. In previous studies, GFR did not change significantly in any type of thalassemia (major, intermediate, and minor) [1, 24, 25]. In the study of Nickavar et al., eGFR was insignificantly increased in thalassemic patients, which was probably secondary to anemia and decrease in systemic vascular resistance in addition to elevated renal plasma flow [1]. Both tubular dysfunction and renal glomerular disorder may occur in patients with thalassemia. These disorders are probably due to decreased production of adenosine triphosphatase, oxidative stress, lipid peroxidation, prostaglandin secretion imbalance, hyperdynamic cardiovascular system, increased renal blood flow and glomerular hyperfiltration [13, 25, 26]. Proximal tubular disorder has been reported in 13–60% of patients with all types of thalassemia. Increased urinary excretion of sodium, calcium, phosphorus, magnesium, uric acid, N-acetyl glucosamine, beta 2-microglobulin, retinol binding protein, and glucose in association with decreased urinary osmolality are symptoms of renal tubular dysfunction in thalassemic patients [9, 12]. Increased protein excretion is one of the most common clinical manifestations of renal involvement, which is seen in about 70% of patients with thalassemia with renal impairment [12, 27].
It has been suggested that renal tubular disorder may be secondary to anemia [14]. Changing the cell function due to reducing the oxygen supply to renal tubular cells can be a major cause of this phenomenon. Empirical evidence has shown that anemia, can lead to renal hypoxia [28]. The proximal tubular disorder has been reported in patients with iron deficiency anemia in Özçay et al.'s study [29]. Therefore, tubular iron loading or red blood cell hemolysis toxins are a potential cause of impaired proximal tubular function. In addition, increased iron exchange due to mild hemolysis of microcytic erythrocytes with a significant increase in LDH level may be another factor associated with proximal tubular injury in patients with TM [14]. Therefore, tubular iron or toxins loading due to red blood cell hemolysis is a potential cause of impaired proximal tubular function.
Studies on the function of all patients with TM are very rare. Renal tubular dysfunction in adults with TM can be due to hemolysis, reduced lifespan of erythrocytes, tubular Iron deposit, oxidative lipid peroxidation, and toxins produced from erythrocytes [14, 30]. Tubular dysfunction has also been reported in patients with iron deficiency anemia [29]. But we believe the mechanisms described for tubular injury in TM are not logically acceptable. TM is not usually associated with iron overload. They do not have chronic hemolysis or severe anemia that explains tissue hypoxemia.
Since the renal disease is common and is progressing to the advanced stages silently, screening for renal disease is essential and in case of lack of screening and intervention, the patients suffer from a range of symptoms from non-clinical signs of renal damage to death. On the other hand, creatinine is a non-susceptible marker. So, in this study, NGAL biomarker was used to investigate the probability of tubular injury in patients with TM. In our study, although a difference was found between NGAL in healthy and TM patients, but it did not reach statistical significance. However, elevated plasma NGAL (> 179 ng/mL) was observed in patients with TM. Also, the mean plasma NGAL was significantly higher in TM patients with at least one renal parameters. NGAL is a 21 kilodalton protein of the lipocalin family and is a biomarker for acute renal damage, but is also a new diagnostic tool for the chronic renal disease that reflects the continuous damage of tubular cells in the kidney [31]. This protein is normally released in small quantities by different cells outside the kidney. Therefore, the origin of NGAL production in patients with chronic renal disease is still controversial, and the effect of its tubular origin in comparison with the external production level has not yet been determined [32]. NGAL has been shown to be associated with morphological changes and albuminuria in patients with primary renal disease [33].
There are some limitations to this study. The main limitation of the study was the lack of performing some tests of renal dysfunction, such as urine electrophoresis and N-acetyl-beta-D-glucosaminidase (GcnA), a marker of renal tubular dysfunction. Also, we did not adapt urine materials to diet because the excretion fraction of urinary salt can change based on diet. A cohort study in thalassemic patients and evaluating the changes in renal tubular and glomerular activity over a period of time is recommended.
According to the results of this study, renal tubular dysfunction is prevalent in patients with TM. Also, we found that increased plasma NGAL level can be considered as the beginning of renal tubular injury in patients with TM. Laboratory measurements of the renal function in these patients at certain time points can prevent further complications. Regarding the prevalence of renal tubular dysfunction in patients with TM and unawareness of the patients about it, a periodic evaluation of renal function can prevent progression of renal dysfunction by identifying them early.
The trial data used to support the findings of this study are available from the corresponding author upon request.
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We thank all patients involved in the β-thalassemia minor renal tubular dysfunction management and treatment.
This study was supported by the Vice-chancellor for research of Babol University of Medical Sciences.
Social Determinants of Health Research Center, Health Research Institute, Babol University of Medical Sciences, Babol, Iran
Mohsen Vakili Sadeghi & Roghayeh Akbari
Students Research Committee, Babol University of Medical Sciences, Babol, IR, Iran
Maryam Mirghorbani
Department of Internal Medicine, Ayatollah Rouhani Hospital, Keshavarz Boulevard, Babol, Mazandaran, Iran
Roghayeh Akbari
Mohsen Vakili Sadeghi
MVS conceived and designed the study protocol. All authors participated in collecting data. RA wrote the manuscript draft and all authors approved the submitted version of the manuscript.
Correspondence to Roghayeh Akbari.
Written informed consent was obtained from the patients prior to study. The Ethics Committee of Babol University of Medical Sciences approved the study protocol. All methods were carried out in accordance with relevant guidelines and regulations.
Written informed consent was obtained from the patients for publication.
The authors declare no conflicts of interest
Sadeghi, M.V., Mirghorbani, M. & Akbari, R. β-Thalassemia minor & renal tubular dysfunction: is there any association?. BMC Nephrol 22, 404 (2021). https://doi.org/10.1186/s12882-021-02602-9
β-Thalassemia minor
Renal tubular dysfunction
Fraction excretion
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CommonCrawl
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\begin{document}
\title{Rational Heap Games} \author{Urban Larsson,\\ Mathematical Sciences, \\
Chalmers University of Technology and University of Gothenburg} \maketitle \begin{abstract} We study variations of classical combinatorial games on two finite heaps of tokens, a.k.a. \emph{subtraction games}. Given non-negative integers $p_1,q_1, p_2,q_2$, where $p_1q_2 > q_1p_2$, $p_1>0$ and $q_2>0$, two players alternate in removing $(m_1,m_2)\ne (0,0)$ tokens from the respective heaps, where the allowed ordered pairs of non-negative integers are given by a certain move set $(m_1,m_2)\in\mathcal{M}$. There is a restriction imposed on the allowed heap sizes $(X, Y)$, they must satisfy $Xq_1\le Yp_1$ and $Yp_2\le Xq_2$. A player who cannot move loses and the other player wins. For a certain restriction of these games, namely where each allowed move option $(m_1,m_2)$ is of the form $(sp_1+tp_2,sq_1+tq_2)$, for some ordered pair of non-negative integers $(s,t)\ne (0,0)$, we show that all games have equivalent outcomes via a certain surjective map to a canonical subtraction game. Other interests in our games are various interactions with classical combinatorial games such as \emph{Nim {\rm and} Wythoff Nim}. \end{abstract}
\section{Introduction}\label{S:1} We study \emph{impartial} \emph{subtraction} games on two \emph{heaps of tokens} \cite{G1966, DR2010, L1, L2, LHF2011, L2012, LW}. For a background, see also \cite{BCG1982}. There are two players who obey the same rules and alternate in moving. We follow the \emph{normal play} convention, meaning that a player who cannot move loses and the other player wins.
In general an impartial game consists of a \emph{position}, which contains information of whose turn it is and describes the given state of the game, and a \emph{ruleset}, which decides what move options there are for a given position. Sometimes the term ``game'' is adapted to mean only the ruleset, other times the term ``position'' encompasses the usual meaning of a game, that is whenever the ruleset is understood. Here we study so-called \emph{invariant} games \cite{G1966, DR2010, L1, L2, LHF2011, L2012, LW}, where the rules in general do not depend on the given position. For example, in \emph{2-heap Nim} \cite{B1902} the players remove tokens from precisely one of two finite heaps, at least one token and at most a whole heap. The game is invariant in the sense that, independent of the size of a heap, a given number of tokens can be removed provided the heap contains at least this number of tokens. As for all impartial games, this game has a \emph{perfect winning strategy}, here: the second player to move wins if and only if the two heaps have the same non-negative number of tokens.
In general, for an impartial game $G$ without drawn moves, the \emph{outcome}, of a given position, is either a \emph{previous player} win ($\mathcal{P}$) or a \emph{next player} win ($\mathcal{N}$). It belongs to $\mathcal{P}=\mathcal{P}(G)$, if none of its options is in $\mathcal{P}$. Otherwise it belongs to $\mathcal{N}$. This gives a recursive characterization of all $\mathcal{P}$-positions of a given game $G$, starting with the \emph{terminal} positions in $\mathcal{T}(G)\subseteq \mathcal{P}$, from which no move is possible.
\subsection{Rational heap games} Let \[ Q:=\left( \begin{array}{cc}
p_1 & q_1 \\
p_2 & q_2 \\ \end{array} \right),\] where $p_1>0, q_1, p_2, q_2>0$ are given non-negative integer \emph{game constants}, with $\det Q = p_1q_2 - q_1p_2 > 0$. We let the allowed \emph{positions} or \emph{heap-sizes} be represented by ordered pairs of non-negative integers, bounded by `rational slopes' as given by the set \begin{align}\label{2} \mathcal{B}_Q:=\left\{(X,Y)\mid Xq_1\le Yp_1\text{ and }Yp_2\le Xq_2\right\}. \end{align} In particular, for the special cases $q_1=p_2=0$ all combinations of heap-sizes are allowed; we omit the index and denoted this set simply by \begin{align}\label{B} \mathcal{B}:=\left\{(X,Y)\mid 0\le X, 0\le Y\right\}. \end{align} Also, we let $\mathcal{B}':=\mathcal{B}\setminus \{(0,0)\}$. Following \cite{DR2010,LHF2011,L2012}, a two heap \emph{subtraction game}, $G=G(\mathcal{M})$, is defined via a given set of ordered pairs of non-negative integers $\mathcal{M} \subseteq \mathcal{B}'$. A legal move from $(X,Y)\in \mathcal{B}$ is to $(X-m_1, Y-m_2)$ for some $(m_1, m_2)\in \mathcal{M}$ and provided $(X-m_1,Y-m_2)\in \mathcal{B}$.
Let $(X,Y)\in \mathcal{B}_Q$. In general, each allowed move for a \emph{$\mathcal{B}_{\cal{Q}}$-subtraction game} is of the form \begin{align}\label{3} (X,Y)\rightarrow (X-m_1, Y-m_2), \end{align} where $(m_1,m_2)$ belongs to a given set of ordered pairs of non-negative integers $\mathcal{M}\subseteq\mathcal{B}'$, and provided $(X-m_1,Y-m_2)\in \mathcal{B}_{\cal{Q}}$. Our main interest in this paper is the following subset of the $\mathcal{B}_{\cal{Q}}$-subtraction games. For a purpose that will become clear later (in Lemma \ref{L:5}) we will alter the notation somewhat.
In a \emph{$\cal{Q}$-subtraction game} $G_Q=G_Q(\mathcal{M})$ we require that $(m_1,m_2)$ in (\ref{3}) is of the form $m_1 = p_1s + p_2t$ and $m_2 = q_1s + q_2t$ where $(s,t)\in \mathcal{M}\subseteq \mathcal{B}'$. Hence for these games it is $(s,t)$ which belongs to ``the set of allowed moves'' $\mathcal{M}$ (and not $(m_1,m_2)$). Thus, a typical move in $G_Q(\mathcal{M})$ is \begin{align}\label{4} (X,Y)\rightarrow (X-p_1s-p_2t, Y-q_1s-q_2t), \end{align} where $(s,t)\in \mathcal{M}$ and $(X-p_1s-p_2t, Y-q_1s-q_2t)\in \mathcal{B}_Q$. As before, whenever a $\cal{Q}$-subtraction game is a subtraction game we write simply $G(\mathcal{M})$, that is whenever \[ Q=\left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\ \end{array} \right)\] and the allowed heap sizes are as given by $\mathcal{B}$ in (\ref{B}).
Let us define a surjective map $\varphi_Q$, which takes as input a position in $\mathcal{B}_Q$ and produces as output a position in $\mathcal{B}$, \begin{align}\label{varphi} \varphi_Q(X,Y) = \left(\left\lfloor \frac{Xq_2-Yp_2}{\det Q}\right\rfloor,\left\lfloor\frac{Yp_1-Xq_1}{\det Q}\right\rfloor\right), \end{align} by $q_1X/p_1\le Y\le q_2X/p_2$. Our main result is that any $\cal{Q}$-subtraction game $G_Q:=G_Q(\mathcal{M})$ is ``$\varphi_Q$-equivalent'' to the subtraction game $G:=G(\mathcal{M})$ in the following sense.
\begin{Thm}\label{T:1} Given game constants $p_i, q_i$, suppose that $(X,Y)\in \mathcal{B}_Q$. Then $(X,Y)\in \mathcal{P}(G_Q)$ if and only if $\varphi_Q(X,Y)\in \mathcal{P}(G)$. This is equivalent to $(A,B)\in \mathcal{P}(G)$ if and only if, for all $(x,y)\in \mathcal{T}_Q$, $(x+Ap_1+Bp_2, y+Aq_1+Bq_2)\in \mathcal{P}(G_Q)$. \end{Thm}
By this result we make the following definition: the $\cal{Q}$-subtraction games $G_{Q}(\mathcal{M})$ and $G_{R}(\mathcal{L})$ are \emph{$\varphi$-equivalent} if $\mathcal{M}=\mathcal{L}$. The \emph{$\mathcal{M}$-canonical game} is the subtraction game $G(\mathcal{M})$.
In Section \ref{S:2} we prove Theorem \ref{T:1}. A reader who wishes to study some examples before plunging into the proof of the main theorem should skip to Section \ref{S:3}, where we illustrate Theorem \ref{T:1} via generalizations of Nim and Wythoff Nim. This discussion is continued in Section \ref{S:4} with some open questions where we relate a certain ``splitting behavior'' of Wythoff type $Q$-subtraction games to similar $\mathcal{B}_{\cal{Q}}$-subtraction games.
\section{Proof of Theorem \ref{T:1}}\label{S:2} A generic game has several terminal positions from which no move is possible.
\begin{Lemma}\label{L:1} Given $\mathcal{M}\subseteq \mathcal{B}'$, the set \begin{align*} \mathcal{T}_Q := \left\{(x,y)\mid p_1(y-q_2)<q_1(x-p_2) \text{ and } p_2(y-q_1)>q_2(x-p_1)\right\}\subseteq \mathcal{B}_Q \end{align*}
is a subset of all \emph{terminal positions} of the $\cal{Q}$-subtraction game $G_Q(\mathcal{M})$. In particular $\mathcal{T}_Q = \mathcal{T}(G_Q)$ is the set of all terminal positions, if and only if $\{(0,1),(1,0)\}\subseteq \mathcal{M}$. Also $\det Q=|\mathcal{T}_Q|$. \end{Lemma} \noindent{\bf Proof.} Since, by assumption $(x,y)\in \mathcal{T}_Q\subset \mathcal{B}_Q$, we need to show that $(x-sp_1-tp_2, y-sq_1-tq_2)\not\in \mathcal{B}_Q$ for all positive integers $s$ and $t$. Hence, by definition of $\mathcal{B}_Q$, we need to show that $(x-sp_1-tp_2)q_1> (y-sq_1-tq_2)p_1$ and that $(x-sp_1-tp_2)q_2 < (y-sq_1-tq_2)p_2$ for all positive integers $s$ and $t$, but this is clear since, by definition of $\mathcal{T}_Q$, it holds for $s=1$ and $t=1$, and the game constants satisfy $p_1q_2 > q_1p_2$ .
The set $\mathcal{T}_Q$ is the complete set of terminal positions of $G_Q(\mathcal{M})$ if $$\{(0,1),(1,0)\}~\subseteq~\mathcal{M}$$ since $(x,y)\not\in \mathcal{T}_Q$ gives $p_1(y-q_2)\ge q_1(x-p_2)$ or $p_2(y-q_1)\le q_2(x-p_1)$. Hence $(x-p_2, y-q_2)\in \mathcal{B}_Q$ or $(x-p_1, y-q_1)\in \mathcal{B}_Q$ respectively. For ``only if'', suppose that $(0,1)\not\in \mathcal{M}$, then $(x+p_2,y+q_2)\in \mathcal{T}(G_Q)\setminus \mathcal{T}_Q$. The other case is similar.
$\Box$\\
The next two lemmas discuss how the positions in $\mathcal{B}_Q$ can be viewed as linear translations of those in $\mathcal{T}_Q$.
\begin{Lemma}\label{L:2} Let $(x,y)\in \mathcal{T}_Q$. Then $(x+Ap_1+Bp_2,y+Aq_1+Bq_2)\in \mathcal{B}_Q$ if and only if $(A,B)\in \mathcal{B}$. \end{Lemma}
\noindent{\bf Proof.} What is required is to show that $$(x+Ap_1+Bp_2)q_1\le (y+Aq_1+Bq_2)p_1$$ and \begin{align}\label{second} (x+Ap_1+Bp_2)q_2\ge (y+Aq_1+Bq_2)p_2 \end{align} if and only if both $A$ and $B$ are non-negative, assuming that $(x,y)\in \mathcal{T}_Q\subset \mathcal{B}_Q$. But, as in the proof of Lemma \ref{L:1}, the ``if''-part is immediate by $p_1q_2 > q_1p_2$.
Hence, suppose that $A<0$. By definition there is no move from any $\cal{Q}$-subtraction game from $(x,y)\in \mathcal{T}_Q$. Hence, by negativity of $A$, we must have $(x+Ap_1)q_2 < (y+Aq_1)p_2$, which contradicts (\ref{second}). The case $B<0$ is similar.
$\Box$\\
\begin{Lemma}\label{L:3} If $(X,Y)\in \mathcal{B}_Q$, then $(X,Y) = (x+Ap_1+Bp_2, y+Aq_1+Bq_2)$, for some unique $(x, y)\in \mathcal{T}_Q$ and some unique non-negative integers $A$ and $B$. \end{Lemma}
\noindent{\bf Proof.} Suppose that $(X,Y) = (x+Ap_1+Bp_2, y+Aq_1+Bq_2)=(x'+A'p_1+B'p_2, y'+A'q_1+B'q_2)$, with $(x',y')\in \mathcal{T}_Q$, $x\ge x', y\ge y'$ and non-negative integers $A'$ and $B'$. Suppose that $x>x'$. This gives $x-x'=(A'-A)p_1+(B'-B)p_2>0$ which implies $y-y' = (A'-A)q_1+(B'-B)q_2 > 0$ which is impossible by Lemma \ref{L:1}. Hence $(A'-A)p_1=(B-B')p_2$ and $(A'-A)q_1=(B-B')q_2$ which gives $q_1p_2=p_1q_2$ which is impossible.
Next, let us find $x,y,A,B$ such that $(X, Y) = (x+Ap_1+Bp_2, y+Aq_1+Bq_2)$. We have that $Xq_1 \le Yp_1$ and $Yp_2 \le Xq_2$. Hence, by $q_1p_2 < p_1q_2$, there is a largest non-negative $B$ such that $(X - Bp_2)q_1 \le (Y - Bq_2)p_1$ and a largest non-negative $A$ such that $(Y - Aq_1)p_2 \le (X - Ap_1)q_2$. Then $(X-Ap_1-Bp_2,Y-Aq_1-Bq_2)\in \mathcal{T}_Q$ defines $(x,y)\in \mathcal{T}_Q$ by our choices of $A$ and $B$.
$\Box$\\
\begin{Rem}\label{R:1}
Following Lemma \ref{L:3}, we say that (the game) $(X,Y)=(x+Ap_1+Bp_2, y+Aq_1+Bq_2)$ belongs to the \emph{$(x,y)$-class}, where $(x,y)\in \mathcal{T}_Q$. Then, by $(X,Y) = (A,B)Q$, $(X,Y)$ belongs to the $(0,0)$-class if and only if the associated restriction of $\varphi_Q$ is $\varphi_Q|_{(0,0)}(X,Y)=(X,Y)Q^{-1} = (A,B)$. \end{Rem}
We get the following consequence of the above lemmas for our $Q$-subtraction games.
\begin{Lemma}\label{L:5} Let $\mathcal{M}\subseteq \mathcal{B}'$. Suppose that there is a move of the form $(A,B)\rightarrow (A-s,B-t)$ in the subtraction game $G(\mathcal{M})$, then for any given game constants $p_i,q_i$ and for each $(x,y)\in \mathcal{T}_Q$, there is a move in the $Q$-subtraction game $G_Q(\mathcal{M})$ of the form \begin{align}\label{moveQ} &(x+Ap_1+Bp_2,y+Aq_1+Bq_2)\rightarrow \notag\\&(x+(A-s)p_1+(B-t)p_2,y+(A-s)q_1+(B-t)q_2). \end{align} Suppose on the other hand that there is a move in $G_Q$ from $(X,Y)\in \mathcal{B}_Q$ via $(s,t)\in \mathcal{M}$. Then there is a corresponding move $(A,B)\rightarrow (A-s,B-t)$ in $G$ where $(X,Y)=(x+Ap_1+Bp_2,y+Aq_1+Bq_2)$ for some $(x,y)\in \mathcal{T}_Q$. \end{Lemma}
Thus, with notation as in Lemma \ref{L:5} and Remark \ref{R:1}, each game in the $(x,y)$-class ends in $(x,y)$ if $\{(0,1),(1,0)\}\subseteq \mathcal{M}$.
Let us restate and prove our main theorem, where the function $\varphi_Q$ is as in (\ref{varphi}).
\begin{thm} Given game constants $p_i, q_i$, suppose that $(X,Y)\in \mathcal{B}_Q$. Then $(X,Y)\in \mathcal{P}(G_Q)$ if and only if $\varphi_Q(X,Y)\in \mathcal{P}(G)$. This is equivalent to $(A,B)\in \mathcal{P}(G)$ if and only if, for all $(x,y)\in \mathcal{T}_Q$, $(x+Ap_1+Bp_2, y+Aq_1+Bq_2)\in \mathcal{P}(G_Q)$. \end{thm}
\noindent{\bf Proof.} We begin with the second part. Suppose that $(A,B)\in \mathcal{P}(G)$. Then, none of its options is in $\mathcal{P}(G)$. We need to prove that none of the options in the $Q$-subtraction game $G_Q$ from $(x+Ap_1+Bp_2, y+Aq_1+Bq_2)$ is in $\mathcal{P}(G_Q)$. We have that, for all $(s,t)\in \mathcal{M}$ such that $A-s\ge 0$ and $B-t\ge 0$, $(A-s,B-t)\in \mathcal{N}(G)$. Then, by Lemma \ref{L:5}, induction gives that $(x+(A-s)p_1+(B-t)p_2, y+(A-s)q_1+(B-t)q_2)\in \mathcal{N}(G_Q)$ if and only if $A-s\ge 0$ and $B-t\ge 0$.
If, on the other hand $(A, B)\in \mathcal{N}(G)$, then there is an option $(A-s, B-t)\in \mathcal{P}(G)$, with $(s, t)\in \mathcal{M}$, and so induction gives that $(x+(A-s)p_1+(B-t)p_2, y+(A-s)q_1+(B-t)q_2)\in \mathcal{P}(G_Q)$.
For the first part, by Lemma \ref{L:3}, if $(X,Y)\in \mathcal{B}_Q$, then $(X,Y) = (x+Ap_1+Bp_2, y+Aq_1+Bq_2)$, for some unique $(x, y)\in \mathcal{T}_Q$ and some unique non-negative integers $A$ and $B$. We plug this into the expression
\begin{align*} \varphi_Q(X,Y) &= \left(\left\lfloor \frac{Xq_2-Yp_2}{\det Q}\right\rfloor,\left\lfloor\frac{Yp_1-Xq_1}{\det Q}\right\rfloor\right)\\ &=\left(\left\lfloor \frac{xq_2-yp_2+A\det Q}{\det Q}\right\rfloor,\left\lfloor\frac{yp_1-xq_1+B\det Q}{\det Q}\right\rfloor\right) &=(A, B), \end{align*} by $0\le \frac{yp_1-xq_1}{\det Q}<1$ and $0\le \frac{xq_2-yp_2}{\det Q}<1$, since $(x,y)\in \mathcal{T}_Q$. But then the first part of the proof gives the result.
$\Box$\\
\section{Examples}\label{S:3} Our first example of a 2-heap subtraction game, 2-heap Nim, was discussed briefly in the introduction.
The game of \emph{Wythoff Nim} \cite{W1907} is also played on two heaps, all the moves in Nim are allowed and also the possibility of removing the same positive number of tokens from both heaps in one and the same move, a number bounded by the number of tokens in the smallest heap.
See Figure \ref{F:Nim} for the $\mathcal{P}$-positions of Nim and Wythoff Nim (\text{W}) respectively. For the latter game, it is known \cite{W1907} that they are of the form:
\begin{align}\label{phi} \mathcal{P}(\text{W})=\{(X,Y),(Y,X)\mid X=\lfloor \phi n\rfloor , Y=\lfloor \phi^2 n \rfloor\}, \end{align} where $n$ ranges over the non-negative integers and $\phi := \frac{1+\sqrt{5}}{2}$ denotes the golden ratio. \begin{figure}
\caption{The red squares represent the initial patterns of $\mathcal{P}$-positions of 2-heap Nim, $(0,0),(1,1),\ldots$ to the left and Wythoff Nim, $(0,0),(1,2),(2,1),(3,5),(5,3),\ldots$ to the right.}
\label{F:Nim}
\end{figure}
\begin{Ex} For our first example of a $Q$-subtraction game, a player may move \begin{align}\label{5} (x,y)\rightarrow (x-p_it, y-q_it), \end{align} for $i\in \{1,2\}$ and any positive integer $t$, provided $(x-p_it,y-q_it)\in \mathcal{B}_Q$.
We denote this game by \emph{Rational Nim} (RN), $(\frac{q_1}{p_1},\frac{q_2}{p_2})$-RN. Here the ratios in the prefix are merely symbols and so for example $(\frac{1}{2},\frac{2}{3})$-RN and $(\frac{2}{4},\frac{4}{6})$-RN denote different games. Note that, given $Q$, this game may be represented simply by the set $\mathcal{M} = \{(0,t),(t,0)\mid t>0\}$ and that the case $p_1=q_2=1$, $p_2=q_1=0$ is the classical game of Nim on 2 heaps. See Figure \ref{F:1} for the $\mathcal{P}$-positions of an RN game. \end{Ex}
An \emph{extension} of a game $G$ has the same set of positions as $G$, contains all the moves in $G$ and possibly some new. Thus a trivial extension of 2-heap Nim is Nim itself. A non-trivial extension is for example Wythoff Nim. Further, a \emph{$Q$-extension} is an extension of a $Q$-subtraction game which is also a $Q$-subtraction game. Thus another way of expressing this is that $G_Q(\mathcal{M}'')=G_Q(\mathcal{M}\cup\mathcal{M}')$ is a $Q$-extension of both $G_Q(\mathcal{M})$ and $G_Q(\mathcal{M}')$. Note that $Q$ is fixed.
\begin{Ex} Given game constants $p_i, q_i$, we denote by \emph{Rational Wythoff Nim} (RW), \emph{$(\frac{q_1}{p_1},\frac{q_2}{p_2})$-\text{RW}}, the following $Q$-extension of $(\frac{q_1}{p_1},\frac{q_2}{p_2})$-RN. In addition to the moves in (\ref{2}), the new moves are of the form $$(x,y)\rightarrow (x-t(p_1+p_2),y-t(q_1+q_2))$$ provided that $(x-t(p_1+p_2),y-t(q_1+q_2))\in \mathcal{B}_Q$. Hence, given $Q$, this game may equivalently be represented by $\mathcal{M}=\{(0,t),(t,0),(t,t)\mid t>0\}$ and the classical game of Wythoff Nim is the game where $p_1=q_2=1$, $p_2=q_1=0$. See Figure \ref{F:2} for the initial $\mathcal{P}$-positions of an RW game. \end{Ex}
\begin{figure}
\caption{The red squares represent the initial $\mathcal{P}$-positions for the game $(\frac{2}{7},\frac{10}{1})$-RN. The two lines represent the bounds for the the ratio of the heap-sizes.}
\caption{The red squares represent all initial $\mathcal{P}$-positions for the game $(\frac{2}{7},\frac{10}{1})$-RW.}
\label{F:1}
\label{F:2}
\end{figure}
Theorem \ref{T:1} gives that $\mathcal{P}(\text{RN})$ is periodic with period $(p_1+p_2,q_1+q_2)$ and $\mathcal{P}(\text{RW})$ is aperiodic similar to $\mathcal{P}(\text{W})$, as described by the following result.
\begin{Cor}\label{C:1} The $\mathcal{P}$-positions of Rational Nim are given by $\mathcal{P}(RN)=\{(x+n(p_1+p_2),y+n(q_1+q_2)\}$, where $(x,y)\in \mathcal{T}_Q$ and where $n$ ranges over the non-negative integers.
The set $\mathcal{P}(\text{RW})$ is given by all positions of the forms \begin{align} (x+p_1\lfloor \phi^2 n\rfloor+p_2\lfloor \phi n\rfloor,y+q_1\lfloor \phi^2 n\rfloor+q_2\lfloor \phi n\rfloor) \end{align} and \begin{align} (x+p_1\lfloor \phi n\rfloor+p_2\lfloor \phi^2 n\rfloor,y+q_1\lfloor \phi n\rfloor+q_2\lfloor \phi^2 n\rfloor), \end{align} where $(x,y)\in \mathcal{T}_Q$ and $n$ is a non-negative integer. \end{Cor}
See also Corollary \ref{C:2} in the next section.
\section{When a game extension splits a set of $\mathcal{P}$-positions}\label{S:4} Is it true that adjoining new moves to an existing impartial game changes its set of $\mathcal{P}$-positions? For the particular games studied in Section \ref{S:3} this is certainly true by Corollary \ref{C:1}, as is also illustrated in Figures $\ref{F:1}$ and $\ref{F:2}$. In fact, we have seen a particular shift of behavior inherited from the relation between Nim and Wythoff Nim. In describing RN's and RW's asymptotic behavior we go from one accumulation point to two accumulation points. Following the terminology in \cite{L1}, we note that the particular $Q$-extension of RN we have introduced, RW, \emph{splits} the `old' set of $\mathcal{P}$-positions of RN into two `new' $\mathcal{P}$ sets for which the ratios of the heap-sizes \emph{converge} to two distinct real numbers. We have the following simple consequence of Corollary~\ref{C:1}.
\begin{Cor}\label{C:2} Let the game constants $p_i,q_i$ be given. Let $\mathcal{P}(\text{RN}) = \{(A_n,B_n)\}$, with the $(A_n,B_n)$s in lexicographical order. Then $$\frac{B_n}{A_n}\rightarrow \frac{q_1+q_2}{p_1+p_2},$$ as $n\rightarrow \infty$. Let $\mathcal{P}(\text{RW}) = \{(A_n^l, B_n^l),(A_n^u,B_n^u)\}$, where the \emph{lower} $\mathcal{P}$-positions are the $(A_n^l,B_n^l)$s in lexicographical order and the \emph{upper} $\mathcal{P}$-positions are the $(A_n^u,B_n^u)$s in lexicographical order for which $$\frac{B_n^l}{A_n^l}< \frac{q_1+q_2}{p_1+p_2}\le \frac{B_n^u}{A_n^u}.$$ Then $$\frac{B_n^l}{A_n^l}\rightarrow \frac{q_1+\phi(q_1+q_2)}{p_1+\phi(p_1+p_2)},$$ $$\frac{B_n^u}{A_n^u}\rightarrow \frac{q_2+\phi(q_1+q_2)}{p_2+\phi(p_1+p_2)},$$ as $n\rightarrow\infty$
\end{Cor}
For another example, take the set $\mathcal{P}((\frac{1}{1},\frac{1}{0})\text{-\text{RW}})$. The first few $\mathcal{P}$-positions are: $(0,0),(1,3),(2,3),(3,8),(5,8),(4,11),(7,11),\ldots .$ For all non-negative integers $n$, by Corollary \ref{C:1} and since the only terminal position is $(0,0)$, $$\mathcal{P}\left(\left(\frac{1}{1},\frac{1}{0}\right)\!-\!\text{RW}\right) = \left\{(\lfloor \phi n\rfloor, \lfloor \phi n\rfloor+\lfloor \phi^2 n \rfloor), (\lfloor \phi^2 n \rfloor, \lfloor \phi n\rfloor+\lfloor \phi^2 n \rfloor)\right\}$$ and thus, the two convergents of the $\mathcal{P}$-positions of this game are $\phi$ and $\phi^2$, whereas for $(\frac{1}{1},\frac{1}{0})$-RN the single convergent is $2$. See also Figure \ref{F3}. This game has a particular interest as being a new simple `restriction' of the game $(1,2)$GDWN in \cite{L1}, the latter which is an extension of Wythoff Nim where the new moves are of the form: in one and the same move remove $t$ tokens from one of the heaps and $2t$ from the other, at most a whole heap. The game $(1,2)$GDWN is conjectured to split the upper (and lower) $\mathcal{P}$-positions of Wythoff Nim from the single convergent $\phi$ to a pair of convergents $1.478\ldots$ and $2.247\ldots$. See also \cite{L2011} for another example where a further split of the $\mathcal{P}$-positions of Wythoff Nim is obtained via a certain \emph{blocking maneuver} on the regular Wythoff Nim moves.
\begin{figure}
\caption{The red squares represent the initial $\mathcal{P}$-positions for the games $(\frac{1}{1},\frac{1}{0})$-RN and $(\frac{1}{1},\frac{1}{0})$-RW respectively.}
\label{F3}
\end{figure}
Let us finish off this paper with some extensions of our sample game in Figure \ref{F:1}, that (unlike the game in Figure \ref{F:2}) are not $Q$-extensions, but indeed $\mathcal{B}_Q$-subtraction games. Our final three extensions of $(\frac{2}{7},\frac{10}{1})$-RN are as follows: adjoin moves of the form \\
\noindent game (a): $(X,Y)\rightarrow (X-4t,Y-6t)$,\\
\noindent game (b): $(X,Y)\rightarrow (X-4t,Y-4t)$,\\
\noindent game (c): $(X,Y)\rightarrow (X-8t,Y-4t)$,\\
for positive integers $t$, bearing in mind that the positions satisfy (\ref{2}). The first two extensions have complicated patterns of $\mathcal{P}$-positions which we do not yet understand, of which at least the first appears to exhibit a similar splitting `behavior' as does $(\frac{2}{7},\frac{10}{1})$-RW, but game (c) has the same set of $\mathcal{P}$-positions as does $(\frac{2}{7},\frac{10}{1})$-RN, which can be proved by elementary methods, see also \cite{FL1991, DFNR2010, L1, L2011} for related results. Hence the answer to the first question in this final section is negative; see also \cite{LW} for a discussion of this question in the context of heap games, computational complexity and algorithmic undecidability.
\section{Discussion}
The purpose of this paper has been to introduce the subject of $\mathcal{B}_Q$-subtraction games and resolve the most basic question, that is of $\varphi_Q$-equivalence of $Q$-subtraction games. As we indicate in Figures \ref{F:4} and \ref{F:5}, the most interesting ``new'' games are the generic $\mathcal{B}_Q$-subtraction games. (For example, could a study of extensions of $(\frac{1}{1},\frac{1}{0})\text{-\text{RW}}$ lead to new revelations of the behavior of $(1,2)$GDWN?) Of course one would like to extend the notion of $\varphi_Q$-equivalence to games on several heaps, which can be done essentially by applying the same ideas as in Section \ref{S:1} and \ref{S:2}. We do not know of any literature on $\mathcal{B}_Q$-subtraction games even on two heaps.\\
\noindent{\bf Acknowledgment.} I thank Ragnar Freij for an interesting discussion regarding generalizations to several heaps.\\\\
\begin{figure}
\caption{The red squares represent the initial $\mathcal{P}$-positions for extension (a) of $(\frac{2}{7},\frac{10}{1})$-RN.}
\label{F:4}
\end{figure}
\begin{figure}
\caption{The red squares represent the initial $\mathcal{P}$-positions for extension (b) of $(\frac{2}{7},\frac{10}{1})$-RN.}
\label{F:5}
\end{figure}
\begin{figure}
\caption{The red squares represent the initial $\mathcal{P}$-positions for extension (c) of $(\frac{2}{7},\frac{10}{1})$-RN.}
\label{F:6}
\end{figure}
\end{document}
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# Understanding the basic concepts of calculus
Calculus is a branch of mathematics that deals with change and motion. It provides us with tools to understand and analyze how things change over time or in relation to other variables. At its core, calculus is about studying rates of change and the accumulation of quantities.
There are two main branches of calculus: differential calculus and integral calculus. Differential calculus focuses on studying rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and finding areas under curves.
In order to understand calculus, it is important to have a solid foundation in algebra and trigonometry. These topics provide the necessary tools and techniques that are used throughout calculus.
One of the fundamental concepts in calculus is the notion of a function. A function is a rule that assigns each input value to a unique output value. It can be thought of as a machine that takes in an input and produces an output.
Functions can be represented in various ways, such as algebraic expressions, graphs, or tables. They can also be classified into different types, such as linear, quadratic, exponential, or trigonometric functions.
The key idea in calculus is to study how functions change. This is done by examining the rate of change of a function at a particular point, which is known as the derivative. The derivative measures how the function is changing with respect to its input.
The derivative of a function can be interpreted geometrically as the slope of the tangent line to the graph of the function at a given point. It represents the instantaneous rate of change of the function at that point.
Another important concept in calculus is the integral. The integral of a function represents the accumulation of the function over a certain interval. It can be thought of as finding the area under the curve of the function.
The integral is the reverse process of differentiation. It allows us to find the original function when we know its rate of change. The fundamental theorem of calculus connects the concepts of differentiation and integration.
In this textbook, we will focus on exploring the power of integration using the fundamental theorem of calculus. This theorem provides a powerful tool for evaluating integrals and solving a wide range of problems.
We will start by understanding the basic concepts of calculus and the definition of integration. Then, we will dive into the fundamental theorem of calculus and its two parts. We will learn how to apply these concepts to solve problems and evaluate definite integrals.
Next, we will explore the practical applications of integration in real-life scenarios. We will see how integration can be used to solve problems in physics, economics, and other fields.
We will also cover more advanced topics, such as integrating equations with multiple variables and the relationship between differentiation and integration.
By the end of this textbook, you will have a deep understanding of the power of integration and how it can be applied to solve a wide range of problems. Let's dive in and explore the fascinating world of calculus!
# The definition of integration
Integration is a fundamental concept in calculus that deals with finding the area under a curve. It is the reverse process of differentiation and allows us to find the original function when we know its rate of change.
The integral of a function represents the accumulation of the function over a certain interval. It can be thought of as summing up infinitely many infinitesimally small rectangles under the curve.
There are two types of integrals: indefinite integrals and definite integrals. An indefinite integral, also known as an antiderivative, gives us a family of functions that have the same derivative. A definite integral, on the other hand, gives us a specific value that represents the area under the curve between two points.
The symbol used to represent integration is the integral sign (∫), and the function being integrated is written after the integral sign. The limits of integration, which specify the interval over which the integration is performed, are written as subscripts and superscripts.
The general form of an indefinite integral is:
$$\int f(x) dx$$
And the general form of a definite integral is:
$$\int_{a}^{b} f(x) dx$$
Where $f(x)$ is the function being integrated, $dx$ represents an infinitesimally small change in $x$, and $a$ and $b$ are the lower and upper limits of integration, respectively.
Integration can be performed using various techniques, such as the power rule, substitution, integration by parts, and partial fractions. These techniques allow us to find the antiderivative of a function and evaluate definite integrals.
# The fundamental theorem of calculus
The fundamental theorem of calculus is a cornerstone of calculus that establishes the relationship between differentiation and integration. It consists of two parts: the first part relates the derivative of an antiderivative to the original function, while the second part allows us to evaluate definite integrals using antiderivatives.
The first part of the fundamental theorem of calculus states that if $f(x)$ is a continuous function on an interval $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$, then the derivative of $F(x)$ is equal to $f(x)$:
$$\frac{d}{dx} \left( \int_{a}^{x} f(t) dt \right) = f(x)$$
This means that if we integrate a function $f(x)$ and then differentiate the resulting antiderivative, we will obtain the original function $f(x)$.
The second part of the fundamental theorem of calculus allows us to evaluate definite integrals. It states that if $f(x)$ is a continuous function on an interval $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$, then the definite integral of $f(x)$ from $a$ to $b$ can be evaluated as the difference between the antiderivative evaluated at the upper limit $b$ and the lower limit $a$:
$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$
This means that instead of directly calculating the area under the curve using geometric methods, we can find an antiderivative of the function and evaluate it at the limits of integration.
# The first part of the fundamental theorem of calculus
The first part of the fundamental theorem of calculus states that if $f(x)$ is a continuous function on an interval $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$, then the derivative of $F(x)$ is equal to $f(x)$:
$$\frac{d}{dx} \left( \int_{a}^{x} f(t) dt \right) = f(x)$$
In other words, if we integrate a function $f(x)$ and then differentiate the resulting antiderivative, we will obtain the original function $f(x)$.
This part of the fundamental theorem of calculus is a powerful result that connects the concepts of integration and differentiation. It allows us to find antiderivatives of functions by reversing the process of differentiation. By finding an antiderivative, we can determine the original function up to a constant.
To understand the first part of the fundamental theorem of calculus, let's consider an example. Suppose we have the function $f(x) = 2x$. We want to find an antiderivative of $f(x)$.
To do this, we can integrate $f(x)$ with respect to $x$:
$$\int f(x) dx = \int 2x dx = x^2 + C$$
where $C$ is the constant of integration. The antiderivative of $f(x)$ is $F(x) = x^2 + C$.
Now, if we differentiate $F(x)$, we should obtain $f(x)$:
$$\frac{d}{dx} (x^2 + C) = 2x$$
As expected, the derivative of $F(x)$ is equal to $f(x)$.
The first part of the fundamental theorem of calculus allows us to find antiderivatives and evaluate definite integrals by connecting the concepts of integration and differentiation. It provides a powerful tool for solving problems in calculus and has numerous applications in various fields.
Suppose we have the function $f(x) = 3x^2$. We want to find an antiderivative of $f(x)$.
To find the antiderivative, we can integrate $f(x)$ with respect to $x$:
$$\int f(x) dx = \int 3x^2 dx = x^3 + C$$
where $C$ is the constant of integration. The antiderivative of $f(x)$ is $F(x) = x^3 + C$.
Now, if we differentiate $F(x)$, we should obtain $f(x)$:
$$\frac{d}{dx} (x^3 + C) = 3x^2$$
As expected, the derivative of $F(x)$ is equal to $f(x)$.
The first part of the fundamental theorem of calculus allows us to find antiderivatives and evaluate definite integrals by connecting the concepts of integration and differentiation. It provides a powerful tool for solving problems in calculus and has numerous applications in various fields.
## Exercise
Find an antiderivative of the function $f(x) = 4x^3$.
### Solution
To find the antiderivative, we can integrate $f(x)$ with respect to $x$:
$$\int f(x) dx = \int 4x^3 dx = x^4 + C$$
where $C$ is the constant of integration. The antiderivative of $f(x)$ is $F(x) = x^4 + C$.
# Applying the first part of the fundamental theorem of calculus to solve problems
The first part of the fundamental theorem of calculus allows us to solve a variety of problems by finding antiderivatives of functions. By reversing the process of differentiation, we can find the original function up to a constant.
One common application of the first part of the fundamental theorem of calculus is finding the area under a curve. Suppose we have a function $f(x)$ and we want to find the area between the curve and the x-axis from $a$ to $b$.
To find this area, we can integrate $f(x)$ with respect to $x$ over the interval $[a, b]$:
$$\int_{a}^{b} f(x) dx$$
The result of this integration will give us the area between the curve and the x-axis from $a$ to $b$.
For example, let's say we have the function $f(x) = x^2$ and we want to find the area under the curve from $x = 0$ to $x = 2$. We can integrate $f(x)$ over this interval:
$$\int_{0}^{2} x^2 dx$$
To evaluate this integral, we can use the power rule for integration:
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
Applying this rule to our integral, we have:
$$\int_{0}^{2} x^2 dx = \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$$
So the area under the curve $f(x) = x^2$ from $x = 0$ to $x = 2$ is $\frac{8}{3}$.
The first part of the fundamental theorem of calculus allows us to solve a variety of problems by finding antiderivatives of functions. It provides a powerful tool for finding areas, evaluating definite integrals, and solving other problems in calculus.
# The second part of the fundamental theorem of calculus
The second part of the fundamental theorem of calculus establishes a connection between differentiation and integration. It states that if we have a function $f(x)$ that is continuous on the interval $[a, b]$ and we define a new function $F(x)$ as the integral of $f(x)$ from $a$ to $x$, then $F(x)$ is differentiable on the interval $(a, b)$ and its derivative is equal to $f(x)$.
In other words, if $F(x) = \int_{a}^{x} f(t) dt$, then $F'(x) = f(x)$.
This means that we can find the derivative of a function by first finding an antiderivative and then differentiating. It provides a way to reverse the process of integration and recover the original function.
For example, let's say we have the function $f(x) = 2x$ and we want to find its antiderivative. We can use the second part of the fundamental theorem of calculus to do this.
We define a new function $F(x)$ as the integral of $f(x)$ from a fixed point $a$ to $x$:
$$F(x) = \int_{a}^{x} 2t dt$$
To find the antiderivative of $f(x)$, we need to find a function whose derivative is equal to $f(x)$. In this case, we can evaluate the integral:
$$F(x) = \left[ t^2 \right]_{a}^{x} = x^2 - a^2$$
So the antiderivative of $f(x) = 2x$ is $F(x) = x^2 - a^2$.
The second part of the fundamental theorem of calculus allows us to find antiderivatives and recover the original function from its derivative. It provides a powerful tool for solving problems in calculus and is a fundamental concept in the study of integration.
# Using the second part of the fundamental theorem of calculus to evaluate definite integrals
The second part of the fundamental theorem of calculus also allows us to evaluate definite integrals. A definite integral is an integral with specified limits of integration, such as $\int_{a}^{b} f(x) dx$.
To evaluate a definite integral using the second part of the fundamental theorem of calculus, we first find an antiderivative of the integrand function. Let's say we have a function $f(x)$ and we want to evaluate the definite integral $\int_{a}^{b} f(x) dx$.
We define a new function $F(x)$ as the integral of $f(x)$ from a fixed point $a$ to $x$:
$$F(x) = \int_{a}^{x} f(t) dt$$
According to the second part of the fundamental theorem of calculus, $F'(x) = f(x)$. This means that $F(x)$ is an antiderivative of $f(x)$.
Using this information, we can evaluate the definite integral by subtracting the values of the antiderivative at the upper and lower limits of integration:
$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$
For example, let's say we want to evaluate the definite integral $\int_{0}^{1} 2x dx$. We can first find the antiderivative of $f(x) = 2x$:
$$F(x) = \int_{0}^{x} 2t dt = \left[ t^2 \right]_{0}^{x} = x^2$$
Then, we evaluate the definite integral:
$$\int_{0}^{1} 2x dx = F(1) - F(0) = 1^2 - 0^2 = 1$$
So, $\int_{0}^{1} 2x dx = 1$.
Using the second part of the fundamental theorem of calculus, we can evaluate definite integrals by finding the antiderivative of the integrand function and subtracting the values of the antiderivative at the upper and lower limits of integration. This provides a powerful method for calculating the area under a curve and solving various problems in calculus.
# Applying integration to real-life scenarios
One common application of integration is in physics, particularly in the calculation of areas and volumes. For example, if you want to find the area of an irregular shape or the volume of a three-dimensional object, you can use integration to break down the shape into infinitesimally small pieces and sum up their areas or volumes.
Another application of integration is in economics, specifically in the calculation of total revenue and total cost. By integrating the revenue and cost functions, you can determine the total profit or loss for a given level of production.
Integration is also used in biology to model population growth and decay. By integrating the rate of change of a population over time, you can determine the total number of individuals in the population at a given time.
In computer science, integration is used in image processing and computer vision to calculate pixel values and perform operations on images. By integrating the intensity values of pixels over a region, you can determine the average intensity or perform tasks such as edge detection.
These are just a few examples of how integration can be applied in real-life scenarios. By understanding the concepts and techniques of integration, you can solve a wide range of problems in various fields and gain a deeper understanding of the world around you.
Let's consider an example to illustrate the application of integration in physics. Suppose you have a curved shape, such as a river bend, and you want to find its area. You can approximate the shape as a series of rectangles and calculate the area of each rectangle. By summing up the areas of all the rectangles, you can get an estimate of the total area of the shape. As you make the rectangles narrower and increase their number, the estimate becomes more accurate. This process is equivalent to integrating the function that describes the shape to find its exact area.
## Exercise
Consider a scenario where a company's revenue function is given by $R(x) = 3x^2 + 5x + 10$, where $x$ represents the number of units sold and $R(x)$ represents the total revenue. Using integration, calculate the total revenue generated when 100 units are sold.
### Solution
To calculate the total revenue, we need to integrate the revenue function $R(x)$ from 0 to 100:
$$\int_{0}^{100} (3x^2 + 5x + 10) dx$$
Integrating term by term, we get:
$$\left[ x^3 + \frac{5}{2}x^2 + 10x \right]_{0}^{100}$$
Evaluating the integral at the upper and lower limits, we get:
$$(100^3 + \frac{5}{2}(100^2) + 10(100)) - (0^3 + \frac{5}{2}(0^2) + 10(0))$$
Simplifying, we find that the total revenue generated when 100 units are sold is:
$$1000000 + 5000 + 1000 = 1006000$$
Therefore, the total revenue generated is $1006000.
# Integrating equations with multiple variables
When integrating equations with multiple variables, the process is similar to integrating single-variable equations. The main difference is that instead of integrating with respect to a single variable, we integrate with respect to one variable while treating the other variables as constants.
To illustrate this, let's consider an example. Suppose we have a function $f(x, y) = x^2 + y^2$ and we want to integrate it with respect to $x$. We can treat $y$ as a constant and integrate $x^2 + y^2$ as if it were a single-variable equation. The result would be $\frac{1}{3}x^3 + y^2x + C$, where $C$ is the constant of integration.
Similarly, if we want to integrate $f(x, y)$ with respect to $y$, we would treat $x$ as a constant and integrate $x^2 + y^2$ with respect to $y$. The result would be $x^2y + \frac{1}{3}y^3 + C$, where $C$ is the constant of integration.
It's important to note that when integrating equations with multiple variables, the order of integration can affect the result. In the example above, integrating with respect to $x$ first and then with respect to $y$ gives a different result than integrating with respect to $y$ first and then with respect to $x$. This is known as the order of integration.
Let's consider a specific example to further illustrate the concept of integrating equations with multiple variables. Suppose we have a function $f(x, y) = 2xy + y^2$ and we want to integrate it with respect to $x$ and then with respect to $y$.
First, we integrate $f(x, y)$ with respect to $x$, treating $y$ as a constant. The result would be $x^2y + xy^2 + C_1$, where $C_1$ is the constant of integration.
Next, we integrate the result with respect to $y$, treating $x$ as a constant. The result would be $\frac{1}{2}x^2y^2 + \frac{1}{3}xy^3 + C_2$, where $C_2$ is the constant of integration.
Therefore, the final result of integrating $f(x, y)$ with respect to $x$ and then with respect to $y$ is $\frac{1}{2}x^2y^2 + \frac{1}{3}xy^3 + C_2$.
## Exercise
Consider the function $f(x, y) = 3x^2y + 2xy^2$.
1. Integrate $f(x, y)$ with respect to $x$ and then with respect to $y$.
2. Integrate $f(x, y)$ with respect to $y$ and then with respect to $x$.
### Solution
1. Integrating $f(x, y)$ with respect to $x$ and then with respect to $y$:
First, integrate $f(x, y)$ with respect to $x$, treating $y$ as a constant. The result would be $\frac{3}{2}x^3y + xy^2 + C_1$, where $C_1$ is the constant of integration.
Next, integrate the result with respect to $y$, treating $x$ as a constant. The result would be $\frac{3}{4}x^3y^2 + \frac{1}{2}xy^3 + C_2$, where $C_2$ is the constant of integration.
Therefore, the final result of integrating $f(x, y)$ with respect to $x$ and then with respect to $y$ is $\frac{3}{4}x^3y^2 + \frac{1}{2}xy^3 + C_2$.
2. Integrating $f(x, y)$ with respect to $y$ and then with respect to $x$:
First, integrate $f(x, y)$ with respect to $y$, treating $x$ as a constant. The result would be $\frac{3}{2}x^2y^2 + \frac{2}{3}xy^3 + C_3$, where $C_3$ is the constant of integration.
Next, integrate the result with respect to $x$, treating $y$ as a constant. The result would be $\frac{1}{2}x^3y^2 + \frac{1}{3}xy^3 + C_4$, where $C_4$ is the constant of integration.
Therefore, the final result of integrating $f(x, y)$ with respect to $y$ and then with respect to $x$ is $\frac{1}{2}x^3y^2 + \frac{1}{3}xy^3 + C_4$.
# Proof of the fundamental theorem of calculus
The fundamental theorem of calculus is a fundamental result in calculus that establishes the relationship between differentiation and integration. It consists of two parts: the first part relates the antiderivative of a function to its definite integral, while the second part provides a method for evaluating definite integrals using antiderivatives.
To understand the proof of the fundamental theorem of calculus, we need to recall some key concepts. First, the derivative of a function measures its rate of change at a specific point. The antiderivative, on the other hand, measures the accumulation of a quantity over an interval.
The proof of the first part of the fundamental theorem of calculus involves showing that the derivative of the antiderivative of a function is equal to the original function. In other words, if we have a function $f(x)$ and its antiderivative $F(x)$, then $F'(x) = f(x)$.
To prove this, we start with the definition of the derivative:
$$F'(x) = \lim_{h \to 0} \frac{F(x + h) - F(x)}{h}$$
We can rewrite this expression using the definition of the antiderivative:
$$F'(x) = \lim_{h \to 0} \frac{\int_{x}^{x+h} f(t) dt}{h}$$
By the properties of integrals, we can split the integral into two parts:
$$F'(x) = \lim_{h \to 0} \frac{\int_{x}^{a} f(t) dt + \int_{a}^{x+h} f(t) dt}{h}$$
where $a$ is a constant. Now, we can rewrite the second integral as:
$$\int_{a}^{x+h} f(t) dt = \int_{a}^{x} f(t) dt + \int_{x}^{x+h} f(t) dt$$
Substituting this back into the expression for $F'(x)$, we get:
$$F'(x) = \lim_{h \to 0} \frac{\int_{x}^{a} f(t) dt + \int_{a}^{x} f(t) dt + \int_{x}^{x+h} f(t) dt}{h}$$
Using the properties of limits and integrals, we can rearrange the terms and simplify the expression:
$$F'(x) = \lim_{h \to 0} \frac{\int_{a}^{x} f(t) dt}{h} + \lim_{h \to 0} \frac{\int_{x}^{x+h} f(t) dt}{h}$$
The first term in the expression is equal to $F(x) - F(a)$, and the second term approaches $f(x)$ as $h$ approaches 0. Therefore, we have:
$$F'(x) = \lim_{h \to 0} \frac{F(x) - F(a)}{h} + f(x)$$
Taking the limit as $h$ approaches 0, we get:
$$F'(x) = f(x)$$
This completes the proof of the first part of the fundamental theorem of calculus.
## Exercise
Prove the first part of the fundamental theorem of calculus for the function $f(x) = 2x^3$.
### Solution
To prove the first part of the fundamental theorem of calculus for the function $f(x) = 2x^3$, we need to find its antiderivative $F(x)$ such that $F'(x) = f(x)$.
Integrating $f(x)$ with respect to $x$, we get:
$$F(x) = \int f(x) dx = \int 2x^3 dx$$
Using the power rule for integration, we can find the antiderivative:
$$F(x) = \frac{2}{4}x^4 + C = \frac{1}{2}x^4 + C$$
where $C$ is the constant of integration.
Taking the derivative of $F(x)$, we have:
$$F'(x) = \frac{d}{dx} \left(\frac{1}{2}x^4 + C\right) = 2x^3$$
Therefore, the antiderivative of $f(x) = 2x^3$ is $F(x) = \frac{1}{2}x^4 + C$, and we have proved the first part of the fundamental theorem of calculus.
# Exploring the relationship between differentiation and integration
Differentiation and integration are two fundamental concepts in calculus that are closely related to each other. While differentiation measures the rate of change of a function, integration measures the accumulation of a quantity over an interval.
The relationship between differentiation and integration is captured by the fundamental theorem of calculus. This theorem states that the derivative of the antiderivative of a function is equal to the original function. In other words, if we have a function $f(x)$ and its antiderivative $F(x)$, then $F'(x) = f(x)$.
This relationship allows us to use differentiation and integration interchangeably to solve problems. For example, if we want to find the area under a curve, we can integrate the function that represents the curve. On the other hand, if we have the antiderivative of a function, we can find the original function by taking its derivative.
Let's consider the function $f(x) = 2x^3$. To find its antiderivative, we can use the power rule for integration:
$$\int f(x) dx = \int 2x^3 dx = \frac{2}{4}x^4 + C = \frac{1}{2}x^4 + C$$
where $C$ is the constant of integration.
Now, if we take the derivative of the antiderivative, we should get back the original function:
$$\frac{d}{dx} \left(\frac{1}{2}x^4 + C\right) = 2x^3$$
As expected, the derivative of the antiderivative is equal to the original function.
## Exercise
Find the antiderivative of the function $f(x) = 4x^2$.
### Solution
To find the antiderivative of the function $f(x) = 4x^2$, we can use the power rule for integration:
$$\int f(x) dx = \int 4x^2 dx = \frac{4}{3}x^3 + C$$
where $C$ is the constant of integration.
# Advanced applications of integration
One important application of integration is in finding the average value of a function over an interval. The average value of a function $f(x)$ over the interval $[a, b]$ is given by the formula:
$$\text{Average value} = \frac{1}{b-a} \int_a^b f(x) dx$$
This formula allows us to find the average value of a function, which can be useful in various contexts such as determining average temperatures, average speeds, or average rates of change.
Suppose we have a function $f(t)$ that represents the temperature in degrees Celsius at different times $t$. We want to find the average temperature over a 24-hour period, from 8:00 AM to 8:00 AM the next day. We can use integration to solve this problem.
Let's say the function $f(t)$ is given by:
$$f(t) = 20 + 5\sin\left(\frac{2\pi}{24}t\right)$$
To find the average temperature over the 24-hour period, we can use the formula:
$$\text{Average temperature} = \frac{1}{24-8} \int_8^{24} (20 + 5\sin\left(\frac{2\pi}{24}t\right)) dt$$
Evaluating this integral will give us the average temperature over the 24-hour period.
## Exercise
Find the average value of the function $f(x) = 3x^2$ over the interval $[1, 5]$.
### Solution
To find the average value of the function $f(x) = 3x^2$ over the interval $[1, 5]$, we can use the formula:
$$\text{Average value} = \frac{1}{5-1} \int_1^5 3x^2 dx$$
Evaluating this integral will give us the average value of the function over the interval $[1, 5]$.
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Textbooks
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The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model
Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation
Global dynamics analysis of a time-delayed dynamic model of Kawasaki disease pathogenesis
Ke Guo , Wanbiao Ma , and Rong Qiang
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083, China
* Corresponding author: Wanbiao Ma
Received May 2020 Revised January 2021 Early access May 2021
Fund Project: The authors were supported by Beijing Natural Science Foundation (No.1202019) and National Natural Science Foundation of China (No.11971055)
Kawasaki disease (KD) is an acute febrile vasculitis that occurs predominantly in infants and young children. With coronary artery abnormalities (CAAs) as its most serious complications, KD has become the leading cause of acquired heart disease in developed countries. Based on some new biological findings, we propose a time-delayed dynamic model of KD pathogenesis. This model exhibits forward$ / $backward bifurcation. By analyzing the characteristic equations, we completely investigate the local stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibria. Our results show that the time delay does not affect the local stability of the inflammatory factors-free equilibrium. However, the time delay as the bifurcation parameter may change the local stability of the inflammatory factors-existent equilibrium, and stability switches as well as Hopf bifurcation may occur within certain parameter ranges. Further, by skillfully constructing Lyapunov functionals and combining Barbalat's lemma and Lyapunov-LaSalle invariance principle, we establish some sufficient conditions for the global stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium. Moreover, it is shown that the model is uniformly persistent if the basic reproduction number is greater than one, and some explicit analytic expressions of eventual lower bounds of the solutions of the model are given by analyzing the properties of the solutions and the range of time delay very precisely. Finally, some numerical simulations are carried out to illustrate the theoretical results.
Keywords: Kawasaki disease, backward bifurcation, global stability, uniform persistence, stability switches, Hopf bifurcation.
Mathematics Subject Classification: Primary: 34K20, 34K18; Secondary: 92B05.
Citation: Ke Guo, Wanbiao Ma, Rong Qiang. Global dynamics analysis of a time-delayed dynamic model of Kawasaki disease pathogenesis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021136
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Figure 1. The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 2.8\in[0, \tau_{1}^{(0)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is locally asymptotically stable
Figure 2. The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 7.1\in(\tau_{1}^{(0)}, \tau_{2}^{(0)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is unstable and periodic oscillations occur
Figure 3. The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 13.6\in(\tau_{2}^{(0)}, \tau_{1}^{(1)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is locally asymptotically stable
Figure 4 (e) is a partial enlarged view of Figure 4 (a) near the inflammatory factors-existent equilibrium $ Q^* $">Figure 4. The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 16\in(\tau_{1}^{(1)}, +\infty) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is unstable and periodic oscillations occur. Figure 4 (e) is a partial enlarged view of Figure 4 (a) near the inflammatory factors-existent equilibrium $ Q^* $
Table 1. Biological meanings of the parameters in model (1) [28]
Parameters Biological meanings
$ \; r $ Proliferation rate of normal endothelial cells (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
$ \; d_{1}\; $ Apoptosis rate of normal endothelial cells (day$ ^{-1} $)
$ \; d_{2}\; $ Hydrolytic rate of endothelial growth factors (day$ ^{-1} $)
$ \; d_{3}\; $ Hydrolytic rate of activated adhesion factors$ / $chemokines (day$ ^{-1} $)
$ \; d_{4}\; $ Hydrolytic rate of inflammatory factors (day$ ^{-1} $)
$ \; k_{1}\; $ The rate of injury of endothelial cells caused by inflammatory factors(pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
$ \; k_{2}\; $ Production rate of endothelial growth factors caused by inflammatory factors (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
$ \; k_{3}\; $ Production rate of activated adhesion factors$ / $chemokines caused by inflammatory factors (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
$ \; k_{4}\; $ Production rate of activated adhesion factors$ / $chemokines caused by endothelial growth factors (day$ ^{-1} $)
$ \; k_{5}\; $ Production rate of inflammatory factors by increasing of abnormally activated immune cells (day$ ^{-1} $)
$ \; k_{6}\; $ Proliferation rate of endothelial cells promoted by endothelial growth factors (day$ ^{-1} $)
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Ke Guo Wanbiao Ma Rong Qiang
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CommonCrawl
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Phenotypic clustering: a novel method for microglial morphology analysis
Franck Verdonk1, 4, 10, 11,
Pascal Roux2,
Patricia Flamant1,
Laurence Fiette1,
Fernando A. Bozza3,
Sébastien Simard2,
Marc Lemaire4,
Benoit Plaud5, 9,
Spencer L. Shorte2,
Tarek Sharshar1, 6, 8, 11,
Fabrice Chrétien†1, 7, 10, 11Email author and
Anne Danckaert†2Email author
†Contributed equally
Accepted: 6 June 2016
Microglial cells are tissue-resident macrophages of the central nervous system. They are extremely dynamic, sensitive to their microenvironment and present a characteristic complex and heterogeneous morphology and distribution within the brain tissue. Many experimental clues highlight a strong link between their morphology and their function in response to aggression. However, due to their complex "dendritic-like" aspect that constitutes the major pool of murine microglial cells and their dense network, precise and powerful morphological studies are not easy to realize and complicate correlation with molecular or clinical parameters.
Using the knock-in mouse model CX3CR1GFP/+, we developed a 3D automated confocal tissue imaging system coupled with morphological modelling of many thousands of microglial cells revealing precise and quantitative assessment of major cell features: cell density, cell body area, cytoplasm area and number of primary, secondary and tertiary processes. We determined two morphological criteria that are the complexity index (CI) and the covered environment area (CEA) allowing an innovative approach lying in (i) an accurate and objective study of morphological changes in healthy or pathological condition, (ii) an in situ mapping of the microglial distribution in different neuroanatomical regions and (iii) a study of the clustering of numerous cells, allowing us to discriminate different sub-populations.
Our results on more than 20,000 cells by condition confirm at baseline a regional heterogeneity of the microglial distribution and phenotype that persists after induction of neuroinflammation by systemic injection of lipopolysaccharide (LPS). Using clustering analysis, we highlight that, at resting state, microglial cells are distributed in four microglial sub-populations defined by their CI and CEA with a regional pattern and a specific behaviour after challenge.
Our results counteract the classical view of a homogenous regional resting state of the microglial cells within the brain. Microglial cells are distributed in different defined sub-populations that present specific behaviour after pathological challenge, allowing postulating for a cellular and functional specialization. Moreover, this new experimental approach will provide a support not only to neuropathological diagnosis but also to study microglial function in various disease models while reducing the number of animals needed to approach the international ethical statements.
Microglial cell morphology
Automated high-content analysis
Sub-population behaviour
Complexity index
Microglial cells are parenchymal tissue macrophages that account for 10 % of brain cells [1] involved in brain immune surveillance and homeostasis. In healthy conditions, they are involved in synaptic development and maintenance [2], neuronal survival [3] and phagocytosis [4] to maintain brain homeostasis. Microglia migrate to interact with other cell types (i.e. astrocyte and neurons) and produce a variety of factors required to induce neural progenitor differentiation [5] and neuronal apoptosis [6]. Their activation is the main component of the neuroinflammatory process, which can result either from direct brain insult or systemic inflammation leading to either neuroprotective [7, 8] or neurotoxic [9, 10] responses. Their activation is characterized by both immunological and morphological changes, including mainly a decrease in ramification up to the amoeboid form (i.e. large non-ramified cells). Hypotheses of an adaptation of their morphology to their specific function within the central nervous system (CNS) are formulated [11, 12] but neither could be actually substantiated due to two boundaries: (i) the difficulty-to-detect subtle morphological variations of microglia and also probably intermediate forms between the categories of activation that are historically described [13, 14] and (ii) the need to couple a very precise morphological single cell approach with a proteomic or transcriptomic study using specific techniques [15] taking into account the regionalization that is specific to the CNS. In fact, since the identification of microglial cells in 1932 by del Rio-Hortega [16], the assessment of microglial morphology allows characterizing their role or their level of activation [17, 18] but it remains challenging. The spectacular complexity of these cells, their morphological variability and their dense distribution within the tissue underlines the importance of a precise morphological characterization in a large number of cells distributed in the different functional CNS regions. The standard nowadays is the use of immunohistochemistry staining using CD45 or ionized calcium-binding adapter molecule 1 (Iba-1) localizing in the monocytic lineage and well expressed in microglia [19]. Those techniques, however, have some limitations. Indeed, since the expression of these markers depends on the intensity of microglial activation, immunostaining can be insufficient for accurately describing a "dendritic-like" or ramified phenotype [20, 21] that constitutes more than 90 % of the microglial cells in young mice [11]. Additionally, it required particular histological techniques such as paraffin embedding that may affect the precision of a morphological study. These limitations might be overcome by the use of transgenic mice, such as an Iba-1GFP/+ [22] or CX3CR1GFP/+ [23] mouse in which brain microglia express spontaneously green fluorescent protein (GFP) respectively under the control of the Iba-1 or the CX3CL1 (fractalkine) receptor locus. These transgenic models allow very precise visualization of the microglial ramifications requiring no immunostaining techniques. These models overcome the technical limits but ask the question of the quantification methods. Usually, quantification methods are based on semi-quantitative scoring and manual counting, making them time consuming, susceptible to inter- or intra-observer variability and imprecise. Furthermore, manual analyses are unable to assess either large numbers of cells or their network organization. Recently, innovative technical and/or mathematical methods have been developed allowing automated acquisition, fractal analysis [13] or segmentation of individual microglial cell shapes [24]. They have enabled to reliably assess the changes in microglial morphology albeit in limited numbers of cells (<70) [25–27]. Because of this limitation, the range of statistical analyses is also restricted while methods are available for analysing large numbers of cells and thus detecting subtle morphological phenotypes and changes therein. These methods are alternatives to conventional statistics and are able to exploit or highlight the major heterogeneity of cell populations in the same tissue, with an underlying organization that cannot be directly observed. Indeed, plasticity, reflected by slight morphological changes, is considered as a major functional property of microglial cells [28]. Using diverse criteria, it becomes now possible to discern precisely these sub-populations and structures by applying a clustering approach [29].
In efforts to develop a method allowing the assessment of the morphology of a large microglial population, and therefore for a better understanding of the microglial behaviour in different developmental, homeostatic and disease contexts, we herein propose an innovative strategy. Our approach is based on the automated acquisition of fluorescence and the measurement of morphological indexes in CX3CR1GFP/+ transgenic mice that allows discriminating microglial sub-populations based on a clustering analysis. To validate this method, we compared clustering analysis to parametric and non-parametric statistical approaches assessing their capacity to detect inter-regional variability and post-stimulation changes in microglial morphology. In conclusion, our approach is extremely efficient, reproducible and accurate.
Male and female C57BL/6 JRj mice purchased from JANVIER LABS and in-house CX3CR1GFP/+ mice aged from 9 to 11 weeks were used for these experiments. In the CX3CR1GFP/+ model, the CX3CL1 receptor gene, the CX3CR1, was knocked-in with a GFP reporter gene [23]. This gene is constitutively turned on in microglial cells and thus allows us to image them selectively using the GFP without any immunochemistry method. Mice were housed in cages in groups of seven, in a temperature- (22 ± 1.5 °C) and humidity-controlled environment, with a 12-h light/dark cycle. Mice were provided with food and water ad libitum according to international guidelines.
Treatment conditions
Two experimental groups were considered. In the lipopolysaccharide (LPS) group (n = 6 by strain), the mice were injected intraperitoneally with 5 mg/kg of LPS from Escherichia coli serotype 055:B5 (Sigma-Aldrich) [30, 31] dissolved in 0.9 % saline. Twenty-four hours later, these mice were killed by cervical dislocation and the brain was collected. In parallel, mice belonging to the control group were not anaesthetized and were also killed by cervical dislocation before brain collection.
Tissue preparation
After cervical dislocation, the brains were immediately removed and cut in a trans-sagittal plane in the inter-hemispheric fissure. Cerebral hemispheres were fixed during 24 h in 4 % buffered formalin (QPath, Labonord SAS, Templemars, France). Following fixation, tissue samples were sliced along a sagittal plane on a calibrated vibratome (VT1000 S, Leica, Germany) into 100-μm-thick free-floating slices. The most medial slices were used for analysis.
Histological analysis and immunohistochemistry
Brain sections of the left hemisphere of C57BL/6 JRj mice were incubated with the rabbit antibody against Iba-1 (Wako Chemicals, Richmond, VA, 1:500) and revealed by the secondary antibody Dy488 (Jackson ImmunoResearch Laboratories, Baltimore, PA). A classical protocol was used: rehydratation, blocking with 20 % goat serum and 0.5 % Triton-X 100 for 2 h, incubation with primary antibody (Dako Diluent buffer, Glostrup, Denmark) overnight at 4 °C followed by incubation with secondary antibody 4 h at room temperature. The stained sections were mounted on slides and coverslipped
Image acquisition and processing
Using a spinning disc confocal system (CellVoyager CV1000, Yokogawa, Japan) with a UPLSAPO 40×/NA 0.9 objective, sample areas were acquired as a square of 10 × 10 fields of view with a depth of 30 μm at 2-μm increments (16 focal depths) generating one volume in four regions of interest: striatum, frontal cortex, hippocampus and cerebellum. These regions were acquired sequentially allowing the coverage of approximately 3 mm2 of tissue per region. Each field corresponds to a matrix of 920 × 920 pixels; the pixel size in X and Y dimensions is 0.19 μm according to the objective. The 488-nm laser was used to excite GFP or detect Iba-1 and thus to image the microglial cells.
Before the shape characterization analysis, focal stacks of each mosaic were reconstructed by combining images from the different focal depths. Each stack was subsequently divided into three 10-μm sub-volumes to allow a two-dimensional (2D) maximum intensity projection analysis (Fig. 1a), consistent with the average size of cell bodies. Mosaic, volume creation and maximum projection processing from confocal images were done using automated free plugins [32] of the ImageJ v1.50 software interface [33].
The characterization of microglial cells by morphological criteria. a Confocal images, representing a sub-part of the analysed image in the frontal cortex region after maximum intensity projection. The individual microglia based on GFP fluorescence appears in white outline. The scale bar equals 50 μm. b Ramification detection based on GFP fluorescence with AcapellaTM software. The segmented ramifications linked to an individual microglia are shown artificially in green, the unattributed ramifications in white. The scale bar equals 50 μm. c The morphological criteria to characterize a microglial cell. The cell body detection (blue) and cytoplasm area (pink) have performed as a starting point to characterize a microglial cell. The complexity index (green) and the covered environment area (CEA in orange) have been deduced from ramification detection. The scale bar equals 10 μm. d Two-dimensional cartography at a single cell resolution. Colours correspond to the range of complexity and CEA with a gradient from a low level of complexity and CEA (yellow) to a high level of complexity and CEA (red). The scale bar equals 50 μm. For illustration, the images are contrast adjusted to aid in visualizing the GFP expression
Image analysis for characterization of microglial cell population
An automatic image analysis was performed consecutively on the three maximum projection mosaics described above, using a custom-designed script developed with the Acapella™ image analysis software (version 2.7, PerkinElmer Technologies, Waltham, USA). This script was subdivided into two subroutines: the first, for automated detection of processes (neurite detection module from Acapella™ [34]), generating morphological characteristics per cell, and the second, for the 2D in situ morphological cartography. The data workflow is illustrated step by step in Additional file 1.
Microglial morphological criteria
Using the custom-designed script cited above, the following morphological criteria could be extracted for each microglial cell (Fig. 1b, c): a set of measured criteria as cell body and cytoplasm area, defined as the cell body area associated with the cytoplasmic area of the primary ramifications, expressed in μm2; branching characteristics such as the total number and length (μm) of ramifications and the number of primary, secondary and tertiary ramifications; and roundness (ratio between surface and perimeter squared of the cell body) and GFP intensity by whole cell.
A second set of calculated criteria extrapolated from the previous ones yielded the complexity index (CI) and the covered environment area (CEA). First, we defined the CI using two different criteria extracted from the Acapella™ script: the number of segments of each cell, a segment being defined as the length of process between two nodes, and the number of its primary ramifications. By dividing these two criteria, we obtain also a mean complexity by primary ramification for each microglial cell (Additional file 2). On the other hand, CEA represents the 2D total surface covered by its ramifications and defined as the area of the polygon formed by linking the extremities of its processes, expressed in μm2. The areal density of microglial cells by region or by brain was calculated by dividing the number of microglial cells selected by the scanned tissue area.
The CI revealed a completely distinct microglial phenotype, the amoeboid cells. The amoeboid or rod cells are characterized by a CI = 1 (no nodes) and are characteristic for activated cells, displaying engulfing, phagocytic properties [11]. Because of their particular role, we distinguish them from the other microglial cells.
Elimination of outliers and redundant cells
We proceeded to the filtration of outliers by size (inferior to 10 μm2 or superior to 500 μm2) and roundness (inferior to 0.7 considered as the limit of a noisy form) to eliminate artefacts due to tissue noise. Using the property of our Acapella™ script to generate in situ 2D cartographies (Additional file 1), cell duplicates on two adjacent sub-volumes were rejected for analysis to avoid data duplication. This 2D localization in situ allowed us also to remove the cells located at the edges of the mosaic reasoning that they may be truncated.
Manual and semi-automated method workflow
To assess the accuracy of the 2D reconstruction of the custom-designed script cited, we compared the criteria measured by the cell using our automated method with the analysis realized manually by three independent experimenters, as a technical benchmark: the cell body area, the number of segments, the number of primary ramifications and the measure of the CEA. The manual method has been performed using Fiji environment (Additional file 3).
In parallel, to assess that the criteria analysis is not software specific, a free semi-automated method has been implemented in a Fiji macro using Skeletonize (2D/3D) and Sholl Analysis plugins [35, 36]. The cell body area, the number of segments and the number of primary ramifications have been extracted by this custom-designed macro.
These two methods were tested and compared with our automated method in two regions of interest, the hippocampus and the cerebellum, and in the two conditions, LPS and control.
Data analysis and statistics
All the data extracted from Acapella™ after elimination of outliers and redundant cells were exploited following two levels: (i) by the brain and (ii) by specific region. We conducted this study using two different statistical methods. The first conventional method consisted in an approach considering only the median or the average for each morphological criterion by animal. The second approach, more powerful and original, presented results using k-means clustering by cell population.
To realize the conventional approach, Prism 6.0 (GraphPad Software Inc.©, USA) was used for statistical analysis by animal and regions of interest. Data were analysed via Mann-Whitney test or Student's t test after being assessed for normality of sample distribution. Inter-sample/inter-region variability was tested by ANOVA Kruskal-Wallis method. Qualitative traits (i.e. clustering phenotype distribution) were analysed with a chi-square (χ 2) test. Spearman coefficient has been expressed to represent the correlation between two sets of data. Statistical significance is shown on the graphs (*p < 0.05; **p < 0.01; ***p < 0.001; ****p < 0.0001). Statistical tests used for each data set are indicated in the figure legends.
The required number of samples per group (n) has been evaluated with pwr.t.test R function [37], with α = 5 % and 1 − β = 90 %.
The clustering analysis
In a second time, Prism data were transferred into JMP® version 11.0 (Statistical Analysis System Institute Inc., USA) for a complete multivariate analysis by cell population. A principal component analysis (PCA) was performed to identify the correlation between the different analysed features. To detect and characterize the sub-population of microglial cells, a k-means clustering method, appropriated for a large set of data, has been applied (k = 4). The statistics of each cluster (mean and frequency) were used to characterize sub-populations and determine their phenotype, later named clusters 1 to 4. For each condition, the amount of microglial cells analysed was about 810 by region, 2870 by brain or 20,000 by group.
Data storage and annotations by in situ 2D cartography
All acquired and analysed mosaics have been imported into the OMERO ("OME Remote Objects technology") image database [38] including visual results in 2D cartographies by phenotyped cell (Fig. 1d and Additional file 1). An open-source script (OMERO.csv) has been used to annotate automatically by textual information (i.e. sex, condition or clinical observation by sample or region appurtenance) our large set of imported images in the database.
Using data extracted from more than 20,000 cells per condition, we performed and compared two types of complementary statistical approaches at an inter-regional and an inter-group level. To validate the acquisition process, the GFP intensity of each microglial cell was measured. No difference in the microglial cell GFP intensity between the two conditions, whatever the brain region, was found (Table 1 and Additional file 4).
Morphological variability study for microglial cells between two groups
GFP intensity
Cell body area
Cytoplasm area
Values are expressed as Mann-Whitney exact p values; significant differences in italics
Conventional statistical approach
For this conventional approach, two statistical analyses were performed: one to observe an inter-region variability (Figs. 2 and 3), the other to detect a difference inter-group by brain region (Tables 1 and 2).
The inter-region variability by morphological criteria. Four regions have been explored: hippocampus (H), frontal cortex (FC), striatum (S) and cerebellum (C) in two different conditions, the control (left column) and the LPS (right column). a Historical parameters to characterize the microglial morphology: the cell body area and the cytoplasm area defined as the cell body area associated with the cytoplasmic area of the primary ramifications in μm2. b Calculated criteria extrapolated from the Acapella™ script: the complexity index (CI) and the covered environment area (CEA), in μm2. Data shown are means ± SD in the control and LPS groups (n = 7 and n = 6, respectively). The scale bars equal 10 μm. ANOVA Kruskal-Wallis test was used to compare the different regions. *p < 0.05, **p < 0.01, ***p < 0.001
Characteristics of the amoeboid population and their inter-region variability. Bar charts represent the characteristics of the amoeboid cell morphology (characterized by a CI = 1, without nodes) and the distribution within the four explored regions: hippocampus (H), frontal cortex (FC), striatum (S) and cerebellum (C) in the two different conditions, the control (left column) and the LPS (right column). a Parameters to characterize the amoeboid cells morphology: the cell body area and the cytoplasm area defined as the cell body area associated with the cytoplasmic area of the primary ramifications in μm2. b Parameters to characterize the amoeboid cell distribution: density calculated by dividing the number of microglial cells selected by the scanned tissue area (3.03 mm2) and frequency as the ratio between the number of amoeboid cells and the total number of microglial cells analysed. Data shown are means ± SD per condition (n = 7, n = 6 for control and LPS, respectively); we used ANOVA Kruskal-Wallis test. **p < 0.01, ***p < 0.001
Morphological variability study for amoeboid between two groups
Microglial cell body area, cytoplasm area and density
Mean cell body and cytoplasm areas of microglial cells in the LPS group brains are significantly higher than in the control group brains (respectively, 149 vs. 74 μm2 for the cytoplasm area in the hippocampus, p = 0.0012, Table 1). We found a high heterogeneity between the mean microglial cell body and cytoplasm areas in the different regions of the brain within a given group (p < 0.01 or p < 0.001). At the opposite, in the context of LPS-induced inflammation, the mean cell body area was homogeneous wherever the region (Fig. 2a). Moreover, in an inter-regional comparison, the mean cytoplasm area was significantly greater in the cerebellum than in the other regions, in either the control or LPS groups (Additional file 4); it did not statistically differ between the hippocampus, frontal cortex and striatum, in both conditions. In an inter-group comparison (Table 1), we observed a significant increase of cell area (body and cytoplasm) in the LPS group; the microglial cell cytoplasm was twice greater after the LPS challenge regardless of the region. In each condition, the microglial density varied significantly among regions, with a higher mean density in the frontal cortex and a lower one in the cerebellum (Additional file 4). The microglial density did not differ between the control and LPS groups, in the total brain or in each region (Table 1).
Complexity index (CI) and covered environment area (CEA)
Both microglial CI, defined as the mean complexity by primary ramification, and CEA, defined as the 2D total surface covered by microglial ramifications, showed no statistical difference between the two different groups. CI and CEA were significantly lower in the cerebellum than in the other regions, in both the control and the LPS group. We did not find any statistical difference between the hippocampus, frontal cortex and striatum regardless of the conditions (Fig. 2b). CI and CEA did not statistically differ between the LPS and control groups, in each region except the cerebellum (Table 1). The cerebellar microglial CI was significantly lower in the LPS group than in the control group (Additional file 4).
Comparison with Iba-1 expression based on morphological criteria by sample
To strengthen our statements about the CX3CR1GFP/+ mice model, we performed the same experiments based on Iba-1 expression in wildtype C57BL/6 mice and we were interested in the same morphological criteria (microglial cell body area, cytoplasm area, cytoplasm intensity, CI and CEA). In the hippocampus, we observed the same differences between the two groups compared to the GFP model with a significant increase of cell area (body and cytoplasm) in the LPS group whereas we observed no difference considering the CI and CEA (Additional file 5). In the cerebellum, despite a trend comparable to what is seen in the GFP mouse model, there is no significant difference in the cell area. This isolated difference may be explained by the smaller number of cells analysed, about 10 times less, linked to the limits of the immunostaining on thick sections.
Amoeboid cells
In each condition, the frequency of amoeboid cells, considered as a particular group of microglial cells with specific morphological characteristics (CI = 1) associated with a specific function, varied significantly among regions, with the lowest frequency in the frontal cortex than in the cerebellum (Fig. 3): control group: 5.64 ± 2.9 % and 14.73 ± 3.11 %, respectively, p = 0.0078; LPS group: 3.13 ± 1.27 % and 17.30 ± 5.53 %, respectively, p = 0.0018 (Additional file 4). The LPS challenge was associated with increased amoeboid cell body and cytoplasm areas among brain regions. There is no difference by region in terms of amoeboid frequencies between the two groups (Table 2 and Additional file 6).
In conclusion, using a conventional statistical approach, we found that (i) in comparison to other regions, cerebellar microglial cells presented a bigger cytoplasm, were less dense and complex, more frequently amoeboid and more responsive to LPS; (ii) the LPS challenge is associated with an increase, such as twice greater, in cell body and cytoplasm areas but not with a decrease in CI and CEA, indicating that there is no evidence for a "deramification" process.
Comparison with benchmark methods
To assess the accuracy of our automated method, after random extraction of an analysed cell subset, three independent experimenters performed manual measures of the selected criteria as cell body area, CI and CEA. Statistical tests carried out by region showed the same trends between the control and LPS groups whatever the method considered (Additional file 7A). It is to be noted that the analysis time per cell is multiplied by 10 between the automated method and the manual method and the number of cells studied by condition is 400 times lower (Additional file 7C).
We also tested whether the results of the analysis were not software specific. Using a semi-automatic analysis with the Fiji software environment on a greater number of cells than in the manual comparison, we did not find any difference in the morphological criteria between the semi-automatic and automatic methods in each region. The results obtained showed a strong correlation between the two methods (Additional files 7B, C).
Cell heterogeneity by condition
Although no difference was observed between the mean CI or CEA of the microglial cells in the two conditions, considering every cell of every brain in each condition, we found a high cell heterogeneity between the samples using Kruskal-Wallis test (p < 0.0001) illustrating the biological variability across individuals (Additional file 8), whatever the morphological criteria tested.
Such variability does not allow us to compare directly populations using classical statistical studies. Moreover, in order to distinguish sub-populations in our large datasets of cells, we pooled all the microglial cells from each group by region and also conducted a new statistical study using a clustering method.
Statistical clustering approach
The principal component analysis (PCA) showed that the CI and CEA did not correlate, allowing proceeding to the cluster analysis based on these indexes. In the control group, based on the whole brain without region discrimination, the rates of the clusters 1, 2, 3 and 4 used to characterize sub-populations by the k-means clustering were 69, 18, 11 and 2 %, respectively (Fig. 4a). The cutoff fixing the high or the low characteristic of one population was set as the average of each morphological criterion in the control group (Fig. 4b) and therefore allowed to discriminate four sub-populations (SP): SP1: low CEA and low CI (−/−); SP2: low CEA and high CI (−/+); SP3: high CEA and low CI (+/−); and SP4: high CEA and high CI (+/+). Considering the whole brain, the proportions of SP did not differ statistically between the control and LPS conditions (Fig. 4b). A contrario, the proportions of these sub-populations varied significantly among the regions in the control groups, as shown in Fig. 5. SP1 was most represented in the cerebellum (92 % of the cells). SP2 represented from 28 % of the cells in the striatum to 15 % in the hippocampus. SP3 is present uniquely in the striatum at a 16 % rate and SP4 from 3 % in the striatum to 11 % in the hippocampus. The regional distribution of the sub-populations varied significantly between the control and LPS conditions (p < 0.0001), with a significant increase in the SP4 in the striatum (from 3 to 46 %) and at the opposite a significant decrease to disappearance of this population in the other areas. The SP3 increases largely in the frontal cortex, hippocampus and cerebellum. The SP1 remains stable except in the hippocampus with an increase up to 80 % of the cell population (Fig. 5).
Approach by clustering to track sub-populations of microglial cells in the whole brain. a The scatter plots illustrate, at a single cell resolution, the CEA and CI characteristics and their frequency by cluster. The symbols "+" and "x" correspond to the centre of each cluster by the control and the LPS condition, respectively. The pie charts show the cluster frequencies by k-means clustering method (k = 4), and no significant difference has been observed between the two conditions using the chi-square test. b Four sub-populations have been defined by the cutoff (dotted lines) fixing the high (+) or the low (−) characteristic of one sub-population in the whole brain (WB). The cutoff was defined as the average of each morphological criterion (CI and CEA) in the control group. The centre of each cluster was plotted in the graph. The pie charts represent the proportions of sub-populations by condition. The same repartitions by sub-population as by cluster were observed
Highlighting sub-populations by region. The pie charts represent the proportions of sub-populations defined by the cutoff previously described in Fig. 4 by region of interest and by condition: in yellow, sub-population with low CEA and low CI (−/−); in orange, sub-population with low CEA and high CI (−/+); in dark orange, sub-population with high CEA and low CI (+/−); and in red, sub-population with high CEA and high CI (+/+). Chi-square test was used to compare the control and the LPS group. ****p < 0.0001
Cluster analysis based on CI and CEA revealed a regional pattern of microglial sub-population with particular responsiveness to LPS. Therefore, we obtained a new frequency of phenotyped cells and hereby defined sub-populations of microglial cells based on particular behaviour.
Impact on the number of needed animals
The impact of a high-content automated approach is also important ethically. It is possible to assess the number of mice required to highlight the same differences as those observed using a classical statistical study and considering only the mean by animal. This evaluation of the number of subjects required (n) is based on the two-sample t test defined by the following formula:
$$ n=2\times t\times \frac{\sigma^2}{{\left({\overline{m}}_A-{\overline{m}}_B\right)}^2} $$
in which \( {\overline{m}}_{A,B} \) are the mean of criteria for two sets of data and σ is the common standard deviation of two samples with the hypothesis that n A = n B .
Considering a power of 90 % and an alpha risk of 5 %, the expected sample size showing exactly the same differences is from 1 to 27 times greater considering the CI depending on the brain area or from 6 to 100 times greater considering the CEA. This sample size is summarized in Table 3.
Expected sample size to obtain significant differences between the two groups using a conventional approach
\( {\overline{m}}_{\kern0.75em \mathrm{CTRL}}\left({\overline{m}}_{\mathrm{LPS}}\right) \)
CEA (μm2)
Cytoplasm area (μm2)
74.4 (149.4)
123.4 (177.0)
Where \( \overline{m} \) is the mean of criteria for the control (LPS) set of data, σ is the common standard deviation of the two groups and n is the expected number of samples to obtain a significant difference
Microglial morphological analysis is the historical technique to describe the microglial cells and also the only way to study these cells within the complex environment of the central nervous system [39]. In the literature, four major microglial phenotypes are usually distinguished based on distinct morphological criteria [11]: ramified (presenting a small cell body and numerous branched ramifications) that constitute about 90 % of the microglial pool between 1 and 3 months in the murine model and are considered to be the microglial "resting" state; primed (bigger cell body but unchanged ramification pattern compared to ramified phenotype); reactive (even bigger cell body, shorter, fewer and thicker ramifications); and amoeboid (two or less processes without any branch). Despite a probable link between morphology and function [40–42] and due to a large panel of slight microglial morphological changes [28], morphological analyses are subject to many hurdles, limiting their strength in terms of objectivity, precision and accuracy leading to subjective interpretations.
The objective of the present study was to develop a new powerful tool and method for describing microglial morphology and for assessing its variability among brain regions in order to further characterize and understand its behaviour in the context of development, homeostasis and disease. We confirmed the added value of our method by studying the changes related to the LPS challenge, a well-characterized model to study inflammation. This method is based on a clustering analysis integrating new developed and automatically acquired morphological indexes, the complexity index (CI) and the covered environment area (CEA), the area covered by microglial ramifications. These two criteria are critical since different experimental studies [43–45] highlighted certain kinetics not only in morphological changes after microglial stimulation including rapid microglial process growth, extension and reorientation towards injury but also in response to healthy central nervous system (CNS) environment [46–48].
For assessing the discriminatory power of our approach, we compared the clustering analysis to parametric and non-parametric statistical approaches. We found that the clustering analysis allowed us to identify different microglial phenotypes that are heterogeneously distributed among brain regions and presenting different behaviours under the LPS challenge. The parametric statistical tests failed to identify these sub-populations.
As a first step, we compared available methods that are manual, qualitative or semi-quantitative to ours. Differences arise in (i) a number of cells analysed 30 times greater than reported by previous studies [25–27], (ii) an objective quantification of the morphological parameters providing a reliability in numerical and resolving power and (iii) providing a significant savings in time with an analysis that is 10 times faster. Second, while the CX3CR1GFP/+ transgenic model is a standard tool to study microglial morphology in in vivo [49, 50] or ex vivo experiments [51, 52], we performed the same experiments based on Iba-1 expression in C57BL/6 mice and observed the same trends than those observed with the GFP model considering the main morphological criteria. Nonetheless, this approach allows to analyse 10 times less cells than the CX3CR1GFP/+ thanks to the immunohistochemistry techniques, confirming the interest of the use of a knock-in model.
The relevance of CI and CEA indexes, easily assessed with automated methods, is confirmed by their ability to identify morphological sub-types with a particular regional distribution and sensitivity to LPS. The lack of collinearity between these two derived indexes indicates that they provide specific morphological information that may reflect particular functional status. One may argue that for a given covered area, hypo- and hyper-ramified microglial cells have different roles.
The last methodological input concerns the analysis by k-means clustering, considered to be a simple but efficient algorithm to define sub-populations of identical phenotyped cells. It divides the data into k clusters, minimizing the squared distance between each data point to the centre of its cluster. The main interests of this algorithm are its speed and ease of interpretation, and it is particularly adapted for identifying sub-groups in a large dataset by cellular heterogeneity recognition [53]. The principal limitations are that it requires an a priori specification of the number of cluster centres and has a strong sensitivity to outliers and noise. Together with Kongsui and colleagues [54] in a recent study interested in the structural alterations of the microglial cells within the prefrontal cortex in rats following LPS injection, we do not find any statistical difference in the mean values of CI or CEA. Kongsui and colleagues raise the hypothesis (i) that microglial process alteration is a later phenomenon or (ii) that a substantially larger group of cells studied associated with improved analytical approaches may reveal differences. The results of our cluster analysis based on these indexes that identify major changes in the global and regional distribution of morphological sub-types support the second hypothesis. This discrepancy clearly illustrates the discriminating power of cluster analysis, when compared to parametric tests.
The regional heterogeneity of microglial morphology and the microglial effect of the LPS challenge have long been studied [1, 55]. In our study, we confirm this regionalization of the microglial distribution with a decrease in density, CI and CEA indexes from the frontal cortex to the cerebellum. Injection of LPS was associated with an increase in the cytoplasmic area and in the proportion of amoeboid cells. Kozlowski and Weimer found the same trend into the cortex in their study in 2012, correlated with an overexpression of Iba-1 and CD68 [24]. In a more original way, we observed a regional susceptibility to LPS thanks to the microglial morphology, as it is also observed at a protein level by various experimental studies [56, 57]. This regionalization of pathophysiological processes at cellular and proteic levels supports the clinical and behavioural specific responses to neurological challenges [58, 59]. A method like ours may contribute to assess the nature of the microenvironmental factors involved in microglial shape and reactivity [60, 61].
Since the cluster analysis enables the assessment of a large population of cells per animal, it dramatically reduces the number of animals needed for testing a hypothesis. For instance, it would have been necessary to sacrifice 6 to 100 times more mice to observe a statistical difference in CI (or CEA) between the LPS- and non-LPS-treated groups, using a parametric test. This is a major ethical advantage and in compliance with the European and American requests.1 Thanks to the large number of cells analysed in the same animal with our automated method, it is also possible to use an accurate statistical approach and consequently to dramatically improve the ethical considerations of experimental works.
Although we have found and confirmed the heterogeneity of microglial cells both in the resting and the inflammatory brain, a functional assay to correlate the morphology with function is still required. Many markers have been described to characterize different microglial activation stages, but the use of these markers only would not have been sufficiently precise, and more parameters are thus needed to further link morphology and microglial function. However, our results further confirm data from the literature as Kozlowski and Weimer showed it [24].
One of the strengths of this approach is that other functional or morphological parameters could be included in the future according to the focus of the experimenters to highlight different behaviour or improve the precision of the study. It is also possible that increasing the number of parameters could lead to the discovery of novel microglial cell states before, during and after pathological conditions.
Moreover, our technique benefits from the ease of accessibility and will allow all labs to use these parameters as a tool to characterize the microglial behaviour and further understand in a standardized manner its role in healthy and diseased condition.
Two main fields could benefit from this approach. First, through the use of optical sectioning microscopy, it is possible and easy with our method to work on a three-dimensional network. The appearance of clusters of complex cells or of similar activation states may confirm what is already highlighted in 2D or, on the contrary, may reveal new behaviours. Second, in subsequent studies, it would be useful to correlate the precise morphology of a cell that we are able to identify with its function, possibly using single cell techniques in situ after morphological analysis within tissue.
Our study presents an automated approach of morphological analysis coupled with a high-content statistical study that allows highlighting sub-populations of microglial cells, even in a healthy condition, counteracting the classical view of a homogenous resting state. These sub-populations present specific behaviour after neuroinflammatory challenge induced by LPS, allowing postulating a cellular specialization specified by its morphology. Moreover, our study confirms the need to work region by region considering this type of cells. This statistical approach may become the cornerstone of any study involving dendritic-shaped cells while seeking the reduction of the number of animals in accordance with the international guidelines for animal welfare.
CEA, Covered environment area; CI, Complexity index; CNS, Central nervous system; GFP, Green fluorescent protein; Iba-1, Ionized calcium binding adapter molecule 1; LPS, Lipopolysaccharide; OMERO, OME Remote Objects technology; PCA, Principal component analysis; SP, Sub-population
Directive 2010/63/EU of the European Parliament and of the Council.
The authors thank Pierre Rocheteau and Daniel Fiole for advice and comments on the manuscript.
This work was supported by the Fondation des Gueules Cassées, the Comité d'Interface SFAR (Société Française d'Anesthésie-Réanimation), SRLF (Société de Réanimation de Langue Française), INSERM and the Fondation Pierre Deniker. Franck Verdonk received financial support from ANRT (Association Nationale de la Recherche et de la Technologie) through an Air Liquide-CIFRE contract 2012/1315. The Imagopole is part of the France BioImaging infrastructure supported by the French National Research Agency (ANR-10-INSB-04-01, "Investments for the future") and is grateful for support from the Conseil de la Region Ile-de-France (programme Sesame 2007, project Imagopole, S.L. Shorte). SS's work was supported by the Wellcome Trust (095931/Z/11/Z; awardee Jason Swedlow, Dundee University and SLS local partner).
All relevant data are within the paper and its additional files. The raw data studied are stored in the OMERO image database not yet accessible from an external account. A set of associated analysis scripts and ImageJ macro can be sent on request to the corresponding authors.
FV and FC conceived and designed the study. FC and LF coordinated the study. PF performed technical support for animal preparation. FV and PR carried out and supervised the image acquisitions. AD conceived, designed and coordinated the image and statistical analysis. SS designed the automatic annotation tools. FV, FC, AD and TS wrote the manuscript. SLS, FAB, BP and ML contributed in the critical revision of the manuscript. All authors read and approved the final manuscript.
All protocols were reviewed by the Institut Pasteur, the competent authority, for compliance with the French and European regulations on Animal Welfare and with Public Health Service recommendations. This project has been reviewed and approved (# 2013-0044) by the Institut Pasteur Ethics Committee (C2EA 89 - CETEA).
Additional file 1: High-content analysis workflow overview, from acquisition to statistics for the seizure of a large number of brain regions of mice, their classification and associated statistical analysis. Acquisition pipeline: four regions of interest were scanned using a spinning disc confocal system (CV1000-Yokagama): striatum (S), frontal cortex (FC), hippocampus (H) and cerebellum (C) representing approximately 13 mm2 of the entire brain surface with a depth of 30 μm. The voxel size is equal to 0.19 × 0.19 × 2.0 μm3, respectively, for the X, Y and Z dimensions by stack. Analysis pipeline: using a macro implemented in ImageJ free software, each data stack was divided into three sub-volumes followed by a maximum projection for two-dimensional (2D) analysis. Each sub-volume (v1 to v3) was analysed by AcapellaTM software to extract morphological criteria for each microglial cell. The original image data and generated in situ 2D cartographies were stored in Image Database for visualization, sharing and clinical annotations. At the end, statistics were done with the data extracted from this analysis pipeline. (TIF 1480 kb)
Additional file 2: The complexity index as new morphological criteria. The complexity index (CI) for each microglial cell was defined by dividing two different criteria: the number of segments of each cell and the number of its primary ramifications. (A) A typical schematic microglial cell presenting a circular cell body area (in red) and some ramified processes. Each of them is composed of one primary ramification (white arrow) and several sub-ramifications separated by nodes (blue arrows). One segment is defined as the length of process between two nodes. Each segment is visible in a single colour. (B) In the left column, individual microglia based on GFP fluorescence appears in white outline. In the right column, schematic representation of the microglial cell characterized by its CI (white text). The scale bars equal 10 μm. (TIF 676 kb)
Additional file 3: Data analysis workflow with manual and semi-automated methods. The manual method has been performed using Fiji environment and selection drawing tools to measure the cell body area, the number of roots, ramifications and the CEA. The semi-automated method has been implemented in a Fiji macro using successively Analyze Particles, Skeletonize (2D/3D) and Sholl Analysis plugins. The scale bars equal 10 μm. (TIF 1987 kb)
Additional file 4: Descriptive statistics for microglial cells by condition. (PDF 13 kb)
Additional file 5: Characterization of microglial cells by morphological criteria based on Iba-1 expression. Two regions have been explored: hippocampus (H) and cerebellum (C) in the control or LPS conditions. The scatter plots illustrate, by analysed cell, the body area (in blue), cytoplasm area (in pink) and intensity (in grey), CI (in green) or CEA (in orange) characteristics for each animal in both groups. Data shown are means ± SD. The scale bars equal 10 μm. The Mann-Whitney test was used to compare the control and LPS groups (respectively, n = 5 and n = 6).*p < 0.05. (TIF 262 kb)
Additional file 6: Descriptive statistics for amoeboid cells by condition. (PDF 53 kb)
Additional file 7: The comparison of different quantitative analysis methods for microglial morphological criteria by cell. (A) The scatter plots illustrate, at a single cell resolution, the body area, CI or CEA characteristics for the automatic or manual method in the control and LPS groups. The Mann-Whitney test was used to compare the control and LPS groups (respectively, n = 5 and n = 6).*p < 0.05. (B) The correlation plots of two morphological criteria between semi-automatic and automatic analysis with cell body area (in blue) and CI (in green) at a single cell resolution. Values indicate the Spearman correlation coefficient. The lines represent the linear regression. (C) The table indicates the number of cells (Nb cells), the cell body area, the CI, the CEA and the mean analysis time per cell for each condition (control and LPS groups) and each region (hippocampus and cerebellum). (TIF 424 kb)
Additional file 8: Quantitative analysis of microglial cell morphology in the whole brain considering CI and CEA. The scatter plots illustrate, at a single cell resolution, the CI or CEA characteristics for each animal in both groups. ANOVA Kruskal-Wallis was used to test the inter-sample heterogeneity. Data shown are means ± SD. ****p < 0.0001. (TIF 438 kb)
Human Histopathology and Animal Models Unit, Infection and Epidemiology Department, Institut Pasteur, Paris, France
Imagopole - CITech, Institut Pasteur, Paris, France
ICU, Instituto de Pesquisa Clínica Evandro Chagas, Fundação Oswaldo Cruz, Rio de Janeiro, Brazil
Air Liquide Santé International, World Business Line Healthcare, Medical R&D, Paris-Saclay Research Center, 1 chemin de la Porte des Loges, Jouy-en-Josas, France
Department of Anaesthesiology and Surgical Intensive Care, Saint-Louis University Hospital of Paris, Paris, France
Department of Intensive Care, Raymond Poincare University Hospital, Garches, France
Laboratoire hospitalo-universitaire de Neuropathologie, Centre Hospitalier Sainte Anne, Paris, France
Versailles Saint Quentin University, Versailles, France
Paris Diderot University, Paris, France
Paris Descartes University, Sorbonne Paris Cité, Paris, France
TRIGGERSEP, F-CRIN Network, Toulouse, France
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CommonCrawl
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Nucleus (order theory)
In mathematics, and especially in order theory, a nucleus is a function $F$ on a meet-semilattice ${\mathfrak {A}}$ such that (for every $p$ in ${\mathfrak {A}}$):[1]
1. $p\leq F(p)$
2. $F(F(p))=F(p)$
3. $F(p\wedge q)=F(p)\wedge F(q)$
Every nucleus is evidently a monotone function.
Frames and locales
Usually, the term nucleus is used in frames and locales theory (when the semilattice ${\mathfrak {A}}$ is a frame).
Proposition: If $F$ is a nucleus on a frame ${\mathfrak {A}}$, then the poset $\operatorname {Fix} (F)$ of fixed points of $F$, with order inherited from ${\mathfrak {A}}$, is also a frame.[2]
References
1. Johnstone, Peter (1982), Stone Spaces, Cambridge University Press, p. 48, ISBN 978-0-521-33779-3, Zbl 0499.54001
2. Miraglia, Francisco (2006). An Introduction to Partially Ordered Structures and Sheaves. Polimetrica s.a.s. Theorem 13.2, p. 130. ISBN 9788876990359.
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The magnetic ray transform on Anosov surfaces
May 2015, 35(5): 1767-1800. doi: 10.3934/dcds.2015.35.1767
On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
Primitivo B. Acosta-Humánez 1, , J. Tomás Lázaro 2, , Juan J. Morales-Ruiz 3, and Chara Pantazi 2,
Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla, Colombia
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain, Spain
Department of Applied Mathematics, Technical University of Madrid, Madrid, Spain
Received September 2013 Revised September 2014 Published December 2014
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.
Keywords: Poincaré problem, rational first integral, Riccati equation, Differential Galois theory, integrating factor, Liouvillian solution., Liénard equation, Darboux theory of integrability.
Mathematics Subject Classification: Primary: 12H05; Secondary: 32S6.
Citation: Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767
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Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
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The ASKAP Variables and Slow Transients (VAST) Pilot Survey
Tara Murphy, David L. Kaplan, Adam J. Stewart, Andrew O'Brien, Emil Lenc, Sergio Pintaldi, Joshua Pritchard, Dougal Dobie, Archibald Fox, James K. Leung, Tao An, Martin E. Bell, Jess W. Broderick, Shami Chatterjee, Shi Dai, Daniele d'Antonio, Gerry Doyle, B. M. Gaensler, George Heald, Assaf Horesh, Megan L. Jones, David McConnell, Vanessa A. Moss, Wasim Raja, Gavin Ramsay, Stuart Ryder, Elaine M. Sadler, Gregory R. Sivakoff, Yuanming Wang, Ziteng Wang, Michael S. Wheatland, Matthew Whiting, James R. Allison, C. S. Anderson, Lewis Ball, K. Bannister, D. C.-J. Bock, R. Bolton, J. D. Bunton, R. Chekkala, A. P Chippendale, F. R. Cooray, N. Gupta, D. B. Hayman, K. Jeganathan, B. Koribalski, K. Lee-Waddell, Elizabeth K. Mahony, J. Marvil, N. M. McClure-Griffiths, P. Mirtschin, A. Ng, S. Pearce, C. Phillips, M. A. Voronkov
Journal: Publications of the Astronomical Society of Australia / Volume 38 / 2021
The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to $\sim\!5$ yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of $\sim\!162$ h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of $0.24\ \mathrm{mJy\ beam}^{-1}$ and angular resolution of $12-20$ arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. The total area of the pilot survey footprint is 5 131 square degrees, covering six distinct regions of the sky. An initial search of two of these regions, totalling 1 646 square degrees, revealed 28 highly variable and/or transient sources. Seven of these are known pulsars, including the millisecond pulsar J2039–5617. Another seven are stars, four of which have no previously reported radio detection (SCR J0533–4257, LEHPM 2-783, UCAC3 89–412162 and 2MASS J22414436–6119311). Of the remaining 14 sources, two are active galactic nuclei, six are associated with galaxies and the other six have no multi-wavelength counterparts and are yet to be identified.
The Evolutionary Map of the Universe pilot survey
Ray P. Norris, Joshua Marvil, J. D. Collier, Anna D. Kapińska, Andrew N. O'Brien, L. Rudnick, Heinz Andernach, Jacobo Asorey, Michael J. I. Brown, Marcus Brüggen, Evan Crawford, Jayanne English, Syed Faisal ur Rahman, Miroslav D. Filipović, Yjan Gordon, Gülay Gürkan, Catherine Hale, Andrew M. Hopkins, Minh T. Huynh, Kim HyeongHan, M. James Jee, Bärbel S. Koribalski, Emil Lenc, Kieran Luken, David Parkinson, Isabella Prandoni, Wasim Raja, Thomas H. Reiprich, Christopher J. Riseley, Stanislav S. Shabala, Jaimie R. Sheil, Tessa Vernstrom, Matthew T. Whiting, James R. Allison, C. S. Anderson, Lewis Ball, Martin Bell, John Bunton, T. J. Galvin, Neeraj Gupta, Aidan Hotan, Colin Jacka, Peter J. Macgregor, Elizabeth K. Mahony, Umberto Maio, Vanessa Moss, M. Pandey-Pommier, Maxim A. Voronkov
Published online by Cambridge University Press: 07 September 2021, e046
We present the data and initial results from the first pilot survey of the Evolutionary Map of the Universe (EMU), observed at 944 MHz with the Australian Square Kilometre Array Pathfinder (ASKAP) telescope. The survey covers $270 \,\mathrm{deg}^2$ of an area covered by the Dark Energy Survey, reaching a depth of 25–30 $\mu\mathrm{Jy\ beam}^{-1}$ rms at a spatial resolution of $\sim$ 11–18 arcsec, resulting in a catalogue of $\sim$ 220 000 sources, of which $\sim$ 180 000 are single-component sources. Here we present the catalogue of single-component sources, together with (where available) optical and infrared cross-identifications, classifications, and redshifts. This survey explores a new region of parameter space compared to previous surveys. Specifically, the EMU Pilot Survey has a high density of sources, and also a high sensitivity to low surface brightness emission. These properties result in the detection of types of sources that were rarely seen in or absent from previous surveys. We present some of these new results here.
The Rapid ASKAP Continuum Survey I: Design and first results
Australian SKA Pathfinder
D. McConnell, C. L. Hale, E. Lenc, J. K. Banfield, George Heald, A. W. Hotan, James K. Leung, Vanessa A. Moss, Tara Murphy, Andrew O'Brien, Joshua Pritchard, Wasim Raja, Elaine M. Sadler, Adam Stewart, Alec J. M. Thomson, M. Whiting, James R. Allison, S. W. Amy, C. Anderson, Lewis Ball, Keith W. Bannister, Martin Bell, Douglas C.-J. Bock, Russ Bolton, J. D. Bunton, A. P. Chippendale, J. D. Collier, F. R. Cooray, T. J. Cornwell, P. J. Diamond, P. G. Edwards, N. Gupta, Douglas B. Hayman, Ian Heywood, C. A. Jackson, Bärbel S. Koribalski, Karen Lee-Waddell, N. M. McClure-Griffiths, Alan Ng, Ray P. Norris, Chris Phillips, John E. Reynolds, Daniel N. Roxby, Antony E. T. Schinckel, Matt Shields, Chenoa Tremblay, A. Tzioumis, M. A. Voronkov, Tobias Westmeier
Published online by Cambridge University Press: 30 November 2020, e048
The Rapid ASKAP Continuum Survey (RACS) is the first large-area survey to be conducted with the full 36-antenna Australian Square Kilometre Array Pathfinder (ASKAP) telescope. RACS will provide a shallow model of the ASKAP sky that will aid the calibration of future deep ASKAP surveys. RACS will cover the whole sky visible from the ASKAP site in Western Australia and will cover the full ASKAP band of 700–1800 MHz. The RACS images are generally deeper than the existing NRAO VLA Sky Survey and Sydney University Molonglo Sky Survey radio surveys and have better spatial resolution. All RACS survey products will be public, including radio images (with $\sim$ 15 arcsec resolution) and catalogues of about three million source components with spectral index and polarisation information. In this paper, we present a description of the RACS survey and the first data release of 903 images covering the sky south of declination $+41^\circ$ made over a 288-MHz band centred at 887.5 MHz.
Pathways to past ways: a positive approach to routeways and mobility
Martin Bell, Jim Leary
Journal: Antiquity / Volume 94 / Issue 377 / October 2020
Published online by Cambridge University Press: 01 September 2020, pp. 1349-1359
Past trackways and the mobility associated with them have long been a neglected topic of research in Britain—a lack of attention attributable to the publication of Alfred Watkins's The old straight track in 1925. Yet, new interdisciplinary approaches, including from the perspectives of social theory and the sciences, can allow us to move forward. Here the authors offer the first steps towards a positive approach to understanding past mobility.
Further studies of neuroangiostrongyliasis (rat lungworm disease) in Australian dogs: 92 new cases (2010–2020) and results for a novel, highly sensitive qPCR assay
Rogan Lee, Tsung-Yu Pai, Richard Churcher, Sarah Davies, Jody Braddock, Michael Linton, Jane Yu, Erin Bell, Justin Wimpole, Anna Dengate, David Collins, Narelle Brown, George Reppas, Susan Jaensch, Matthew K. Wun, Patricia Martin, William Sears, Jan Šlapeta, Richard Malik
Journal: Parasitology / Volume 148 / Issue 2 / February 2021
Print publication: February 2021
The principal aim of this study was to optimize the diagnosis of canine neuroangiostrongyliasis (NA). In total, 92 cases were seen between 2010 and 2020. Dogs were aged from 7 weeks to 14 years (median 5 months), with 73/90 (81%) less than 6 months and 1.7 times as many males as females. The disease became more common over the study period. Most cases (86%) were seen between March and July. Cerebrospinal fluid (CSF) was obtained from the cisterna magna in 77 dogs, the lumbar cistern in f5, and both sites in 3. Nucleated cell counts for 84 specimens ranged from 1 to 146 150 cells μL−1 (median 4500). Percentage eosinophils varied from 0 to 98% (median 83%). When both cisternal and lumbar CSF were collected, inflammation was more severe caudally. Seventy-three CSF specimens were subjected to enzyme-linked immunosorbent assay (ELISA) testing for antibodies against A. cantonensis; 61 (84%) tested positive, titres ranging from <100 to ⩾12 800 (median 1600). Sixty-one CSF specimens were subjected to real-time quantitative polymerase chain reaction (qPCR) testing using a new protocol targeting a bioinformatically-informed repetitive genetic target; 53/61 samples (87%) tested positive, CT values ranging from 23.4 to 39.5 (median 30.0). For 57 dogs, it was possible to compare CSF ELISA serology and qPCR. ELISA and qPCR were both positive in 40 dogs, in 5 dogs the ELISA was positive while the qPCR was negative, in 9 dogs the qPCR was positive but the ELISA was negative, while in 3 dogs both the ELISA and qPCR were negative. NA is an emerging infectious disease of dogs in Sydney, Australia.
The GLEAM 4-Jy (G4Jy) Sample: I. Definition and the catalogue
Sarah V. White, Thomas M. O Franzen, Chris J. Riseley, O. Ivy Wong, Anna D. Kapińska, Natasha Hurley-Walker, Joseph R. Callingham, Kshitij Thorat, Chen Wu, Paul Hancock, Richard W. Hunstead, Nick Seymour, Jesse Swan, Randall Wayth, John Morgan, Rajan Chhetri, Carole Jackson, Stuart Weston, Martin Bell, Bi-Qing For, B. M. Gaensler, Melanie Johnston-Hollitt, André Offringa, Lister Staveley-Smith
Published online by Cambridge University Press: 01 June 2020, e018
The Murchison Widefield Array (MWA) has observed the entire southern sky (Declination, $\delta< 30^{\circ}$ ) at low radio frequencies, over the range 72–231MHz. These observations constitute the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (EGC) (Galactic latitude, $|b| >10^{\circ}$ ) to define the GLEAM 4-Jy (G4Jy) Sample. This is a complete sample of the 'brightest' radio sources ( $S_{\textrm{151\,MHz}}>4\,\text{Jy}$ ), the majority of which are active galactic nuclei with powerful radio jets. Crucially, low-frequency observations allow the selection of such sources in an orientation-independent way (i.e. minimising the bias caused by Doppler boosting, inherent in high-frequency surveys). We then use higher-resolution radio images, and information at other wavelengths, to morphologically classify the brightest components in GLEAM. We also conduct cross-checks against the literature and perform internal matching, in order to improve sample completeness (which is estimated to be $>95.5$ %). This results in a catalogue of 1863 sources, making the G4Jy Sample over 10 times larger than that of the revised Third Cambridge Catalogue of Radio Sources (3CRR; $S_{\textrm{178\,MHz}}>10.9\,\text{Jy}$ ). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I) synthesised beam ( $\sim2$ arcmin at 200MHz), and we label 67% of the sample as 'single', 26% as 'double', 4% as 'triple', and 3% as having 'complex' morphology at $\sim1\,\text{GHz}$ (45 arcsec resolution). We characterise the spectral behaviour of these objects in the radio and find that the median spectral index is $\alpha=-0.740 \pm 0.012$ between 151 and 843MHz, and $\alpha=-0.786 \pm 0.006$ between 151MHz and 1400MHz (assuming a power-law description, $S_{\nu} \propto \nu^{\alpha}$ ), compared to $\alpha=-0.829 \pm 0.006$ within the GLEAM band. Alongside this, our value-added catalogue provides mid-infrared source associations (subject to 6" resolution at 3.4 $\mu$ m) for the radio emission, as identified through visual inspection and thorough checks against the literature. As such, the G4Jy Sample can be used as a reliable training set for cross-identification via machine-learning algorithms. We also estimate the angular size of the sources, based on their associated components at $\sim1\,\text{GHz}$ , and perform a flux density comparison for 67 G4Jy sources that overlap with 3CRR. Analysis of multi-wavelength data, and spectral curvature between 72MHz and 20GHz, will be presented in subsequent papers, and details for accessing all G4Jy overlays are provided at https://github.com/svw26/G4Jy.
The GLEAM 4-Jy (G4Jy) Sample: II. Host galaxy identification for individual sources
Sarah V. White, Thomas M. O. Franzen, Chris J. Riseley, O. Ivy Wong, Anna D. Kapińska, Natasha Hurley-Walker, Joseph R. Callingham, Kshitij Thorat, Chen Wu, Paul Hancock, Richard W. Hunstead, Nick Seymour, Jesse Swan, Randall Wayth, John Morgan, Rajan Chhetri, Carole Jackson, Stuart Weston, Martin Bell, B. M. Gaensler, Melanie Johnston–Hollitt, André Offringa, Lister Staveley–Smith
The entire southern sky (Declination, $\delta< 30^{\circ}$ ) has been observed using the Murchison Widefield Array (MWA), which provides radio imaging of $\sim$ 2 arcmin resolution at low frequencies (72–231 MHz). This is the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we have previously used a combination of visual inspection, cross-checks against the literature, and internal matching to identify the 'brightest' radio-sources ( $S_{\mathrm{151\,MHz}}>4$ Jy) in the extragalactic catalogue (Galactic latitude, $|b| >10^{\circ}$ ). We refer to these 1 863 sources as the GLEAM 4-Jy (G4Jy) Sample, and use radio images (of ${\leq}45$ arcsec resolution), and multi-wavelength information, to assess their morphology and identify the galaxy that is hosting the radio emission (where appropriate). Details of how to access all of the overlays used for this work are available at https://github.com/svw26/G4Jy. Alongside this we conduct further checks against the literature, which we document here for individual sources. Whilst the vast majority of the G4Jy Sample are active galactic nuclei with powerful radio-jets, we highlight that it also contains a nebula, two nearby, star-forming galaxies, a cluster relic, and a cluster halo. There are also three extended sources for which we are unable to infer the mechanism that gives rise to the low-frequency emission. In the G4Jy catalogue we provide mid-infrared identifications for 86% of the sources, and flag the remainder as: having an uncertain identification (129 sources), having a faint/uncharacterised mid-infrared host (126 sources), or it being inappropriate to specify a host (2 sources). For the subset of 129 sources, there is ambiguity concerning candidate host-galaxies, and this includes four sources (B0424–728, B0703–451, 3C 198, and 3C 403.1) where we question the existing identification.
The search for planet and planetesimal transits of white dwarfs with the Zwicky Transient Facility
Keaton J. Bell
Journal: Proceedings of the International Astronomical Union / Volume 15 / Issue S357 / October 2019
Published online by Cambridge University Press: 09 October 2020, pp. 37-40
Planetary materials orbiting white dwarf stars reveal the ultimate fate of the planets of the Solar System and all known transiting exoplanets. Observed metal pollution and infrared excesses from debris disks support that planetary systems or their remnants are common around white dwarf stars; however, these planets are difficult to detect since a very high orbital inclination angle is required for a small white dwarf to be transited, and these transits have very short (minute) durations. The low odds of catching individual transits could be overcome by a sufficiently wide and fast photometric survey. I demonstrate that, by obtaining over 100 million images of white dwarf stars with 30-second exposures in its first three years, the Zwicky Transient Facility (ZTF) is likely to record the first exoplanetary transits of white dwarfs, as well as new systems of transiting, disintegrating planetesimals. In these proceedings, I describe my project strategy to discover these systems using the ZTF data.
Accreting pulsating white dwarfs: Probing heating and rotation
Paula Szkody, Boris Gänsicke, Odette Toloza, Patrick Godon, Edward Sion, Stella Kafka, Keaton Bell, Zachary Vanderbosch, AAVSO observers
The white dwarfs in close, interacting binaries provide a natural laboratory for exploring the effects of heating and angular momentum from the accreting material arriving on the surface from the companion. This study is even more fruitful when it involves a pulsating white dwarf, which allows an exploration of the effects of the accretion on the interior as well as in the atmosphere. The last decade has seen the accomplishment of UV (HST) and optical (ground) studies of several accreting white dwarfs that have undergone a dwarf nova outburst that heated the white dwarf and subsequently returned to its quiescent temperature. The most recent study involves V386 Ser, which underwent its first known outburst in January 2019, after 19 years at quiescence. V386 Ser is unique in that its quiescent pulsation shows a triplet, with spacing indicating a rotation period of 4.8 days, extremely slow for accreting white dwarfs. This paper presents the result of HST ultraviolet spectra obtained 7 months after its outburst that shows the first clear confirmation of shorter period modes being driven following the heating from a dwarf nova outburst.
Longitudinal relationships among depressive symptoms and three types of memory self-report in cognitively intact older adults
Nikki L. Hill, Jacqueline Mogle, Sakshi Bhargava, Tyler Reed Bell, Iris Bhang, Mindy Katz, Martin J. Sliwinski
Journal: International Psychogeriatrics / Volume 32 / Issue 6 / June 2020
Print publication: June 2020
The current study examined whether self-reported memory problems among cognitively intact older adults changed concurrently with, preceded, or followed depressive symptoms over time.
Data were collected annually via in-person comprehensive medical and neuropsychological examinations as part of the Einstein Aging Study.
Community-dwelling older adults in an urban, multi-ethnic area of New York City were interviewed.
The current study included a total of 1,162 older adults (Mage = 77.65, SD = 5.03, 63.39% female; 74.12% White). Data were utilized from up to 11 annual waves per participant.
Multilevel modeling tested concurrent and lagged associations between three types of memory self-report (frequency of memory problems, perceived one-year decline, and perceived ten-year decline) and depressive symptoms.
Results showed that self-reported frequency of memory problems covaried with depressive symptoms only in participants who were older at baseline. Changes in perceived one-year and ten-year memory decline were related to changes in depressive symptoms across all ages. Depressive symptoms increased the likelihood of perceived ten-year memory decline the next year; however, perceived ten-year memory decline did not predict future depressive symptoms. Additionally, no significant temporal relationship was observed between depressive symptoms and self-reported frequency of memory problems or perceived one-year memory decline.
Our findings highlight the importance of testing the unique associations of different types of self-reported memory problems with depressive symptoms.
eC-CLEM: Flexible Multidimensional Registration Software for Correlative Microscopies with Refined Accuracy Mapping
Xavier Heiligenstein, Perrine Paul-Gilloteaux, Martin Belle, Graça Raposo, Jean Salamero
The HPM Live μ–From Live Cell Imaging to High Pressure Freezing in Less than 2 Seconds for Correlative Microscopy Approaches
Xavier Heiligenstein, Martin Belle, Frederic Eyraud, Graca Raposo, Jean Salamero, Jerome Heiligenstein
Low-Frequency Spectral Energy Distributions of Radio Pulsars Detected with the Murchison Widefield Array
Murchison Widefield Array
Tara Murphy, David L. Kaplan, Martin E. Bell, J. R. Callingham, Steve Croft, Simon Johnston, Dougal Dobie, Andrew Zic, Jake Hughes, Christene Lynch, Paul Hancock, Natasha Hurley-Walker, Emil Lenc, K. S. Dwarakanath, B.-Q. For, B. M. Gaensler, L. Hindson, M. Johnston-Hollitt, A. D. Kapińska, B. McKinley, J. Morgan, A. R. Offringa, P. Procopio, L. Staveley-Smith, R. Wayth, C. Wu, Q. Zheng
Published online by Cambridge University Press: 26 April 2017, e020
We present low-frequency spectral energy distributions of 60 known radio pulsars observed with the Murchison Widefield Array telescope. We searched the GaLactic and Extragalactic All-sky Murchison Widefield Array survey images for 200-MHz continuum radio emission at the position of all pulsars in the Australia Telescope National Facility (ATNF) pulsar catalogue. For the 60 confirmed detections, we have measured flux densities in 20 × 8 MHz bands between 72 and 231 MHz. We compare our results to existing measurements and show that the Murchison Widefield Array flux densities are in good agreement.
Using experimental archaeology and micromorphology to reconstruct timber-framed buildings from Roman Silchester: a new approach
Rowena Y. Banerjea, Michael Fulford, Martin Bell, Amanda Clarke, Wendy Matthews
Published online by Cambridge University Press: 09 October 2015, pp. 1174-1188
Determining the internal layout of archaeological structures and their uses has always been challenging, particularly in timber-framed or earthen-walled buildings where doorways and divisions are difficult to trace. In temperate conditions, soil-formation processes may hold the key to understanding how buildings were used. The abandoned Roman town of Silchester, UK, provides a case study for testing a new approach that combines experimental archaeology and micromorphology. The results show that this technique can provide clarity to previously uncertain features of urban architecture.
Heroin on trial: Systematic review and meta-analysis of randomised trials of diamorphine-prescribing as treatment for refractory heroin addiction
John Strang, Teodora Groshkova, Ambros Uchtenhagen, Wim van den Brink, Christian Haasen, Martin T. Schechter, Nick Lintzeris, James Bell, Alessandro Pirona, Eugenia Oviedo-Joekes, Roland Simon, Nicola Metrebian
Journal: The British Journal of Psychiatry / Volume 207 / Issue 1 / July 2015
Published online by Cambridge University Press: 02 January 2018, pp. 5-14
Supervised injectable heroin (SIH) treatment has emerged over the past 15 years as an intensive treatment for entrenched heroin users who have not responded to standard treatments such as oral methadone maintenance treatment (MMT) or residential rehabilitation.
To synthesise published findings for treatment with SIH for refractory heroin-dependence through systematic review and meta-analysis, and to examine the political and scientific response to these findings.
Randomised controlled trials (RCTs) of SIH treatment were identified through database searching, and random effects pooled efficacy was estimated for SIH treatment. Methodological quality was assessed according to criteria set out by the Cochrane Collaboration.
Six RCTs met the inclusion criteria for analysis. Across the trials, SIH treatment improved treatment outcome, i.e. greater reduction in the use of illicit 'street' heroin in patients receiving SIH treatment compared with control groups (most often receiving MMT).
SIH is found to be an effective way of treating heroin dependence refractory to standard treatment. SIH may be less safe than MMT and therefore requires more clinical attention to manage greater safety issues. This intensive intervention is for a patient population previously considered unresponsive to treatment. Inclusion of this low-volume, high-intensity treatment can now improve the impact of comprehensive healthcare provision.
The Murchison Widefield Array Commissioning Survey: A Low-Frequency Catalogue of 14 110 Compact Radio Sources over 6 100 Square Degrees
Natasha Hurley-Walker, John Morgan, Randall B. Wayth, Paul J. Hancock, Martin E. Bell, Gianni Bernardi, Ramesh Bhat, Frank Briggs, Avinash A. Deshpande, Aaron Ewall-Wice, Lu Feng, Bryna J. Hazelton, Luke Hindson, Daniel C. Jacobs, David L. Kaplan, Nadia Kudryavtseva, Emil Lenc, Benjamin McKinley, Daniel Mitchell, Bart Pindor, Pietro Procopio, Divya Oberoi, André Offringa, Stephen Ord, Jennifer Riding, Judd D. Bowman, Roger Cappallo, Brian Corey, David Emrich, B. M. Gaensler, Robert Goeke, Lincoln Greenhill, Jacqueline Hewitt, Melanie Johnston-Hollitt, Justin Kasper, Eric Kratzenberg, Colin Lonsdale, Mervyn Lynch, Russell McWhirter, Miguel F. Morales, Edward Morgan, Thiagaraj Prabu, Alan Rogers, Anish Roshi, Udaya Shankar, K. Srivani, Ravi Subrahmanyan, Steven Tingay, Mark Waterson, Rachel Webster, Alan Whitney, Andrew Williams, Chris Williams
We present the results of an approximately 6 100 deg2 104–196 MHz radio sky survey performed with the Murchison Widefield Array during instrument commissioning between 2012 September and 2012 December: the MWACS. The data were taken as meridian drift scans with two different 32-antenna sub-arrays that were available during the commissioning period. The survey covers approximately 20.5 h < RA < 8.5 h, − 58° < Dec < −14°over three frequency bands centred on 119, 150 and 180 MHz, with image resolutions of 6–3 arcmin. The catalogue has 3 arcmin angular resolution and a typical noise level of 40 mJy beam− 1, with reduced sensitivity near the field boundaries and bright sources. We describe the data reduction strategy, based upon mosaicked snapshots, flux density calibration, and source-finding method. We present a catalogue of flux density and spectral index measurements for 14 110 sources, extracted from the mosaic, 1 247 of which are sub-components of complexes of sources.
CHAPTER VI: REPORTS on DIVISION, COMMISSION, and WORKING GROUP MEETINGS
George Kaplan, Catherine Hohenkerk, Toshio Fukushima, Jean-Eudes Arlot, John A. Bangert, Steven A. Bell, William Folkner, Martin Lara, Elena V. Pitjeva, Sean E. Urban, Jan Vondrak
Journal: Proceedings of the International Astronomical Union / Volume 10 / Issue T28B / August 2013
Published online by Cambridge University Press: 13 August 2015, pp. 77-82
The triennial meeting of Commission 4 was attended by 16 people. All of the presentations from the meeting are provided on the commission website at http://www.iaucom4.org/c4docs.html, so this report provides only summaries.
Radio Continuum Surveys with Square Kilometre Array Pathfinders
Square Kilometre Array
Ray P. Norris, J. Afonso, D. Bacon, Rainer Beck, Martin Bell, R. J. Beswick, Philip Best, Sanjay Bhatnagar, Annalisa Bonafede, Gianfranco Brunetti, Tamás Budavári, Rossella Cassano, J. J. Condon, Catherine Cress, Arwa Dabbech, I. Feain, Rob Fender, Chiara Ferrari, B. M. Gaensler, G. Giovannini, Marijke Haverkorn, George Heald, Kurt Van der Heyden, A. M. Hopkins, M. Jarvis, Melanie Johnston-Hollitt, Roland Kothes, Huib Van Langevelde, Joseph Lazio, Minnie Y. Mao, Alejo Martínez-Sansigre, David Mary, Kim Mcalpine, E. Middelberg, Eric Murphy, P. Padovani, Zsolt Paragi, I. Prandoni, A. Raccanelli, Emma Rigby, I. G. Roseboom, H. Röttgering, Jose Sabater, Mara Salvato, Anna M. M. Scaife, Richard Schilizzi, N. Seymour, Dan J. B. Smith, Grazia Umana, G.-B. Zhao, Peter-Christian Zinn
Published online by Cambridge University Press: 27 March 2013, e020
In the lead-up to the Square Kilometre Array (SKA) project, several next-generation radio telescopes and upgrades are already being built around the world. These include APERTIF (The Netherlands), ASKAP (Australia), e-MERLIN (UK), VLA (USA), e-EVN (based in Europe), LOFAR (The Netherlands), MeerKAT (South Africa), and the Murchison Widefield Array. Each of these new instruments has different strengths, and coordination of surveys between them can help maximise the science from each of them. A radio continuum survey is being planned on each of them with the primary science objective of understanding the formation and evolution of galaxies over cosmic time, and the cosmological parameters and large-scale structures which drive it. In pursuit of this objective, the different teams are developing a variety of new techniques, and refining existing ones. To achieve these exciting scientific goals, many technical challenges must be addressed by the survey instruments. Given the limited resources of the global radio-astronomical community, it is essential that we pool our skills and knowledge. We do not have sufficient resources to enjoy the luxury of re-inventing wheels. We face significant challenges in calibration, imaging, source extraction and measurement, classification and cross-identification, redshift determination, stacking, and data-intensive research. As these instruments extend the observational parameters, we will face further unexpected challenges in calibration, imaging, and interpretation. If we are to realise the full scientific potential of these expensive instruments, it is essential that we devote enough resources and careful study to understanding the instrumental effects and how they will affect the data. We have established an SKA Radio Continuum Survey working group, whose prime role is to maximise science from these instruments by ensuring we share resources and expertise across the projects. Here we describe these projects, their science goals, and the technical challenges which are being addressed to maximise the science return.
VAST: An ASKAP Survey for Variables and Slow Transients
TARA MURPHY, SHAMI CHATTERJEE, DAVID L. KAPLAN, JAY BANYER, MARTIN E. BELL, HAYLEY E. BIGNALL, GEOFFREY C. BOWER, ROBERT A. CAMERON, DAVID M. COWARD, JAMES M. CORDES, STEVE CROFT, JAMES R. CURRAN, S. G. DJORGOVSKI, SEAN A. FARRELL, DALE A. FRAIL, B. M. GAENSLER, DUNCAN K. GALLOWAY, BRUCE GENDRE, ANNE J. GREEN, PAUL J. HANCOCK, SIMON JOHNSTON, ATISH KAMBLE, CASEY J. LAW, T. JOSEPH W. LAZIO, KITTY K. LO, JEAN-PIERRE MACQUART, NANDA REA, UMAA REBBAPRAGADA, CORMAC REYNOLDS, STUART D. RYDER, BRIAN SCHMIDT, ROBERTO SORIA, INGRID H. STAIRS, STEVEN J. TINGAY, ULF TORKELSSON, KIRI WAGSTAFF, MARK WALKER, RANDALL B. WAYTH, PETER K. G. WILLIAMS
Published online by Cambridge University Press: 15 February 2013, e006
The Australian Square Kilometre Array Pathfinder (ASKAP) will give us an unprecedented opportunity to investigate the transient sky at radio wavelengths. In this paper we present VAST, an ASKAP survey for Variables and Slow Transients. VAST will exploit the wide-field survey capabilities of ASKAP to enable the discovery and investigation of variable and transient phenomena from the local to the cosmological, including flare stars, intermittent pulsars, X-ray binaries, magnetars, extreme scattering events, interstellar scintillation, radio supernovae, and orphan afterglows of gamma-ray bursts. In addition, it will allow us to probe unexplored regions of parameter space where new classes of transient sources may be detected. In this paper we review the known radio transient and variable populations and the current results from blind radio surveys. We outline a comprehensive program based on a multi-tiered survey strategy to characterise the radio transient sky through detection and monitoring of transient and variable sources on the ASKAP imaging timescales of 5 s and greater. We also present an analysis of the expected source populations that we will be able to detect with VAST.
The North of England Survey of Twin and Multiple Pregnancy
Svetlana V. Glinianaia, Judith Rankin, Stephen N. Sturgiss, Martin P. Ward Platt, Danielle Crowder, Ruth Bell
Journal: Twin Research and Human Genetics / Volume 16 / Issue 1 / February 2013
The population-based Northern Survey of Twin and Multiple Pregnancy (NorSTAMP, formerly the Multiple Pregnancy Register) has collected data since 1998 on all multiple pregnancies in North of England (UK) from the earliest point of ascertainment in pregnancy. This paper updates recent developments to the NorSTAMP and presents some early mortality data from the first 10 years of data collection (1998–2007). Since 2005, mothers have been asked to give explicit consent for their identifiable data to be held by the survey, in line with changing guidance and legal frameworks for identifiable data. In 2009, regional standards of care for multiple pregnancies were developed, agreed, and disseminated. During 1998–2007, 4,865 twin maternities (pregnancies with at least one live birth or stillbirth) were registered, with an average twinning rate of 14.9 per 1,000 maternities. The overall stillbirth and neonatal mortality rates in twins were 18.0/1,000 births and 23.0/1,000 live births respectively. Stillbirth and neonatal mortality rates were significantly higher in monochorionic than dichorionic twins: 44.4 versus 12.2 per 1,000 births (relative risk [RR] 3.6, 95% Confidence Intervals [CI] 2.6–5.1), and 32.4 versus 21.4 per 1,000 live births (RR 1.5, 95% CI 1.04–2.2) respectively. There was no significant improvement during this period in either stillbirth or neonatal mortality rates in either chorionicity group. This population-based survey is an important source of data on multiple pregnancies, which allows monitoring of trends in multiple birth rates and pregnancy losses, providing essential information to support improvements in clinical care and for epidemiological research.
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Optimum apodization profile for chirped fiber Bragg gratings based chromatic dispersion compensator
Y. T. Aladadi1,
A. F. Abas1,2 &
M. T. Alresheedi1
In this paper, we optimized the apodization profile to improve the dispersion compensation performance of the chirped fiber Bragg gratings (CFBGs). Half tanh half uniform (HTHU), half exponential half uniform (HEHU), and half hamming half uniform (HHHU) apodization profiles were evaluated at 2000 nm wavelength. At this wavelength, Hollow-Core Photonic Crystal Fiber (HC-PCF) will be the targeted fiber for the future implementation. In this work, our aim is to obtain the dispersion compensator design with minimum average group delay ripple (GDR) and maximum Full Width Half Maximum (FWHM) bandwidth. The result shows that the best FWHM bandwidth is obtained by using HTHU profile that is approximately 96.27 %. In term of GDR, all apodization profiles show similar performance.
Exploring 2000 nm transmission window in optical communication system is one of important solutions to increase the bandwidth. To explore this transmission window, several measures need to be taken in optimizing dispersion compensation for this window. Hollow-Core Photonic Crystal Fiber (HC-PCF) was introduced as the transmission fiber, which is dominated by waveguide dispersion [1]. The Group Velocity Dispersion (GVD) of HC-PCF can be compensated at a given wavelength by scaling the fiber dimensions. However, this method is costly and complex. The dispersion slope of HC-PCFs is very steep. Consequently the range of zero dispersion wavelengths is very small. This means that without Chromatic Dispersion Compensator (CDC), HC-PCF can only support a small number of Dense Wavelength Division Multiplexing (DWDM) channels. One of the methods to compensate HC-PCF's GVD is to use apodized Chirped Fiber Bragg Gratings (CFBG) [2]. Several researches focused on optimizing the apodization profile to improve the performance of CDC at 1.55 μm wavelength [3–5]. Selection of the best parameters for the CFBG such as grating length, chirp period and index variation is very important. Apodization is a crucial procedure to reduce the group delay ripple that influences by chirped FBG dispersion characteristics [2, 6, 7]. The worst effect of apodization is losses in the bandwidth, which is very crucial.
In this paper, a simulation by using Matlab is conducted to evaluate three apodization profiles namely Half Tanh Half Uniform (HTHU), Half Exponential Half Uniform (HEHU), and Half Hamming Half Uniform (HHHU) profiles. The average group delay ripple and the bandwidth's variation are monitored to determine the optimum CDC performance at 2000 nm wavelength.
In order to design a CFBG with specified reflectivity and bandwidth, we use an objective reflectivity response as a reference for the designing process. We set 100 % reflectivity, and 0.8 nm Full Width Half Maximum (FWHM) bandwidth as the target as shown in Eq. 1:
$$ {R}_{Obj}=\left\{\begin{array}{c}\hfill 1\kern0.35em 2019.6\ nm\le \lambda \le 2020.4\ nm\hfill \\ {}0\kern6.55em elsewhere\hfill \end{array}\right. $$
The optimum value of refractive index modulation amplitude, Δn, CFBG length, \( {L}_{\mathit{\mathsf{g}}} \), and the chirp parameter, C, is obtained by using the error estimation equation (Eq. 2) where error, E is defined as
$$ E={\displaystyle \sum_{i=1}^N}{\left({R}_{Obj}\left({\lambda}_i\right)-{R}_{Sim}\left({\lambda}_i\right)\right)}^2=0,\ i=1,2,\dots, N $$
where λ i is the discrete set of wavelengths, and R sim is the simulated reflectivity.
Referring to Eq. (2), our simulation shows that the parameters that give the minimum E are \( {L}_{\mathit{\mathsf{g}}}=108\ mm \), Δn = 9e − 5 and C = − 0.23e − 1 nm/cm.
Figure 1 shows the simulated reflectivity response (blue) of a non-apodized CFBG, and the corresponding group delay (red). It is shown that the reflectivity nearly reaches 100 % and the 3-dB bandwidth is about 0.8 nm. The slope of the group delay indicates the amount of chromatic dispersion that can be compensated by our CDC that is about −1500 ps/nm. The value of group delay ripple (GDR) is too large, which is around ± 30 ps. Therefore, there is an urgent need to reduce GDR while maintaining the bandwidth as optimum as possible.
The simulated reflectivity (blue), the group delay (red)
To suppress the GDR, several apodization profiles have been tested [8]. In this paper, we optimize the following apodization profiles in which we consider the bandwidth to be the effective element as GDR.
Half Tanh- half uniform profile (HTHU):
$$ {\overline{\delta n}}_{eff}\ (z) = \left\{\begin{array}{c}\hfill \frac{ \tan {h}^2\left(2a\frac{z}{L_g}\right)}{tan{h}^2(a)},\ 0\le z < \frac{L_g}{2}\hfill \\ {}\ 1\kern5.65em ,\ \frac{L_g}{2}\le z\le {L}_g\hfill \end{array}\ \right. $$
Half Exponential-half uniform profile (HEHU):
$$ {\overline{\delta n}}_{eff}\ (z)=\left\{\begin{array}{c}\hfill exp\left(-a\frac{{\left(z-\frac{L_g}{2}\right)}^4}{{L_g}^4}\right), 0\le z < \frac{L_g}{2}\hfill \\ {}1\kern8.3em ,\ \frac{L_g}{2}\le z\le {L}_g\hfill \end{array}\right. $$
Half Hamming-half uniform profile (HHHU):
$$ {\overline{\delta n}}_{eff}\ (z) = \left\{\begin{array}{c}\hfill \frac{\left(1+ acos\left(\pi \frac{2z-{L}_g}{L_g}\right)\right)}{\left(1+a\right)},\ 0\le z < \frac{L_g}{2}\hfill \\ {}1\kern9.35em ,\ \frac{\ {L}_g}{2}\le z\le {L}_g\hfill \end{array}\right. $$
Sharpness parameter a is used to control the sharpness of the apodization profile.
Reflectivity and group delay vary with the refractive index change, \( {\overline{\delta n}}_{eff}\ (z) \). Eq. 3, Eq. 4 and Eq. 5, show that the sharpness parameter a is the variable that determines the value of \( {\overline{\delta n}}_{eff}\ (z) \). Therefore, it can also be used to optimize the reflectivity and GDR. Varying parameter α will vary both reflectivity FWHM bandwidth, and GDR. However, minimizing GDR will be accompanied by the loss of reflectivity bandwidth. Figure 2 (a) and (b) respectively show the effect of implementing HTHU profile (Eq. 3) with α = 1 to the reflectivity bandwidth and GDR. It can be noticed that the bandwidth reduces significantly as the targeted GDR is achieved. This trade-off shows that the optimization of the apodization profile is very important to obtain optimum performance.
Apodized and unapodized (a) reflectivity and (b) group delay
Selecting the optimum point of α solves a tradeoff between the maximum FWHM bandwidth and minimum GDR. Thus, we rely on a new method to manage this conflict. This method depends on the relationship between the normalized apodized mean GDR and the apodized FWHM bandwidth. Equations (6) and (7) represent the normalized GDR, F ripple and normalized FWHM bandwidth, F bandwidth , where A apod and A unapod are the apodized and unapodized peak to peak GDR amplitude respectively. BW apod and BW unapod are the apodized and unapodized FHWM bandwidth respectively.
$$ {F}_{ripple}=\frac{A_{apod}(a)}{A_{unapod}} $$
$$ {F}_{bandwidth}=\frac{B{W}_{apod}(a)}{B{W}_{unapod}} $$
The relationship between F ripple and F bandwidth for the HEHU apodization profile (Eq. 4) are shown in Fig. 3.
Group delay ripple factor ratio versus bandwidth factor ratio for HEHU apodization (Eq. 4)
To obtain the optimum value of α, the distance d which is defined by Eq. (8) is evaluated.
$$ d(a)=\sqrt{\left({\left({F}_{ripple}(a)\right)}^2+\right(1-{\left({F}_{bandwidth}(a)\right)}^2} $$
Distance d is measured from the point F ripple = 0 and F bandwidth = 1, to any point on the curve, as depicted in Fig. 3.
The optimum point of α can be obtained at the minimum value of d, denoted as d min . Figure 4 (a), Fig. 4 (b) and Fig. 4 (c) show the values of d at different value of sharpness parameter for different apodization profile. d 1 , d 2 and d 3 are the shortest distance for HTHU, HEHU and HHHU profiles respectively. These results occur at a = 13 for HTHU profile, a = 47 for HEHU and a = 1 for HHHU (d 1 = 0.1209, d 2 = 0.1935, d 3 = 0.2556).
The distance, d, versus the sharpness parameter, a for (a) HTHU profile, (b) HEHU profile, (c) HHHU profile
Figure 5 shows the relationship between F bandwidth and F ripple for HTHU, HEHU and HHHU apodization profiles during the optimization process. All apodization profiles manage to reach the minimum value of F ripple of around 0.12. In contrast, different values of bandwidth are achieved for different apodization profile. The FWHM bandwidth for HTHU profile is the closest to the targeted value (0.8 nm), which is approximately 96.27 %. HEHU profile obtained 85.56 % and 77.71 % is recorded for HHHU profile. Therefore, we decided that HTHU is the best among the 3 optimized apodization profiles, and is the best to be used in the design of the CFBG.
Group delay ripple factor ratio versus bandwidth factor ratio for three apodization functions
Figure 6 shows the reflectivity response of CFBG and corresponding group delay after applying the HTHU apodization profile with α = 13. In comparison to Fig. 1 (unapodized case), the FWHM bandwidth is reduced by around 0.03 nm. This reduction is acceptable as it may not significantly affect the characteristic of the system. Figure 7 and Fig. 8 show the reflectivity responses and corresponding group delay after applying HEHU and HHHU apodization profiles respectively. The values of α are 47 and 1 for HEHU and HHHU apodization profiles respectively. The percentage bandwidth loss in case of HEHU and HHHU are 14.5 % (0.115 nm) and 22 % (0.18 nm) respectively. On the other hand, with the best optimized apodization profiles, the GDR is reduced to the minimum extent. These results show that our target to design a CFBG that can become the CDC with optimum performance has been achieved.
The simulated reflectivity (blue color), the group delay (red color) with optimized HTHU apodization profile
The simulated reflectivity (blue color), the group delay (red color) with optimized HEHU apodization profile
The simulated reflectivity (blue color), the group delay (red color) with optimized HHHU apodization profile
HTHU profile is found to be the best apodization profile in designing high performance CDC. The optimization method used in this work is a good technique to evaluate and optimize the apodization profile. Our results show that by using HTHU profile, CDC working at 2000 nm wavelength range can be realized with optimum performance. Apart from the tested wavelength, with minimum modification, this optimization method can also be used for other bands. This report may become very important for the CDC researchers especially those who have interest in exploring the future 2000 nm bands.
Russell, PSJ: Photonic-crystal fibers. J Lightwave Technol. 24, 4729–4749 (2006)
Ennser, K, Zervas, MN, Laming, R: Optimization of apodized linearly chirped fiber gratings for optical communications. Quantum Electron. 34, 770–778 (1998)
Mohammed, NA, Ali, TA, Aly, MH: Performance optimization of apodized FBG-based temperature sensors in single and quasi-distributed DWDM systems with new and different apodization profiles. AIP Advances. 3, 122–125 (2013)
Zhang, H: A novel method of optimal apodization selection for chirped fiber Bragg gratings. Optik Int J Light Electron Optics. 125, 1646–1649 (2014)
Osuch, T, Markowski, K, Jedrzejewski, K: Numerical model of tapered fiber Bragg gratings for comprehensive analysis and optimization of their sensing and strain-induced tunable dispersion properties. Appl Optics. 54, 5525–5533 (2015)
Fernandez, P, Aguado, J, Blas, J, Duran, R, de Miguel, I, Duran, J, Lorenzo, R, Abril, E: Analysis and optimisation of the apodisation sharpness for linearly chirped dispersion compensating gratings. IEE Proc Optoelectron. 151, 69–73 (2004)
Williams, J, Bennion, I, Doran, N: The design of in-fiber Bragg grating systems for cubic and quadratic dispersion compensation. Optics Commun. 116, 62–66 (1995)
Erdogan, T: Fiber grating spectra. Lightwave Technol. 15, 1277–1294 (1997)
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for supporting this work through RG-1437-008 Research Group Fund.
Department of Electrical Engineering, College of Engineering, King Saud University, Riyadh, 11421, Saudi Arabia
Y. T. Aladadi, A. F. Abas & M. T. Alresheedi
Wireless and Photonics Research Center, Department of Computer and Communication Systems Engineering, Faculty of Engineering, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia
A. F. Abas
Y. T. Aladadi
M. T. Alresheedi
Correspondence to A. F. Abas.
YTA formulated minimum distance optimization equation and carried out all the simulations. AFA proposed the concept of the study, involves in data analysis and overall manuscript preparation. MTA formulated the error estimation equation and all optimized apodization profiles. All authors read and approved the final manuscript.
Aladadi, Y.T., Abas, A.F. & Alresheedi, M.T. Optimum apodization profile for chirped fiber Bragg gratings based chromatic dispersion compensator. J. Eur. Opt. Soc.-Rapid Publ. 12, 6 (2016). https://doi.org/10.1186/s41476-016-0006-8
Chromatic dispersion compensator
Chirped FBG
HC-PCF
Apodization
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CommonCrawl
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Trance opera—Spente le Stelle • be dramatic • more quotes And whatever I do will become forever what I've done. • Wislawa Szymborska • don't rehearse • more quotes Feel the vibe, feel the terror, feel the pain • Hooverphonic • Mad about you, orchestrally. • more quotes Love itself became the object of her love. • Jonathan Safran Foer • count sadnesses • more quotes listen; there's a hell of a good universe next door: let's go. • e.e. cummings • go there • more quotes Twenty — minutes — maybe — more. • Naomi • choose four words • more quotes Tango is a sad thought that is danced. • Enrique Santos Discépolo • think & dance • more quotes Lips that taste of tears, they say, are the best for kissing. • Dorothy Parker • get cranky • more quotes Here we are now at the middle of the fourth large part of this talk. • Pepe Deluxe • get nowhere • more quotes This love loves love. It's a strange love, strange love. • Liz Fraser • find a way to love • more quotes This love's a nameless dream. • Cocteau Twins • try to figure it out • more quotes Sun is on my face ...a beautiful day without you. • Royskopp • be apart • more quotes syncopation & accordion • Cafe de Flore (Doctor Rockit) • like France, but no dog poop • more quotes Thoughts rearrange, familiar now strange. • Holly Golightly & The Greenhornes • break flowers • more quotes And she looks like the moon. So close and yet, so far. • Future Islands • aim high • more quotes Where am I supposed to go? Where was I supposed to know? • Violet Indiana • get lost in questions • more quotes Poetry is just the evidence of life. If your life is burning well, poetry is just the ash • Leonard Cohen • burn something • more quotes In your hiding, you're alone. Kept your treasures with my bones. • Coeur de Pirate • crawl somewhere better • more quotes music + dance + projected visuals • Nosaj Thing • marvel at perfect timing • more quotes Safe, fallen down this way, I want to be just what I am. • Cocteau Twins • safe at last • more quotes Drive, driven. Gave, given. • Yello • Give me a number of games. • more quotes Without an after or a when. • Papercut feat. Maiken Sundby • can you hear the rain? • more quotes I'm not real and I deny I won't heal unless I cry. • Cocteau Twins • let it go • more quotes Embrace me, surround me as the rush comes. • Motorcycle • drift deeper into the sound • more quotes What do the trees know. • Laleh • sway, sway, sway • more quotes
home>Music Video — Max Cooper's Aleph 2>Video walkthrough
data visualization + art
Yearning for dimensions? ·
My animation of 5 dimensions sets the stage for Max Cooper's track Ascent from his new album Unspoken Words.
Another mathy collaboration with Max!
Infinity with Max Cooper — In Six Minutes
menu video walkthrough set theory math making of the video photos resources
1 · Oh my God, it's full of numbers
2 · One, two, three, infinity
3 · Glitches in the story
4 · Hilbert's Grand Hotel
5 · New bijections await
6 · Last stop before switching infinity trains
7 · Beyond infinity — other infinities
8 · Power set of natural numbers
9 · Cardinality of the continuum
10 · Aleph 2
The video starts straight at number 1 and go to infinity (and beyond). Let's go!
The walkthrough includes key mathematical details behind the theory of transfinite numbers. The story is aimed at a math-enthusiast audience. For even more details for the hardy and curious, I cover some advanced concepts in set theory, such as beth numbers and ordinal numbers.
This math may cause you to lose sleep. I hope it does, as many things in life are worth staying awake for.
Take care — I'm not a set theoretician. Please report errors, technical shortcomings and places where you feel the explanations are lacking.
The video begins humbly is with the first natural number, 1.
▲ We begin counting the natural numbers, whose cardinality is `|\mathbb{N}| = \aleph_0`. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
What follows are all the other natural numbers—everyone has their favourite. I like 17.
All the natural numbers make up a set, $\mathbb{N}$. $$\mathbb{N} = \{ 1, 2, 3...\}$$
We never run out of these numbers—to get the next one just take the previous and add one. This process of adding one to get the next number is called the successor function. For now, this is easy but we'll see below an application of this that's a little trickier.
How big is $\mathbb{N}$? In set theory, the size of a set is referred to as its cardinality and denoted by $|\mathbb{N}|$. At this point you might simply want to answer "It's infinitely large!" and be done with it.
Hang around, though. This is just where the story gets interesting.
To acknowledge the fact that your mind might glitch while investigating infinity, you'll see that the music video glitches as well.
▲ The first glitch is at frame 1272. The alephs stay for the next two frames.
▲ At frame 1,275 the alephs begin to decay into set theory symbols..
▲ At frame 1,276 the decay ends with all characters becoming periods and disappearing.
As the video progresses, the frequency and intensity of the glitching is increased and new set theory symbols are added. Many of the glitches are triggered by the snare drum. As the percussion becomes more intense, so does the glitching.
The video ends with a return to the counting of natural numbers which then glitches into a blank screen and only `\aleph` remains.
There is a magical place called the Grand Hotel. It has an infinite number of rooms, each indexed by a natural number.
▲ Hilbert's Grand Hotel of infinite rooms with no vacancy yet infinite capacity. Adapted from Grand Budapest Hotel.
Despite the fact that this hotel is always completely full, it can always accommodate another guest. To do this, we ask everyone to move to room $n+1$ thus freeing up room $1$. We have solved the problem at the cost of annoying infinitely many people.
But wait. There's infinitely more.
The hotel can always accommodate an additional infinite number of guests. Everyone is asked to move to room $2n$ thus freeing up all odd-numbered rooms. Since there's an infinite number of these rooms, we've just doubled the capacity!
I don't mean to suggest that I loved you the best
I can't keep track of each fallen robin
I remember you well in the Chelsea Hotel
That's all, I don't even think of you that often
— Leonard Cohen, Chelsea Hotel No. 2
The craziness is only beginning. If an infinite number of busses show up, each with an infinite number of guests, guess what? Yup, they can all be accommodated.
First, denote the $i$th guest in the $j$th bus by $s_n = (i,j)$. Since $i \in \mathbb{N}$ and $j \in \mathbb{N}$ then, we can enumerate $s_n$ as $s_1, s_2, s_3, ...$ Then, the formula for assigning guests to rooms is to assign guest $i$ from bus $j$ to room $n$. In this scheme, the hotel itself is treated as a bus indexed by $j = 0$.
This is sometimes called "Hilbert's Paradox" but it's not really a paradox. Rather, it's a demonstration that we should not expect intuition about finite quantities to carry over to the behaviour of infinite quantities.
Case in point, in Hilbert's hotel "there is a guest in every room" does not imply that "no more guests can be accommodated".
Your mind may be in revolt at this moment. "Surely, there are fewer even numbers than natural numbers, since even numbers are a proper subset of natural numbers!"
First, in a hotel with finite number of rooms, you would be correct.
Second, your confusion puts you in good company. Gregor Cantor (1845-1918), the mathematician who was the first to develop a theoretical framework of infinite quantities, suffered recurrent nervous breakdowns.
Remember the concept of the cardinality of a set? Cantor assigned the number $\aleph_0$ (named so after the first letter, aleph, in the Hebrew alphabet) to represent the cardinality of the naturals.
He called the family of numbers, in which $\aleph_0$ is the first, transfinite numbers . Today 'infinite' is more commonly used to refer to these numbers.
For two sets to have the same cardinality a special condition has to be met. We say that $|X| = |Y|$ if and only if we can demonstrate a function $f(X) \mapsto Y$ that is one-to-one, meaning that $f(x \in X)$ maps to a unique $y \in Y$, and onto, meaning that every $y \in Y$ is mapped to by some $x \in X$.
▲ Injective, surjective and bijective functions. The presence of a bijection between $X$ and $Y$ implies $|X| = |Y|$.
In other words, to show that two sets have the same cardinality, we just have to find a bijection. Here is one between the even numbers and natural numbers $$ f(\{2,4,6,8,...\} \mapsto \mathbb{N}) : x \mapsto x/2 $$
To show that $f$ is a bijection we need to show that $f$ is injective and surjective. It is injective (one-to-one) since any two distinct even numbers $x \ne y$ are sent to distinct values $x/2 \ne y/2$. It is surjective (onto) since every natural number $x$ has an even number $2x$ that is mapped to it.
There are lots of sets that have the same cardinality as the naturals: odd numbers, even numbers, prime numbers. Any infinite subset of naturals has the same cardinality as the naturals.
The next chapter in the video brings us to the observation that since the naturals are a subset of the integers, $\mathbb{Z}$ and since both are infinite, both have to have the same cardinality.
▲ A bijection between the naturals and integers demonstrating that both have the same cardinality. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
The video shows a bijection between the two sets, $f(\mathbb{N}) \mapsto \mathbb{Z}$ defined as follows: $f(0) \mapsto 0$ and $f(2k) \mapsto k$, $f(2k-1) \mapsto -k$ for $k \in \mathbb{N}$. Even naturals are sent to positive integers and odd naturals are sent to negative integers. Each natural has a unique integers (injective) and all integers are covered (surjective).
For example, the positive integer $n$ is mapped from $2n \in \mathbb{N}$ (e.g. $22 \mapsto 11$) and the negative integer $-n$ from $-(2n-1)$ (e.g. $23 \mapsto -11$).
The next part of the video shows many more possible bijections.
▲ Various bijections between the naturals and integers. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
The first column on the right of the naturals is the bijection described above and others take the form $$f(x \in \mathbb{N}) = \begin{cases} 0, & \text{if $x = 1$} \\k, & \text{if $x$ is the $k$th number that passes the rule $g(x)$} \\-k, & \text{if $n$ is the $k$th number that fails the rule $g(x)$} \end{cases} $$
For our first bijection, the rule was $g(x) \stackrel{?}{=} 2k$, which checks whether $x$ is even.
The second column on the right of the naturals uses the rule $g(x) \stackrel{?}{=} 4k$, which checks whether $x$ is a multiple of 4. Thus, $4 \mapsto 1$, $8 \mapsto 2$, $12 \mapsto 3$ and so on. All naturals that are not multiples of 4 are sent to negative integers.
Columns on the left of the naturals use an odd multipler in the $g(x)$ rule and additionally flip the sign of the integer to which $f(x)$ maps. This was done so that I could have an equal balance of white positive integers in the columns on the right and white negative integers in the columns on the left.
▲ Various bijections between the naturals and integers. The music is speeding up. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
As we sample more natural numbers, pages of bijections update faster and faster, in waves of numbers that decay into symbols used in set theory.
Because I could not stop for Death —
He kindly stopped for me —
The Carriage held but just Ourselves —
And Immortality.
Since then — 'tis Centuries — and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity —
— Emily Dickinson
We ramp up the complexity in the video by showing a bijection between the natural numbers and rational numbers, $\mathbb{Q}$, which are all fractions of the form $\mathbb{Q} = \{ q = x/y, x,y \in \mathbb{N}_0, y \ne 0 \}$.
First, we create a table in which the cell in row $x$ and column $y$ is assigned the rational number $x/y$.
▲ Building a table of rational numbers. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
To create the bijection, we assign each rational to a natural number as we traverse the table in a zig-zag fashion.
▲ A bijection between the naturals and rationals. We traversing the table of rationals in a zig-zag manner. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
We snake our way from the upper-left corner ($1/1$) to the bottom-right ($78/31$), which is assigned 2418.
▲ Almost done traversal for this screen. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Given that rational numbers can be considered as two-dimensional naturals, $\mathbb{Q} = \mathbb{N}^2 = { \mathbb{N} \times \mathbb{N} }$, the same traversal argument can be used to show that all higher-dimensional spaces of naturals also have the same cardinality as the naturals, $|\mathbb{N}^k| = |\mathbb{N}|$. The bijection construction is the same as in the $k=2$ case above, except that now we're snaking across a higher dimensional space. When $k$ itself is infinite, we have the scenario of infinite guests in infinite busses arriving at Hilbert's Grand Hotel.
In our story so far, we have shown bijections between $\mathbb{N}$ and $\mathbb{Z}$ and $\mathbb{Q}$ we have proven that all these sets have the same cardinality $|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| = \aleph_0$.
To summarize, the naturals are considered to be infinite but countable and any set that has a bijection with the naturals is also countable.
Our discussion has brought us to infinity, $\aleph_0$. What lies beyond?
The first hint that there is indeed something beyond lies in the proof that the real numbers, $\mathbb{R}$ are not countable: there is no bijection between the naturals and reals. If we try to pair up naturals and reals we'll always run out of naturals.
Real numbers are continuous quantities that, for example, can measure the distance along a line. They include the naturals, integers, rationals as well as irrationals, which include numbers like $\sqrt{2}$, which cannot be written as a fraction, and numbers like $\pi$, which are transcendentals and not solutions to polynomial equations.
Cantor's demonstration that $|\mathbb{R}| > |\mathbb{N}|$ is the next part of the story. The proof is by contradiction and applies to the unit interval $ [0,1) = \{ x \, | \, 0 \le x \lt 1 \}$. First, suppose that there is a bijection $f(\mathbb{N}) \mapsto [0,1)$. This implies that for each $n \in \mathbb{N}$ there is some associated $r \in [0,1)$. We write down this assignment—for each natural, we pair up a natural with a real from the unit interval. Obviously, this list goes on forever in both the vertical and horizontal direction.
▲ The start of Cantor's diagonal proof that the reals are more numerous than the naturals. We begin by assuming that we can pair up every real with a natural number. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
We don't know what the exact assignment is, so the numbers in the story are only representative. Typically, the proof is written out symbolically with each natural $n_i$ assigned to a series of digits $n_{i1}n_{i2}n_{i3} ... $.
Because this assignment is a bijection it is a surjection and every real number from the unit interval appears somewhere in the list. If we could demonstrate that there is a real number from $[0,1)$ that doesn't appear in the list, we would have a contradiction and the assumption that a bijection exists would be invalid.
We do so as follows. We transform the first digit of the first real to $x \mapsto x + 1 \, \text{mod} \, 10$. In other words $0 \mapsto 1$, $1 \mapsto 2$ and $9 \mapsto 0$.
We do the same for the second digit of the second number, the third digit of the third number, and so on.
▲ We alter the digits on the diagonal to create a real that is nowhere in the list. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
This creates a new real, shown here without the leading $0.$
▲ The demonstration of a real that is not mapped to by any natural proves that there is no bijection between naturals and reals and therefore that the reals are more numerous than the naturals. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
By construction, this real is nowhere in our list of reals. It can't be—it's different from each of the reals in at least one digit. It's different from the first number in the first digit, the second number in the second digit, the third number in the third digit and so on. But it's obviously in the unit interval. A contradiction.
If we write the numbers in the unit interval in binary $ [0,1) = \{ 0.b_1b_2b_3,... \, | \, b \in \{0,1\} \}$ we can use the fact that $b_i$ is indexed by $\mathbb{N}$ to realize that there are $2^{|\mathbb{N}|}$ such binary numbers, since at each position we have two choices ($0$ or $1$). And because $|\mathbb{N}| = \aleph_0$ we have $$ |\mathbb{R}| = 2^{|\mathbb{N}|} = 2^{\aleph_0} $$
Given a set $X$, the power set is the set of all subsets of $X$, including the empty set.
The next part of the story builds up the power set of naturals, $\mathbb{P}(\mathbb{N})$.
▲ We create power sets of the naturals, which are all combinations of natural numbers. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
For example, for $X = \{1\}$ the power set has two elements, the empty set $\{\}$ and the whole set $\{1\}$. We write $ \mathbb{P}(X) = \{\{\},\{1\}\} $.
For $X = \{1,2\}$ the power set has four elements, the empty set $\{\}$, each of the naturals on their own $\{1\}$ and $\{2\}$ and the whole set $\{1,2\}$. We write $ \mathbb{P}(X) = \{\{\},\{1\},\{2\},\{1,2\}\} $.
In general, the power set of $\{1,2,3,...,n\}$ has cardinality $2^n$.
If we look closely at the $ 2^{\aleph_0} $, we can interpret it as the cardinality of the power set of naturals. This is because for each natural, of which there are $ \aleph_0 $ we have two choices: put it in the subset or not.
As the video continues, the power set elements appear faster and faster. The braces form hypnotising patterns.
▲ Hypnotic braces in power set elements of naturals. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Because the reals are continuous quantities, the number of reals is (wonderfully) called the cardinality of the continuum. I wouldn't turn down the job of cardinal of the continuum.
Given what we learned about power sets of naturals above, we can write $$ |\mathbb{R}| = | \mathbb{P}(\mathbb{N}) | = 2^{\aleph_0} $$
With Cantor's diagonal proof, we know that $|\mathbb{R}| > \aleph_0$ but we don't know how much larger. Cantor therefore proposed the Continuum Hypothesis which stated that whatever the size of $2^{\aleph_0}$ was, it was a distinct kind of infinite number and, importantly, the next smallest infinite number after $\aleph_0$.
A consequence of this theorem is that there is no set $X$ for which $$ \aleph_0 \lt |X| \lt 2^{\aleph_0} $$
meaning that there is no set that is larger than the naturals but smaller than the reals.
The Continuum Hypothesis also implies that the cardinality of the continuum is the next number ($\aleph_1$) in the hierarchy of transfinite cardinals, $$ |\mathbb{R}| = 2^{\aleph_0} = \aleph_1 $$
From it, we also get that the cardinality of the power set of an infinite set is the next transfinite cardinal. In other words, for sets $\mathbb{N}, \mathbb{P}(\mathbb{N}), \mathbb{P}(\mathbb{P}(\mathbb{N})), ...$ the cardinalities are $\aleph_0, \aleph_1, \aleph_2, ...$ And in general, $$ \aleph_{\alpha+1} = 2^{\aleph_\alpha} $$
The Continuum Hypothesis is thus far unproven.
At this point, we arrive at the third infinity in the video—the cardinality of the power set of reals is $ | \mathbb{P}({\mathbb{R}}) | = \aleph_2 $.
▲ Power set elements of real numbers. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Elements of the power sets of naturals and reals continue to flash. If the Continuum Hypothesis is true, their cardinality is $\aleph_1$ and $\aleph_2$ respectively.
▲ Power set elements of naturals (left) and reals (right). The size of these sets is separated by one step in the hierarchy of Aleph numbers, if the Continuum Hypothesis is true. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
The music grows in intensity and the scene deteriorates into set theory symbols.
▲ Music intensifies beyond intensity. Symbols appear and disappear. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
The story brings us back to where we started from: the list of naturals. These pick up where we left off and continue counting.
▲ Back to naturals. What a relief. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
These too decay in a jitter of symbols
▲ Glitches and blips. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
with soon nothing but symbols left
▲ It's time for all this to be over. Symbols decay. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Suddently $\aleph$ appears.
▲ Aleph appears. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
And while everything else decays,
▲ Aleph remains. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
▲ Aleph decays. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
We are reminded of where we started, how far we've gone and how many more infinites are left to go.
How many ages hence
Shall this our lofty scene be acted over,
In states unborn and accents yet unknown!
And so, in our yearning for the infinite, we return to basic counting and find that we never understood it well in the first place.
news + thoughts
Cell Genomics cover
Mon 16-01-2023
Our cover on the 11 January 2023 Cell Genomics issue depicts the process of determining the parent-of-origin using differential methylation of alleles at imprinted regions (iDMRs) is imagined as a circuit.
Designed in collaboration with with Carlos Urzua.
▲ Our Cell Genomics cover depicts parent-of-origin assignment as a circuit (volume 3, issue 1, 11 January 2023). (more)
Akbari, V. et al. Parent-of-origin detection and chromosome-scale haplotyping using long-read DNA methylation sequencing and Strand-seq (2023) Cell Genomics 3(1).
Browse my gallery of cover designs.
▲ A catalogue of my journal and magazine cover designs. (more)
Science Advances cover
Thu 05-01-2023
My cover design on the 6 January 2023 Science Advances issue depicts DNA sequencing read translation in high-dimensional space. The image showss 672 bases of sequencing barcodes generated by three different single-cell RNA sequencing platforms were encoded as oriented triangles on the faces of three 7-dimensional cubes.
More details about the design.
▲ My Science Advances cover that encodes sequence onto hypercubes (volume 9, issue 1, 6 January 2023). (more)
Kijima, Y. et al. A universal sequencing read interpreter (2023) Science Advances 9
Regression modeling of time-to-event data with censoring
If you sit on the sofa for your entire life, you're running a higher risk of getting heart disease and cancer. —Alex Honnold, American rock climber
In a follow-up to our Survival analysis — time-to-event data and censoring article, we look at how regression can be used to account for additional risk factors in survival analysis.
We explore accelerated failure time regression (AFTR) and the Cox Proportional Hazards model (Cox PH).
▲ Nature Methods Points of Significance column: Regression modeling of time-to-event data with censoring. (read)
Dey, T., Lipsitz, S.R., Cooper, Z., Trinh, Q., Krzywinski, M & Altman, N. (2022) Points of significance: Regression modeling of time-to-event data with censoring. Nature Methods 19.
Music video for Max Cooper's Ascent
Tue 25-10-2022
My 5-dimensional animation sets the visual stage for Max Cooper's Ascent from the album Unspoken Words. I have previously collaborated with Max on telling a story about infinity for his Yearning for the Infinite album.
I provide a walkthrough the video, describe the animation system I created to generate the frames, and show you all the keyframes
▲ Frame 4897 from the music video of Max Cooper's Asent.
The video recently premiered on YouTube.
Renders of the full scene are available as NFTs.
Gene Cultures exhibit — art at the MIT Museum
I am more than my genome and my genome is more than me.
The MIT Museum reopened at its new location on 2nd October 2022. The new Gene Cultures exhibit featured my visualization of the human genome, which walks through the size and organization of the genome and some of the important structures.
▲ My art at the MIT Museum Gene Cultures exhibit tells shows the scale and structure of the human genome. Pay no attention to the pink chicken.
Annals of Oncology cover
Wed 14-09-2022
My cover design on the 1 September 2022 Annals of Oncology issue shows 570 individual cases of difficult-to-treat cancers. Each case shows the number and type of actionable genomic alterations that were detected and the length of therapies that resulted from the analysis.
▲ An organic arrangement of 570 individual cases of difficult-to-treat cancers showing genomic changes and therapies. Apperas on Annals of Oncology cover (volume 33, issue 9, 1 September 2022).
Pleasance E et al. Whole-genome and transcriptome analysis enhances precision cancer treatment options (2022) Annals of Oncology 33:939–949.
▲ My Annals of Oncology 570 cancer cohort cover (volume 33, issue 9, 1 September 2022). (more)
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CommonCrawl
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\begin{document}
\author{Valentin Blomer}
\address{Mathematisches Institut, Bunsenstr. 3-5, 37073 G\"ottingen, Germany} \email{[email protected]}
\author{Rizwanur Khan} \address{Department of Mathematics, University of Mississippi, University, MS 38677, USA} \email{[email protected]}
\title{Twisted moments of $L$-functions and spectral reciprocity}
\thanks{First author partially supported by the DFG-SNF lead agency program grant BL 915/2-1 and BL 915/2-2.}
\keywords{Spectral reciprocity, moments of $L$-functions, amplification, subconvexity}
\begin{abstract} A reciprocity formula is established that expresses the fourth moment of automorphic $L$-functions of level $q$ twisted by the $\ell$-th Hecke eigenvalue as the fourth moment of automorphic $L$-functions of level $\ell$ twisted by the $q$-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for $L$-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic $L$-functions. \end{abstract}
\subjclass[2010]{Primary: 11M41, 11F72}
\setcounter{tocdepth}{2} \maketitle
\maketitle
\section{Introduction}
\subsection{Spectral reciprocity} Let $q, \ell$ be two distinct odd primes. Gau{\ss}' celebrated law of quadratic reciprocity expresses the quadratic Legendre symbol $(\ell/q)$ in terms of $(q/\ell)$. This is a very remarkable statement, because it connects the arithmetic of two unrelated finite fields $\Bbb{F}_{q}$ and $\Bbb{F}_{\ell}$. Hilbert re-interpreted quadratic reciprocity as a local-to-global statement: the local Hilbert symbols have to satisfy a global consistency relation (their product equals 1). The present paper establishes a reciprocity formula between the spectrum of the Laplacian on two different arithmetic hyperbolic surfaces $\Gamma_0(q) \backslash \Bbb{H}$ and $\Gamma_0(\ell)\backslash \Bbb{H}$ featuring a product of $L$-functions of total degree 8. One of the key steps involves the additive reciprocity formula \eqref{rec} below, which is used as a vehicle to shift information between different places of $\Bbb{Q}$, so that again a local-to-global principle works in the background. We will go into more detail later and proceed by describing our main result more precisely.
Let $F$ be an automorphic form for the group ${\rm SL}_3(\Bbb{Z})$ and in the following summations let generally denote by $f$ a cusp form for a congruence subgroup of ${\rm SL}_2(\Bbb{Z})$. Then roughly speaking, we will prove a formula of the shape $$\sum_{f \text{ of level } q} L(s, f \times F) L(w, f) \lambda_f(\ell) \rightsquigarrow \sum_{f \text{ of level } \ell} L(s', f\times F) L(w', f) \lambda_f(q)$$ where \begin{equation}\label{new} \textstyle s' = \frac{1}{2}(1 + w - s), \quad w' = \frac{1}{2}(3s + w - 1). \end{equation}
In particular, specializing $s = w = 1/2$ and $F = {\tt E}_0$ the minimal parabolic Eisenstein series with trivial spectral parameters whose Hecke eigenvalues are the ternary divisor function $\tau_3(n)$, we obtain a reciprocity formula for the twisted fourth moment \begin{equation}\label{twisted} \sum_{f \text{ of level } q} L(1/2, f)^4 \lambda_f(\ell) \rightsquigarrow \sum_{f \text{ of level } \ell} L(1/2, f )^4 \lambda_f(q). \end{equation} In addition to its aesthetic features linking arithmetic data on different congruence quotients of the upper half plane, this formula is useful for applications, because we can trade the level of the forms for the twisting Hecke eigenvalue. We will demonstrate this is in the next subsection. We also mention that such a formula is implicit in the work of K\i ral and Young \cite{KY}, although in a coarser and less conceptual form. After having stated the main result in Theorem \ref{thm1} below, we will also discuss other versions of reciprocity of central $L$-values. \\
For the rest of the paper let $q, \ell$ be two positive integers, $s, w$ two complex numbers with positive real part, and $F$ a cuspidal or non-cuspidal automorphic form for the group ${\rm SL}_3(\Bbb{Z})$ with Fourier coefficients $A(n_1, n_2)$.
We need to (a) parametrize the spectrum of level $q$ and identify suitable test functions on the various components, (b) define correction factors at the ramified primes $p \mid q\ell$, and (c) add additional ``main terms''.
Irreducible automorphic representations of $Z(\Bbb{A}_{\Bbb{Q}}) {\rm GL}_2(\Bbb{Q})\backslash {\rm GL}_2(\Bbb{A}_{\Bbb{Q}})/K_0(q)$ with $K_0(q) = \{\left(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix}\right)\in {\rm GL}_2(\widehat{\Bbb{Z}}) \mid c\equiv 0 \, (\text{mod } q \widehat{\Bbb{Z}})\}$ can be generated by three types of functions:
\begin{itemize}
\item Cuspidal holomorphic newforms $f$ of weight $k = k_f \in 2\Bbb{N}$, level $q' \mid q$ and Hecke eigenvalues $\lambda_f(n) \in \Bbb{R}$; we denote the set of such forms by $\mathcal{B}_{\text{hol}}(q)$ and the set of newforms of exact level $q$ by $\mathcal{B}^{\ast}_{\text{hol}}(q)$;
\item Cuspidal Maa{\ss} newforms $f$ of spectral parameter $t = t_f\in \Bbb{R} \cup [-i \vartheta, i \vartheta]$, level $q' \mid q$ and Hecke eigenvalues $\lambda_f(n) \in \Bbb{R}$, where at the current state of knowledge $\vartheta = 7/64$ can be taken \cite{KiS} (though $\vartheta = 0$ is expected); we denote the set of such forms by $\mathcal{B}(q)$ and the set of newforms of exact level $q$ by $\mathcal{B}^{\ast}(q)$;
\item Unitary Eisenstein series $E_{t, \chi}$ for $\Gamma_0(q)$, where $t \in \Bbb{R}$ and $\chi$ is a primitive Dirichlet character of conductor $c_{\chi}$ satisfying $c_{\chi}^2 \mid q$. Their $n$-th Hecke eigenvalue is $\lambda_{t, \chi}(n) = \sum_{ab = n} \chi(a)\bar{\chi}(b) (b/a)^{it}$ for $(n, q) = 1$. \end{itemize}
We write $\mathcal{T}_0 := (\Bbb{R} \cup [-i \vartheta, i \vartheta] ) \times 2\Bbb{N}$
and consider the set of functions $\mathfrak{h} = (h, h^{\text{hol}}) : \mathcal{T}_0 \rightarrow \Bbb{C}$, where \begin{equation}\label{weakly}
h(-t) = h(t) \ll (1 + |t|)^{-15}, \,\, t\in \Bbb{R} \cup [-i \vartheta, i \vartheta], \quad \quad h^{\text{hol}}(k) \ll k^{-15}, \,\,k \in 2\Bbb{N}. \end{equation}
We call such functions \emph{weakly admissible}. We will define in Section \ref{admissible} the notion of an \emph{admissible} function $\mathfrak{h}$ which imposes additional analytic conditions on $\mathfrak{h}$; see Lemma \ref{lem1} for explicit examples.
For a cuspidal (holomorphic or Maa{\ss}) newform $f$ or an Eisenstein series $E_{t, \chi}$ and $w \in \Bbb{C}$ define \begin{equation}\label{Lambda} \Lambda_f(\ell; w) := \sum_{ab = \ell} \frac{\mu(a)}{a^w} \lambda_f(b), \quad \Lambda_{(t, \chi)}(\ell; w) := \sum_{ab = \ell} \frac{\mu(a)}{a^w} \lambda_{t, \chi}(b). \end{equation}
We also need local correction factors $\tilde{L}_q(s, w, f \times F)$ resp.\ $\tilde{L}_q(s, w, E_{t, \chi} \times F)$ that are defined in \eqref{local-q-cusp} and \eqref{local-q-eis} and satisfy the following properties.
\begin{lemma}\label{lem2} For $f \in \mathcal{B}^{\ast}(q) \cup \mathcal{B}^{\ast}_{\text{\rm hol}}(q)$ we have
$ \tilde{L}_q(s, w, f \times F) =1$.
For arbitrary $f \in \mathcal{B}(q) \cup \mathcal{B}_{\text{\rm hol}}(q)$, $t \in \Bbb{R}$, $\chi$ a primitive Dirichlet character of conductor $c_{\chi}$ with $c_{\chi}^2 \mid q$, $\varepsilon > 0$ and $\Re s, \Re w \geqslant 1/2$ we have the uniform bound \begin{equation}\label{local-bound}
\tilde{L}_q(s, w, f \times F), \,\, \tilde{L}_q(s, w, E_{t, \chi} \times F) \ll_{\varepsilon} q^{\theta +\varepsilon}.
\end{equation}
\end{lemma}
Here and henceforth, $\theta$ (not to be confused with $\vartheta$) denotes an admissible exponent towards the Ramanujan conjecture for $F$. At the current state of knowledge, $\theta \leqslant 5/14$ (cf.\ \cite{BB}) is known in general, and for $F = {\tt E}_0$ we have $\theta = 0$. \\
For a weakly admissible function $\mathfrak{h} = (h, h^{\text{hol}})$, complex numbers $s, w$ with $\Re s, \Re w \geqslant 1/2$ and coprime integers $q, \ell$ we define the twisted spectral mean values \begin{equation}\label{moment} \begin{split} & \mathcal{M}^{\text{Maa{\ss}}, \pm}_{q, \ell}(s, w; \mathfrak{h}) := \frac{\phi(q)}{q^2} \sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast}(d_0)} \epsilon_f^{(1 \mp 1)/2}\frac{L(s, f \times F) L(w, f)}{L(1, \text{Ad}^2 f)}\tilde{L}_q(s, w, f \times F)\frac{ \Lambda_f(\ell; w)}{\ell^{w}} h(t_f),\\ & \mathcal{M}^{\text{hol}}_{q, \ell}(s, w; \mathfrak{h}) := \frac{\phi(q)}{q^2} \sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast}_{\text{hol}}(d_0)} \frac{L(s, f \times F) L(w, f)}{L(1, \text{Ad}^2 f)}\tilde{L}_q(s, w, f \times F) \frac{ \Lambda_f(\ell; w)}{\ell^{w}} h^{\text{hol}}(k_f),\\
& \mathcal{M}^{\text{Eis}}_{q, \ell}(s, w; \mathfrak{h}) := \frac{\phi(q)}{q^2}\sum_{\chi : c_{\chi}^2 \mid q} \int_{\Bbb{R}} \frac{L(s + it, F \times \chi) L(s - it, F \times \bar{\chi}) L(w + it, \chi)L(w - it, \bar{\chi})}{L(1 + 2 it, \chi^2)L(1 - 2it, \bar{\chi}^2)} \\ & \quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad \tilde{L}_q(s, w, E_{t, \chi} \times F) \frac{\Lambda_{t, \chi}(\ell; w)}{\ell^{w}} h(t) \frac{dt}{2\pi}, \end{split} \end{equation} where $\epsilon_f \in \{\pm 1\}$ is the parity of the Maa{\ss} form $f \in \mathcal{B}(q)$ and $\Re s, \Re w \not= 1$ in the expression for $\mathcal{M}^{\text{Eis}}_{q, \ell}(s, w; \mathfrak{h})$. The Dirichlet series expansions of $L(s, \text{Ad}^2 f)$ and $L(s, f \times F)$ in $\Re s> 1$ (which serve as a definition for these functions) are given in \eqref{Lad} and \eqref{L23}. We write \begin{equation}\label{moment-together} \mathcal{M}^{\pm}_{q, \ell}(s, w, \mathfrak{h}) = \mathcal{M}^{\text{Maa{\ss}}, \pm}_{q, \ell}(s, w; \mathfrak{h}) + \mathcal{M}^{\text{Eis}}_{q, \ell}(s, w; \mathfrak{h}) + \delta_{\pm = +} \mathcal{M}^{\text{hol}}_{q, \ell}(s, w; \mathfrak{h}). \end{equation}
Absolute convergence follows from \eqref{weakly}, Weyl's law, the convexity bound for the respective $L$-functions and the lower bounds \eqref{hl}. It is not hard to see (cf.\ Lemma \ref{analyticcont}) that under suitable circumstances $\mathcal{M}^{\text{Eis}}_{q, \ell}(s, w; \mathfrak{h})$ for $\Re s, \Re w > 1$ can be continued to an $\varepsilon$-neighbourhood of $\Re s, \Re w \geqslant 1/2$, and this continuation equals $\mathcal{M}^{\text{Eis}}_{q, \ell}(s, w; \mathfrak{h})$ for $1/2 < \Re s, \Re w <1$ plus some polar terms.
\begin{theorem}\label{thm1} Let $\mathfrak{h}$ be an admissible function, $(q, \ell) = 1$ and let $s, w\in \Bbb{C}$ be such that \begin{equation}\label{final-region} 1/2 \leqslant \Re s \leqslant \Re w < 3/4.
\end{equation}
Let $F$ be either cuspidal or the minimal parabolic Eisenstein series\footnote{More general Eisenstein series could be treated in the same way, but we do not have applications in this case.} ${\tt E}_0$. Then \begin{equation}\label{final-formula} \begin{split} \mathcal{M}^{+}_{q, \ell}(s, w; \mathfrak{h}) = \mathcal{N}_{q, \ell}(s, w; \mathfrak{h}) + \sum_{\pm} \mathcal{M}^{\pm}_{\ell, q}(s', w'; \mathscr{T}_{s', w'}^{\pm}\mathfrak{h}), \end{split} \end{equation} where $s', w'$ are as in \eqref{new}, the ``main term'' $\mathcal{N}_{q, \ell}(s, w; \mathfrak{h})$ and the integral transform $\mathscr{T}^{\pm}_{s', w'}\mathfrak{h}$ are given in \eqref{main-term-final} resp.\ \eqref{final-trafo}. The function $\mathscr{T}^{\pm}_{s', w'}\mathfrak{h}$ is weakly admissible, and for $\varepsilon > 0$ we have
\begin{equation}\label{required-bound}
\mathcal{N}_{q, \ell}(s, w; \mathfrak{h})\ll_{s, w, \mathfrak{h}, \varepsilon} \ell^{\theta - 1 + \varepsilon} + q^{\theta - 1 + \varepsilon}.
\end{equation}
\end{theorem}
In the special case $q = \ell = 1$, $F = {\tt E}_0$, a beautiful formula for the fourth moment was envisaged by Kuznetsov in the late 1980s and completed by Motohashi \cite{Mo}. This formula, however, has different features. Our proof decomposes the fourth moment as $4 = 3+1$, whereas Kuznetsov and Motohashi use a decomposition of the form $4=2+2$. While more symmetric on the surface, their setup leads to
convergence problems that are overcome by considering a carefully designed \emph{difference} of the holomorphic and the Maa{\ss} spectrum. In particular, everywhere positive test functions $\mathfrak{h}$ as in Lemma \ref{lem1}c) are in general excluded. (As long as $\ell = q = 1$, one can exploit the fact that there are no holomorphic forms of small weight to circumvent this problem, but for more general values of $q$ and $\ell$ this device is not available.) This is probably the reason why this formula seems to not have been used for applications.
Our formula should be seen as a contribution to the rich theory of automorphic forms on ${\rm GL}(2) \times {\rm GL}(4)$ in the special situation where the ${\rm GL}(4)$ function is an isobaric $3+1$-sum. A corresponding reciprocity formula for $\mathcal{M}^-_{q, \ell}(s, w; \mathfrak{h})$ could also be established, in fact in an analytically simpler fashion (cf.\ \cite[end of Section 2]{Mo}). The structure of the argument indicates that a linear combination of these two formulae is an exact involution; it would be very interesting to provide a formal proof of this statement.
A spectral reciprocity formula in the \emph{archimedean} aspect -- i.e.\ the level is kept fixed, but the spectral parameter is varying -- for generic automorphic forms $F$ on ${\rm GL}(4)$ with an additional twist by the parity $$ \sum_{t_f \leqslant T} \epsilon_f L(1/2, f \times F) \rightsquigarrow T \sum_{t_f \ll 1} \epsilon_f L(1/2, f \times \tilde{F})$$ (where $\tilde{F}$ denotes the dual form) was considered by a different technique in \cite{BLM} along with applications to non-vanishing of $L$-functions. (In our set-up this would correspond to the easier, but excluded case $q= \ell$ in which case the Hecke eigenvalue is essentially the root number, at least as long as $q$ is squarefree.) Finally we mention that there is ``lower dimensional'' version of reciprocity for certain $L$-functions on ${\rm GL}(1) \times {\rm GL}(2)$:
$$\sum_{\chi \text{(mod }q)} |L(1/2, \chi)|^2 \chi(\ell) \rightsquigarrow \sum_{\chi \text{(mod }\ell)} |L(1/2, \chi)|^2 \chi(q).$$
This was first observed by Conrey \cite{Co}, and extended by Young \cite{Yo} and Bettin \cite{Be}. It would be very interesting to investigate if there is a ``master formula'' that contains all previously mentioned spectral reciprocity formulae as special cases.
\subsection{Applications}\label{app} The most interesting applications of our reciprocity formula concern the twisted fourth moment \eqref{twisted}, which has a long history in the theory of $L$-functions. The investigation started with seminal work of Duke-Friedlander-Iwaniec \cite{DFI}, who used their work on the binary additive divisor problem to obtain\footnote{This is not stated explicitly, but at least for prime level follows very easily from their Corollary 2, for instance.} \begin{equation}\label{DFI-bound}
\sum_{f } L(s, f)^4 \lambda_f(\ell) \ll_{k, s, \varepsilon} q^{1+\varepsilon} \ell^{-1/2} + q^{11/12+\varepsilon} \ell^{3/4} \end{equation} for $\Re s = 1/2$, where the sum is taken over holomorphic newforms of fixed weight $k$ and prime level $q$. Together with the amplification method they deduced the first subconvexity result \begin{equation}\label{subconvex} L(s, f) \ll_{k, s} q^{\frac{1}{4} - \delta}, \quad \delta < 1/192, \quad \Re s = 1/2 \end{equation} for this family. The then newly developed machinery was taken up by Kowalski-Michel-VanderKam \cite{KMV}, who derived a precise asymptotic formula for the left hand side of \eqref{DFI-bound} in the case of weight $k=2$ and prime level $q$ with error term $O_{\varepsilon}(q^{11/12+\varepsilon} \ell^{3/4})$. This allowed them to deduce, via mollification, various non-vanishing results for central $L$-values. Quite recently, Balkanova and Frolenkov \cite{BF} improved the error term somewhat. On heuristic grounds, one would expect an error term $O(q^{1/2+\varepsilon})$, and one would expect that the limit of current machinery should be an error term $O(q^{1/2+\varepsilon} \ell^{1/2} )$, at least under the Ramanujan conjecture. This is precisely what we prove: \begin{theorem}\label{thm2} Let $q$ be prime, $\varepsilon > 0$, $(\ell, q) = 1$ and $\mathfrak{h} = (h, h^{\text{{\rm hol}}})$ an admissible function. Then $$\sum_{f \in \mathcal{B}^{\ast}(q)} \frac{L(1/2, f)^4 }{L(1, \text{{\rm Ad}}^2 f)} \lambda_f(\ell) h(t_f) + \sum_{f \in \mathcal{B}^{\ast}_{\text{{\rm hol}}}(q)} \frac{L(1/2, f )^4 }{L(1, \text{{\rm Ad}}^2 f)} \lambda_f(\ell) h^{\text{{\rm hol}}}(k_f)
\ll_{\mathfrak{h}, \varepsilon} (q \ell)^{\varepsilon}\left(q \ell^{-1/2} + q^{1/2 + \vartheta} \ell^{1/2}\right).$$
\end{theorem} A slightly more general result is given in Proposition \ref{prop-new} in Section \ref{sec-new}. The bound is optimal for $\ell$ up to size $q^{1/2}$ (modulo the Ramanujan conjecture). Since this is the length of an approximate functional equation for $L(1/2, f)$, we can sum this bound trivially to obtain a bound for the fifth moment. \begin{theorem}\label{cor1} Let $q$ be prime, $\varepsilon > 0$. Then $$ \sum_{ f \in \mathcal{B}^{\ast} (q) } L(1/2, f)^5 e^{-t_f^2} \ll_{\varepsilon} q^{1+ \vartheta + \varepsilon}.$$ \end{theorem} For holomorphic forms of weight $k \in \{4, 6, 8, 10, 14\}$ an analogous result was recently obtained by K\i ral and Young \cite{KY} in an impressive 80-page tour de force argument. Our version for Maa{\ss} forms comes as a special case of a more general framework.
Since $L(1/2, f)$ is non-negative \cite{KaS}, Theorem \ref{cor1} implies immediately the strong subconvexity bound \begin{equation}\label{1/5subconvex} L(1/2, f) \ll_{\varepsilon} q^{(1+\vartheta + \varepsilon)/5}. \end{equation} This is an example of ``self-amplification'', where the fourth moment is amplified by a fifth copy of the $L$-function itself. Working with a traditional amplifier, however, is a more robust technique, since one can exploit positivity more strongly. This gives
\begin{theorem}\label{cor2} Let $q$ be squarefree with $(6, q) = 1$, $\varepsilon > 0$ and $f \in \mathcal{B}^{\ast}_{\text{\rm hol}}(q) \cup \mathcal{B}^{\ast}(q)$. Then $$L(1/2, f) \ll_{\varepsilon} q^{\frac{1}{4} - \frac{1 - 2\vartheta}{24}+\varepsilon}$$ where the implicit constant depends on the archimedean parameter of $f$, i.e.\ $t_f$ or $k_f$. \end{theorem} Note that this holds both for Maa{\ss} forms and for holomorphic forms of arbitrary (fixed) weight, including weight $k=2$. The assumptions on $q$ could be weakened with more work. While asymptotically weaker than \eqref{1/5subconvex} for $\vartheta\rightarrow 0$, with the current value of $\vartheta = 7/64$ the exponent $0.217$ is numerically a little stronger than $0.221$ in \eqref{1/5subconvex} and appears to be the current record. The generic $1/24$-saving, which is 16.6 percent of the convexity bound, is the natural limit in situations where the Iwaniec-Sarnak amplifier based on the relation $\lambda_f(p)^2 = 1+ \lambda_f(p^2)$ is applied, cf.\ for instance the corresponding bound \cite{IS} in the sup-norm problem. As we cannot exclude the possibility of many small Hecke eigenvalues, a more efficient amplifier is not available unconditionally, but an optimal amplifier (whose existence is potentially a much deeper problem than the subconvexity problem discussed here) would yield the bound $$L(1/2, f) \ll_{\varepsilon} q^{\frac{1}{4} - \frac{1 - 2\vartheta}{16}+\varepsilon}$$ of Burgess quality, which again marks the limit of current technology.
\subsection{The methods}
The general strategy of the proof is simple to describe. We apply the Petersson and/or Kuznetsov spectral summation formula, followed by the ${\rm GL}(3)$ Voronoi formula. In this way, the Kloosterman sums become ${\rm GL}(1)$-exponentials of the form $e(n \bar{d}/c)$, to which we can apply the additive reciprocity formula \begin{equation}\label{rec}
e\Bigl(\frac{n\bar{d}}{c}\Bigr)= e\left(-\frac{n\bar{c}}{d}\right) e\left(\frac{n}{cd}\right).
\end{equation}
Now we reverse all transformations, i.e.\ we apply the ${\rm GL}(3)$ Voronoi formula in the other direction and the Kuznetsov formula backwards to arrive at a ``dualized'' spectral sum. The application of \eqref{rec} makes this procedure non-involutory and yields the desired reciprocity formula. Note that in our application $c$ will be divisible by $q$ and $d$ by $\ell$, so the formula \eqref{rec} moves information from the primes dividing $q$ to the primes dividing $\ell$ as well as the infinite prime, because the right most exponential is treated as an archimedean weight function.
Experience has shown that this type of simple back-of-an-envelope heuristics can lead to enormous technical challenges. In this paper we have tried to present an approach that exploits structural features (including higher rank tools) as much as possible and circumvents many of the well-known technical problems.
On the analytic side, we avoid stationary phase arguments and intricate asymptotic analysis completely, but let instead analytic continuation (along with Stirling's formula) do the work for us. We do not start with an approximate functional equation (which also relieves us from any root number considerations), but work always with infinite Dirichlet series in the region of absolute convergence and postpone analytic continuation to the center of the critical strip to the last moment.
On the arithmetic side, we design the set-up carefully to take care of the combinatorial difficulties of higher rank Hecke algebras. This was already a key device in \cite{BLM}. The ${\rm GL}(3)$ Voronoi formula (or -- roughly equivalently -- three instances of Poisson summation in the Eisenstein case) is conceptually well-understood, but introduces in practice a whole alphabet of auxiliary variables for additional divisibility and coprimality conditions. The multiple Dirichlet series that we introduce in Section \ref{multiple} is tailored to the features of the ${\rm GL}(3)$ Hecke algebra. The combinatorial difficulties do not completely disappear in this way; a shadow remains in the rather complicated local correction factors $\tilde{L}_q(s, w, f \times F)$ and $\tilde{L}_q(s, w, E_{t, \chi} \times F)$, but the advantage of our approach is complete symmetry before and after the application of the arithmetic reciprocity formula \eqref{rec}. We would like to emphasize the strength of genuine higher rank tools (such as combinatorial and analytic properties of the ${\rm GL}(3)$ Voronoi formula) even for applications that involve only ${\rm GL}(2)$ objects (such as Theorems \ref{thm2} -- \ref{cor2}), and it seems that the present paper is the first instance where higher rank tools are used for the analysis of the twisted fourth moment.
One of the major problems in earlier approaches \cite{KMV, KY} is the presence of very complicated main terms coming from various sources, the most difficult being the various zero frequencies in the multiple applications of the Poisson summation formula. These terms are complicated by nature, but in our approach they arise in a very simple and conceptual way twice as a single residue of a certain multiple Dirichlet series.
It may be instructive to compare the general strategy with other options used in earlier works. As mentioned before, the analysis of \cite{DFI} and \cite{KMV} is based on a decomposition $4 = 2+2$ and a treatment of the binary additive divisor problem. The formula of Kuznetsov and Motohashi \cite{Mo} also decomposes $4 = 2+2$ and dualizes both ${\rm GL}(2)$ components. Unfortunately this leads to an essentially self-dual deadlock situation that is resolved -- as mentioned earlier -- by subtracting the holomorphic spectrum from the Maa{\ss} spectrum in a carefully designed way. K\i ral and Young \cite{KY} instead use the decomposition $5=3+1+1$ in the context of the fifth moment. They first dualize one of the ${\rm GL}(1)$ components, apply arithmetic reciprocity \eqref{rec} and then dualize the ${\rm GL}(3)$ component. A major technical difficulty is the fact that this leads to Kloosterman sums with multiplicative inverses in the arguments, so that the Kuznetsov formula at general cusps for possibly non-squarefree levels is required. In contrast, our approach never touches the ${\rm GL}(1)$ components, but dualizes the ${\rm GL}(3)$ component twice. This more symmetric set-up avoids complications with the Kuznetsov formula and makes a classical amplification procedure possible, which seems to be hard to implement in the work of \cite{KY}. The idea of coupling a $4=3+1$ structure along with arithmetic reciprocity \eqref{rec} goes back to important work of X.\ Li \cite{Li} and was first used for varying levels by the second author in \cite{Kh}.
\subsection{A roadmap} Sections \ref{basic} -- \ref{voronoi-formula} contain essentially known material, in particular the Kuznetsov formula and the Voronoi summation formula tailored in suitable form for later reference. An important technical ingredient in Section \ref{kuz-formula} is a detailed Fourier expansion of a complete orthonormal basis (including oldforms) in terms of Hecke eigenvalues. Lemma \ref{final-decay} will be used at the very end of the argument to verify that the integral transform $\mathscr{T}^{\pm}_{s', w'}\mathfrak{h}$ is weakly admissible. To treat holomorphic functions of small weight as in Theorem \ref{cor2}, we use a combination of the Petersson and the Kuznetsov formula and certain special functions that are also introduced in Section \ref{kuz-formula}.
Section \ref{multiple} features a multiple Dirichlet series in three variables that contains both ${\rm GL}(3)$ Fourier coefficients and Kloosterman sum. It is carefully designed as the combinatorial hinge between the Kuznetsov formula and the Voronoi formula, so that both formulas can be applied without extra technical complications. If $F= {\tt E}_0$ is an Eisenstein series, we take some time to compute the Laurent expansion of the triple pole in one of the variables that will contribute to the main term $\mathcal{N}_{q, \ell}(s, w; \mathfrak{h})$ in Theorem \ref{thm1}. These are related to the ``fake main terms'' in \cite{KY}. After a somewhat involved computation it turns out that the Laurent coefficients (as a function of the two other variables) have a beautiful formulation as a quotient of Riemann zeta-functions and their derivatives.
Section \ref{admissible} gives a precise definition of admissible functions and establishes some technical properties of these functions. Section \ref{int-trafo} is also of analytic nature and studies in detail a certain integral transform that will later become the central part of the transform $\mathscr{T}_{s', w'}^{\pm}$ in \eqref{final-trafo}. The key point here is to obtain analytic continuation and suitable decay properties which is achieved by careful contour shifts.
Section \ref{prelim-formula} features a prototype of the reciprocity formula based on the triad Voronoi-reciprocity-Voronoi. At this point we still work with points $s, w$ whose real part is sufficiently large. The use of Kuznetsov formula at the beginning and at the end to obtain the full 5-step procedure outlined in the previous subsection, as well as the analytic continuation to $\Re w \geqslant \Re s \geqslant 1/2$, is postponed to Section \ref{proof1} where the proof of Theorem \ref{thm1} is completed. The necessary local computations to deal with the ramified primes $p \mid \ell q$ are provided in Section \ref{local}. It is then a simple task to derive Theorems \ref{thm2} -- \ref{cor2} in the final two sections.
Finally we mention that while the abstract procedure Kuznetsov-Voronoi-reciprocity-Voronoi-Kuznetsov is completely symmetric, there are certain differences on a technical level before and after the reciprocity formula. In the latter case, some local factors at primes $p \mid \ell q$ and the archimedean place converge absolutely only slightly to the right of $\Re w \geqslant \Re s \geqslant 1/2$. To get the desired analytic continuation, some extra maneuvers are necessary, as can be seen in the proof of Lemma \ref{pole2} and in particular in Lemma \ref{sec-prop} that will be applied both in Sections \ref{prelim-formula} and \ref{proof1}.
\subsection{Notation and conventions} Throughout, the letter $\varepsilon$ denotes an arbitrarily (and sufficiently) small positive real number, not necessarily the same at each occurrence. All implied constants may depend on $\varepsilon$ (where applicable), but this is suppressed from the notation. We will frequently encounter multiple sums and integrals of holomorphic functions in one or more variables. All expressions of this type will be absolutely convergent by which we mean in addition locally uniformly convergent in the auxiliary variables, so that they represent again holomorphic functions. Also implied constants depending on complex variables are always understood to depend locally uniformly on them.
By an $\varepsilon$-neighbourhood of a strip $c_1 \leqslant \Re s \leqslant c_2$ we mean the open set $c_1 -\varepsilon < \Re s < c_2 + \varepsilon$, and similarly for multidimensional tubes. Occasionally we will encounter meromorphic functions in multidimensional tubes which will be holomorphic outside a finite set of polar divisors given by affine hyperplanes. By $v_p$ we denote the usual $p$-adic valuation. We write $a \mid b^{\infty}$ to mean that $a$ divides some power of $b$. We write $\Bbb{N}_0 = \Bbb{N}\cup \{0\}$.
\textbf{Acknowledgements.} The authors would like to thank Matthew Young and Guohua Chen as well as the referees for useful comments and suggestions that helped improving and correcting various aspects of teh paper.
\section{Basic analysis}\label{basic}
\subsection{Mellin transform} Throughout this paper we denote by \begin{equation}\label{mellin1} \widehat{W}(s) = \int_0^{\infty} W(x) x^s \frac{dx}{x} \end{equation}
the Mellin transform of a function $W : [0, \infty) \rightarrow \Bbb{C}$ of class $C^J$ for some $J \in \Bbb{N}_0$ satisfying $x^j W^{(j)}(x) \ll_{J, a, b} \min(x^{-a}, x^{-b})$ for some $-\infty < a < b < \infty$ and all $0 \leqslant j \leqslant J $. The function $\widehat{W}$ is then (initially) defined in $a < \Re s < b$ as an absolutely convergent integral and satisfies $\widehat{W}(s) \ll (1 + |s|)^{-J}$ in this region (in some cases it may be continued meromorphically to a larger region).
The inverse Mellin transform of a function $\mathcal{W}$ that is holomorphic in a strip containing $a \leqslant \Re s \leqslant b$ and bounded by $\mathcal{W}(s) \ll (1 +|s|)^{-r}$ for some $r > n \in \Bbb{N}$, is given by \begin{equation}\label{mellin2} \widecheck{\mathcal{W}}(x) = \int_{(c)} \mathcal{W}(s) x^{-s} \frac{ds}{2\pi i}, \end{equation} where here and in the following $(c)$ denotes the line $\Re s = c$ with $a \leqslant c \leqslant b$. We have \begin{equation}\label{mellin} x^j\widecheck{\mathcal{W}} {}^{(j)}(x) \ll \min(x^{-a}, x^{-b}), \end{equation} for $j = 0, 1, \ldots, n-1$.
\subsection{The gamma function} We recall the reflection, recursion and duplication formula $$\Gamma(s)\Gamma(1-s)= \pi \sin(\pi s)^{-1}, \quad \Gamma(s+1) = s\Gamma(s), \quad \Gamma(s)\Gamma(s+1/2) = \sqrt{\pi} 2^{1-2s} \Gamma(2s).$$
For fixed $\sigma \in \Bbb{R}$, real $|t| \geqslant 3$, and any $M > 0$ we recall Stirling's formula \begin{equation}\label{stir}
\Gamma(\sigma + it) = e^{-\frac{\pi}{2}|t|} |t|^{\sigma-\frac{1}{2}} \exp\left(i t \log \frac{|t|}{e}\right)g_{\sigma, M}(t) + O_{\sigma, M}(|t|^{-M}), \end{equation} where \begin{equation}\label{stir1}
g_{\sigma, M}(t) = \sqrt{2\pi} \exp\left(\frac{\pi}{4}(2\sigma-1) i \, \text{sgn}(t) \right) + O_{\sigma, M}\left(|t|^{-1}\right) \end{equation} and also \begin{equation}\label{above}
|t|^j g^{(j)}_{\sigma, M}(t) \ll_{j, \sigma, M} 1 \end{equation} for all fixed $j \in \Bbb{N}_0$. This implies in particular bounds of the shape
$$\frac{\Gamma(as + b)}{\Gamma(cs + d)} \ll_{a, b, c, d, \sigma} (1+ |t|)^{(a-c)\sigma + b-d}$$
for $a, b, c, d \in \Bbb{R}$, $s = \sigma + it$ and $\min_{n \in \Bbb{N}_0} |as + b + n| \geqslant 1/10$.
We define $\Gamma_{\Bbb{R}}(s) = \pi^{-s/2} \Gamma(s/2)$ and for $j \in \{0, 1\}$ we write $$ G_j(s) = \frac{\Gamma_{\Bbb{R}}(s + j)}{\Gamma_{\Bbb{R}}(1-s + j)} = 2(2\pi)^{-s}\Gamma(s) \begin{cases} \cos(\pi s/2), & j = 0,\\ \sin(\pi s/2), & j = 1,\end{cases}$$ where the last identity follows from the reflection and duplication formula of the gamma function. We also need the linear combination \begin{equation}\label{Gpm} G^{\pm}(s) = \frac{1}{2}G_0(s) \pm \frac{i}{2} G_1(s) = \Gamma(s) (2\pi)^{-s} \exp(\pm i\pi s/2), \end{equation}
which by \eqref{stir} and \eqref{above} satisfies for $|t| \geqslant 3$
the bound \begin{equation}\label{bound-g}
G^{\pm}(s) \ll (1 + |s|)^{\Re s - 1/2} e^{-\pi \max(0, \pm \Im s)/2} \end{equation} and the asymptotic formula \begin{equation}\label{asymp-g}
G^{\pm}(\sigma + it) = |t|^{\sigma - 1/2} \exp\Bigl(i t \log \frac{|t|}{2\pi e}\Bigr) \tilde{g}_{\sigma, M}(t) + O_{\sigma, M}\left(|t|^{-M}\right) \end{equation} with $\tilde{g}_{\sigma, M}$ satisfying \eqref{above}. Note that $G^{\pm}(\sigma + it)$ is exponentially decaying for $\pm t \rightarrow \infty$ by \eqref{bound-g}.
From the theory of hypergeometric integrals we quote the following formula \cite[(2.2.1.2)]{PBM}: if $a, b, c, d\in \Bbb{C}$ satisfy $\Re(c+d)- 1> \Re(a+b) > 0$, then \begin{equation}\label{hyper} \int_{(\sigma)} \frac{\Gamma(a + s) \Gamma(b-s)}{\Gamma(c+s)\Gamma(d-s)} \frac{ds}{2\pi i} = \frac{\Gamma(a+b)\Gamma(c+d-1-a-b)}{\Gamma(c-a)\Gamma(d-b)\Gamma(d+c-1)} \end{equation} where the integral on the left is absolutely convergent and the path of integration separates the poles, i.e.\ $\Re b > \sigma > -\Re a$, for instance $\sigma = \frac{1}{2}(b-a)$.
\subsection{Integration by parts} We quote a useful integration by parts lemma \cite[Lemma 8.1]{BKY}, which follows immediately from \cite[(8.6)]{BKY}.
\begin{lemma} \label{integrationbyparts}
Let $Y \geqslant 1$, $X, Q, U, R > 0$, $B \in \Bbb{N}$,
and suppose that $w$
is a smooth function with support on some interval $[\alpha, \beta]$, satisfying \begin{equation*} w^{(j)}(t) \ll X U^{-j} \end{equation*} for $0 \leqslant j \leqslant B$. Suppose $H$
is a smooth function on $[\alpha, \beta]$ such that \begin{equation*}
|H'(t)| \gg R, \quad
H^{(j)}(t) \ll Y Q^{-j} \,\, \text{for } j=2, 3, \dots, B. \end{equation*} Then \begin{equation*} I = \int_{\Bbb{R}} w(t) e^{i H(t)} dt
\ll_B (\beta - \alpha) X \left[(QR/\sqrt{Y})^{-B/2} + (RU)^{-B/2}\right]. \end{equation*} \end{lemma}
\section{The Kuznetsov formula}\label{kuz-formula}
Let $N \in \Bbb{N}$ and write \begin{equation}\label{0}
N \nu(N) := [\Gamma_0(1) : \Gamma_0(N)], \quad \text{i.e.} \quad \nu(N) = \prod_{p \mid N} \left(1 + \frac{1}{p}\right). \end{equation} We equip $\Gamma_0(N) \backslash \Bbb{H}$ with the inner product \begin{equation}\label{inner} \langle f, g\rangle := \int_{\Gamma_0(N)\backslash \Bbb{H}} f(z) \bar{g}(z) \frac{dx\, dy}{y^2}. \end{equation} The Kuznetsov/Petersson formulae require a sum over an orthonormal basis of automorphic forms of a given level $N$. While vectors in different representation spaces of $Z(\Bbb{A}_{\Bbb{Q}}) {\rm GL}_2(\Bbb{Q})\backslash {\rm GL}_2(\Bbb{A}_{\Bbb{Q}})/K_0(N)$ are always orthogonal, such a space may contain more than one $L^2$-normalized vector of given $K_{\infty}$-type. Classically this corresponds to the existence of oldforms. In the following we give explicit formulae for the Fourier coefficients of a complete orthonormal basis.
\subsection{Fourier expansion of Eisenstein series} The unitary Eisenstein spectrum of $\Gamma_0(N)$ can be parametrized by a continuous parameter $s = 1/2 + it$ together with pairs $(\chi, M)$, where $\chi$ is a primitive Dirichlet character of conductor $c_{\chi}$ and $M \in \Bbb{N}$ satisfies $c_{\chi}^2 \mid M \mid N$. For our purposes this adelic parametrization is more convenient than the classical parametrization by cusps. For fixed $t$ and $\chi$ the Eisenstein series for various $M$ belong to the same representation space. The corresponding spectral decomposition is proved in \cite{GJ}, and the Fourier expansion of these Eisenstein series is computed explicitly in \cite[Section 5]{KL} (see \cite[Section 2.7]{BH} for similar calculations over number fields). In the following, we quote from \cite{KL}. In the notation of \cite{KL} we have $M = \prod p^{i_p}$. We define as in \cite[(5.22)]{KL} \begin{equation}\label{1} \mathfrak{n}^2(M) := \frac{1}{M} \prod_{\substack{p \mid N\\ p \nmid (M, N/M)}} \frac{p}{(p+1)} \prod_{ p \mid (M, N/M)} \frac{p-1}{p+1} = \frac{1}{M} \tilde{ \mathfrak{n}}^2(M), \end{equation} say. Next, as in \cite[Section 5.5]{KL} we define \begin{equation*} N_1 = \prod_{p \mid N/M} p^{v_p(N)}.
\end{equation*}
The normalized Eisenstein series $E_{\chi, M, N}(z, s)$ of level $N$ corresponding to $(\chi, M)$ has the Fourier expansion $$E_{\chi, M, N}(z, 1/2 + it) = \rho^{(0)}_{\chi, M, N}(t, y) + \frac{2 \pi^{1/2 + it} y^{1/2}}{\Gamma(1/2 + it)} \sum_{n \not= 0} \rho_{\chi, M, N}(n, t) K_{it}(2 \pi |n| y) e(nx),$$ where for $n \not= 0$ we have
$$\rho_{\chi, M, N}(n, t) = \frac{C(\chi, M)}{(N \nu(N))^{1/2}\mathfrak{n}(M) L^{(N)}(1 + 2it, \chi^2)} \cdot \frac{|n|^{it}}{M^{1 + 2it}} \sum_{\substack{c \mid n \\ (c, N_1) = 1}} \frac{\chi(c)}{c^{2it}} \sum_{\substack{d \, (\text{mod }M)\\ (d, M) = 1}} \chi(d) e\left(\frac{d n/c}{M}\right),$$
for a constant $|C(\chi, M)| = 1$. As usual, the superscript $L^{(N)}$ denotes that the Euler factors at primes dividing $N$ are omitted. This follows from \cite[(5.32), (5.34)]{KL} after re-normalizing by $\mathfrak{n}(M)$, cf.\ \cite[(5.22)]{KL}, and taking into account that the adelic inner product in \cite{KL} differs from \eqref{inner} by a factor \eqref{0}. We write \begin{equation}\label{3}
M = c_{\chi} M_1 M_2, \quad \text{where} \quad (M_2, c_{\chi}) = 1, \quad M_1 \mid c_{\chi}^{\infty}, \end{equation}
so that $c_{\chi} \mid M_1$ and $(M_1, M_2) = 1$. Then by the Chinese Remainder Theorem the $d$-sum equals $$r_{M_2}(n/c) \sum_{\substack{d \, (\text{mod } c_{\chi}M_1)\\ (d, c_{\chi})= 1}} \chi(d) e\left(\frac{d \overline{M_2} (n/c)}{c_{\chi} M_1}\right) = \delta_{cM_1 \mid n} r_{M_2}(n/c)\chi(M_2) \bar{\chi}\left(\frac{n}{cM_1}\right)M_1 \sum_{\substack{d \, (\text{mod } c_{\chi}) \\ (d, c_{\chi})= 1}} \chi(d) e\left(\frac{d}{c_{\chi}}\right), $$
where the Gau{\ss} sum on right hand side has absolute value $c_{\chi}^{1/2}$ and \begin{equation}\label{rama}
r_M(n) = \sum_{\substack{d \, (\text{mod } M)\\ (d, M) = 1}} e\left(\frac{dn}{M}\right) = \sum_{d \mid (n, M)} d \mu\left(\frac{M}{d}\right)
\end{equation}
is the Ramanujan sum. Recalling \eqref{1}, we conclude
$$\rho_{\chi, M, N}(n, t) = \frac{C(\chi, M, t)|n|^{it}}{(N \nu(N))^{1/2}\tilde{ \mathfrak{n}}(M) L^{(N)}(1 + 2it, \chi^2)} \left(\frac{M_1}{M_2}\right)^{1/2} \sum_{\substack{cM_1 \mid n \\ (c, N_1) = 1}} \frac{\chi(c)}{c^{2it}} r_{M_2}(n/c)\bar{\chi}\left(\frac{n}{cM_1}\right) $$
where $|C(\chi, M, t)| = 1$. We note that the condition $(c, N_1) = 1$ is equivalent to $(c, N/M) = 1$. Finally we insert the expression on the right of \eqref{rama} for the Ramanujan sum getting the final formula
\begin{equation}\label{rho-eis}
\rho_{\chi, M, N}(n, t) = \frac{C(\chi, M, t)|n|^{it}}{(N \nu(N))^{1/2}\tilde{ \mathfrak{n}}(M) L^{(N)}(1 + 2it, \chi^2)} \left(\frac{M_1}{M_2}\right)^{1/2} \sum_{\delta \mid M_2} \delta \mu(M_2/\delta) \bar{\chi}(\delta) \sum_{\substack{cM_1\delta f = n \\ (c, N/M) = 1}}\frac{\chi(c)}{c^{2it}} \bar{\chi}(f)
\end{equation}
with the notation \eqref{0}, \eqref{1}, \eqref{3}. In particular, using the third bound of \eqref{hl} below, we find for $t \in \Bbb{R}$ that
\begin{equation}\label{rho-bound}
\rho_{\chi, M, N}(n, t)\ll ((1 + |t|)N|n|)^{\varepsilon} \frac{(M_1M_2)^{1/2}}{N^{1/2}} \leqslant ((1 + |t|)N|n|)^{\varepsilon} .
\end{equation}
\subsection{Fourier expansion of cusp forms} The cuspidal spectrum is parametrized by pairs $(f, M)$ of $\Gamma_0(N)$-normalized newforms $f$ of level $N_0 \mid N$ and integers $ M \mid N/N_0$. This comes from orthonormalizing the set $\{z \mapsto f(Mz) : M \mid N/N_0\}$ (which belongs to the same representation space) by Gram-Schmidt\footnote{Of course, there are many ways to apply the Gram-Schmidt procedure. For concreteness, we choose the one used in \cite{BM} and referenced below; this fixes our basis uniquely}. If $f$ is a Maa{\ss} form, we write the Fourier expansion of the pair $(f, M)$ as
$$ (2 \cosh(\pi t) y)^{1/2} \sum_{n \not= 0} \rho_{f, M, N}(n) K_{it}(2 \pi |n| y) e(nx).$$ A standard Rankin-Selberg computation shows (verbatim as in \cite[(2.8) - (2.9)]{Bl}) that for $M = 1$, i.e.\ for newforms, one has \begin{equation}\label{rho1}
|\rho_{f, 1, N}(1)|^2 = \frac{1}{L(1, \text{Ad}^2 f)N\nu(N)} \prod_{p \mid N_0} \left(1 - \frac{1}{p^2}\right) \end{equation} and $\rho_{f, 1, N}(n) = \rho_{f,1, N}(1) \lambda_f(n)$ for $n \in \Bbb{N}$. Here we use the notation \begin{equation}\label{Lad} L(s, \text{Ad}^2 f) = \zeta^{(N)}(2s) \sum_n \frac{\lambda_f(n^2)}{n^s} \end{equation} (which may differ from the corresponding Langlands $L$-function by finitely many Euler factors).
For general $M$, the Fourier coefficients of the orthonormal basis were computed in\footnote{The conditions $(d, N_0) = 1$ in the definition of $r_f$ and $(b, N_0) = 1$ in the definitions of $\alpha$ and $\beta$ were erroneously missing there, see {\tt www.uni-math.gwdg.de/blomer/corrections.pdf}.} \cite[Lemma 9]{BM}. Define \begin{displaymath} \begin{split} & r_{f}(c) = \sum_{b \mid c} \frac{\mu(b) \lambda_f(b)^2}{b} \Bigl(\sum_{\substack{d \mid b\\ (d, N_0) = 1}}\frac{1}{d} \Bigr)^{-2}, \quad \alpha(c) = \sum_{\substack{b \mid c \\ (b, N_0) = 1}} \frac{\mu(b)}{b^2}, \quad \beta(c) = \sum_{\substack{b \mid c\\ (b, N_0) = 1} } \frac{\mu^2(b)}{b^2},\end{split} \end{displaymath} define $\mu_f(c)$ as the Dirichlet series coefficients of $L(s, f)^{-1} $ and let $$\xi'_{f}(M, d) = \frac{\mu(M/d) \lambda_f(M/d)}{r_{f}(M)^{1/2} \beta(M/d)}, \quad \xi''_{f}(M, d) = \frac{\mu_f(M/d) }{r_{f}(M)^{1/2} \alpha(M)^{1/2}}.$$ Write $M = M_1 M_2$ where $M_1$ is squarefree, $M_2$ is squarefull and $(M_1, M_2)=1$. Then \begin{equation}\label{xi-arithmetic} \xi_{f}(M, d) := \xi_{f}'(M_1, (M_1, d)) \xi_{f}''(M_2, (M_2, d)) \ll M^{\varepsilon}(M/d)^{\vartheta} \end{equation}
and \begin{equation}\label{rho-cusp}
\rho_{f, M, N}(n) = \frac{1}{L(1, \text{Ad}^2 f)^{1/2}(N\nu(N))^{1/2}} \prod_{p \mid N_0} \left(1 - \frac{1}{p^2}\right)^{1/2} \sum_{d \mid M} \xi_{f}(M, d) \frac{d}{M^{1/2}} \lambda_f(n/d)
\end{equation}
for $n \in \Bbb{N}$ with the convention $\lambda_f(n) = 0$ for $n \not \in \Bbb{Z}$. For $-n \in \Bbb{N}$ we have $\rho_{f, M, N}(n) = \epsilon_f \rho_{f, M, N}(-n)$ where $\epsilon_f \in \{\pm 1\}$ is the parity of $f$. Using a standard Rankin-Selberg bound as well as the standard lower bound \eqref{hl} below, we obtain \begin{equation}\label{rho-cusp-bound}
\sum_{n \leqslant x} |\rho_{f, M, N}(n) |^2 \ll \frac{1}{N}(Nx(1 + |t_f|))^{\varepsilon} x . \end{equation}
If $f$ is a holomorphic newform of weight $k$ and level $N_0 \mid N$, we write the Fourier expansion of the pair $(f, M)$ as $$\left(\frac{2\pi^2}{\Gamma(k)}\right)^{1/2}\sum_{n > 0} \rho_{f, M, N}(n) (4\pi n)^{(k-1)/2} e(nz).$$
If $M=1$, then \eqref{rho1} remains true (cf.\ e.g.\ \cite[(2.1)]{BKY}), and so does \eqref{rho-cusp} for $n \in \Bbb{N}$ and arbitrary $M \mid N/N_0$ as well as \eqref{rho-cusp-bound} with $k_f$ in place of $(1 + |t_f|)$. For negative $n$ we define $\rho_{f, M, N}(n) = 0$.
\subsection{Versions of the Kuznetsov formula} For $x > 0$ we define the integral kernels \begin{displaymath} \begin{split} & \mathcal{J}^+(x, t) := \frac{\pi i}{ \sinh(\pi t)} (J_{2 it}(4 \pi x) - J_{-2it}(4 \pi x)), \\ & \mathcal{J}^-(x, t) := \frac{\pi i}{ \sinh(\pi t)} (I_{2 it}(4 \pi x) - I_{-2it}(4 \pi x)) = 4 \cosh(\pi t) K_{2it}(4\pi x),\\ & \mathcal{J}^{\text{hol}}(x, k) := 2\pi i^k J_{k-1}(4\pi x) = \mathcal{J}^+(x, (k-1)/(2i)) , \quad k \in 2\Bbb{N}.
\end{split}
\end{displaymath}
For future reference we record the Mellin transforms of these kernels:
\begin{equation}\label{mellin-j} \begin{split}
\widehat{\mathcal{J}^+(., t)}(u) & = \frac{\pi i (2\pi)^{-u} }{2\sinh(\pi t)} \left(\frac{\Gamma(u/2 + it)}{ \Gamma(1 - u/2 + it)} - \frac{\Gamma(u/2 - it)}{ \Gamma(1 - u/2 - it)} \right) \\ & = (2\pi)^{-u} \Gamma(u/2 + it)\Gamma(u/2 - it) \cos(\pi u /2), \\
\widehat{\mathcal{J}^-(., t)}(u) & = (2\pi)^{-u} \Gamma(u/2 + it)\Gamma(u/2 - it) \cosh(\pi t),\\
\widehat{\mathcal{J}^{\text{hol}}(., k)}(u) & = i^k(2\pi)^{-u} \pi \Gamma( (u + k-1)/2) \Gamma( (1+k-u)/2)^{-1}.
\end{split}
\end{equation}
These formulae follow from \cite[17.43.16 \& 18]{GR} together with the reflection formula for the gamma function.
Let $h$ be an even function $h$ satisfying $h(t) \ll (1+|t|)^{-2-\delta}$ for $t \in \Bbb{R} \cup [-i\vartheta, i\vartheta]$, and let $h^{\text{hol}} : 2\Bbb{N} \rightarrow \Bbb{C}$ be a function satisfying $h^{\text{hol}}(k) \ll k^{-2-\delta}$ for some $\delta > 0$.
Then for $q\in \Bbb{N}$, $n, m \in \Bbb{Z} \setminus \{0\}$ we define
\begin{equation*} \begin{split} & \mathcal{A}_q^{\text{Maa{\ss}}}(n, m; h) := \sum_{q_0M \mid q} \sum_{f \in \mathcal{B}^{\ast}(q_0)} \rho_{f, M, q}(n) \overline{ \rho_{f, M, q}(m)} h(t_f),\\ & \mathcal{A}_q^{\text{Eis}}(n, m; h) := \sum_{c_{\chi}^2\mid M \mid q} \int_{\Bbb{R}} \rho_{\chi, M, q}(n, t) \overline{ \rho_{\chi, M, q}(m, t)} h(t) \frac{dt}{2\pi},\\ & \mathcal{A}_q^{\text{hol}}(n, m; h^{\text{hol}}) := \sum_{q_0M \mid q} \sum_{ f \in \mathcal{B}_{\text{hol}}^{\ast}(q_0) } \rho_{f, M, q}(n) \overline{ \rho_{f, M, q}(m)} h^{\text{hol}}(k_f).
\end{split} \end{equation*}
(Note that by definition $A_q^{\text{hol}}(n, m; h^{\text{hol}}) = 0$ if $n$ or $m$ are negative.) If in addition $h$ is holomorphic in an $\varepsilon$-neighbourhood of $|\Im t| \leqslant 1/2$ and still satisfies the bound $h(t) \ll (1+|t|)^{-2-\delta}$ in this region, then for $n, m \in \Bbb{N}$ the Bruggeman-Kuznetsov formula states\footnote{Classical references parametrize the Eisenstein series by cusps, but the same proof works of course with the spectral expansion of Eisenstein series given by \cite{GJ}.} (e.g.\ \cite[Theorem 16.3]{IK}, cf.\ \cite[(16.19)]{IK} for the normalization there)
\begin{equation}\label{kuz1} \begin{split}
\mathcal{A}_q^{\text{Maa{\ss}}}(n, m; h) + \mathcal{A}_q^{\text{Eis}}(n, m; h)
& = \delta_{n, m} \int_{-\infty}^{\infty} h(t) \frac{ t \tanh(\pi t) dt}{2\pi^2} + \sum_{q \mid c} \frac{S(n, m, c)}{c}\mathscr{K}h \Bigl(\frac{\sqrt{nm}}{c}\Bigr), \end{split} \end{equation} where \begin{equation}\label{H}
\mathscr{K}h(x) := \int _{-\infty}^{\infty} \mathcal{J}^+(x, t) h(t) t \tanh(\pi t) \frac{dt}{2\pi^2}
\end{equation}
and $S(m, n, c) \ll (m, n, c)^{1/2} \tau(c) c^{1/2}$ is the Kloosterman sum. Absolute convergence of the $c$-sum in \eqref{kuz1} follows from shifting the contour in \eqref{H} to $\Im t = \pm 3/8$, say.
The formula \eqref{kuz1} is complemented by the Petersson formula \cite[Proposition 14.5]{IK}
\begin{equation}\label{pet}
\begin{split} & \mathcal{A}_q^{\text{hol}}(n, m; \delta_{k = k_0}) = \frac{k_0-1}{2\pi^2} \Bigg(\delta_{n, m} + 2\pi i^{-k_0} \sum_{q\mid c} \frac{S(m, n, c)}{c} J_{k_0-1}\Big(\frac{4\pi \sqrt{mn}}{c}\Big)\Bigg)
\end{split}
\end{equation} for $k_0 \in 2\Bbb{N}$, $n, m \in \Bbb{N}$. Sometimes it is useful to apply the Petersson formula \eqref{pet} and the Kuznetsov formula \eqref{kuz1} simultaneously. With this in mind, for a pair of functions $\mathfrak{h} = (h, h^{\text{hol}})$ satisfying the above conditions, we define \begin{equation}\label{kast} \mathscr{K}^{\ast} \mathfrak{h}(x) := \mathscr{K} h(x) + \sum_{k\in 2\Bbb{N}}i^{-k} \frac{k-1}{\pi}h^{\text{hol}}(k) J_{k-1}(4\pi x). \end{equation} Then
\begin{equation}\label{kuz-all} \begin{split} \mathcal{A}_q(n, m; \mathfrak{h}) & := \mathcal{A}_q^{\text{Maa{\ss}}}(n, m; h) + \mathcal{A}_q^{\text{Eis}}(n, m; h)+ \mathcal{A}_q^{\text{hol}}(n, m; h^{\text{hol}}) \\
& = \delta_{n, m} \mathscr{N}\mathfrak{h} + \sum_{q \mid c} \frac{S(n, m, c)}{c}\mathscr{K}^{\ast}\mathfrak{h} \Bigl(\frac{\sqrt{nm}}{c}\Bigr), \end{split} \end{equation} for $n, m \in \Bbb{N}$ with $$\mathscr{N}\mathfrak{h} := \int_{-\infty}^{\infty} h(t) t \tanh(\pi t) \frac{dt}{2\pi^2} +\sum_{k\in 2\Bbb{N}} \frac{k-1}{2\pi^2} h^{\text{hol}}(k).$$
Conversely, if $H \in C^3((0, \infty))$ satisfies\footnote{There are various assumptions in the literature, e.g. \cite[(16.38)]{IK}, \cite[Theorem 2]{Ku}, \cite[(2.4.6)]{Mo}. We follow the latter, although the precise exponents make no difference for our argument.} $x^j H^{(j)}(x) \ll \min(x, x^{-3/2})$ for $0 \leqslant j \leqslant 3$ and $n, m, q \in \Bbb{N}$, then we have \cite[Theorem 16.5]{IK}
\begin{equation}\label{kuz2} \begin{split} \sum_{q \mid c} &\frac{S(\pm n, m, c)}{c}H \left(\frac{\sqrt{nm}}{c}\right) = \mathcal{A}_q(\pm n, m; \mathscr{L}_{\pm} H)
\end{split} \end{equation} where \begin{equation}\label{Hback} \begin{split} \mathscr{L}_{\pm}H = (\mathscr{L}^{\pm}H, \mathscr{L}^{\text{hol}}H), \quad \mathscr{L}^{\diamondsuit}H = \int_{0}^{\infty} \mathcal{J}^{\diamondsuit}(x, .) H(x) \frac{dx}{x}. \end{split} \end{equation} for $\diamondsuit \in \{+, -, \text{hol}\}$.
The formulas \eqref{kuz-all} and \eqref{kuz2} are inverses to each other, and so are the corresponding integral transforms: for $H \in C^2((0, \infty))$ with $H^{(j)}(x) \ll \min(x^{1/2}, x^{-5/2})$ for $j = 0, 1, 2$ we have the Sears-Titchmarsh inversion formula (cf.\ \cite[(4.9)]{ST} or \cite[(A.4)]{Ku}) \begin{equation}\label{inv} \mathscr{K}^{\ast} \mathscr{L}_+ H = \mathscr{K}^{\ast}(\mathscr{L}^+ H, \mathscr{L}^{\text{hol}}H) = H.
\end{equation}
To treat holomorphic cusp forms of small weight, we use the following special functions, borrowed from \cite[Section 2]{BHM}. For integers $3 < b < a$ with $a \equiv b$ (mod 2) let \begin{equation}\label{defHab} H(x)= H_{a, b}(x) = i^{b-a} J_a(4\pi x) (4\pi x)^{-b}. \end{equation}
By \cite[(2.21)]{BHM} we have
$$\mathscr{L}^{+}H(t) = \frac{b!}{2^b} \prod_{j=0}^b \left(t^2 + \Big(\frac{a+b}{2} - j\Big)^2\right)^{-1}, \quad \mathscr{L}^{\text{hol}}H(k) = \frac{b!}{2^b} \prod_{j=0}^b \left(\Big(\frac{(1-k)i}{2}\Big)^2 + \Big(\frac{a+b}{2} - j\Big)^2\right)^{-1}.$$
Obviously we have \begin{equation}\label{pos}
\mathscr{L}^{+}H(t) >0, \quad \mathscr{L}^{+}H(t) \asymp (1 + |t|)^{-2b-2}, \quad t \in \Bbb{R} \cup [-i\vartheta, i \vartheta] \end{equation} and \begin{equation*}
\mathscr{L}^{\text{hol}}H(k) > 0 \,\,\, \text{for}\,\,\, 2 \leqslant k \leqslant a-b, \quad |\mathscr{L}^{\text{hol}}H(k)| \asymp k^{-2b-2}.
\end{equation*} We choose a constant $c(a, b)$ such that \begin{equation}\label{hposhol} h_{\text{pos}}^{\text{hol}} (k) := \mathscr{L}^{\text{hol}}H_{a, b}(k) + \delta_{k > a-b} c(a, b) k^{-2b-1} > 0 \end{equation} for all $k \in 2\Bbb{N}$, and we put $h_{\text{pos}} := \mathscr{L}^{+}H_{a, b}$.
Then by \eqref{pos} and \eqref{hposhol}, the pair $\mathfrak{h}_{\text{pos}} = (h_{\text{pos}},h^{\text{hol}}_{\text{pos}} )$ (which depends on $a, b$, but this is suppressed from the notation) is strictly positive on the entire spectrum $\mathcal{T}_0$, and we have by \eqref{inv}, \eqref{defHab} and \eqref{hposhol} the identity
\begin{equation}\label{pos-id}
\mathscr{K}^{\ast} \mathfrak{h}_{\text{pos}}(x) = i^{b-a} J_a(4\pi x) (4\pi x)^{-b} + \sum_{a-b < k \in 2\Bbb{N}} \frac{i^{-k}(k-1)}{\pi}\frac{c(a, b)}{k^{2b+1}} J_{k-1}(4\pi x).
\end{equation}
For future reference we state the following lemma, for which we recall the notation \eqref{mellin2}.
\begin{lemma}\label{final-decay} Let $0 \leqslant \vartheta \leqslant 7/64$ and $A \geqslant 5$ an integer. Let $\Phi$ be a function that is holomorphic in $-2\vartheta-\varepsilon < \Re u < A$ for some $\varepsilon > 0$ and satisfies $\Phi(u) \ll (1 + |u|)^{-A}$ in this region. \\
{\rm a)} We have $$\mathscr{L}^{\pm}\widecheck{\Phi}(t) \ll_A (1 + |t|)^{-A}, \quad t \in \Bbb{R} \cup [-i\vartheta, i\vartheta],\quad\quad \mathscr{L}^{\text{{\rm hol}}}\widecheck{\Phi}(k) \ll_A k^{-A}, \quad k \in 2\Bbb{N}.$$
{\rm b)} If $0 \leqslant \tau < 1$ and $\Phi$ is in addition meromorphic in $\Re u \geqslant -2\tau - \varepsilon$ with finitely many poles at $u_1, \ldots, u_n \in \Bbb{C}$, then $\mathscr{L}^{\pm}\widecheck{\Phi}(t)$ has meromorphic continuation to $|\Im t| < \tau$ with poles at most at $t = \pm i u_j/2$, $j = 1, \ldots, n$. \end{lemma}
\textbf{Proof.} a) Using the definitions \eqref{Hback} and \eqref{mellin2} and exchanging integrals, we have for $t \in \Bbb{R} \cup [-i\vartheta, i\vartheta]$ that \begin{equation}\label{start} \mathscr{L}^{\pm}\widecheck{\Phi}(t) = \int_{(2\vartheta + \varepsilon/2)}\widehat{ \mathcal{J}^{\pm}(., t)}(u) \Phi(-u) \frac{du}{2\pi i} \end{equation}
as an absolutely convergent integral (by Stirling's formula), where $ \widehat{\mathcal{J}^{\pm}(., t)}(u)$ is given in \eqref{mellin-j}. This contour is within the region of holomorphicity of $\Phi(-u)$, but to the right of the poles of $\widehat{\mathcal{J}^{\pm}(., t)}$ for $t \in \Bbb{R} \cup [-i\vartheta, i\vartheta]$. To deduce the required bound, we may assume without loss of generality that $|t| \geqslant 1$. We shift the contour to the left to $\Re u = -A + 1/2$. On the way we pick up poles at $u = -2n \pm 2 it$, $n \in \{0, 1, \ldots, [\frac{1}{2}(A-\frac{1}{2})]\}$ with residues of the shape
$$\pm 2 (2\pi)^{2n \pm 2 it} \cosh(\pi t) \Gamma(-n \mp 2it) \Phi(2n \pm 2it) \ll (1 + |t|)^{-A-1/2}$$ with various sign combinations. We estimate the remaining integral by Stirling's formula \eqref{stir} as
$$\ll \int_{(-A+1/2)} \big((1 + |\Im u + 2t|)(1 + |\Im u - 2t|)\big)^{-(2A+1)/4} (1 + |u|)^{-A} |du| \ll (1 + |t|)^{-A-1/2},$$ which completes the proof for $\diamondsuit \in \{+, -\}$.
The holomorphic case is similar (and a little easier). Here we have nothing to show for $k \leqslant A$, and for $k \geqslant A$, the contour shift to the left $\Re u = -A+1 \in \Bbb{N}$ does not produce any poles. The remaining integral can be estimated in the same way using the stronger bound $$ \widehat{ \mathcal{J}^{\text{hol}}(., k)}(-A+1 + iw) \ll \frac{\Gamma((-A + k + i w)/2)}{\Gamma( (k + A- iw)/2) } \ll (k + |w|)^{-A}$$ for $A \in \Bbb{N}$ by the repeated application of the recursion formula for the gamma function.
b) We fix some $t_0 \in \Bbb{C}$ with $|\Im t_0| < \tau$. By assumption we can bend the contour in \eqref{start} to be to the right of $2t_0$ (within the region of meromorphicity of $\Phi$), but the unbounded part still runs at real part equal\footnote{to ensure absolute convergence since a priori we have no growth condition for $|\Im t| > 2\vartheta + \varepsilon$ available.} to $2\vartheta + \varepsilon/2$. We may pick up poles with residue $\widehat{\mathcal{J}^{\pm}(., t)}(-u_j)$, which are meromorphic functions with poles at most at $t = \pm i u_j/2$, while the remaining integral is holomorphic in a neighbourhood of $t_0$.
\section{The ${\rm GL}(3)$ Voronoi summation formula}\label{voronoi-formula}
Let $F$ be a cuspidal automorphic form for the group ${\rm SL}_3(\Bbb{Z})$. As in the introduction, let $\theta \leqslant 5/14$ be an admissible bound for the Ramanujan-Petersson conjecture for $F$. We denote its archimedean Langlands parameters by $\mu = (\mu_1, \mu_2, \mu_3)$ with \begin{equation}\label{mu}
\mu_1 + \mu_2 + \mu_3 = 0,\quad |\Re \mu_j| \leqslant \theta, \end{equation}
and we denote its Fourier coefficients by $A(n, m)$ as in \cite{Go}. They satisfy $A(n, m) = \overline{A(m, n)}$, \begin{equation}\label{hecke} A(n, m) = \sum_{d \mid (n, m)} \mu(d) A(n/d, 1) A(1, m/d) \end{equation} (as follows from \cite[Theorem 6.4.11]{Go} and M\"obius inversion), and the Rankin-Selberg bound \begin{equation}\label{gl3-RS}
\sum_{n m^2 \leqslant x} |A(n, m)|^2 \ll x. \end{equation} Individually, we only know \begin{equation}\label{indiv}
A(n, m) \ll (nm)^{\theta+\varepsilon}.
\end{equation}
We will always regard $F$ as fixed, and all implied constants may depend on $F$, in particular on $\mu$, but we suppress this from the notation.
The Voronoi summation formula \cite[Theorem 1.8]{MS} states the following: for $c, d, m \in \Bbb{Z}$ with $(c, d) = 1$, $c, m > 0$ and a smooth compactly supported weight function $w : \Bbb{R}_{>0} \rightarrow \Bbb{C}$ we have \begin{equation}\label{Voronoi-ms} \sum_{n} A(m, n) e\left(\frac{n \bar{d}}{c}\right) w(n) = c\sum_{\pm} \sum_{n_1 \mid cm} \sum_{n_2 > 0} \frac{A(n_2, n_1)}{n_2n_1} S( md, \pm n_2, mc/n_1)W^{\pm} \left(\frac{ n_2n_1^2}{c^3m}\right), \end{equation} where $$W^{\pm}(x) = \int_{(1)} x^{-s}\mathcal{G}_{\mu}^{\pm} (s+1) \widehat{w}(-s) \frac{ds}{2\pi i} $$ with \begin{equation}\label{defgmu} \begin{split} \mathcal{G}_{\mu}^{\pm}(s) & = \frac{1}{2} \prod_{j=1}^3G_0(s+\mu_j) \pm \frac{1}{2i} \prod_{j=1}^3G_1(s+\mu_j)\\ & = 4(2\pi)^{-3s} \prod_{j=1}^3 \Gamma(s + \mu_j)\Bigg(\prod_{j=1}^3 \cos\Big(\frac{\pi(s + \mu_j)}{2}\Big) \pm \frac{1}{i} \prod_{j=1}^3\sin\Big(\frac{\pi(s + \mu_j)}{2}\Big)\Bigg) . \end{split} \end{equation}
(To compare this with the formula in \cite{MS}, write $w = w_{\text{even}} + w_{\text{odd}}$ with $w_{\text{even}} (x)= \frac{1}{2}( w(x) + w(-x))$, $w_{\text{odd}} (x)= \frac{1}{2}( w(x) - w(-x))$, and observe that $M_{0}w_{\text{even}} = M_{1}w_{\text{odd}} = \widehat{w}$.) The following lemma summarizes the analytic properties of $\mathcal{G}_{\mu}^{\pm}$.
\begin{lemma}\label{lemG} The functions $\mathcal{G}_{\mu}^{\pm}$ are meromorphic on $\Bbb{C}$ with poles at $s= -n - \mu_k$, $n \in \Bbb{N}_0$, $k \in \{1, 2, 3\}$. In particular, they are holomorphic in $\Re s > \theta$. Away from poles they satisfy the bound\footnote{Recall that all implied constants may depend on $\mu$.} \begin{equation}\label{bound-G}
\mathcal{G}_{\mu}^{\pm}(s) \ll_{\Re s} (1 + |\Im s|)^{3 \Re s - \frac{3}{2}} e^{-\pi \max(0, \pm \Im s/2)}. \end{equation} In particular, $\mathcal{G}_{\mu}^{\pm}(s)$ is exponentially decaying for $\pm \Im s \rightarrow \infty$.
For $| t| \geqslant 3$ sufficiently large we have the asymptotic formula \begin{equation}\label{asymp-G} \begin{split}
\mathcal{G}_{\mu}^{\pm}(\sigma + i t) = |t|^{3\sigma - \frac{3}{2}}
& \exp\left( 3i t\log \frac{|t |}{2\pi e}\right) w_{\sigma, M, \mu}(t) + O_{\sigma, M}\left(|t|^{-M}\right) \end{split} \end{equation} with
$$|t|^j w_{\sigma, M, \mu}^{(j)}(t) \ll_{j, \sigma, M} 1$$ for all $j, M \in \Bbb{N}_0$. Finally, for $n \in \Bbb{N}_0$ and $\epsilon_2 \in \{\pm 1\}$ we have \begin{equation}\label{new-formula}
\sum_{\epsilon_1 \in \{\pm 1\}} e^{\epsilon_1 i\pi n/2} \mathcal{G}_{-\mu}^{-\epsilon_1}(1-v + n) \mathcal{G}_{\mu}^{\epsilon_1\epsilon_2}(v) = \frac{1+\epsilon_2}{2} (2\pi)^{-3 n} \prod_{k=1}^3 \prod_{j=1}^n (v-j+\mu_k). \end{equation} \end{lemma}
\textbf{Proof.} Equations \eqref{bound-G} and \eqref{asymp-G} follow directly from from \eqref{mu} and Stirling's formula \eqref{stir}. For the proof of \eqref{new-formula} we use the recursion and reflection formula of the gamma function to see that $\mathcal{G}_{-\mu}^{-\epsilon_1}(1-v + n) \mathcal{G}_{\mu}^{\epsilon_1\epsilon_2}(v) $ equals \begin{displaymath} \begin{split}
16(2\pi)^{-3-3n} &\Big[ \prod_{k=1}^3 \frac{\pi \prod_{j=1}^n (-v+j-\mu_k) }{ \sin(\pi(v+\mu_k))} \Big]
\Big[ \prod_{k=1}^3 \cos\left(\frac{\pi(v+\mu_k)}{2}\right) +\frac{\epsilon_1\epsilon_2}{i} \prod_{k=1}^3 \sin\left(\frac{\pi(v+\mu_k)}{2}\right)\Big] \\ & \Big[\prod_{k=1}^3 \cos\left(\frac{\pi(1-v+n-\mu_k)}{2}\right) - \frac{\epsilon_1}{i} \prod_{k=1}^3 \sin\left(\frac{\pi(1-v+n-\mu_k)}{2}\right)\Big]. \end{split} \end{displaymath} Hence $e^{\epsilon_1 i \pi n/2}\mathcal{G}_{-\mu}^{-\epsilon_1}(1-v + n) \mathcal{G}_{\mu}^{\epsilon_1\epsilon_2}(v) $ equals \begin{displaymath} \begin{split}
2(2\pi)^{ -3n} &\Big[ \prod_{k=1}^3 \frac{ \prod_{j=1}^n (-v+j-\mu_k) }{ \sin(\pi(v+\mu_k))} \Big]
\Big[ \prod_{k=1}^3 \cos\left(\frac{\pi(v+\mu_k)}{2}\right) +\frac{\epsilon_1\epsilon_2}{i} \prod_{k=1}^3 \sin\left(\frac{\pi(v+\mu_k)}{2}\right)\Big] \\ & (-1)^n \Big[\epsilon_1 i \prod_{k=1}^3 \cos\left(\frac{\pi( v+\mu_k)}{2}\right) + \prod_{k=1}^3 \sin\left(\frac{\pi(v+\mu_k)}{2}\right)\Big]. \end{split} \end{displaymath} Summing over $\epsilon_1 \in \{\pm 1\}$, we can drop all terms depending linearly on $\epsilon_1$, and \eqref{new-formula} follows easily from the addition theorem of the $\sin$-function.
\\
We rewrite the summation formula in terms of Dirichlet series as follows. For positive integers $c, d, m$ with $(c, d) = 1$ let \begin{equation}\label{defPhi} \Phi(c, \pm d, m; v) := \sum_{n > 0} A(m, n) e\left(\pm \frac{n \bar{d}}{c}\right) n^{-v}. \end{equation} By \eqref{hecke} -- \eqref{indiv} this is absolutely convergent in $\Re v > 1$ and satisfies the uniform bound \begin{equation}\label{phibound}
\Phi(c, \pm d, m; v) \ll \alpha(m) := m^{\varepsilon}\max_{d \mid m} |A(d, 1)|, \quad \Re v \geqslant 1+\varepsilon.
\end{equation} Moreover, let
\begin{equation}\label{defXi}
\Xi(c, \pm d, m; v) := c \sum_{n_1 \mid cm} \sum_{n_2 > 0} \frac{A(n_2, n_1)}{n_2n_1} S(\pm md, n_2, mc/n_1) \left(\frac{ n_2n_1^2}{c^3m}\right)^{-v}.
\end{equation} This is absolutely convergent in $\Re v > 0$ and satisfies the uniform bound \begin{equation}\label{xibound} \Xi(c, \pm d, m; v) \ll (mc^3)^{1/2 + \Re v+\varepsilon} , \quad \Re v \geqslant \varepsilon, \end{equation} using Weil's bound for Kloosterman sums and again \eqref{hecke} -- \eqref{indiv}.
By converting additive characters into multiplicative characters we see that the functions $\Phi(c, \pm d, m; .) $ and $\Xi(c, \pm d, m; .)$ are, up to finitely many Euler factors at primes dividing $m$ that are holomorphic in $\Re v > \theta$ resp.\ $\Re v \geqslant \theta-1$, linear combinations of $L$-functions corresponding to $F \times \chi$ for Dirichlet characters $\chi$. This shows that $\Phi(c, \pm d, m; .)$ is of finite order and analytic in (an $\varepsilon$-neighbouhood of) $\Re v \geqslant 1/2$ and $\Xi(c, \pm d, m; .)$ is of finite order and analytic in $\Re v \geqslant -1/2$.
The Voronoi formula is equivalent to the vector-valued functional equation \begin{equation}\label{func1} \left(\begin{matrix}\Phi(c, d, m; v) \\ \Phi(c, -d, m; v)\end{matrix}\right) = \left(\begin{matrix} \mathcal{G}_{\mu}^+(1-v) & \mathcal{G}_{\mu}^-(1-v) \\ \mathcal{G}_{\mu}^-(1-v) & \mathcal{G}_{\mu}^+(1-v) \end{matrix}\right) \left(\begin{matrix} \Xi(c, d, m; -v) \\ \Xi(c, -d, m; -v)\end{matrix}\right). \end{equation}
Inverting the ``scattering matrix'' (using \eqref{new-formula} with $n=0$), we obtain \begin{equation}\label{func2} \left(\begin{matrix}\Xi(c, d, m; v) \\ \Xi(c, -d, m; v)\end{matrix}\right) = \left(\begin{matrix} \mathcal{G}_{-\mu}^-(-v) & \mathcal{G}_{-\mu}^+(-v) \\ \mathcal{G}_{-\mu}^+(-v) & \mathcal{G}_{-\mu}^-(-v) \end{matrix}\right) \left(\begin{matrix} \Phi(c, d, m; -v) \\ \Phi(c, -d, m; -v)\end{matrix}\right). \end{equation} In particular, both $\Xi(c, \pm d, m; .) $ and $\Phi(c, \pm d, m; .)$ are of finite order and entire.
By the Phragm\'en-Lindel\"of principle, \eqref{phibound}, \eqref{xibound} and \eqref{bound-G} we obtain \begin{equation}\label{xi} \begin{split}
& \Phi(c, d, m; v) \ll_{\Re v} \alpha(m)\left(mc^3 (1 + |\Im v|)^3\right)^{ \max\left(\frac{1}{2} - \Re v, \frac{1}{2} (1 - \Re v), 0\right)+\varepsilon},\\
&\Xi(c, d, m; v) \ll_{\Re v} \alpha(m) \left(mc^3\right)^{ \max\left(\frac{1}{2} + \Re v, \frac{1}{2} (1 + \Re v), 0\right)+\varepsilon} (1 + |\Im v|^3)^{\max(0, -\Re v-\frac{1}{2}, - \frac{1}{2}\Re v)+\varepsilon}. \end{split} \end{equation}
The Voronoi formula is a consequence of the functional equation of the twisted $L$-functions $L(s, F \times \chi)$ and relations in the unramified ${\rm GL}(3)$ Hecke algebra. Therefore the functional equations \eqref{func1} and \eqref{func2} must continue to hold if $F$ is an Eisenstein series, except that in this case the functions $\Xi(c, \pm d, m; .) $ and $\Phi(c, \pm d, m; .)$ are not necessarily entire any more (since $L(s, F \times \chi)$ will have a pole if $F$ is Eisenstein and $\chi$ is trivial). We are only interested in the case when $F= E_0$ is the minimal parabolic Eisenstein series with trivial parameters $\mu= (0, 0, 0)$. In this case, $A(n, 1) = A(1, n) = \tau_3(n)$ is the ternary divisor function, and in general $A(n, m)$ is given by \eqref{hecke}. As above, we see that $\Phi(c, \pm d, m; .)$ can only have a (triple) pole at $v=1$ in $\Re v \geqslant 1/2$ and $\Xi(c, \pm d, m; .) $ can only have a (triple) pole at $v=0$ in $\Re v \geqslant -1/2$. (A classical Voronoi formula for ${\tt E}_0$ analogous to \eqref{Voronoi-ms} with extra polar terms has been worked out in \cite{Li2}.) The corresponding Laurent expansions are computed in Lemma \ref{lem6} and \ref{pole1} below.
\section{A multiple Dirichlet series}\label{multiple}
Fix $\ell, q \in \Bbb{N}$ and $s, w, u \in \Bbb{C}$. If
\begin{equation}\label{variables}
\begin{split}
&\Re (w+u/2)> 1,
\quad \Re(3s + u/2) > 2, \quad \Re (3s - u/2) > 4, \quad \Re u < -1/2, \end{split}
\end{equation}
we define
\begin{equation}\label{defD}
\mathcal{D}^{\pm}_{q, \ell}(s, u, w) := \sum_{(c, d) = 1} \sum_{\substack{q \mid mc\\ \ell \mid md}} \frac{ \Xi(c, \pm d, m; -1 + s + u/2)}{c^{3s+u/2 -1} m^{s+w}d^{w+u/2}}
\end{equation}
and
\begin{equation}\label{deftildeD} \widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w) := \sum_{(c, d) = 1} \sum_{\substack{q \mid mc\\ \ell \mid md}} \frac{\Phi(c, \pm d, m; 1 - s - u/2)}{c^{3s + u/2 - 1}m^{w+s} d^{w + u/2}} \end{equation}
with $\Phi$ as in \eqref{defPhi} and $\Xi$ as in \eqref{defXi}, satisfying the functional equations
\begin{equation}\label{Dfunc}
\begin{split}
& \mathcal{D}^{\pm}_{q, \ell}(s, u, w) = \mathcal{G}^{\mp}_{-\mu}(1-s-u/2) \widetilde{\mathcal{D}}^{+}_{q, \ell}(s, u, w) + \mathcal{G}^{\pm}_{-\mu}(1-s-u/2) \widetilde{\mathcal{D}}^{-}_{q, \ell}(s, u, w),\\ & \widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w) = \mathcal{G}^{\pm}_{\mu}(s+u/2) \mathcal{D}^{+}_{q, \ell}(s, u, w) + \mathcal{G}^{\mp}_{\mu}(s+u/2) \mathcal{D}^{-}_{q, \ell}(s, u, w)
\end{split}
\end{equation}
by \eqref{func2} and \eqref{func1}. Recall that the numerator of \eqref{defD} and \eqref{deftildeD} is holomorphic in the region \eqref{variables} if $F$ is cuspidal.
If $F = {\tt E}_0$, the only polar divisor of $\mathcal{D}^{\pm}_{q, \ell}(s, u, w)$ in $\Re(-1+s+u/2) \geqslant -1/2$ can occur at $-1+s+u/2 = 0$, and the only polar divisor of $\tilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w)$ in $\Re(1-s-u/2) \geqslant 1/2$ can be at $1-s-u/2 = 1$. We will see in a moment in Lemmas \ref{pole1} and \ref{pole2} that the Laurent expansions at these poles are independent of the $\pm$ sign. Since $\mathcal{G}_{\mu}^+(s) + \mathcal{G}^-_{\mu}(s)$ has a triple zero at $s=1$, the functional equations \eqref{Dfunc} imply that no other polar divisors of $\mathcal{D}^{\pm}_{q, \ell}(s, u, w)$ and $\widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w)$ can exist in the domain \eqref{variables} of definition.
By \eqref{xi} the triple sums in \eqref{defD} and \eqref{deftildeD} are absolutely convergent: the first condition in \eqref{variables} ensures absolute convergence of the $d$-sum, the other three conditions ensure absolute convergence of the $c$-sum; the $m$-sum requires $\Re(s+w) > 1$, $\Re(w - \frac{1}{2}u) > 1/2$ and $\Re (\frac{1}{2}s + w - \frac{1}{4}u) > 1$ for absolute convergence, which also follows from \eqref{variables}. Again by \eqref{xi} we have the bounds
\begin{equation}\label{boundD}
\begin{split}
& \mathcal{D}^{\pm}_{q, \ell}(s, u, w) \ll_{s, w} (1 + |u|)^{3\max(0, \frac{1}{2} - \Re (s + \frac{1}{2}u), \frac{1}{2} - \Re( \frac{1}{2}s + \frac{1}{4}u))+\varepsilon},\\
& \widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w) \ll_{s, w} (1 + |u|)^{3\max(0, -\frac{1}{2} + \Re (s + \frac{1}{2}u), \Re( \frac{1}{2}s + \frac{1}{4}u))+\varepsilon}
\end{split}
\end{equation}
for $s, u, w$ satisfying \eqref{variables} (away from the pole if $F = {\tt E}_0$).
If in addition $\Re(s + u/2) > 1$, we can insert the definition \eqref{defXi} into \eqref{defD}. This gives the following alternative representation.
\begin{lemma}\label{lem6} Let $s, u, w \in \Bbb{C}$ satisfy
\begin{equation}\label{lem1-cond}
\Re(w + u/2) > 1, \quad \Re(s + u/2) > 1, \quad \Re u < -1/2.
\end{equation}
Then
\begin{equation}\label{alt}
\mathcal{D}^{\pm}_{q, \ell}(s, u, w) = \sum_{\ell \mid r} \sum_{q \mid n_1c} \sum_{n_2} \frac{A(n_2, n_1)S(\pm r, n_2, c)}{n_2^{s + u/2 }n_1^{2s} c^{1-u}r^{w+u/2}}.
\end{equation}
\end{lemma}
\textbf{Proof.} The conditions \eqref{lem1-cond} imply \eqref{variables}. By absolute convergence we can re-arrange the sums to see that $\mathcal{D}^{\pm}_{q, \ell}(s, u, w)$ equals
\begin{displaymath} \begin{split}
& \sum_{ d} \sum_{\ell \mid md} \frac{1}{(md)^{w+u/2}} \sum_{n_1} \sum_{\substack{q \mid mc\\ (c, d) = 1\\ \frac{n_1}{(n_1, m)} \mid c}} \frac{1}{(mc)^{1-u}}\sum_{n_2} \frac{A(n_2, n_1)}{n_2^{s + u/2 } n_1^{2s + u-1}} S\left(\pm md, n_2, \frac{mc}{n_1}\right) \\ &= \sum_{ d} \sum_{\ell \mid md} \frac{1}{(md)^{w+u/2}} \sum_{fg = m} \sum_{(\nu_1, g) = 1} \sum_{\substack{q \mid mc\\ (c, d) = 1\\ \nu_1 \mid c}}\frac{1}{(mc)^{1-u}} \sum_{n_2} \frac{A(n_2, \nu_1f)}{n_2^{s + u/2 } (\nu_1f)^{2s + u-1}} S\left(\pm md, n_2, \frac{gc}{\nu_1}\right) \\ & = \sum_{ d} \sum_{ \ell \mid fgd}\frac{1}{(fgd)^{w+u/2}} \sum_{(\nu_1, g) = 1} \sum_{\substack{ (\nu_1 \gamma, d) = 1\\ q \mid fg\nu_1\gamma}} \frac{1}{(fg\nu_1 \gamma)^{1-u}}\sum_{n_2} \frac{A(n_2, \nu_1f)S(\pm fgd, n_2, g\gamma ) }{n_2^{s + u/2 } (\nu_1f)^{2s + u-1}} .
\end{split} \end{displaymath} There is a bijection between integer quintuples $$(d, f, g, \nu_1, \gamma) \quad \text{satisfying} \quad \ell \mid fgd, \,q \mid fg\nu_1\gamma, \,(\nu_1, g) = 1 , \,(\nu_1\gamma, d) = 1$$ and integer triples $$(n_1, r, c) \quad \text{satisfying} \quad q\mid n_1 c, \, \ell \mid r$$ given by $$(n_1, r, c) = (\nu_1f, f gd , g\gamma)$$ with inverse map $$(d, f, g, \nu_1, \gamma) = \left(\frac{r}{(n_1c, r)}, (n_1, r), \frac{(n_1c, r)}{(n_1, r)}, \frac{n_1}{(n_1, r)}, \frac{c(n_1, r)}{(n_1c, r)} \right). $$ This shows the desired formula \eqref{alt}.\\
Next we compute the Laurent expansion of $\mathcal{D}^{\pm}_{q, \ell}(s, u, w)$ at $u = 2-2s$ if $F = {\tt E}_0$. We write \begin{equation}\label{resR} \mathcal{D}_{q, \ell}^{\pm}(s, u, w) = \sum_{j=1}^3 \frac{\mathcal{R}_{q,\ell;j}(s, w)}{(u - (2-2s))^j} + O(1). \end{equation} We will see in the proof of the next lemma that $\mathcal{R}_{q,\ell; j}(s, w)$ is independent of the $\pm$ sign.
\begin{lemma}\label{pole1} Let $F = {\tt E}_0$ and suppose $(q, \ell) = 1$. Then the Laurent coefficients $\mathcal{R}_{q, \ell;j}(s, w)$ are meromorphic in $\Bbb{C}\times \Bbb{C}$. In an $\varepsilon$-neighbourhood of the region $\Re w \geqslant \Re s\geqslant 1/2$, the function \begin{equation}\label{hol} \big((s - \textstyle\frac{1}{2})(w-s)(s+w-1)\big)^4\mathcal{R}_{q, \ell;j}(s, w) \end{equation}
is holomorphic and bounded by
$O_{s, w} \left((q\ell)^{-1+\varepsilon}\right)$.
\end{lemma}
\textbf{Proof.} In the region \eqref{lem1-cond} we have (recall \eqref{hecke} and $A(n, 1) = A(1, n) = \tau_3(n)$) \begin{equation}\label{dql} \begin{split} \mathcal{D}_{q, \ell}^{\pm}(s, u, w) & = \sum_{\ell \mid r} \sum_{q \mid n_1 c} \sum_{d, b_1, b_2, b_3} \frac{\tau_3(n_1) \mu(d) S(\pm r, db_1b_2b_3, c)}{n_1^{2s} d^{3s + u/2}(b_1 b_2 b_3)^{s + u/2} c^{1-u} r^{w+u/2}}\\ & =
\sum_{\ell \mid r} \sum_{q \mid n_1c} \sum_{d} \sum_{ b_1, b_2, b_3\, (\text{mod }c)} \frac{\tau_3(n_1)\mu(d) S(\pm r, db_1b_2b_3, c)}{n_1^{2s} d^{3s + u/2} c^{1+3s+u/2} r^{w+u/2}}
\prod_{j=1}^3\zeta\left(s + u/2, \frac{b_j}{c}\right) \end{split} \end{equation} where \begin{equation}\label{hurwitz} \zeta(s, \alpha) = \sum_{(n+\alpha) > 0} \frac{1}{(n+\alpha)^s} = \frac{1}{s-1} - \psi(\alpha) - \gamma(\alpha)(s-1) + \ldots \end{equation} is the Hurwitz zeta function, and $\psi$, $\gamma$ are suitable functions denoting the Taylor coefficients. We note that \begin{equation}\label{averagezeta} \begin{split} \sum_{b \, (\text{mod } m)} \zeta(s, b/m)& = m^s \zeta(s)\\ & = \frac{m}{s-1} + m(\log m + \gamma) + \left(\frac{1}{2}m \log^2 m + \gamma m \log m - m \gamma_1\right)(s-1) + \ldots \end{split} \end{equation} for Euler's constant $\gamma = 0.577\ldots$ and another constant $\gamma_1 \in \Bbb{R}$.
Inserting \eqref{hurwitz} into \eqref{dql}, we compute the Laurent coefficients as
$$\mathcal{R}_{q ,\ell;j}(s, w) =
\sum_{\ell \mid r} \sum_{q \mid n_1c} \sum_{d} \sum_{ b_1, b_2, b_3\, (\text{mod }c)} \frac{\tau_3(n_1) \mu(d) S(\pm r, db_1b_2b_3, c)}{n_1^{2s} d^{2s + 1} c^{2+2s} r^{w+1-s}}
\rho_j\left(\frac{b_1}{c}, \frac{b_2}{c}, dcr\right)$$
where
\begin{equation}\label{rho}
\begin{split}
& \rho_1(n_1, n_2, m) = 8, \quad \rho_2(n_1, n_2, m) = -12\psi(n_1) - 4\log m, \\
& \rho_3(n_1, n_2, m) = 6\psi(n_1)\psi(n_2) - 3\gamma(n_1) + (\log m)^2 + 6 \psi(n_1) \log m.
\end{split}
\end{equation}
We open the Kloosterman sum and evaluate the $b_3$-sum getting \begin{displaymath} \begin{split}
\mathcal{R}_{q,\ell; j}(s, w)& =
\sum_{\ell \mid r} \sum_{q \mid n_1c} \sum_{d} \sum_{\substack{ b_1, b_2\, (\text{mod }c)\\ c\mid db_1b_2}} \frac{\tau_3(n_1) \mu(d) S(\pm r, 0, c)}{n_1^{2s}d^{2s + 1} c^{1+2s} r^{w+1-s}}
\rho_j\left(\frac{b_1}{c}, \frac{b_2}{c}, dcr\right)\\ & =
\sum_{q \mid n_1c} \sum_{d} \sum_{a \mid c} \sum_{\ell \mid ra}\frac{\mu(c/a)}{a^{w-s}} \sum_{\substack{ b_1, b_2\, (\text{mod }c)\\ c\mid db_1b_2}} \frac{\tau_3(n_1) \mu(d)
}{n_1^{2s}d^{2s + 1} c^{1+2s} r^{w+1-s}}\rho_j\left(\frac{b_1}{c}, \frac{b_2}{c}, dca r\right). \end{split} \end{displaymath} At this point it is clear that the $\pm$-sign plays no role. We order this by $\beta_i = (b_i, c)$ getting \begin{equation}\label{1new} \mathcal{R}_{q,\ell; j}(s, w) =
\sum_{q \mid n_1c}\sum_{d} \sum_{a \mid c} \sum_{\ell \mid ra}\sum_{\substack{\beta_1, \beta_2 \mid c\\ c \mid d \beta_1\beta_2}} \frac{\tau_3(n_1) \mu(d) \mu(c/a)
}{n_1^{2s}a^{w-s} d^{2s + 1} c^{1+2s} r^{w+1-s}}R_j\left(\frac{c}{\beta_1}, \frac{c}{\beta_2}, dca r\right) \end{equation} where \begin{equation*} \begin{split} & R_3(n_1, n_2, m) = 8\phi(n_1)\phi(n_2), \\ & R_2(n_1, n_2, m) = -12 \Psi^{\ast}(n_1)\phi(n_2) - 4 \phi(n_1)\phi(n_2) \log m, \\ & R_1(n_1, n_2, m) = 6\Psi^{\ast}(n_1) \Psi^{\ast}(n_2) - 3 G^{\ast}(n_1)\phi(n_2) + \phi(n_1)\phi(n_2) (\log m)^2 + 6\Psi^{\ast}(n_1)\phi(n_2) \log m \end{split} \end{equation*} with $$\Psi^{\ast}(n) = \sum_{\substack{b \, (\text{mod } n)\\ (b, n) = 1}} \psi(b/n), \quad G^{\ast}(n) = \sum_{\substack{b \, (\text{mod } n)\\ (b, n) = 1}} \gamma(b/n).$$
Let \begin{equation}\label{open} \phi_j(n) = \sum_{ab= n} \mu(a)b (\log b)^j \end{equation} denote the Dirichlet series coefficient of $(-1)^j \zeta^{(j)}(s-1)/\zeta(s)$. In particular, $\phi_0 = \phi$ is the Euler phi-function. Then $$\Psi(m) := \sum_{b\, (\text{mod }m)} \psi(b/m) = -m(\gamma + \log m)$$ by \eqref{averagezeta}, so that by M\"obius inversion $$\Psi^{\ast}(n) = \sum_{ab = n} \mu(a) \Psi(b) = -\gamma \phi_0(n)- \phi_1(n). $$ Similarly, we have $$G(m) := \sum_{b\, (\text{mod }m)} \gamma(b/m) = - \frac{1}{2} m \log^2 m-\gamma m \log m + m \gamma_1,$$ so that $$G^{\ast}(n) = \sum_{ab = n} \mu(a) G(b) = \gamma_1 \phi_0(n) - \gamma \phi_1(n) - \frac{1}{2} \phi_2(n) .$$ Altogether, $$R_j(n_1, n_2, m) = \sum_{\nu+\mu+\kappa \leqslant 3-j} C_{\nu, \mu, \kappa, j}\phi_{\nu}(n_1) \phi_{\mu}(n_2) (\log m)^{\kappa}$$ for certain constants $C_{\nu, \mu, \kappa, j}$. Substituting back into \eqref{1new} we obtain a linear combination of
\begin{equation*}
\sum_{q \mid n_1c} \sum_{d} \sum_{a \mid c} \sum_{\ell \mid r a} \sum_{\substack{\beta_1, \beta_2 \mid c\\ c \mid d \beta_1\beta_2}} \frac{\tau_3(n_1)\mu(d) \mu(c/a)
}{n_1^{2s}a^{w-s} d^{2s + 1} c^{1+2s} r^{w+1-s}}\phi_{\nu}(c/\beta_1) \phi_{\mu}(c/\beta_2) (\log dca r)^{\kappa}
\end{equation*} with $\nu + \mu + \kappa \leqslant 3-j$.
We make several changes of variables. We write $\beta_i \gamma_i = c$, switch to the co-divisor $\gamma_i$ and open the $\phi_{\nu}$-functions as in \eqref{open}, writing $\gamma_i = a_ib_i$. Next we replace $c$ with $ac$ and recast the previous display as $$ \sum_{q \mid n_1a c} \sum_{d} \sum_{\substack{a_1b_1, a_2b_2 \mid a c\\ a_1b_1a_2b_2 \mid acd }} \sum_{\ell \mid ra} \frac{\tau_3(n_1)\mu(d) \mu(c) \mu(a_1)\mu(a_2) b_1b_2 }{n_1^{2s}(c d)^{2s + 1} a^{1+s+w} r^{w+1-s}} (\log b_1)^{\nu} (\log b_2)^{\mu} (\log da^2c r)^{\kappa}. $$
We introduce the following generalized function \begin{equation}\label{defZ} Z_{q, \ell}(x_1, x_2, x_3) := \sum_{q \mid n_1a c} \sum_{d} \sum_{\substack{a_1b_1, a_2b_2 \mid a c\\ a_1b_1a_2b_2 \mid acd }} \sum_{\ell \mid ra} \frac{\tau_3(n_1)\mu(d) \mu(c) \mu(a_1)\mu(a_2) b_1b_2 }{n_1^{2s}(c d)^{2s + 1} a^{1+s+w} r^{w+1-s}} b_1^{x_1} b_2^{x_2} (da^2c r)^{x_3} \end{equation} so that $\mathcal{R}_{q, \ell; j}(s, w)$ is a linear combination of \begin{equation*}
\frac{\partial^{\nu}}{\partial x_1^{\nu}}\frac{\partial^{\mu}}{\partial x_2^{\mu}}\frac{\partial^{\kappa}}{\partial x_3^{\kappa}}Z_{q, \ell}(x_1, x_2, x_3)|_{x_1 = x_2 = x_3 = 0} \end{equation*} with $\nu + \mu + \kappa = 3 - j \leqslant 2$. The function $Z_{q, \ell}(x_1, x_2, x_3)$ can be written as an Euler product, and for a generic prime $p \nmid \ell q $ the $p$-Euler factor equals $$(1 - p^{-2s})^{-3}(1 - p^{-(1+w-s-x_3)})^{-1} \sum_{ \alpha = 0}^{\infty} \sum_{\alpha_1, \alpha_2, \gamma, \delta = 0}^1\sum_{\substack{\alpha_1+\beta_1, \alpha_2+\beta_2\leqslant \alpha+\gamma\\ \alpha_1 + \beta_1 + \alpha_2 + \beta_2 \leqslant \alpha+ \gamma + \delta}} \frac{(-1)^{\delta+\gamma + \alpha_1 + \alpha_2}p^{ \beta_1(1+x_1) + \beta_2(1+x_2)}}{p^{(\gamma + \delta)(2s+1-x_3) + \alpha(1 + s + w - 2x_3) }} .$$ This can be expressed in closed form using geometric series, for instance by distinguishing the 5 disjoint cases (i) $\alpha = \gamma = 0$, (ii) $\alpha + \gamma \geqslant 1$, $\alpha_1 = \alpha_2 = \beta_1 = \beta_2 = 0$, (iii) $\alpha + \gamma \geqslant 1$, $\alpha_1 = \beta_1 = 0$, $\alpha_2 + \beta_2 \geqslant 1$, (iv) $\alpha + \gamma \geqslant 1$, $\alpha_1 + \beta_1 \geqslant 1$, $\alpha_2 + \beta_2 = 0$ and (v) $\alpha + \gamma \geqslant 1$, $\alpha_1 + \beta_1 \geqslant 1$, $\alpha_2 + \beta_2 \geqslant 1$. After a lengthy, but completely straightforward computation we obtain the beautiful expression
\begin{equation}\label{euler}
\frac{(1 - p^{-2s})^{-3}(1 - p^{-(1+w-s-x_3)})^{-1}(1-p^{-(s+w-x_1-2x_3)})^{-1}(1-p^{-(s+w-x_2-2x_3)})^{-1}}{(1 - p^{-(1+s+w-2x_3)})^{-1}(1 - p^{-(2s - x_1-x_3)})^{-1}(1 - p^{-(2s - x_2-x_3)})^{-1}}.
\end{equation} The Euler factor at primes $p \mid q\ell$ can again be computed by a more complicated, but finite computation with geometric series, and it is clear that it is a rational function in $p^{-s}$, $p^{-w}$ and $p^{-x_j}$, $j = 1, 2, 3$, in particular meromorphic. Since the product of \eqref{euler} over all primes is a quotient of zeta-functions, it follows that $\mathcal{R}^{\pm,j}_{q, \ell}(s, w)$ is a meromorphic function.
In an $\varepsilon$-neighbourhood of the region $\Re w \geqslant \Re s \geqslant 1/2$ and $x_1 = x_2 = x_3 = 0$, the Euler factors at primes $p \mid \ell q$ converge absolutely. In particular, they are holomorphic, and we see after taking derivatives and putting $x_1 = x_2 = x_3 = 0$ that \eqref{hol} is holomorphic. To get the desired bound, we estimate the Euler factors at $p \mid q$ trivially as
$$4\underset{\nu_1 + \alpha + \gamma \geqslant v_p(q)}{\sum_{ \nu_1, \alpha, \rho = 0}^{\infty} \sum_{\gamma, \delta = 0}^1} \sum_{\beta_1 + \beta_2 \leqslant \alpha + \delta + \gamma} \frac{(\nu_1+2)(\nu_1+1)/2 }{p^{\nu_1 +2(\alpha + \delta+\gamma)+\rho - \beta_1 - \beta_2 + O(\varepsilon)}} \ll (p^{v_p(q)})^{-1+O(\varepsilon)}.$$ The same argument with the condition $\alpha + \rho \geqslant v_p(\ell)$ instead of $\nu_1 + \alpha + \gamma \geqslant v_p(q)$ applies for Euler factors at $p \mid \ell$. This completes the proof of the lemma. \\
We also need to study the Laurent expansion of the companion function \begin{equation}\label{tildeD} \widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w) = \sum_{j=1}^3 \frac{\tilde{\mathcal{R}}_{q,\ell; j}(s, w)}{(u + 2s)^j} + O(1) \end{equation} at $s + u/2 = 0$ in the case when $F = {\tt E}_0$ (again the Laurent coefficients are independent of the $\pm$ sign). \begin{lemma}\label{pole2} Let $F = {\tt E}_0$ and $(q, \ell) = 1$. Then the Laurent coefficients $\tilde{\mathcal{R}}_{q, \ell; j}(s, w)$ are meromorphic in $\Bbb{C}\times \Bbb{C}$. In an $\varepsilon$-neighbourhood of the region $\Re w \geqslant \Re s\geqslant 1/2$, the function \begin{equation}\label{hol1} \big((s - \textstyle\frac{1}{2})(w-s)(s+w-1)\big)^3\tilde{\mathcal{R}}_{q, \ell;j}(s, w) \end{equation}
is holomorphic and bounded by
$O_{s, w}\left((\ell q)^{\varepsilon} q^{-1}\right)$.
\end{lemma}
\textbf{Proof.} The proof is similar, so we can be brief. We have \begin{displaymath} \begin{split} \Phi(c, \pm d, m, v)& = \sum_n A(m, n) e\left(\pm \frac{n \bar{d}}{c}\right) n^{-v} = \sum_{r_1r_2 = m} \frac{\mu(r_1) A(r_2, 1)}{r_1^v} \sum_n \frac{A(1, n) e(\pm n r_1 \bar{d}/c)}{n^v}\\ & = \sum_{r_1r_2 = m} \frac{\mu(r_1) \tau_3(r_2)}{r_1^v} \frac{1}{c^{3v}} \sum_{a_1, a_2, a_3 \, (\text{mod }c)} e\left( \pm \frac{a_1a_2a_3 r_1 \bar{d}}{c}\right) \prod_{j=1}^3\zeta\left(v, \frac{a_j}{c}\right), \end{split} \end{displaymath} so that $$\widetilde{\mathcal{D}}^{\pm}_{q, \ell}(s, u, w) = \sum_{(c, d) = 1} \sum_{\substack{q \mid r_1r_2c\\ \ell \mid r_1r_2d}} \frac{\tau_3(r_2) \mu(r_1) }{r_1^{1+w-u/2}r_2^{s+w}c^{2-u} d^{w + u/2} }\sum_{a_1, a_2, a_3 \, (\text{mod }c)} e\left( \pm \frac{a_1a_2a_3 r_1 \bar{d}}{c}\right) \prod_{j=1}^3\zeta\left(1-s-\frac{u}{2}, \frac{a_j}{c}\right) .$$
We conclude $$ \tilde{\mathcal{R}}_{q,\ell;j}(s, w) = \sum_{(c, d) = 1} \sum_{\substack{q \mid r_1r_2c\\ \ell \mid r_1r_2d}} \frac{\tau_3(r_2) \mu(r_1) }{r_1^{1+w+s}r_2^{s+w}c^{2+2s} d^{w -s} }\sum_{a_1, a_2, a_3 \, (\text{mod }c)} e\left( \pm \frac{a_1a_2a_3 r_1 \bar{d}}{c}\right)\tilde{\rho}_j\left(\frac{a_1}{c}, \frac{a_2}{c}, \frac{d}{r_1c^{2}}\right)$$ with $\tilde{\rho}_j(n_1, n_2, m) = (-1)^j \rho_j(n_1, n_2, m)$ as in \eqref{rho}. We evaluate the $a_3$-sum getting $$ \tilde{\mathcal{R}}_{q,\ell;j}(s, w) = \sum_{(c, d) = 1} \sum_{\substack{q \mid r_1r_2c\\ \ell \mid r_1r_2d}} \frac{\tau_3(r_2)\mu(r_1) }{r_1^{1+w+s}r_2^{s+w}c^{1+2s} d^{w -s} }\sum_{\substack{a_1, a_2 \, (\text{mod }c)\\ c \mid a_1a_2r_1}} \tilde{\rho}_j\left(\frac{a_1}{c}, \frac{a_2}{c}, \frac{d}{r_1c^{2}}\right).$$ Again we order by $\alpha_i = (a_i, c)$ getting $$\tilde{\mathcal{R}}_{q,\ell;j}(s, w) = \sum_{(c, d) = 1} \sum_{\substack{q \mid r_1r_2c\\ \ell \mid r_1r_2d}} \frac{\tau_3(r_2)\mu(r_1) }{r_1^{1+w+s}r_2^{s+w}c^{1+2s} d^{w -s} } \sum_{\substack{\alpha_1, \alpha_2 \mid c\\ c \mid \alpha_1\alpha_2 r_1}} \tilde{R}_j\left(\frac{\alpha_1}{c}, \frac{\alpha_2}{c}, \frac{d}{r_1c^{2}}\right)$$ with $$\tilde{R}_j(n_1, n_2, m) = (-1)^j R_j(n_1, n_2, m) = \sum_{\nu+\mu+\kappa \leqslant 3-j} \tilde{C}_{\nu, \mu, \kappa, j}\phi_{\nu}(n_1) \phi_{\mu}(n_2) (\log m)^{\kappa}$$ for certain constants $\tilde{C}_{\nu, \mu, \kappa, j}$. Removing the coprimality condition $(c, d) = 1$ by M\"obius inversion and changing variables similarly as in the preceding proof we are left with terms of the form \begin{displaymath} \begin{split}
& \sum_{\substack{q \mid r_1r_2b c\\ \ell \mid r_1r_2b d}} \frac{\tau_3(r_2)\mu(r_1) \mu(b) }{r_1^{1+w+s}r_2^{s+w}c^{1+2s} b^{1+w+s}d^{w -s} } \sum_{\substack{\alpha_1, \alpha_2 \mid b c\\ \alpha_1\alpha_2 \mid b c r_1}} \phi_{\nu}(\alpha_1) \phi_{\mu}(\alpha_2) (\log d(b r_1)^{-1} c^{-2})^{\kappa}. \end{split} \end{displaymath} As before we define the generalized function \begin{equation}\label{tildeZ} \tilde{Z}_{q, \ell}(x_1, x_2, x_3) = \sum_{\substack{q \mid r_1r_2b c\\ \ell \mid r_1r_2b d}} \sum_{\substack{a_1b_1, a_2b_2 \mid b c\\ a_1b_1a_2b_2 \mid b c r_1}} \frac{\tau_3(r_2)\mu(r_1) \mu(b) \mu(a_1) \mu(a_2)b_1b_2 }{r_1^{1+w+s}r_2^{s+w}c^{1+2s} b^{1+w+s} d^{w -s} } b_1^{x_1} b_2^{x_2} ( d (b r_1)^{-1} c^{-2})^{x_3}. \end{equation} We compute the generic Euler factor at primes $p \nmid \ell q$ to be \begin{equation*}
\frac{(1 - p^{-s-w})^{-3}(1 - p^{-(w-s-x_3)})^{-1}(1-p^{-(2s-x_1+2x_3)})^{-1}(1-p^{-(2s-x_2+2x_3)})^{-1}}{(1 - p^{-(s+w+x_3-x_1)})^{-1}(1 - p^{-(s+w + x_3-x_2)})^{-1}(1 - p^{-(2s+1+2x_3)})^{-1}}.
\end{equation*} This is the same computation with different exponents after renaming the variables $(a, c, d, n_1, r) \rightarrow (r_1, r_2, b, c, d)$ in \eqref{defZ}. As before, we conclude that $\tilde{\mathcal{R}}_{q, \ell;j}(s, w)$ is meromorphic.
The estimation of \eqref{hol1} requires slightly more care, because the region $\Re w \geqslant \Re s \geqslant 1/2$, $x_1 = x_2 = x_3 = 0$ just fails to be inside the region of absolute convergence of the individual Euler factors. However, the only problem is caused by the $d$-sum in \eqref{tildeZ}.
Let first $p\mid q$. Summing over $d$ (restricted to powers of $p$) first, the same computation as before shows that the $p$-Euler factor equals $(1 - p^{-(w-s-x_3)})^{-1}$ times a holomorphic expression that is bounded by $\ll (p^{v_p(q)})^{-1+\varepsilon}$ in an $\varepsilon$-neighbourhood of $\Re w \geqslant \Re s \geqslant 1/2$, $x_1 = x_2 = x_3 = 0$.
Suppose now that $p\mid \ell$. We split the sum into $v_p(\ell) + 1$ terms with $p^{\lambda} \mid d$ for $0 \leqslant \lambda \leqslant v_p(\ell)$. In each of these we sum over $d$ first, and then estimate the rest as before. This gives an expression of the form $p^{\lambda(w-s-x_3)}(1 - p^{-(w-s-x_3)})^{-1}$ times an absolutely convergent sum that is uniformly bounded in an $\varepsilon$-neighbourhood of $\Re w \geqslant \Re s \geqslant 1/2$, $x_1 = x_2 = x_3 = 0$.
This yields the desired bound for \eqref{hol1} and completes the proof.
\section{Admissible functions}\label{admissible}
We call a function $H : \Bbb{R}_{> 0} \rightarrow \Bbb{C}$ \emph{admissible of type $(A, B)$} for some $A, B>5$ if it is of one of the following two types: \begin{itemize}
\item {[first kind]} we have $H(x) = \mathscr{K}h(x)$ (with $\mathscr{K}$ as in \eqref{H}) where $h$ is even and holomorphic in $|\Im t| < A$, such that $h(t) \ll (1 + |t|)^{-B-2}$ and $h$ has zeros at $\pm (n+1/2)i$, $n \in \Bbb{N}_0$, $n+1/2 < A$; \item {[second kind]} we have \begin{equation}\label{secondtype}
H(x) = \alpha_0 J_a(4\pi x)(4\pi x)^{-b} + \sum_{a-b < k \in 2\Bbb{N}} \alpha_k J_{k-1}(4\pi x) \end{equation}
for a constants $\alpha_0 \in \Bbb{C}$, $a, b \in \Bbb{N}$ with $a - b \geqslant A$, and $\alpha_k \ll k^{-B-2}$. \end{itemize}
If $\{H_j\} = \{\mathscr{K}h_j\}$ is a family of admissible functions of type $(A, B)$ (of the first kind), we call it \emph{uniformly admissible of type $(A, B)$} if the implied constant in the required bound $h_j(t) \ll (1 + |t|)^{-B-2}$
can be chosen independently of $j$.
We call a weakly admissible (i.e.\ satisfying \eqref{weakly}) pair $\mathfrak{h} = (h, h^{\text{hol}})$ \emph{admissible} if $h(t)$ is holomorphic in an $\varepsilon$-neighbourhood of $|\Im t| \leqslant 1/2$ and
$ \mathscr{K}^{\ast}\mathfrak{h}$ (with $\mathscr{K}^{\ast}$ as in \eqref{kast}) is admissible of type\footnote{We made no effort to optimize the requirements on $\mathfrak{h}$; most likely the number 500 could be reduced to 50, say, with no additional effort.} (500, 500), i.e.\ of one of the above two kinds. Such a pair satisfies in particular the assumptions of \eqref{kuz-all}. We call a family of pairs of the special shape $\{\mathfrak{h}_j = (h_j, 0)\}$ uniformly admissible if the family $\{h_j\}$ is uniformly admissible of type $(500, 500)$.
From now on, we agree on the \textbf{convention} that all implied constants may depend on admissible weight functions $\mathfrak{h}$ where applicable; however, if $\{\mathfrak{h}_j = (h_j, 0)\}$ is a uniformly admissible family, implied constants can always be chosen independently of $j$ (this applies also to \eqref{required-bound}, for instance). We will consider a uniformly admissible family only once in this paper, in the proof of Theorem \ref{cor1}.
\begin{lemma}\label{lem1} A pair $\mathfrak{h} = (h, h^{\text{{\rm hol}}})$ is admissible if \\ {\rm a)} $\mathfrak{h} = (0, \delta_{k = k_0})$ for some $k_0 > 500$;\\
{\rm b)} $\mathfrak{h} = (h, 0)$, where $h$ is even and holomorphic in $|\Im t| < 500$, such that $h(t) \ll (1 + |t|)^{-502}$ and $h$ has zeros at $\pm (n+1/2)i$, $n \in \Bbb{N}_0$, $n+1/2 < 500$;\\ {\rm c)} $\mathfrak{h} = (h_{\text{{\rm pos}}}, h_{\text{{\rm pos}}}^{\text{{\rm hol}}})$ as in \eqref{pos-id} with $a = 1000$, $b = 400$. This function is strictly positive on $\mathcal{T}_0 = (\Bbb{R} \cup [-i\vartheta, i \vartheta]) \times 2\Bbb{N}$. \end{lemma} This is clear from the definition, cf.\ also \eqref{pos}. A typical function satisfying the hypotheses of Lemma \ref{lem1}b) is $$h(t) = e^{-t^2} \prod_{n \leqslant 500} \big(t^2 + (n+ \textstyle \frac{1}{2})^2\big).$$
We need two technical properties of admissible functions that are presented in the following two lemmas.
\begin{lemma}\label{admis1} Let $H$ be an admissible function of type $(A, B)$. \\ {\rm a)} The Mellin transform $\widehat{H}$ is holomorphic in $\Re u > -A$ and in this region bounded by \begin{equation}\label{assumption-c}
\widehat{H} (u) \ll_{\Re u} (1 + |u|)^{\Re u - 1} . \end{equation} {\rm b)} For any $M > 0$, we have the asymptotic formula \begin{equation}\label{asymp-j}
\widehat{H}(\sigma + it) = |t|^{\sigma - 1} \exp\left(it \log \frac{|t|}{4\pi e}\right) j_{\sigma}(t) + O_{\sigma}(|t|^{\sigma-1 -B}) \end{equation}
for $\sigma > -A$, $|t| \geqslant 30$ and a smooth function $j_{\sigma}$ satisfying for all $\nu\leqslant B$ the bound \begin{equation}\label{ad3}
|t|^{\nu} j^{(\nu)}_{\sigma}(t) \ll_{\sigma} 1. \end{equation} \end{lemma}
As per our convention, if $\{H_j\} = \{\mathscr{K}h_j\}$ is uniformly admissible of type $(A, B)$, then all implied constants in the previous lemma can be chosen independently of $j$. \\
\textbf{Proof.} We treat the two kinds of admissible functions separately.
1) Suppose that $H = \mathscr{K}h$ is of the first kind. For $ \Re u > 0$ we have by \eqref{mellin-j} and \eqref{H} that
\begin{equation}\label{mel}
\begin{split}
\widehat{H}(u)
& = \frac{i}{4\pi} \int_{-\infty}^{\infty} (2\pi)^{-u}\left( \frac{\Gamma(u/2 + i\tau)}{\Gamma(1 - u/2 + i\tau)} - \frac{\Gamma(u/2 - i\tau)}{\Gamma(1 - u/2 - i\tau)}\right) \frac{h(\tau) \tau}{\cosh(\pi \tau)} \, d\tau\\
& = \frac{i}{2\pi} \int_{-\infty}^{\infty} (2\pi)^{-u} \frac{\Gamma(u/2 + i\tau)}{\Gamma(1 - u/2 + i\tau)} \frac{h(\tau) \tau}{\cosh(\pi \tau)} \, d\tau ,
\end{split}
\end{equation}
as an absolutely convergent integral.
We shift the contour to the line $\Im \tau = -A + 1/8$. This does not cross poles of $1/\cosh(\pi \tau)$ because of the zeros of $h$. In this way see that the integral is holomorphic in $ \Re u> -2A + 1/4$ and bounded by $O((1+|u|)^{\Re u-1})$, with a uniform implied constant if $\{\mathscr{K}h_j\}$ is a uniform family. This confirms a).
Moreover, for $u = \sigma + it$ with $\sigma > -A$, $|t| \geqslant 30$ and $r= (\sigma-1)/2$ (say) we have \begin{displaymath} \begin{split}
&\widehat{H}(\sigma+it) |t|^{-\sigma+1} \exp\left(-it \log\frac{|t|}{4\pi e}\right) \\
&= \frac{i}{2\pi} \int_{-\infty}^{\infty} (2\pi)^{-\sigma} |t|^{-\sigma-1} \exp\left(-it \log \frac{|t|}{2 e}\right) \frac{\Gamma(\frac{1}{2}\sigma - r + i(\frac{1}{2}t + \tau))}{\Gamma(1 - \frac{1}{2}\sigma - r + i(\tau - \frac{1}{2}t))} \frac{h(\tau+ir) (\tau+ir)}{\cosh(\pi(\tau + ir))} d \tau,
\end{split}
\end{displaymath}
having shifted the contour in \eqref{mel} to imaginary part $r$ (as we need to obtain the analytic continuation to $\Re u = \sigma$).
We want to show that except for an error of $O(|t|^{-B})$, this function satisfies \eqref{ad3} for $\nu \leqslant B$ (and fixed $r$ and $\sigma$). By trivial estimates using $h(t) \ll (1 + |t|)^{-B-2}$, the portion $|\tau | > \frac{1}{4}|t|$ contributes $O(|t|^{-B})$. For the remaining portion we can insert Stirling's formula \eqref{stir} for the gamma quotient. The error term in \eqref{stir} can be bounded by any negative power of $|t|$, and the main term gives us a phase $\exp(i \phi_{\tau}(t))$ with
$$ \phi_{\tau}(t) = -t \log \frac{|t|}{2 e}+ \frac{1}{2}t \log\frac{|t^2 - (2\tau)^2|}{(2e)^2} + \tau \log\Big| \frac{t+2\tau}{t -2\tau}\Big|$$
satisfying
$$ \phi_{\tau}'(t) = \frac{1}{2} \log\Big|\frac{t^2 - (2\tau)^2}{t^2}\Bigr| \ll \frac{|\tau|}{|t|}, \quad \phi^{(j)}_{\tau}(t) \ll \frac{|\tau|}{|t|^{j}}\,\,\, (j\geqslant 2)$$
uniformly in $|\tau| \leqslant \frac{1}{4}|t|$. Now we differentiate under the integral sign, and since
$$|t|^{\nu} \frac{d^{\nu}}{dt^{\nu}} \left( |t|^{-\sigma-1} \exp\left(-it \log \frac{|t|}{2 e}\right) \frac{\Gamma(\frac{1}{2}\sigma - r + i(\frac{1}{2}t + \tau))}{\Gamma(1 - \frac{1}{2}\sigma - r + i(\tau - \frac{1}{2}t)) \cosh(\pi (\tau + ir))}\right) \ll 1 + |\tau|,$$
and $\int_{\Bbb{R}} |h(\tau + ir)| (1 + |\tau|)^2 d\tau \ll 1$, we obtain \eqref{ad3} as desired. This confirms b).
2) If $H$ is of the form \eqref{secondtype}, then \begin{equation}\label{second} \widehat{H}(u) = \alpha_0 \frac{\Gamma(\frac{1}{2}(a+ u -b))}{2(2\pi)^{u}2^b\Gamma(\frac{1}{2}(2+a-u+b ))} + \sum_{a-b < k \in 2\Bbb{N}} \alpha_k \frac{\Gamma(\frac{1}{2}(k-1 + u ))}{2(2\pi)^{u}\Gamma(\frac{1}{2}(1+k-u ))}, \end{equation}
and a) is clear. By Stirling's formula, the first term on the right hand side satisfies \eqref{asymp-j} and \eqref{ad3}. To treat the second term, we write for $k \in 2\Bbb{N}$ and $u = \sigma +it$ the last fraction in the form
$$\frac{\Gamma(\frac{1}{2}(k-1 + u ))}{ \Gamma(\frac{1}{2}(1+k-u ))} =\frac{\Gamma(\frac{1}{2}(1+\sigma + it))}{\Gamma(\frac{1}{2}(3 - \sigma - it))} \prod_{n=1}^{\frac{k}{2} - 1} \frac{k-1 + \sigma- 2n + it}{k-1 -\sigma - 2n - it}. $$ We use Stirling's formula for the gamma fraction on the right hand side. In order to verify \eqref{asymp-j} and \eqref{ad3}, it remains to show that
\begin{equation}\label{verify} \sum_{k\in 2\Bbb{N}} \Big| \alpha_k t^{\nu} \frac{d^{\nu}}{dt^{\nu}} \prod_{n=1}^{k/2 - 1} \left(\frac{k-1 + \sigma- 2n + it}{k-1 -\sigma - 2n - it} \right)\Big| \ll 1 \end{equation} for $\nu \leqslant B$. For $\nu \geqslant 1$ we have
$$\frac{d^{\nu}}{dt^{\nu}} \left(\frac{k-1 + \sigma- 2n + it}{k-1 -\sigma - 2n - it} \right) = 2i^{\nu} \nu! \frac{k-1-2n}{(k-1-2n-\sigma - i t)^{\nu+1}} \ll_{\sigma} \frac{k}{(1+|t|)^{\nu+1}},$$
and so by Leibniz' rule the left hand side of \eqref{verify} is bounded by $\sum_k |\alpha_k| k^{\nu} \ll 1$ for $ \nu \leqslant B$.
\begin{lemma}\label{sec-prop} Let $c, d \in \Bbb{C}$, $x \in \Bbb{R} \setminus \Bbb{N}_0$. Let $H$ be an admissible function of type $(A, B)$. Suppose that $2x + \Re c > -A$, $\Re d > \theta+ \max(x, (x+1)/2)$. Then $$\mathscr{I}^{\pm}(c, d) := \int_{(x)} G^{\pm}(\xi) \widehat{H} (c+2\xi) \mathcal{G}_{-\mu}^{\mp}(d-\xi) \frac{d\xi}{2\pi i}$$ with $G^{\pm}$ as in \eqref{Gpm} and $\mathcal{G}^{\pm}_{\mu}$ as in \eqref{defgmu} is holomorphic for $\Re(c+3d) < 2$ and has a meromorphic continuation to $\Re(c + 3d) < 3$ with a simple polar divisor at most at $c+3d = 2$.\end{lemma}
\textbf{Proof.} We recall that the spectral parameter $\mu = (\mu_1, \mu_2, \mu_3)$ of the automorphic form $F$ satisfies $|\Re \mu_j|\leqslant \theta \leqslant 5/14$. Throughout the proof we always assume that $c$ and $d$ satisfy \begin{equation}\label{cond-lemma}
2x + \Re c > -A, \quad \Re d > \theta+ \max(x, (x+1)/2).
\end{equation} The first condition ensures that we are in the domain of holomorphicity of $\widehat{H}$, the condition $\Re d > \theta+ x$ ensures that $ \mathcal{G}_{-\mu}^{\mp}(d-\xi)$ is holomorphic, and the condition $\Re d > \theta+ (x+1)/2$ will be needed later.
Holomorphy in $\Re(c+3d) < 2$ follows immediately from \eqref{bound-g}, \eqref{bound-G} and Lemma \ref{admis1}a, but for $\Re(c+3d) \geqslant 2$, the integral fails to converge absolutely. Fix any $\beta > 0$. Clearly it suffices to continue the restricted integral with the compact region $|\Im \xi| \leqslant D := 2\beta + 2\max_j |\Im \mu_j| + 2$ removed, meromorphically to the region $|\Im d| \leqslant \beta$ and $\Re(c + 3d) < 3$.
To this end, we will approximate the integrand for $ |\Im \xi| \geqslant D$ by a simpler expression. The error in this approximation will decay more quickly by an additional power of $1/|\xi|$, which buys us absolute convergence in $\Re(c + 3d) < 3$. We will then complete the simplified integral by re-inserting the portion $|\Im \xi| \leqslant D$ and evaluate it explicitly by \eqref{hyper}.
In the following we treat only the case $\mathscr{I}^+(c, d)$ and drop the superscript, the other case is analogous.
Using the definitions \eqref{Gpm} and \eqref{defgmu}, we have
$$ G^{+}(\xi)
\mathcal{G}_{-\mu}^{-}(d-\xi) = \Gamma(\xi) (2\pi)^{ 2\xi} \Bigl( \prod_{j=1}^3\Gamma(d - \xi -\mu_j) \Bigr)\sum_{\nu = -3}^4 \gamma_{\nu}(d) e^{i \frac{\pi}{2} \nu \xi}. $$
for certain holomorphic functions $\gamma_{\nu}(d)$ with $\gamma_{4}(d) = (2\pi)^{-3d}$
(that in general depend also on the spectral parameter $\mu$ which we suppressed from the notation). The terms $-3 \leqslant \nu \leqslant 3$ lead to exponentially decaying integrals and can easily be continued homomorphically to $\Re(c + 3d)$ arbitrarily large.
For notational simplicity let us write $d_j := d - \mu_j$. We have
$$\prod_{j=1}^3\Gamma(d_j - \xi ) = \Gamma(1 - \xi) \Gamma(d_1 + d_2-1 - \xi) \frac{\Gamma(d_1 - \xi) \Gamma(d_2 - \xi)}{ \Gamma(1 - \xi) \Gamma(d_1 + d_2-1 - \xi)} \frac{\pi}{\sin(\pi(d_3 - \xi)) \Gamma(1 - d_3 + \xi)}$$
by the reflection formula, and by another application of the reflection formula we see that the
term corresponding to $\nu = 4$ equals
\begin{equation*}
(2\pi)^{2\xi } \frac{\Gamma(d_1 + d_2 - 1 - \xi)}{\Gamma(1 - d_3 + \xi) } \frac{\pi^2 (2\pi)^{-3d} e^{2\pi i \xi}}{\sin(\pi \xi) \sin(\pi (d_3 - \xi))} \frac{\Gamma(d_1 - \xi) \Gamma(d_2 - \xi)}{ \Gamma(1 - \xi) \Gamma(d_1 + d_2-1 - \xi)}.
\end{equation*}
The basic observation is now that the last gamma fraction is asymptotically constant as $|\Im \xi| \rightarrow \infty$. More precisely, by \eqref{stir} -- \eqref{stir1}, the difference
$$ \frac{\Gamma(d_1 - \xi) \Gamma(d_2 - \xi)}{ \Gamma(1 - \xi) \Gamma(d_1 + d_2-1 - \xi)} - 1,
$$
is holomorphic in $|\Im d| < \beta$, $|\Im \xi| > D$ and bounded by $O_{d, \Re \xi}(|\xi|^{-1})$. Consequently, the difference
$$ \frac{\pi^2 (2\pi)^{-3d} e^{2\pi i \xi}}{\sin(\pi \xi) \sin(\pi (d_3 - \xi))} \frac{\Gamma(d_1 - \xi) \Gamma(d_2 - \xi)}{ \Gamma(1 - \xi) \Gamma(d_1 + d_2-1 - \xi)} - \begin{cases} 0, & \Im \xi > D,\\ 4\pi^2 (2\pi)^{-3d} e^{i\pi d_3},
& \Im \xi < - D,\end{cases}$$
is holomorphic in $|\Im d| < \beta$ (because we stay away from zeros of the $\sin$-function) and bounded by $O_{d, \Re \xi}(|\xi|^{-1})$. Since
$$ (2\pi)^{2\xi } \frac{\Gamma(d_1 + d_2 - 1 - \xi)}{\Gamma(1 - d_3 + \xi) }
\widehat{H} (c+2\xi) |\xi|^{-1} \ll_{\Re \xi, d, \Re c} |\xi|^{\Re(c + 3d) - 4}$$
by \eqref{assumption-c}, this portion can be continued to $\Re (c+3d) < 3$ as an absolutely (and locally uniformly in $d$) convergent integral over $|\Im \xi| > D$. Hence we are left with
$$4\pi^2 (2\pi)^{-3d} e^{i\pi d_3} \int_{\substack{\Re \xi = x\\ |\Im \xi| > D}} \widehat{H} (c+2\xi) (2\pi)^{2\xi } \frac{\Gamma(d_1 + d_2 - 1 - \xi)}{\Gamma(1 - d_3 + \xi) } (1 - \text{sgn}(\Im \xi)) \frac{d\xi}{2\pi i},$$
and of course the constant in front of the integral plays no role. In order to remove the unwanted factor $ 1 - \text{sgn}(\Im \xi)$, we use that
$$-i\cdot \text{sgn}(\Im \xi) - \frac{\Gamma(1 - d_3 + \xi) \Gamma(d_1 + d_2 - 1/2 - \xi)}{\Gamma(3/2 - d_3 + \xi)\Gamma(d_1 + d_2 -1 - \xi)} $$
is holomorphic in $|\Im d - \beta| < 1$ for $|\Im \xi| > D$ and bounded by $O_{d, \Re \xi}(|\xi|^{-1})$,
which follows again from \eqref{stir} -- \eqref{stir1}. Hence our job is reduced to continuing
$$\int_{\substack{\Re \xi = x\\ |\Im \xi| > D}} \widehat{H} (c+2\xi) (2\pi)^{2\xi } \frac{\Gamma(d_1 + d_2 - 1+ \kappa - \xi)}{\Gamma(1 + \kappa - d_3 + \xi) } \frac{d\xi}{2\pi i}$$
for $\kappa \in \{0, 1/2\}$, and we may now re-insert the compact interval $|\Im \xi|
leq D$. At this point the condition $\Re d > \theta+ (x+1)/2$ comes handy, since the completed contour is still to the right of all poles.
Thus we let
$$\mathscr{I}_{\kappa}(c, d) := \int_{(x)} \widehat{H} (c+2\xi) (2\pi)^{2\xi } \frac{\Gamma(d_1 + d_2 - 1+ \kappa - \xi)}{\Gamma(1 + \kappa - d_3 + \xi) } \frac{d\xi}{2\pi i}$$
and treat the two kinds of admissible functions separately.
If $H = \mathscr{K}h$ is of the first kind, then by \eqref{mel} we have $$\mathscr{I}_{\kappa}(c, d )= (2\pi)^{-c-1} i \int_{\Im \tau =-A+1/2 } \int_{(x)} \frac{\Gamma(d_1 + d_2 - 1+\kappa - \xi)}{\Gamma(1+\kappa - d_3 + \xi) } \frac{\Gamma(\frac{c}{2}+\xi + i\tau)}{\Gamma(1-\frac{c}{2} - \xi + i\tau)} \frac{d\xi}{2\pi i} \frac{h(\tau) \tau}{\cosh(\pi \tau)} \, d\tau,$$ where the exchange of integration is easily justified by absolute convergence and we are allowed to shift the $\tau$-contour since $H$ is admissible of type $(A, B)$. The conditions \eqref{cond-lemma} ensure that the arguments of the gamma factors in the numerator are to the right of all poles.
By \eqref{hyper} and \eqref{mu}, we can evaluate the $\xi$-integral getting $$\mathscr{I}_{\kappa}(c, d ) = (2\pi)^{-c-1} i \int_{\Im \tau =-A+1/2 } \frac{\Gamma(d_1+d_2 - 1+\kappa + \frac{c}{2} + i\tau)\Gamma( 2-c-3d)}{\Gamma(2-3d)\Gamma(1-c)\Gamma( 1+\kappa -d_3 - \frac{c}{2} + i\tau )} \frac{h(\tau) \tau}{\cosh(\pi \tau)} \, d\tau.$$ This expression is meromorphic in $\Re (c+3d) < 3$ (intersected with \eqref{cond-lemma}) with its only polar divisor at $c+3d = 2$. (Note that \eqref{cond-lemma} implies that the argument of the first gamma factor in the numerator has positive real part.)
If $H$ is of the second kind, we argue similarly: by \eqref{second}, $\mathscr{I}_{\kappa}(c, d)$ equals \begin{displaymath} \begin{split}
\frac{1}{2} \int_{(x)}& \Bigl[ \alpha_0 \frac{(2\pi)^{-c}\Gamma(\xi + \frac{1}{2}(a -b+c))}{2^b \Gamma(-\xi+\frac{1}{2}(2+a+b-c ))} + \sum_{a-b < k \in 2\Bbb{N}} \alpha_k \frac{(2\pi)^{-c} \Gamma(\xi+\frac{1}{2}(k-1 +c ))}{\Gamma(-\xi + \frac{1}{2}(1+k -c))}\Bigr] \frac{\Gamma(d_1 + d_2 - 1+ \kappa - \xi)}{\Gamma(1 + \kappa - d_3 + \xi) } \frac{d\xi}{2\pi i}. \end{split} \end{displaymath} The first term in parentheses yields an integral that is absolutely convergent in $\Re (c+3d) < 2 + b$. For the second term we exchange sum and integration and evaluate the $\xi$-integral explicitly by \eqref{hyper}, which provides the meromorphic continuation to $\Re (c+3d) < 3$ with a polar divisor only at $c+3d = 2$.
\section{An integral transform}\label{int-trafo}
Fix ${\tt s}, {\tt w}\in \Bbb{C}$ and suppose that $H$ is admissible of type $(A, B)$.
Then for all ${\tt u}$ satisfying
\begin{equation}\label{cond}
\begin{split}
&\textstyle -A < \Re(3{\tt s} - {\tt w} + {\tt u} -3),
\quad \theta < \Re ({\tt s} + \frac{1}{2}{\tt u}) < \frac{1}{2}, \quad \Re (\frac{1}{2} {\tt u} + {\tt w}) > 0
\end{split}
\end{equation}
the integral transforms \begin{equation}\label{trafo} \begin{split} (\mathscr{V}_{{\tt s}, {\tt w}}^{\pm} \widehat{H})({\tt u}) := \int_{\mathcal{C}} \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} + 2\xi) &\Big[G^+(\xi) \mathcal{G}_{-\mu}^-(1 - {\tt s} - \textstyle\frac{1}{2} {\tt u} - \xi)\mathcal{G}_{\mu}^{\mp}({\tt s} + \frac{1}{2}{\tt u}) \\ &+ G^-(\xi) \textstyle\mathcal{G}_{-\mu}^+(1 - {\tt s} - \frac{1}{2} {\tt u}- \xi)\mathcal{G}_{\mu}^{\pm}({\tt s} + \frac{1}{2}{\tt u})\Big] \displaystyle\frac{d\xi}{2\pi i} \end{split} \end{equation} with $G^{\pm}$ as in \eqref{Gpm} and $\mathcal{G}^{\pm}_{\mu}$ as in \eqref{defgmu} define an absolutely convergent integral. Indeed, the first condition in \eqref{cond} implies that the argument of $\widehat{H}$ is in the region of holomorphicity as provided in Lemma \ref{admis1}a, while the third condition ensures, by Lemma \ref{lemG} and \eqref{assumption-c}, absolute convergence. It is important to note that this condition is independent of $\xi$ and in particular not affected by possible contour shifts. Finally, the second condition in \eqref{cond} ensures that as ${\tt u}$ varies no poles are crossed. We would like to continue $(\mathscr{V}_{{\tt s}, {\tt w}}^{\pm} \widehat{H})({\tt u})$ holomorphically to a larger region of $({\tt u}, {\tt s}, {\tt w})$ and also obtain good bounds in terms of ${\tt u}$. All implied constants in the following may depend on ${\tt s}, {\tt w}, A, B, \mu$.
We remark already at this point that this integral will come up later with the following slightly deformed contour consisting of four line segments
\begin{equation}\label{contour}
\mathcal{C} = \textstyle (-\frac{3}{5} -i\infty, -\frac{3}{5} - i] \cup [-\frac{3}{5} - i, \frac{1}{10}] \cup [\frac{1}{10}, -\frac{3}{5} + i] \cup [-\frac{3}{5} + i, -\frac{3}{5} + i\infty).
\end{equation}
The first condition in \eqref{cond} has enough elbow room to make sure that the argument of $\widehat{H}$ is still in the region of holomorphicity.
\begin{lemma}\label{lem9} Let ${\tt s}, {\tt w}\in \Bbb{C}$, $A, B > 2$, suppose that $H$ is admissible of type $(A, B)$ and suppose that the set ${\tt u}$ satisfying \eqref{cond} is non-empty. Then $(\mathscr{V}_{{\tt s}, {\tt w}}^{\pm} \widehat{H})({\tt u})$ can be extended holomorphically to all $({\tt u}, {\tt s}, {\tt w}) \in \Bbb{C}^3$ satisfying \begin{equation}\label{region} \begin{split}
&\textstyle \theta < \Re ({\tt s} + \frac{1}{2}{\tt u}) < \frac{1}{2}A + \frac{1}{2}\Re({\tt s} - {\tt w}), \quad \Re( \frac{1}{2} {\tt u} + {\tt w}) > 0
\end{split} \end{equation} and satisfies\footnote{Again, if $\{H_j \}$ is uniformly admissible of type $(A, B)$, the implied constants can be chosen independently of $j$.}
\begin{equation}\label{boundJ} \begin{split} (\mathscr{V}_{{\tt s}, {\tt w}}^{\pm} \widehat{H})({\tt u}) \ll
(1 + |u|)^{3 (|\Re{\tt s} | + |\Re{\tt u} |+ |\Re{\tt w} |) - \min(\frac{1}{8} B, \frac{1}{4} A)}
\end{split}
\end{equation}
in this region. Moreover, $(\mathscr{V}_{{\tt s}, {\tt w}}^{\pm} \widehat{H})({\tt u})$ can be extended meromorphically to \begin{equation}\label{mero} \begin{split}
&\textstyle -1+ \theta < \Re ({\tt s} + \frac{1}{2}{\tt u}) < \frac{1}{4}A + \frac{1}{4}\Re({\tt s} - {\tt w}), \quad \Re( \frac{1}{2} {\tt u} + {\tt w}) > -1
\end{split} \end{equation} with poles at most at ${\tt u}/2 + {\tt w} = 0$ and $ {\tt u}/2 + {\tt s} + \mu_j = 0$, $j = 1, 2, 3$.
\end{lemma}
\textbf{Proof.} Before we start with the proof, we use \eqref{new-formula} to compute explicitly the residue of the integrand in \eqref{trafo} at $\xi = -n$, $n \in \Bbb{N}_0$, as \begin{equation}\label{resi} \begin{cases} 0 , & \pm = +,\\ \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} - 2n) (2\pi)^{-2n} (n!)^{-1} \prod_{j=1}^n \prod_{k=1}^3({\tt s} + \frac{1}{2}{\tt u} +\mu_k - j),& \pm = -. \end{cases} \end{equation}
The proofs of \eqref{boundJ} and \eqref{mero} need a slightly different treatment depending on the sign, and we start with $(\mathscr{V}_{{\tt s}, {\tt w}}^{-} \widehat{H})({\tt u})$. Let initially $({\tt u}, {\tt s}, {\tt w})$ be in the region \eqref{cond}. We straighten the $\xi$-contour and shift it to the far left to $\Re \xi = -A_1 \not\in \Bbb{N}$. Since $ \Re ({\tt s} + {\tt u}/2) > 0$, this does not leave the domain of holomorphicity of $\widehat{H}$ provided
\begin{equation}\label{a1} \Re({\tt s} - {\tt w} - 1 -2A_1) > -A.
\end{equation} We pick up possible poles at $\xi = -n$, $n \in \Bbb{N}_0$, $n_0 < A_1$, whose residues are by \eqref{resi} holomorphic functions in ${\tt u}, {\tt s}, {\tt w}$ in the region \eqref{a1}. The remaining integral is holomorphic in the region \begin{equation}\label{newregion} \textstyle \theta < \Re ({\tt s} + \frac{1}{2}{\tt u}) < 1 - \theta + A_1, \quad \Re (\frac{1}{2}{\tt u} + {\tt w}) > 0. \end{equation}
This gives a holomorphic continuation of $(\mathscr{V}_{{\tt s}, {\tt w}}^{-} \widehat{H})({\tt u})$ in $({\tt u}, {\tt s}, {\tt w})$ to the intersection of the regions \eqref{a1} and \eqref{newregion}. Choosing $A_1 = \frac{1}{2}(A + \Re ({\tt s}- {\tt w}) - 1) - 1/100$, we obtain a region containing \eqref{region}.
The same argument combined with Lemma \ref{sec-prop} shows that $(\mathscr{V}_{{\tt s}, {\tt w}}^{-} \widehat{H})({\tt u})$ has a meromorphic continuation to
$$\textstyle -1 + \theta < \Re ({\tt s} + \frac{1}{2}{\tt u}) < 1 - \theta + \frac{1}{2}A_1, \quad \Re (\frac{1}{2}{\tt u} + {\tt w}) > -1$$
with poles at most at ${\tt u}/2 + {\tt w} = 0$ and $ {\tt u}/2 + {\tt s} + \mu_j = 0$, $j = 1, 2, 3$.
Now we fix ${\tt s}$ and ${\tt w}$ and proceed to estimate $(\mathscr{V}_{{\tt s}, {\tt w}}^{-} \widehat{H})({\tt u})$ for ${\tt u}$ satisfying \eqref{region} with the aim of establishing \eqref{boundJ}. We write $v = \frac{1}{2} \Im{\tt u}$ and may assume that $v > 0$ is sufficiently large (in terms of $\mu$), for if $|v|$ is bounded there is nothing to prove, and the case of negative $v$ is similar. We may then focus our attention to the second term in \eqref{trafo}, since by \eqref{bound-G} the first term, which contains a factor $\mathcal{G}^+_{\mu}({\tt s} + \frac{1}{2} {\tt u})$, is exponentially decaying as $v \rightarrow +\infty$.
We shift the $\xi$-contour back to the far right, to $\Re \xi = A_2 > 0$, say, making sure that this line does not cross poles. This cancels all residues from \eqref{resi} that we picked up earlier, but introduces potentially new residues at $1 - {\tt s} - \frac{1}{2} {\tt u} - \xi = -\mu_j-n$, $n \in \Bbb{N}_0$, $j \in \{1,2, 3\}$. We do not compute them explicitly, but observe that a priori the residues must be meromorphic in $({\tt u}, {\tt s}, {\tt w})$; since we know already that $(\mathscr{V}_{{\tt s}, {\tt w}}^{-} \widehat{H})({\tt u})$ is holomorphic, their joint contribution must be holomorphic, too. Moreover, for $1 - {\tt s} - \frac{1}{2} {\tt u} - \xi = O(1)$ but off the poles, the integrand in \eqref{trafo} is \begin{equation*}
\ll v^{O(1)} e^{-\pi v/2}
\end{equation*}
by \eqref{bound-g} and \eqref{bound-G}; by Cauchy's theorem this bound also holds for the residues at $1 - {\tt s} - \frac{1}{2} {\tt u} - \xi = -\mu_j-n$, which is in agreement with \eqref{boundJ}. This time we used the exponential decay of $G^-(\xi)$ in the second term of \eqref{trafo} under our current assumption $v > 0$.
It remains to estimate the remaining integral on the line $\Re \xi = A_2$. We split the integral smoothly into two pieces as follows. Let
$w : \Bbb{R} \rightarrow [0, 1]$ be a smooth function that is constantly 1 on $[-1, 1]$ and vanishes outside $[-2, 2]$. Then it suffices to estimate $$I^-_1 := \int_{(A_2)} \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} + 2\xi) G^-(\xi) \mathcal{G}_{-\mu}^+(1 - {\tt s} - \textstyle\frac{1}{2} {\tt u} - \xi)\mathcal{G}_{\mu}^{-}({\tt s} + \frac{1}{2} {\tt u}) \displaystyle w\left(\frac{\Im \xi}{V}\right)\frac{d\xi}{2\pi i}$$ and $$I^-_2 := \int_{(A_2)} \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} + 2\xi) G^-(\xi) \mathcal{G}_{-\mu}^+(1 - {\tt s} - \textstyle\frac{1}{2} {\tt u} - \xi)\mathcal{G}_{\mu}^{-}({\tt s} + \frac{1}{2} {\tt u}) \displaystyle\left(1 - w\left(\frac{\Im \xi}{V}\right)\right) \frac{d\xi}{2\pi i},$$ where $1 \leqslant V \leqslant v$ will be chosen in a moment.
We estimate $I_1^-$ trivially by \eqref{assumption-c}, \eqref{bound-g} and \eqref{bound-G} getting \begin{equation*} I^-_1 \ll v^{\Re(3{\tt s} - {\tt w} - 2 + {\tt u} - A_2) } V^{A_2 + \frac{1}{2}}. \end{equation*} Since we are free to choose $A_2$ as large as we wish, this is admissible for \eqref{boundJ} provided $V \leqslant v^{1-\delta}$ for some fixed $\delta > 0$, which we assume from now on. For the estimation of $I_2^-$, we first observe that we can restrict to the branch $ \Im \xi > 0$, as the branch $\Im \xi < 0$ can be estimated trivially by \begin{equation}\label{exp-sr}
\ll v^{O(1)} e^{-\frac{1}{2}\pi V},
\end{equation}
which is admissible for \eqref{boundJ} provided $V > v^{\delta}$ for some fixed $\delta > 0$. The treatment of the remaining case $\Im \xi > 0$, which we assume from now on, is the only point where properties \eqref{asymp-j} and \eqref{ad3} of $\widehat{H}$ are required.
We insert the asymptotic formula \eqref{asymp-j} along with the asymptotic formulae \eqref{asymp-g} and \eqref{asymp-G}. The error terms corresponding to \eqref{asymp-g} and \eqref{asymp-G} save arbitrarily many powers of $V> v^{\delta}$ which is admissible for \eqref{boundJ}. The error term corresponding to \eqref{asymp-j} contributes at most \begin{equation}\label{E1} \int_{x \geqslant V} (v+x)^{- \Re( {\tt w} + \frac{1}{2} {\tt u}) - A_2 - \frac{1}{2} - B} x^{A_2 - \frac{1}{2}} v^{3\Re({\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2}}dx \ll v^{\Re (3{\tt s} + {\tt u} - {\tt w}) - \frac{3}{2} - B}. \end{equation} For the main terms,
we obtain an integral of the shape \begin{displaymath} \begin{split} \mathcal{G}_{\mu}^+({\tt s} + \textstyle \frac{1}{2} {\tt u})&\displaystyle \int_0^{\infty} (x + v )^{-\frac{1}{2} - A_2 - \Re {\tt w} - \frac{1}{2} \Re {\tt u}} x^{A_2 - \frac{1}{2}} \omega(x) e^{i\phi(x)}
\left(1 - w\left( \frac{x}{V}\right)\right) dx \end{split} \end{displaymath} with $x^j \omega^{(j)} (x) \ll_j 1$ for all $j \in \Bbb{N}_0$ and \begin{equation}\label{phi}
\phi(x) = x \log \displaystyle\frac{|x|}{2\pi e} - (x + v) \displaystyle\log \frac{|x + v|}{2\pi e}. \end{equation}
We compute $$\phi'(x) = \log\Big| \frac{x} {x+v}\Bigr|,
\quad \phi^{(j)}(x) \ll \frac{|v|(|v| + |x|)^{j-2}}{|x|^{j-1} |v+x|^{j-1}}, \quad j \geqslant 2. $$
We attach another smooth partition of unity and decompose the integral smoothly into dyadic pieces supported on $Z< x < 4Z$ with $Z \geqslant V$. For each piece we can apply Lemma \ref{integrationbyparts} with
$$\beta-\alpha \asymp U = Q = Z, \quad Y = \frac{vZ}{v + Z}, \quad R = \frac{v}{v + Z}, \quad X = ( v + Z)^{-\frac{1}{2} - A_2 - \Re {\tt w} - \frac{1}{2} \Re {\tt u}} Z^{A_2 - \frac{1}{2}}$$
(note that for $ Z \asymp x \geqslant v^{\delta}$ we have $\log x/(x+v) \asymp v/(v+Z)$) and bound it by
$$\ll v^{3(\Re {\tt s} + \frac{1}{2} \Re {\tt u} - \frac{1}{2})}( v + Z)^{-\frac{1}{2} - A_2 - \Re {\tt w} - \frac{1}{2} \Re {\tt u}} Z^{A_2 + \frac{1}{2}} \Bigl(\sqrt{\frac{Zv}{v + Z}}\Bigr)^{-B/2}.$$
We estimate $Zv/(v + Z) \gg \min(v, Z) \gg V$ and bound the previous display by
$$\ll v^{3(\Re{\tt s} + \frac{1}{2} \Re {\tt u} - \frac{1}{2})} V^{-\frac{1}{4}B} Z^{-\Re({\tt w} + \frac{1}{2} {\tt u})}.$$
Summing this over $Z = 2^{\nu}$, we get a convergent sum by the last condition in \eqref{region}, and choosing $V > v^{3/4}$, we obtain a total bound of
\begin{equation}\label{E2}
v^{3(\Re{\tt s} + \frac{1}{2} \Re {\tt u} - \frac{1}{2}) - \frac{3}{16}B}.
\end{equation}
We proceed to treat $(\mathscr{V}_{{\tt s}, {\tt w}}^{+} \widehat{H})({\tt u})$, which similar, but slightly simpler. Let initially $({\tt u}, {\tt s}, {\tt w})$ be in the region \eqref{cond}. We straighten the $\xi$-contour and shift it to the far left to $\Re \xi = -A_1 \not\in \Bbb{N}$ satisfying \eqref{a1}. By \eqref{resi} we do not pick up any poles. As before this gives analytic continuation to the region \eqref{region} and meromorphic continuation to the region \eqref{mero} with poles at most at ${\tt u}/2 + {\tt w} = 0$ and $ {\tt u}/2 + {\tt s} + \mu_j = 0$, $j = 1, 2, 3$. In the present case, we do not shift back to the right, but estimate the integral on the line $\Re \xi = -A_1$. We continue to write $v = \frac{1}{2} \Im {\tt u} > 0$ and assume that $v$ is sufficiently large. We focus now on the first term in \eqref{trafo}, since the second is exponentially decreasing in $v$. Again we split the $\xi$-integral into two pieces and consider
$$I^+_1 := \int_{(-A_1)} \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} + 2\xi) G^+(\xi) \mathcal{G}_{-\mu}^-(1 - {\tt s} - \textstyle\frac{1}{2} {\tt u} - \xi)\mathcal{G}_{\mu}^{-}({\tt s} + \frac{1}{2} {\tt u}) \displaystyle w\left(\frac{v+\Im \xi}{V}\right)\frac{d\xi}{2\pi i}$$ and $$I^+_2 := \int_{(-A_1)} \widehat{H}(3{\tt s} - {\tt w} - 1 + {\tt u} + 2\xi) G^+(\xi) \mathcal{G}_{-\mu}^-(1 - {\tt s} - \textstyle\frac{1}{2} {\tt u} - \xi)\mathcal{G}_{\mu}^{-}({\tt s} + \frac{1}{2} {\tt u}) \displaystyle\left(1 - w\left(\frac{v+\Im \xi}{V}\right)\right) \frac{d\xi}{2\pi i},$$ that is, we distinguish between $\Im ({\tt u} + 2\xi)$ small or large. As before we assume $v^{\delta} < V < v^{1-\delta}$. We estimate $I_1^+$ trivially getting \begin{equation}\label{err1} I_1^+ \ll v^{3\Re({\tt s} + \frac{1}{2} {\tt u} - \frac{1}{2})} v^{-A_1 - \frac{1}{2}} V^{\frac{3}{2} - \frac{1}{2}\Re{\tt u} - \Re{\tt w} +A_1} . \end{equation}
For the estimation of $I_2^+$ we can restrict ourselves to the branch $v + \Im \xi < -V$, for the other branch satisfies \eqref{exp-sr}. Again we now use the asymptotic formula \eqref{asymp-j} which contributes an error term of at most \begin{equation}\label{err2} \begin{split}
&\int_{-\infty}^{-V-v} |v+x|^{-\Re( \frac{1}{2}{\tt u} + {\tt w}) - \frac{1}{2} + A_1-B} |x|^{-A_1 - \frac{1}{2}} v^{3\Re({\tt s} + \frac{1}{2}{\tt u}) - \frac{3}{2}} dx\\ & \ll v^{\Re (3{\tt s} - {\tt u} - {\tt w}) - \frac{3}{2} - B} + v^{3\Re( {\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2} - A_1 - \frac{1}{2}} V^{-\Re( \frac{1}{2}{\tt u} + {\tt w}) + \frac{1}{2} + A_1 - B}. \end{split} \end{equation} The main term produces an integral of the shape \begin{displaymath} \begin{split} \mathcal{G}_{\mu}^-({\tt s} + \textstyle \frac{1}{2} {\tt u})&\displaystyle \int_{-\infty}^{0} (x + v )^{-\frac{1}{2} + A_1 - \Re( {\tt w} + \frac{1}{2} {\tt u})} x^{-A_1 - \frac{1}{2}} \omega(x) e^{i\phi(x)}
\left(1 - w\left( \frac{v+x}{V}\right)\right) dx \end{split} \end{displaymath}
with the same phase function \eqref{phi} as before and $|x+v|^j \omega^{(j)}(x) \ll_j 1$. We restrict the integral to smooth dyadic ranges $|x+v| \asymp Z \geqslant V$, so $|x| \asymp Z + v$.
By the same application of Lemma \ref{integrationbyparts} each such dyadic region can be bounded by $$Z^{\frac{1}{2} + A_1 - \Re( {\tt w} + \frac{1}{2} {\tt u})} (Z+v)^{-A_1 - \frac{1}{2}} v^{3\Re({\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2}} \Bigl(\sqrt{\frac{Zv}{v+Z}}\Bigr)^{-B/2}\ll v^{3\Re({\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2}} Z^{- \Re( {\tt w} + \frac{1}{2} {\tt u})} V^{-\frac{1}{4}B},$$
and summing this over $Z = 2^{\nu}$, we obtain a total bound of
\begin{equation}\label{err3}
v^{3\Re({\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2}} V^{-\frac{1}{4}B}.
\end{equation} We now choose $V = v^{1/2}$, $A_1 = \frac{1}{2} (A + \Re( s - w) - 2)$ in agreement with \eqref{a1}. Then the bounds \eqref{err1}, \eqref{err2}, \eqref{err3} become $O(v^E)$ with \begin{displaymath} \begin{split} E = \max& \Big(\textstyle \frac{1}{4} \Re(11{\tt s} + 5{\tt u} - {\tt w} - 3) - \frac{1}{4} A , \,\,\Re (3{\tt s} - {\tt u} - {\tt w}) - \frac{3}{2} - B, \\ & \textstyle\frac{1}{4}(\Re(11{\tt s} + 5{\tt u} - {\tt w} - 5) - \frac{1}{4}A - \frac{1}{2}B, \,\, 3\Re({\tt s} + \frac{1}{2} {\tt u}) - \frac{3}{2} - \frac{1}{8}B \Big). \end{split} \end{displaymath}
Combining this with the bounds \eqref{E1} and \eqref{E2}, we complete the proof of \eqref{boundJ}.
\section{A preliminary reciprocity formula}\label{prelim-formula}
As outlined in the introduction, Theorem \ref{thm1} is a consequence of the five-step procedure Kuznetsov-Voronoi-reciprocity-Voronoi-Kuznetsov. In the following proposition we consider the middle triplet Voronoi-reciprocity-Voronoi.
For $s, w \in \Bbb{C}$ with $\Re s, \Re w > 3/2$, $q, \ell \in \Bbb{N}$ coprime and a function $H$ satisfying $H(x) \ll x^{2/3}$, we define the absolutely convergent expression
$$\mathcal{E}^{\pm}_{q, \ell}(s, w; H) := \sum_{\ell \mid r} \sum_{q \mid n_1c} \sum_{n_2} \frac{A(n_2, n_1)S(\pm r, n_2, c)}{n_2^{s }n_1^{2s} c r^{w}}H\left(\frac{\sqrt{rn_2}}{c}\right). $$
\begin{prop}\label{prelim} Let $H$ be an admissible function of type $(500, 500)$. Let $3/2 < \Re s < 2$, $4 < \Re w < 5$ and suppose that $q, \ell \in \Bbb{N}$ are coprime. Then \begin{equation}\label{rec-prelim} \mathcal{E}_{q, \ell}^{+}(s, w; H) = \mathcal{N}^{(1)}_{q, \ell}(s, w; H) - \mathcal{N}^{(2)}_{q, \ell}(s, w; H) + \sum_{\pm} \mathcal{E}_{\ell, q}^{\pm}\Big(s', w'; \widecheck{\big(\mathscr{V}^{\pm}_{s', w'}\widehat{H}\big)}\Big) \end{equation} where $s', w'$ are as in \eqref{new}, the right hand side employs the notation \eqref{mellin1}, \eqref{mellin2} and \eqref{trafo}, and the ``main terms'' $\mathcal{N}^{(1)}_{q, \ell}(s, w; H)$ and $ \mathcal{N}^{(2)}_{q, \ell}(s, w; H)$ are given in \eqref{res1} and \eqref{N2}; they vanish if $F$ is cuspidal, and if $F={\tt E}_0$, they have meromorphic continuation to an $\varepsilon$-neighbourhood of $\Re w \geqslant \Re s \geqslant 1/2$ and satisfy the bounds
\begin{equation}\label{boundsN}
\begin{split} & \big((s - \textstyle\frac{1}{2})(w-s)(s+w-1)\big)^4\mathcal{N}^{(1)}_{q, \ell}(s, w; H) \ll_{s, w} (q\ell)^{-1+\varepsilon}, \\
& \big((s - \textstyle\frac{1}{2})(w-s)(s+w-1)\big)^6\mathcal{N}^{(2)}_{q, \ell}(s, w; H) \ll_{s, w} (\ell q)^{\varepsilon} \ell^{-1}
\end{split}
\end{equation}
Moreover, the function $\mathscr{V}^{\pm}_{s', w'}\widehat{H}$ is holomorphic in \begin{equation}\label{prop-1} 0 < \Re s < 2, \quad 0 < \Re w < 5, \quad \max(2\theta - 2 \Re s', -2\Re w') < \Re u < 15 \end{equation} and in this region bounded by \begin{equation}\label{prop-2}
\mathscr{V}^{\pm}_{s', w'}\widehat{H}(u) \ll_{s, w} (1 + |u|)^{-15}.
\end{equation} In addition, it is meromorphic in \begin{equation}\label{mero2} 0 < \Re s < 2, \quad 0 < \Re w < 5, \quad \max(-2+2\theta - 2 \Re s', -2-2\Re w') < \Re u < 15 \end{equation} with poles at most at $u \in \{-2w', -2s' - 2\mu_1, -2s' - 2\mu_2, -2s' - 2\mu_3\}$.
\end{prop}
\textbf{Proof.} We first note that if $H$ is admissible of type $(A, B)$, then by Lemma \ref{admis1}a) and \eqref{mellin} we have $H(x) \ll x^{2/3}$, so that $\mathcal{E}^+_{q, \ell}(s, w; H)$ makes sense. By Mellin inversion and \eqref{alt} we have $$ \mathcal{E}^+_{q, \ell} (s, w; H) = \int_{(-1 )} \widehat{H}(u) \mathcal{D}^+_{q, \ell}(s, u, w) \frac{du}{2\pi i}.$$ By \eqref{boundD} and \eqref{assumption-c} the $u$-integral is absolutely convergent provided that $$ \Re u < 0, \quad \Re(3 s - u/2) > 3, \quad \Re(3 s + u/2) > 3/2,$$ which is automatically satisfied, provided \eqref{variables} holds. We shift the $u$-contour to $\Re (s + u/2) = -2/3$ (for the moment any negative number would suffice, but later we need $\Re (s + u/2) < -1/2$). This is still in agreement with \eqref{variables}. On the way we may pick up a pole at $u = 2 - 2s$, which contributes \begin{equation}\label{res1} \mathcal{N}^{(1)}_{q, \ell}(s, w; H) := \underset{u = 2-2s}{\text{res}} \widehat{H}(u) \mathcal{D}^+_{q, \ell}(s, u, w) =\begin{cases} \sum_{j = 1}^3 \frac{1}{(j-1)!}\mathcal{R}_{q, \ell;j}(s, w) \widehat{H}^{(j-1)}(2-2s), & F = {\tt E}_0,\\ 0, & F \text{ cuspidal}\end{cases} \end{equation} with the notation as in \eqref{resR}. Lemma \ref{pole1} provides analytic continuation of this term as well as the first bound in \eqref{boundsN}.
Having shifted the contour to $\Re (s + u/2) = -2/3$, we can insert the first functional equation in \eqref{Dfunc} and apply the definition \eqref{defPhi} since we are in the region of absolute convergence. In this way we conclude \begin{equation}\label{int} \begin{split} \mathcal{E}_{q, \ell}^{+}(s, w; H) = \mathcal{N}^{(1)}_{q, \ell}(s, w; H) + & \int_{(-\frac{4}{3}- 2\Re s)} \widehat{H}(u) \sum_{\pm} \mathcal{G}_{-\mu}^{\mp}(1 - s - u/2) \\ & \sum_{(c, d) = 1} \sum_{\substack{\ell \mid md \\ q \mid mc}} \sum_n \frac{A(m, n) e(\pm n \bar{d}/c)}{c^{3s+u/2 -1} m^{s+w} d^{w+u/2} n^{1 - s - u/2}} \frac{du}{2\pi i}. \end{split} \end{equation} At this point we insert artificially a factor $$1 = e\left(\mp \frac{n}{cd}\right) e\left(\pm \frac{n}{cd}\right) = e\left(\mp \frac{n}{cd}\right) \int_{\mathcal{C}} G^{\pm}(\xi)\left(\frac{n}{cd}\right)^{-\xi} \frac{d\xi}{2\pi i},$$ where $G^{\pm}$ and $\mathcal{C}$ were defined in \eqref{Gpm} resp.\ \eqref{contour}.
The contour is designed so that the integral is absolutely convergent, but the contour is to the right of the pole at $\xi = 0$. By the reciprocity formula \eqref{rec} the integral in \eqref{int} equals \begin{displaymath} \begin{split}
\int_{(-\frac{4}{3}-2\Re s)} \int_{ \mathcal{C}}\widehat{H}(u)\sum_{\pm} G^{\pm}(\xi) \mathcal{G}_{-\mu}^{\mp}(1 - s - u/2) \sum_{(c, d) =1} \sum_{\substack{q \mid mc \\ \ell \mid md}} \frac{\Phi(d, \mp c, m; 1 - s - u/2 + \xi) }{c^{3s+u/2 -1} m^{s+w}d^{w+u/2} (cd)^{-\xi} } \frac{d\xi}{2\pi i} \frac{du}{2\pi i}, \end{split} \end{displaymath} and the entire expression is still absolutely convergent. Here we used that $\Re(1 - s - u/2 + \xi) \geqslant 16/15 > 1$ on the entire $\xi$-contour, so that we are in the region of absolute convergence of $\Phi$. It is now convenient to interchange the $u$- and $\xi$-integration and to replace the straight $u$-contour with a polygonal contour $\mathcal{C}(\xi)$ (depending on $\xi$) such that $\Re(1 - s - u/2 + \xi) = 16/15$. The resulting expression is still absolutely convergent. We now introduce the new variables \begin{equation}\label{new-var} s' = \textstyle\frac{1}{2}(1 - s + w), \quad w' = \frac{1}{2}(3s+w-1),\quad u' = 3s-w -1 + u - 2\xi. \end{equation} This has the following effect: the exponents $(3s + \frac{1}{2}u - 1- \xi, s+w, w+\frac{1}{2}u - \xi)$ of $(c, m, d)$ become $(w'+\frac{1}{2}u', s'+w', 3s' + \frac{1}{2}u' - 1)$, and the contour $C(\xi)$ given by $\Re(1 - s - u/2 + \xi) = 16/15$ becomes $s' + \frac{1}{2} u' = -\frac{1}{15}$ (independently of $\xi$). Recalling the definition \eqref{deftildeD}, we can recast the integral in \eqref{int} as \begin{displaymath} \begin{split}
\int_{ \mathcal{C}} \int_{( - \frac{2}{15} - 2\Re s')} \widehat{H}(3s' - w' - 1 + u' + 2\xi) \sum_{\pm} G^{\pm}(\xi) \mathcal{G}_{-\mu}^{\mp}(1 - s' - \textstyle\frac{1}{2}u'-\displaystyle\xi) \widetilde{\mathcal{D}}^{\mp}_{\ell, q}(s', u', w') \frac{du'}{2\pi i} \frac{d\xi}{2\pi i}. \end{split} \end{displaymath} Now we shift the $u'$-integral the right to $\Re (s' + u'/2) = 1/2$. Again we may pick up a pole with residue $$\mathcal{N}^{(2)}_{q, \ell}(s, w; H) := \int_{\mathcal{C}}\underset{u'= -2s'}{\text{res}} \widehat{H}(3s' - w' - 1 + u' + 2\xi) \sum_{\pm} G^{\pm}(\xi) \mathcal{G}_{-\mu}^{\mp}(1 - s' - \textstyle\frac{1}{2}u'-\displaystyle\xi) \widetilde{\mathcal{D}}^{\mp}_{\ell, q}(s', u', w') \frac{d\xi}{2\pi i} $$ (which counts negative because of the right shift).
If $F$ is cuspidal, this vanishes, and if $F = {\tt E}_0$ , then by \eqref{tildeD} it equals \begin{equation}\label{N2}
\sum_{j=1}^3 \tilde{\mathcal{R}}_{\ell, q;j}(s', w') \sum_{\nu_1 + \nu_2 = j-1} \left( - \frac{1}{2}\right)^{\nu_2} \sum_{\pm} \int_{(1/10)} G^{\pm}(\xi) \widehat{H}^{(\nu_1)}(-2s + 2\xi) (\mathcal{G}_{(0, 0, 0)}^{\mp})^{(\nu_2)}(1 -\displaystyle\xi) \frac{d\xi}{2\pi i},
\end{equation} where we straightened the contour $\mathcal{C}$. Note that the conditions $\Re w \geqslant \Re s \geqslant 1/2$ and $\Re w' \geqslant \Re s' \geqslant 1/2$ are equivalent and $(s' - 1/2)(w'-s')(s' +w' - 1) = (s-1/2)(w-s)(s+w-1)$, so Lemma \ref{pole2} and Lemma \ref{sec-prop} with $x = 1/10$, $c = -2s$, $d=1$ provide meromorphic continuation of this term as well as the second bound in \eqref{boundsN}. (Note that $\ell$ and $q$ are interchanged relative to Lemma \ref{pole2} and the $\xi$-integral contributes at most a triple pole at $s = 1/2$.)
Having shifted the contour to $\Re(s' + u'/2) = 1/2$, we apply the second functional equation in \eqref{Dfunc} and recall the definition \eqref{trafo} to obtain $$\mathcal{E}_{q, \ell}^{+}(s, w; H) = \mathcal{N}^{(1)}_{q, \ell}(s, w; H) - \mathcal{N}^{(2)}_{q, \ell}(s, w; H) +\sum_{\pm} \int_{(1 - 2\Re s')} (\mathscr{V}^{\pm}_{s', w'}\widehat{H})(u') \mathcal{D}_{\ell, q}^{\pm}(s', u', w') \frac{du'}{2\pi i}.$$ Lemma \ref{lem9} gives us analytic continuation and decay conditions for $\mathscr{V}^{\pm}_{s', w'}\widehat{H}$, so that we can continue to shift the contour to the right into the region of absolute convergence of $ \mathcal{D}_{\ell, q}^{\pm}(s', u', w')$ to, say, $\Re(s' + u'/2) = 3/2$. We write $ \mathcal{D}_{\ell, q}^{\pm}(s', u', w') $ in terms of its Dirichlet series representation \eqref{alt}, and by Mellin inversion we obtain the formula \eqref{rec-prelim}.
Now we observe that for $0 < \Re s < 2$, $0 < \Re w < 5$ we have $ -1/2 < \Re s', \Re w' < 5$.
By \eqref{region} -- \eqref{boundJ} with $A, B \geqslant 500$, $|\Re {\tt s}|, |\Re{\tt w}| \leqslant 5$,
we obtain the holomorphicity of $\mathscr{V}_{s', w'}^{\pm}\widehat{H}$ in the region \eqref{prop-1} and the bound \eqref{prop-2}. In particular, if $3/2 < \Re s < 2$ and $4 < \Re w < 5$, then $\Re s' \geqslant 3/2$ and $\Re w' \geqslant 7/2$, so that \eqref{prop-1} and \eqref{prop-2} in combination with \eqref{mellin} imply that $\widecheck{\big(\mathscr{V}^{\pm}_{s', w'}\widehat{H}\big)}(x) \ll x^{2/3}$, so that the rightmost term in \eqref{rec-prelim} makes sense. Meromorphic continuation of $\mathscr{V}^{\pm}_{s', w'}\widehat{H}(u)$ to \eqref{mero2} and the location of poles follows from the statement containing \eqref{mero}. This completes the proof. \\
We end this section by relating $\mathcal{E}^{\pm}(s, w; H)$ to spectral sums. By \eqref{kuz-all} we have for $\Re s, \Re w > 3/2$, $q, \ell$ coprime and $\mathfrak{h}$ admissible that \begin{equation}\label{formula1} \sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}_{d_2}(r, n_2, \mathfrak{h}) = \sum_{\ell \mid n_2}\sum_{n_1} \frac{A(n_2, n_1)}{n_2^{s+w} n_1^{2s}} \mathscr{N}\mathfrak{h} + \mathcal{E}_{q, \ell}^+(s, w; \mathscr{K}^*\mathfrak{h}). \end{equation}
Conversely, if ${\tt H}$ satisfies $x^j{\tt H}^{(j)}(x) \ll \min(x, x^{-3/2})$ for $0 \leqslant j \leqslant 3$ and if in addition $\widehat{{\tt H}}$ is holomorphic in $-2\vartheta-\varepsilon <\Re u < 5$ and satisfies $\widehat{{\tt H}}(u) \ll (1 + |u|)^{-5}$, say, then by
\eqref{kuz2} for $\Re s', \Re w' > 3/2$ we have \begin{equation}\label{formula2} \mathcal{E}_{\ell, q}^{\pm}(s', w'; {\tt H}) = \sum_{d_1d_2 = \ell} \sum_{q \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^{s'}n_1^{2s'}r^{w'}} \mathcal{A}_{d_2}(\pm r, n_2, \mathscr{L}^{\pm}{\tt H}), \end{equation} where by \eqref{rho-bound}, \eqref{rho-cusp-bound}, \eqref{gl3-RS}, Lemma \ref{final-decay}a and Weyl's law, the various sums in \eqref{formula1} and \eqref{formula2} are absolutely convergent. The next section is devoted to relating the left hand side of \eqref{formula1} and the right hand side of \eqref{formula2} to $L$-functions.
\section{Local factors}\label{local}
\subsection{Local computations} For a prime $p$ let $\alpha_{f, \nu}(p)$ ($\nu = 1, 2$), $\alpha_{F, j}(p)$ ($j = 1, 2, 3$) denote the Satake parameters of $f$ and $F$ at $p$ satisfying \begin{equation}\label{satake3}
\alpha_{F, 1}(p) \alpha_{F, 2}(p) \alpha_{F, 3}(p)
= 1, \quad \alpha_{f, 1}(p) \alpha_{f, 2}(p) = \begin{cases} 1, & p \nmid \text{cond}(f).\\ 0, & p \mid \text{cond}(f). \end{cases}
\end{equation}
We have \begin{equation*} \lambda_f(p^{\nu}) = \frac{\alpha_{f, 1}(p)^{\nu+1} - \alpha_{f, 2}(p)^{\nu+1} }{\alpha_{f, 1}(p) - \alpha_{f, 2}(p)} \end{equation*} and \begin{equation}\label{satake2}
A(p^{\nu}, p^{\mu}) = \det\left(\begin{matrix} \alpha_{F, 1}(p)^{\nu+\mu+2} & \alpha_{F, 2}(p)^{\nu+\mu+2} & \alpha_{F, 3}(p)^{\nu+\mu+2}\\ \alpha_{F, 1}(p)^{\mu+1} & \alpha_{F, 2}(p)^{\mu+1} & \alpha_{F, 3}(p)^{\mu+1}\\1 & 1 & 1\end{matrix}\right) V_F(p)^{-1} \end{equation} where $V_F(p)= \det\left(\begin{matrix} \alpha_{F, 1}(p)^{ 2} & \alpha_{F, 2}(p)^{ 2} & \alpha_{F, 3}(p)^{2}\\ \alpha_{F, 1}(p) & \alpha_{F, 2}(p) & \alpha_{F, 3}(p)\\1 & 1 & 1\end{matrix}\right); $
in particular \begin{equation}\label{satake4} \begin{split} & \lambda_f(p) = \alpha_{f, 1}(p) + \alpha_{f,2}(p), \quad A(p, 1) = \alpha_{F, 1}(p)+ \alpha_{F, 2}(p)+ \alpha_{F, 3}(p), \\
& A(1, p) = \alpha_{F, 1}(p) \alpha_{F, 2}(p) + \alpha_{F, 1}(p) \alpha_{F, 3}(p) + \alpha_{F, 2}(p) \alpha_{F, 3}(p).
\end{split}
\end{equation}
Note that \eqref{satake2} remains formally true for $\nu$ or $\mu = -1$ if we define $A(p^{\nu}, p^{\mu}) = 0$ in this case. Let $$L_p(p^{-s}, f \times F) = \prod_{j=1}^3\prod_{\nu=1}^2 \left(1 - \frac{\alpha_{F, j}(p) \alpha_{f, \nu}(p)}{p^s}\right)^{-1}$$ denote the local Rankin-Selberg factor at $p$ and correspondingly \begin{equation*} L(s, f \times F) := \prod_p L_p(p^{-s}, f \times F) \end{equation*} in $\Re s > 1$ (this may differ from the corresponding Langlands $L$-function by finitely many Euler factors).
We start with the following combinatorial lemma.
\begin{lemma}\label{eulerlemma} For $p$ prime and $\Re s> \theta + \vartheta$, the following identities hold: \begin{displaymath} \begin{split} &\sum_{\nu \in \Bbb{N}_0} \frac{A(p^{\nu}, 1)\lambda_f(p^{\nu})}{p^{\nu s}} = \begin{cases} \left(1 - \frac{A(1, p)}{p^{2s}} + \frac{\lambda_f(p)}{p^{3s}}\right) L_p(p^{-s}, f \times F), & p \nmid \text{{\rm cond}}(f),\\ L_p(p^{-s}, f \times F), &p \mid \text{{\rm cond}}(f),\end{cases} \\ &\sum_{\nu \in \Bbb{N}_0} \frac{A(p^{\nu+1}, 1)\lambda_f(p^{\nu})}{p^{\nu s}} = \left(A(p, 1) - \frac{A(1, p)\lambda_f(p)}{p^{s}} + \frac{\lambda_f(p^2)}{p^{2s}}\right)L_p(p^{-s}, f \times F),\\ &\sum_{\nu_1, \nu_2 \in \Bbb{N}_0} \frac{A(p^{\nu_2}, p^{\nu_1+1}) \lambda_f(p^{\nu_2})}{p^{s(\nu_2 + 2\nu_1)}} = \left(A(1, p) - \frac{\lambda_f(p)}{p^s}\right) L_p(p^{-s}, f \times F), \quad p \nmid \text{{\rm cond}}(f).\\
\end{split} \end{displaymath}
\end{lemma}
\textbf{Proof.} A simple computation based on \eqref{satake2} and geometric series shows the power series identity \begin{displaymath} \begin{split} \sum_{\nu \in \Bbb{N}_0} A(p^{\nu}, 1) \lambda_f(p^{\nu}) X^{\nu} &= \big[1 - ( \alpha_{F, 1}(p) \alpha_{F, 2}(p) + \alpha_{F, 1}(p) \alpha_{F, 3}(p) +\alpha_{F, 2}(p) \alpha_{F, 3}(p))\alpha_{f, 1}(p) \alpha_{f, 2}(p) X^2 \\ & + \alpha_{F, 1}(p) \alpha_{F, 2}(p) \alpha_{F, 3}(p)(\alpha_{f, 1}(p)+ \alpha_{f, 2}(p) )\alpha_{f, 1}(p) \alpha_{f, 2}(p) X^3\big] L_p(X, f \times F) \end{split} \end{displaymath} which by \eqref{satake3} and \eqref{satake4} with $X = p^{-s}$ gives the first formula. The other two are proved in the same way.\\
\textbf{Remarks:} 1) Note that $\theta+\vartheta \leqslant 5/14 + 7/64 < 1/2$. This is useful numerical coincidence, but $f \times F$ is known to correspond to an automorphic representation on ${\rm GL}(6)$ \cite{KiSh}, therefore such a relation follows at every place from general bounds towards the Ramanujan conjecture on ${\rm GL}(n)$ \cite{LRS}. \\ 2) We conclude from the first formula that the Dirichlet series expansion of $L(s, f \times F)$ in $\Re s > 1$ is given by \begin{equation}\label{L23} \sum_{\substack{n, m\\ (\text{cond}(f) , m) = 1}} \frac{A(n, m) \lambda_f(n)}{n^s m^{2s}}. \end{equation} This holds also with $(F, \lambda_f)$ replaced with $(E_{t, \chi}, \lambda_{t, \chi})$.\\
We need a similar technical lemma that we apply later for the contribution of Eisenstein series.
\begin{lemma}\label{euler2} Let $M, d, g_1, g_2, q \in \Bbb{N}$ with $g_1 \mid g_2$ and $d, M \mid q$. Then the series $$\sum_{\substack{c, f, n \mid q^{\infty}\\ (c, g_1) = (n, g_2) = 1}} \frac{A(cfM, nd) }{c^u f^v n^{u+v}} \prod_{p \mid q} \prod_{j=1}^3 \left(1 - \frac{\alpha_{F, j}(p) }{p^u}\right) \left(1 - \frac{\alpha_{F, j}(p) }{p^v}\right), $$ initially defined for $\Re u, \Re v > \theta$ as an absolutely convergent series, has a holomorphic extension to an $\varepsilon$-neighbourhood of $\Re u, \Re v \geqslant 0$, $\Re (u+v) \geqslant 1/2$ and is bounded by $O(q^{\varepsilon} (dM)^{\theta})$ in this region.
\end{lemma}
\textbf{Proof.} The sum factorizes into a product of $p\mid q$, and it suffices to consider each factor separately. Let $m = v_p(M)$, $k=v_p(d)$, and put $X = p^{-u}$, $Y = p^{-v}$. The local $p$-factor equals $$E(p) := \sum_{\substack{\beta, \gamma, \delta \in \Bbb{N}_0\\ (p^{\beta}, g_1) = (p^{\delta}, g_2) = 1}} A( p^{\beta+\gamma + m}, p^{\delta + k} )X^{\beta} Y^{\gamma} (XY)^{\delta} \prod_{j=1}^3 (1 - \alpha_{F, j}(p) X)(1 - \alpha_{F, j}(p) Y).$$ By \eqref{hecke} we have
$$A( p^{\beta+\gamma + m}, p^{\delta + k} ) = A( p^{\beta+\gamma + m}, 1) \overline{A(p^{\delta + k}, 1 )} - A( p^{\beta+\gamma + m-1}, 1) \overline{A(p^{\delta + k-1}, 1 )}$$
(with the above convention $A(p^{-1}, 1) = 0$). We treat the first summand on the right hand side, the second one is similar. Depending on whether (i) $p \mid g_1$, (ii) $p \mid g_2$, but $p \nmid g_1$ or (iii) $p \nmid g_2$, we need to compute one, two or three geometric series. In cases (i) and (ii) we obtain
$$E(p) = \overline{{A}(p^{k}, 1) }\sum_{i= 0}^2 \sum_{j=0}^3 \frac{P_{i, j}(\alpha_{F, 1}(p),\alpha_{F, 2}(p), \alpha_{F, 3}(p))}{ V_F(p)}
X^i Y^j$$
where $P_{i, j}$ is a homogeneous polynomial of degree $3 +m +i+j$.
Since $E(p)$ is an entire function in $\alpha_{F, j}$ for $X, Y$ sufficiently small, each $P_{i, j}$ must be divisible $V_F(p)$, and we obtain $$\overline{{A}(p^{k}, 1) } \sum_{i= 0}^2 \sum_{j=0}^3 \tilde{P}_{i, j}(\alpha_{F, 1},\alpha_{F, 2}, \alpha_{F, 3})
X^i Y^j$$ with a homogeneous polynomial $\tilde{P}_{i, j}$
of degree $m +i+j$. In case (iii) a similar argument shows $$E(p) = \sum_{i= 0}^2 \sum_{j=0}^2 \frac{ \tilde{Q}_{i, j}(\alpha_{F, 1}(p),\alpha_{F, 2}(p), \alpha_{F, 3}(p)) \tilde{R}_{i, j}(\overline{\alpha_{F, 1}(p)},\overline{\alpha_{F, 2}(p)}, \overline{\alpha_{F, 3}(p)}) }{(1 - \overline{\alpha_{F, 1}(p)}XY)(1 - \overline{\alpha_{F, 2}(p)}XY)(1 - \overline{\alpha_{F, 3}(p)}XY)}
X^i Y^j$$
for homogeneous polynomials $\tilde{Q}_{i, j}$, $\tilde{R}_{i, j}$
of degrees $m +i+j$ and $k+i+j$ respectively.
Since $\max_j |\alpha_{F, j}(p)|
\leqslant p^{\theta}$, we can continue each $E(p)$ to an $\varepsilon$-neighbourhood of $\Re u, \Re v \geqslant 0$, $\Re (u+v) \geqslant 1/2$ and bound it by $O(p^{\theta(k+m) + \varepsilon})$ in this region.
This completes the proof. \\
Finally, using the formula for Bump's double Dirichlet series \cite[Proposition 6.6.3]{Go}, we have \begin{equation}\label{bump-double}
\sum_{\ell \mid n_2}\sum_{n_1} \frac{A(n_2, n_1)}{n_2^{s+w} n_1^{2s}} = \frac{L(s+w, F) L(2s, \bar{F})}{\ell^{s+w}\zeta(3s+w)} \sum_{n_1, n_2 \mid \ell^{\infty}} \frac{A(\ell n_2, n_1)}{n_2^{s+w} n_1^{2s}}\Biggr(\sum_{n_1, n_2 \mid \ell^{\infty}} \frac{A( n_2, n_1)}{n_2^{s+w} n_1^{2s}}\Biggl)^{-1}, \end{equation} which provides analytic continuation of the left hand side, initially defined in $\Re s, \Re w > 1/2$ to the region $\Re (3s + w) > 1$, $\Re s, \Re w > \theta$ (with polar divisors at most at $s = 1/2$, $s + w = 1$ if $F = {\tt E}_0$), and it also provides the bound \begin{equation}\label{bound-bump}
O_{s, w}(\ell^{-\Re (s+ w)+ \theta + \varepsilon})
\end{equation} in this region,
away from polar divisors.
\subsection{The cuspidal case} We start by considering \begin{displaymath} \begin{split} &\sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}^{\text{Maa{\ss}}}_{d_2}(\pm r, n_2,h) =\sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast}(d_0)} \epsilon_f^{(1 \mp 1)/2} h(t_f) \mathcal{S}_{q, \ell}(s, w; f) \end{split} \end{displaymath} with $$ \mathcal{S}_{q, \ell}(s, w; f) := \sum_{d_1d_2 = q}\sum_{ M \mid d_2/d_0} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1) \rho_{f, M, d_2}( r) \overline{\rho_{f, M, d_2}(n_2)}}{n_2^sn_1^{2s}r^w} $$ for $f \in \mathcal{B}^{\ast}(d_0)$. We insert \eqref{rho-cusp} and obtain \begin{displaymath} \begin{split} & \frac{L(1, \text{Ad}^2f)}{\prod_{p \mid d_0}(1 -p^{-2})} \mathcal{S}_{q, \ell}(s, w; f) \\ = & \sum_{d_1d_2 = q}\sum_{\ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \sum_{d_0M \mid d_2} \sum_{\delta_1, \delta_2 \mid M}\frac{\xi_{f}(M, \delta_1) \xi_{f}(M, \delta_2) }{d_2\nu(d_2)} \frac{\delta_1\delta_2}{M} \frac{A(n_2, n_1)\lambda_{f}(r/\delta_1) \lambda_{f}(n_2/\delta_2)}{n_2^{s }n_1^{2s} r^{w}}\\
= & \frac{1}{q \ell^w } \sum_{d_1d_2 = q} \sum_{Md_0 \mid d_2} \sum_{\delta_1, \delta_2 \mid M} \frac{ \xi_{f}(M, \delta_1) \xi_{f}(M, \delta_2) \delta_1^{1-w}\delta_2^{1-s} d_1^{1-2s} }{M \nu(d_2) } \sum_{ r} \sum_{(n_1, d_2) = 1} \sum_{n_2}\frac{A(n_2\delta_2, n_1d_1)\lambda_{f}(\ell r) \lambda_{f}(n_2)}{n_2^{s }n_1^{2s} r^{w}}.
\end{split}
\end{displaymath} We open $\lambda_f(\ell r)$ using the Hecke relations and recognize the $r$-sum as $\Lambda_f(\ell; w) L(w, f)$ with the notation as in \eqref{Lambda}. We also recognize the $n_1, n_2$-sum as $L(s, f \times F)$ up to Euler factors at primes dividing $q$. Hence
\begin{displaymath} \begin{split} \mathcal{S}_{q, \ell}(s, w; f) = \frac{\phi(q)}{q^2} \frac{L(w, f) L(s, f \times F)}{ \ell^w L(1, \text{Ad}^2f)} \Lambda_f(\ell; w) \tilde{L}_q(s, w, f \times F)
\end{split}
\end{displaymath} with \begin{equation}\label{local-q-cusp}
\begin{split} \tilde{L}_q(s, w, f\times F) = & \frac{q}{\phi(q)}\prod_{p \mid q} \prod_{j=1}^3 \prod_{\nu=1}^2 \left(1 - \frac{\alpha_{F, j}(p) \alpha_{f, \nu}(p)}{p^s}\right) \sum_{d_1d_2 = q} \sum_{Md_0 \mid d_2} \sum_{\delta_1, \delta_2 \mid M} \delta_1^{1-w}\delta_2^{1-s} d_1^{1-2s} \\ &\frac{ \xi_{f}(M, \delta_1) \xi_{f}(M, \delta_2) }{M \nu(d_2) } \sum_{\substack{(n_1, d_2) = 1\\ n_1 \mid q^{\infty}}} \sum_{n_2 \mid q^{\infty}}\frac{A(n_2\delta_2, n_1d_1) \lambda_{f}(n_2)}{n_2^{s }n_1^{2s} }\prod_{p \mid d_0}(1 -p^{-2}).
\end{split}
\end{equation}
In particular, using the notation \eqref{moment} we have
\begin{equation}\label{moment1}
\sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}^{\text{Maa{\ss}}}_{d_2}(\pm r, n_2, h) = \mathcal{M}_{q, \ell}^{\text{Maa{\ss}}, \pm}(s, w; (h, h^{\text{hol}}))
\end{equation} for any $h^{\text{hol}}$. Estimating trivially (using \eqref{indiv} and \eqref{xi-arithmetic}), we obtain
$$\tilde{L}_q(s, w, f\times F) \ll q^{\varepsilon} \sum_{d_1d_2 = q} \sum_{Md_0 \mid d_2} \sum_{\delta_1, \delta_2 \mid M} (\delta_1\delta_2)^{1/2} \left(\frac{M^2}{\delta_1\delta_2}\right)^{\vartheta} \frac{1}{M} (d_1\delta_2)^{\theta} \ll q^{\theta + \varepsilon}$$ for $\Re s, \Re w \geqslant 1/2$. This proves \eqref{local-bound} when $f$ is Maa{\ss}. For $f \in \mathcal{B}^{\ast}(q)$ we have $d_0 = q$, hence $d_2=q$, $d_1 = M = \delta_1 = \delta_2 = 1$, and the $n_1, n_2$-sum over powers of primes dividing $q$ in \eqref{local-q-cusp} can be computed explicitly using the first formula in Lemma \ref{eulerlemma}. In this way one obtains $\tilde{L}_q(s, w, f\times F) = 1$ for Maa{\ss} newforms $f$ of level $q$. With slightly more computational effort and the the other two formulae in Lemma \ref{eulerlemma} one shows \begin{equation}\label{extension} \tilde{L}_q(1/2, 1/2, f\times F) = \frac{\phi(d_0)}{qd_0} \prod_{p \mid \frac{q}{d_0}} \left(\frac{2 + A(1, p) + A(p, 1)}{1 + \lambda_f(p)p^{-1/2} + p^{-1}}-1\right) \end{equation} if $q$ is squarefree and $f$ is a newform of level $d_0 \mid q$.
The same computation holds verbatim for $f \in \mathcal{B}^{\ast}_{\text{hol}}(d_0)$, which completes the proof Lemma \ref{lem2} in the cuspidal case.
\subsection{The Eisenstein case} Similarly as in the previous subsection we consider \begin{displaymath} \begin{split} &\sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}^{\text{Eis}}_{d_2}(\pm r, n_2, h) = \sum_{d_0^2 \mid q} \sum_{\substack{\chi \text{ (mod } d_0)\\ \text{primitive}}} \int_{\Bbb{R}} \mathcal{S}_{q, \ell}(s, w; (t, \chi)) h(t) \frac{dt}{2\pi} \end{split} \end{displaymath} with $$ \mathcal{S}_{q, \ell}(s, w; (t, \chi)) := \sum_{d_1d_2 = q} \sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{\substack{ (M_2, d_0) = 1 \\ d_0 M_1M_2 \mid d_2}} \sum_{\ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)\rho_{\chi, d_0M_1M_2, d_2}(r, t) \overline{\rho_{\chi, d_0M_1M_2, d_2}(n_2, t)}}{n_2^{s }n_1^{2s} r^{w}}$$ for $t\in \Bbb{R}$ and $\chi$ a primitive Dirichlet character modulo $d_0$. We define \begin{equation}\label{lchitd2} L(\chi, t, d_2) := \prod_{p \mid d_2} \left(\left(1 - \frac{\chi^2(p)}{p^{1 + 2it}}\right) \left(1 - \frac{\bar{\chi}^2(p)}{p^{1 - 2it}}\right)\right)^{-1}, \end{equation}
insert \eqref{rho-eis} and recast $ |L(1 + 2it, \chi^2)|^2\mathcal{S}_{q, \ell}(s, w; (t, \chi))$ as \begin{displaymath} \begin{split}
\sum_{d_1d_2 = q} &\frac{L(\chi, t, d_2) }{d_2\nu(d_2)} \sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{\substack{ (M_2, d_0) = 1 \\ d_0 M_1M_2 \mid d_2}} \sum_{\delta_1, \delta_2 \mid M_2} \frac{M_1\delta_1\delta_2 \mu(M_2/\delta_1) \mu(M_2/\delta_2)\bar{\chi}(\delta_1) \chi(\delta_2) }
{ \tilde{\mathfrak{n}}(d_0M_1M_2)^2M_2} \\
& \sum_{(n_1, q) = d_1} \sum_{\substack{c_1,f_1\\ (c_1, \frac{d_2}{d_0M_1M_2}) = 1}} \sum_{\substack{\ell \mid c_2f_2\\ (c_2, \frac{d_2}{d_0M_1M_2}) = 1}} \frac{A(c_1f_1M_1\delta_1, n_1)\bar{\chi}(c_1f_2) \chi(c_2f_1) (c_2f_2M_1\delta_2)^{it} (c_1f_1M_1\delta_1)^{-it} }{(c_2/c_1)^{2it}(c_1f_1M_1\delta_1)^{s }n_1^{2s} (c_2f_2 M_1 \delta_2)^{w}}. \end{split} \end{displaymath} We slightly re-write this as \begin{displaymath} \begin{split} \frac{1}{q} \sum_{d_1d_2 = q} &\frac{d_1^{1-2s}}{ \nu(d_2)}L(\chi, t, d_2) \sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{\substack{ (M_2, d_0) = 1 \\ d_0 M_1M_2 \mid d_2}} \sum_{\delta_1, \delta_2 \mid M_2} \frac{M_1^{1-s-w}\delta_1^{1-s-it}\delta_2^{1-w+it} \mu(M_2/\delta_1) \mu(M_2/\delta_2) }{ \tilde{\mathfrak{n}}(d_0M_1M_2)^2M_2} \\
& \bar{\chi}(\delta_1) \chi(\delta_2) \sum_{(n_1, d_2) = 1} \sum_{\substack{c_1,f_1\\ (c_1, \frac{d_2}{d_0M_1M_2}) = 1}} \sum_{\substack{\ell \mid c_2f_2\\ (c_2, \frac{d_2}{d_0M_1M_2}) = 1}} \frac{A(c_1f_1M_1\delta_1, n_1d_1)\bar{\chi}(c_1f_2) \chi(c_2f_1) }{(c_2f_1/c_1f_2)^{it}(c_1f_1 )^{s }n_1^{2s} (c_2f_2 )^{w}}. \end{split} \end{displaymath} Next we dissolve the condition $\ell \mid c_2f_2$ by using the formula $$\sum_{x \mid ab} F(a, b) = \sum_{r_1 r_2 r_3 \mid x} \mu(r_3) \sum_{a, b} F(r_1 r_3 a, r_2 r_3 b)$$ for any function $F$ (as long as the $a, b$-sum is absolutely convergent), which can be seen by first summing over $b \equiv 0$ (mod $x/(x, a)$), then sorting the $a$-sum by the gcd of $x$ and $a$ and finally removing the coprimality condition by M\"obius inversion. In this way we see that the previous display equals \begin{displaymath} \begin{split} \frac{1}{q \ell^w}& \sum_{\lambda_1\lambda_2\lambda_3 = \ell} \frac{\mu(\lambda_3)\chi(\lambda_1 \overline{\lambda_2})}{\lambda_3^w(\lambda_1/\lambda_2)^{it}} \sum_{d_1d_2 = q} \frac{d_1^{1-2s}}{ \nu(d_2)}L(\chi, t, d_2)\sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{\substack{ (M_2, d_0) = 1 \\ d_0 M_1M_2 \mid d_2}} \\ & \sum_{\delta_1, \delta_2 \mid M_2} \frac{M_1^{1-s-w}\delta_1^{1-s-it}\delta_2^{1-w+it} \mu(M_2/\delta_1) \mu(M_2/\delta_2) \bar{\chi}(\delta_1) \chi(\delta_2) }{ \tilde{\mathfrak{n}}(d_0M_1M_2)^2M_2} \\
& \sum_{(n_1, d_2) = 1} \sum_{\substack{c_1,f_1\\ (c_1, \frac{d_2}{d_0M_1M_2}) = 1}} \sum_{\substack{c_2,f_2\\ (c_2, \frac{d_2}{d_0M_1M_2}) = 1 }} \frac{A(c_1f_1M_1\delta_1, n_1d_1)\bar{\chi}(c_1f_2) \chi(c_2f_1) }{(c_2f_1/c_1f_2)^{it}(c_1f_1 )^{s }n_1^{2s} (c_2f_2 )^{w}}. \end{split} \end{displaymath} We recognize the $\lambda_1, \lambda_2, \lambda_3$-sum as $\Lambda_{(t, \chi)}(\ell; w)$, defined in \eqref{Lambda}, and the $c_1, c_2, f_1, f_2$-sum as $L(w + i t, \chi) L(w-it, \bar{\chi}) L(s + it, F \times \chi) L(s - it, F \times \bar{\chi}) $ up to Euler factors at primes dividing $q$. Thus, by brute force, we write the previous display as $$\frac{\phi(q)}{ q^2\ell^w}\Lambda_{(t, \chi)}(\ell; w) L(w + i t, \chi) L(w-it, \bar{\chi}) L(s + it, F \times \chi) L(s - it, F \times \bar{\chi})\tilde{L}_q(s, w; E_{t, \chi} \times F),$$ where $\tilde{L}_q(s, w; E_{t, \chi} \times F)$ is defined by
\begin{equation}\label{local-q-eis} \begin{split}
& \frac{q}{\phi(q)}\sum_{d_1d_2 = q} \frac{d_1^{1-2s}}{ \nu(d_2)}L(\chi, t, d_2) \sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{\substack{ (M_2, d_0) = 1 \\ d_0 M_1M_2 \mid d_2}} \sum_{\delta_1, \delta_2 \mid M_2} \frac{ \mu(M_2/\delta_1) \mu(M_2/\delta_2) \bar{\chi}(\delta_1) \chi(\delta_2) }{ \tilde{\mathfrak{n}}(d_0M_1M_2)^2M_2} \\ & M_1^{1-s-w}\delta_1^{1-s-it}\delta_2^{1-w+it} \sum_{\substack{(n_1, d_2) = 1 \\ n_1 \mid q^{\infty}}} \sum_{\substack{c_1 f_1\mid q^{\infty} \\ (c_1, \frac{d_2}{d_0M_1M_2}) = 1}} \frac{A(c_1f_1M_1\delta_1, n_1d_1)\bar{\chi}(c_1 ) \chi(f_1) }{(f_1/c_1)^{it}(c_1f_1 )^{s }n_1^{2s} } \\ & \prod_{p \mid \frac{d_2}{d_0M_1M_2}} \left(1 - \frac{\chi(p)}{p^{w + it}}\right) \left(1 - \frac{\bar{\chi}(p)}{p^{w - it}}\right)\prod_{p \mid q} \prod_{j=1}^3 \left(1 - \frac{\alpha_{F, j}(p) \chi(p)}{p^{s+it}}\right) \left(1 - \frac{\alpha_{F, j}(p) \bar{\chi}(p)}{p^{s-it}}\right)
\end{split} \end{equation} with the notation \eqref{lchitd2}. In particular, we can write \begin{equation}\label{moment2} \sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}^{\text{Eis}}_{d_2}(\pm r, n_2, h) = \mathcal{M}_{q, \ell}^{\text{Eis}}(s, w; (h, h^{\text{hol}})) \end{equation}
for any $h^{\text{hol}}$. For $\Re s, \Re w \geqslant 1/2$, $t \in \Bbb{R}$ we estimate trivially (using \eqref{indiv}) \begin{displaymath} \begin{split}
\tilde{L}_q(s, w; E_{t, \chi} \times F) \ll q^{\varepsilon} \sum_{d_1d_2 = q} \sum_{d_0 \mid M_1 \mid d_0^{\infty}} \sum_{ d_0 M_1M_2 \mid d_2} \sum_{\delta_1, \delta_2 \mid M_2} \frac{ (\delta_1\delta_2)^{1/2 }}{M_2} (M_1 \delta_1 d_1)^{\theta} \ll q^{\theta + \varepsilon} \end{split} \end{displaymath} confirming \eqref{local-bound} in the case of Eisenstein series. For an application in the next section we will also need analytic continuation of $\tilde{L}_q(s, w; E_{t, \text{triv}} \times F) $ to certain complex values of $t$ for the trivial character $\chi = \text{triv}$ modulo 1, i.e.\ the constant function with value 1.
\begin{lemma}\label{cor-eis} The functions $\tilde{L}_q(s, w; E_{\pm (1-s)/i, \text{{\rm triv}}} \times F) $, $\tilde{L}_q(s, w; E_{\pm (1-w)/i, \text{{\rm triv}}} \times F) $, initially defined in $\Re s, \Re w > 1$ as absolutely convergent series, have meromorphic continuation to an $\varepsilon$-neighbourhood of $\Re s, \Re w \geqslant 1/2$ with polar divisors at most at $s = 1/2$, $ w = 1/2$ and satisfy the bounds \begin{equation}\label{bound-cor-eis}
\begin{cases}(s - 1/2) \tilde{L}_q(s, w; E_{\pm (1-s)/i,\text{{\rm triv}}} \times F) \\(w - 1/2) \tilde{L}_q(s, w; E_{\pm (1-w)/i, \text{{\rm triv}}} \times F) \end{cases} \ll_{s, w} q^{\theta + \varepsilon }
\end{equation} for $1/2 -\varepsilon \leqslant \Re s, \Re w < 1$. \end{lemma}
\textbf{Proof.} The possible polar divisors at $s = 1/2$ or $w = 1/2$ come from $L(\text{triv}, t, d_2)$ at $t = \pm(1-s)/i$ or $\pm (1-w)/i$ defined in \eqref{lchitd2}. An application of Lemma \ref{euler2} shows that the rest can be continued holomorphically to an $\varepsilon$-neighbourhood of $\Re s, \Re w \geqslant 1/2$. For $it \in \{\pm (1-s), \pm(1-w)\}$ and $\delta_1, \delta_2 \mid M_2$ we have
$|\delta_1^{1-s-it} \delta_2^{1-w+it}| \leqslant \max(M_2^{2 - 2\min(\Re s, \Re w)}, M_2^{|\Re w - \Re s|})$, so that \eqref{bound-cor-eis} follows from \eqref{local-q-eis} and
$$q^{\varepsilon} \sum_{d_1d_2 = q} \sum_{d_0 \mid M_1 } \sum_{ d_0 M_1M_2 \mid d_2} \sum_{\delta_1, \delta_2 \mid M_2} \frac{M_2^{2 - 2\min(\Re s, \Re w)} + M_2^{|\Re w - \Re s|}}{M_2} (M_1 \delta_1d_1)^{\theta} \ll q^{\theta + \varepsilon }$$ for $1/2 -\varepsilon \leqslant \Re s, \Re w < 1$. This completes the proof.\\
We end this section by combining \eqref{moment1} (and the corresponding formula for the holomorphic case) and \eqref{moment2} with \eqref{moment-together} to obtain \begin{equation}\label{moment-comb}
\sum_{d_1d_2 = q} \sum_{ \ell \mid r} \sum_{(n_1, q) = d_1} \sum_{n_2} \frac{A(n_2, n_1)}{n_2^sn_1^{2s}r^w} \mathcal{A}_{d_2}(\pm r, n_2, \mathfrak{h}) = \mathcal{M}^{\pm}_{q, \ell}(s, w; \mathfrak{h}). \end{equation}
\section{Proof of Theorem \ref{thm1}}\label{proof1}
We have now prepared the scene for a quick proof of Theorem \ref{thm1}. Let initially be $3/2 < \Re s < 2$, $4 < \Re w < 5$ and let $\mathfrak{h} = (h, h^{\text{hol}})$ be admissible. Then by definition, $\mathscr{K}^{\ast}\mathfrak{h}$ is admissible of type $(500, 500)$, and moreover $3/2 < \Re s' < 2$, $7/2 < \Re w' < 5$. Combining \eqref{rec-prelim}, \eqref{formula1}, \eqref{formula2}, \eqref{moment-comb}, we obtain \begin{equation}\label{weobtain} \begin{split} & \mathcal{M}^{+}_{q, \ell}(s, w; \mathfrak{h}) = \sum_{\ell \mid n_2}\sum_{n_1} \frac{A(n_2, n_1)}{n_2^{s+w} n_1^{2s}} \mathscr{N}\mathfrak{h} - \sum_{j \in \{1, 2\}} (-1)^j \mathcal{N}^{(j)}_{q, \ell}(s, w; \mathscr{K}^*\mathfrak{h}) + \sum_{\pm} \mathcal{M}^{\pm}_{\ell, q}\big(s', w', \mathscr{T}^{\pm}_{s', w'} \mathfrak{h}\big)
\end{split} \end{equation} where \begin{equation}\label{final-trafo}
\mathscr{T}_{s', w'}^{\pm} \mathfrak{h} := \mathscr{L}^{\pm} \widecheck{\big(\mathscr{V}^{\pm}_{s', w'}\widehat{\mathscr{K}^*\mathfrak{h}}\big)}
\end{equation}
using the notation \eqref{kast}, \eqref{trafo}, \eqref{Hback} and \eqref{mellin1}, \eqref{mellin2}. Note, however, that we cannot simply insert the various formulas into each other to compute the transform on the right of \eqref{final-trafo}, because there is a process of analytic continuation in Lemma \ref{lem9}. By \eqref{prop-1}, \eqref{prop-2} and \eqref{mellin} we conclude that \eqref{formula2} is indeed applicable under the current assumption $3/2 < \Re s' < 2$, $7/2 < \Re w' < 5$.
Write $\mathscr{T}_{s', w'}^{\pm} \mathfrak{h} = ({\tt h}_{s', w', \pm}, {\tt h}^{\text{hol}}_{s', w', \pm})$. Combining \eqref{prop-1} -- \eqref{prop-2} with Lemma
\ref{final-decay}a, we conclude that $\mathscr{T}_{s', w'}^{\pm} \mathfrak{h}$ is weakly admissible as in \eqref{weakly}
provided that $$\Re s' > \theta + \vartheta, \quad \Re w' > \vartheta, \quad 0 < \Re s < 2, \quad 0 < \Re w < 5$$ (clearly ${\tt h}_{s', w', \pm}(t)$ is even in $t$). Since $\theta + \vartheta < 1/2$, this includes in particular the region $1/2 \leqslant \Re s \leqslant \Re w < 3/4$. Moreover, combining \eqref{mero2} with Lemma
\ref{final-decay}b, we see that ${\tt h}_{s', w', \pm}(t)$ is meromorphic in an $\varepsilon$-neighbourhood of $|\Im t | \leqslant 1/2$ with poles at most at $\pm it \in \{w', s' + \mu_1, s'+\mu_2, s' + \mu_3\}$. We will need this observation in a moment after having proved the next lemma.
It remains to continue all terms in \eqref{weobtain} to a domain containing \eqref{final-region}. The analytic continuation of the cuspidal contribution of the terms $\mathcal{M}^{+}_{q, \ell}(s, w; \mathfrak{h}) $ and $\mathcal{M}^{\pm}_{\ell, q}(s', w', \mathscr{T}^{\pm} \mathfrak{h})$ is clear.
For the Eisenstein contribution, we appeal to the following lemma.
\begin{lemma}\label{analyticcont} Let $\mathfrak{h} = (h, h^{\text{{\rm hol}}})$ be weakly admissible, and suppose that $h$ has a meromorphic continuation to an $\varepsilon$-neighbourhood of $|\Im t | \leqslant 1/2$ with at most finitely many poles. Then the term $ \mathcal{M}^{\text{{\rm Eis}}}_{q, \ell}(s, w; \mathfrak{h}) $, initially defined in $\Re s, \Re w > 1$ continues meromorphically to an $\varepsilon$-neighbourhood of $\Re w, \Re s \geqslant 1/2$ with at most finitely many polar divisors. If $1/2 \leqslant \Re s, \Re w < 1$, its analytic continuation is given by $ \mathcal{M}^{\text{{\rm Eis}}}_{q, \ell}(s, w; \mathfrak{h}) + R_{q, \ell}(s, w; \mathfrak{h})$ where $R_{q, \ell}(s, w; \mathfrak{h})$ is defined as $$\frac{\phi(q)}{q^2} \sum \underset{\substack{t = \pm i(1-s)\\ t = \pm i(1-w)}}{\text{{\rm res}}} (\pm i) \frac{L(s + it, F)L(s - it, F) \zeta(w + it) \zeta(w - it)}{\zeta(1 + 2it)\zeta(1 - 2it)} \tilde{L}_q(s, w, E_{t, \text{{\rm triv}}} \times F) \frac{\Lambda_{t, \text{{\rm triv}}}(\ell; w)}{\ell^w} h(t).$$
\end{lemma}
\textbf{Proof.} For $t \in \Bbb{R}$ choose $0 < \sigma(t) < 1/4$ in a continuous way such that $L(1 - 2 \sigma + 2it, \chi) \not= 0$ for $0 \leqslant \sigma < 2\sigma(t)$ and all primitive Dirichlet characters $\chi$ of conductor $c_{\chi}$ such that $c_{\chi}^2 \mid q$ and in addition $h(t - i\sigma)$ is pole-free for $0 \leqslant \sigma < 2\sigma(t)$. Let initially $1 < \Re s < 1 + \sigma(\Im s)$, $1 < \Re w < 1 + \sigma(\Im w)$. In the defining integral of $ \mathcal{M}^{\text{{\rm Eis}}}_{q, \ell}(s, w; \mathfrak{h}) $ we shift the $t$-contour down to $\Im t = -\sigma(\Re t)$. We pick up a pole at $w-it = 1$ if $\chi = \text{triv}$ and a (triple) pole at $s - it = 1$ if in addition $F = {\tt E}_0$. The remaining integral is holomorphic in $1- \sigma(\Im s) < \Re s < 1 + \sigma(\Im s)$, $1- \sigma(\Im w) < \Re w < 1 + \sigma(\Im w)$. Now choosing $s, w$ with $1- \sigma(\Im s) < \Re s < 1 $, $1- \sigma(\Im w) < \Re w < 1 $, we shift the $t$-contour back to $\Im t = 0$ picking up a pole at $w + it = 1$ if $\chi = \text{triv}$ and a (triple) pole at $s + it = 1$ if in addition $F = {\tt E}_0$. This proves the desired formula for $1- \sigma(\Im s) < \Re s < 1$, $1- \sigma(\Im w) < \Re w < 1$ , but then it follows for $s, w$ in an $\varepsilon$-neighbourhood of $\Re w, \Re s \geqslant 1/2$ by analytic continuation, using Lemma \ref{cor-eis} for the term $ \tilde{L}_q(s, w, E_{t, \text{{\rm triv}}} \times F) $. This completes the proof. \\
Note that this lemma is applicable both for the admissible function $\mathfrak{h}$ and for $\mathscr{T}_{s', w'}^{\pm} \mathfrak{h}$, since the latter satisfies the assumption of meromorphic continuation. We recall that condition \eqref{final-region} implies $1/2 \leqslant \Re s' \leqslant \Re w' < 1$. We conclude that the analytic continuation of $ \mathcal{M}^{\text{{\rm Eis}}}_{q, \ell}(s, w; \mathfrak{h}) $ and $\mathcal{M}^{\text{Eis}}_{\ell, q}\big(s', w', \mathscr{T}^{\pm}_{s', w'} \mathfrak{h}\big)$ infers two extra main terms, and we obtain the reciprocity formula \eqref{final-formula} with \begin{equation}\label{main-term-final} \begin{split}
& \mathcal{N}_{q, \ell}(s, w; \mathfrak{h}) =\\
& \sum_{\ell \mid n_2}\sum_{n_1} \frac{A(n_2, n_1)}{n_2^{s+w} n_1^{2s}} \mathscr{N}\mathfrak{h} - \sum_{j \in \{0, 1\}}(-1)^j \mathcal{N}^{(j)}_{q, \ell}(s, w; \mathscr{K}^*\mathfrak{h}) - R_{q, \ell}(s, w; \mathfrak{h}) + R_{\ell, q}(s', w'; \mathscr{T}_{s', w'}^{\pm} \mathfrak{h}).
\end{split} \end{equation} A (meromorphic) continuation of these terms follows from
\eqref{bump-double}
and Proposition \ref{prelim} and Lemma \ref{analyticcont}, and the bounds \eqref{bound-cor-eis}, \eqref{bound-bump}, \eqref{boundsN} confirm \eqref{required-bound}, away from polar divisors, in an $\varepsilon$-neighbourhood of \eqref{final-region}. While the individual main terms may have polar lines, their joint contribution must be holomorphic, because the rest of terms in the reciprocity formula are holomorphic. Hence the various (possible) polar divisors must cancel, so that by standard complex analysis (e.g.\ Cauchy's integral formula in the $s$ and $w$ variable) the estimates \eqref{required-bound} are valid in the entire region \eqref{final-region}. This completes the proof.
\section{Proof of Theorem \ref{thm2}}\label{sec-new}
Before we start with the proof, we recall that a standard application of the large sieve (cf.\ e.g.\ \cite[Theorm 7.35]{IK}) shows that \begin{equation}\label{largesieve}
\sum_{f \in \mathcal{B}^{\ast}(q)} L(1/2, f)^4 |h(t_f)|+
\sum_{f \in \mathcal{B}_{\text{hol}}^{\ast}(q)} L(1/2, f)^4 |h^{\text{hol}}(k_f) | \ll q^{1+\varepsilon}
\end{equation}
for any $q \in \Bbb{N}$ (not necessarily prime), whenever $\mathfrak{h} = (h, h^{\text{hol}})$ satisfies \eqref{weakly}. Moreover, by another standard application of the large sieve (\cite[Theorem 7.34]{IK}) we have
\begin{equation}\label{largesieve2}
\sum_{c_{\chi}^2 \mid q} \int_{-\infty}^{\infty} |L(1/2 + it, \chi)|^8 |h(t)| dt \leqslant \sum_{c_{\chi}\leqslant q^{1/2} } \int_{-\infty}^{\infty} |L(1/2 + it, \chi)|^8 |h(t)| dt \ll q^{1+\varepsilon}.
\end{equation}
We will also frequently use the standard bounds \begin{equation}\label{hl} \begin{split}
(q(1 + |t_f|))^{-\varepsilon} & \ll L(1, \text{Ad}^2f) \ll (q(1 + |t_f|))^{\varepsilon}, \quad f \in \mathcal{B}^{\ast}(q),\\
(q k_f)^{-\varepsilon} &\ll L(1, \text{Ad}^2f) \ll (q k_f)^{\varepsilon}, \quad\quad\quad\,\,\,\,\, f \in \mathcal{B}_{\text{hol}}^{\ast}(q),\\
(q(1+|t|))^{-\varepsilon} & \ll |L(1 + 2 it, \chi)|, \quad\quad\quad\quad\quad\quad\quad t \in \Bbb{R}, c_{\chi}^2 \mid q. \end{split} \end{equation} Finally we recall that the central values $L(1/2, f)$ for $f \in \mathcal{B}^{\ast}(q) \cup \mathcal{B}^{\ast}_{\text{hol}}(q)$ are non-negative \cite{KZ, KaS}, for arbitrary $q \in \Bbb{N}$.
Now let $F = {\tt E}_0$ be the minimal parabolic Eisenstein series with trivial spectral parameters. By Theorem \ref{thm1} with $\theta = 0$ we have for $(q, \ell) = 1$, $\mathfrak{h}$ admissible that \begin{equation}\label{moeb}
\sum_{a b = \ell} \mathcal{M}^+_{q, a}(1/2, 1/2, \mathfrak{h}) \left(\frac{a}{b }\right)^{\frac{1}{2}} \ll (\ell q)^{\varepsilon}\sum_{a b = \ell} \left(\frac{a}{b}\right)^{\frac{1}{2}} \Bigr(\frac{1}{a} + \frac{1}{q} + \sum_{\pm}\big |\mathcal{M}^{\pm}_{a, q}(1/2, 1/2; \mathscr{T}_{1/2, 1/2}^{\pm}\mathfrak{h})\big|\Bigr).
\end{equation} For $\Lambda_f(a; w)$ as in \eqref{Lambda} we have $$ \sum_{a b = \ell} \frac{\Lambda_f(a; 1/2)}{a^{1/2}} \left(\frac{a}{b}\right)^{1/2} = \lambda_f(\ell).$$ Now let $q$ be squarefree. Then we can use \eqref{largesieve2} with $q = 1$ together with \eqref{local-bound} and the last bound in \eqref{hl} to estimate the Eisenstein contribution in \eqref{moment}. In this way we see that
\begin{displaymath}
\begin{split} \sum_{a b = \ell} \mathcal{M}^+_{q, a}(1/2, 1/2, \mathfrak{h})& \left(\frac{a}{b }\right)^{1/2} = \frac{\phi(q)}{q^2} \sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast} (d_0)} \frac{L(1/2, f)^4}{L(1, \text{Ad}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) \lambda_f(\ell) h(t_f) \\ & + \frac{\phi(q)}{q^2} \sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast}_{\text{hol}}(d_0)} \frac{L(1/2, f)^4}{L(1, \text{Ad}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) \lambda_f(\ell) h^{\text{hol}}(k_f) + O(q^{-1} \ell^{\varepsilon}).
\end{split}
\end{displaymath}
On the other hand, again by \eqref{moment}, \eqref{moment-together}, \eqref{Lambda} with $q$ in place of $\ell$ and \eqref{local-bound} with $\theta = 0$ and $a$ in place of $q$ we have
\begin{displaymath}
\begin{split}
\mathcal{M}^{\pm}_{a, q}(1/2, 1/2; &\mathscr{T}_{1/2, 1/2}^{\pm}\mathfrak{h}) \ll a^{\varepsilon-1} q^{\vartheta - 1/2+\varepsilon} \left(\sum_{a_0 \mid a}\sum_{f\in \mathcal{B}^{\ast}(a_0)} \frac{|L(1/2, f)|^4}{L(1, \text{Ad}^2 f)} (1+|t_f|)^{-15} \right.\\
& \left. + \sum_{a_0 \mid a}\sum_{f\in \mathcal{B}^{\ast}_{\text{hol}}(a_0)} \frac{|L(1/2, f)|^4}{L(1, \text{Ad}^2 f)} k_f^{-15} + \sum_{\chi: c_{\chi}^2 \mid a} \int_{\Bbb{R}} \frac{L(1/2 + it, \chi)|^8}{|L(1 + 2it, \chi)|^2}(1+|t|)^{-15} \frac{dt}{2\pi} \right).
\end{split}
\end{displaymath} By \eqref{largesieve} -- \eqref{hl} and the non-negativity of $L(1/2, f)$ we conclude $$ \mathcal{M}^{\pm}_{a, q}(1/2, 1/2; \mathscr{T}_{1/2, 1/2}^{\pm}\mathfrak{h}) \ll q^{\vartheta-1/2+\varepsilon} a^{\varepsilon},$$ so that the right hand side of \eqref{moeb} is $\ll (q\ell)^{\varepsilon} ( q\ell^{-1/2} +q^{1/2 + \vartheta}\ell^{1/2})$. We have shown \begin{prop}\label{prop-new} For $q$ squarefree, $\varepsilon > 0$, $(\ell, q) = 1$ and $\mathfrak{h}$ admissible, we have \begin{equation}\label{squarefree} \begin{split} &\sum_{d_0 \mid q} \sum_{f \in \mathcal{B}^{\ast} (d_0)} \frac{L(1/2, f)^4}{L(1, \text{{\rm Ad}}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) \lambda_f(\ell) h(t_f) \\ & + \sum_{d_0 \mid q}\sum_{f \in \mathcal{B}^{\ast}_{\text{{\rm hol}}}(d_0)} \frac{L(1/2, f)^4}{L(1, \text{{\rm Ad}}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) \lambda_f(\ell) h^{\text{{\rm hol}}}(k_f) \ll (q\ell)^{\varepsilon}\left( \frac{q}{\ell^{1/2}} +q^{1/2 + \vartheta}\ell^{1/2}\right). \end{split} \end{equation} \end{prop}
If $q$ is prime, by Lemma \ref{lem2} we have
$\tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) = 1$ for
$f \in \mathcal{B}^{\ast}(q) \cup \mathcal{B}_{\text{hol}}^{\ast}(q)$ and
$\tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) \ll q^{\varepsilon}$ for $f \in \mathcal{B}(1) \cup \mathcal{B}_{\text{hol}}(1)$.
The contribution of the level 1 forms in \eqref{squarefree} is $O((q\ell)^{\varepsilon} \ell^{\vartheta })$. This completes the proof of Theorem \ref{thm2}.
\section{Proofs of Theorems \ref{cor1} and \ref{cor2}}
\subsection{Proof of Theorem \ref{cor1}} Let $f \in \mathcal{B}^{\ast}(q)$. By an approximate functional equation\footnote{This is the only point where an approximate functional equation is used explicitly although it is implicit in \eqref{largesieve}.}, cf.\ \cite[Theorem 5.3]{IK}, we have \begin{equation}\label{approx} L(1/2, f) = (1 + \omega_f)\sum_{\ell} \frac{\lambda_f(\ell)}{\ell^{1/2}} W_f\Bigl(\frac{\ell}{q^{1/2}}\Bigr) \end{equation}
where $\omega_f \in \{\pm 1\}$ is the root number and we can choose $$W_f(x) =\frac{1}{2\pi i} \int_{(\varepsilon)} \frac{L_{\infty}(1/2 + s, f)}{L_{\infty}(1/2, f)} \frac{G_f(s)}{G_f(0)} x^{-s} \frac{ds}{s}$$ with $L_{\infty}(s, f) = \pi^{-s}\textstyle \Gamma\left(\frac{1}{2}( s + it_f)\right)\Gamma\left(\frac{1}{2}( s - it_f)\right)$ and $$G_f(s) = \prod_{j=0}^{1000} \prod_{\epsilon_1, \epsilon_2 \in \{\pm 1\}} \left( \frac{1}{2} + \epsilon_1 s + i \epsilon_2 t_f + j\right).$$
The choice for the particular weight function $G_f$ will become clear in a moment. Since $L(1/2, f) \geqslant 0$ with equality if $\omega_f = -1$, we can get rid of the root number in \eqref{approx} and obtain by \eqref{hl} that
\begin{displaymath} \begin{split}
\sum_{f \in \mathcal{B}^{\ast}(q)} &L(1/2, f)^5 e^{- t_f^2} \ll \sum_{f \in \mathcal{B}^{\ast}(q)} L(1/2, f)^4 e^{- t_f^2} \frac{(q ( 1+ |t_f|))^{\varepsilon}}{L(1, \text{Ad}^2f)}\cdot 2\frac{\lambda_f(\ell)}{\ell^{1/2}} W_f\Bigl(\frac{\ell}{q^{1/2}}\Bigr)\\
& =
\frac{2q^{\varepsilon}}{2\pi i} \int_{(1)} \pi^{-s} q^{s/2} \sum_{\ell} \ell^{-1/2-s} \sum_{f\in \mathcal{B}^{\ast}(q)} \frac{L(1/2, f)^4}{L(1, \text{Ad}^2 f)} \lambda_f(\ell) \frac{L_{\infty}(1/2 + s, f)}{L_{\infty}(1/2, f)} \frac{G_f(s)}{G_f(0)} e^{- t_f^2} ( 1+ |t_f|)^{\varepsilon} \frac{ds}{s}.
\end{split}
\end{displaymath}
Shifting the contour to the far right, we can truncate the $\ell$-sum at $q^{1/2+\varepsilon}$ at the cost of an error $O(q^{-10})$. Note that now automatically $(q, \ell) = 1$. Having done this, we shift the contour back to $\Re s = \varepsilon$, and by the exponential decay of $$\frac{L_{\infty}(1/2 + s, f)}{L_{\infty}(1/2, f)} e^{-t_f^2}$$ along vertical lines (uniformly in $t_f$) we can truncate the contour at $|\Im s| \leqslant (\log q)^2$ with the same error. Hence
$$ \sum_{f \in \mathcal{B}^{\ast}(q)} L(1/2, f)^5 e^{- t_f^2} \ll q^{\varepsilon} \max_{|\tau| \leqslant (\log q)^2} \sum_{\ell \leqslant q^{1/2 + \varepsilon}} \frac{1}{\ell^{1/2}} \Bigl| \sum_{f\in \mathcal{B}^{\ast}(q)} \frac{L(1/2, f)^4}{L(1, \text{Ad}^2 f)} \lambda_f(\ell)h_{\tau}(t_f) \Bigr| + q^{-10}$$ where
$$h_{\tau}(t_f) = \frac{L_{\infty}(1/2 + \varepsilon + i\tau, f)}{L_{\infty}(1/2, f)} \frac{G_f(\varepsilon + i\tau)}{G_f(0)} e^{- t_f^2} ( 1+ |t_f|)^{\varepsilon}.$$ By construction, the family
$\{\mathfrak{h}_{\tau} =(h_{\tau}, 0) : |\tau| \leqslant (\log q)^2\}$ is uniformly admissible. Theorem \ref{thm2} now yields the desired bound.
\subsection{Proof of Theorem \ref{cor2}} Let $f_0 \in \mathcal{B}^{\ast}(q) \cup \mathcal{B}^{\ast}_{\text{hol}}(q)$. Let $q^{1/100} < L < q$ be a parameter to be determined later in terms of $q$. In the following all implied constants may depend on the archimedean parameter of $f_0$, and $p$ denotes a prime number. For $f \in \mathcal{B}(q) \cup \mathcal{B}_{\text{hol}}(q)$ we choose the following amplifier
$$A_f := \Bigl|\sum_{ \substack{p \leqslant L\\ p \nmid q} } \lambda_f(p) x(p)\Bigr|^2 + \Bigl|\sum_{\substack{p \leqslant L\\ p \nmid q}} \lambda_f(p^2) x(p^2)\Bigr|^2, \quad x(n) = \text{sgn}(\lambda_{f_0}(n)).$$
Then \begin{equation}\label{lowerbound}
A_{f_0} = \Bigl(\sum_{\substack{ p\leqslant L\\ p \nmid q} } |\lambda_{f_0}(p) |\Bigr)^2 + \Bigl(\sum_{\substack{ p\leqslant L\\ p \nmid q} } |\lambda_{f_0}(p^2)| \Bigr)^2 \geqslant \frac{1}{2} \Bigl(\sum_{\substack{ p\leqslant L\\ p \nmid q} } |\lambda_{f_0}(p)| + |\lambda_{f_0}(p^2)| \Bigr)^2 \gg \frac{L^2}{\log L} \end{equation} by the prime number theorem and the Hecke relation $\lambda_{f_0}(p)^2 = 1 + \lambda_{f_0}(p^2)$. On the other hand, $$A_f = \sum_{\substack{ p \leqslant L \\ p \nmid q}} (x(p)^2 + x(p^2)^2)+ \sum_{ \substack{p_1, p_2 \leqslant L \\ p_1p_2 \nmid q}} \left(x(p_1)x(p_2) + \delta_{p_1 = p_2} x(p_1^2) x(p_2^2)\right) \lambda_f(p_1p_2) + \sum_{ \substack{p_1, p_2 \leqslant L \\ p_1p_2 \nmid q}}x(p_1^2) x(p_2^2) \lambda_f(p_1^2p_2^2).$$
Now suppose that $q$ is squarefree and $(6, q) = 1$. Then by \eqref{extension} with $A(1, p) = A(p, 1) = \tau_3(p) = 3$ and $|\lambda_f(p)| \leqslant p^{\vartheta} + p^{-\vartheta}$ we see that $$\tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) > 0, \quad f \in \mathcal{B}(q) \cup \mathcal{B}_{\text{{\rm hol}}}(q).$$ We employ now the admissble function $\mathfrak{h}_{\text{pos}} = (h_{\text{pos}}, h_{\text{pos}}^{\text{hol}})$ with $a = 1000$, $b = 400$, the
non-negativity of central values and \eqref{hl} to conclude
\begin{displaymath}
\begin{split}
A_{f_0} L(1/2, f_0)^4 \ll & q^{\varepsilon} \sum_{f \in \mathcal{B}^{\ast}(q)} \frac{L(1/2, f)^4 }{L(1, \text{{\rm Ad}}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) A_f h_{\text{pos}}(t_f)\\
& + q^{\varepsilon} \sum_{f \in \mathcal{B}^{\ast}_{\text{{\rm hol}}}(q)} \frac{L(1/2, f )^4 }{L(1, \text{{\rm Ad}}^2 f)} \tilde{L}_q(1/2, 1/2, f \times {\tt E}_0) A_f h_{\text{pos}}^{\text{hol}}(k_f) .
\end{split}
\end{displaymath} We insert the lower bound \eqref{lowerbound} on the left hand side and the exact formula for $A_f$ on the right hand side. By Proposition \ref{prop-new} with $\ell \in\{ 1, p_1p_2, p_1^2p_2^2\}$, we obtain $$L^2 L(1/2, f)^4 \ll q^{\varepsilon}(q L + q^{1/2 + \vartheta} L^4).$$ Choosing $L = q^{(1 - 2\vartheta)/6}$ completes the proof.
\end{document}
|
arXiv
|
\begin{document}
\title[EuclidNets]{EuclidNets: An Alternative Operation for Efficient Inference of Deep Learning Models}
\author[1]{\fnm{Xinlin} \sur{Li}}\email{[email protected]}
\author[2]{\fnm{Mariana} \sur{Parazeres}}\email{[email protected]}
\author[2]{\fnm{Adam} \sur{Oberman}}\email{[email protected]}
\author[1]{\fnm{Alireza} \sur{Ghaffari}}\email{[email protected]}
\author[2]{\fnm{Masoud} \sur{Asgharian}}\email{[email protected]}
\author*[1,3]{\fnm{Vahid} \sur{Partovi~Nia}}\email{[email protected]}
\affil*[1]{\orgdiv{Huawei Noah's Ark Lab}, \orgname{Montreal Research Centre}, \orgaddress{\street{7101 Park Avenue}, \city{Montreal}, \postcode{H3N 1X9}, \state{QC}, \country{Canada}}}
\affil[2]{\orgdiv{Department of Mathematics and Statistics}, \orgname{McGill University}, \orgaddress{\street{805 Sherbrooke Street West}, \city{Montreal}, \postcode{H3A 0B9}, \state{QC}, \country{Canada}}}
\affil[3]{\orgdiv{Department of Mathematics and Industrial Engineering}, \orgname{Polytechnique Montreal}, \orgaddress{\street{2500 Chem. Polytechnique}, \city{Montreal}, \postcode{H3T 1J4}, \state{QC}, \country{Canada}}}
\abstract{With the advent of deep learning application on edge devices, researchers actively try to optimize their deployments on low-power and restricted memory devices. There are established compression method such as quantization, pruning, and architecture search that leverage commodity hardware. Apart from conventional compression algorithms, one may redesign the operations of deep learning models that lead to more efficient implementation. To this end, we propose EuclidNet, a compression method, designed to be implemented on hardware which replaces multiplication, $xw$, with Euclidean distance $(x-w)^2$. We show that EuclidNet is aligned with matrix multiplication and it can be used as a measure of similarity in case of convolutional layers. Furthermore, we show that under various transformations and noise scenarios, EuclidNet exhibits the same performance compared to the deep learning models designed with multiplication operations. }
\keywords{Deep learning compression, Euclidean distance, Convolutional Neural Network, Hardware efficient algorithms}
\maketitle
\section{Introduction}\label{sec1}
While majority of deep neural networks are trained on GPUs, they are increasingly being deployed on edge devices, such as mobile devices. These edge devices require
to compress the architecture for a given hardware design (e.g. GPU or lower precision chips) due to memory and power constraints \cite{benmeziane2021comprehensive, cheng2017survey}. Moreover, application specific hardware are being designed to accommodate the deployment of deep learning models. Thus, designing efficient deep learning architectures that are efficient for the deployment (i.e. \emph{inference}) has become a new challenge in the deep learning community.
The combined problem of hardware and deep learning model design is complex, and the precise measurement of efficiency is both device and model specific. This is because researchers have to take into account various efficiency factors such as latency, memory footprint, energy consumption. Here we deliberately oversimplify the problem in order to make it tractable, by addressing a fundamental element of hardware cost. Knowing that power consumption is directly related to the chip area in a digital circuit, we use the chip area required to implement an arithmetic operation on a hardware as a surrogate to measure the efficiency of a deep learning model. While this is very coarse, and full costs will depend on other aspects of hardware implementation, it nevertheless represents a fundamental unit of cost in hardware design \cite{hennessy2011computer}.
In a deep learning model, weights are multiplied by inputs, hence on of the fundamental operations in deep learning models is multiplication $S_{{\mathrm{conv}}}(x,w) = wx$. In our work, we replace multiplication with the EuclidNet operator, \begin{equation}\label{eq: euclid}
S_{{\mathrm{euclid}}}(x,w) = -\frac{1}{2}\|x-w\|_2^2. \end{equation} which combines a difference with a square operator. We will refer to the family of deep learning models that use equation \eqref{eq: euclid} as EuclidNets. These models compromise between standard multiplicative models and AdderNets\cite{chen2020addernet}, which remove multiplication entirely, but at the cost of a significant loss of accuracy and difficult training procedure. Replacing multiplication with square can potentially reduce the computation cost. The feature representation of each of the architectures is illustrated in Figure~\ref{fig:feature}. EuclidNets can be implemented on 8-bit precision without loss of accuracy as demonstrated in Table~\ref{tab: quant}.
The square operator is cheaper than multiplication and it can also be implemented using look up tables \cite{de2009large}. In \cite{baluja2018no,covell2019table}, authors prove that replacing look up table can replace actual float computing, while works such as LookNN in \cite{razlighi2017looknn} take the first step in designing hardware for look up table use. On a low precision hardware, we can compute $S_{\mathrm{euclid}}$ for about half the cost of computing $S_{\mathrm{conv}}$.
Furthermore, using EuclidNets, the deep learning model does not lose expressivity, as explained ins Section \ref{sec:theory}. To summarize, we make the following contributions: \begin{itemize}
\item We design a deep learning architecture based on replacing the multiplication $S_{\mathrm{conv}}(x,w) = wx$ by the squared difference equation \eqref{eq: euclid}. We show that using square operator can potentialy reduce the hardwaer cost.
\item These deep learning models are just as expressive as convolutional neural networks. In practice, they have comparable accuracy (drop of less than 1 percent on ImageNet on ResNet50 going from full precision convolutional to 8-bit EuclidNets).
\item We show theoretically and empirically that EuclidNets have the same behaviour compared to convolutional neural network in the case that the input is transformed (e.g. linear transformation) or affected by noise (e.g. Guassian noise).
\item We provide an easy approach to train EuclidNets using homotopy. \end{itemize}
\begin{figure}
\caption{Feature representation of traditional convolution with $S(x,w) = xw$ (left), AdderNet $S(x,w) = -\|x-w\|_1$ (middle), EuclidNet $S(x,w) = -\frac{1}{2}\|x-w\|_2^2$ (right).}
\label{fig:feature}
\end{figure}
\begin{table}[h]
\caption{Euclid-Net Accuracy with full precision and 8-bit quantization: Results on ResNet-20 with Euclidian similarity for CIFAR10 and CIFAR100, and results on ResNet-18 for ImageNet. Euclid-Net achieves comparable or better accuracy with 8-bit precision, compared to the conventional full precision convolutional neural network.
}\label{tab: quant}
\centering
\begin{tabular}{ccc ccc}
\multirow{3}{*}{\textbf{Network}} &
\multirow{3}{*}{\textbf{Quantization}} &
\multirow{3}{*}{\textbf{Chip Efficiency}} &
\multicolumn{3}{c}{\textbf{Top-1 accuracy}} \\
&&& CIFAR10 & CIFAR100& ImageNet \\
\hline
\multirow{2}{*}{$S_{{\mathrm{conv}}}$} &
Full precision & \ding{55} & 92.97
& 68.14 & 69.56 \\%69.8 \\
& 8-bit & \ding{51} & 92.07 & 68.02 & 69.59 \\
\multirow{2}{*}{$S_{{\mathrm{euclid}}}$} &
Full precision & \ding{55} & 93.32 & 68.84 & 69.69 \\
& 8-bit & \ding{51} & 93.30 & 68.78 & 68.59 \\
\multirow{2}{*}{$S_{{\mathrm{adder}}}$} &
Full precision & \ding{55} & 91.84 & 67.60 & 67.0 \\
& 8-bit & \ding{51} & 91.78 & 67.60 & 68.8 \\
\multirow{1}{*}{BNN} &
1-bit & \ding{51} & 84.87 & 54.14 & 51.2 \\ \end{tabular}
\end{table}
\section{Context and related work}
Compressing deep learning models comes at the costs of accuracy loss, and increasing training time (to a greater extent on quantized networks) \cite{frankle2018lottery, cheng2018model}. Part of the accuracy loss comes simply from decreasing model size, which is required for mobile and edge devices \cite{wu2019machine}. Some of the most common deep learning compression methods include pruning \cite{reed1993pruning}, quantization \cite{guo2018survey}, knowledge distillation \cite{hinton2015distilling}, and efficient design \cite{iandola2016squeezenet,howard2017mobilenets,zhang2018shufflenet,tan2019efficientnet}. Between the compression methods, the most prominent approach is low bit quantization \cite{guo2018survey}. In this case, the inference can speed up with lowering bit size, at the cost of
accuracy drop and longer training times. In the extreme quantization, such as binary networks, operations have negligible cost at inference but exhibits a considerable accuracy drop \cite{Hubara_BNN}.
Here we focus on a small sub-field of compression, that optimizes mathematical operations in a deep learning model. This approach can be combined successfully with other conventional compression methods, such as quantization \cite{xu2020kernel} and pruning \cite{reed1993pruning}.
On the other hand, knowledge distillation \cite{hinton2015distilling} consists of transferring information from a larger teacher network to a smaller student network. The idea is easily extended by thinking of information transfer between different similarity measures, which \cite{xu2020kernel} explores in the context of AdderNets. Knowledge distillation is an uncommon training procedure and requires extra implementation effort. However, EuclidNet preserves the accuracy without knowledge distillation. We suggest a straightforward training using a smooth transition between common convolution and Euclid operation using homotopy.
\section{Similarity and Distances}
\subsection{Inner Products versus Distances}
Consider an intermediate layer of a deep learning model with input $x\in{\mathbb{R}}^{H\times W \times c_{{\mathrm{in}}}}$ and output $y~\in~{\mathbb{R}}^{H\times W \times c_{{\mathrm{out}}}}$ where $H,W$ are the dimensions of the input feature, and $c_{{\mathrm{in}}}, c_{{\mathrm{out}}}$ the number of input and output channels, respectively. For a standard convolutional network, we represent the input to output transformation via weights $w~\in~{\mathbb{R}}^{d\times d\times c_{{\mathrm{in}}}\times c_{{\mathrm{out}}}}$ as \begin{equation}\label{eq: layer} y_{mnl} = \sum_{i = m}^{m+d} \sum_{j=n}^{n+d} \sum_{k = 0}^{c_{{\mathrm{in}}}} x_{ijk} w_{ijkl}
\end{equation} Setting $d=1$ reduces the equation \eqref{eq: layer} to a fully-connected layer. We can abstract the multiplication of the weights $w_{ijkl}$ by $x_{ijkl}$ in the equation above by using a similarity measure $S:{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}$. The original convolutional layer corresponds to $$ S_{{\mathrm{conv}}}(x,w) = xw. $$ In our work, we replace $S_{\mathrm{conv}}$ with $S_{\mathrm{euclid}}$, given by equation \eqref{eq: euclid}. A number of works have also replaced the multiplication operator in deep learning models. The most relevant work is the AdderNet \cite{chen2020addernet}, which uses \begin{equation}\label{eq: adder}
S_{{\mathrm{adder}}}(x,w) = -\|x-w\|_1. \end{equation} to replace multiplication by $\ell_1$ norm, i.e. summation of the absolute value of differences. This operation can be implemented very efficiently on a custom hardware, knowing that subtraction and absolute value of different $n$-bit integers cost $\mathcal{O}(n)$ gate operations, compared to $\mathcal{O}(n^2)$ for multiplication i.e. $S_{\mathrm{conv}}(x,w) = xw$. However, AdderNet comes with a significant loss in accuracy, and is difficult to train.
\subsection{Other Similarity Measures}
The idea of replacing multiplication operations to save resources within the context of neural networks dates back to 1990s. Equally motivated by computational speed-up and hardware logic minimization, authors of \cite{dogaru1999comparative} defined perceptrons that use the synapse similarity, \begin{equation}\label{eq: comp_syn}
S_{{\mathrm{synapse}}}(x,w) = \sign(x)\cdot \sign(w) \cdot \min(\|x\|,\|w\|), \end{equation} which is cheaper than multiplication in terms of hardware complexity.
Equation \eqref{eq: comp_syn} has not been experimented with modern deep learning models and datasets. Moreover, in \cite{akbas2015multiplication} a slight variation is introduced which is also a multiplication-free operator, \begin{equation}\label{eq: mf}
S_{{\mathrm{mfo}}}(x,w) = \sign(x)\cdot\sign(w)\cdot(\|x\|+\|w\|)). \end{equation} Note that both equations \eqref{eq: comp_syn} and \eqref{eq: mf} use $\ell_1$-norm. Also note that in \cite{mallah2018multiplication}, the updated design choice allows contributions from both operands $x$ and $w$. Furthermore, in \cite{afrasiyabi2018non}, the similarity in image classification on CIFAR10 is studied. Other applications of equation \eqref{eq: mf} are studied in \cite{badawi2017multiplication, pan2019additive}.
In \cite{you2020shiftaddnet}, the similarity operation is further combined with a bit-shift, leading to an improved accuracy with negligible added hardware cost. However, the accuracy results for AdderNet appear to be lower than those reported in \cite{chen2020addernet}. Another follow-up work uses knowledge distillation to further improve the accuracy of AdderNets \citep{xu2020kernel}.
Instead of simply replacing the similarity on the summation, there is also the possibility to replace the full expression of equation \eqref{eq: layer} as, for example, proposed in \cite{limonova2020resnet,limonova2020bipolar}, by approximating the activation of a given layer with an exponential term. Unfortunately, these methods only lead to speed-up in certain cases and, in particular, they do not improve CPU inference time. Moreover, the reported accuracy on the benchmark problems is also lower than the typical baseline.
In \cite{mondal2019dense}, authors used three layer morphological neural networks for image classification. Morphological neural networks were introduced in 1990s in \cite{davidson1990theory, ritter1996introduction} and use the notion of erosion and dilation to replace equation \eqref{eq: layer}:
\begin{align*}
\mbox{Erosion}(x,w) &= \min_j S(x_j, w_j) = \min_j (x_j - w_j), \\
\mbox{Dilation}(x,w) &= \max_j S(x_j, w_j) = \max_j (x_j + w_j).
\end{align*} The authors proposed two methods by stacking layers to expand networks, but they admitted the possibility of over-fitting and difficult training issues, casting doubt on scalability of the method.
\section{Theoretical Justification}\label{sec:theory} This section provides some theoretical ground for the connections among AdderNets, EuclidNets, and conventional convolution. \subsection{Equivalence with Multiplication}\label{sec:theory_align}
Euclidean distance has a close tie with multiplication and hence, it can replace the multiplications in convolution and linear layers. Here, we delve into the details of this claim a bit more.
Let us consider Euclidean distance between the two vectors ${\mathbf{x}}$ and ${\mathbf{w}}$ as $\norm{{\mathbf{x}}-{\mathbf{w}}}=({\mathbf{x}}-{\mathbf{w}}){^\top}({\mathbf{x}}-{\mathbf{w}})$ where ${\mathbf{x}}$ and ${\mathbf{w}}$ are the vectors of inputs and weights respectively. Moreover, ${\mathbf{x}}$ and ${\mathbf{w}}$ are vectors of random variables, so it is of interest to study the expected value of the EuclidNet operation first, \begin{equation}
-\frac 1 2 \mathbb{E} \norm{{\mathbf{x}}-{\mathbf{w}}} = -\frac 1 2 \mathbb{E}\norm {\mathbf{x}} - \frac 1 2 \mathbb{E}\norm {\mathbf{w}} + \mathbb{E}({\mathbf{x}}{^\top}{\mathbf{w}}) .
\label{eq:euclid_expect} \end{equation} In other words \eqref{eq:euclid_expect}, convlution similarity measure, i.e. the inner product ${\mathbf{x}}{^\top}{\mathbf{w}}$, is embedded in EuclidNet form. However, the result is biased with two extra terms i.e. $-\frac 1 2 \mathbb{E}\norm{\mathbf{x}}$ and $-\frac 1 2 \mathbb{E}\norm{\mathbf{w}}$. Thus we may conclude that Euclidean distance is aligned with multiplication shifted by two bias terms. The induced bias by the EuclidNet operation remains controlled in both training or inference, most deep learning models use some sort of normalization mechanism such as batch norm, layer norm, and weight norm.
Euclidean distance also has a close relationship with cosine similarity. Let us define $S_{\mathrm{cos}}$ as
\begin{equation}
S_{\mathrm{cos}}({\mathbf{x}},{\mathbf{w}}):= \frac{{\mathbf{x}}^\top {\mathbf{w}}}{\rootnorm {\mathbf{x}} \rootnorm{\mathbf{w}}}.
\label{eq:cosine_sim} \end{equation} It is easy to see that in the case of having a normalization mechanism (i.e. $\rootnorm {\mathbf{x}}=\rootnorm {\mathbf{w}}=1$) the cosine similarity and Euclid similarity become equivalent \begin{eqnarray}
S_{\mathrm{euclid}}({\mathbf{x}},{\mathbf{w}})= S_{\mathrm{cos}}({\mathbf{x}},{\mathbf{w}})-1 &\mathrm{\quad s.t.}& \rootnorm {\mathbf{x}} = \rootnorm {\mathbf{w}}=1 .
\label{eq:cosine_distance} \end{eqnarray}
Moreover, Euclidean norm is a transitive similarity measure since it satisfies the following inequality
\begin{equation}
\|{\mathbf{x}}-{\mathbf{w}}\|_2 \geq\lvert~\|{\mathbf{x}}\|_2-\|{\mathbf{w}}\|_2 ~\rvert .
\label{eq:inv_triangle} \end{equation}
It is noteworthy to mention that this transitivity holds for p-norms (i.e. $\|\mathbf{a}\|_p= (\sum_i \|{a}_i\|^p)^{\frac{1}{p}}$). This means that the AdderNet \cite{chen2020addernet} operator is also transitive. According to equation \eqref{eq:cosine_distance}, however, the only norm that has such a close relationship with the cosine similarity is Euclidean norm. This is the distinguishing feature of the EuclidNets that while they are distance based, and hence enjoy the transitivity property in measuring similarity, their performance is also completely aligned with those based on Cosine similarity.
\subsection{Expressiveness of EuclidNets} Deep learning models that use the EuclidNet operation are just as expressive as those using multiplication. Note the polarization identity, \[ S_{\mathrm{conv}}(x,w) = S_{\mathrm{euclid}}(x,w) - S_{\mathrm{euclid}}(x,0) - S_{\mathrm{euclid}}(0,w) \] which means that any multiplication operation can be expressed using only Euclid operations.
\subsection{Hardware cost}
Traditionally, hardware developers use smaller multipliers to create larger multipliers \cite{de2009large}. They use various methods of multiplier tiling or divide and conquer to form larger multiplier. Karatsuba algorithm and its generalization \cite{weimerskirch2006generalizations} is among the most known algorithms to implement large multipliers. Here we show that Euclidean distance can be potentially implemented with fewer multipliers in hardware.
Karatsuba algorithm is a form of divide and conquer algorithm to perform $n-$bit multiplication using $m-$bit multipliers. Let us assume $a$ and $b$ are $n-$bit integer numbers and they can be re-written using two $m-$bits partitions \begin{align} \nonumber
&a = a_1 \times 2^m + a_2,\\ \nonumber
&b = b_1 \times 2^m + b_2.\\
\label{eq:karatsuba_parts} \end{align} In the case of multiplication, we have \begin{align} \nonumber
&ab = (a_1 \times 2^m + a_2) (b_1 \times 2^m + b_2)\\ \nonumber
&~~~= 2^{2m}a_1b_1+2^m a_1b_2+2^m a_2b_1+a_2b_2,\\
\label{eq:mult} \end{align} which is comprised of \textit{three} additions and \textit{four} $m-$bits multiplications. However for the squaring operation, we have \begin{align} \nonumber
&a^2 = (a_1 \times 2^m + a_2) (a_1 \times 2^m + a_2)\\ \nonumber
&~~~= 2^{2m}a_1^2+2^{m+1} a_1a_2+a_2^2,\\
\label{eq:square} \end{align} which is comprised of \textit{two} additions and \textit{three} $m-$bits multiplications. Thus, the squaring operation can be cheaper in hardware. Also note that such divide and conquer techniques are used commonly in designing accelerator on FPGA targets.
\section{Training EuclidNets}
Training EuclidNets are much easier compared to other similarity measures such as AdderNets. This makes EuclidNet attractive for complex tasks such as image segmentation, and object detection where training compressed networks are challenging and causes large accuracy drops. However, EuclidNets are more expensive than AdderNets when using floating-point number format, however, their quantization is easy since, unlike AdderNets, they behave similar to traditional convolution to a great extent. In another words EuclidNets are easy to quantize.
While training a deep learning model using EuclidNets, it is more appropriate to use the identity \begin{equation}
S_{{\mathrm{euclid}}}(x,w) = -\frac {x^2}{2} - \frac{w^2}{2} + x w, \end{equation} that is more appropriate for GPUs that are optimized for inner product computations. As such, training EuclidNets does not require additional CUDA kernel implementation unlike AdderNets \citep{cuda}. The official implementation of AdderNet \citep{chen2020addernet} reflects order of $20\times$ slower training than the traditional convolution on PyTorch. This is specially problematic for large deep learning models and complex tasks since even traditional convolution training takes few days or even weeks. EuclidNet training is about $2\times$ slower in the worst case and their implementation is natural in deep learning frameworks such as PyTorch and Tensorflow.
\begin{table}[h]\label{tab: times}
\caption{Time (seconds) and maximum training batch-size that can fit in a signle GPU \textit{Tesla V100-SXM2-32GB}, during ImageNet training. In parenthesis is the slowdown with respect to the $S_{conv}$ baseline.
We do not show times for AdderNet, which is much slower than both, because it is not implemented in CUDA
}
\centering \begin{tabular}{cc l l cc}
\multirow{2}{*}{\textbf{Model}} & \multirow{2}{*}{\textbf{Method}} & \multicolumn{2}{l}{\textbf{ Maximum Batch-size}} & \multicolumn{2}{l}{\textbf{Time per step}} \\
& & \multicolumn{1}{l}{\textbf{\begin{tabular}[c]{@{}c@{}} power of 2\end{tabular}}} & \multicolumn{1}{l}{\textbf{integer}} & \textbf{Training} & \textbf{Testing} \\ \hline
\multirow{2}{*}{ResNet-18} & $S_{{\mathrm{conv}}}$ & 1024 & 1439 & 0.149 & 0.066 \\
& $S_{{\mathrm{euclid}}}$ & 512 & 869 ($1.7\times$) & 0.157 ($1.1\times$) & 0.133 ($2\times$) \\ \hline
\multirow{2}{*}{ResNet-50} & $S_{{\mathrm{conv}}}$ & 256 & 371 & 0.182 & 0.145 \\
& $S_{{\mathrm{euclid}}}$ & 128 & 248 ($1.5\times$) & 0.274 ($1.5\times$) & 0.160 ($1.1\times$) \\ \hline \end{tabular}
\end{table}
A common method in training neural networks is fine-tuning, that means initializing with weights trained on different data but with a similar nature. Here, we introduce the idea of using a weight initialization from a model trained on a related similarity measure.
Rather than training from scratch, we wish to fine-tune EuclidNet starting from accurate CNN weights. This is achieved by an ``architecture homotopy" where we change hyperparameters to convert a regular convolution to an EuclidNet operation \begin{equation} S(x,w; \lambda_k) = xw - \lambda_k\frac{x^2 + w^2}{2},\qquad \mbox{ with }\lambda_k = \lambda_0 + \frac{1 - \lambda_0}{n} \cdot k, \label{eq: homotopy} \end{equation} where $n$ is the total number of epochs and $0<\lambda_0<1$ is the initial transition phase. Note that $S(x,w,0) = S_{{\mathrm{conv}}}(x,w)$ and $S(x,w,1) =S_{{\mathrm{euclid}}}(x,w)$ and equation \eqref{eq: homotopy} is a convex combination of these two similarities. One may interpret $\lambda_k$ as a scheduler for the homotopy, similar to the way learning rate is scheduled in training a deep learning model. We found that a linear scheduling as shown in equation \eqref{eq: homotopy} is empirically effective.
Transformations like equation \eqref{eq: homotopy} are commonly used in scientific computing \cite{allgower2003introduction}. The idea of using homotopy in training neural networks can be traced back to \cite{chow1991homotopy}. Recently, homotopy was used in deep learning in the context of activation functions \citep{pathak2019parameter,cao2017hashnet, mobahi2016training,farhadi2020}, loss functions \citep{gulcehre2016mollifying}, compression \citep{chen2019efficient} and transfer learning \citep{bengio2009curriculum}. Here, we use homotopy in the context of transforming operations of a deep learning model.
Fine-tuning method in equation \eqref{eq: homotopy} is inspired by continuation methods in partial differential equations. Assume $S$ is a solution to a differential equation with the initial condition $S(x,0) = S_0(x)$. In certain situations, solving this differential equation for $S(x,t)$ and then evaluating at $t=1$ might be easier than solving directly for $S_1$. One may think of this homotopy method as an evolution for deep learning model weights. At time zero the deep learning model consists of regular convolutional layers, but they gradually transform to Euclidean layers.
The homotopy method can also be interpreted as a sort of of knowledge distillation. Whereas knowledge distillation methods tries to match a student network to a teacher network, the homotopy can be seen as a slow transformation from the teacher network into a student network. Figure \ref{fig: homotopy} demonstrates the idea. Interestingly, problems that have been solved with homotopy have also been tackled by knowledge distillation \citep{hinton2015distilling,chen2019efficient,yim2017gift, bengio2009curriculum}.
\begin{figure}
\caption{Training schema of EuclidNet using Homotopy, i.e. transitioning from traditional convolution $S(x,w)=xw$ towards EuclidNet $S(x,w)=-\frac{1}{2} |x-w|^2$ through equation \eqref{eq: homotopy}.}
\label{fig: homotopy}
\end{figure}
\section{Experiments}\label{sec:Experiments}
To illustrate performance of the EuclidNets, We apply our proposed method on image classification tasks. We also test our trained deep learning model under different transformations on the input image and compare the accuracy to standard convolutional networks.
\subsection{CIFAR10}\label{sec: cifar10}
First, we consider the CIFAR10 dataset, consisting of $32\times32$ RGB images with 10 possible classifications \citep{krizhevsky2009learning}. We normalize and augment the dataset with random crop and random horizontal flip. We consider two ResNet models \cite{he2015deep}, ResNet-20 and ResNet-32.
We train EuclidNet using the optimizer from \cite{chen2020addernet}, which we will refer to as AdderSGD, to evaluate EuclidNet under a similar setup. We use initial learning rate $0.1$ with cosine decay, momentum $0.9$, batch size 128 and weight decay $5\times 10^{-4}$. We follow \cite{chen2020addernet} in setting the learning-rate scaling parameter $\eta$. For traditional convlutional network, we use the same hyper-parameters with stochastic gradient descent optimizer.
The details of classification accuracy is provided in Table \ref{tab: cifar10}. We consider two different weight initialization for EuclidNets. First, we initialize the weights randomly and second, we initialize them with pre-trained on a convolutional network. The accuracy for EuclidNets has negligible accuracy loss compared to the standard ResNets. We see that for CIFAR10 training from scratch achieves even a higher accuracy, while initializing with convolution network and using linear homotopy training improves it even further.
\begin{table}[h] \caption{Results on CIFAR10. The initial learning rate is adjusted for non-random initialization.
} \label{tab: cifar10} \centering
\begin{tabular}{ccccccc}
\multirow{2}{*}{Model} & \multirow{2}{*}{Similarity} & \multirow{2}{*}{Initialization} & \multirow{2}{*}{Homotopy} & \multirow{2}{*}{Epochs} & \multicolumn{2}{c}{Top-1 accuracy} \\
& & & & & CIFAR10 & CIFAR100 \\ \hline
\multirow{4}{*}{ResNet-20} & $S_{{\mathrm{conv}}}$ & Random & None & 400 & 92.97 & \textbf{69.29} \\
& \multirow{3}{*}{$S_{{\mathrm{euclid}}}$} & Random & None & 450 & {93.00} & 68.84 \\
& & \multirow{2}{*}{Conv} & None & 100 & 90.45 & 64.62 \\
& & & Linear & 100 & \textbf{93.32}
& 68.84
\\ \hline
\multirow{4}{*}{ResNet-32} & $S_{{\mathrm{conv}}}$ & Random & None & 400 & \textbf{ 93.93} & 71.07 \\
& \multirow{3}{*}{$S_{{\mathrm{euclid}}}$} & Random & None & 450 & 93.28 & \textbf{71.22} \\
& & \multirow{2}{*}{Conv} & None & 150 & 91.28 & 66.58 \\
& & & Linear & 100 & 92.62 & 68.42 \\ \hline
\end{tabular}
\end{table}
EuclidNets can become unstable during the training, despite careful choice of the optimizer. Figure \ref{fig: train_comparison} shows a comparison of the EuclidNet training with a standard convolutional network. As it can be seen in the Figure \ref{fig: train_comparison}, fine-tuning the EuclidNets directly from convolutional networks' weights is more stable than training from scratch. Also observe that when we train EuclidNets from scrach, accuracy is lower but the convergence is faster. Finally, using homotopy in the training procedure, the accuracy is improved. Note that the pre-trained convolution weights are commonly available in the most of neural compression tasks, so initializing EuclidNets with pre-trained convolution is a commonplace procedure in optimizing deep learning models for inference.
\begin{figure}
\caption{Evolution of testing accuracy during training of ResNet-20 on CIFAR10, initialized with random weights, or initialized from convolution pre-trained network. Initializing from a pretrained convolution network speeds up the convergence. EuclidNet is harder to train compared with convolution network when both initialized from random weights.}
\label{fig: train_comparison}
\end{figure}
EuclidNets are not only faster to train compared to other norm based similarity measures, but also stand superior in terms of accuracy. AdderNet performs slightly worse in terms of accuracy and also is much slower to train. The accuracy is significantly lower for the synapse \cite{dogaru1999comparative} and the multiplication-free \cite{akbas2015multiplication} operators. Table \ref{tab: sim_comparison} demonstrates a top-1 accuracy comparison of different methods. The reported results on AdderNet are from \cite{xu2020kernel}. Note that although for AdderNet in \cite{xu2020kernel}, authors used knowledge distillation to close the gap with the full precision, it still falls short compared with EuclidNet.
\begin{table}
\caption{Full precision results on ResNet-20 for CIFAR10 for different multiplication-free similarities.}
\label{tab: sim_comparison}
\centering
\begin{tabular}{ c c c c c c}
\multirow{1}{4em}{\textbf{Similarity}} & \multirow{1}{3em}{$S_{{\mathrm{conv}}}$} & \multirow{1}{3em}{$S_{{\mathrm{euclid}}}$} & $S_{{\mathrm{adder}}}$ & $S_{{\mathrm{mfo}}}$ & $S_{{\mathrm{synapse}}}$ \\
\hline
\textbf{Accuracy} & 92.97
& \textbf{93.00}
& 91.84
& 82.05 & 73.08
\\
\end{tabular} \end{table}
Training a quantized $S_{{\mathrm{euclid}}}$ is very similar to convolutional neural networks. This allows a wider use of such models for lower resource devices. Quantization of the EuclidNets to 8bits keeps the accuracy drop within the range of one percent \citep{wu2020integer} similar to traditional convolutional neural networks. Table \ref{tab: quant} shows 8-bit quantization of EuclidNet where the accuracy drop remains negligible. Furthermore, training EuclidNets on CIFAR100 dataset exhibits a negligible accuracy drop when the weights are initialized with pre-trained standard model weights.
\subsection{ImageNet}
Next, we consider testing EuclidNet classifier on ImageNet \cite{imagenet_cvpr09} which is known to be a challenging classification task comparing to CIFAR10. We trained our baseline convolutional neural network with standard augmentations of random resized crop and horizontal flip and normalization. We consider ResNet-18 and ResNet-50 models with the same hyper-parameters as those used in Section \ref{sec: cifar10}.
Table \ref{tab: in} shows top-1 and top-5 classification accuracy of ImageNet dataset. As shown in Table \ref{tab: in}, the accuracy of EuclidNet when it is trained from scratch is lower than the baseline emphasizing the importance of homotopy training. We believe that the accuracy drop with no homotopy is because the hyper-parameter tuning is harder for large datasets such as ImageNet. This means that even though there exists hyper-parameters that achieve equivalent accuracy with random initialization, however it is too difficult to find them. Thus, it is much easier to use the existing hyper-parameters of traditional convolutional neural network, and use homotopy to smoothly transfer the weights to wights that are suitable for EuclidNets.
\begin{table}[h]
\centering
\caption{Full precision results on ImageNet. Best result for each model is in bold.}\label{tab: in}
\scalebox{0.8}{
\begin{tabular}{ccccccc}
Model & Similarity & Initialization & Homotopy & Epochs & \multicolumn{1}{l}{Top-1 Accuracy} & \multicolumn{1}{l}{Top-5 Accuracy} \\ \hline
\multirow{6}{*}{ResNet-18} & $S_{{\mathrm{conv}}}$ & Random & None & 90 & 69.56 & 89.09 \\ \cline{2-7}
& \multirow{5}{*}{$S_{{\mathrm{euclid}}}$} & Random & None & 90 & 64.93 & 86.46 \\ \cline{3-7}
& & \multirow{4}{*}{Conv} & None & 90 & 68.52 & 88.79 \\ \cline{4-7}
& & & \multirow{3}{*}{Linear} & 10 & 65.36 & 86.71 \\
& & & & 60 & 69.21 & 89.13 \\
& & & & 90 & \textbf{ 69.69} & \textbf{ 89.38} \\ \hline
\multirow{6}{*}{ResNet-50} & $S_{{\mathrm{conv}}}$ & Random & None & 90 & 75.49 & 92.51 \\ \cline{2-7}
& \multirow{5}{*}{$S_{{\mathrm{euclid}}}$} & Random & None & 90 & 37.89 & 63.99 \\ \cline{3-7}
& & \multirow{4}{*}{Conv} & None & 90 & 75.12 & 92.50 \\ \cline{4-7}
& & & \multirow{3}{*}{Linear} & 10 & 70.66 & 90.10 \\
& & & & 60 & 74.93 & 92.52 \\
& & & & 90 & \textbf{ 75.64} & \textbf{ 92.86} \\ \hline
\end{tabular}
} \end{table}
\subsection{Transformation and blurring}
Here we provide empirical evidence that Euclidean norm is aligned with the multiplication. First, we show that EuclidNets perform as well as standard convolutional neural networks in the case of \textit{pixel transform}. Second, we show that when the image is blurred with Guassian noise, EuclidNets closely follow the behaviour of the convolutional neural networks.
\subsubsection{Pixel transformation} We define pixel transformation of an image as \begin{equation} \mathbf{I_T} = a\mathbf{I}+b, \label{eq:transform} \end{equation} where $\mathbf{I}$ is a tensor representing the original image, scalars $a$ and $b$ are transformation parameters, and $\mathbf{I_T}$ is the transformed image. Note that in \eqref{eq:transform}, $a$ controls the contrast and $b$ controls the brightness of the image. Such transformations are widely used in various stages of the imaging systems for instance in color correction, and gain-control (ISO).
Figure \ref{fig:transform} shows the accuracy of the standard ResNet-18 and EuclidNet ResNet-18 when the input image is affected by the pixel transformation of equation \eqref{eq:transform}. We can see that when changing $a$ and $b$, EuclidNet ResNet-18 closely follow the accuracy of the standard ResNet-18.
\begin{figure}
\caption{Accuracy of CIFAR10 classification affected by pixel transformation for a standard ResNet-18 (left) and EuclidNet ResNet-18 (right).}
\label{fig:transform}
\end{figure} \begin{figure}
\caption{Accuracy of CIFAR10 classification affected by Guassian noise for a standard ResNet-18 (left) and EuclidNet ResNet-18 (right).}
\label{fig:noise}
\end{figure}
\subsubsection{Gaussian Blurring}
Additive noise can be injected to an image in different stages of the imaging system due to faulty equipments or environmental conditions. We tested EuclidNet when the input image is affected by a Guassian additive noise. Figure \ref{fig:noise} demonstrates comparison of the standard ResNet-18 and EuclidNet ResNet-18 for different noise intensities i.e. $\sigma$ and kernel sizes. This experiment is done for classification of the CIFAR10 dataset. We can see that EuclidNet ResNet-18 closely follow the behaviour of the standard ResNet-18 in the case of different kernel sizes and noise intensities.
\section{Conclusion}
EuclidNets are a class of deep learning models in which the multiplication operator is replaced with the Euclidean distance. They are designed to be implemented on application specific hardware, with the idea that subtraction and squaring are cheaper than multiplication when designing efficient hardware for inference. Furthermore, in contrast to other efficient architectures that are difficult to train in low precision, EuclideNets are easily trained in low precision.
EuclidNets can be initialized with pre-trained weights of the standard convolutional neural networks and hence, the training procedure of EuclidNets using homotopy is considered as a fine tuning of convolutional networks for inference. The homotopy method further improves training in such scenarios and training using this method sometimes surpass regular convolution accuracy. .
\begin{appendices}
\end{appendices}
\end{document}
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arXiv
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Indeterminate system
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions).[1] In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable[2]), but that property does not extend to nonlinear systems (e.g., the system with the equation $x^{2}=1$).
An indeterminate system by definition is consistent, in the sense of having at least one solution.[3] For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.
Examples
The following examples of indeterminate systems of equations have respectively, fewer equations than, as many equations as, and more equations than unknowns:
${\text{System 1: }}x+y=2$
${\text{System 2: }}x+y=2,\,\,\,\,\,2x+2y=4$
${\text{System 3: }}x+y=2,\,\,\,\,\,2x+2y=4,\,\,\,\,\,3x+3y=6$
Conditions giving rise to indeterminacy
In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point), which in turn excludes the possibility of having more than one solution. On the other hand, if the rank of the augmented matrix exceeds (necessarily by one, if at all) the rank of the coefficient matrix, then the equations will jointly contradict each other, which excludes the possibility of having any solution.
Finding the solution set of an indeterminate linear system
Let the system of equations be written in matrix form as
$Ax=b$
where $A$ is the $m\times n$ coefficient matrix, $x$ is the $n\times 1$ vector of unknowns, and $b$ is an $m\times 1$ vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all $x$ vectors generated by[4]
$x=A^{+}b+[I_{n}-A^{+}A]w$
where $A^{+}$ is the Moore–Penrose pseudoinverse of $A$ and $w$ is any $n\times 1$ vector.
See also
• Indeterminate equation
• Indeterminate form
• Indeterminate (variable)
• Linear algebra
• Simultaneous equations
• Independent equation
• Identifiability
References
1. "Indeterminate and Inconsistent Systems: Systems of Equations". TheProblemSite.com. Retrieved 2019-12-02.
2. Gustafson, Grant B. (2008). "Three Possibilities (of a Linear System)" (PDF). math.utah.edu. Retrieved 2019-12-02.{{cite web}}: CS1 maint: url-status (link)
3. "Consistent and Inconsistent Systems of Equations | Wyzant Resources". www.wyzant.com. Retrieved 2019-12-02.
4. James, M., "The generalised inverse", Mathematical Gazette 62, June 1978, 109–114.
Further reading
• Lay, David (2003). Linear Algebra and Its Applications. Addison-Wesley. ISBN 0-201-70970-8.
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Wikipedia
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Methodology article
Identifying rare variants for quantitative traits in extreme samples of population via Kullback-Leibler distance
Yang Xiang1,2,3,
Xinrong Xiang4 &
Yumei Li ORCID: orcid.org/0000-0003-1755-45041,2,3
The rapid development of sequencing technology and simultaneously the availability of large quantities of sequence data has facilitated the identification of rare variant associated with quantitative traits. However, existing statistical methods depend on certain assumptions and thus lacking uniform power. The present study focuses on mapping rare variant associated with quantitative traits.
In the present study, we proposed a two-stage strategy to identify rare variant of quantitative traits using phenotype extreme selection design and Kullback-Leibler distance, where the first stage was association analysis and the second stage was fine mapping. We presented a statistic and a linkage disequilibrium measure for the first stage and the second stage, respectively. Theory analysis and simulation study showed that (1) the power of the proposed statistic for association analysis increased with the stringency of the sample selection and was affected slightly by non-causal variants and opposite effect variants, (2) the statistic here achieved higher power than three commonly used methods, and (3) the linkage disequilibrium measure for fine mapping was independent of the frequencies of non-causal variants and simply dependent on the frequencies of causal variants.
We conclude that the two-stage strategy here can be used effectively to mapping rare variant associated with quantitative traits.
Thanks to the rapid development of sequencing technology and the lowering of sequencing costs in the last decade, the availability of large quantities of sequence data provides an unprecedented opportunity for researchers to investigate the role of rare variants in complex traits [1,2,3,4]. But due to the low minor allele frequency (MAF < 5%) and thus resulting in weak linkage disequilibrium (LD) with nearby markers, detecting rare variant (RV) association with complex traits faces great challenges [5,6,7,8]. One challenge is that detection of rare causal variants with traditional designs usually requires a large sample, which will be the high cost [3, 6]. Thus cost-effective design should be considered to reduce sample size. Another challenge is that the statistical power with test statistics of single-marker tests is generally low in genetic association studies of rare variants with more moderate or weak genetic effects [8,9,10]. To date many statistical methods have been developed for rare variant association analysis, including burden tests [11,12,13], variance-component tests [14, 15], series of sequence kernel association tests [10, 16, 17]. Any of these methods has relative perfect performance in special scenario, but none of them can overwhelm others in all scenarios [8, 9], especially for quantitative traits.
In fact, rare variant association analysis in the past several years mainly focused on the qualitative trait. Only a few statistical methods have been developed for the quantitative trait [13, 18,19,20,21]. One approach for rare variant association analysis of quantitative traits is the linear regression model. However, most regression-based methods rely on the normality assumption of the phenotype [8, 21]. Another commonly used approach adopts phenotype extreme selection design where one can transform the quantitative trait association study into case-control association study of qualitative traits by treating the upper extreme as cases and the lower extreme as controls in a strategy using extreme phenotype [22,23,24,25]. Extreme phenotypes of a quantitative trait are generally considered to be more informative. Moreover, a smaller sample size for extreme-phenotype sampling than that for random sampling is needed to achieve similar power [23, 24].
In this report, we use phenotype extreme selection design and Kullback-Leibler distance (KL-distance) [26] to propose a simple statistic method to identify rare variants for quantitative traits. Two stages strategies are adopted in our analysis where association analysis and fine mapping will be done in the first stage and the second stage, respectively. This method will compare the frequency distributions of rare variant in two extreme phenotypes based on KL-distance. Our method has three features: (1) it has increasing power with the stringency of the sample selection, (2) it is affected slightly by non-causal variants and the opposite effect variants.in the first stage for association analysis, and (3) it is not depend on the frequencies of non-causal variants and just dependent on the frequencies of causal variants in the second stage for fine mapping. Through simulation studies, we investigate the performance of the proposed method and compare it with three commonly used methods of the burden test [12], the sequence kernel association test (SKAT) [17], and the optimal test that combines SKAT and the burden test (SKAT-O) [10].
Type I error rate and power for association analysis
Table 1 exhibits the estimated type I error rates of the statistic TKL for the extreme sample with sample-selection threshold value of 20, 10, and 5% and with sample size of 1000 and 1500. It can be seen that, under various genetic parameters, type I error rates of TKL are not appreciably different from the nominal alpha levels, which indicates the validity of the statistic TKL.
Table 1 Estimated type I error rates of the statistic TKL
Figure 1 shows the results of power for 9 scenarios when sample sizes are 1000 and 1500. It is found that the power of the statistic TKL with the sample size of 1500 is nearly 0.20 larger than that with the sample size of 1000, indicating that the power of the statistic TKL significantly increase with the increasing of the sample size. It can be seen that, under the same sample size, the powers of the statistic TKL with the low 5% samples and the up 5% samples are highest and the powers with the low 20% samples and the up 20% samples are lowest, which indicates that the powers of the statistic TKL increase with the stringency of the sample selection. It is observed from scenarios {1, 2, 3} that, when rare variant effects are in the same direction, the powers of the statistic TKL increase with the increasing of the number of causal variants. The same above conclusions are observed when 80% causal variants have positive effect and 20% causal variants have negative effect (scenarios {4, 5, 6}) and when there is the same number of causal variants with positive effects and negative effects (scenarios {7, 8, 9}). By comparing the powers under scenarios {1, 4, 7} with 10 causal variants, the powers under scenarios {2, 5, 8} with 20 causal variants, and the powers under scenarios {3, 6, 9} with 50 causal variants, we found that, when the number of causal variants with negative effect increases, the power of the statistic TKL decreases slightly. From Fig. 1, we can observe that, among four statistics of the TKL, the burden test, the SKAT, and the SKAT-O, the power of TKL is higher than that of other three statistics. The burden test, the SKAT, and the SKAT-O are severely affected by the number of non-causal variants and the opposite effect variants, especially when there are the same number of opposite effect variants. Although non-causal variants and the opposite effect variants affect the power of the TKL, the impact is slight. For example, when the sample size is 1500 and the number of causal variants is 50 for 10% sample-selection threshold value (B2), as the number of variants with negative effect increases from zero to 25, the powers of the burden test, the SKAT, and the SKAT-O decrease from ~ 0.80, ~ 0.79, and ~ 0.84 to ~ 0.23, ~ 0.63, and ~ 0.74, respectively, with the decline rate of 71.2, 20.2, and 12.0%. Nevertheless, when the number of variants with negative effect is 25, the TKL still achieves ~ 0.83 power, with the decline rate of just 7% comparing to ~ 0.90 when the number of variants with negative effect is zero.
Empirical power of four statistics from the extreme samples with 20% threshold value a, 10% threshold value b, and 5% threshold value c when the sample sizes are 1000 (a1, b1, c1) and 1500 (a2, b2, c2) at a 0.05 significance level
Power for fine mapping
In fine mapping study, the QTL can be located by the maximum value of the measure lKL. So we sample 10 times from each of 100 simulation populations where each sample includes 750 individuals with the up-extreme phenotype of Y > U and 750 individuals with the lower-extreme phenotype with Y < L. For each sample, we calculate the value of the measure lKL for each variant. In order to guard against noisy distributions of the measure lKL, we adopt the 5-point moving-average method to determine the maximum value. We count the number (here, we denote it B) of the maximum values that locate at variant 10 or variant 11. Then the probability that the maximum values of lKL locate at variant 10 or variant 11 is B/1000. We refer this value as the power of lKL, which measure the likelihood of fine mapping the QTL. Table 2 shows the results of the power for lKL. It can be seen that the power of lKL for fine mapping under dominant model is highest and the power of lKL for fine mapping under recessive model is lowest. The power of lKL increases with increasing of the heritability h2 of the causal variant and the stringency of the sample selection. For example, power of lKL under dominant model with the heritability h2 of 0.01 is 0.52, 0.62, and 0.67 at 20, 10, and 5% sample-selection threshold value, respectively; power of lKL under dominant at 5% sample-selection threshold value increase from 0.67 to 0.83 with the heritability h2 of the causal variant increasing from 0.01 to 0.10. We also investigate the effect of different sample sizes (e.g., 2n =1000, 1500, and 2000). As expected, power of lKL increases with the increasing sample size (data not shown). In order to assess the performance of lKL, we compare it with two LD measures l [27] and pexcess [28] with case-control design using extreme samples. Table 2 also lists the powers for l and pexcess. We found that the powers of lKL and l are nearly the same and higher than those of pexcess.
Table 2 The power of the QTL fine mapping for three LD measures by use of five-point moving average
In this report, we present a robust approach to identify rare variant of quantitative traits. The proposed approach adopts phenotype extreme selection design and KL-distance method. We use a two-stage strategy in our analysis where the first stage is association analysis and the second stage is fine mapping of QTL if the first stage is positive result. We propose a statistic TKL for association analysis and a LD measure lKL for fine mapping. Simulation studies present the performance of the proposed method. We found that the power of the TKL increases with the stringency of the sample selection and the increasing of the number of causal variants. The TKL here has higher power for association analysis than three existing statistics. Meanwhile, the impact of non-causal variants and the opposite effect variants on the TKL is slight. The LD measure lKL for fine mapping in the second stage has a good feature of not dependence on the frequencies of non-causal variants and just dependence on the frequencies of causal variants. These results show that our method can be used to detect rare variant associated with quantitative traits. At the same time, we found that the proposed method can be easily extended to case-control study by treating cases and controls as samples with upper extreme phenotype and lower extreme phenotype, respectively.
In rare variant association analysis, in order to achieve high statistical power of tests, usually a large sample with high sequencing costs is needed. Thus less costly sequencing design is preferred in rare variant association study. For quantitative traits, extreme phenotypes are generally considered to be more informative because of rare causal variants enriched among them. One can use a smaller sample size for extreme-phenotype sampling to achieve similar power as that for random sampling [23, 24]. Moreover, because extreme phenotypes of quantitative traits relative to human health are of primary clinical significance and thus data set can be obtained easily for subjects with extreme phenotypic values, using extreme phenotype samples in association analysis will make our study useful and practical. Here we use KL-distance to construct the statistics TKL to measure the difference between two probability distributions of rare variants in two extreme populations. Based on the principle that the larger TKL value is, the more dissimilar two probability distributions of rare variants, the statistics TKL can be used as a test statistic to quantify the magnitude of association between the variants and the quantitative trait in the first stage of association analysis. We found that the statistic TKL here for association analysis has higher power than three existing statistics of the burden test, the SKAT, and the SKAT-O. Moreover, whereas increasing the number of non-causal variants and the opposite effect variants result in decreasing severely the powers of the burden test, the SKAT, and the SKAT-O, non-causal variants and the opposite effect variants affect slightly on the TKL. The TKL has relatively stable power with small change range under various parameters set.
In the second stage of fine mapping, lKL is essentially a measure of LD between the variant and the QTL. Although LD between rare variant and QTL maybe weak [24], the maximum value across all rare variants can be usually found to identify the causal variant (QTL). The measure lKL here has a good performance of just dependence on the frequency of the causal variant. In practice, not dependence on the frequency of the non-causal variant can eliminate"noise" and even bias introduced by varying frequencies of non-causal variants. In our early works, we proposed the LD measure l for mapping common variant of the QTL [27]. The performance of the measure l for mapping rare variant is unknown. We found from theory analysis that the two LD measures lKL and l are parallel and have the same performance, that is, both of them can quantify LD between the variant and the QTL and do not depend on frequencies of non-causal variants. The difference between them is that the measure lKL here is based on KL-distance and the measure l is based on entropy theory. Another LD measure for fine mapping is pexcess [28]. The pexcess is originally developed for fine mapping common variant of qualitative trait. We compare the performance of these three LD measures for fine mapping rare variant of quantitative traits using extreme samples. We found from theory analysis and simulation study that lKL is superior to pexcess.
It is noted that, in practice, we do not know how many causal variants there are in the region established through association analysis at first stage. Although we considered a region having only a single causal variant, our method works for the general case with a region consisting of multiple causal variants. In fact, when there is a region linked to a quantitative trait has multiple causal variants, we can detect all causal variants using following steps: (1) lKL is used to mapping a causal variant with the maximum value of lKL; (2) TKL is used to do association analysis for all variants except the causal variant detected in (1). If the association analysis result is positive, then return to (1). All causal variants will be found when the association analysis result is negative. It should be noted that we use the permutation procedure to assess the statistical significance of the statistic TKL for association analysis. Permutation procedure may need more computing time to conduct simulation. But with the development of high-performance computing, computing time may not be a problem in our study. In addition, it can be seen that our method involves only rare variants. A phenotype may affected by common variants or both common variants and rare variants. So our further work will involve extensive field for common variants or both common variants and rare variants.
The statistic TKL is affected slightly by non-causal variants and the opposite effect variants. The power of the TKL for association analysis of rare variants increases with the stringency of the sample selection for quantitative traits. Extreme phenotypes allow TKL to achieve higher power than three commonly used methods. The LD measure lKL for fine mapping is independent of the frequencies of non-causal variants and just dependent on the frequencies of causal variants.
In this study, all datasets were publically available and no research requiring ethics approval was conducted.
We consider an interesting gene region with k biallelic variants and assume that each variant has a minor allele m with the MAF Pm and a normal allele M with the allele frequency PM (Pm + PM = 1). The variants are indexed by i (i = 1, ..., k). The index i may or may not correspond to the variant orders. Let Xi be minor allele count at ith variant carried by a subject. Assume that there is a quantitative trait Y: Y = β0 + G + ε, where \( G=\underset{i=1}{\overset{k}{\Sigma}}{\beta}_i{X}_i \), β0 is the mean baseline value, and ε is residual due to random environmental effects. Without loss of generality, we assume β0 =0 and ε~N(0, σ2). To simplify our presentation, we use a measure with a superscript "U" to indicate a measure in the upper extreme population that has phenotypic values of the quantitative trait Y > U (U is an upper-threshold value, chosen from the continuous distribution of the study quantitative trait). We also use a measure with a superscript "L" to indicate a measure in the low extreme population that has phenotypic values of the quantitative trait Y < L (L is an low-threshold value, chosen from the continuous distribution of the study quantitative trait). Assume NU and NL subjects are sequenced with k variants in the upper extreme population and in the low extreme population, respectively, which are indexed by j (j = 1,..., NU/ NL). Denote \( {X}_{ij}^{\mathrm{U}} \) and \( {X}_{ij}^{\mathrm{L}} \) as the number of copies "m" for jth subject at ith variant in the upper extreme population and in the low extreme population, respectively. Then the frequencies of Pm and PM at ith variant in the upper extreme population and in the low extreme population, denoted as \( {p}_{mi}^{\mathrm{U}} \), \( {p}_{Mi}^{\mathrm{U}} \), and \( {p}_{mi}^{\mathrm{L}} \), \( {p}_{Mi}^{\mathrm{L}} \), respectively, are estimated as follows:
\( {p}_{mi}^{\mathrm{U}}=\frac{\underset{j=1}{\overset{N^{\mathrm{U}}}{\Sigma}}{X}_{ij}^{\mathrm{U}}}{2{N}^{\mathrm{U}}} \), \( {p}_{Mi}^{\mathrm{U}}=1-{p}_{mi}^{\mathrm{U}} \), \( {p}_{mi}^{\mathrm{L}}=\frac{\underset{j=1}{\overset{N^{\mathrm{L}}}{\Sigma}}{X}_{ij}^{\mathrm{L}}}{2{N}^{\mathrm{L}}} \), and \( {p}_{Mi}^{\mathrm{L}}=1-{p}_{mi}^{\mathrm{L}} \).
A statistic test for association analysis in the first stage
In the first stage, we propose a statistic test for association analysis. We define a k-dimensional random vector \( {\tilde{p}}_m={\left({\tilde{p}}_{m1},\cdots, {\tilde{p}}_{mk}\right)}^T \) as the proportion of the minor allele m among all k variants, where \( {\tilde{p}}_{mi}=\frac{\underset{j}{\Sigma}{X}_{ij}}{\underset{i=1}{\overset{k}{\Sigma}}\underset{j}{\Sigma}{X}_{ij}} \) and Xij is the number of copies "m" for jth subject at ith variant. In the upper extreme population and in the low extreme population, the k-dimensional random vector of the proportion of the minor allele m are denoted as \( {\tilde{p}}_m^{\mathrm{U}}={\left({\tilde{p}}_{m1}^{\mathrm{U}},\cdots, {\tilde{p}}_{mk}^{\mathrm{U}}\right)}^T \) and \( {\tilde{p}}_m^{\mathrm{L}}={\left({\tilde{p}}_{m1}^{\mathrm{L}},\cdots, {\tilde{p}}_{mk}^{\mathrm{L}}\right)}^T \), respectively, where \( {\tilde{p}}_{mi}^{\mathrm{U}}=\frac{\underset{j=1}{\overset{N^{\mathrm{U}}}{\Sigma}}{X}_{ij}^{\mathrm{U}}}{\underset{i=1}{\overset{k}{\Sigma}}\underset{j=1}{\overset{N^{\mathrm{U}}}{\Sigma}}{X}_{ij}^{\mathrm{U}}} \) and \( {\tilde{p}}_{mi}^{\mathrm{L}}=\frac{\underset{j=1}{\overset{N^{\mathrm{L}}}{\Sigma}}{X}_{ij}^{\mathrm{L}}}{\underset{i=1}{\overset{k}{\Sigma}}\underset{j=1}{\overset{N^{\mathrm{L}}}{\Sigma}}{X}_{ij}^{\mathrm{L}}} \) (i = 1, 2, ⋯, k). We compare the two probability distributions \( {\tilde{p}}_m^{\mathrm{U}} \) and \( {\tilde{p}}_m^{\mathrm{L}} \) using the KL-distance which is defined as in Kullback & Leibler [26], here, we denote it the statistic TKL:
$$ {\mathrm{T}}_{KL}=H\left({\tilde{p}}_m^{\mathrm{U}},{\tilde{p}}_m^{\mathrm{L}}\right)=\frac{1}{2}\left(\underset{i=1}{\overset{k}{\Sigma}}{\tilde{p}}_{mi}^{\mathrm{U}}\cdot \log \frac{{\tilde{p}}_{mi}^{\mathrm{U}}}{{\tilde{p}}_{mi}^{\mathrm{L}}}+\underset{i=1}{\overset{k}{\Sigma}}{\tilde{p}}_{mi}^{\mathrm{L}}\cdot \log \frac{{\tilde{p}}_{mi}^{\mathrm{L}}}{{\tilde{p}}_{mi}^{\mathrm{U}}}\right) $$
It is easy to find the relationship between TKL and the frequencies of Pm and PM as follows:
$$ {\mathrm{T}}_{\mathrm{KL}}=\frac{1}{2}\left(\underset{i=1}{\overset{k}{\Sigma}}\frac{p_{mi}^{\mathrm{U}}}{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{U}}}\cdot \log \left(\frac{p_{mi}^{\mathrm{U}}}{p_{mi}^{\mathrm{L}}}\cdot \frac{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{L}}}{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{U}}}\right)+\underset{i=1}{\overset{k}{\Sigma}}\frac{p_{mi}^{\mathrm{U}}}{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{U}}}\cdot \log \left(\frac{p_{mi}^{\mathrm{L}}}{p_{mi}^{\mathrm{U}}}\cdot \frac{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{U}}}{\underset{i=1}{\overset{k}{\Sigma}}{p}_{mi}^{\mathrm{L}}}\right)\right) $$
TKL is the mean between two KL-distances where one is the KL-distance between \( {\tilde{p}}_m^{\mathrm{U}} \) and \( {\tilde{p}}_m^{\mathrm{L}} \) and the other is the KL-distance between \( {\tilde{p}}_m^{\mathrm{L}} \) and \( {\tilde{p}}_m^{\mathrm{U}} \). KL-distance provides a non-symmetric measure of how big of the difference between two probability distributions are. The KL-distance is always non-negative and equal to 0 only if two distributions are identical. It can be seen that TKL is a non-negative and symmetric measure of the two probability distributions \( {\tilde{p}}_m^{\mathrm{U}} \) and \( {\tilde{p}}_m^{\mathrm{L}} \). So, TKL can be used as a statistic to quantify the magnitude of association between the variants and the quantitative trait: a much larger TKL value will be observed under the alternative hypothesis of association compared to that under the null hypothesis of no association.
A KL-distance index for fine mapping of QTL in the second stage
Assume that a region linked to a quantitative trait has already been established through association analysis at first stage. In order to simplify our presentation, we assume that this region contains only a causal variant with a minor allele a (with frequency pa) and a normal allele A (with frequency pA = 1 − pa), here, we call it the quantitative trait locus (QTL). We consider the quantitative trait Y = GQ + ε, where GQ is the genotypic value at the QTL and ε~N(0, σ2). We hope to fine map this region by calculating the linkage disequilibrium (LD) measure between the QTL and a variant. We still use KL-distance to construct this measure through comparing the probability distributions of allele m and M at a variant in the upper extreme population and in the low extreme population. Following the previous symbols, let Pm and PM be the frequencies of allele m and M at a variant. From Eq. (1), we have
$$ H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right)=\frac{1}{2}\left({p}_m^{\mathrm{U}}\cdot \log \frac{p_m^{\mathrm{U}}}{p_m^{\mathrm{L}}}+{p}_M^{\mathrm{U}}\cdot \log \frac{p_M^{\mathrm{U}}}{p_M^{\mathrm{L}}}+{p}_m^{\mathrm{L}}\cdot \log \frac{p_m^{\mathrm{L}}}{p_m^{\mathrm{U}}}+{p}_M^{\mathrm{L}}\cdot \log \frac{p_M^{\mathrm{L}}}{p_M^{\mathrm{U}}}\right) $$
From Appendix, \( H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right) \) can be asymptotically expressed as a function of LD (δam) between the QTL and the variant:
$$ H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right)\approx \frac{\delta_{ma}^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_m\cdot {p}_M} $$
Assume that there is an initial complete association between the variant allele m and the QTL allele a, at the 0th generation when the allele a is initially introduced into the study population. Let \( {\delta}_{ma}^{(0)} \) be the initial complete LD between the allele a and m at the 0th generation, \( {\delta}_{ma}^{(0)}={p}_M\cdot {p}_a \). After n generations, the LD between the allele m and a is \( {\delta}_{ma}^{(n)}={\left(1-\theta \right)}^n{\delta}_{ma}^{(0)}={\left(1-\theta \right)}^n\cdot {p}_M\cdot {p}_a \) [29], where, θ is the recombination between the QTL and the variant. Then we have
$$ H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right)\approx \frac{\delta_{am}^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_M\cdot {p}_m}=\frac{{\left(1-\theta \right)}^{2n}{p}_a^2\cdot {p}_M^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_M\cdot {p}_m} $$
Now we define a LD measure, here, we denote it as lKL, as follows:
$$ {l}_{KL}=\frac{p_m}{p_M}H\ \left(\left\{{p_m}^{\mathrm{U}},{p_M}^{\mathrm{U}}\right\},\left\{{p_m}^{\mathrm{L}},{p_M}^{\mathrm{L}}\right\}\right)\approx \frac{1}{2}{\left(1-\theta \right)}^{2n}{p}_a^2\cdot {b}^2 $$
where b = bU − bL. It can be seen that lKL is a decreasing function of the recombination θ and reaches its maximum at θ = 0. So we can use lKL to find the variant closest to the QTL and thus fine map the QTL. Notice from Eq. (6) that lKL is independent of the frequency of the variant, just only dependent on the frequency of the QTL.
Simulation for association analysis
To evaluate the performance of the test statistic TKL, we perform a series of simulation studies. We consider k = 100 variants with MAF values of causal variants determined by a uniform distribution U (0.001, 0.01) and MAF values of non-causal variants determined by a uniform distribution U (0.001, 0.05). The genotype data are simulated similar to those in Wang and Elston [30]. We first generate haplotypes for k variants based on a latent variable Z = (Z1, ⋯, Zk) from a multivariate normal distribution with covariance structure cov(Zi, Zj) = 0.4∣i − j∣ between any two latent components. Then we combine two haplotypes to obtain the genotype value for each individual Xi = (Xi1, ⋯, Xik). A phenotype Y under the null hypothesis of no association is generated using the model Y = ε with ε~N(0, 1) (β1 = ⋯ = βk = 0). Under the alternative hypothesis of association, we randomly chose s variants as causal variants while other k-s variants as non-causal variants having βj = 0. Here, we let s = 10, 20, 50 in which 10, 20%, or 50% of rare variants were causal. For causal variants under the alternative hypothesis, we set βj = c ⋅ log10(pmi) as used in Lee et al. [10], where c is 0.6, 0.3, and 0.2 for different values of s and different direction of the effects of causal variants. We consider 9 scenarios in the simulation study with the parameter values detailed in Table 3. We conduct 1000 simulations for each scenario. In each simulation, we select three extreme sample strategies, the low 20% and the up 20%, the low 10% and the up 10%, and, the low 5% and the up 5%, each of which consists of 2 N individuals including N individuals in an upper sample and N individuals in a lower sample. The statistical significance is assessed by a permutation procedure. We first calculate the value of the data-based statistic TKL for each simulation. Then we permute the "upper sample" and "lower sample" labels with equal probability and recalculate the statistic TKL for 1000 times. The estimated P value is then the proportion of permutation-based statistics that are larger than the data-based statistic in 1000 permutations for each simulation. For a given significance level α, the power/type I error rate is estimated as the proportion of rejecting the null hypothesis when p-value ≤ α in 1000 simulations. In order to compare the performance of the test statistic TKL with the existing methods, we also obtain the power for the burden, SKAT, and SKAT-O tests using case-control design with the same samples as for the test statistic TKL.
Table 3 The parameter values for power study
Simulation for fine mapping
To assess the performance of the LD measure lKL in fine mapping rare causal variants of quantitative traits, we conduct a simulation study using the method similar to those described in our early work [27, 28]. We consider a genetic region that has 21 variants, where only a variant locating at the middle of variant 10 and variant 11 is causal variant (that is, the QTL). The MAF of the causal variant is set to be 0.01(pa =0.01) and the MAFs for 20 other variants are uniformly determined with values ranging from 0.001 to 0.05. Other parameters in simulation include the ratio d/v (here, v and d are the genotypic values for individuals with genotypes aa and Aa, respectively), the thresholds L and U, the heritability (h2) of the causal variant, and the sample size (2 N) [31]. We let the ratio d/v be − 1, 0, and 1 which correspond to recessive, additive, and dominant models, respectively. Once the parameter values are chosen, a population with the effective size of 15,000 is simulated starting from the 0th generation, with an initial complete association between the minor allele a at the causal variant and m at other variants [P(m|a) = 1]. The population then evolved for 50 generations under random mating and genetic drift. A hundred populations are simulated for analyses.
All data generated or analyzed during this study are included in this published article.
RV:
Rare variant
KL-distance:
Kullback-Leibler distance
SKAT:
Sequence kernel association test
SKAT-O:
The optimal test that combines SKAT and the burden test
Quantitative trait locus
HW:
Hardy-Weinberg
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This work was supported by the Foundation of Hunan Double First-rate Discipline Construction Projects of Bioengineering.
School of Mathematics and Computational Science, Huaihua University, Huaihua, Hunan, 418008, People's Republic of China
Yang Xiang & Yumei Li
Key Laboratory of Research and Utilization of Ethnomedicinal Plant Resources of Hunan Province, Huaihua University, Huaihua, 418008, China
Key Laboratory of Hunan Higher Education for Western Hunan Medicinal Plant and Ethnobotany, Huaihua University, Huaihua, 418008, China
School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, People's Republic of China
Xinrong Xiang
Yang Xiang
Yumei Li
XY developed the statistical method and wrote the manuscript. XXR developed the statistical method and revised the manuscript. LYM conceived the idea, designed the study, and revised the manuscript. All authors have read and approved the final version of the manuscript.
Correspondence to Yumei Li.
Firstly, we calculate \( {p}_m^{\mathrm{U}}\cdot \log \frac{p_m^{\mathrm{U}}}{p_m^{\mathrm{L}}} \). Under the assumption of random mating and, thus, Hardy-Weinberg (HW) equilibrium holding in the population, we can get.
\( {p}_m^{\mathrm{L}}= pr\left(m|Y<L\right)= pr\left(m,Y<L\right)/{\varphi}_L \), where φL = pr(Y < L)
$$ {\displaystyle \begin{array}{l} pr\left(m,Y<L\right)= pr\left(m, aa,Y<L\right)+ pr\left(m, aA,s<T\right)+ pr\left(m, AA,Y<L\right)\\ {}={p}_{ma}\cdot {p}_a\cdot pr\left(Y<L| aa\right)+\left({p}_{ma}\cdot {p}_A+{p}_{mA}\cdot {p}_a\right)\cdot pr\left(Y<L| aA\right)+{p}_{mA}\cdot {p}_A\cdot pr\left(Y<L| AA\right)\\ {}={p}_{ma}\cdot {p}_a\cdot {\phi}_{11}+\left({p}_{ma}\cdot {p}_A+{p}_{mA}\cdot {p}_a\right)\cdot {\phi}_{12}+{p}_{mA}\cdot {p}_A\cdot {\phi}_{22}\\ {}={a}_1\cdot {p}_{ma}+{a}_2\cdot {p}_{mA}\end{array}} $$
where ϕ11 = pr(Y < L| aa), ϕ12 = pr(Y < L| aA), ϕ22 = pr(Y < L| AA), a1 = (ϕ11 ⋅ pa + ϕ12 ⋅ pA)/φL, a2 = (ϕ22 ⋅ pA + ϕ12 ⋅ pa)/φL.
Note that.
pma = pm ⋅ pa + δma, pmA = pm ⋅ pA + δmA, δma = − δmA and a1 ⋅ pa + a2 ⋅ pA = 1, here δm~ is the measure of LD between variant allele m and the QTL allele ~ and is defined as δ~m = p~m − p~ ⋅ pm, where pm~ is the frequency of haplotype m ~ .
$$ {\displaystyle \begin{array}{l}{p}_m^L={a}_1\cdot \left({p}_m\cdot {p}_a+{\delta}_{ma}\right)+{a}_2\cdot \left({p}_m\cdot {p}_A+{\delta}_{mA}\right)={p}_m+{\delta}_{ma}\left({a}_1-{a}_2\right)\\ {}={p}_m+{b}_L\cdot {\delta}_{ma}\end{array}} $$
$$ {b}_L={a}_1-{a}_2 $$
(Where)
Similarly, we can get \( {p}_M^L={p}_M+{b}_L\cdot {\delta}_{Ma} \), \( {p}_m^U={p}_m+{b}_U\cdot {\delta}_{ma} \) and \( {p}_M^U={p}_M+{b}_U\cdot {\delta}_{Ma} \), where bU = c1 − c2, c1 = (γ11 ⋅ pa + γ12 ⋅ pA)/φU, c2 = (γ22 ⋅ pA + γ12 ⋅ pa)/φU, φU = pr(Y > U),
γ11 = pr(Y > U| aa), γ12 = pr(Y > U| aA), γ22 = pr(Y > U| AA), c1 = (γ11 ⋅ pa + γ12 ⋅ pA)/φU, c2 = (γ22 ⋅ pA + γ12 ⋅ pa)/φU.
$$ {\displaystyle \begin{array}{l}{p}_m^{\mathrm{U}}\cdot \log \frac{p_m^{\mathrm{U}}}{p_m^{\mathrm{L}}}={p}_m^{\mathrm{U}}\cdot \log {p}_m^{\mathrm{U}}-{p}_m^{\mathrm{U}}\cdot \log {p}_m^{\mathrm{L}}=\left({p}_m+{b}_U\cdot {\delta}_{ma}\right)\cdot \log \frac{p_m\cdot \left(1+{b}_U\cdot {\delta}_{ma}/{p}_m\right)}{p_m\cdot \left(1+{b}_L\cdot {\delta}_{ma}/{p}_m\right)}\\ {}=\left({p}_m+{b}_U\cdot {\delta}_{ma}\right)\cdot \left[\log \left(1+{b}_U\cdot {\delta}_{ma}/{p}_m\right)-\log \left(1+{b}_L\cdot {\delta}_{ma}/{p}_m\right)\right]\\ {}\left[ by\ \log \left(1+x\right)\approx x-{x}^2/2\right]\\ {}\approx \left({p}_m+{b}_U\cdot {\delta}_{ma}\right)\cdot \left[\left(\frac{b_U\cdot {\delta}_{ma}}{p_m}-\frac{{b_U}^2\cdot {\delta_{ma}}^2}{2{p_m}^2}\right)-\left(\frac{b_L\cdot {\delta}_{ma}}{p_m}-\frac{{b_L}^2\cdot {\delta_{ma}}^2}{2{p_m}^2}\right)\right]\\ {}=\left({p}_m+{b}_U\cdot {\delta}_{ma}\right)\cdot \frac{\left({b}_U-{b}_L\right)\cdot {\delta}_{ma}}{p_m}\cdot \left(1-\frac{\left({b}_U+{b}_L\right)\cdot {\delta}_{ma}}{2{p}_m}\right)\end{array}} $$
$$ {p}_M^{\mathrm{U}}\cdot \log \frac{p_M^{\mathrm{U}}}{p_M^{\mathrm{L}}}\approx \left({p}_M+{b}_U\cdot {\delta}_{Ma}\right)\cdot \frac{\left({b}_U-{b}_L\right)\cdot {\delta}_{Ma}}{p_M}\cdot \left(1-\frac{\left({b}_U+{b}_L\right)\cdot {\delta}_{Ma}}{2{p}_M}\right) $$
$$ {p}_m^{\mathrm{L}}\cdot \log \frac{p_m^{\mathrm{L}}}{p_m^{\mathrm{U}}}\approx \left({p}_m+{b}_L\cdot {\delta}_{ma}\right)\cdot \frac{\left({b}_L-{b}_U\right)\cdot {\delta}_{ma}}{p_m}\cdot \left(1-\frac{\left({b}_L+{b}_U\right)\cdot {\delta}_{ma}}{2{p}_m}\right) $$
$$ {\displaystyle \begin{array}{l}H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right)=\frac{1}{2}\left({p}_m^{\mathrm{U}}\cdot \log \frac{p_m^{\mathrm{U}}}{p_m^{\mathrm{L}}}+{p}_M^{\mathrm{U}}\cdot \log \frac{p_M^{\mathrm{U}}}{p_M^{\mathrm{L}}}+{p}_m^{\mathrm{L}}\cdot \log \frac{p_m^{\mathrm{L}}}{p_m^{\mathrm{U}}}+{p}_M^{\mathrm{L}}\cdot \log \frac{p_M^{\mathrm{L}}}{p_M^{\mathrm{U}}}\right)\\ {}\approx \frac{1}{2}\left[\frac{{\left({b}_U-{b}_L\right)}^2\cdot {\delta_{ma}}^2}{p_m}+\frac{{\left({b}_U-{b}_L\right)}^2\cdot {\delta_{Ma}}^2}{p_M}\right]\\ {}=\frac{{\delta_{ma}}^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_m\cdot {p}_M}\end{array}} $$
$$ H\left(\left\{{p}_m^{\mathrm{U}},{p}_M^{\mathrm{U}}\right\},\left\{{p}_m^{\mathrm{L}},{p}_M^{\mathrm{L}}\right\}\right)\approx \frac{{\delta_{ma}}^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_m\cdot {p}_M}=\frac{{\left(1-\theta \right)}^{2n}{p}_a^2\cdot {p}_M^2\cdot {\left({b}_U-{b}_L\right)}^2}{2{p}_m\cdot {p}_M} $$
Then, we have
$$ {l}_{KL}=\frac{p_m}{p_M}H\ \left(\left\{{p_m}^{\mathrm{U}},{p_M}^{\mathrm{U}}\right\},\left\{{p_m}^{\mathrm{L}},{p_M}^{\mathrm{L}}\right\}\right)\kern0.5em \approx \frac{1}{2}\ {\left(\ 1-\theta \right)}^{2n}{p}_a^2\cdot {b}^2 $$
where b = bU − bL.
Xiang, Y., Xiang, X. & Li, Y. Identifying rare variants for quantitative traits in extreme samples of population via Kullback-Leibler distance. BMC Genet 21, 130 (2020). https://doi.org/10.1186/s12863-020-00951-2
Quantitative trait
Association analysis
Extreme phenotype
|
CommonCrawl
|
Differential item functioning of the SF-12 in a population-based regional joint replacement registry
Iraj Yadegari1,
Eric Bohm3,
Olawale F. Ayilara1,2,
Lixia Zhang1,2,
Richard Sawatzky4,
Tolulope T. Sajobi5 &
Lisa M. Lix ORCID: orcid.org/0000-0001-8685-32121,2
Joint replacement, an increasingly common procedure amongst older adults, can substantially improve health-related quality of life (HRQoL). However, differential item functioning (DIF) may affect the accurate interpretation of differences in HRQoL amongst patients with different demographic and health status characteristics but the same underlying (i.e., latent) level of the investigated construct. This study tested for DIF in pre-operative SF-12 physical health (PH) and mental health (MH) sub-scale items amongst patients undergoing total hip arthroplasty (THA) and total knee arthroplasty (TKA).
Data were from a population-based joint replacement registry from the Canadian province of Manitoba. TKA and THA patients who had surgery between 2009 and 2015 and completed a pre-operative assessment were included. DIF was tested using the multiple indicators multiple causes (MIMIC) method with sex, age group, body weight status, and presence of multiple comorbid conditions (i.e., multimorbidity) as covariates. Analyses were stratified by joint type.
The study cohort included 8820 patients; 42.1% underwent THA, 57.3% were female, 32.7% were 70+ years, and 52.8% were obese. For each sub-scale, four of the six items exhibited DIF in both THA and TKA groups. Differences in the covariate effect estimates for DIF and No-DIF models on the MH latent variable were largest for age and body weight status for the THA group, and for sex and multimorbidity for the TKA group. All of the differences were small for PH. Multimorbidity had the strongest association with PH and age and sex had the strongest association with MH in the DIF models.
Demographic and health status characteristics influenced SF-12 PH and MH item responses in joint replacement populations, although the size of the effects were not large for PH. We recommend testing and adjusting for DIF effects to ensure comparability of HRQoL measures in joint replacement populations.
Joint replacement is an increasingly common procedure; rates of total hip and knee arthroplasty (THA/TKA) are increasing worldwide [1, 2]. THA and TKA can positively impact the health-related quality of life (HRQoL) of patients, resulting in substantial improvements in functional abilities and reductions in pain [3, 4]. There is strong interest worldwide in the incorporation of patient-reported outcome measures (PROMs) into joint replacement registries for monitoring appropriateness of care, improvements in health status, and health system performance [5]. The International Society of Arthroplasty Registries has convened working groups to evaluate and advise on best practices in the selection, administration, and interpretation of PROMs for joint replacement registries [6, 7].
Measurement validity and reliability are key considerations in the interpretation of patient responses on PROMs. An important validity criterion relevant to group comparisons is that the scoring of PROMs must be free from the effects of differential item functioning (DIF), which arises when patients with the same underlying level of the latent trait that the PROM is intended to measure do not interpret a PROM's items in the same way [8]. DIF results in different item response probabilities for individuals with similar observed characteristics [9]. If DIF is present, then observed group differences will at least partially reflect something other than the latent construct, such as different interpretations of the item(s). DIF can result in biased between-group comparisons because the response patterns may reflect attributes other than those that the instrument is intended to measure.
Brief general-purpose PROMs, such as the 12-item Short Form Survey (i.e., SF-12), are advantageous to administer to joint replacement patients because they facilitate comparisons across patient populations while reducing participant response burden at pre- and post-operative measurement occasions. The SF-12 has undergone comprehensive psychometric evaluations of its reliability and validity [10].
Although DIF has been tested in other measurement instruments [9, 11,12,13], only a few studies have investigated DIF for the SF-12. DIF has been detected for the SF-12 in population-based data [14, 15]; a study in the general population revealed DIF effects by age, sex, and level of education [14]. However, DIF has not been thoroughly investigated in specific populations, such as in joint replacement populations.
The goal of our study was to test for DIF on the SF-12 physical health (PH) and mental health (MH) sub-scale items in a joint replacement population. We considered demographic characteristics in addition to health status characteristics in assessing DIF; the latter have recently been examined as potential contributors to DIF in PROMs for patients with osteoarthritis [16] and joint pain [17].
Data were from the Winnipeg Regional Health Authority Joint Replacement Registry; the Health Authority is the largest health region in the province of Manitoba, Canada and has a population of more than 700,000 residents. The province has a single-payer health care system that provides necessary hospital, medical and surgical services to all individuals eligible to receive health services. The Registry captures more than 90% of the joint replacement procedures conducted within the health region and more than three-quarters of all procedures in the entire province.
The Registry was initiated in 2004 with partial capture of information on all joint replacement surgeries; this was expanded to full mandatory capture of information in 2005. The Registry has been described in detail elsewhere [18]; it contains patient demographics, comorbid conditions, surgical technique, implant details, and complications. Both general and condition-specific HRQoL measures are included in the Registry. The former includes the SF-12 and the latter includes the Oxford Hip and Knee scores [19, 20]. Pre-operative data capture occurs in the pre-admission clinic under the guidance of a clinic nurse. Post-operative data are collected via a mail-out questionnaire conducted by Registry staff. Data entry is undertaken by the hospital medical records department for hospital stay characteristics and by Registry staff for PROMs. All data are collected via standardized instruments and the process of data collection and entry is overseen by Registry staff for all hospital sites.
The study cohort included all individuals who underwent THA or TKA between April 1, 2009, and March 31, 2015 and for whom complete pre-operative data were available. All patients from one hospital were excluded in 2011 because pre-operative questionnaires were not distributed that year.
The SF-12 (version 2) is a general-purpose instrument consisting of 12 items that comprise eight sub-domains [21]: physical functioning, role physical, bodily pain, general health, vitality, social functioning, role-emotional, and mental health. The eight sub-domain scores can be weighted and summarized into MH and PH sub-scale scores. According to this model, the items from the physical functioning, role-physical, bodily pain, and general health sub-domains are indicators of PH while vitality, social functioning, role-emotional, and mental health items are indicators for MH. Assessments of construct validity using latent variable models has confirmed this measurement structure [21, 22], although correlations of residual errors for items associated with PH and MH latent variables has been observed [21,22,23].
Covariates used to describe the study cohort and examine potential DIF sources for the SF-12 included sex, age group, body weight status, and multimorbidity, the presence of two or more chronic conditions [24]. Age was classified as 60 years or less (reference category), 61 to 70 years, and greater than 70 years; the dummy variables AGE1 (0 if age ≤ 60 and 1 otherwise) and AGE2 (0 if age ≤ 70 and 1 otherwise) were created to represent these age categories. Body weight status was based on body mass index (BMI), which was calculated from measured height and weight (kg/m2) captured by clinic nurses; it was categorized as underweight or normal weight (BMI ≤ 25.0; reference category), overweight (25.0 < BMI ≤ 30.0), and obese (BMI > 30.0) [25, 26]. The dummy variables of BMI1 (0 if BMI ≤ 25.0 and 1 otherwise) and BMI2 (0 if BMI ≤ 30.0 and 1 otherwise) were created to represent these categories.
Information about 14 chronic conditions was captured from a self-report questionnaire administered by clinic staff at the pre-operative occasion; individuals were classified as having multimorbidity if they had at least two of these chronic conditions. A single dummy variable COMORB (1 = presence of 2+ comorbid conditions and 0 otherwise) was defined.
The analyses were conducted for patients with complete information (i.e., no missing data) on all SF-12 items. Descriptive analyses were conducted using frequencies and percentages. All analyses were stratified by joint type.
A variety of methods have been used to detect DIF including logistic regression [27], item response theory (IRT) models [28, 29], and the multiple indicators multiple causes (MIMIC) model [30,31,32]. IRT and MIMIC models can be applied to binary and ordinal item responses, and are flexible to incorporate one or more latent constructs. In addition, the MIMIC is flexible to allow for the specification of dependencies between item residuals [23, 33]. Consequently, we adopted the MIMIC model to test for uniform DIF.
We constructed baseline models for MH and PH sub-scales based on the hypothesized measurement structure of the SF-12, in which the PH and MH items have no cross loading items (Additional file 1: Figures S1 and S2). The baseline models included two correlated residuals (items P2 and P3, P4 and P5) for the PH sub-scale and two correlated residuals (items M1 and M2, M3 and M5) for the MH sub-scale [21, 23] and confirmed by the assessment of fit measures, which demonstrated poorer overall fit when these residuals were not correlated.
In a MIMIC model with m items and k covariates, the latent response for the ith item (i = 1,…, m) is regressed on the latent variable F and the covariate vector Ζ,
$$ {y}_i^{\ast }=\backslash {\mathrm{uplambda}}_iF+{\boldsymbol{\beta}}_i^{\prime}\mathbf{Z}+\backslash {\mathrm{upvarepsilon}}_i, $$
where εi is the error term, λi is the factor loading, and \( {\boldsymbol{\beta}}_i^{\prime }=\left(\backslash {\mathrm{upbeta}}_{i1}\dots \backslash {\mathrm{upbeta}}_{ik}\right) \) is the vector of the effects of covariates on the latent response \( {y}_i^{\ast } \). The latent response is scored via a threshold model
$$ {y}_i=c,\kern0.5em if\kern0.5em {\tau}_{i(c)}<{y}_i^{\ast}\le {\tau}_{i\left(c+1\right)}, $$
for categories c = 0, 1, 2, …, C – 1, where τi(0) = − ∞ and τi(C) = + ∞. Thus, yi is a polytomous variable which takes discrete values 0, 1, …, C – 1. In addition, the latent factor is regressed on the covariates via
$$ F=\backslash {\mathrm{upgamma}}^{\prime}\mathbf{Z}+\eta, $$
where η is the error term and is independent of Z, and γ′ = (γ1, … , γk) is a vector of regression coefficients that describe between group differences in F (Fig. 1). These formulations enable us to estimate and test \( {\boldsymbol{\beta}}_{\boldsymbol{i}}^{\prime } \) conditional on F. If \( {\boldsymbol{\beta}}_{\boldsymbol{i}}^{\prime}\boldsymbol{\ne}\mathbf{0} \), there is a significant direct effect from the covariates to the latent response \( {y}_i^{\ast } \) which means that DIF exists in the ith item [34, 35].
Example of MIMIC model permitting DIF for the ith item. The dashed arrow from each covariate to the ith item represents the direct effect; γk = regression coefficient for the effect of the kth covariate on the latent variable; λi = the regression coefficient for the latent variable and the ith item; βik = regression coefficient showing the effect of the kth covariate on the ith item; εi = measurement error for the ith item; η = residual for the latent variable
There were four primary steps in the DIF analysis. First, unidimensionality of the measurement scales was assessed. Next, anchor items were selected. Then, each item was assessed for DIF. Finally after adjustment for DIF, the contributions of the covariates and items to the final DIF model was assessed.
In the first step the unidimensionality assumption, which implies that all sub-scale items measure a single latent construct, was examined by applying a single-factor model with an oblique rotation to the polychoric correlation matrix for the items for each of the MH and PH sub-scales. To make a decision about unidimensionality, we used two criteria: (a) the existence of only one eigenvalue greater than one, and (b) a large value for the ratio of the first to second eigenvalues (i.e., r > 4) [36]. We used several criteria to evaluate the goodness-of-fit of a single-factor model. We considered the model to be a reasonable fit to the data if it had a small root mean square error of approximation (i.e., RMSEA < 0.06), a large comparative fit index (i.e., CFI > 0.95), a large Tucker-Lewis Index (i.e., TLI > 0.95), and a small weighted root mean square residual (i.e., WRMR < 1.0) [37,38,39].
In the second step, we selected anchor items (i.e., DIF-free items). At least one anchor item must be selected to define the latent construct on which the groups are compared. We used the following method to select the anchor item(s). First, for each sub-scale, a single-factor model was fit to the data; it included direct effects of the covariates on the latent variable but no direct effects between the covariates and the sub-scale items. This was the base model. Next, a series of single-factor models were fit to the data that added direct effects from the covariates; there was one model for each sub-scale item. A χ2 difference test was used to compare the models with and without the direct effects. The item(s) with the smallest χ2 statistics was(were) selected as the anchor item(s) [40]. Note that this process was applied to the data for all cohort members so that the same anchor items were selected for both THA and TKA patients. This facilitated the interpretation of the study findings because the same item(s) served as reference points for all analyses. We confirmed the same anchor items in separate analyses for THA and TKA patients.
In the third step, item purification was conducted to identify the items affected by DIF. First, a full model was fit to the data that included direct effects from covariates to all sub-scale items except the anchor item(s). Next, we fit a series of reduced models that excluded direct effects from the covariates to each item; this was done one item at a time. A χ2 difference test was used to compare these nested models using DIFFTEST for the robust weighted least square estimation method (i.e., WLSMV) in Mplus (https://www.statmodel.com/chidiff.shtml). A large χ2 difference statistic implies uniform DIF is present for the item.
The fourth step was to fit a model that included direct effects from the covariates to all DIF items (i.e., the items for which DIF was identified in the previous step) and direct effects of the covariates on the latent variable [9, 31]. This model was used to obtain parameter estimates of direct effects of the covariates on the PH and MH sub-scale items. The total effect of DIF was measured via the relative difference between standardized coefficient estimates for the DIF and No-DIF models (i.e., difference in standardized estimates divided by the standardized estimates for the No-DIF model). A difference in standardized coefficients of 0.20 was considered as small, 0.50 as moderate, and 0.80 or greater as large [41]. Estimates of the total effects (i.e., direct and indirect effects) of the covariates on the individual sub-scale items were also produced.
We used an approach based on dominance analysis [42] and Nagelkerke's coefficient of determination [43,44,45] to assess the relative importance of both individual items and covariates in the DIF models. Specifically, an item's importance in the final DIF model was estimated based on its contribution (i.e., direct effects from the covariates to the item) conditional on the contributions of the other items. To measure the item's importance, a full model was fit to the data that include direct effects of the covariates on all DIF items identified in the previous step, as well as direct effects of the covariates on the latent variable. Next, we fit a series of reduced models that excluded direct effects from the covariates to each DIF item; we did this one item at a time. The importance of each DIF item was assessed using an adaptation of Nagelkerke's coefficient of determination,
$$ {R}^2=\left(1-{e}^{-\left(\Delta {\chi}^2\right)/N}\right)/\left(1-{e}^{-{\chi}_R^2/N}\right), $$
where N is the total sample size, \( {\chi}_R^2 \) is the chi-square test statistic for the reduced model, and Δχ2 is the scaled difference in χ2 test statistics for the reduced and full models. The statistic R2 is equal to Nagelkerke's coefficient of determination if we replace \( {\chi}_R^2 \) with −2 Log(LR) and Δχ2 with −2 Log(LR/LF) in maximum likelihood estimation, where LR and LF are the likelihood of the reduced and full models, respectively. An item was more important than all other items if it had the largest R2 amongst all items.
The importance of a covariate in the final DIF model was measured by its contribution (i.e., direct effects from the covariate to all DIF items), conditional on the contribution of the other model covariates. We used a similar approach to that described above to measure covariate importance in the final DIF model. First, a full model was fit to the data that include direct effects from all covariates to the DIF items, as well as direct effects of the covariates on the latent variable. Next, a series of reduced models were fit to the data that excluded the effect of each covariate; this was done one covariate at a time. Using the adapted Nagelkerke coefficient of determination, we measured the importance of each covariate. A covariate was more important than all other covariates if it had the largest R2 amongst all of the covariates.
All analyses were conducted using Mplus software, version 8. In all analyses, the latent factor mean was constrained to zero and its variance was fixed to one.
The study cohort included 8820 patients with complete information on all SF-12 items at the pre-operative occasion. Overall, 42.1% patients had THA in the observation period.
For the THA group (Table 1), 53.4% were female, one-third (33.7%) were 60 years of age or younger, and one-third (33.6%) were more than 70 years of age. Overweight and obese individuals accounted for 37.9% and 41.8% of the THA group, respectively. Slightly more than one-half of THA patients had multimorbidity. The most common chronic conditions were hypertension, other (i.e., secondary) osteoarthritis, and back pain.
Table 1 Frequency (%) of demographic and health status characteristics of study cohort, stratified by type of joint replacement
For the TKA group, 60.2% were female and approximately one-third (30.0%) were 60 years of age or younger and one-third (32.0%) were more than 70 years of age. Overweight and obese individuals accounted for 28.7% and 60.8% of patients, respectively. Multimorbidity was identified in almost two-thirds of TKA patients (62.2%) and the most common chronic conditions were the same as for the THA group.
The frequencies of responses to the MH and PH sub-scale items are reported in Table 2 for the entire cohort. For the MH sub-scale, close to half (46.4%) of patients responded "A little of the time" to M3 ("felt calm and peaceful"), while 36.0% of patients responded "None of the time" to M2 ("less careful than usual"). Furthermore, for the PH sub-scale, more than half (58.9%) of patients respond "Yes, limited a lot" to P2 ("moderate activities") while 74.4% of patients respond "Yes, limited a lot" to P3 ("climbing several flights of stairs").
Table 2 Frequencies (%) of responses to the SF-12 mental health (MH) and physical health (PH) sub-scale items
With respect to multimorbidity, 19.0% of individuals in the cohort had no chronic conditions and 22.5% had a single chronic condition. Almost one-quarter (24.2%) had two chronic conditions, and the remainder had three or more chronic conditions.
Exploratory factor analysis revealed that for both the PH and MH sub-scales there existed only one eigenvalue with a value greater than one. The ratio of the first to second eigenvalues was larger than four, except for the PH sub-scale in the TKA group where it was only slightly less than this criterion (r = 3.95). In addition, the second eigenvalue was similar in size to the third eigenvalue in both groups and for both sub-scales. Therefore, it was reasonable to accept unidimensionality of the MH and PH sub-scales for both the THA and TKA groups.
In the baseline model for the PH and MH sub-scales, two correlated residuals for the PH sub-scale (items P2 and P3, P4 and P5) and four correlated residuals for the MH sub-scale (items M1 and M2, M3 and M5) were considered for inclusion based on the empirical results and previous research [21, 23]. Adding residual correlations for these items resulted in an acceptable model fit (Table 3). Specifically, the single-factor model fit to the MH sub-scale items for the THA group had RMSEA = 0.05, CFI = 1.00, TLI = 1.00, and WRMR = 0.62 and for the TKA group it had RMSEA = 0.03, CFI = 1.00, TLI = 1.00, and WRMR = 0.38 for the TKA group. The single-factor model fit to the PH sub-scale items had RMSEA = 0.03, CFI = 1.00, TLI = 1.00, and WRMR = 0.61 for the THA group. For the TKA group, this model had RMSEA = 0.02, CFI = 1.00, TLI = 1.00, and WRMR = 0.52.
Table 3 Goodness-of-fit statistics for the SF-12 mental health (MH) and physical health (PH) sub-scales with and without correlated residual variances, stratified by type of joint replacement
We selected anchor items empirically for each sub-scale. We selected items M5 ("Felt downhearted and depressed") and M6 ("Social limitations") for the MH sub-scale and items P2 ("Moderate activity") and P6 ("Have pain with normal work)" for the PH sub-scale, because they had the smallest χ2 statistics. Specifically, the χ2 statistics had values of 96.9, 157.1, 150.9, 160.1, 17.7, and 26.4 for items M1 to M6, respectively and values of 719.6, 41.0, 136.9, 62.2, 99.2, and 43.5 for items P1 to P6, respectively. .
Then we tested all non-anchor items for uniform DIF. For the MH sub-scale, all of the χ2 difference tests produced statistically significant results for both the TKA and THA groups, suggesting uniform DIF was present in all non-anchor items (see Table 4). Furthermore, the χ2 difference tests suggested the presence of uniform DIF in all non-anchor items for the PH sub-scale for both the THA and TKA groups.
Table 4 Tests for differential item functioning on the SF-12 mental health (MH) and physical health (PH) sub-scale items, stratified by type of joint replacement
Table 5 provides estimates of the direct effects of the covariates on the latent variables. As well, differences in the estimates when there were direct effects from the covariates to the items (i.e., DIF model) versus the case when there were no direct effects from covariates to the items (i.e., No-DIF model) are provided. As Table 5 reveals, in both the THA and TKA groups, the PH and MH latent variables were always negatively associated with the covariates COMORB and BMI2 and positively associated with sex. This indicates that patients with multimorbidity had smaller PH and MH latent variable scores on average, relative to other patients; obese patients had smaller PH and MH latent variable scores relative to non-obese patients, and the PH and MH latent variable scores for men were always larger than those for women. Almost all of the estimates were statistically significant in both the DIF and No-DIF models. The relative differences revealed that the largest effects of the covariates on the MH latent variable were observed for the age and body weight status covariates in the THA group. The majority of the standardized differences indicate small effects; the exceptions were for the covariates AGE2 and BMI1 for the THA group, which were moderate in size. For the PH latent variable, all of the covariates had smaller relative difference statistics than for the MH latent variable in both the TKA and THA groups.
Table 5 Regression model estimates for covariate associations with SF-12 mental health (MH) and physical health (PH) latent variable scores
Adjustment for DIF resulted in changes in the estimates of the total effects for most of the SF-12 MH and PH sub-scale items (Additional file 2: Tables S1 and S2). For the MH sub-scale items, in general the largest relative differences in total effect estimates for the DIF and no-DIF models were associated with the age and body weight status covariates; these differences were generally larger in size for the THA group than for the TKA group. For the PH sub-scale items, the relative differences in total effect estimates for the DIF and no-DIF models were all small, except for item P1 ("General health") in the THA group.
In the final DIF model for the PH sub-scale, multimorbidity and age had the largest and smallest relative importance, respectively as judged by the modified coefficient of determination (Fig. 2). This finding was consistent for both the THA and TKA groups. In the final DIF model for the MH sub-scale, sex and age had the largest importance for the THA group while multimorbidity and age had the largest R2 statistics for the TKA group.
Importance of the covariates in the final DIF models. a Importance analysis for the PH sub-scale. b Importance analysis for the MH sub-scale
Finally, the relative importance analyses were conducted for all sub-scale items (Fig. 3). For the PH sub-scale, item P1 ("General health") had the largest contribution to the final DIF model while item P4 ("Accomplished less, physical") had the smallest contribution. For the MH sub-scale, item M4 ("Have a lot of energy") for the THA group and items M4 and M2 ("Less careful than usual") for the TKA group had the largest contributions to the final DIF model.
Importance of the items in the final DIF models. a Importance of the PH sub-scale items. b Importance of the MH sub-scale items
This study tested for DIF in the PH and MH sub-scale items of the SF-12 across demographic and health status characteristics for patients having joint replacement surgery. We focused on responses given prior to surgery, as this is when health status measures (i.e., body weight status and presence of comorbid conditions) were collected, and also because pre-surgery assessments are an essential reference point for assessing the magnitude of post-surgery improvements [46]. The responses given by patients on the SF-12 items have some consistency with previous research, which has shown, for example, that older patients are more likely to report problems with moderate activities and climbing several flights of stairs [47, 48].
Our findings suggest that multimorbidity is not only a source of DIF but also had the largest contribution to the DIF model for the PH sub-scale in the relative importance analysis. Other recent studies have shown a strong association between SF-12 PH sub-scale scores and multimorbidity [49], although this covariate has not been explored for its effect in DIF analyses. At the same time, the differences in estimates of the effect of comorbidity on the MH and PH latent variables between the No-DIF and DIF models were generally small.
Item M4 from the MH sub-scale and item P1 from the PH sub-scale were associated with the largest contributions to the final DIF models. Adjustment for DIF did not change the direction of the association between the covariates and the PH and MH scores. This result was consistent with other findings in the literature for the SF-12 and also for the SF-36 [9, 14].
While this study investigated DIF in a population for which PROMs are of significant value for assessing surgical outcomes, further research is warranted. Since DIF may change from pre-surgery to post-surgery occasions, future studies might explore response shift [50], a change in an individual's values, internal standards, and conceptualization of QOL over time, in joint replacement populations. Research conducted to date [51,52,53] has identified the presence of response shift in patients undergoing total knee replacement. As well, we only tested for uniform DIF in MH and PH. The MIMIC model cannot easily be used to investigate the presence of non-uniform DIF, which involves testing interactions between covariates and latent variables on the item responses. Specification of interaction terms assumes normally distributed covariates [32]. Thus, there is opportunity for opportunities to investigate new approaches to test for non-uniform DIF in MIMIC models. Finally, the generalizability of the study findings regarding the measurement model fit to the data and presence of DIF should be explored in other joint replacement populations.
In summary, this study suggests the existence of DIF in population-based SF-12 data for joint replacement patients. PH and MH sub-scale scores may not be comparable across sub-groups defined by demographic and health status variables without considering the effects of DIF. Moreover, this study has provided evidence that having more than one chronic condition may be a source of DIF; multimorbidity should therefore be explored further in studies about DIF in other populations. At the same time, associations between the latent construct and the covariates revealed generally small differences between the DIF and no-DIF models, indicating that the effect of DIF on the latent construct was not substantial in either THA or TKA patients.
DIF should be given routine consideration in the analysis of PROMs because it can impact the interpretation of group differences. Measurement equivalence is essential to ensure accurate assessments of patient health; inaccurate assessment can result in incorrect estimates of the magnitude of group differences and can impact on clinical decision making about the effectiveness of interventions, such as THA and TKA, on patient's perceptions of their own health.
There are a few methods to address the presence of DIF in PROMs data, although no method is recognized as the optimal solution [54]. Removing DIF items from the SF-12 is likely to effect the validity and accuracy of this measure. Replacing DIF items with equivalent items that do not exhibit DIF is conditional on having a resource of known DIF-free items. Examining items for DIF prior to conducting analyses on the SF-12 and adjusting for DIF before comparing sub-groups may be a reasonable solution, although it can also affect the comparability of scores across populations. Sensitivity analyses, in which analyses of PROMs are conducted after accounting for DIF and then not accounting for DIF, is a feasible approach for researchers to adopt in practice.
Data used in this article were derived from administrative health data as a secondary source. The data were provided under specific data sharing agreements only for the approved use. The original source data are not owned by the researchers and as such cannot be provided to a public repository. The original data source and approval for use has been noted in the acknowledgments of the article. Where necessary and with appropriate approvals, source data specific to this article or project may be reviewed with the consent of the original data providers, along with the required privacy and ethical review bodies.
COMORB:
DIF:
Differential item functioning
MH:
PROM:
Patient-reported outcome measure
THA:
Total hip arthroplasty
TKA:
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Access to the data was provided by the Winnipeg Regional Health Authority. The results and conclusions are those of the authors and no official endorsement by the data providers is intended or should be inferred.
Funding for this study was provided by the Canadian Institutes of Health Research (grant # MOP-142404). LML was supported by a Research Chair from Research Manitoba during the period of the study. RS is supported by a Tier 2 Canada Research Chair in Patient-Reported Outcomes.
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada
Iraj Yadegari, Olawale F. Ayilara, Lixia Zhang & Lisa M. Lix
Department of Community Health Sciences, University of Manitoba, S113-750 Bannatyne Avenue, Winnipeg, MB, R3E 0W3, Canada
Olawale F. Ayilara, Lixia Zhang & Lisa M. Lix
Department of Surgery, University of Manitoba, Winnipeg, MB, Canada
Eric Bohm
School of Nursing, Trinity Western University, Langley, BC, Canada
Richard Sawatzky
Department of Community Health Sciences, University of Calgary, Calgary, AB, Canada
Tolulope T. Sajobi
Iraj Yadegari
Olawale F. Ayilara
Lixia Zhang
Lisa M. Lix
IY, OA, and LZ conducted the analysis. LML, EB, RS, TTS, and IY conceived the study and prepared the analysis plan. IY and LML prepared the draft manuscript. All authors reviewed and approved the final version of the manuscript.
Correspondence to Lisa M. Lix.
This study received ethical approval from the University of Manitoba Health Research Ethics Board. Consent was not received from study participants; this was a retrospective population-based cohort study that used secondary data and therefore obtaining consent was not practicable.
Figure S1. Baseline model for the SF-12 mental health sub-scale. Figure S2. Baseline model for the SF-12 physical health sub-scale. (PDF 100 kb)
Table S1. Total effects of covariates on the SF-12 mental health sub-scale items for differential item functioning (DIF) and No-DIF models, stratified by type of joint replacement. Table S2. Total effects of covariates on the SF-12 physical health sub-scale items for differential item functioning (DIF) and No-DIF models, stratified by type of joint replacement. (PDF 189 kb)
Yadegari, I., Bohm, E., Ayilara, O.F. et al. Differential item functioning of the SF-12 in a population-based regional joint replacement registry. Health Qual Life Outcomes 17, 114 (2019). https://doi.org/10.1186/s12955-019-1166-1
Measurement bias
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CommonCrawl
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The Nature of High Reynolds Number Turbulence
Inertial-Range Dynamics and Mixing
Timetable (HRTW02)
Monday 29th September 2008 to Friday 3rd October 2008
08:30 to 09:55 Registration
09:55 to 10:00 David Wallace - welcome
10:00 to 10:30 Curvature of vortex tubes and sheets in turbulence
Chair: C Doering
Weak structures of vorticity are studied as they evolve essentially passively as a result of the induced velocity field of the large-scale turbulence. Such objects are possibly an important component of the inertial and dissipation ranges of turbulence. Earlier work [1} along these lines concentrated on the effects of accumulated strain, apropos of Lagrangian chaos, on localized packets of vorticity. In particular it was shown that time-averaging the self-energy spectrum of such a structure leads to a fractional power law for the energy spectrum. In this paper we consider the effects of strain on the geometry of vortex tubes and sheets and concentrate on regions of high curvature, as an extension of other recent work [2]. In the case of a vortex tube, we find that a region of high curvature on the tube is likely to develop around a point where the vorticity vector is orthogonal to the principal axis of maximum strain. For vortex sheets, a region of high curvature is likely to occur where the above-mentioned principal axis is normal to the sheet. The strain tensor in question is that corresponding to the future deformation of a material element. We explore the notion that such high-curvature structures play a role in establishing the energy spectrum in inertial-range turbulence. References 1. A. Leonard 2002," Interaction of localized packets of vorticity with turbulence" in Proc. IUTAM Symp. on Tubes, Sheets, and Singularities in Fluid Dynamics (Kluwer) 201-210. 2. A. Leonard 2008, "The universal structure of high curvature regions of material lines in chaotic flows", submitted.
INI 1
10:30 to 11:00 Emerging symmetries and condensates in inverse cascades
I shall review symmetry aspects of turbulent inverse cascades in fluid mechanics, optics and plasma physics. Inverse cascades are generally scale invariant in distinction from most direct cascades. It has been recently discovered that in many cases scale scale invariance can be promoted to a more general and powerful conformal invariance. Namely, two-dimensional isolines of different fields (vorticity, temperature) belong to the remarkable class of curves called Schramm-Loewner evolution (SLE). I shall briefly review the properties of such curves. That discovery brings unexpected connections between fluid mechanics, mathematics (SLE was a subject of the recent Fields medal), theory of critical phenomena and quantum field theory. I then describe the appearance of spectral condensates (modes coherent across the whole system) due to inverse cascades. I briefly discuss the possible role of condensates in atmospheric physics, including the controversy on the energy flux at meso-scales. I shall describe initial findings on how condensates break symmetries established in inverse cascades.
11:00 to 11:30 Coffee
11:30 to 12:00 Kinetic theory representation for turbulence modeling and computation
Chair: T Gotoh
One of the most common approximations in turbulence for the averaged effect of small scales is by the so called eddy viscosity modeling. That is, one approximates the Reynolds stress as a linear function of the local rate of strain of the averaged flow field. The proportionality constant is referred to as an eddy viscosity. This concept was first proposed over a century ago. It stems from an analogy for small eddy interactions with collisions of molecules resulting in Newtonian fluid constitutive relations. This approximation has made enormous impact particularly in computational fluid dynamics for turbulent flows. Many theoretical works were also developed since then, with various successes, in order to analytically derive such a functional relationship. However, unlike molecular interactions in a fluid, one of the apparent criticisms or difficulties in this analogy is the lack of scale separations between averaged fluid motions and fluctuating eddies. In this presentation, the speaker will give a somewhat provocative argument in favor of such analogy, provided that this concept be expanded in a generalized kinetic theory framework. In such an expanded framework, the analogy between eddy interactions and molecular collisions has a broader physical validity, while the eddy viscosity approximation is its consequence in the very long wave length limit. The kinetic theory representation itself needs not depend on scale separations. Using such an expanded analogy, one can also draw similarities between turbulent flow phenomena to that of non-Newtonian fluid flows in micro/nano scales. Nevertheless, other than phenomenological argument, as far as the speaker is aware, so far there have been little theoretical attempts to produce such a kinetic theory for turbulence on a concrete footing via first principle, if its physical soundness is acceptable.
12:00 to 12:30 Poster adverts
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 14:30 SV Nazarenko ([Warwick])
Bottleneck crossover between classical and quantum superfluid turbulence
Chair: H Chen
We consider superfluid turbulence near absolute zero, which consists of a polarized tangle of mutually interacting vortex filaments with quantized vorticity. This system exhibits a classical K41-type cascade at the scales greater than the intervortex separation l, and Kelvin wave turbulence at the scales
14:30 to 15:00 Homogeneous isotropic turbulence with polymer additives
We investigate the effects of polymer additives on flows that display homogeneous, isotropic turbulence by extensive direct numerical simulations (DNS) of (a) a shell model and (b) the Navier-Stokes equation coupled to an equation for the polymer-conformation tensor (the FENE-P model). Our simulations show that the addition of polymers to such flows leads to dissipation reduction in both decaying and statistically steady turbulence; this dissipation reduction is the analogue of drag reduction in wall-bounded flows. Our numerical results agree well with recent experimental results. In particular, we find that polymers decrease the energy of the turbulent fluid at intermediate length scales but increase it at small length scales; a scale-dependent viscosity provides a natural means of understanding our results. Preliminary studies of the multiscaling of structure functions, in the presence of polymer additives, and shock-capturing schemes for this problem are also discussed.
15:00 to 15:30 Tea and Posters
15:30 to 16:00 Quantum turbulence -from superfluid helium to atomic Bose-Einstein condensates-
Chair: R Pandit
Quantum turbulence (QT) was discovered in superfluid in the 1950s, and the research has tended toward a new direction since the mid 90s. The similarities and differences between quantum and classical turbulence have become an important area of research in low temperature physics. QT is comprised of quantized vortices that are definite topological defects, being expected to yield a model of turbulence that is much simpler than the classical model. The general introduction of the issue is followed by a description of the dynamics of quantized vortices. After reviewing the modern research trends on QT, we focus on the energy spectrum of QT at very low temperatures. The numerical simulation of QT by the Gross-Pitaevskii model shows that energy is transferred through the Richardson cascade of quantized vortices and the spectrum obeys the Kolmogorov law. Then we discuss QT in trapped atomic Bose-Einstein condensates (BECs). Under the combined rotations around two axes, a BEC cloud develops to a turbulent state and the energy spectrum obeys the Kolmogorov law. Ref. M. Tsubota, cond-mat/ 08062737
16:00 to 16:30 The spectrum of the vortex line density in a turbulent superfluid
Over the last twenty years, experiments and numerical simulations revealed strong similarities between Superfluid turbulence and Navier-Stokes turbulence. In particular, most measurements of integral quantities -such as the head loss in a pipe- or fluctuating ones -such as the velocity in the inertial range- could be interpretated by analogy with classical turbulence. Recently, measurements of the local fluctuations of the vortex lines density in superfluid 4He near 1.5 K have been reported. Over more than one decade of inertial range, the power spectrum was fitted with a f^(-5/3) power law. Remarkably, such a scaling differs from the one found in Navier-Stokes turbulence for the enstrophy, or for the absolute value of vorticity, which are often considered as classical counterparts of the vortex line density. We argue that this observation could be interpreted as the signature of a passive advection of the superfluid vortex lines by the largest scales of the flow.
Roche P.-E. & Barenghi C. , EPL 81: 36002 (2008)
http://crtbt.grenoble.cnrs.fr/helio/GROUP/infa.html
17:00 to 18:00 Cryogenic turbulence
Chair: J Jimenez
An overview of our knowledge of turbulence in both helium 4 and superfluid helium will be presented. The results will be cast in historical perspective and supplemented by numerical and theoretical ideas of the last fifty years.
18:00 to 18:30 Welcome Wine Reception
18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
09:30 to 10:00 Anisotropic pressure and acceleration spectra in uniform shear flow
Chair: GL Eyink
According to the local isotropic hypothesis, the small scale physical quantities such as velocity, temperature and pressure fluctuations are to be universal in any kind of turbulent flow. At this stage, the question is not whether this assumption is correct or not, but seems to be how the large scale anisotropy lost its information as the scale becomes small. In this talk, the anisotropic effect on inertial-range quantities are directly checked following the formula presented by Ishihara, Yoshida and Kaneda (P.R.L.,vol.88,154501,2002), in which the velocity correlation spectrum was uniquely determined by the rate of strain tensor of mean flow, the energy dissipation of per unit mass, and the two-non dimensional constants. This idea is applied to the pressure field in the uniform shear flow, and the shear effect on pressure and pressure gradient (acceleration) is studied experimentally up to the Reynolds number based on Taylor micro scale is 800. The results show the excellent agreement with the prediction by theory, and the universal trend of anisotropic spectra was observed.
10:00 to 10:30 Intermittency in imperfect multiplicative cascades
The standard multifractal cascade model assumes both a Markovian self-similar multiplicative cascade, and locality, in the sense that point properties depend only on their immediate neighbourhoods. Relaxing the second condition leads to more general cascades in which a point property v_{n+1} depends both on the local previous cascade step v_n, and on the global variance v'_n. The first contribution models a local breakdown process, while the second represents the effect of the background perturbations. There are two stochastic multipliers, one for each term, and they are characterised empirically for experimental high-Reynolds number experimental turbulence. General conditions are derived for such an imperfect multiplicative cascade to be intermittent, in the sense of creating unbounded high-order flatness factors after many steps. The experimental values are such as to be most likely intermittent, but they may not reach true multifractal distributions, and power laws for the structure functions, until extremely large Reynolds numbers.
10:30 to 11:00 J Peinke ([Oldenburg])
New insights into turbulence
We present a more complete analysis of measurement data of fully developed, local isotropic turbulence by means of the estimations of Kramers- Moyal coefficients, which provide access to the joint probability density function of increments for n- scales \cite{JFM}. In this contribution we report on new findings based on this technique and based on the investigation of many different flow data over a big range of Re numbers.
In particular we show:
- An improved method to reconstruct from given data the underlying stochastic process in form of a Fokker-Planck equation, which includes intermittency effects, will be shown.
- It is shown that a new length scale, for turbulence can be defined, which corresponds to a memory effect in the cascade dynamics. This coherence length can be seen as analogue to the mean free path length of a Brownian Motion. For length scales larger than this coherence length the complexity of turbulence can be treated as a Markov process. We show that this Einstein- Markovian coherence length is closely related to the Taylor micro-scale.
- It is shown that the stochastic process of a cascade will change with the Re-number and its universal or non-universal behavior with changing large scale boundary conditions will be discussed.
- For longitudinal and transversal velocity increments we present the reconstruction of the two dimensional stochastic process equations, which shows that the cascade evolves differently for the longitudinal and transversal increments. A different "speed" of the cascade for these two components can explain the reported difference for these components. The rescaling symmetry is compatible with the Kolmogorov constants and the Karman equation and give new insight into the use of extended self similarity (ESS) for transverse increments.
- A method is presented which allows to reconstruct time series from a estimated stochastic process evolving in scale. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The original and reconstructed time series coincide with respect to the unconditional and conditional probability densities. Therefore the method proposed here is able to generate artificial time series with correct n-point statistics.
11:00 to 11:30 Coffee and Posters
11:30 to 12:00 On extension of the formalism MPDFA and its application to the analyses of DNS 4096$^3$ conducted by Kaneda and Ishihara
Chair: J Peinke
Our original theoretical framework, named Multi-fractal Probability Density Function Analysis (MPDFA), has been extended successfully in order to make it possible to analyze a series of probability density functions (PDFs), extracted from experiments and numerical simulations, with arbitrarily changed measuring areas or distances for various physical quantities representing intermittent behavior characterizing fully developed turbulence.
MPDFA is a unified self-consistent approach for the systems with large deviations, which has been constructed based on the Tsallis-type distribution function following the assumption raised by Frisch and Parisi that the singularities due to the scale invariance of the Navier-Stokes equation for high Reynolds number distribute themselves multifractal way in real physical space. MPDFA can be said as a generalization of the log-normal model. It was shown that MPDFA derives the log-normal model when one starts with the Boltzmann-Gibbs distribution function instead of Tsallis-type distribution function.
As a test of the validity of the extension, we analyzed the PDFs for energy transfer rates and for energy dissipation rates extracted by Kaneda and Ishihara group at Nagoya University from their DNS 4096$^3$.
In this talk, we will present mainly on the theoretical extension of MPDFA and its validity. The detailed analyses of PDFs out of DNS 4096$^3$ and the physical outcomes from them will be given at our poster presentation of this workshop.
http://www.px.tsukuba.ac.jp/home/tcm/arimitsu/Marseilles04.pdf - Journal of Physics: Conference Series {\bf 7} (2005) 101--120.
http://www.px.tsukuba.ac.jp/home/tcm/arimitsu/Roman%202.pdf - Anomalous Fluctuation Phenomena in Complex Systems: Plasma Physics, Bio-Science and Econophysics (Special Review Book for Research Signpost), eds. C.~Riccardi and H.E.~Roman (Transworld Research Network, Kerala, India, 2008) in press.
12:00 to 12:30 The structure of the velocity and passive scalar mixing in a multiple opposed jets reactor
We document an experimental investigation of a confined chamber in which fluid is injected through two sets of 16 opposed jets that issue from top/bottom boundary porous planes. The investigated Reynolds numbers, based on injection velocity and jet diameter are up to 28,000. The high Reynolds numbers and impinging configuration of the flow produce very intense turbulence levels and a turbulence with zero mean velocity in the central region of the reactor. The analysis is done for basically two geometries: opposed jets with strong backflow, and opposed jets with very slow backflow. Particles Image Velocimetry (PIV) measurements in different planes allowed for a characterization of the mean and fluctuating velocity fields. Fluid recirculation in the reactor creates annular shear layers. The central region of the reactor includes stagnation regions, where mean vertical velocity gradients are very strong with low local mean velocity values, leading to high rms-to-mean velocity ratios. Such gradients are responsible for considerable kinetic-energy production, that sharply peaks in the central region. A particular attention is paid to the determination of the small-scales characteristics (energy dissipation rate) in different points of the flow, which is done using both PIV (using indirect methods, i.e. inertial-range information) and LDV (Laser Doppler Anemometry). Inertial-range behaviour is discussed, in terms of second and third-order structure functions, and a critical comparison with classical (forced) flows is done.
A passive scalar (Sc=1.3) is injected in the flow, in an alternate sequence (Z=0 and Z=1) among each two opposed jets and measured using Laser-Induced Fluorescence. Scalar structure functions are investigated, as well as the extent to which isotropy and homogeneity are adequate approximations for this flow. Flow visualisations exhibit very sharp scalar gradients at the frontier among two opposed jets. The dynamics of these regions closely follows that dictated by the back-and-forth motion developed in the central region, due to the opposed jets instabilities. Thus, the scalar is directly injected at the level of small scales, whereas the velocity field itself is injected over a whole range of scales. This directly leads to a very effective mixing in the stagnation region of the opposed jets.
14:00 to 14:30 J Schumacher ([TU Ilmenau])
Statistics and growth rates of high-amplitude vorticity events in turbulence
Chair: M Farge
Fluid turbulence is often characterized as a tangle of many intermittent vortices embedded in regions of straining motion. Although there have been many experimental and numerical studies on the evolution of isolated intense vortices, pairs of them or on the kinematics of ensembles of randomly distributed vortices, not much is known about their dynamic evolution in a fully turbulent flow. We present a high-resolution numerical simulation that monitors the formation and time evolution of high-amplitude vorticity regions. In order to identify and follow these events, we track the turbulence fields in local Cartesian frames of reference which move with Lagrangian tracers through the fluid. The local enstrophy -a measure of vorticity- shows temporal growth compatible on average with a classical prediction by Howarth and von Karman (1938). It remains well below a recently predicted rigorous upper bound for the enstrophy growth (Lu and Doering, 2008). However locally, enstrophy growth rates are detected that go beyond the mean trend and approach the predicted global bound. Related Links * http://www.tu-ilmenau.de/tsm - Homepage of Theoretical Fluid Mechanics Group
14:30 to 15:00 T Ishihara ([Nagoya])
Statistics of two-point velocity difference in high-resolution direct numerical simulations of turbulence in a periodic box
Statistics of two-point velocity difference are studied by analyzing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to $4096^3$ grid points. The DNS consist of two series of runs; one is with $k_{max}\eta\sim 1$ (Series 1) and the other is with $k_{max}\eta\sim 2$ (Series 2), where $k_{max}$ is the maximum wavenumber and $\eta$ the Kolmogorov length scale. The maximum, time-averaged, Taylor-microscale Reynolds number $R_\lambda$ in Series 1 is about 1145, and it is about 680 in Series 2. Particular attention is paid to the possible Reynolds number ($Re$) and $r$ dependence of the statistics, where $r$ is the distance between two points. The statistics include the probability distribution functions (PDFs) of velocity differences and the longitudinal and transversal structure functions. DNS data suggest that the PDFs of the longitudinal velocity difference at different values of Re but the same values of $r/L$, where $L$ is the integral length scale, overlap well with each other when r is in the inertial subrange and when using the same method of forcing at large scales. The similar is also the case for the transversal velocity difference. The tails of the PDFs of normalized velocity differences ($X$'s) are well approximated by such a function as $\exp(-A|X|^a)$, where $a$ and $A$ depend on $r$, and where $a$ becomes $\approx 1$ in the inertial subrange. Analysis shows that the scaling exponents of the $n$th-order longitudinal and transversal structure functions are not sensitive to $Re$ but sensitive to the large-scale anisotropy and non-stationarity, and suggests that nevertheless their difference is a decreasing function of $Re$.
15:30 to 16:00 CVS filtering to study turbulent mixing
Chair: A Leonard
Coherent Vortex Simulation (CVS) is based on the wavelet filtered Navier-Stokes equations, where at each instant the turbulent flow is split into two orthogonal contributions: the coherent flow made of vortices which is kept, and the incoherent flow made of the background fluctuations which is discarded. The CVS filter is based on an orthogonal wavelet decomposition of the vorticity field where only the wavelet coefficients whose modulus is larger than a given threshold are kept. The value of the threshold depends only on the total enstrophy and on the numerical resolution used to represent the flow. The CVS filter has already been applied to 2D [Farge, Schneider and Kevlahan in Phys. Fluids 11(8) 1999] and 3D [Farge, Pellegrino and Schneider in Phys. Rev. Lett. 87(55) 2001, Farge et al. in Phys. Fluids 15(10) 2003, Okamoto et al. in Phys. Fluids 19 2007] turbulent flows, where it has been shown that only few wavelet coefficients (from 0.7% for 256^2 up to 2.6% for 2046^3 resolution) are sufficient to represent the coherent flow which preserves the vorticity and velocity PDFs, the energy spectrum and the nonlinear transfers all along the inertial range. We will analyze the time evolution of a decaying homogeneous isotropic turbulent flow, by applying the CVS filter at each time step of a Direct Numerical Simulation (DNS). We will compare the Eulerian and Lagrangian mixing properties of the total, coherent and incoherent flows by studying how they advect a passive tracer and many particles during several eddy turn-over times. We will quantify the mixing properties of coherent and incoherent flows and show that efficient mixing is due to the transport by vortices, while the incoherent contribution is much weaker and only diffusive. Related Links * http://wavelets.ens.fr - Web
16:00 to 16:30 Energy cascade in turbulent flows: quantifying effects of Reynolds number and local and nonlocal interactions
The classical Kolmogorov theory of three-dimensional turbulence is based on the concept of the energy transfer from larger to progressively smaller scales of motion. The theory postulates that bulk of the energy transfer in the inertial range of turbulence occurs between scales of similar size, a process known as the local energy cascade. The locality allows to postulate that after multiple cascade steps the small scale dynamics become universal, i.e., independent of particulars of large scales that are determined by geometry, boundary conditions, and forces causing a flow. Yet despite its central role in the Kolmogorov theory the locality assumption cannot be easily verified, neither analytically nor experimentally. This is because the energy transfer is a result of interactions among different scales of motion originating from the nonlinear term in the Navier-Stokes equation that couples all scales. Relevant questions have been productively addressed for the first time using databases generated in large scale numerical simulations. We revisit and extend previous work and use such databases to compute detailed energy exchanges between scales of motion obtained by decomposing numerical velocity fields using banded filters, and investigate how the detailed transfers contribute to the global quantities such as the classical energy transfer, the energy flux, and the subgrid-scale transfer. We address two questions in detail. First, for the purposes of quantitative analyzes, various definitions of scales of motion can be used. This non-uniqueness leads to the possibility, raised in the literature on the subject, that properties of the energy transfer deduced from such analyzes can be qualitatively affected by the employed scale definitions. We address this question by computing detailed energy exchanges between different scales of motion defined by decomposing velocity fields using three specific filters: sharp spectral, Gaussian, and tangent hyperbolic. Second, we quantify the locality of the energy transfer and address a persistent controversy concerning the role of nonlocal interactions in the energy transfer process, i.e., the role of much larger scales than those transferring energy. The analysis of detailed interactions reveals that the individual nonlocal contributions are always large but significant cancellations lead to the global quantities asymptotically dominated by the local interactions. The detailed locality functions are computed and their behavior compared with the asymptotic scaling laws valid for infinite Reynolds numbers turbulence. Apart from an intellectual challenge of clarifying these issues, obtained results have bearing on practical questions of turbulence modeling that will also be addressed in the talk.
16:30 to 17:00 Physical-space decimation and constrained large Eddy simulation
Traditional decimation theory of fluid turbulence was proposed by Kraichnan and the analysis was carried out in the Fourier space. It has been shown that the low-order decimation theory leads to the direct-interaction-approximation, while the high-order decimation theory can include the effect of intermittency. In this talk, we propose a physical-space decimation method which can be used for large-eddy-simulation. In particular, we propose to impose physical constraints in the dynamic procedure of the dynamic subgrid-scale (SGS) stress model in large eddy simulation, and to calculate the SGS model coefficients using a constrained variation. One simple constraint for fluid turbulence in both physical and Fourier space decimation models is the conservation of energy across the inertial range. Numerical simulations of forced and decaying isotropic turbulence demonstrate that the constrained dynamic mixed model predicts the energy evolution and the SGS energy dissipation well. The constrained SGS model also shows a strong correlation with the real stress and is able to capture the energy backscatter, manifesting a desirable feature of combining the advantages of dynamics Smagorinsky and mixed models. It should be mentioned that all previous LES models do not satisfy underlying physical constraints. We have also extended the constrained LES to helical, passive-scalar and intermittent systems.
17:10 to 17:40 Extraction of the hierarchical energy spectrum in forced turbulence
Chair: JA Domaradzki
Properties of the energy spectrum in turbulent flows are studied using the DNS data for forced homogeneous isotropic and shear turbulence. The perturbation expansion about the Kolmogorov -5/3 energy spectrum yields the hierarchical spectrum, in which the -7/3 spectrum induced by the fluctuation of the dissipation rate is added. Averaging conditioned on the temporal variations of dissipation rate is applied to the ensemble of the energy spectra. The base steady spectrum fits -5/3 power, but its deviatoric part exhibits a fitting with -7/3 power. The role of the -7/3 power spectrum in generation of energy cascade is elucidated by examining the temporal variations of the energy spectrum and transfer function. The cascade process is divided into two phases. The energy contained in the low-wavenumber range in Phase 1 is transferred to the high wavenumbers in Phase 2 with switchover of the sign in the -7/3 power component. In Phase 1, a very large gain in the energy transfer function occurs at the scale corresponding to the integral length. The vortex sheet whose lateral length is comparable to the scale of this input is created, and many Mode 3 or 2 spiral vortices (LSV) (Horiuti & Fujisawa 2008) are detected in Phase 1. These LSVs are converted to Mode 1 in Phase 2. The statistics are compared in Phases 1 and 2. The turbulent energy is larger in Phase 1 than in Phase 2. The moderately large dissipation rate dominates in Phase 2, but the dissipation field is more intermittent in Phase 1. Averaging conditioned on the dissipation and enstrophy indicates that the regions of extreme dissipation and enstrophy possess a significant degree of overlap in space in Phase 1. These extreme events occur along the spiral sheet of Mode 3 LSV which is strained and stretched by the tube in the core.
17:40 to 18:00 Y Li ([Sheffield])
Lagrangian evolution of non-Gaussianity and material deformation in restricted Euler dynamics
Small-scale intermittency in fluid turbulence refers to the infrequent but strong bursts in the signals of small scale parameters. These bursts display highly non-Gaussian statistics, and its prediction poses serious challenges to turbulence research. Based on the restricted Euler approximation, and following the recent idea of tetrad dynamics, we derive a simple system of equations for the short-time Lagrangian evolution of velocity and passive scalar increments. The system reproduces several important intermittency trends observed in turbulence. A generalization to rotating turbulence shows that system captures some qualitative effects of rotation. An analytic solution to the material deformation in restricted Euler dynamics, obtained following the same idea, is also presented.
Wednesday 1st October 2008
09:00 to 09:30 On the large-scale structure of two-dimensional turbulence
Chair: G Falkovich
We consider freely-decaying, two-dimensional, isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k->0)=Ik^3 , where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k->0)=Lk , where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E~Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k->0)=Jk^-1, where J, like L, is an invariant. However, while E~Jk^-1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E~Jk^-1 spectra. The constraint imposed by the conservation of energy is particularly important,ruling out E~Jk^-1 spectra for certain classes of initial conditions. We illustrate these ideas with some direct numerical simulations.
09:30 to 10:00 Resolving the cascade bottleneck in vortex-line turbulence
Both in many superfluid experimental situations and simulations of a 3D hard-core interaction model, it is found that the vortex line length in superfluid turbulence decays in a manner consistent with classical turbulence. Two decay mechanisms have been proposed, Kelvin wave emission along lines and phonon radiation at small scales. It has been suggested that both would require a Kelvin wave cascade, which theory says cannot reach the smallest scales due to a bottleneck. In this presentation we will discuss a new approach using a recent quaterionic formulation of the Euler equations, coupled with the local induction approximation. Without the extra quaterionic terms It can be shown that if there are sharp reconnections, the above scenario occurs. But with the extra terms, the direction of propagation of nonlinear waves is reversed, there is a cascade to the smallest scales that could create phonons, and the paradox can be resolved.
10:00 to 10:30 The absence of bottleneck in the Lagrangian-averaged model for incompressible magnetohydrodynamics
In order to better understand the small scale dynamics of geophysical and astrophysical flows with huge Reynolds numbers, numerical modeling is an invaluable tool but it needs to be assessed against experimental and observational data as well as direct numerical simulations (DNS) at high resolution. In this context, we study the properties of the Lagrangian-averaged magnetohydrodynamics (MHD) $\alpha-$model, LAMHD hereafter; this model can be viewed as a norm-preserving filtering of the primitive MHD equations. Among its advantages is the fact that the LAMHD formulation preserves the basic properties of MHD, e.g. the Alfv\'en theorem of flux conservation, and invariants such as the total energy, the cross-correlation between the velocity and magnetic field and magnetic helicity, albeit in a modified (H_1) form. LAMHD has been tested in two and three space dimensions and is found to behave satisfactorily, for example reproducing the threshold for dynamo action at moderately low magnetic Prandtl numbers P_M, as encountered in the liquid core of the Earth, the solar convection zone or in liquid metals in the laboratory (where P_M is the ratio of viscosity to magnetic diffusivity). Here we demonstrate that, for the case when there is initially quasi-equipartition between the velocity and the magnetic field and with a magnetic Prandtl number equal to unity, the model reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes $\alpha-$model is somewhat more limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. The LAMHD model is thus shown to be capable of leading to large reductions in required numerical degrees of freedom for a given set of kinetic and magnetic Reynolds number. Specifically, we find a reduction factor of approx 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Taylor Reynolds number approx 1700. The DNS having been stopped at the peak of dissipation of total energy, the run was pursued using LAMHD. We thus also report on preliminary explorations of the decaying dynamics of that high Reynolds number MHD flow at late times using the LAMHD model.
10:30 to 11:00 Kinetic turbulence: a nonlinear route to dissipation through phase space
This talk will describe a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (possibly arbitrarily small) degree of collisionality. The smallness of the collision rate is compensated by the emergence of small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring in the gyrocentre space and in velocity space. Scaling relations for the corresponding fluctuation spectra are derived. An estimate for the collisional cutoff is provided. The relevance of these results to understanding the dissipation-range turbulence in the solar wind and the electrostatic microturbulence in fusion plasmas is discussed. Related Links * http://arxiv.org/abs/0806.1069 - preprint
11:30 to 11:50 On enstrophy dissipation in 2D turbulence
Chair: S Nazarenko
We consider dissipation of enstrophy, one half the integral of squared vorticity, in 2D incompressible, turbulent flows at very high Reynolds number. We prove rigorously that, if fully developed turbulence is to be modeled mathematically by irregular (weak) solutions of the 2D Euler equations in the limit of vanishing viscosity, then there is no dissipation as long as the initial enstrophy is finite. We also provide examples of dissipative flows when the initial enstrophy is infinite. Our analysis is inspired by work of G. Eyink. This is joint work with Helena and Milton Lopes.
11:50 to 12:10 Thresholds for the formation of satellites in two--dimensional vortices
We examine the evolution of a two--dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m=2 added to it. If the perturbation is weak then the vortex returns to an axisymmetric state and the non--zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.
We consider the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine--scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical--layer equations numerically. These calculations are supported by simulations of the full Navier--Stokes equations using a family of profiles based on the tanh function.
12:10 to 12:30 MD Bustamante ([Warwick])
Capturing reconnection in Navier-Stokes and resistive MHD dynamics
In this work, the phenomena of vortex reconnection in Navier-Stokes (and magnetic reconnection in MHD), of importance in fully developed turbulence, are studied from the point of view of the Eulerian-Lagrangian representation. This representation is interpreted as a full characterization of fluid motion using only particle description. New generalized equations of motion for the Weber-Clebsch potentials associated to this representation are derived. We perform direct numerical simulations in order to confirm the validity of the paradigm proposed by Constantin where particles will diffuse anomalously in the space -and time- vicinity of reconnection events. For Navier-Stokes, the generalized formalism captures the intense reconnection of vortices of the Boratav, Pelz and Zabusky flow, in agreement with the previous study by Ohkitani and Constantin. For MHD, the new formalism is used to detect magnetic reconnection in several flows: the 3D Arnold, Beltrami and Childress (ABC) flow and the (2D and 3D) Orszag-Tang vortex. It is concluded that periods of intense activity in the magnetic enstrophy are correlated with periods of increasingly frequent resettings. Finally, the positive correlation between the sharpness of the increase in resetting frequency and the spatial localization of the reconnection region is discussed. Related Links * http://arxiv.org/abs/0804.3602v1 - ArXiv Preprint of Paper
14:00 to 14:30 JC Vassilicos (Imperial College London)
Non-universality of the turbulence dissipation constant in homogeneous isotropic turbulence and the universal relations which account for it
Chair: J Schumacher
The dimensionless dissipation constant of homogeneous isotropic turbulence is equal to the third power of a number which reflects the number of large-scale eddies multiplied by a function of Reynolds number. This function of Reynolds number may tend to a constant in the limit of very high Reynolds number as a result of an eventual balance between a slow growth of the range of viscous length-scales and the increasing non-Gaussianity of the small scales. However, when the turbulence is generated by fractal grids this function of Reynolds number is inversely proportional to the Reynolds number for a very wide range of Taylor length-based Reynolds number up to about 1000 even though the turbulence energy spectrum has a well-defined -5/3 range. See Mazellier, N. & Vassilicos, J.C. 2008 The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology. Phys. Fluids 20, 015101 Seoud, R.E. & Vassilicos, J.C. 2007 Dissipation and decay of fractal-generated turbulence. Phys. Fluid 19, 105108
14:30 to 15:00 M Oberlack ([Fachgebiet für Strömungsdynamik])
Scaling law of fractal-generated turbulence and its derivation from a new scaling group of the multi-point correlation equation
Investigating the multi-point correlation equations for the velocity and pressure fluctuations in the limit of homogeneous turbulence a new scaling symmetry has been discovered. Interesting enought this property is not shared with the Euler or Navier-Stokes equations from which the multi-point correlation equations have orginally emerged. This was first observed for parallel wall-bounded shear flows (see Khujadze, Oberlack 1994, TCFD (18)) though there this property only holds true for the two-point equation. Hence, in a strict sense there it is broken for higher order correlation equations. Presently using this extended set of symmetry groups a much wider class of invariant solutions or turbulent scaling laws is derived for homogeneous turbulence. In particular, we show that the experimentally observed specific scaling properties of fractal-generated turbulence (see Vassilicos etal.) fall into this new class of solutions. This is in particular a constant integral and Taylor length scale downstream of the fractal grid and the exponential decay of the turbulent kinetic energy along the same axis. These particular properties can only be conceived from multi-point equations using the new scaling symmetry since the two classical scaling groups of space and time are broken for this specific case. Hence, extended statistical scaling properties going beyond the Euler and Navier-Stokes have been clearly observed in experiments for the first time.
15:30 to 16:00 How deep should one go to get the inertial range right?
Chair: S Chen
Theoretical and empirical arguments will be presented to show that the grid resolution in direct numerical simulations ought to be finer than previously thought. Error estimates for poorer resolutions will be presented. By going deep in the dissipation region, it may be possible to recover inertial range properties at finite Reynolds numbers.
16:00 to 16:30 Kolmogorov 4/5 law, nonlocality and sweeping decorrelation hypothesis
Results of experiments at high Reynolds numbers - both field an airborne - are used to validate an equivalent form of the 4/5 Kolmogorov, which demonstrate one of important aspects of non-locality of turbulent flows in the inertial range and stand in contradiction with the sweeping decorrelation hypothesis understood as statistical independence between large and small scales. This is supported by a set of exact purely kinematic relations also validated experimentally.
16:30 to 17:00 T Tatsumi ([Kyoto])
Statistical mechanics of fluid turbulence based on the cross-independence closure hypothesis
Statistical theory of turbulence is presented, which deals with homogeneous isotropic turbulence and inhomogeneous turbulent flows, their large-scale structures and small-scale similarities on an equal footing, using the "cross-independence closure hypothesis" proposed by Tatsumi(2001) for closing the Lundgren-Monin equations(1967) for the multi-point velocity distributions. Homogeneous isotropic turbulence at large Reynolds numbers is shown to be governed by the closed set of the one- and two-point velocity distributions. The distributions are expressed as the universal inertial-normal distributions associated with their own energy-dissipation rates as only parameters. Only exception from this universality is the longitudinal velocity-difference distribution, which is given by the local non-normal distributions in the inertial and viscous subranges. These theoretical results are discussed in comparison with the existing experimental and numerical results. Inhomogeneous turbulent flows at large Reynolds numbers are shown to be governed by the closed set of equations for the mean velocity and the one- and two-point velocity distributions. These equations have eminent feature that the effect of the mean flow is limited to the lage-scale components of turbulence. This feature is expected to largely simplify the formalism of shear-flow turbulence just like the 'boundary layer' in laminar flows.
17:10 to 17:40 Modelling two-time velocity correlations for prediction of both Lagrangian and Eulerian statistics
Chair: M Oberlack
More information on two-point two-time velocity correlations are needed for a better prediction of turbulent dispersion as well as radiated noise using an acoustic analogy. Conceptual aspects will be emphasized and not applications. Only isotropic turbulence will be considered, although many applications are developped in our team towards strongly anisotropic turbulence, mainly in rotating, stably stratified and/or MHD flows.
A simple synthetic model of isotropic turbulence is firstly considered, using a random superposition of Fourier modes : This is the KS (Kinematic simulation) following Kraichnan and Fung et al. Unsteadiness of velocity field realizations is mimicked using temporal frequencies, which are expressed in term of a prescribed energy spectrum and the wavenumber. Even if the orientation of the wavevector is randomly chosen, the link of the temporal frequency to the wavenumber is deterministic in the simpler version of the KS model. Although such a model was relevant for several applications, it is dramatically questioned for the evaluation of two-time velocity correlations. It is shown that spurious oscillations are generated, and that it is needed to model the temporal frequencies as random Gaussian variables with a standard deviation of the same order of magnitude as their mean value. Further applications to noise radiation are touched upon, in order to illustrate dominant (Lagrangian) `straining' or dominant (Eulerian) `sweeping' effects, according to the scale under consideration.
The role of a typical time-scale for the decorrelation of triple velocity correlations is then recalled and discussed in the classical `triadic closures' from the Orszag and Kraichnan's legacy, such as EDQNM, DIA and semi-Lagrangian more sophisticated variants.
Finally, these different concepts (diffusive and/or dispersive eddy dampings, straining or sweeping processes) are applied to a recent closure theory of weakly compressible isotropic turbulence. A Gaussian kernel for the decorrelation of triple velocity correlations was shown to give much better results than the classical exponential kernel inherited from EDQNM in the incompressible case. A new explanation is given in accordance with the renormalization of the acoustic wave frequency by a pure random term with zero mean but with a standard deviation of the same order of the eddy damping term formerly used in EDQNM. This analysis can be related to the concept of Kraichnan's random oscillator, recently revisited by Kaneda (2007), with a connection to the much simpler KS problem firstly presented (see also the monograph `homogeneous turbulence dynamics' by Pierre Sagaut and Claude Cambon, just published in Cambridge University Press.)
19:30 to 23:00 Conference Dinner at St Catharine's College
Thursday 2nd October 2008
09:00 to 09:30 Lagrangian velocity statistics in turbulence: theory, experiments and numerics
Chair: JF Pinton
A detailed comparison between experimental and numerical data of Lagrangian velocity structure functions in turbulent flows is presented. Thanks to the integration of information coming from experimental and numerical data, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers can be achieved. Intermittency changes if measured close to the Kolmogorov time scales or at larger time lags. A quantitative comparison with prediction from multifractal theory for Lagrangian turbulence will also be presented. These results shed some new insight on the relevance of vortex filaments for the statistics of tracers and/or heavy/light particles in turbulence.
09:30 to 10:00 Experimental results on the dynamics of tracers and inertial particles in highly turbulent flows
We report measurements on the statistics of two particle dispersion, acceleration, and velocity structure functions for tracers and inertial particles. We will especially discuss large, neutrally buoyant and small, heavy particles. The experiments are conducted in the center of von Karmann mixing flows at high Reynolds numbers using direct Lagrangian particle tracking. We will show that single time, single particle statistics are not sensitive to particle inertial for both particles; however, we observed an inertial effect on the two-time, or the two-particle statistics. We will also show that the preferential concentration (increase of the radial distribution function) measured in the center of the apparatus is caused by a decrease of the average particle number density as a function of the distance from the center of the apparatus, which is the statistical stationary point of the flow field. The work has been conducted with Mathieu Gibert and Haitao Xu.
10:00 to 10:30 CR Doering
Statistically stationary stirring of a scalar sustained by steady sources and sinks
Stirring generally enhances mixing, aiding molecular diffusivity by amplifying scalar gradients. Scalars sustained by steady sources and sinks, however, may best be mixed by flows with optimal transport properties. In this talk we describe differences between transient and steady state mixing and discuss implications for the concept of effective (eddy) diffusion.
10:30 to 11:00 GL Eyink ([Johns Hopkins])
Turbulent Lagrangian dynamics of vortex and magnetic-field line
We do not understand the laws of motion of vortex and magnetic-field lines in high-Reynolds-number turbulent flows. The current lore is self-contradictory. On the one hand, vortex/magnetic-field lines are often assumed to wander and elongate nearly as material lines in the limit of small viscosity/resistivity, and thus also to intensify, as a consequence of the Kelvin/Alfvén theorems. On the other hand, the topology of the lines is assumed to be continuously altered by viscous/resistive reconnection, implying breakdown of those same theorems. We discuss experimental and numerical evidence that these laws are both sometimes observed and sometimes violated in high-Reynolds-number turbulence. Unfortunately, we have no rational criterion to say when the Kelvin/Alfvén theorems or the Helmholtz laws of ``frozen-in'' motion should hold and when they should not. The problem has grown more perplexing with the theoretical discovery of "spontaneous stochasticity" for Lagrangian particle evolution in a Kolmogorov inertial range. As a consequence of the forgetting of initial separations in Richardson pair-diffusion, Lagrangian trajectories are not unique and must be replaced with random distributions of trajectories in the limit of small viscosity. This result presents a major crisis to our understanding of the turbulent dynamics of vortex/magnetic-field lines. As a possible resolution, we discuss a conjectured generalization of the Kelvin/Alfvén theorems, namely, that they survive as "backward martingales" of the spontaneous stochastic flows at high Reynolds-number. This conjectured relation provides a precise mathematical framework for the theory of turbulent reconnection. We discuss current rigorous results related to the conjecture and also important questions for investigation by experiment and simulation.
11:30 to 12:00 Intermittency and scaling of passive scalar convected by isotropic steady turbulence under the uniform mean scalar gradient
Chair: L Biferale
It has been more convincing that passive scalar in turbulence is more intermittent than the turbulent velocity field itself, implying that the small scales of the passive scalar are more affected by the large scale conditions. In order to get more precise knowledge about the scaling behavior of the passive scalar for various Reynolds (Peclet) numbers and large scale conditions, we have performed very high resolution direct numerical simulations (DNSs) of the passive scalar turbulence with or without uniform mean scalar gradient up to $2048^3$ grid points and $R_\lambda\approx 600$, and analysed the various statistical functions. Turbulent velocity field was statistically in a steady and isotropic state by Gaussian random force applied at large scales. Fundamental statistics such as the spectra of the kinentic energy, pressure, scalar variance, scalar-velocity flux were examined, especially in their scaling behavior. It is found that although curves of the kinetic energy and scalar spectra are well collapsed onto a single curve when the Kolmogorov variables are used, while the others are not as well as the former, suggesting need of more elaborated scaling. The scaling of the velocity structure functions is consistent with the existing data of experiments and DNSs, while the scaling of the passive scalar is not convincing and difficult to reach definite conclusion. When the isotropic random injection for the passive scalar is applied at large scales (Case R), each curve of the local scaling exponent at a given order has one local minimum and maximum point, unlike the velocity case, and plateau is not wide enough to precisely determine the scaling exponents. On the other hand, when the uniform mean scalar gradient is applied (Case G), the curves of the local scaling exponents of the isotropic sector are found to have well developed plateau, and their plateau levels are smaller than those of Case R, meaning stronger intermittency for the Case of G. Crossover of the velocity and scalar structure functions is also examined. The crossover of the transverse velocity structure functions is found to be very similar to that of the passive scalar. We seek the reason for the above differences and similarities.
12:00 to 12:30 PK Yeung (Georgia Institute of Technology)
Local flow structure and Reynolds number dependence of Lagrangian statistics in direct numerical simulations of homogeneous turbulence
Reynolds number dependence including the effects of intermittency is a crucial issue in the study of Lagrangian statistics and in how information from direct numerical simulations can be useful for stochastic modeling. Intermittency in the form of localized regions of intense straining or rotation is, for example, expected to influence how rapidly a fluid particle undergoes acceleration, and how multiple diffusing fluid particles move apart from each other. An effective approach to delineate such effects is to compute Lagrangian statistics conditioned on energy dissipation rate, enstrophy, or pseudo-dissipation following fluid particle trajectories. In this talk we shall illustrate these issues via recent results from simulations of isotropic turbulence at Reynolds numbers sufficiently high for observing inertial range behavior in the Eulerian (but not necessarily Lagrangian) frame. We also discuss research directions in the near future, including flows of greater complexity, and the promise of simulations at ever-increasing grid resolution that rapid advances in computing power are expected to make feasible.
14:00 to 14:30 Relative dispersion and Richardson's constant
Chair: E Bodenschatz
This talk will describe some very recent analysis of Direct Numerical Simulation results for turbulent relative dispersion over a wide range of Reynolds numbers. We will start with some background discussion of the nature and significance of relative dispersion and of the role of Kolmogorov's similarity theory, leading to the introduction of Richardson's constant as a fundamental parameter of relative dispersion. Although it is of great fundamental and practical significance, Richardson's constant has not been well-quantified, and model estimates for it range from 0.01 to 4. We will describe first a traditional analysis of relative dispersion data, concluding that this approach does not yield a good estimate for Richardson's constant even at the highest Reynolds number currently available. We then use a modified version of a new approach developed by Ott & Mann (JFM, 422, 207, (2000)) to show that a well-defined Richardson scaling range exists in our data. We estimate Richardson's constant over a range of Reynolds numbers showing that it decreases weakly with Reynolds number to an asymptotic value at large Reynolds number of 0.55 - 0.57.
14:30 to 15:00 Dynamics of inertial particles in fully developed turbulence
We focus on acceleration of inertial particles, using experimatal measurements and numerical studies. In the experiment, particles are optically tracked in a turbulent flow of water using an Extended Laser Doppler Velocimetry technique. The probability density functions (PDF) of particle accelerations and their auto-correlation in time are computed. Numerical results are obtained from a direct numerical simulation in which a suspension of passive pointwise particles is tracked, with the same finite density and the same response time as in the experiment. We observe that many effects cannot be accounted for by point-particle models. We show that a much better description is achieved when one includes Faxen corrections in the particle's equation of evolution.
15:30 to 16:00 Mixing due to Rayleigh-Taylor instability
Chair: PK Yeung
Rayleigh-Taylor instability occurs when a dense fluid rests on top of a light fluid in a gravitational field. It also occurs in an equivalent situation, in the absence gravity, where there is a pressure gradient normal to a interface between fluids of different density such that the direction of acceleration is from the light to the heavy fluid. This situation occurs in Inertial Confinement Fusion Implosions (ICF), see for exapmle Amemdt et al [1].
There have been a number of successful experiments on mixing due to Rayleigh-Taylor instability, for example Dimonte [2] and Dalziel [3]. However, it is impractical to perform the "perfect" experiment and experimental diagnostics are necessarily limited. High-resolution Large Eddy Simulation (LES) can now be used to greatly add to our understanding of the mixing processes and this is the subject of the talk. The numerical technique used,the TURMOIL code, was first used for Rayleigh-Taylor mixing by Youngs[4]. A MILES approach is used because of the need to treat discontinuities in the flow e.g. the initilal density discontinuity and shock waves (in some applications). The high Reynolds case is of most interest where it is assumed that the bulk properties of the turbulent zone are independent of the Reynolds number. It is argued that LES (rather that DNS) is then an appropriate technique. Mesh convergence, or near-mesh convergence, will be demonstrated for key statistical averages.
Results are discussed for a range of situations-(a) Rayleigh-Taylor mixing at a plane boundary, (b) three-layer Raleigh-Taylor mixing and (c) mixing in a spherical implosion (a simplified version of an ICF implosion).The three cases are illustrated in figs 1,2 and 3 in the attached file. Two main aspects of the mixing process will be discussed. Firstly the influence of initial conditions. It is argued that loss of memory of initial conditions is unlikely to occur in experimental situations. The initial conditions have a significant effect on the overall width of the mixing zone - an important issue for engineering models. It would very difficult to obtain corresponding results experimentally because of the lack of control and the difficulty in measuring initial conditions. Secondly, the internal structure of the turbulent mixing zone will also be discussed, in particular the dissipation of both turbulence kinetic energy and of density fluctuations. For the internal structure results are more universal and less dependent on the initial conditions.
1. P. Amendt et al., "Indirect-drive noncryogenic double-shell ignition targets for the National Ignition Facility: Design and analysis", Physics of Plasmas, 9, p2221, (2002). 2. G Dimonte & M Schneider, "Density ratio dependence of Rayleigh-Taylor mixing for sustained and impulsive acceleration histories", Physics of Fluids, 12, p304 (2000) 3. S.B.Dalziel, "Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability", J. Fluid Mech. 399, p1,
16:00 to 16:30 K Schneider ([Universite de Provence, Marseille])
Lagrangian acceleration in confined 2d turbulent flow
A Lagrangian study of two-dimensional turbulence for two different geometries, a periodic and a confined circular geometry, is presented to investigate the influence of solid boundaries on the Lagrangian dynamics. It is found that the Lagrangian acceleration is even more intermittent in the confined domain than in the periodic domain. The flatness of the Lagrangian acceleration as a function of the radius shows that the influence of the wall on the Lagrangian dynamics becomes negligible in the center of the domain and it also reveals that the wall is responsible for the increased intermittency. The transition in the Lagrangian statistics between this region, not directly influenced by the walls, and a critical radius which defines a Lagrangian boundary layer, is shown to be very sharp with a sudden increase of the acceleration flatness from about 5 to about 20. Related Links * http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.3139v1.pdf - PDF file of a preprint to appear in PRL
16:30 to 17:00 CM Casciola ([Sapienza Università di Roma])
Clustering of inertial particles in shear flows
Recently, clustering of inertial particles in turbulence has been thoroughly analyzed for statistically homogeneous and isotropic flows. The most striking result concerns the singular behavior exhibited by the radial distribution function under proper resonance conditions, showing clustering below the Kolmogorov scale. Since anisotropy is strongly depleted through the inertial range, the advecting field anisotropy may be expected in-influential for the small scale features of particle configurations. By addressing direct numerical simulations (DNS) of a statistically steady particle-laden homogeneous shear flow, we find instead that the small scales of the particle distribution are strongly affected by the geometry of velocity fluctuations at large scales. The proper statistical tool is the angular distribution function of particle pairs (ADF). Its anisotropic component may develop a singularity whose strength quantifies the anisotropy of the small scale clustering. The data provide evidence that the process is essentially anisotropic, even in the range of scales where isotropization of velocity statistics already occurred. Possible implications and connections of the above findings for turbophoresis in wall bounded shear flows will be briefly outlined using DNS data of particle laden turbulent pipe flows as example.
17:10 to 17:40 D Thomson (Met Office)
The behaviour of particle pairs in kinematic simulations
Chair: B Sawford
The way pairs of particles separate is an important aspect of turbulent mixing which has often been explored using the technique of kinematic simulation. However kinematic simulation is not like real turbulnce in that the Fourier modes are independent and the smaller eddies are not advected (or 'swept') by the large eddies. Our aim here is to explore this aspect of kinematic simulation both theoretically and numerically. The fact that the small eddies are not swept by the large eddies, but the particles in the flow are so swept, means that particle pairs are swept through the smaller eddies by the large eddies. This is expected to alter the time scale on which the relative velocity of the particles fluctuates. A simple argument then shows that the mean square separation of pairs is expected to grow, not as t cubed as expected following Richardson, but as t to the sixth power. This is confirmed in numerical simulations where we add a mean flow to the kinematic flow field to exaggerate the problem caused by lack of sweeping (with the eddies not being advected by the mean flow). Without the mean flow the situation is more complex with a significant contribution to the separation process from locations where the velocity is small and where there is no sweeping issue. This leads to a separation growing like t to the power 9/2. The time dependence of the kinematic flow field can also lead to a wider range of behaviours. The work described here is not especially new (we published the main idea in 2005) but it remains controversial and we hope the talk will generate some discussion of the ideas involved.
17:40 to 18:10 R Rubinstein ([NASA Langley])
Closure theories for inhomogeneous turbulence
Although Kraichnan formulated the Direct Interaction Approximation and the Test-Field model for general problems of inhomogeneous turbulence, the resulting equations, requiring repeated multiple integrations over the flow domain, are both difficult to understand and difficult to apply in practice; in the homogeneous case, triad interactions provide the key to unraveling the physics of the approximation. The goal of this work is to formulate the closure theory of some special inhomogeneous problems with comparable simplicity. It is done by decomposing the inhomogeneous problem into a set of coupled quasi-homogeneous problems, each of which admits a simple formulation. The formalism will be applied to the problem of weakly inhomogeneous turbulence, where previous heuristic theories have been incomplete. The same formalism applies to problems admitting scaling transformations; it will be applied to give a simple formulation of the problem of turbulence in a half-space.
Friday 3rd October 2008
09:40 to 10:10 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements
Chair: JM Chomaz
Several existing sets of smaller-scale ocean and atmospheric data appear to display Kolmogorov-Obukov-Corrsin inertial ranges in horizontal spectra for length scales up to at least a few hundred meters. It is argued here that these data are inconsistent with the assumptions for these inertial range theories. Instead, it is hypothesized that the dynamics of stratified turbulence explain these data. If valid, these dynamics may also explain the behavior of strongly stratified flows in similar dynamic ranges of other geophysical flows.
10:10 to 10:40 Vertical dispersion by stratified turbulence
An analytical relation is derived for the growth of the vertical mean square displacement of fluid particles in stratified turbulence. A number of numerical simulations are carried out to test the analytical relation. The comparison shows good agreement between the analytical and the numerical results.
10:40 to 11:00 Divergent-rotational modes and passive scalars in stratified turbulence
Strongly stratified turbulent flows are anisotropic but have three-dimensional dynamics with a forward energy cascade as shown by Lindborg (2006). Simulations with hyperviscosity also revealed an inertial range with a horizontal k^-5/3-spectrum. We have continued this work and examined the features of divergent and rotational modes, and of passive scalars in the inertial range of stratified turbulence.
The Helmholtz decomposition of the velocity into a rotational and divergent part have been used to study vortices and internal waves in stratified flows. In flows with mainly vertically oriented vortices the rotational part dominates while the divergent part dominates when there are mainly internal waves. The timescale ratio of the 'fast' waves and the 'slow' horizontal vortical motions can be estimated as the vertical Froude number F_v = u/Nl_v, where u is a horizontal velocity scale, l_v a verti cal length scale, N the Brunt-Vaisala frequency. It is often assumed that F_v = 0 in strongly stratified flows suggesting separate time scales of vortices and waves, and consequently weak interactions. However, scaling analysis suggests l_v = u/N in stratified turbulence, i.e. F_v = 1, which was supported by DNS (Brethouwer et al. 2007). In stratified turbulence the divergent and rotational modes may thus have similar timescales leading to strong nonlinear interactions between rotational and dive rgent modes. This implies that forcing in either divergent or rotational modes may have minor effects on the inertial range dynamics of stratified turbulence. Results of simulations with different stratification and resolution show that the energy of divergent and rotational modes hav e the same magnitude in the inertial range when large-scale rotational modes are forced, see Lindborg & Brethouwer (2007). In the simulations with forcing of divergent modes, small horizontal wave numbers and one vertical wave number are forced which introduces a vertical length scale. Results show that the inertial range spectra are very similar in simulations with forcing of rotational modes and divergent modes, suggesting similar dynamics, if in the latter simulations the large-scale dynamics o beys F_v = 1, but deviations are found when this condition is not fulfilled. The next subject is a passive scalar in stratified turbulence. The Obukhov-Corrsin theory for the one-dimensional spectrum of the variance of a passive scalar in the Kolmogorov inertial range of turbulence predicts a k^-5/3 slope. The second-order structure function according to the same theory has a r^2/3 power-law range. Measurements of horizontal spectra and structure functions of passive scalars (e.g. ozone) in the mesoscale range of the middle atmosphere are consistent with this theory. This is remarkable because the mesoscale range is strongly stratified and does not resemble Kolmogorov turbulence. We have carried out numerical simulations of stratified turbulence with large-scale forcing, hyperviscosity
11:30 to 12:00 Transition in energy spectrum for forced stratified turbulence
Chair: J Riley
Energy spectrum for forced stably stratified turbulence is investigated numerically by solving the 3D Navier-Stokes equations under the Boussinesq approximation with stochastic forcing applied to the largest velocity scales. Using pseudo-spectral simulations with 1024^3 grid points, we could verify the transition in the vortex (horizontal) spectrum (as a function of horizontal wave number) from $k_{\perp}^{-3}$ to $k_{\perp}^{-5/3}$. Meanwhile the wave spectra shows $k_{\perp}^{-2}$ for the large scales, and $k_{\perp}^{-5/3}$ for the small scales. According to Carnevale {\it et.~al.}, the transition wave number is understood as the Ozmidov scale with a correction by the coefficients of the buoyancy spectrum, $E(k) =\alpha N^2k^{-3}$, and the Kolmogorov spectrum, $E(k)=C_K\epsilon^{2/3} k^{-5/3}$. By equating these spectra, $k_b \sim (\alpha/C_K)^{3/4}\sqrt {N^3/ \epsilon}$ is obtained for the transition wavenumber. Our calculation shows, however, that the vortex spectra at large scales seem to have the same slope irrespective of stratification, which implies a possibility of a different mechanism for producing the $k_{\perp}^{-3}$ spectrum. We will discuss possibility that the spectrum corresponds to two-dimensional turbulence. Referece: Carnevale,G.F. {\it et.~al}: 2001 J.~Fluid Mech. {\bf 427} 205--239.
12:00 to 12:30 Non-Oberbeck-Boussinesq effects in Rayleigh-Benard convection
The problem of Rayleigh-Benard convection is commonly analyzed within the so-called Oberbeck-Boussinesq (OB) approximation, in which the fluid properties are assumed to be temperature independent, apart from the density for which a linear temperature dependence is assumed. Under normal conditions, i.e., small temperature differences between the bottom and top plate, this approximation is rather good. However, in order to achieve ever larger Rayleigh numbers for given cell height and fluid properties the temperature difference is quite frequently increased to such an extent that the OB approximation has to be expected to fail. Non-Oberbeck-Boussinesq (NOB) effects on the mean center cell temperature, the Nusselt number Nu, and the Reynolds number Re then have to be expected at the largest Rayleigh numbers. We report on our recent experimental, theoretical, and numerical results on these NOB corrections. For water and glycerol they are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient: With increasing NOBness, for water and glycerol Nu goes down and the center temperature goes up, whereas for ethane gas in general Nu goes up and the center temperature goes down. However, for ethane close to the critical point the main origin of NOB corrections lies in the strong temperature dependence of the isobaric thermal expansion coefficient, namely in the nonlinear temperature dependence of the density, leading to NOB corrections which presently cannot be described by our extended Prandtl-Blasius boundary layer theory. Related Links * http://pof.tnw.utwente.nl/ - Web page Phyiscs of Fluids group Twente
14:00 to 14:30 Y Zhou ([LLNL])
Scaling criteria for high Reynolds and Peclet number turbulent flow, scalar transport, mixing, and heat transfer
Chair: C Cambon
Very high Reynolds (Re) and Peclet (Pe) number turbulent flows are commonly encountered in engineering, geophysical and astrophysical applications. In comprehensive statistical flow experiments or corresponding direct numerical simulations of high Re and Pe number turbulent flow, scalar transport, mixing, and heat transfer the energetic excitation influences of the entire range of dynamic spatial scales combining both velocity fluctuations and passive scalar variances must be considered together. However, direct computational simulations or experiments directed to the very high Re and Pe flows of practical interest commonly exceed the resolution possible using current or even foreseeable future super computer capability or spatial, temporal and diagnostic technique limitations of current laboratory facilities. Pragmatic considerations and practical needs promote use of statistical flow data bases developed from direct numerical simulations or experiments at the highest Re and Pe levels achievable within the currently available facility limitations. Unfortunately the obtainable levels are lower than those associated with the flows of practical interest. Moreover, at present, there is no metric to indicate whether and how much of the fully resolved physics of the flow of interest has been captured within the facilities available to the investigator. This talk presents metric criteria based on establishing a smaller subset of the total range of dynamic scale interactions that will still faithfully reproduce all of the essential, theoretically significant, influences of the complete range of scale interactions associated with the flows of practical interest. The present work leads to the identification of the minimum significant Re flow and Pe field that a researcher must attain in direct simulation or experiment (hereafter called the minimum state). These threshold criteria levels are minimum values to be attained in experiments or direct simulations which assure that the energy-containing scales of the flows ? and scalar fields under investigation are not contaminated by the (non-universal) velocity dissipation and scalar diffusivity inertial range scale limits.
14:30 to 15:00 Numerical study of 3D Rayleigh-Taylor turbulence
The Rayleigh-Taylor turbulence in 3D space is numerically studied via the Boussinesq approximation along the same line as the 2D numerical study by Celani et al. (2006 Phys.Rev.Lett. 96 134504). A comparison with the phenomenology proposed by Chertkov (2003 Phys.Rev.Lett. 91 115001), in particular deviation from the phenomenology (intermittency), will be presented.
15:00 to 15:30 Tea
15:30 to 16:00 Some remarks on the dissipative properties of homogenous and isotropic turbulence
Chair: E Lindborg
Dissipation of kinetic energy plays a key role in understanding the statistical properties of turbulence. In this talk, we shall review some recent results obtained by performing high resolution numerical simulations of homogenous and isotropic turbulence. In particular, we present an exhaustive investigation of the statistics of velocity gradients along the trajectories of neutral tracers and of heavy/light particles advected by an homogeneous and isotropic turbulent flow. We propose a Lagrangian rephrasing of the Refined Kolmogorov Similarity Hypothesis (RKSH) and test its validity along the particle trajectories. We also show that for homogenous and isotropic compressible turbulence, there is no statistical differences in the statistical properties of inertial range intermittency due either to the slight compressibility or to the different dissipative mechanism.
16:00 to 16:30 CF Barenghi ([Newcastle])
Reconnection of superfluid vortex bundles
Using the vortex filament model and the Gross Pitaevskii nonlinear Schroedinger equation, we show that bundles of quantised vortex lines in helium~II are structurally robust and can reconnect with each other maintaining their identity. We discuss vortex stretching in superfluid turbulence and show that, during the bundle reconnection process, a large amount of Kelvin waves is generated, in agreement with the finding that helicity is produced by nearly singular vortex interactions in classical Euler flows.
16:30 to 16:35 Closing Remarks
17:40 to 18:10 Scale-invariance in three-dimensional isotropic turbulence
We present a critical review of the Kolmogorov (1941) theory of isotropic turbulence, with particular reference to the `2/3' power-law for the second-order structure function (and the corresponding `-5/3' law for the energy spectrum). We begin by noting that the recent resolution of an associated paradox allows the inertial range to be defined in terms of the scale-invariance of the energy flux (David McComb, J. Phys. A: Math. Theor. 41, 075501 (2008)), thus permitting the Kolmogorov arguments to be presented independently of concepts such as localness which are themselves counter-intuitive when interpreted in terms of vortex-stretching. If this approach is pursued further, then a simple phenomenological analysis shows that we can regard turbulence as a statistical field theory possessing one nontrivial fixed point (corresponding to the top of the inertial range) and two trivial fixed points at the origin and infinity, respectively, in wavenumber space. Using this framework, various schools of criticism, ranging from the original criticism by Landau (1959), through `intermittency corrections' to present-day analogies with the theory of critical phenomena, with the introduction of `anomalous exponents', are analysed. In particular, we examine the conflict between the recent work of Lundgren (2002), which uses mathematical arguments to show that the `2/3' law must be asymptotically true in the limit of infinite Reynolds numbers; and that of Falcovich which uses mathematical arguments to show that the `2/3' law is incompatible with the observed values for higher-order moments. We conclude by attempting to put forward a picture in which various long-standing contentious issues may be seen as either resolved or at least potentially resolvable.
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Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs
June 2017, 7(2): 305-345. doi: 10.3934/mcrf.2017011
Exact controllability of linear stochastic differential equations and related problems
Yanqing Wang 1, , Donghui Yang 2, , Jiongmin Yong 3, and Zhiyong Yu 4,,
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
School of Mathematics and Statistics, School of Information Science and Engineering, Central South University, Changsha 410075, China
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
School of Mathematics, Shandong University, Jinan 250100, China
∗ Corresponding author: Zhiyong Yu.
Received January 2017 Published April 2017
Fund Project: the National Natural Science Foundation of China (11471192, 11371375, 11526167), the Fundamental Research Funds for the Central Universities (SWU113038, XDJK2014C076), the Nature Science Foundation of Shandong Province (JQ201401), the Natural Science Foundation of CQCSTC (2015jcyjA00017), China Postdoctoral Science Foundation and Central South University Postdoctoral Science Foundation, and NSF Grant DMS-1406776.
A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.
Keywords: Controlled stochastic differential equation, $L^p$-exact controllability, observability inequality, norm optimal control problem.
Mathematics Subject Classification: Primary: 93B05, 93E20; Secondary: 60H10.
Citation: Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011
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Yanqing Wang Donghui Yang Jiongmin Yong Zhiyong Yu
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\begin{definition}[Definition:Quasimetric]
A '''quasimetric''' on a set $X$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:
{{begin-axiom}}
{{axiom | n = \text M 1
| q = \forall x \in A
| m = \map d {x, x} = 0
}}
{{axiom | n = \text M 2
| q = \forall x, y, z \in A
| m = \map d {x, y} + \map d {y, z} \ge \map d {x, z}
}}
{{axiom | n = \text M 4
| q = \forall x, y \in A
| m = x \ne y \implies \map d {x, y} > 0
}}
{{end-axiom}}
Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset.
The difference between a '''quasimetric''' and a metric is that a '''quasimetric''' does not insist that the distance function between distinct elements is commutative, that is, that $\map d {x, y} = \map d {y, x}$.
\end{definition}
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The critical exponent for fast diffusion equation with nonlocal source
Chunxiao Yang ORCID: orcid.org/0000-0002-0203-47441,
Linghua Kong2,
Yingxue Wu1 &
Qing Tian1
This paper considers the Cauchy problem for fast diffusion equation with nonlocal source \(u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}\), which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\), namely, any solution of the problem blows up in finite time whenever \(1< p\le p_{c}\), and there are both global and non-global solutions if \(p>p_{c}\).
In this paper, we study the following Cauchy problem of fast diffusion parabolic equation with a nonlinear nonlocal source:
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}, &(x,t)\in\mathbb{R}^{n}\times(0,T), \\ u(x,0)=u_{0}(x), & x\in \mathbb{R}^{n}, \end{array}\displaystyle \right . \end{aligned}$$
where the spatial dimension \(n\ge1\), the coefficients m, p, q, r satisfy \(\max\{0,1-\frac{2}{n}+r\}< m<1\), \(p> 1\), \(q\ge1\), \(0\le r<\frac{2}{n}\), and the initial data \(u_{0}(x)\) is a nontrivial nonnegative continuous function.
The quasilinear parabolic equations involving a nonlocal term originate in the phenomena of diffusion about concentration of some Newtonian fluids or the density of some biological species and heat transfer in a special medium with nonlocal source (see [2, 3] and the references therein). In the past three decades, various nonlocal mathematical models were established to describe many physical phenomena (see [1, 4,5,6,7,8,9] and references therein). At the same time, many important results have appeared on the blow-up problem for a nonlinear parabolic equation with nonlocal source (see [2, 6, 8,9,10,11] and references therein), and for nonlocal nonlinear diffusion equations [12, 13]. However, most of efforts have been devoted in bounded domains, there were few researches for the Cauchy problems (see [1, 14, 15]).
It is well known that the classical Cauchy problem
$$\begin{aligned} u_{t}=\Delta u+u^{p} \quad\text{in } \mathbb{R}^{n}\times(0,T) \end{aligned}$$
possesses the critical exponent \(1+\frac{2}{n}\) [16,17,18,19], that is to say, any nontrivial solution blows up in finite time if \(1< p\le1+\frac {2}{n}\), whereas global and non-global solutions coexist if \(p>1+\frac{2}{n}\), depending on the size of initial data. From then on, the Fujita phenomenon has been observed for many nonlinear PDEs (see surveys [20, 21] and references therein).
The study for the Cauchy problem of nonlocal nonlinear parabolic equation was proposed by Galaktionov et al. [1], in which it was proved that the Cauchy problem (1.1) with \(m=1\) has a critical Fujita exponent, and Wang et al. [15] obtained similar results by other methods. Recently, Zhou [14] considered the global and non-global existence of solutions for (1.1) with \(m>1\).
The present paper investigates a fast diffusion parabolic equation (1.1) (\(\max\{0,1-\frac{2}{n}\}< m<1\)) with a nonlocal source, and establishes the critical Fujita exponent \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\). Comparing with the known result for the parallel problem with a local source
$$\begin{aligned} u_{t}=\Delta u^{m}+u^{p} \quad \mbox{in } \mathbb{R}^{n}\times(0,T), \end{aligned}$$
the critical Fujita exponent was obtained in [22, 23] and shown to be \(p_{c}=1+\frac{2m}{n}\).
In the rest of the paper, we always let u be a solution to (1.1), and \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\). The main results are stated in the following theorems.
For \(1< p\leq p_{c}\), there are no global nontrivial solutions to (1.1).
For \(p>p_{c}\), there are both global and non-global solutions to (1.1).
This paper is organized as follows. Section 2 concerns the non-global solution to prove Theorem 1.1. Section 3 deals with the global existence to prove Theorem 1.2. And Sect. 4 shows in what ways the parameter q of the nonlocal source affects the behavior of solutions in the fast diffusion problem (1.1).
Non-global solutions
This section mainly applies the test function method (refer to [15, 22]) to prove that any solution of (1.1) must blow up in finite time for \(1< p\le p_{c}\). Introducing the test function
$$\begin{aligned} \varphi_{k}(x)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}}\mathrm{e}^{-k|x|^{2}} \end{aligned}$$
for some \(k>0\), we can simply verify that
$$\begin{aligned} \int_{\mathbb{R}^{n}}\varphi_{k}(x)\,dx=1,\qquad \bigl\Vert \varphi_{k}(x) \bigr\Vert _{L^{\infty}}= \biggl(\frac{k}{\pi} \biggr)^{\frac {n}{2}},\qquad \Delta\varphi_{k}(x)\ge-2kn \varphi_{k}(x). \end{aligned}$$
$$\begin{aligned} F(t)= \int_{\mathbb{R}^{n}}u(x,t)\varphi_{k}(x)\,dx. \end{aligned}$$
It is sufficient to show that \(F(t)\) blows up in finite time as \(1< p\le p_{c}\) to deal with Theorem 1.1.
Proof of Theorem 1.1
Firstly, we consider the case of \(1< p< p_{c}\). Multiplying equation (1.1) by \(\varphi_{k}(x)\) and integrating by parts in \(\mathbb{R}^{n}\), we get
$$\begin{aligned} F'(t)={}& \int_{\mathbb{R}^{n}}u_{t}\varphi_{k}\,dx \\ ={}& \int_{\mathbb{R}^{n}}\Delta u^{m}\varphi_{k}\,dx+ \biggl( \int_{\mathbb {R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}} u^{r+1}\varphi _{k}\,dx \\ \ge{}& {-}2kn \int_{\mathbb{R}^{n}} u^{m}\varphi_{k}\,dx+ \Vert \varphi_{k} \Vert _{L^{\infty}}^{-\frac{p-1}{q}} \biggl( \int_{\mathbb{R}^{n}}u^{q}\varphi_{k}\,dx \biggr)^{\frac {p-1}{q}} \int_{\mathbb{R}^{n}} u^{r+1}\varphi_{k}\,dx. \end{aligned}$$
Using Jensen's inequality for \(m<1\), \(q>1\), and \(r>0\),
$$\begin{aligned} F'(t)\ge {}&{-}2knF^{m}(t)+ \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}F^{p+r}(t) \\ ={}&F^{p+r}(t) \biggl( \biggl(\frac{k}{\pi} \biggr)^{-\frac {n(p-1)}{2q}}-2knF^{-(p+r-m)}(t) \biggr). \end{aligned}$$
$$\begin{aligned} F(t)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}}, \end{aligned}$$
we obtain
$$\begin{aligned} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}>4knF^{-(p+r-m)}(t), \end{aligned}$$
$$\begin{aligned} F'(t)\ge \frac{1}{2} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}F^{p+r}(t). \end{aligned}$$
$$F(t)\geq \biggl(F^{-(p+r-1)}(0)-\frac{p+r-1}{2} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}t \biggr)^{-\frac{1}{p+r-1}}. $$
Obviously, \(F(t)\) blows up for any nonnegative initial data as \(t\to T=\frac{2F^{-(p+r-1)}(0)}{p+r-1} (\frac{k}{\pi} )^{\frac{n(p-1)}{2q}}\). In the following, we show that
$$\begin{aligned} F(0)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}} \end{aligned}$$
is a sufficient condition to prove condition (2.2). If not, there exists some τ, such that
$$\begin{aligned} F(\tau)= \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}}, \end{aligned}$$
$$\begin{aligned} F(t)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}},\quad t\in[0,\tau). \end{aligned}$$
This implies \(F'(\tau_{0})<0\) for some \(\tau_{0}\in(0,\tau)\), which contradicts \(F'(t)\ge0\), \(t\in(0,\tau)\). Thereby, to prove that a solution of (1.1) blows up in finite time, we only show (2.4) is true for any nonnegative nontrivial initial data \(u_{0}(x)\). Since \(\frac{n}{2}<\frac{2q+n(p-1)}{2q(p+r-m)}\) which was derived by \(1< p< p_{c}\), there exists a \(k>0\) small enough, such that
$$F(0)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}\mathrm {e}^{-k|x|^{2}}u_{0}(x)\,dx> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}. $$
Next, we consider the case of \(p=p_{c}\). Supposing a solution of (1.1) is global for any \(t\ge0\), it holds that
$$\begin{aligned} F(t)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}{\mathrm{ e}}^{-k|x|^{2}}u(x,t)\,dx\le \bigl( \pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}. \end{aligned}$$
That is, if (2.5) is not true, namely \(F(t_{1})> (\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} )^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}\) for some \(t_{1}>0\), then the solution \(u(x,t)\) must blow up in finite time by the above proof. The condition \(p=p_{c}\) means \(\frac{n}{2}=\frac{2q+n(p-1)}{2q(p+r-m)}\), and (2.5) can be rewritten as
$$\begin{aligned} \int_{\mathbb{R}^{n}}{\mathrm{ e}}^{-k|x|^{2}}u(x,t)\,dx\le\pi^{\frac{n}{2}} \bigl(\pi^{\frac {n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}\quad \text{for } t>0. \end{aligned}$$
Without loss of generality, assuming \(u_{0}(x)\) has compact support in \(\mathbb{R}^{n}\), we get that \(u(x,t)\in L(\mathbb{R}^{n})\) for any fixed \(t>0\) (see [24]). By Lebesgue Dominated Convergence Theorem, as \(k\to0\) in (2.6),
$$\begin{aligned} \int_{\mathbb{R}^{n}}u(x,t)\,dx\le\pi^{\frac{n}{2}} \bigl( \pi^{\frac {n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}. \end{aligned}$$
Integrating equation (1.1) on \(\mathbb{R}^{n}\times[0,t]\), we have
$$\begin{aligned} \int_{\mathbb{R}^{n}}u(x,t)\,dx- \int_{\mathbb{R}^{n}}u_{0}(x)\,dx= \int_{0}^{t} \biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb {R}^{n}}u^{r+1}\,dx\,dt. \end{aligned}$$
$$\begin{aligned} \int_{0}^{t} \biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int _{\mathbb{R}^{n}}u^{r+1}\,dx\,dt\le \int_{\mathbb{R}^{n}}u(x,t)\,dt\le \pi^{\frac{n}{2}} \bigl( \pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}} \end{aligned}$$
as \(u_{0}(x,t)\ge0\). This implies that
$$\begin{aligned} \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int _{\mathbb{R}^{n}}u^{r+1}\,dx\,dt< +\infty. \end{aligned}$$
On the other hand, from [25] we know that there exists \(\delta>0\) such that the solution of (1.1) satisfies
$$\begin{aligned} u(x,\tau)>\delta \bigl(1+B \vert x \vert ^{2} \bigr)^{-\frac{1}{1-m}} \end{aligned}$$
for \(B=\frac{(1-m)\alpha\delta^{1-m}}{2mn}\) and some \(\tau>0\). Setting
$$\underline{u}(x,t)=\delta(1+t)^{-\alpha} \bigl(1+B \vert x \vert ^{2}(1+t)^{-\frac {2\alpha}{n}} \bigr)^{-\frac{1}{1-m}} $$
with \(\alpha=\frac{n}{2-n(1-m)}\), it is simple to verify
$$\begin{aligned} \underline{u}(x,t)\le u(x,t+\tau) \quad\text{for } x\in \mathbb{R}^{n}, t>0. \end{aligned}$$
$$\begin{aligned}& \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx \biggr)^{\frac {p-1}{q}} \int_{\mathbb{R}^{n}}u^{r+1}(x,t)\,dx\,dt \\& \quad\ge \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}(x,t+\tau)\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}u^{r+1}(x,t+\tau)\,dx\,dt \\& \quad\ge \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}\underline{u}^{q}(x,t)\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}\underline{u}^{r+1}(x,t)\,dx\,dt \\& \quad= B^{-\frac{p+q-1}{2q}}\delta^{p+r} \biggl( \int_{\mathbb{R}^{n}}\bigl(1+ \vert \xi \vert ^{2} \bigr)^{-\frac{q}{1-m}}\,d\xi \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}\bigl(1+ \vert \xi \vert ^{2} \bigr)^{-\frac{r+1}{1-m}}\,d\xi \int_{0}^{\infty}(1+t)^{-1}\,dt \\& \quad=+\infty, \end{aligned}$$
since \(-\alpha(p+r-1)+\frac{\alpha(p-1)}{q}=-1\) for \(p=p_{c}\) and \(\xi=\sqrt{B}x(1+t)^{-\frac{\alpha}{n}}\). This contradicts (2.8), and so our assumption that the solution of (1.1) globally exist for \(t>0\) is not true, which proves Theorem 1.1 with \(p=p_{c}\). □
Coexistence of global and non-global solutions
This section mainly deals with the global solution for the case of \(p>p_{c}\) to derive Theorem 1.2.
Firstly, we show that the solution of (1.1) must blow up in finite time for large initial data \(u_{0}(x)\). The proof of Theorem 1.1 means that \(u(x,t)\) does not exist globally, provided \(u_{0}\) satisfies
$$\begin{aligned} \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}\mathrm{e}^{-k|x|^{2}} u_{0}(x)\,dx> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p+1)}{2q(p+r-m)}}. \end{aligned}$$
For any fixed \(k=k_{0}>0\), we can choose large \(u_{0}(x)\) to fulfil condition (3.1).
Next, we prove that the solution of (1.1) exists globally for any small initial data \(u_{0}(x)\). Let
$$\bar{u}=(t+1)^{-\beta} \bigl(D_{1}+D_{2} \vert x \vert ^{2}(t+1)^{-\beta(1-m)-1} \bigr)^{-\frac{1}{1-m}}, $$
where \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\), and \(D_{1},D_{2}>0\) are to be determined. We demonstrate that ū is a global supersolution of (1.1) for suitable \(D_{1}\) and \(D_{2}\). Setting
$$Z=D_{1}+D_{2} \vert x \vert ^{2}(t+1)^{-\beta(1-m)-1}=:D_{1}+D_{2}z, $$
with \(z=|x|^{2}(t+1)^{-\beta(1-m)-1}\), we have
$$\begin{aligned}& \bar{u}_{t}-\Delta \bar{u}^{m}- \biggl( \int_{\mathbb{R}^{n}}\bar{u}^{q}\,dx \biggr)^{\frac {p-1}{q}}\bar{u}^{r+1} \\& \quad=(t+1)^{-\beta-1}Z^{-\frac{1}{1-m}-1} \biggl[-\beta Z+\frac{D_{2}(\beta-\beta m+1)}{1-m}z+ \frac{2mD_{2}n}{1-m}Z- \frac{4mD_{2}^{2}}{(1-m)^{2}}z \\& \qquad{}-(t+1)^{-\beta r+1} \biggl( \int_{\mathbb{R}^{n}}(t+1)^{-\beta q} \bigl(D_{1}+D_{2} \vert y \vert ^{2}(t+1)^{-1-\beta(1-m)} \bigr)^{-\frac{q}{1-m}}\,dy \biggr) ^{\frac{p-1}{q}}Z^{-\frac{r}{1-m}+1} \biggr] \\& \quad=:(t+1)^{-\beta-1}Z^{-\frac{1}{1-m}-1}G(Z). \end{aligned}$$
For \(\max\{0,1-\frac{2}{n}+r\}< m<1\), \(q\ge1\), \(r\ge0\) implying \(\frac{2q}{1-m}\ge\frac{2}{1-m}>n\), there exists a constant \(C>0\) such that
$$\begin{aligned}& \int_{\mathbb{R}^{n}}(t+1)^{-\beta q} \bigl(D_{1}+D_{2} \vert y \vert ^{2}(t+1)^{-1-\beta(1-m)} \bigr)^{-\frac{q}{1-m}}\,dy \\& \quad= \int_{\mathbb{R}^{n}}(t+1)^{-\beta q+\frac{n+n\beta(1-m)}{2}} \bigl(D_{1}+D_{2} \vert w \vert ^{2} \bigr)^{-\frac{q}{1-m}}\,dw \\& \quad\le C(t+1)^{-\beta q+\frac{n+n\beta(1-m)}{2}}. \end{aligned}$$
Substituting the above inequity into the expression of \(G(Z)\) in (3.2), and using \(D_{2}z=Z-D_{1}\), \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\), we have
$$\begin{aligned} G(Z)\ge{}&{-}\beta Z+\frac{D_{2}(\beta-\beta m+1)}{1-m}z+\frac{2mD_{2}n}{1-m}Z- \frac{4mD_{2}^{2}}{(1-m)^{2}}z \\ &-C(t+1)^{-\beta(p+r-1)+\frac{n+n\beta(1-m)}{2q}(p-1)+1}Z^{-\frac {r}{1-m}+1} \\ ={}& \biggl(-\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}-\frac {4mD_{2}}{(1-m)^{2}} \biggr)Z \\ &- \biggl(\frac{\beta-\beta m+1}{1-m} -\frac{4mD_{2}}{(1-m)^{2}} \biggr)D_{1}-CZ^{-\frac{r}{1-m}+1} \\ =:{}&F(Z). \end{aligned}$$
To describe \(F(Z)\ge0\) for some \(D_{1}\) and \(D_{2}\), we have to show (i) \(F(D_{1})\ge0\) and (ii) \(F'(Z)\ge0\) for \(Z\ge D_{1}\).
(i) \(F(D_{1})= (-\beta+\frac{2mDn}{1-m} )D_{1}-CD_{1}^{-\frac {r}{1-m}+1}\ge 0\) is equivalent to
$$\begin{aligned}& D_{1}^{-\frac{r}{1-m}}\le\frac{1}{C} \biggl(-\beta+ \frac {2mD_{2}n}{1-m} \biggr), \end{aligned}$$
$$\begin{aligned}& D_{2}>\frac{\beta(1-m)}{2mn}. \end{aligned}$$
(ii) By simple computation, \(F'(Z)=-\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}}-C (1-\frac{r}{1-m} )Z^{-\frac{r}{1-m}}\). If \(1-\frac{r}{1-m}\le0\), condition (ii) is ensured by
$$\begin{aligned} -\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}}> 0. \end{aligned}$$
If \(1-\frac{r}{1-m}> 0\), condition (ii) is ensured by (3.6) and
$$\begin{aligned} D_{1}^{-\frac{r}{1-m}}\le\frac{1-m}{C(1-m-r)} \biggl(-\beta+\frac{\beta -\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}} \biggr). \end{aligned}$$
Inequalities (3.5) and (3.6) require
$$\begin{aligned} \frac{\beta(1-m)}{2mn}< D_{2}< \frac{1-m}{2m(2-n(1-m))}. \end{aligned}$$
Due to \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\) and \(p>p_{c}\), we can choose some \(D_{2}>0\) that fulfils (3.8). For such \(D_{2}\), choose \(D_{1}>0\) large enough to satisfy (3.4) and (3.7).
In conclusion, ū is a global supersolution to problem (1.1) with small initial data \(u_{0}(x)\le\bar{u}(x,0)=(D_{1}+D_{2}|x|^{2})^{-\frac{1}{1-m}}\). □
This paper shows that the model (1.1) possesses critical Fujita exponent \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\) in Theorems 1.1 and 1.2, and we find that the coefficient q of the nonlocal term affects the critical Fujita exponent. It's easy to see that \(p_{c}\) is decreasing in q with \(\lim_{q\to\infty}p_{c}=m+\frac{2}{n}-r\) and \(\lim_{q\to1}p_{c}=\infty\). That is to say, the scope \(1< p\leq p_{c}\) for the blow-up of any nontrivial solutions will be enlarged as q is decreasing, and any nontrivial solution of (1.1) will blow up when \(p>1\) and \(q=1\). Refer to Fig. 1.
Critical Fujita exponent curve in q–p plane
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 11501438, 11501076), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2018JM1035), and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (No. 20170607).
School of Science, Xi'an University of Architecture and Technology, Xi'an, P.R. China
Chunxiao Yang
, Yingxue Wu
& Qing Tian
School of Science, Dalian Ocean University, Dalian, P.R. China
Linghua Kong
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All authors contributed equally to this work. They all read and approved the final version of the manuscript.
Correspondence to Chunxiao Yang.
Yang, C., Kong, L., Wu, Y. et al. The critical exponent for fast diffusion equation with nonlocal source. Bound Value Probl 2019, 164 (2019) doi:10.1186/s13661-019-1282-1
Critical exponents
Fast diffusion
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CommonCrawl
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\begin{document}
\title{Looking for symmetric Bell inequalities} \author{Jean-Daniel Bancal$^1$, Nicolas Gisin$^1$, Stefano Pironio$^2$ \\[0.5em] {\it $^1$Group of Applied Physics, University of Geneva,} \\ {\it 20 rue de l'Ecole-de-M\'edecine, CH-1211 Geneva 4, Switzerland}\\ {\it $^2$Laboratoire d'Information Quantique, Universit\'e Libre de Bruxelles, Belgium}} \date{\today} \maketitle
\begin{abstract} Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate $238885$ new Bell inequalities and $1085$ new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-Massar-Popescu type. \end{abstract}
\section{Introduction}
Already discovered by Boole in the theory of logic and probabilities as ``conditions of possible experience'' \cite{Pitowsky}, Bell inequalities found a new dimension with the work of John Bell who showed that quantum physics could violate these conditions in some situations, highlighting what is now known as quantum nonlocality \cite{Bell}.
Complete set of Bell inequalities are known only for setups involving small numbers of parties, measurement settings, and measurement outcomes. This may already be sufficient for various applications, such as exhibiting the nonlocality of a noisy quantum state in a real experiment \cite{Aspect}, or establishing the security of a device-independent quantum key distribution protocol \cite{Pironio09,masanes}. But the simplest inequalities are not always optimal. For instance, certain inequalities with large number of measurement settings are much more resistant to the detection inefficiencies than the CHSH inequality \cite{Vertesi09,Brunner}, or are violated by quantum states that do not violate the CHSH inequality \cite{Vertesi,Collins}. This motivates a search for Bell inequalities involving more parties, measurements, or outcomes. But finding all Bell inequalities pertaining to a given experimental setup is in general a hard task \cite{Pitowski,Werner}, and a complete search with current-day techniques is not feasible in most instances.
It is instructive, however, to realize that many useful Bell inequalities, like CHSH \cite{CHSH}, Mermin \cite{Mermin} or CGLMP \cite{CGLMP} to cite just a few, can be written in a form that is invariant under any permutation of the parties. Symmetric inequalities are also attractive because they are likely to be easier to handle. Motivated by this observation, we show here how to exploit a symmetric version of the full Bell polytope to generate all symmetric Bell inequalities. This symmetrized polytope is much easier to handle than the full Bell polytope. In particular, for Bell experiment with binary settings and outcomes, the number of extremal points of the symmetrized polytope only grows polynomially with the number of parties, which is an exponential gain compared to the general situation.
Our method for finding symmetric inequalities is not restricted to Bell inequalities, but can be applied to any set of inequalities characterizing a correlation polytope, like for instance the Svetlichny inequalities, which allow to test for genuinely multipartite nonlocality~\cite{Svetlichny}. We present in the next section our approach from this general perspective. We then apply in Section~3 our method to several examples for which listing all (non-symmetric) inequalities is computationally intractable with present-day techniques. In the Bell scenario with two parties, two settings, and four outcomes, we find in particular facet inequalities that are not of the CGLMP form, answering an open question raised by Gill~\cite{gill}.
\section{General setting}
Let ($n$, $m$, $k$) denote a Bell experiment where $n$ parties can choose one out of $m$ possible measurement settings that each yield one out of $k$ possible outcomes\footnote{More general Bell scenarios with different number of measurement settings $m_i$ and outcomes $k_i$ for each party~$i$ are also possible, but since we will consider situations that are symmetric under permutations of the parties, we choose $m_i=m$, $k_i=k$ for all $i$.}. The statistics of the observed results are described by the joint conditional probability distributions (also called correlations\footnote{Throughout this paper, the term ``correlations" refers to probability distributions of the form \eqref{eq:P}. It should not to be confused with ``correlator" or ``correlation functions", such as $E(s_1,s_2)=p(r_1=r_2|s_1,s_2)-p(r_1\neq r_2|s_1,s_2)$.}) \begin{equation}\label{eq:P}
p(r_1,\ldots,r_n|s_1,\ldots,s_n) \end{equation} where $s_i\in\{1,\ldots,m\}$ denotes the measurement setting of party $i$ and $r_i\in\{1,\ldots,k\}$ denotes the corresponding measurement result.
Note that in general the $N=m^nk^n$ probabilities (\ref{eq:P}) are not all independent but satisfy linear constraints, such as the normalization conditions or the no-signalling conditions. We are thus actually interested in an affine subspace of $\mathbb{R}^N$ of dimension $d$, where $d=m^n(k^n-1)$ for normalized correlations, and $d=(m(k-1)+1)^n-1$ for correlations that satisfy in addition the no-signalling conditions \cite{Collins,liftings}. We suppose in the following that a proper parametrization has been introduced so that the joint distributions (\ref{eq:P}) can be identified with points $p$ in $\mathbb{R}^d$.
We are interested in whether a given $p$ belongs to some special subset $P\subseteq \mathbb{R}^d$. In this work, we consider sets $P$ which are polytopes\footnote{In general, it might also be interesting to consider sets that are not polytopes~\cite{npa,npa2}, nor even convex~\cite{Branciard}.}. A polytope can be described by the list ${V}$ of its vertices (or extremal points) $v\in {V}$, and any point $p\in P$ can be written as \begin{equation} p=\sum_{v\in {V}} \rho_v v\,, \end{equation} where $\rho_v$ are positive and normalized weights: $\rho_v\geq 0$ for all $v$ and $\sum_{v\in {V}} \rho_v = 1$. In the case of the Bell polytope, for instance, the extremal points are the deterministic local strategies, corresponding to the prior assignment of an outcome $r_{i,s_i,v}$ to each measurement setting $s_i$. There are thus $k^{nm}$ different vertices $v$, corresponding to joint probability distributions of the form \begin{equation}\label{eq:localstrategy}
p(r_1,\ldots,r_n|s_1,\ldots,s_n)=\begin{cases} 1&\text{if $r_i=r_{i,s_i,v}$ for all $i=1,\ldots,n$}\\ 0&\text{otherwise}\,. \end{cases} \end{equation}
Alternatively to its representation in term of vertices (the $V$-repesentation), a polytope $P$ can also be described, by the Farkas-Minkowski-Weyl theorem, as the intersection of finitely many half-spaces (the $H$-representation). A half-space is defined by an inequality $h\cdot p=\sum_{i=1}^d h_i\,p_i\leq h_0$ specified by the couple $(h, h_0)\in \mathbb{R}^{d+1}$. We say that an inequality $(h,h_0)$ is valid for the polytope $P$, if $h\cdot p\leq h_0$ for all points $p$ in $P$. The Farkas-Minkowski-Weyl theorem states that a polytope can be characterized by a finite set of valid inequalities. That is, a point $p$ belongs to $P$ if and only~if \begin{equation}\label{hs} h\cdot p\leq h_0\quad\text{ for all } ( h,h_0)\in H\,, \end{equation} where $H$ is some finite set in $\mathbb{R}^{d+1}$. This description is particularly appropriate when willing to show that a point does not belong to the polytope, since it is sufficient to exhibit the violation of a single one of the inequalities (\ref{hs}). In the case of the Bell polytope, these inequalities are known as Bell inequalities.
A complete and minimal representation of a polytope in the form (\ref{hs}) is provided by the set of facets of the polytope. An inequality $(f,f_0)\in H$ defines a facet if its associated hyper-plane $f\cdot p=f_0$ intersects the boundary of the polytope in a set of dimension $d-1$, i.e., if there exists $d$ affinely independent points
of $P$ satisfying $f\cdot p=f_0$. The set $\hat f=\{p\mid p\in P, \,f\cdot p=p_0\}$ then corresponds to the facet defined by $(f,f_0)$.
Finding all the Bell inequalities corresponding to an experimental configuration $(n,m,k)$ thus amounts to determining the facets (minimal H-representation) of a polytope when given its extremal points (V-representation). This conversion problem is well known and there exists several available algorithms to solve it \cite{polytopeSoft}. However, when $n$, $m$, or $k$ are too large, the associated polytope becomes too complex to be handled by these algorithms. It might then be appropriate to focus the search on a subclass of all facet inequalities. We explain how this can be done for symmetric inequalities in the next section.
\subsection{Focusing on symmetric facets}
Some facet inequalities $(f,f_0)$ of a correlation polytope $P$ can be written in a way that is invariant under permutations of the parties. This is the case for instance for the CHSH inequality, which can be written in the CH form as \begin{equation}\label{chsh} p(a_1)+p(b_1)-p(a_1b_1)-p(a_1b_2)-p(a_2b_1)+p(a_2b_2)\geq 0
\end{equation} where we write $p(a,b|x,y)$ for $p(r_1,r_2|s_1,s_2)$\footnote{From now on, we write $p(a,b,c\ldots|x,y,z,\ldots)$ for $p(r_1,r_2,r_3,\ldots|s_1,s_2,s_3,\ldots)$.} and define $p(a_x)=p(a=1|x)$, $p(b_y)=p(b=1|y)$, $p(a_xb_y)=p(a=1,b=1|x,y)$. In the following we will call such facets \emph{symmetric facets}\footnote{Note that a symmetric inequality need not appear in a symmetric form when written in any of its equivalent forms under relabeling of measurement settings and outcomes. For instance : $p(a_1)+p(b_2)-p(a_1b_1)-p(a_1b_2)+p(a_2b_1)-p(a_2b_2)\geq 0$ is equivalent to the CH inequality (\ref{chsh}) up to relabeling of the settings and outcomes, but is not invariant under the exchange of party $1$ and $2$.}. If one is interested in finding only the symmetric Bell inequalities relevant to a given scenario $(n,m,k)$, then it is possible to restrict the space in which to search for them. This is what we show now.
Let $G$ be the set of permutations of $\{1,\ldots,n\}$. Given a permutation $\pi\in G$ let
\begin{equation} p(a_1,\ldots,a_n|x_1,\ldots,x_n)\mapsto p(a_{\pi(1)},\ldots a_{\pi(n)}|x_{\pi(1)},\ldots,x_{\pi(n)})\,. \end{equation} be its action on the joint probability distributions. This permutation induces a transformation $\pi\,:\,\mathbb{R}^d\mapsto\mathbb{R}^d\,:\, p\mapsto \pi p$ in the vector space $\mathbb{R}^d$ in which the correlations $p$ are represented, which by abuse of language we denote by the same symbol~$\pi$. Note that the correlation polytopes that we consider here are evidently invariant under such permutations, i.e., $\pi P=P$, and any vertex $v\in V$ is mapped into another vertex $\pi v\in V$.
Given a facet-inequality $(f,f_0)$ of $P$, let $(\pi f,f_0)$ be its image under $\pi$. Note that with this definition the facet $\hat f=\{p\mid p\in P,\,f\cdot p=f_0\}$ is mapped onto $\pi\hat f=\{\pi p\mid p\in P,\,f\cdot p=f_0\}$ since $\{p\mid p\in P, (\pi f) \cdot p=f_0\}=\{p\mid p\in P, f \cdot (\pi^{\dagger} p)=f_0\}=\{\pi p\mid p\in P,\, f\cdot p=f_0\}$, where we used the fact that the transformations $\pi$ are unitary, i.e., $\pi^\dagger=\pi^{-1}$.
We say that a facet $(f,f_0)$ is \emph{symmetric} if $\pi f =f$ for all $\pi\in G$.
Consider now the symmetrizing map \begin{equation}
\Pi=\frac1{|G|}\sum_{\pi\in G}\pi \end{equation} and let $\tilde \Pi$ denote the projection on the symmetric affine subspace $S$ of $\mathbb{R}^d$ of dimension $d_s=\text{dim}(S)$. We suppose to simplify the presentation that the origin of $\mathbb{R}^d$ is contained in $S$ (this can always be achieved by a proper translation of the correlation vectors $p$), so that $S$ is actually a linear subspace of $\mathbb{R}^d$. An arbitrary vector $p\in \mathbb{R}^d$ can then be written as $p=p_s+ p_{t}$ where $p_s=\tilde \Pi p$ is the projection of $p$ on the symmetric subspace $S$ and $p_t=(1-\tilde \Pi)p$ on the complementary space $T= (1-\tilde \Pi)\mathbb{R}^d$. Similarly an arbitrary inequality $(f,f_0)$ can be written as $(f_s\oplus f_t, f_0)$. A symmetric inequality then takes the form $(f_s\oplus 0,f_0)$ where ${0}$ denotes the null vector in $\mathbb{R}^{d-d_s}$. Given an inequality $(f_s,f_{0})\in \mathbb{R}^{d_s+1}$ defined in the symmetric subspace $S$, we denote its \emph{symmetric extension} as the inequality $(f,f_0)=(f_s\oplus {0},f_{0})$ defined in the full space $\mathbb{R}^d$.
Let $P_s=\tilde \Pi P=\{p_s\mid p\in P\}$ be the projection of the polytope $P$ on the symmetric subspace.
Any vertex $w\in P_s$ is necessarily the projection $w=v_s$ of some vertex $v\in P$. Indeed, suppose that $w$ is the projection of a non-extremal point $p=\sum_iq_iv_i\in P$, $\sum_i q_i=1$, $0<q_i<1$. Then it needs to be the projection of every $v_i$ as well: $w=\tilde\Pi p =\sum_iq_i\tilde\Pi v_i = w$ implies $w=\tilde\Pi v_i$ since $w$ is a vertex. Note, however, that every vertex $v\in P$ does not necessarily induce a vertex $v_s\in P_s$ when projected on the symmetric subspace. The symmetrized polytope $P_s$ has thus in general less vertices than the original polytope. Moreover it is defined in a space of smaller dimension $d_s<d$ than the full space $\mathbb{R}^d$. It is thus in general easier to determine the facets of the symmetrized polytope $P_s$ than those of the full polytope $P$. The following theorem shows that determining the facets of this symmetrized polytope is sufficient to find all symmetric facet inequalities of the full polytope $P$ (see also Figure~1). \begin{theorem*}\label{thm} Let $(f_s,f_{0})$ be a facet inequality for the polytope $P_s\in\mathbb{R}^{d_s}$. Then its symmetric extension $(f_s\oplus 0,f_0)\in\mathbb{R}^{d+1}$ defines a valid inequality for the full polytope $P\in\mathbb{R}^d$. Moreover all symmetric facet inequalities of the full polytope $P$ are the symmetric extension of some facet of the symmetrized polytope $P_s$. \end{theorem*} \begin{proof} The symmetric extension $(f,f_0)=(f_s\oplus 0,f_0)$ is valid for $P$ if $f\cdot p\leq f_0$ is satisfied by all points $p\in P$. But this immediately follows from the fact that $f\cdot p = ( f_s\oplus {0})\cdot (p_s\oplus p_t) = f_s \cdot p_s\leq f_{0}$ and the fact that $f_s\cdot p_s\leq f_0$ is valid for all $p_s\in P_s$.
Now, let $(g,g_0)=(g_s\oplus 0, g_0)$ be a symmetric facet of $P$. Clearly, it is the symmetric extension of the inequality $(g_s,g_0)\in\mathbb{R}^{d_s}$, which is valid for $P_s$. Moreover, $(g_s,g_0)$ defines a facet of $P_s$ as there exist $d_s$ affinely independent points in $P_s$ that saturate it. Indeed, since $(g, g_0)$ defines a facet of $P$, there exists $d$ affinely independent points $p$ in $P$ that satisfy $(g_s\oplus 0)\cdot p=g_0$. These points are of the form $p=p_s\oplus p_t$, where $p_t$ can clearly be arbitrary. Since the complementary space $T$ is of dimension $d-d_s$, there must therefore be at least $d-(d-d_s)=d_s$ affinely independent points of the form $p_s\oplus 0$ that saturate the inequality $(g_s\oplus 0,g_0)$. These points obviously define $d_s$ affinely independent points in $P_s$ that saturate the inequality $(g_s,g_0)$. \end{proof} \begin{figure}
\caption{(a) Example of a polytope $P$ in the vector space $\mathbb{R}^3$. (b) Subspace $S$ symmetric under the exchange of coordinates $e_1$ and $e_2$. $P_s$ (grey) is the projection of the polytope onto this subspace. (c) $f_s$ and $g_s$ are two facets of $P_s$, and $f$ and $g$ are their symmetric extensions to the whole space $\mathbb{R}^3$. $f$ is a symmetric facet of the original polytope $P$, whereas $g$ is just a valid inequality for $P$.}
\label{fig:fig1}
\end{figure} Note however that the converse of the theorem is not true, as illustrated in Figure~\ref{fig:fig1}: facets of $P_s$ do not necessarily extend to facets of the polytope $P$ in the general space $\mathbb{R}^d$. We show in Section 2.3 how it is nevertheless possible to take advantage of such inequalities to generate new (not necessarily symmetric) facet inequalities for the original polytope $P$.
\subsection{Illustration on the $(2,2,2)$ Bell scenario}
We now illustrate in detail the above approach on the $(2,2,2)$ Bell scenario. This scenario is characterized by 16 probabilities $p(ab|xy)$, where $x\in\{1,2\}$ denote the measurement setting of Alice and $a\in\{1,2\}$ the corresponding outcome, and where similarly $y\in\{1,2\}$ and $b\in\{1,2\}$ denote Bob's measurement setting and outcome. These probabilities satisfy normalization \begin{equation}
\sum_{a,b=1,2} p(ab|xy)=1\quad\text{ for all } x,y=1,2, \end{equation} and no-signalling \begin{eqnarray}
&&p(a|x)\equiv\sum_{b=1,2} p(ab|xy)\quad \text{ for all } a,x,y=1,2\nonumber\\
&&p(b|y)\equiv\sum_{a=1,2} p(ab|xy)\quad \text{ for all } b,x,y=1,2\,. \end{eqnarray}
In total, only $8$ of the 16 probabilities $p(ab|xy)$ are therefore independent and we can represent the correlations $p$ as elements of $\mathbb{R}^8$. For specificity, we choose the following parametrization \begin{equation}\label{eq:nosigparam} p=\left[p(a_1),p(a_2),p(b_1),p(b_2),p(a_1b_1),p(a_1b_2),p(a_2b_1),p(a_2b_2)\right]\,, \end{equation}
where $p(a_x)=p(1|x)$, $p(b_y)=p(1|y)$ and $p(a_xb_y)=p(11|xy)$.
The Bell-local polytope is described by 16 vertices \begin{equation}\label{det222} v_{a_1a_2b_1b_2}=\left[a_1,a_2,b_1,b_2,a_1b_1,a_1b_2,a_2b_1,a_2b_2\right]\,, \end{equation} where $a_1,a_2,b_1,b_2\in\{1,2\}$ specifies the deterministic assignment of an outcome to each measurement setting.
The group $G$ of permutation of two parties contains 2 elements: the identity $\openone$ and the permutation $\pi$ acting as follows on a vector $p$, \begin{equation} \pi p=\left[p(b_1),p(b_2),p(a_1),p(a_2),p(a_1b_1),p(a_2b_1),p(a_1b_1),p(a_2b_2)\right]\,. \end{equation} The symmetrizing map projecting on the space $S$ of symmetric correlations is thus equal to $\Pi=\frac12(\openone+\pi)$, while the map projecting on the complementary space $T$ is $\openone-\Pi=\frac12(\openone-\pi)$. Arbitrary correlations $p$ can thus be decomposed into a symmetric and an asymmetric part $ p=p_s\oplus p_t, $ where \begin{eqnarray} p_s &=& \frac{p+\pi p}{2}\nonumber \\ &=&\left[\frac{p(a_1)+p(b_1)}{2}, \frac{p(a_2)+p(b_2)}{2}, \frac{p(a_1)+p(b_1)}{2}, \frac{p(a_2)+p(b_2)}{2},\right. \\ &&\left.\quad p(a_1b_1),\frac{p(a_1b_2)+p(a_2b_1)}{2},\frac{p(a_1b_2)+p(a_2b_1)}{2},p(a_2b_2)\right]\,,\nonumber \end{eqnarray} and \begin{eqnarray} p_t &=& \frac{p-\pi p}{2}\nonumber \\ &=&\left[\frac{p(a_1)-p(b_1)}{2}, \frac{p(a_2)-p(b_2)}{2}, \frac{-p(a_1)+p(b_1)}{2}, \frac{-p(a_2)+p(b_2)}{2},\right.\\ &&\quad \left.0,\frac{p(a_1b_2)-p(a_2b_1)}{2},\frac{-p(a_1b_2)+p(a_2b_1)}{2},0\right]\,.\nonumber \end{eqnarray} Note that the symmetric part $p_s$ lives in a 5-dimensional subspace of $\mathbb{R}^8$ and can thus be expressed in an appropriate basis as \begin{equation}\label{sym5} p_s = \left[\frac{p(a_1)+p(b_1)}{2}, \frac{p(a_2)+p(b_2)}{2},p(a_1b_1),\frac{p(a_1b_2)+p(a_2b_1)}{2},p(a_2b_2)\right] \end{equation} Similarly, $p_t$ lives in a 3-dimensional space of $\mathbb{R}^8$ and can be decomposed in a proper basis as \begin{equation} p_t = \left[\frac{p(a_1)-p(b_1)}{2}, \frac{p(a_2)-p(b_2)}{2},\frac{p(a_1b_2)-p(a_2b_1)}{2}\right]\,. \end{equation}
The projection of the 16 deterministic points (\ref{det222}) on the symmetric subspace defined by (\ref{sym5}) are given by \begin{equation}\label{dets222} v_{s;a_1a_2b_1b_2}=\left[\frac{a_1+b_1}{2},\frac{a_2+b_2}{2},a_1b_1,\frac{a_1b_2+a_2b_1}{2},a_2b_2\right]\,. \end{equation} Note that some vertices of the original polytope are projected onto the same point of the symmetric polytope. For instance, $v_{s;1112}=v_{s;1211}$. In total, it can be verified that the set defined by (\ref{dets222}) contains only 10 extremal points.
We thus have reduced the original 8-dimensional polytope defined by 16 vertices to a 5-dimensional polytope with 10 vertices. Applying to this symmetrized polytope a standard algorithm performing the transformation from the V-representation to the H-representation, we find 4 different types (up to relabeling of settings and outcomes) of facets of the symmetric polytope:
\begin{align} &P(a_1b_1)\geq0&\nonumber\\ &P(a_1b_2)+P(a_2b_1)\geq0&\nonumber\\ &P(a_1)+P(b_1)-2P(a_1b_1)\geq0&\\ &P(a_1)+P(b_1)-P(a_1b_1)-P(a_1b_2)-P(a_2b_1)+P(a_2b_2)\geq0&\nonumber \end{align} We recognize the first inequality as the positivity condition for the joint probabilities and the last one as the CHSH inequality (written in the CH form as in equation \eqref{chsh}). These two classes of inequalities define symmetric facets of the full polytope. The two other inequalities are valid inequalities for the full local polytope, but do not correspond to facets (although they are facets of the symmetrized polytope, as the inequality $g$ in Figure~\ref{fig:fig1}). Note that in this (2,2,2) Bell scenario, the two only types of facet inequalities of the full polytope (the positivity condition and the CHSH inequality) can be written in a symmetric way. Hence in this simple case finding the facets of the symmetrized polytope is sufficient to generate all Bell inequalities.
\subsection{Generating facet inequalities from valid inequalities}\label{sousIne}
As we mentioned earlier, and was illustrated above, facets of the symmetrized polytope $P_s$ can correspond to inequalities which are not facets of the original polytope $P$. These inequalities are nonetheless valid inequalities which are satisfied by all points in $P$ and which might be violated by points that do not violate any of the symmetric facets of $P$. These inequalities may be used to generate new (non-symmetric) facets of $P$.
There exist various deterministic or heuristic algorithms which may generate new facet inequalities starting from a valid (not facet-defining) inequality, see for instance \cite{Pal}. Here, we describe a procedure that can be used whenever the starting valid inequality corresponds to a high-dimensional face of $P$, i.e., when the number of affinely independent vertices that saturate the inequality is large. In this case, it is possible to find all the facets that contain this high-dimensional face by completing the list of vertices with all possible combination of vertices that do not saturate the inequality, as detailed by the following algorithm:
\begin{enumerate} \item Let $(f,f_0)$ be a valid inequality for $P$ and let $W=\{v\mid f\cdot v=f_0\}$ be the set of vertices saturating this inequality. \item Let $\text{dim}(W)$ denote the number of affinely independent points in $W$. \begin{itemize} \item If $\text{dim}(W)=d$, then $(f,f_0)$ is a facet of $P$. \item If $\text{dim}(W)<d$, let $U=\{v\mid \text{dim}(W\cup v)>\text{dim}(W)\}$.\\ For every $u\in U$, let $(g,g_0)$ be the hyperplane passing through the points in $U$. If $(g,g_0)$ is a valid inequality for $P$, it now defines a face of $P$ of dimension $\text{dim}(W)+1$; in this case, go back to point 1 with $(g,g_0)$ as a starting inequality. \end{itemize} \end{enumerate}
\section{Applications}
We now illustrate our method in several situations for which generating the complete set of facet inequalities using standard polytope software \cite{polytopeSoft} is too time-consuming to be feasible. Due to the large number of inequalities that we have found, we only explicitly write a few of them here. Complete lists of all the facet inequalities that we generated are posted on the website \cite{website}. Our results are summarized in Table I.
Note that we list here only inequalities that belong to different equivalence classes, where two inequalities are considered equivalent if they are related by a relabeling of parties, measurement settings, or measurement outputs, or if they correspond to two different liftings of the same lower-dimensional inequality \cite{liftings}. In appendix A, we introduce a parametrization of the correlation space that naturally induces several invariants for each equivalence class. These invariants are easily computed and are useful to determine quickly wether two inequalities are equivalent (two equivalent inequalities have equal invariants).
\begin{table}[b]\label{tab:table}
\begin{tabular}{|r|c|c|c|c|c|c|}
\hline {Bell scenario} & {$d$} & {$d_s$} & {$|V|$} & {$|V_s|$} & \# symmetric inequalities & \# symmetric facets \\ \hline (2,2,4)&48&27&256&136&29&12\\ (2,4,2)&24&14&256&136&90&55\\ (4,2,2)& 80 & 14& 256 & 35 & 627 & 392\\ (5,2,2)& 242 & 20 & 1024 & 56 & $>$238464 & 238464\\
Correlators (3,3,2)& 27 & 10 & 512 & 40 & 44 & 20\\ Svetlichny (3,2,2)& 56 & 14& 2944 & 132 & 1204 & 1087\\\hline \end{tabular}
\caption{Summary of our numerical results. For each scenario we give the dimension of the space the polytope lives in as well as its number of extremal vertices, both before ($d$, $|V|$) and after projection on the symmetric subspace ($d_s,|V_s|$). We give the number of inequivalent symmetric inequalities valid for $P$ (but not necessarily facet-defining) obtained by resolving the symmetric polytope $P_s$ and the number of those inequalities that are facet defining.} \end{table}
\subsection{(2,2,4)}
We first consider bipartite experiments with two measurement settings per sites and $4$ possible outcomes. Note that the case $(2,2,3)$ was completely solved by Kaszlikowski et al. \cite{KKCZO} and Collins et al. \cite{CGLMP}, who showed that all facets of the (2,2,3) polytope either correspond to the positivity of probabilities, the CHSH inequality, or the CGLMP inequality \cite{CGLMP}. The CGLMP inequality was introduced for any number of outcomes $k\geq 3$ in \cite{CGLMP}, and Gill raised the question \cite{gill} whether all non-trivial facet inequalities of $(2,2,k)$ are of this form.
Using our method, we found that the Bell polytope corresponding to $(2,2,4)$ contains 12 inequivalent symmetric Bell inequalities. Among them, 8 involve the four possible outcomes in a nontrivial way, i.e., they correspond to genuine 4-outcome inequalities that cannot be seen as liftings of inequalities with lower numbers of outcomes, these inequalities are listed in Appendix B. The list of these 8 inequalities contains the CGLMP inequality, but surprisingly, it also contains 7 inequalities that are inequivalent to it, thus answering in the negative Gill's question~\cite{gill}.
\subsection{(2,4,2)}
We now consider a bipartite scenario involving 4 settings with binary outcomes. The simpler case (2,3,2) was solved in \cite{CHSH,Froissart,Collins} and contains a single new inequality besides the positivity constraints and the CHSH inequality.
With $4$ settings, we could use our method to find all of the 90 inequivalent facets of the symmetrized polytope. Among these, 55 turn out to be facets of the (2,4,2) full local polytope, there are thus in total 55 symmetric inequalities for this scenario. Most of them were already known (see \cite{Brunner, Pal, Avis}), but we could not find the two following ones in the litterature, given here in the notation of \cite{Collins} :
\begin{narrow}{-0.8cm}{0cm} \begin{equation}
S_{(2,4,2)}^{51}=\begin{array}{c||cccc}
& -1 & -2 & -2 & -2\\ \hline \hline -1 &-3 & 3 & 2 & 2\\ -2 & 3 & 2 &-1 &-1\\ -2 & 2 &-1 &-1 & 3\\ -2 & 2 &-1 & 3 & 0 \end{array} \leq 0\ , \ \
S_{(2,4,2)}^{52}=\begin{array}{c||cccc}
& 0 & -2 & -2 & -2\\ \hline \hline
0 &-3 & 2 &-2 & 1\\ -2 & 2 & 0 & 2 & 2\\ -2 &-2 & 2 & 4 &-1\\ -2 & 1 & 2 &-1 & 1 \end{array} \leq 0 \end{equation}
\end{narrow}
\subsection{(4,2,2) and (5,2,2)}
Since our method takes advantage of the symmetry between parties, we expect that it will be particularly useful for multipartite Bell scenarios. Indeed for the Bell scenario (n,2,2), corresponding to $n$ parties with binary settings and outcomes, the full local polytope has $4^n$ vertices and is embedded in a space of dimension $3^n-1$. The symmetrized polytope, on the other hand, has at most $\frac16(n+1)(n+2)(n+3)$ vertices and is embedded in a space of dimension $d_s=\frac12n(n+3)$. These quantities are polynomial in $n$ and represent an exponential advantage with respect to the general, non-symmetric situation. Note that it therefore follows that it is possible using linear programming to decide in polynomial time in $n$ if a given symmetric correlation vector $p$ is local.
The case (3,2,2) was already completely solved in \cite{Sliwa}. For (4,2,2) we found a total of 627 inequivalent symmetric inequalities, of which 392 are facet-defining. These facet inequalities correspond to the positivity conditions and to 391 genuinely 4-partite inequalities. Amongst them, the following one is quite interesting, as it can be violated by a 4-partite W state with measurements lying in the x-y plane (contrary to the 3-partite case where no inequality is known that can be violated by a $W$-state with measurements lying in the x-y plane):
\begin{equation} \begin{split}\label{4partin} I_W =& - p(a_1b_1) + p(a_1b_1c_1) + p(a_1b_1c_2) - p(a_2b_2c_2) - p(a_1b_1c_1d_1) \\ &- p(a_1b_1c_1d_2) - p(a_1b_1c_2d_2) + p(a_1b_2c_2d_2) + p(a_2b_2c_2d_2) + \text{sym} \leq 0.\\ \end{split} \end{equation}
The notation ``$\text{sym}$'' stands for the symmetric terms that are missing in (\ref{4partin}), such as $p(a_1c_1),p(a_1b_2c_1)$, etc. If we consider the $W$ state $|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle$ and measurements in the $x-y$ plane at an angle $\phi$ with respect to the $x$ axis, and set \begin{align} \phi_{A_1} &= \phi_{C_1} = 0 & \phi_{B_1}&=\phi_{D_1}=\arccos\frac14-2\arcsin\frac14\nonumber\\ \phi_{A_2} &= \phi_{C_2} = \arccos\frac14 & \phi_{B_2}&=\phi_{D_2}=-2\arcsin\frac14 \end{align} we find a value $I_W=1/16>0$.
For (5,2,2), we found 238464 inequivalent symmetric facets. Note that all of these inequalities, except the positivity of the probabilities, are truly 5-partite ones (i.e., they do not correspond to lifting of inequalities involving less parties). Among these inequalities, 9 of them involve only full (5-partite) correlators and were already given in \cite{Werner}.
\subsection{Correlation inequalities for (3,3,2)}
We considered also a tripartite scenario with 3 binary measurements per party. Since in this case even the symmetrized polytope is quite time-consuming to solve, we made a further restriction by considering only ``full-correlator" inequalities, which can be written using only terms of the form $\langle A_xB_yC_z\rangle=p(a+b+c=0|x,y,z)-p(a+b+c=1|x,y,z)$. This corresponds to performing a projection of the polytope on the subspace defined by \begin{equation} \langle A_i\rangle=\langle B_i\rangle=\langle C_i\rangle=\langle A_iB_j\rangle=\langle A_iC_j\rangle=\langle B_iC_j\rangle=0\ \quad \text{for all } i,j=1,2,3. \end{equation} We obtained 40 inequalities in this way, 18 of which are facets of the full original polytope that truly involve 3 inputs per party. Using the method presented in section \ref{sousIne}, we found 13 supplementary facet inequalities, all of which involve again full-correlators only.
\subsection{Svetlichny inequalities for (3,2,2)}
To illustrate that our method is not restricted to Bell-local polytope, but can address any correlation polytope, we consider the Svetlichny polytope for 3 parties \cite{Svetlichny}, which characterizes true tripartite nonlocality \cite{Seevinck,Collins2,Bancal}.
In a Svetlichny model, two of the three subsystems are allowed to communicate once the measurement settings have been chosen. There are thus three types of Svetlichny vertices $v_{AB/C}$, $v_{AC/B}$, $v_{BC/A}$, depending on which pairs of parties are allowed to communicate. A vertex of the form, e.g. $v_{AB/C}$, corresponding to a deterministic strategy where outcomes $\alpha(x,y)$ and $\beta(x,y)$ are jointly determined for party 1 and 2, and an outcome $\gamma(z)$ is assigned to party 3. This defines a joint distribution of the form \begin{equation}\label{eq:sverstrategy}
p(a,b,c|x,y,z)=\begin{cases} 1&\text{if $a=\alpha(x,y), b=\beta(x,y), c=\gamma(z)$}\\ 0&\text{otherwise}\,. \end{cases} \end{equation} Such probability points do not satisfy the no-signalling conditions. For binary settings and outcomes, the Svetlichny polytope thus lives in a vector space of dimension $d=56$. The subspace which is symmetric under the exchange of the three parties, however, has only dimension $d_s=14$.
This great reduction in the space dimension, together with a reduction in the number of extremal points (see Table 1), allowed us to find all symmetric Svetlichny inequalities. After projection on the no-signaling space\footnote{Indeed, we are only interested in whether these inequalities are violated by quantum correlations, which satisfy the no-signalling conditions.} a total of 1087 facet symmetric Svetlichny inequalities were found.
Interestingly, there are only two symmetric Svetlichny inequalities that involve only full-triparite correlation terms: the original Svetlichny inequality \cite{Svetlichny} and the following one: \begin{equation} \begin{split} I_{Corr}=&- \langle A_1 B_1 C_1\rangle + \langle A_1 B_1 C_2\rangle + \langle A_1 B_2 C_1\rangle - 3\langle A_1 B_2 C_2\rangle\\ &+ \langle A_2 B_1 C_1\rangle - 3\langle A_2 B_1 C_2\rangle - 3\langle A_2 B_2 C_1\rangle - 3\langle A_2 B_2 C_2\rangle \leq 10 \end{split} \end{equation} This last inequality can be violated by quantum states, for instance by GHZ states having a visibility larger than $95.68\%$.
The following inequality is also interesting: \begin{equation} I_{GHZ}=-3P(a_2)+P(a_1b_2)-P(a_1b_1c_1)-P(a_1b_2c_2)+7P(a_2b_2c_2)+\text{sym} \leq 0. \end{equation} where ``$\text{sym}$'' stands for the missing symmetric terms. It can be shown \cite{Svet10} that it is violated by every GHZ-like state of the form \begin{equation}\label{eq:GHZ} \ket{GHZ}=\cos\theta\ket{000}+\sin\theta\ket{111}. \end{equation}
\section{Outlook}
Motivated by the number of interesting Bell inequalities that are invariant under permutations of the parties, we introduced a method to list all symmetric inequalities. This method works even in cases where solving the full correlation polytope is computationally intractable. Our method can also be used as a starting point to generate more general, non-symmetric inequalities using algorithms such as the one described in section \ref{sousIne} or in \cite{Pal}.
Our method allowed us to find a number of new Bell inequalities. But a new problem is now at sight: so many different inequalities are generated, even for simple situations, that it is difficult to find which ones are the most interesting. Evidence of this problem was already put forward in \cite{Werner} and in \cite{Avis} where it was shown that the number of inequivalent Bell inequalities increases very quickly with the number of parties or measurement settings. New insights are thus necessary in order to classify these inequalities and understand which ones are the most relevant.
For simple Bell scenarios, such as (2,2,2), (2,2,3) or (2,3,2) \emph{all} facet inequalities happen to be symmetric inequalities. What is the proportion of symmetric inequalities in more complicated scenarios? Are there other useful symmetries or properties that can be exploited to generate more inequalities?
\section{Acknowledgments}
We thank K. F. P\'al and T. V\'ertesi for useful remarks. This work was supported by the Swiss NCCR Quantum Photonics, the European ERC-AG QORE, and the Brussels-Capital region through a BB2B grant.
\appendix \section*{Appendix A: Correlators and inequality invariants}
When dealing with Bell inequalities, it is useful to check quickly if two inequalities are equivalent under relabeling of parties, measurement settings, or outcomes. We were confronted with this problem when classifying the inequalities that we derived here. In this appendix, we introduce a parametrization of the correlation space which naturally induces several invariants for each equivalence class. Two inequalities that are equivalent have equal invariants.
Let us consider a $(n,m,k)$ Bell scenario and let $I\subseteq\{1,\ldots,n\}$ be a subset of the $n$ parties. Define $p(a_I|x)$ as the probability that these $|I|$ parties obtain outputs $a_I=a_{i_1},\ldots,a_{i_{|I|}}$ given that measurements $x=x_1,\ldots,x_n$ have been made on all parties. Note that with this notation, the no-signalling condition is expressed as $p(a_I|x)=p(a_I|x_I)$, where $x_I=x_{i_1},\ldots,x_{i_{|I|}}$.
Define now single-party ``correlators" $E(a_i|x)$ by \begin{equation}\label{eq:E}
E(a_i|x)=kp(a_i|x)-1\,, \end{equation}
and define by induction multipartite correlators $E(a_I|x)$ through \begin{equation}\label{eq:E2}
E(a_{I_1}, a_{I_2} | x) = E(a_{I_1}|x)*E(a_{I_2}|x)\,, \end{equation}
where the $\ast$ operation is just the usual multiplication, except when acting on two probabilities, in which case it satisfies $p(a_{I_1}|x)*p(a_{I_2}|x)=p(a_{I_1},a_{I_2}|x)$. The no-signalling condition is then expressed as
\begin{equation} E(a_I|x)=E(a_I|x_I)\,.
\end{equation} The normalization condition on the probabilities $p(a_I|x)$ imply, on the other hand, that for any $i\in I$ \begin{equation}\label{eq:normalisation} \begin{split}
\sum_{a_i=0}^{k-1}E(a_I|x)&=\sum_{a_i=0}^{k-1}E(a_i|x)*E(a_{I\backslash i}|x)\\
&=\left[\sum_{a_i=0}^{k-1}\left(kp(a_i|x)-1\right)\right]*E(a_{I\backslash i}|x)=0\,. \end{split} \end{equation}
The correlators $E(a_I|x)$ are in one-to-one correspondence with the probabilities $p(a_I|x)$ and thus represent an alternative parametrization of the correlation space. Note that in the case of binary outcomes ($k=2$), $E(a_I|x_I)$ coincides with the usual definition of a correlation function. The definitions \eqref{eq:E} and \eqref{eq:E2} thus represent a possible generalisation of correlation function to more outcomes.
With the notation that we just introduced, a generic Bell inequality in the no-signalling space takes the form \begin{equation}\label{incor}
\sum_{I,x_I,a_I} c(a_I,x_I) E(a_I|x_I) \leq c(0) \end{equation}
where $c(a_I,x_I)$ are the coefficients of the inequality. A property of the correlators $E(a_I|x_I)$ is that the white noise yields $E(a_I|x_I)=0$. The resistance to noise of a an inequality written in the form (\ref{incor}) is thus directly given by the ratio of the local bound to the violation.
A relabeling of parties, measurement settings, or measurement outcomes simply amounts to rearrange the order of the coefficients of the inequality. Note, however, that because of the normalization conditions (\ref{eq:normalisation}), the basis of the correlation space that we chose is overcomplete. The coefficients $c(a_I,x_I)$ are thus not uniquely defined: adding $\sum_{a_i}E(a_I|x_I)=0$ for some $i\in I$ to the inequality (\ref{incor}) does not change the inequality itself, but does change its coefficients. To compare two inequalities, we must therefore ensure first that they are written in some standard way.
The freedom that we have in adding terms of the form $\lambda(i,a_{I\setminus i},x_I)\sum_{a_i}E(a_I|x_I)=0$ to (\ref{incor}) corresponds to define new coefficients for the inequality in the following manner \begin{equation}\label{eq:cprime} c'(a_I,x_I)= c(a_I,x_I) +\sum_{i\in I} \lambda(i,a_{I\setminus i},x_I)\,. \end{equation}
We show now that requiring the inequality coefficients $c'(a_I,x_I)$ to satisfy the relation \begin{equation}\label{eq:conditioncoeffs} \sum_{a_i=0}^{k-1}c'(a_I,x_I)=0\quad \text{for all }i\in I\,. \end{equation} allows to define them uniquely.
\begin{proposition*} There exist values of $\lambda(i,a_{I\setminus i},x_I)$ such that the newly defined coefficients $c'(a_I,x_I)$ satisfy relation \eqref{eq:conditioncoeffs}. Moreover for all such $\lambda$'s, the $c'(a_I,x_I)$ are the same, they are thus unique. \end{proposition*}
\begin{proof} Since equations \eqref{eq:cprime} and \eqref{eq:conditioncoeffs} apply independently on every subset $I$ of the parties, and on every inputs $x_I$, we omit these indices, writing for instance $c(a_I)$ instead of $c(a_I,x_I)$ to lighten the notation. Moreover, all sums on the outputs $a_i$ go from $0$ to $k-1$ and all sums on the parties $i$ run on $I$, so we also omit these bounds in the proof.
The existence of the $\lambda$'s can be shown by directly checking that the following formula is of the form \eqref{eq:cprime}, and satisfies \eqref{eq:conditioncoeffs}. Let \begin{equation}\label{eq:cprimeformula} c'(a_I)=\left[\left(\openone - \frac1k \sum_{a_1}\right)\circ\ldots\circ\left(\openone - \frac1k \sum_{a_n}\right)\right]c(a_I) \end{equation} where we used the notation $f + \sum_a f + \sum_b f = [\openone + \sum_a + \sum_b]f$ for any function $f$ and the $\circ$ composition satisfies $\openone \circ \openone = \openone$, $\openone \circ \sum_a = \sum_a \circ \openone = \sum_a$, $\sum_{a} \circ \sum_{b} = \sum_{a,b}$ and distributes over addition. To get this expression from \eqref{eq:cprime} one possible choice of lambdas is: \begin{equation}\label{eq:lambdas} \begin{split} \lambda(1,a_{I\setminus 1}) &= -\frac1k \sum_{a_1} c(a_I)\\ \lambda(2,a_{I\setminus 2}) &= -\frac1k \sum_{a_2} c(a_I) + \left(\frac1k\right)^2 \sum_{a_1,a_2} c(a_I)\\ &\ldots \end{split} \end{equation} and equation \eqref{eq:conditioncoeffs} is satisfied: \begin{equation} \begin{split} \sum_{a_i}c'(a_I)&=\left[\sum_{a_i}\circ\left(\openone - \frac1k \sum_{a_1}\right)\circ\ldots\circ\left(\openone - \frac1k \sum_{a_n}\right)\right]c(a_I)\\ &=\left[\ldots\circ\left(\sum_{a_i} - \sum_{a_i}\right)\circ\ldots\right]c(a_I)=0. \end{split} \end{equation}
Now to show the unicity of the $c'$ coefficients, we notice that equation \eqref{eq:conditioncoeffs} is a non-homogeneous linear system of equations in the $\lambda$ variables, which we can write as: \begin{equation}\label{eq:linsys} \sum_{a_j}\sum_{i} \lambda(i,a_{I\setminus i}) = -\sum_{a_j} c(a_I) \end{equation} Every solution of this system can thus be written as $\lambda=\lambda_p+\lambda_v$ where $\lambda_p$ is a particular solution of the equation (as given by equation \eqref{eq:lambdas} for instance) and $\lambda_v$ is a solution of the homogeneous system, where the right-hand side of equation \eqref{eq:linsys} is replaced by zero. Thus every $c'(a_I)$ that satisfies equation \eqref{eq:conditioncoeffs} can be written as \begin{equation}\label{eq:cpf} c'(a_I) = c(a_I) + \sum_{i}\lambda_p(i,a_{I\setminus i}) + \sum_{i}\lambda_v(i,a_{I\setminus i}). \end{equation}
Now we show that the last term of equation \eqref{eq:cpf} is zero for every solution $\lambda_v$ of the homogeneous counterpart of system \eqref{eq:linsys}, which implies that the coefficients $c'$ are uniquely defined. For this, consider the following expression: \begin{equation} Z=\left[\left(\openone-\frac1k\sum_{a_1}\right)\circ\ldots\circ\left(\openone-\frac1k\sum_{a_n}\right)\right]\lambda_v(i,a_{I\setminus i}). \end{equation} It is clearly zero, since it contains the term \begin{equation} \left[\left(\openone-\frac1k\sum_{a_i}\right)\right]\lambda_v(i,a_{I\setminus i})=\lambda_v(i,a_{I\setminus i})-\frac1k k \lambda_v(i,a_{I\setminus i})=0. \end{equation} On the other hand, we have that \begin{align} \sum_i Z &= \left[\left(\openone-\frac1k\sum_{a_1}\right)\circ\ldots\circ\left(\openone-\frac1k\sum_{a_n}\right)\right]\sum_i\lambda_v(i,a_{I\setminus i})\\ &=\left[\left(\openone-\frac1k\sum_{a_1}\right)\circ\ldots\circ\left(\openone-\frac1k\sum_{a_{n-1}}\right)\right]\sum_i\lambda_v(i,a_{I\setminus i})\nonumber\\ &\ \ \ -\left[\left(\openone-\frac1k\sum_{a_1}\right)\circ\ldots\circ\left(\openone-\frac1k\sum_{a_{n-1}}\right)\right]\frac1k\sum_{a_n}\sum_i\lambda_v(i,a_{I\setminus i})\label{eq:justbefore}\\ &=\left[\left(\openone-\frac1k\sum_{a_1}\right)\circ\ldots\circ\left(\openone-\frac1k\sum_{a_{n-1}}\right)\right]\sum_i\lambda_v(i,a_{I\setminus i})\\ \end{align} where the second term in \eqref{eq:justbefore} vanishes by definition of $\lambda_v(i,a_{I\setminus i})$. Repeating iteratively the above step, we find eventually that \begin{equation} \sum_i Z = \sum_i\lambda_v(i,a_{I\setminus i}). \end{equation} This, combined with the fact that $Z=0$, implies the desired result. \end{proof}
We showed that the coefficients of an inequality can be defined in a standard and unique way by requiring them to satisfy the constraints (\ref{eq:conditioncoeffs}). Now, since a relabeling of parties, measurement settings, or outcomes can only rearrange the coefficients $c'(a_I,x_I)$ without changing their value, the ordered lists of coefficients $c'(a_I,x_I)$ for $|I|=0,1,\ldots,n$ provide $n+1$ invariants for each equivalence class. For instance the local bound $c(0)$ is an (easily checkable) invariant. In general requiring that two inequalities have their ordered list of coefficients identical does not guarantee that they are equivalent, but in the symmetric $(4,2,2)$ scenario for instance, all the $391$ classes of inequalities that we generated had different lists.
\section*{Appendix B: List of symmetric inequalities with two inputs and four outcomes.}
The following inequalities are given in the notation of \cite{Collins}.
\subsection*{Case with 3 outcomes for the first setting and 4 outcomes for the second one.}
\begin{narrow}{-2cm}{0cm} \begin{equation} \begin{split}
S_{(2,2,(3,4))}^{2}=\begin{array}{c||cc|ccc}
& -1 & -1 & 0 & 0 & 0 \\ \hline \hline -1 & 0 & 1 & 0 & 1 & 1 \\ -1 & 1 & 0 & 1 & 0 & 1 \\ \hline
0 & 0 & 1 & 0 & -1 & -1 \\
0 & 1 & 0 & -1 & 0 & -1 \\
0 & 1 & 1 & -1 & -1 & -1 \\ \end{array} \leq 0\\ \end{split} \end{equation} \end{narrow}
\subsection*{Case with 4 outcomes for both settings.}
\begin{narrow}{ 0.6cm}{0cm} \begin{equation} \begin{aligned}
&S_{(2,2,4)}^{1}=\begin{array}{c||ccc|ccc} &-1&-1&-1&0&0&0\\ \hline\hline -1&0&0& 1& 1& 1& 1\\ -1&0& 1& 1& 1& 1&0\\ -1& 1& 1& 1& 1&0&0\\ \hline 0& 1& 1& 1&-1&-1&-1\\ 0& 1& 1&0&-1&-1&0\\ 0& 1&0&0&-1&0&0\\ \end{array}\leq0, &
&S_{(2,2,4)}^{2}=\begin{array}{c||ccc|ccc} &-1&-1&0&-1&0&0\\ \hline\hline -1&0& 1&0& 1&0& 1\\ -1& 1&0&0& 1& 1&0\\ 0&0&0&-1& 1&0&0\\ \hline -1& 1& 1& 1& 1&0&0\\ 0&0& 1&0&0&0&-1\\ 0& 1&0&0&0&-1&0\\ \end{array}\leq0\\
&S_{(2,2,4)}^{3}=\begin{array}{c||ccc|ccc} &-1&-1&0&-1&-1&0\\ \hline\hline -1&0&0& 1& 1& 1&0\\ -1&0& 1&0&1&0&1\\ 0& 1&0&0&-1&-1&0\\ \hline -1& 1&1&-1&1&2&0\\ -1& 1&0&-1&2&2&0\\ 0&0&1&0&0&0&-1\\ \end{array}\leq 0, &
&S_{(2,2,4)}^{4}=\begin{array}{c||ccc|ccc} &-1&-1&0&-1&-1&0\\ \hline\hline -1&-1&1&-1&1&2&1\\ -1&1&1&0&1&0&1\\ 0&-1&0&-1&1&1&0\\ \hline -1&1&1&1& 1&0&0\\ -1&2&0&1&0& 1&-1\\ 0&1&1&0&0&-1&-1\\ \end{array}\leq 0\\
&S_{(2,2,4)}^{5}=\begin{array}{c||ccc|ccc} &-1&-1&-1&0&0&0\\ \hline\hline -1&0& 1& 1&0&0& 1\\ -1& 1&0& 1&0& 1&0\\ -1& 1& 1&0& 1&0&0\\ \hline 0&0&0& 1&0&-1&-1\\ 0&0& 1&0&-1&0&-1\\ 0& 1&0&0&-1&-1&0\\ \end{array}\leq0, &
&S_{(2,2,4)}^{6}=\begin{array}{c||ccc|ccc} &-1&-1&-1&-1&0&0\\ \hline\hline -1&0& 1& 1&0& 1&0\\ -1& 1& 1&0&1&0&1\\ -1& 1&0&-1&2&0&1\\ \hline -1&0&1&2&1&-1&0\\ 0&1&0&0&-1&0&-1\\ 0&0&1&1&0&-1&-1\\ \end{array}\leq 0\\
&S_{(2,2,4)}^{7}=\begin{array}{c||ccc|ccc} &-1&-1&-1&0&0&0\\ \hline\hline -1&0& 1& 1&0&0& 1\\ -1& 1&0&0& 1& 1&0\\ -1& 1&0& 1& 1&0&0\\ \hline 0&0& 1& 1&-1&0&-1\\ 0&0& 1&0&0&0&-1\\ 0& 1&0&0&-1&-1&0\\ \end{array}\leq0, &
&S_{(2,2,4)}^{8}=\begin{array}{c||ccc|ccc} &-2&-1&-1&0&0&0\\ \hline\hline -2& 2&0& 1&0& 1& 2\\ -1&0& 1& 1& 1&0&0\\ -1& 1& 1&0&0& 1&0\\ \hline 0&0& 1&0&-1&-1&0\\ 0& 1&0& 1&-1&0&-1\\ 0& 2&0&0&0&-1&-2\\ \end{array}\leq0 \end{aligned} \end{equation} \end{narrow}
\end{document}
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arXiv
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Non-Hausdorff manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.
Examples
Line with two origins
The most familiar non-Hausdorff manifold is the line with two origins,[1] or bug-eyed line. This is the quotient space of two copies of the real line $\mathbb {R} \times \{a\}$ and $\mathbb {R} \times \{b\}$ obtained by identifying points $(x,a)$ and $(x,b)$ whenever $x\neq 0.$
An equivalent description of the space is to take the real line $\mathbb {R} $ and replace the origin $0$ with two origins $0_{a}$ and $0_{b}.$ The subspace $\mathbb {R} \setminus \{0\}$ retains its usual Euclidean topology. And a local base of open neighborhoods at each origin $0_{i}$ is formed by the sets $(U\setminus \{0\})\cup \{0_{i}\}$ with $U$ an open neighborhood of $0$ in $\mathbb {R} .$
For each origin $0_{i}$ the subspace obtained from $\mathbb {R} $ by replacing $0$ with $0_{i}$ is an open neighborhood of $0_{i}$ homeomorphic to $\mathbb {R} .$[1] Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of $0_{a}$ intersect every neighbourhood of $0_{b}.$ It is however a T1 space.
The space is second countable.
The space exhibits several phenomena that don't happen in Hausdorff spaces:
• The space is path connected but not arc connected. In particular, to get a path from one origin to the other one can first move left from $0_{a}$ to $-1$ within the line through the first origin, and then move back to the right from $-1$ to $0_{b}$ within the line through the second origin. But it is impossible to join the two origins with an arc, which is an injective path; informally, if one moves first to the left, one has to eventually backtrack and move back to the right.
• The intersection of two compact sets need not be compact. For example, the sets $[-1,0)\cup \{0_{a}\}$ and $[-1,0)\cup \{0_{b}\}$ are compact, but their intersection $[-1,0)$ is not.
• The space is locally compact in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure. So the space is not a regular space, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.
The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.[2]
Line with many origins
The line with many origins[3] is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set $S$ with the discrete topology and taking the quotient space of $\mathbb {R} \times S$ that identifies points $(x,\alpha )$ and $(x,\beta )$ whenever $x\neq 0.$ Equivalently, it can be obtained from $\mathbb {R} $ by replacing the origin $0$ with many origins $0_{\alpha },$ one for each $\alpha \in S.$ The neighborhoods of each origin are described as in the two origin case.
If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set $A=[-1,0)\cup \{0_{\alpha }\}\cup (0,1]$ is the set $A\cup \{0_{\beta }:\beta \in S\}$ obtained by adding all the origins to $A$, and that closure is not compact. From being locally Euclidean, such a space is locally compact in the sense that every point has a local base of compact neighborhoods. But the origin points don't have any closed compact neighborhood.
Branching line
Similar to the line with two origins is the branching line.
This is the quotient space of two copies of the real line
$\mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}$
with the equivalence relation
$(x,a)\sim (x,b)\quad {\text{ if }}\;x<0.$
This space has a single point for each negative real number $r$ and two points $x_{a},x_{b}$ for every non-negative number: it has a "fork" at zero.
Etale space
The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)[4]
Properties
Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff).
See also
• List of topologies – List of concrete topologies and topological spaces
• Locally Hausdorff space
• Separation axiom – Axioms in topology defining notions of "separation"
Notes
1. Munkres 2000, p. 227.
2. Gabard 2006, Proposition 5.1.
3. Lee 2011, Problem 4-22, p. 125.
4. Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer-Verlag. p. 164. ISBN 978-0-387-90894-6.
References
• Baillif, Mathieu; Gabard, Alexandre (2006), Manifolds: Hausdorffness versus homogeneity, arXiv:math.GN/0609098v1, Bibcode:2006math......9098B
• Gabard, Alexandre (2006), A separable manifold failing to have the homotopy type of a CW-complex, arXiv:math.GT/0609665v1, Bibcode:2006math......9665G
• Lee, John M. (2011). Introduction to topological manifolds (Second ed.). Springer. ISBN 978-1-4419-7939-1.
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
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Abstract: U13.00006 : Upgrades for an improved measurement of the EDM of $^{\mathrm{\mathbf{225}}}$\textbf{Ra}
Tenzin Rabga
(Argonne National Lab, Michigan State University)
Kevin Bailey
(Argonne National Lab)
Matthew R. Dietrich
John P. Greene
Roy J. Holt
Wolfgang Korsch
(University of Kentucky)
Zheng-Tian Lu
(University of Science and Technology of China)
Peter Mueller
Tom P. O'Connor
Steven Fromm
(Michigan State University)
Roy Ready
Jaideep T. Singh
If charge conjugation (C), parity (P) and time-reversal (T) symmetries, collectively form a good symmetry of nature, CPT, then T-violating phenomena would also violate CP. An Electric Dipole Moment (EDM) would violate time-reversal symmetry, and therefor EDMs provide a sensitive way for probing CP-violation that might explain the abundance of matter over anti-matter in the Universe. The $^{\mathrm{225}}$Ra atom (t$_{\mathrm{1/2}} \quad =$ 15 days, I $=$ 1/2) is a particularly attractive candidate for an EDM search in diamagnetic atoms due to its octupole deformed nuclear structure, nearly degenerate parity doublet ground state, and a large mass, that make it sensitive to T-violating interactions in the nuclear sector. Our latest measurement limits the atomic EDM of $^{\mathrm{225}}$Ra to be less than 1.4x10$^{\mathrm{-23}}$ e-cm (95{\%} C.L). Further experimental upgrades are being implemented including an electric field upgrade to enhance the EDM sensitivity and STIRAP for an improved spin precession detection scheme. With these upgrades in place our EDM sensitivity should increase by nearly two orders of magnitude and allow us to substantially improve constraints on certain T-violating processes within the nucleus.~This work is supported by the U.S. DOE, Office of Science, Office of Nuclear Physics, under contract DE-AC02-06CH11357 and the Michigan State University.
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CommonCrawl
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\begin{document}
\begin{center} \LARGE Bounding sums of the M\"obius function over arithmetic progressions
\normalsize Lynnelle Ye
\end{center}
\begin{abstract} Let $M(x)=\sum_{1\le n\le x}\mu(n)$ where $\mu$ is the M\"obius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+\epsilon})$ for all $\epsilon>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the M\"obius function over arithmetic progressions, of the form \[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}\mu(n). \] They were able to show that assuming the Generalized Riemann Hypothesis, $M(x;q,a)$ satisfies \[ M(x;q,a)\ll_{\epsilon}\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+\epsilon}\right) \] for all $q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$, with $a$ such that $(a,q)=1$, and $\epsilon>0$.
In this paper, we improve Halupczok and Suger's work to obtain the same bound for $M(x;q,a)$ as Soundararajan's bound for $M(x)$ (with a $1/2$ in the exponent of $\log x$), with no size or divisibility restriction on the modulus $q$ and residue $a$. \end{abstract}
\section{Introduction}
The Riemann Hypothesis (RH) is a conjecture about the zeros of the Riemann zeta function, a meromorphic function $\zeta:\mathbb{C}\to\mathbb{C}$ defined as follows: for $s\in\mathbb{C}$ with $\Re(s)>1$, we let \[ \zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}. \] For $\Re(s)\le1$, $\zeta(s)$ is the value of the unique analytic continuation of this function to the rest of the complex plane. The function $\zeta$ is holomorphic everywhere except at $s=1$, where it has a simple pole. For $\Re(s)>1$, we can also write $\zeta(s)$ in its Euler product representation \[ \zeta(s)=\prod_{p\text{ prime}}\left(1-\frac1{p^s}\right)^{-1}. \]
The Riemann zeta function satisfies an important functional equation relating it to the $\Gamma$ function, which is defined by $\Gamma(s)=\int_0^{\infty}x^{s-1}e^{-x}dx$ for $s\in\mathbb{C}$ such that $\Re(s)>0$ and analytically continued to the rest of the complex plane. The functional equation is \[ \pi^{-s/2}\Gamma\left(\frac s2\right)\zeta(s)=\pi^{-(1-s)/2}\Gamma\left(\frac{1-s}2\right)\zeta(1-s). \]
The $\Gamma$ function is nonzero everywhere, and it has simple poles exactly at the nonpositive integers. From this and the above equation, we can see that $\zeta(s)=0$ when $s$ is a negative even integer. From the Euler product of $\zeta$, we may compute for $\Re(s)>1$ that \[ \frac1{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s} \] where $\mu(n)$ is the multiplicative function, called the M\"obius function, defined by \[
\mu(n) =
\begin{cases}
0 & \text{if } p^2|n \text{ for some prime } p, \\
1 & \text{if } n = p_1\dotsb p_{2k} \text{ for distinct primes } p_i, \\
-1 & \text{if } n=p_1\dotsb p_{2k-1} \text{ for distinct primes } p_i.
\end{cases} \] Since $\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$ is absolutely convergent for $\Re(s)>1$, we know $\zeta(s)\neq0$ for $\Re(s)>1$, and hence from the functional equation that $\zeta(s)\neq0$ for $\Re(s)<0$, except for the previously mentioned negative even integers. So all other zeros of $\zeta$ lie in the strip $0\le\Re(s)\le1$ (referred to as the critical strip) and are known as the ``nontrivial zeros''.
The behavior of the zeros of the Riemann zeta function is deeply connected to the behavior of the primes in $\mathbb{N}$. Let $\pi(x)$ be the number of primes no greater than $x$; then the Prime Number Theorem, which states that $\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}=1$, follows from the fact that $\zeta$ has no zeros on the line $\Re(s)=1$. More specific information about the distribution of primes follows from more specific information about the zeros of $\zeta$, such as would be given by the Riemann Hypothesis.
\begin{conj}[Riemann Hypothesis] All nontrivial zeros of $\zeta(s)$ lie on the line $\Re(s)=\frac12$, called the critical line. \end{conj}
From the fact that $\zeta$ has no zeros on the line $\Re(s)=1$, the more specific version of the Prime Number Theorem as proven by Hadamard and de la Vall\'ee Poussin is that \[ \pi(x)=\int_2^x\frac1{\log t}dt+O(x\exp(-a\sqrt{\log x})) \] for some $a>0$. Assuming the Riemann Hypothesis, this would be improved to \[ \pi(x)=\int_2^x\frac1{\log t}dt+O(\sqrt{x}\log x). \]
The Mertens function $M(x)$ is defined by $M(x)=\sum_{1\le n\le x}\mu(n)$ where $\mu$ is the M\"obius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+\epsilon})$ for all $\epsilon>0$ (see~\cite{iwakow}, for example, for details). There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. Landau (\cite{landau}) showed in 1924 that RH implies $M(x)=O(x^{1/2+\epsilon})$ with $\epsilon\ll \log\log\log x/\log\log x$, and Titchmarsh (\cite{titchmarsh}) in 1927 improved this to $\epsilon\ll 1/\log\log x$. In 2009, Maier and Montgomery (\cite{maimont}) gave a much improved bound of \[ M(x)\ll\sqrt{x}\exp\left(C(\log x)^{39/61}\right). \] Finally, in~\cite{sound}, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \]
(where Soundararajan set $c$ to $14$, though Balazard and de Roton decrease it to $5/2+\epsilon$ in~\cite{baladero}). Gonek has conjectured (see Ng in~\cite{ng} for details) that the true range of $|M(x)|$ goes up precisely to the much smaller $O\left(\sqrt{x}(\log\log\log x)^{5/4}\right)$.
Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the M\"obius function over arithmetic progressions, of the form \[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}\mu(n). \] This requires a generalized version of the Riemann Hypothesis, appropriately called the Generalized Riemann Hypothesis (GRH). The GRH deals with Dirichlet $L$-functions, defined by \[ L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s} \] and analytically continued to the complex plane, where $\chi$ is a certain multiplicative function called a Dirichlet character. There are $\varphi(q)$ distinct characters $\chi$ for each modulus $q$, each with range contained in the $q$th roots of unity, satisfying \[ \frac1{\varphi(q)}\sum_{\chi}\chi(n)\overline{\chi}(a)= \begin{cases} 1 & \text{ if } n\equiv a\pmod{q} \\ 0 & \text{ otherwise.} \end{cases} \] Like $\zeta(s)$, $L(s,\chi)$ has a functional equation relating it to $\Gamma$, which looks like \[ \left(\frac q{\pi}\right)^{s/2}\Gamma\left(\frac{\kappa+s}2\right)L(s,\chi) =\epsilon(\chi)\left(\frac q{\pi}\right)^{(1-s)/2}\Gamma\left(\frac{\kappa+1-s}2\right)L(1-s,\overline{\chi}) \] where $\kappa$ is equal to $0$ if $\chi(-1)=1$ and $1$ if $\chi(-1)=-1$. Consequently its zeros also have a similar structure.
\begin{conj}[Generalized Riemann Hypothesis] For every $\chi$, all nontrivial zeros of $L(s,\chi)$ lie on the line $\Re(s)=\frac12$. \end{conj} As with the consequences of RH for the distribution of primes, GRH would imply, among other things, similar consequences for the distribution of primes in arithmetic progressions.
Halupczok and Suger showed that assuming GRH, given $\epsilon>0$, the bound \[ M(x;q,a)\ll_{\epsilon}\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+\epsilon}\right) \] holds for all progressions $a\pmod{q}$ with $q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$ and $(a,q)=1$.
In this paper, we improve Halupczok and Suger's work to obtain a bound of strength equal to that of Soundararajan, with no restriction on the modulus $q$ or the residue $a$. Our main theorem is as follows. \begin{thm} \label{main} Assuming GRH, given $\epsilon>0$, the bound \[ M(x;q,a)\ll_{\epsilon}\sqrt{x/d}\exp\left((\log(x/d))^{1/2}(\log\log(x/d))^{3+\epsilon}\right) \] holds uniformly for all progressions $a\pmod{q}$, where $d=\gcd(a,q)$. \end{thm}
Since $|\mu(n)|\le1$, the trivial bound on $M(x;q,a)$ is $x/q$. Our bound therefore remains nontrivial whenever $q\le x^{1/2-\epsilon}$.
In Section~\ref{prelim}, we provide preliminaries on the explicit formula for sums over zeros of $L$-functions, consequent bounds on the deviation from average of the number of zeros in an interval of the critical line, and the large sieve. In Section~\ref{pointcount}, we bound the number of well-separated ordinates on the critical line with an unusual accumulation of zeros of $L(s,\chi)$ nearby. In Section~\ref{Lfuncbds}, we give bounds for $L(s,\chi)$ depending on the number of zeros near $s$. In Section~\ref{finalthm}, we prove Theorem~\ref{main}.
We assume GRH for the rest of this paper.
\section{Preliminaries} \label{prelim}
First consider the case $(a,q)=1$. Using the Dirichlet characters $\chi\pmod{q}$ and Perron's formula, we may write \begin{align*} M(x;q,a) &=\frac1{\varphi(q)}\sum_{\chi\pmod{q}}\overline{\chi}(a)\sum_{n\le x}\chi(n)\mu(n) \\ &=\frac1{\varphi(q)}\sum_{\chi\pmod{q}}\frac{\overline{\chi}(a)}{2\pi i}\int_{1+1/\log x-i\lfloor x\rfloor}^{1+1/\log x+i\lfloor x\rfloor}\frac{x^s}{sL(s,\chi)}ds+O(\log x). \end{align*} More generally, let $a=bd,q=rd$ with $(b,r)=1$. We have \[ M(x;q,a) = \sum_{\substack{n\le x\\ n\equiv bd\pmod{rd}}}\mu(n) = \sum_{\substack{m\le x/d\\ m\equiv b\pmod{r}}} \mu(dm)=\mu(d)\sum_{\substack{m\le x/d\\ m\equiv b\pmod{r}\\ (d,m)=1}} \mu(m) \] so the sum we are interested in is really just \[ \sum_{\substack{m\le x/d\\ m\equiv b\pmod{r}\\ (d,m)=1}} \mu(m)=\frac1{\varphi(r)}\sum_{\chi\pmod{r}}\overline{\chi}(b)\sum_{\substack{m\le x/d\\ (d,m)=1}}\chi(m)\mu(m) \] \[ =\frac1{\varphi(r)}\sum_{\chi\pmod{r}}\frac{\overline{\chi}(b)}{2\pi i}\int_{1+1/\log(x/d)-i\lfloor x/d\rfloor}^{1+1/\log(x/d)+i\lfloor x/d\rfloor}\frac{(x/d)^s}{sL(s,\chi)l_d(s,\chi)}ds+O(\log(x/d)) \] where \[
l_d(s,\chi)=\prod_{p|d}\left(1-\frac{\chi(p)}{p^s}\right). \] We are now concerned with bounding the average value of the integrand $\frac{(x/d)^s}{sL(s,\chi)l_d(s,\chi)}$ over all $\chi$, as this suffices to bound $M(x;q,a)$. We first require the following lemmas. The first constructs smooth approximations to the characteristic function of an interval.
\begin{lem} \label{charfnapprox} Let $\chi_{[-h,h]}$ be the characteristic function of the interval $[-h,h]$. There are even entire functions $F_+$ and $F_-$ depending on $h$ and $\Delta$ which satisfy the following properties:
1. $F_-(u)\le\chi_{[-h,h]}(u)\le F_+(u)$ for $u\in\mathbb{R}$.
2. $\int_{-\infty}^{\infty}|F_{\pm}(u)-\chi_{[-h,h]}(u)|du\le\frac1{\Delta}$.
3. $\hat{F}_{\pm}(x)=0$ for $|x|\ge\Delta$, and $\hat{F}_{\pm}(x)=\frac{\sin(2\pi hx)}{\pi x}+O(1/\Delta)$.
4. If $z=x+iy$ with $|z|\ge2h$ then $|F_{\pm}(z)|\ll\frac{e^{2\pi\Delta|y|}}{(\Delta|z|)^2}$. \end{lem}
\begin{proof} Such functions were originally constructed by Selberg (\cite{selberg}) using Beurling's approximation to the sign function. They are discussed in detail in~\cite{vaaler}. We will just state the relevant formulas. Let \[ K(z)=\frac{(\sin \pi z)^2}{(\pi z)^2},\hspace{0.2in} H(z)=\left(\frac{\sin\pi z}{\pi}\right)^2\left(\sum_{n=-\infty}^{\infty}\frac{\text{sgn}(n)}{(z-n)^2}+\frac2z\right), \] where $\text{sgn}(x)$ is the sign function, with values $1$ for $x>0$, $-1$ for $x<0$, and $0$ for $x=0$. Then the functions $F_{\pm}$ are \[ F_{\pm}(z)=\frac12(H(\Delta(z+h))\pm K(\Delta(z+h))+H(\Delta(h-z))\pm K(\Delta(h-z))). \] \end{proof}
Next we need the following form of the explicit formula for sums over zeros of $L$-functions.
\begin{lem} \label{explicitformula}
Let $\chi$ be a primitive character $\pmod{q}$. Let $f(s)$ be analytic in the strip $|\Im(s)|\le1/2+\epsilon$ for some $\epsilon>0$, real-valued on the real line, and such that $|f(s)|\ll(1+|s|)^{-1-\delta}$ for some $\delta>0$. Then if $\rho=1/2+i\gamma$ runs over the nontrivial zeroes of $L(s,\chi)$, we have \begin{align*} \sum_{\rho}f(\gamma)=&\frac1{2\pi}\hat{f}(0)\log\frac{q}{\pi}+(1-\kappa)f\left(\frac1{2i}\right) +\frac1{2\pi}\int_{-\infty}^{\infty}f(u)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du \\ &-\frac1{2\pi}\sum_{n=2}^{\infty}\frac{\Lambda(n)\chi(n)}{\sqrt{n}}\left(\hat{f}\left(\frac{\log n}{2\pi}\right)+\hat{f}\left(-\frac{\log n}{2\pi}\right)\right) \end{align*} where $\kappa$ is equal to $0$ if $\chi(-1)=1$ and $1$ if $\chi(-1)=-1$. \end{lem}
\begin{proof} This is a specialization of Theorem 5.12 in~\cite{iwakow}. We reproduce the proof here.
We begin by proving a version of this identity for Mellin transforms. This states that for a $C^{\infty}$ function $\varphi:(0,+\infty)\to\mathbb{C}$ with compact support and Mellin transform $\tilde{\varphi}(s)=\int_0^{\infty}\varphi(x)x^{s-1}dx$, and setting $\psi(x)=x^{-1}\varphi(x^{-1})$, we have \begin{align*} \sum_{\rho}\tilde{\varphi}(\rho)= &\varphi(1)\log\frac{q}{\pi}+(1-\kappa)\tilde{\varphi}(1)+\frac1{2\pi}\int_{-\infty}^{\infty}\tilde{\varphi}(u)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du\\ &-\sum_n(\Lambda(n)\chi(n)\varphi(n)+\Lambda(n)\overline{\chi}(n)\psi(n)). \end{align*} To obtain this, we recall that the functional equation for $L(s,\chi)$ takes the form \[ \left(\frac q{\pi}\right)^{s/2}\Gamma\left(\frac{\kappa+s}2\right)L(s,\chi) =\epsilon(\chi)\left(\frac q{\pi}\right)^{(1-s)/2}\Gamma\left(\frac{\kappa+1-s}2\right)L(1-s,\overline{\chi}) \] from which \[ \log\frac{q}{\pi}+\frac12\frac{\Gamma'}{\Gamma}\left(\frac{\kappa+s}2\right)+\frac12\frac{\Gamma'}{\Gamma}\left(\frac{\kappa+1-s}2\right)+\frac{L'}{L}(s,\chi)+\frac{L'}{L}(1-s,\overline{\chi})=0. \] We multiply this equation by $\tilde{\varphi}(s)$, integrate along the line $\Re(s)=2-\delta$, and divide by $2\pi i$. The first term becomes $\varphi(1)\log q$ by Mellin inversion. The line of integration for the second and third terms (with $\Gamma'/\Gamma$) may be shifted to $\Re(s)=1/2$ while picking up a single pole at $s=1$ if $\kappa=0$ and nothing otherwise (giving the term $(1-\kappa)\tilde{\varphi}(1)$), after which the two terms in the integrand are conjugates and cancel to give $\tilde{\varphi}(u)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)$ for $s=1/2+iu$. We shift the line of integration for the fourth and fifth terms (with $L'/L$) to $\Re(s)=-\delta$, picking up a pole with residue $\tilde{\varphi}(\rho)$ for each zero $\rho$ of $L$ on the $1/2$-line. Finally, the remaining integrals may be written as $\sum_n(\Lambda(n)\chi(n)\varphi(n)+\Lambda(n)\overline{\chi}(n)\psi(n))$ by Mellin inversion.
The Fourier transform version follows from setting $\varphi(x)=\frac{1}{2\pi}x^{-1/2}\hat{f}\left(\frac{\log x}{2\pi}\right)$, $x=e^{2\pi y}$. \end{proof}
Let $\chi$ be a primitive character $\pmod{q}$ and let $N(t,\chi)$ be the number of zeros $1/2+i\gamma$ of $L(s,\chi)$ lying in the interval $0\le\gamma\le t$ on the critical line, so that the ``average'' value of $N(t+h,\chi)-N(t-h,\chi)$ is $\frac h{\pi}\log\frac{qt}{2\pi}$. We now apply the explicit formula to our characteristic function approximations $F_{\pm}$ to estimate the deviation of the actual value of $N(t+h,\chi)-N(t-h,\chi)$ from the average as a Dirichlet polynomial.
\begin{lem} \label{zerostoprimesum} Let $t\ge25$, $\Delta\ge2$, and $0<h\le\sqrt{t}$. Let $\chi$ be a primitive character $\pmod{q}$. Then \[ N(t+h,\chi)-N(t-h,\chi)-\frac h{\pi}\log\frac{qt}{2\pi}\le\frac{\log(qt)}{2\pi\Delta} -\frac1{\pi}\Re\left(\sum_{p\le e^{2\pi\Delta}}\frac{\chi(p)\log p}{p^{1/2+it}}\hat{F}_+\left(\frac{\log p}{2\pi}\right)\right)+O(\log\Delta) \] and \[ N(t+h,\chi)-N(t-h,\chi)-\frac h{\pi}\log\frac{qt}{2\pi}\ge-\frac{\log(qt)}{2\pi\Delta} -\frac1{\pi}\Re\left(\sum_{p\le e^{2\pi\Delta}}\frac{\chi(p)\log p}{p^{1/2+it}}\hat{F}_-\left(\frac{\log p}{2\pi}\right)\right)+O(\log\Delta) \] where $F_{\pm}$ are the functions from Lemma~\ref{charfnapprox}. \end{lem}
\begin{proof} Let $f(s)=F_{\pm}(s-t)$, so that $\hat{f}(x)=\hat{F}_{\pm}(x)e^{-2\pi ixt}$. We have \[ \sum_{\rho}F_-(\gamma-t)\le N(t+h,\chi)-N(t-h,\chi)=\sum_{\rho}\chi_{[-h,h]}(\gamma-t)\le\sum_{\rho}F_+(\gamma-t). \] Since $F_{\pm}$ is real and even on the real line, $\hat{F}_{\pm}$ is also real and even. Hence \[ \hat{f}\left(\frac{\log n}{2\pi}\right)+\hat{f}\left(-\frac{\log n}{2\pi}\right)=n^{-it}\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)+n^{it}\hat{F}_{\pm}\left(-\frac{\log n}{2\pi}\right)=\Re\left(\frac{1}{n^{it}}\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)\right) \] so that using the explicit formula from Lemma~\ref{explicitformula}, we find \begin{align*} \sum_{\rho}F_{\pm}(\gamma-t)= &\frac1{2\pi}\hat{F}_{\pm}(0)\log\frac{q}{\pi}+(1-\kappa)F_{\pm}\left(\frac1{2i}-t\right)-\frac1{\pi}\Re\left(\sum_{n=2}^{\infty}\frac{\Lambda(n)\chi(n)}{n^{1/2+it}}\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)\right) \\ &+\frac1{2\pi}\int_{-\infty}^{\infty}F_{\pm}(u-t)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du. \end{align*} We have $F_{\pm}\left(\frac1{2i}-t\right)\ll\frac1{\Delta^2t^2}$ from Property 4 of Lemma~\ref{charfnapprox}. Also, as shown in~\cite{goldgon}, we have that \[ \frac1{2\pi}\int_{-\infty}^{\infty}F_{\pm}(u-t)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du=\frac1{2\pi}\log\frac t2\hat{F}_{\pm}(0)+O(1). \] We reproduce the proof here, using Stirling's approximation for $\Gamma'/\Gamma$. From Property 4 of Lemma~\ref{charfnapprox}, we have \[ \int_{t+4\sqrt{t}}^{\infty}F_{\pm}(u-t)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du\ll\int_{t+4\sqrt{t}}^{\infty}\frac{\log(u+2)}{\Delta^2(u-t)^2}du \ll\frac{\log t}{\sqrt{t}} \] and similarly for the integral from $-\infty$ to $t-4\sqrt t$, while \begin{align*} \int_{t-4\sqrt t}^{t+4\sqrt t}F_{\pm}(u-t)\Re\frac{\Gamma'}{\Gamma}\left(\frac{1/2+\kappa+iu}2\right)du &= \int_{t-4\sqrt t}^{t+4\sqrt t}F_{\pm}(u-t)\left(\log\frac u2 + O(1/u)\right)du \\ &= \int_{t-4\sqrt t}^{t+4\sqrt t}F_{\pm}(u-t)\left(\log\frac t2 + O(1/t)\right)du \\ &= \int_{-\infty}^{\infty}F_{\pm}(u-t)\log\frac t2du+O\left(\frac{\log(t/2)+2h}{t}\right)\\ &= \left(\log\frac t2\right)\hat{F}(0)+O(1). \end{align*} Finally, since $\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)=0$ for $n\ge e^{2\pi\Delta}$, we have \[ \Re\left(\sum_{n=2}^{\infty}\frac{\Lambda(n)\chi(n)}{n^{1/2+it}}\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)\right) =\Re\left(\sum_{n\le e^{2\pi\Delta}}\frac{\Lambda(n)\chi(n)}{n^{1/2+it}}\hat{F}_{\pm}\left(\frac{\log n}{2\pi}\right)\right) \] \[ =\Re\left(\sum_{p\le e^{2\pi\Delta}}\frac{\Lambda(p)\chi(p)}{p^{1/2+it}}\hat{F}_{\pm}\left(\frac{\log p}{2\pi}\right)\right) +\Re\left(\sum_{p\le e^{\pi\Delta}}\frac{\Lambda(p)\chi(p)}{p^{1+2it}}\hat{F}_{\pm}\left(\frac{\log p}{\pi}\right)\right)+O(1) \] \[ =\Re\left(\sum_{p\le e^{2\pi\Delta}}\frac{\Lambda(p)\chi(p)}{p^{1/2+it}}\hat{F}_{\pm}\left(\frac{\log p}{2\pi}\right)\right)+O(\log\Delta). \] \end{proof}
Finally, we need the following version of the large sieve inequality.
\begin{defn}
A set of points $\{s_r^{\chi}=\sigma_r^{\chi}+it_r^{\chi}\}$, where $r=1,\dotsc,R_{\chi}$ and $\chi$ ranges over the characters $\pmod{q}$, is well-separated if, for any given $\chi$, we have $|t_i^{\chi}-t_j^{\chi}|\ge1$ for all $i\neq j$. \end{defn}
\begin{prop} \label{sieve} Let $A(s,\chi)=\sum_{p\le N}a(p)\chi(p)p^{-s}$ be a twisted Dirichlet polynomial. Let $T$ be large and let $\{s_r^{\chi}\}$ be a set of well-separated points with $T<t_1^{\chi}<t_2^{\chi}<\dotsb<t_{R_{\chi}}^{\chi}\le 2T$ for all $\chi$ and $\sigma_r^{\chi}\ge\alpha$. Then for any $k$ with $N^k\le qT$, we have \[
\sum_{\chi}\sum_{r=1}^{R_{\chi}}|A(s_r^{\chi},\chi)|^{2k}\ll \varphi(q)T(\log T)^2k!\left(\sum_{p\le N}|a(p)|^2p^{-2\alpha}\right)^k. \] \end{prop}
\begin{proof}
The proof follows that of Proposition 4 in~\cite{maimont}; it is identical except for the inclusion of a summation over characters. Let \[ B(s,\chi)=A(s,\chi)^k=\sum_{n\le N^k}c_n\chi(n)n^{-s} \] and let $D_r(s)$ denote the disk of radius $r$ centered at $s$. By the mean-value property of harmonic functions and the Cauchy-Schwarz inequality, we have \[
|B(s,\chi)|^2\le\frac{(\log T)^2}{\pi}\iint_{D_{1/\log T}(s)}|B(x+iy,\chi)|^2dxdy \] for all $s$. Since the disks $D_{1/\log T}(s_r^{\chi})$ are disjoint and lie in the half-strip $\sigma\ge\alpha-1/\log T$, between the lines $t=T-1$ and $t=2T+1$, we can write \[
\sum_{\chi}\sum_{r=1}^{R_{\chi}}|B(s_r^{\chi},\chi)|^2\ll(\log T)^2\int_{\alpha-1/\log T}^{\infty}\sum_{\chi}\int_{T-1}^{2T+1}|B(\sigma+it,\chi)|^2dtd\sigma. \] By the proof of Theorem 6.4 of~\cite{mont}, specifically Equation 6.14, we have \[
\sum_{\chi}\int_{T-1}^{2T+1}|B(\sigma+it,\chi)|^2dt=\varphi(q)(T+O(N^k/q))\sum_n\frac{|c_n|^2}{n^{2\sigma}} \] which, using the condition that $N^k\le qT$, plugging back into the previous inequality, and integrating with respect to $\sigma$, gives \[
\sum_{\chi}\sum_{r=1}^{R_{\chi}}|B(s_r^{\chi},\chi)|^2\ll\varphi(q)T(\log T)^2\sum_n\frac{|c_n|^2}{n^{2\alpha}\log n}\le\varphi(q)T(\log T)^2\sum_n\frac{|c_n|^2}{n^{2\alpha}} \] (since $c_1=0$). If $n$ has prime factorization $p_1^{e_1}\dotsb p_m^{e_m}$ with $e=e_1+\dotsb e_m$, then we can explicitly write \[ c_n=\binom{e}{e_1,\dotsc,e_m}\prod_{i=1}^ma(p_i)^{e_i} \] so \begin{align*}
\sum_n\frac{|c_n|^2}{n^{2\alpha}} &=\sum_n\binom{e}{e_1,\dotsc,e_m}^2\prod_{i=1}^m\frac{|a(p_i)|^{e_i}}{p_i^{2k_i\alpha}} \\
&\le k!\sum_n\binom{e}{e_1,\dotsc,e_m}\prod_{i=1}^m\frac{|a(p_i)|^{e_i}}{p_i^{2k_i\alpha}} = k!\left(\sum_{p\le N}|a(p)|^2p^{-2\alpha}\right)^k \end{align*} and we are done. \end{proof}
\section{Point count bounds} \label{pointcount}
In this section, we use Lemma~\ref{zerostoprimesum} and Lemma~\ref{sieve} to upper-bound the frequency with which an ordinate $t$ may have an abnormal number of zeros of $L(s,\chi)$ near it. We introduce some quick notation.
\begin{defn} \label{range}
Let \[
a(T,q)=\sqrt{\log q}(\log\log(qT))^2,\hspace{0.2in} b(T,q)=\frac{\log(qT)}{\log\log(qT)}. \] \end{defn}
We will also need the following elementary inequality.
\begin{lem} \label{elem1} Let $a(T,q)\le V\le b(T,q)$, $\eta=1/\log V$, $k=\left\lfloor\frac{V}{1+\eta}\right\rfloor$. Then we have \[ k(\log(k\log\log qT)-2\log(\eta V)\le -V\log\left(\frac{V}{\log\log qT}\right)+2V\log\log V+V. \] \end{lem}
\begin{proof} This calculation is Proposition 14 in~\cite{baladero}. We reproduce the proof here. Since $k\le V$, we have\[\log(k\log\log(qT))-2\log(\eta V)\le-\log\left(\frac{V}{\log\log(qT)}\right)+2\log\log V\le0\]but also $k\ge\frac{V}{1+\eta}-1\ge V(1-\eta)$, so\[k(\log(k\log\log(qT))-2\log(\eta V))\le V(1-\eta)\left(-\log\left(\frac{V}{\log\log(qT)}\right)+2\log\log V\right)\]\[\le-V\log\left(\frac{V}{\log\log(qT)}\right)+2V\log\log V+\eta V\log V=-V\log\left(\frac{V}{\log\log(qT)}\right)+2V\log\log V+V.\] \end{proof}
All our statements about the distribution of zeros of $L(s,\chi)$ assume that $\chi$ is primitive, but of course Perron's formula for $M(x;q,a)$ includes the imprimitive characters $\pmod{q}$ as well. To deal with this, we just note that there is not much difference between the behavior of $\chi$ and $\chi_1$ where $\chi_1$ is the primitive character inducing $\chi$.
\begin{lem} \label{nonprimerr} For any character $\chi\pmod{q}$, let $\chi_1$ be the primitive character inducing $\chi$ and $q_1$ be the conductor of $\chi$. \begin{enumerate} \item For any twisted Dirichlet polynomial $A(s,\chi)=\sum_{p\le N}a(p)\chi(p)p^{-s}$ and $\sigma\ge1/2$, we have \[
|A(\sigma+it,\chi_1)-A(\sigma+it,\chi)|\ll(\max_pa(p))\frac{\sqrt{\log q}}{\log\log q}. \]
\item If $l_d(s,\chi)=\prod_{p|d}\left(1-\frac{\chi(p)}{p^s}\right)$, then we have \[
\log|l_d(s,\chi)|=O\left(\frac{\sqrt{\log d}}{\log\log d}\right) \] and, as a consequence, \[
\log|L(s,\chi)| - \log|L(s,\chi_1)| = O\left(\frac{\sqrt{\log q}}{\log\log q}\right). \] \end{enumerate} \end{lem}
\begin{proof} \begin{enumerate} \item We can write \[
|A(\sigma+it,\chi_1)-A(\sigma+it,\chi)|=\left|\sum_{p|q}\frac{a(p)\chi_1(p)}{p^{\sigma+it}}\right|\le\max_pa(p)\sum_{p|q}\frac1{p^{1/2}}. \] We then have \[
\sum_{p|q}\frac1{p^{1/2}}\le\sum_{p\le2\log q}\frac1{p^{1/2}}\ll\frac{\sqrt{\log q}}{\log\log q}. \]
\item We can write \[
\log|l_d(s,\chi)|=\sum_{p|d}\log\left|1-\frac{\chi(p)}{p^s}\right|\le \sum_{p|d}\frac1{p^{1/2}}=O\left(\frac{\sqrt{\log d}}{\log\log d}\right) \] as in part 1, and \[
L(s,\chi)=L(s,\chi_1)\prod_{p|q}\left(1-\frac{\chi_1(p)}{p^s}\right)=L(s,\chi_1)l_q(s,\chi_1). \] \end{enumerate} \end{proof}
We are now ready to formulate the analog for Dirichlet $L$-functions of the key estimate in Soundararajan's proof.
\begin{defn} \label{vtypical} Let $q\in\mathbb{N}$ and let $\chi$ be a character $\pmod{q}$ induced by primitive $\chi_1\pmod{q_1}$. Let $T>e$, $0<\delta\le1$, and $a(T,q)\le V\le b(T,q)$.
An ordinate $t\in[T,2T]$ is $(V,\delta,\chi)$-typical of order $T$ if it satisfies the following conditions:
i. Let $y=(qT)^{1/V}$. For all $\sigma\ge\frac12$, we have \[
\left|\sum_{n\le y}\frac{\chi_1(n)\Lambda(n)}{n^{\sigma+it}\log n}\frac{\log(y/n)}{\log y}\right|\le2V. \]
ii. Every sub-interval of $(t-1,t+1)$ of length $\frac{2\delta\pi V}{\log(qT)}$ contains at most $(1+\delta)V$ ordinates of zeroes of $L(s,\chi)$.
iii. Every sub-interval of $(t-1,t+1)$ of length $\frac{2\pi V}{\log V\log(qT)}$ contains at most $V$ ordinates of zeroes of $L(s,\chi)$. \end{defn}
Note that this definition differs from the one used in~\cite{halupsu}, in that it sets $y=(qT)^{1/V}$ rather than $T^{1/V}$. When there is no risk of confusion, we will shorten ``$(V,\delta,\chi)$-typical'' to ``$V$-typical''.
Our next two propositions give an upper bound for the size of any set of well-separated $V$-atypical ordinates. First we bound the size of any set of well-separated ordinates with a given accumulation of zeros in intervals of given size centered on the ordinates.
\begin{prop} \label{zerocount} Let $\chi$ be a character $\pmod{q}$ with conductor $q_1$, $T$ be large, $h$ be such that $0<h\le\sqrt T$, $a(T,q)\le V\le b(T,q)$, and $\{t_r^{\chi}\}$ be well-separated ordinates with $T<t_1^{\chi}<t_2^{\chi}<\dotsb<t_{R_{\chi}}^{\chi}\le 2T$ for all $\chi$, such that for all $\chi$ and $r$ we have \[ N(t_r^{\chi}+h)-N(t_r^{\chi}-h,\chi)-\frac{h}{\pi}\log\frac{q_1t_r^{\chi}}{2\pi}\ge V+O(1). \] Let $R=\sum_{\chi}R_{\chi}$. Then \[ R\ll\varphi(q) T\exp\left(-V\log\frac{V}{\log\log(qT)}+2V\log\log V+O(V)\right). \] \end{prop}
\begin{proof} By Lemma~\ref{zerostoprimesum}, we have for all $\Delta\ge2$ that \begin{align*} V+O(1) &\le N(t_r^{\chi}+h)-N(t_r^{\chi}-h,\chi)-\frac{h}{\pi}\log\frac{q_1t_r^{\chi}}{2\pi} \\
&\le\frac{\log(2qT)}{2\pi\Delta}+\left|\frac1{\pi}\sum_{p\le e^{2\pi\Delta}}\frac{\chi_1(p)\log p}{p^{1/2+it_r^{\chi}}}\hat{F}_+\left(\frac{\log p}{2\pi}\right)\right|+O(\log\Delta). \end{align*}
Let $a(p)=\frac{\log p}{\pi}\hat{F}_+\left(\frac{\log p}{2\pi}\right)$, so that $|a(p)|\le4$ and \[
\left|\sum_{p\le e^{2\pi\Delta}}\frac{a(p)\chi_1(p)}{p^{1/2+it_r^{\chi}}}\right|\ge V-\frac{\log(2qT)}{2\pi\Delta}+O(\log\Delta) \] which implies by Lemma~\ref{nonprimerr} that \[
\left|\sum_{p\le e^{2\pi\Delta}}\frac{a(p)\chi(p)}{p^{1/2+it_r^{\chi}}}\right|\ge V-\frac{\log(2qT)}{2\pi\Delta}+O\left(\log\Delta+\frac{\sqrt{\log q}}{\log\log q}\right). \] Let $\eta=1/\log V$ and $\Delta=\frac{(1+\eta)\log(qT)}{2\pi V}$. We have $\exp(2\pi\Delta)=(qT)^{(1+\eta)/V}$ and $\log\Delta\ll\log\log(qT)\le\sqrt{V}$. Also \[ \eta V=\frac{V}{\log V}\ge\frac{2\sqrt{\log q}(\log\log(qT))^2}{\log\log q+4\log\log\log(qT)}\ge\sqrt{\log q} \] so for $q$ sufficiently large, we have \[ \frac{\sqrt{\log q}}{\log\log q}\le\frac{\sqrt{\log q}}{100}\le\frac1{100}\eta V. \] On the other hand, for small $q$, we also have \[ \frac{\sqrt{\log q}}{\log\log q}\le\frac1{100}\eta V \] for $T$ sufficiently large. Therefore, as long as $qT$ is sufficently large, we have \[ V-\frac{\log(2qT)}{2\pi\Delta}+O\left(\log\Delta+\frac{\sqrt{\log q}}{\log\log q}\right)\ge V-\frac{V\log(2qT)}{(1+\eta)\log(qT)}+\frac{\eta V}{100}+O(\sqrt{V}) \] \[ \ge\frac{\eta V}{1+\eta}-\frac{\log 2}{(1+\eta)\log\log T}+\frac{\eta V}{100}+O(\sqrt{V})\ge\frac12\eta V. \]
Let $k=\left\lfloor\frac{V}{1+\eta}\right\rfloor$, so that in fact $\exp(2\pi\Delta)^k\le T$ and we can apply Proposition~\ref{sieve}. This gives \[
R\left(\frac12\eta V\right)^{2k}\le\sum_{\chi}\sum_r\left|\sum_{p\le(qT)^{(1+\eta)/V}}\frac{a(p)}{p^{1/2+it_r^{\chi}}}\right|^{2k}\ll\varphi(q) T(\log T)^2(Ck\log\log(qT))^k. \] Using Lemma~\ref{elem1}, this implies \[ R\ll\varphi(q)T\exp\left(-V\log\frac{V}{\log\log(qT)}+2V\log\log V+O(V)\right). \] \end{proof}
We now translate this bound into one for $V$-atypical ordinates.
\begin{prop} \label{vtypicalbd} Let $\chi$ be a character $\pmod{q}$, let $T$ be large, let $a(T,q)\le V\le b(T,q)$, and let $\{t_r^{\chi}\}$ be well-separated $V$-atypical ordinates with $T<t_1^{\chi}<t_2^{\chi}<\dotsb<t_{R_{\chi}}^{\chi}\le 2T$ for all $\chi$. Let $R=\sum_{\chi}R_{\chi}$. Then \[ R\ll T\exp\varphi(q)\left(-V\log\left(\frac{V}{\log\log(qT)}\right)+2V\log\log V+O(V)\right). \] \end{prop}
\begin{proof} Let $R_1$ be the number of $t_r^{\chi}$ in the list failing criterion (i). For such $t_r^{\chi}$ there is a $\sigma_r^{\chi}\ge\frac12$ such that \[
\left|\sum_{n\le y}\frac{\chi_1(n)\Lambda(n)}{n^{\sigma_r^{\chi}+it}\log n}\frac{\log(y/n)}{\log y}\right|>2V. \] The sum over $n=p^{\alpha}$ with $\alpha\ge2$ is \[
\left|\sum_{p^{\alpha}\le y,\alpha\ge2}\frac{\chi_1(p^{\alpha})}{p^{\sigma_r^{\chi}+it}}\frac{\log(y/n)}{\log y}\right| \le\sum_{p\le\sqrt y}\frac1{p}+\sum_{p^{\alpha}\le y,\alpha\ge3}\frac1{p^{\alpha/2}} \] \[ \ll\log\log y\ll\log\log(qT)\le\sqrt{V}. \] The error $\frac{\sqrt{\log q}}{\log\log q}$ from subtracting off the contribution from imprimitive characters is a small fraction of $V$ for $qT$ sufficiently large, as argued in the proof of Proposition~\ref{zerocount}, so it suffices to count $t_r^{\chi}$ such that \[
\left|\sum_{p\le y}\frac{\chi(p)}{p^{\sigma_r+it}}\frac{\log(y/p)}{\log y}\right|\ge V. \] Since $y^k\le qT$ for all $k\le V$, applying Proposition~\ref{sieve} gives \[
R_1V^{2k}\le\sum_{\chi}\sum_{r\le R}\left|\sum_{p\le y}\frac{\chi(p)}{p^{\sigma_r+it}}\frac{\log(y/p)}{\log y}\right|^{2k} \ll \varphi(q)T(\log T)^2k!\left(\sum_{p\le y}\frac{\log^2(y/p)}{p\log^2y}\right)^k \] and we have \[ \sum_{p\le y}\frac{\log^2(y/p)}{p\log^2y}\le\sum_{p\le y}\frac1p\ll\log\log y\ll\log\log(qT), \] so \[ R_1\ll \varphi(q)T\exp\left(-V\log\frac{V}{\log\log(qT)}+O(V)\right). \]
Next let $R_2$ be the number of $t_r^{\chi}$ in the list failing criterion (ii). For each such $t_r$ there is some $t_r'$ with $|t_r-t_r'|\le1$ and \[ N\left(t_r'+\frac{\pi\delta V}{\log(qT)},\chi\right)-N\left(t_r'-\frac{\pi\delta V}{\log(qT)},\chi\right)>(1+\delta)V \] from which \[ N\left(t_r'+\frac{\pi\delta V}{\log(qT)},\chi\right)-N\left(t_r'-\frac{\pi\delta V}{\log(qT)},\chi\right)-\frac{\delta V}{\log(qT)}\log\left(\frac{q_1t_r'}{2\pi}\right)\ge V+O(1). \] Applying Proposition~\ref{zerocount} on the sets $\{t_{3s+j}^{\chi}\}$ for $j=0,1,2$ gives the desired bound on $R_2$. The computation for $R_3$, the number of $t_r^{\chi}$ failing criterion (iii), is effectively the same. \end{proof}
We will additionally find it convenient to obtain the following absolute bound on the amount by which $N(t+h,\chi)-N(t-h,\chi)$ can deviate from average. This is an analog of Theorem 1 of~\cite{goldgon}, and is also proven as Propositon 7 of~\cite{halupsu}.
\begin{prop} \label{goldgonanalog} Let $qt$ be sufficiently large, $0<h\le\sqrt{t}$, and $\chi$ be a primitive character $\pmod{q}$. Then \[
\left|N(t+h,\chi)-N(t-h,\chi)-\frac{h}{\pi}\log\frac{qt}{2\pi}\right|\le\frac{\log(qt)}{2\log\log(qt)}+\left(\frac12+o(1)\right)\frac{\log(qt)\log\log\log(qt)}{(\log\log(qt))^2}. \] \end{prop}
\begin{proof} We apply Proposition~\ref{zerostoprimesum} with $\delta=\frac1{\pi}\log\frac{\log(qt)}{\log\log(qt)}$. Since \[
\left|\frac1{\pi}\Re\left(\sum_{p\le e^{2\pi\Delta}}\frac{\chi(p)\log p}{p^{1/2+it}}\hat{F}_+\left(\frac{\log p}{2\pi}\right)\right)\right| \ll\sum_{p\le e^{2\pi\Delta}}\frac1{\sqrt p}\ll\frac{e^{\pi\Delta}}{\Delta}, \] we can compute \[
\left|N(t+h,\chi)-N(t-h,\chi)-\frac{h}{\pi}\log\frac{qt}{2\pi}\right| \] \[ \le\frac{\log(qt)}{2(\log\log(qt)-\log\log\log(qt))}+O\left(\frac{\log(qt)/\log\log(qt)}{\log\log(qt)-\log\log\log(qt)}\right) \] \[ =\frac{\log(qt)}{2\log\log(qt)}\sum_{k=0}^{\infty}\left(\frac{\log\log\log(qt)}{\log\log(qt)}\right)^k+O\left(\frac{\log(qt)}{(\log\log(qt))^2}\right) \] \[ =\frac{\log(qt)}{2\log\log(qt)}+\frac{\log(qt)\log\log\log(qt)}{2(\log\log(qt))^2}(1+o(1)). \] \end{proof}
The above bound allows us to state, for a given $t$, a choice of $V$ for which $t$ is guaranteed to be $V$-typical.
\begin{prop} \label{extremev} For $V$ between $(1/2+o(1))\log(qT)/\log\log(qT)$ and $\log(qT)/\log\log(qT)$, all ordinates $t\in[T,2T]$ are $V$-typical of order $T$. \end{prop}
\begin{proof} We have \[ f(u)=\sum_{n\le u}\frac{\Lambda(n)\chi_1(n)}{\sqrt{n}\log n}\ll\frac{\sqrt{u}}{\log u} \] and consequently \[ \sum_{n\le y}\frac{\Lambda(n)\chi_1(n)}{\sqrt{n}\log n}\log\left(\frac yn\right)=\sum_{n\le y}\frac{\Lambda(n)\chi_1(n)}{\sqrt{n}\log n}\sum_{n\le j\le y}\log\left(\frac{j+1}j\right) \] \[ =\sum_{j\le y}\sum_{n\le j}\frac{\Lambda(n)\chi_1(n)}{\sqrt{n}\log n}\log\left(\frac{j+1}j\right)=\sum_{j\le y}\int_j^{j+1}\sum_{n\le j}\frac{\Lambda(n)\chi_1(n)}{\sqrt{n}\log n}\frac{du}{u} \] \[ =\int_1^yf(u)\frac{du}u\ll\frac{\sqrt{y}}{\log y}. \] We conclude that \[
\left|\sum_{n\le y}\frac{\chi_1(n)\Lambda(n)}{n^{\sigma_r+it}\log n}\frac{\log(y/n)}{\log n}\right|\ll\frac{\sqrt{y}}{(\log y)^2}=\frac{V^2(qT)^{1/V}}{(\log(qT))^2}\le\frac{\log(qT)}{(\log\log(qT))^2}=o(V). \] The other two follow directly from Proposition~\ref{goldgonanalog}. \end{proof}
\section{$L(s,\chi)$ bounds} \label{Lfuncbds}
Our next goal is to find lower bounds for $L(s,\chi)$ given that $s$ has $V$-typical imaginary part, which we will obtain from bounds for the logarithmic derivative $\frac{L'}{L}(s,\chi)$. Let $F(s,\chi)=\sum_{\rho}\Re\left(\frac{1}{s-\rho}\right)$ where $\rho$ runs over the nontrivial zeroes of $L(s,\chi)$. We can write $L'/L$ in terms of $F$.
\begin{prop} \label{logderinF} Let $\chi$ be a primitive character $\pmod{q}$, $T$ be sufficiently large, $\frac12\le\sigma\le2$, $T\le t\le 2T$, and $\sigma$ be such that $L(\sigma+it,\chi)\neq0$. Then \[ \Re\frac{L'}{L}(\sigma+it,\chi)=F(\sigma+it,\chi)-\frac12\log(qT)+O(1). \] \end{prop}
\begin{proof} We start with Equation 12.17 of~\cite{davenport}, which states that for primitive $\chi$, \[ \frac{L'}{L}(s,\chi)=-\frac12\log\frac q{\pi}-\frac12\frac{\Gamma'}{\Gamma}\left(\frac{s+\kappa}2\right)+B(\chi)+\sum_{\rho}\left(\frac1{s-\rho}+\frac1{\rho}\right) \] where $\Re(B(\chi))=-\sum_{\rho}\Re(1/\rho)$. From Stirling's formula for $\Gamma'/\Gamma$, we have \begin{align*} \Re\frac{L'}{L}(s,\chi) &=-\frac12\log\frac q{\pi}-\frac12\Re\frac{\Gamma'}{\Gamma}\left(\frac{\sigma+it+\kappa}2\right)+\Re(B(\chi))+\sum_{\rho}\Re\left(\frac1{\sigma+it-\rho}+\frac1{\rho}\right) \\
&=-\frac12\log\frac q{\pi}-\frac12\log|\sigma+it+\kappa|+O(|\sigma+it+\kappa|^{-1})+F(\sigma+it,\chi) \\ &=F(\sigma+it,\chi)-\frac12\log(qT)+O(1). \end{align*} \end{proof}
We also need $L'/L$ in terms of a sum over primes together with another sum over zeros of $L$.
\begin{prop} \label{eqnwithz} Let $\chi$ be a primitive character $\pmod{q}$ and $y\ge1$. Let $z\in\mathbb{C}$ be such that $\Re(z)\ge0$ and $T\le\Im(z)\le2T$, and assume that $z$ is not a pole of $\frac{L'}{L}(s,\chi)$. Then \[ \sum_{n\le y}\frac{\chi(n)\Lambda(n)}{n^z}\log\left(\frac yn\right)=-\frac{L'}{L}(z,\chi)\log y-\left(\frac{L'}{L}\right)'(s,\chi)-\sum_{\rho}\frac{y^{\rho-z}}{(\rho-z)^2}+O(T^{-1}). \] \end{prop}
\begin{rem} This is Proposition 13 of~\cite{halupsu}, with the condition $y\le T$ removed. \end{rem}
\begin{proof} Proposition 12 in~\cite{halupsu} states that for $\Re(z)>1/2$, \[ \sum_{n\le y}\frac{\chi(n)\Lambda(n)}{n^z}\log\left(\frac yn\right)=-\frac{L'}{L}(z,\chi)\log y-\left(\frac{L'}{L}\right)'(z,\chi)-\sum_{\rho}\frac{y^{\rho-z}}{(\rho-z)^2}-\sum_{n\ge0}\frac{y^{-2n-\kappa-z}}{(z+2n+\kappa)^2}. \] The proof is as follows. We begin with the identity \[ \frac1{2\pi i}\int_{c-i\infty}^{c+i\infty}-\frac{L'}{L}(z+w,\chi)\frac{y^w}{w^2}dw=\sum_{n\le y}\frac{\Lambda(n)\chi(n)}{n^z}\log\frac yn, \]
which is valid for $\Re(z)>1/2$ and $c>1/2$, and can be obtained by expanding $-\frac{L'}{L}(z+w,\chi)$ as a Dirichlet series and integrating term by term. Then we shift the line of integration to $\Re(w)=-2N-1-\kappa-\Re(z)$ where $N$ is a positive integer, picking up the residue $-\frac{L'}{L}(z,\chi)\log y-\left(\frac{L'}{L}\right)'(z,\chi)$ from the pole at $0$ and the sums $-\sum_{\rho}\frac{y^{\rho-z}}{(\rho-z)^2}$ and $-\sum_{0\le n\le2N+\kappa}\frac{y^{-2n-\kappa-z}}{(z+2n+\kappa)^2}$ from the nontrivial and trivial zeroes of $L$ respectively. Since\\ $\frac{L'}{L}(s,\chi)\ll\log(q|s|)$ for $\Re(s)\le-1$ excluding circles of radius $1/2$ around the trivial zeroes (see page 116 of~\cite{davenport}), the integral $\int_{c'-i\infty}^{c'+i\infty}-\frac{L'}{L}(z+w,\chi)\frac{y^w}{w^2}dw$ approaches $0$ as $N$ approaches infinity, giving the identity above.
Finally we note that since $\Re(-2n-\kappa-z)\le0$ for $n\ge0,\kappa\in\{0,1\}$, and $\Re(z)\ge0$, it is clear that \[
\left|\sum_{n\ge0}\frac{y^{-2n-\kappa-z}}{(z+2n+\kappa)^2}\right|\le\sum_{n\ge0}\frac1{(T+2n+\kappa)^2}\ll\frac1T. \]
\end{proof}
From the two previous propositions, we can write an explicit expression for a lower bound for $\log|L(\sigma+it,\chi)|$.
\begin{prop} \label{initineq} Let $\chi$ be a character $\pmod{q}$ induced by $\chi_1\pmod{q_1}$. Let $T$ be sufficiently large and $t\in[T,2T]$. Then we have for all $\frac12\le\sigma\le2$ and $y\ge2$ that \[
\log|L(\sigma+it,\chi)|\ge\Re\left(\sum_{n\le y}\frac{\Lambda(n)\chi_1(n)}{n^{\sigma+it}\log n}\frac{\log(y/n)}{\log y}\right)-\left(1+\frac{y^{1/2-\sigma}}{(\sigma-1/2)\log y}\right)\frac{F(\sigma+it,\chi)}{\log y} \] \[ +O\left(\frac{\sqrt{\log q}}{\log\log q}\right). \] \end{prop}
\begin{proof} First assume $\chi$ is primitive. We integrate the equation from Proposition~\ref{eqnwithz} from $z=\sigma+it$ to $z=2+it$. This gives \[ \sum_{n\le y}\Lambda(n)\chi(n)\log\frac yn\left(\frac{n^{-2-it}}{-\log n}-\frac{n^{-\sigma-it}}{-\log n}\right) \] \[ =(-\log L(2+it,\chi)+\log L(\sigma+it,\chi))\log y-\frac{L'}{L}(2+it,\chi)+\frac{L'}{L}(\sigma+it,\chi) \] \[ -\sum_{\rho}\int_{\sigma}^2\frac{y^{\rho-u-it}}{(\rho-u-it)^2}du+O(T^{-1}) \] or, rearranging and absorbing small terms into the error term, \[ (\log y)\log L(\sigma+it,\chi)=\sum_{n\le y}\frac{\Lambda(n)\chi(n)}{n^{\sigma+it\log n}}\log\frac yn-\frac{L'}{L}(\sigma+it,\chi)+\sum_{\rho}\int_{\sigma}^2\frac{y^{\rho-u-it}}{(\rho-u-it)^2}du+O(\log y). \] We divide by $\log y$, take real parts, and use Proposition~\ref{logderinF} to turn this into \[
\log|L(\sigma+it,\chi)|=\Re\sum_{n\le y}\frac{\Lambda(n)\chi(n)}{n^{\sigma+it}\log n}\frac{\log(y/n)}{\log y}-\frac{F(\sigma+it)}{\log y}+\frac{\log(qT)}{2\log y} \] \[ +\frac1{\log y}\Re\sum_{\rho}\int_{\sigma}^2\frac{y^{\rho-u-it}}{(\rho-u-it)^2}du+O(1) \] where \[
\left|\sum_{\rho}\int_{\sigma}^2\frac{y^{\rho-u-it}}{(\rho-u-it)^2}du\right| \le\sum_{\rho}\frac1{|\rho-\sigma-it|^2}\int_{\sigma}^2y^{1/2-u}du \] \[
\le\frac{y^{1/2-\sigma}}{\log y}\sum_{\rho}\frac1{|\rho-\sigma-it|^2}=\frac{y^{1/2-\sigma}}{(\sigma-1/2)\log y}F(\sigma+it,\chi). \] This gives the inequality for $\chi$ primitive. Otherwise we have \[
\log|L(s,\chi)|
=\log|L(s,\chi_1)|+O\left(\frac{\sqrt{\log q}}{\log\log q}\right) \] by Lemma~\ref{nonprimerr}. \end{proof}
Now we assume that $t$ is $V$-typical and get two explicit bounds for $\log|L(\sigma+it,\chi)|$, depending on the size of $\sigma$. The first is for $\sigma$ not too close to $1/2$, and is like Proposition 15 of~\cite{halupsu}, with the condition $T\ge q$ removed.
\begin{prop} \label{largesglogbound} Let $\chi$ be a character $\pmod{q}$. Let $T$ be sufficiently large, $a(T,q)\le V\le b(T,q)$, and $t\in[T,2T]$ be $(V,\delta,\chi)$-typical of order $T$. Then for $\sigma$ such that $\frac12+\frac{V}{\log(qT)}\le\sigma\le2$ and some $C>0$, we have \[
\log|L(\sigma+it,\chi)|\ge-C\left(\frac{V}{\delta}+\sqrt{\frac{\log q}{\log\log q}}\right). \] \end{prop}
\begin{proof} For $y=(qT)^{1/V}$, we have \[ \frac{y^{1/2-\sigma}}{(\sigma-1/2)\log y}\le\frac{\exp\left(-\frac{V}{\log(qT)}\cdot\frac{\log qT}{V}\right)}{\frac{V}{\log(qT)}\frac{\log qT}{V}} =\frac{\exp(-1)}{1}<1 \] so by Proposition~\ref{initineq}, given that $t$ is $(V,\delta,\chi)$-typical, we have \[
\log|L(\sigma+it,\chi)|\ge-2V-2\frac{V}{\log(qT)}F(\sigma+it,\chi)+O\left(\sqrt{\frac{\log q}{\log\log q}}\right). \] The proof of Proposition 15 in~\cite{halupsu} states that $F(\sigma+it,\chi)=O\left(\frac{\log qT}{\delta}\right)$; we reproduce this computation here. For $0\le n\le N=\left\lfloor\frac{\log(qT)}{4\pi\delta V}\right\rfloor$, let $I_n$ be the set of zeros $\rho=1/2+i\gamma$ such that \[
\frac{2\pi n\delta V}{\log(qT)}\le|t-\gamma|\le\frac{2\pi(n+1)\delta V}{\log(qT)}. \] Note that $\sum_{n=0}^N\frac{a}{a^2+c^2n^2}\le\frac1a+\int_0^{\infty}\frac{a}{a^2+c^2t^2}dt=\frac1a+\frac{\pi}{2c}$. Then since $t$ is $V$-typical, we use Property 2 of Definition~\ref{vtypical} to get \[ \sum_{\gamma\in I_n}\Re\left(\frac1{\sigma+it-1/2-i\gamma}\right) = \sum_{\gamma\in I_n}\frac{\sigma-1/2}{(\sigma-1/2)^2+(t-\gamma)^2} \] \[ \le2(1+\delta)V\sum_{n=0}^N\frac{\sigma-1/2}{(\sigma-1/2)^2+\left(\frac{2\pi n\delta V}{\log(qT)}\right)^2} \le 2(1+\delta)V\left(\frac1{\sigma-1/2}+\frac{\log(qT)}{4\delta V}\right) \] \[ \le2(1+\delta)V\left(\frac{\log(qT)}V+\frac{\log(qT)}{4\delta V}\right)=O(\log(qT)/\delta). \]
For the remaining zeros, we note that if $|t-\gamma|\ge1/2$ and $1/2\le\sigma\le2$ then \[ \frac{(\sigma-1/2)^2+(t-\gamma)^2}{1+(t-\gamma)^2}\ge\frac15\ge\frac{\sigma-1/2}{8} \] so citing Equation 16.3 of~\cite{davenport}, we have \[
\sum_{|t-\gamma|\ge1/2}\Re\left(\frac1{\sigma+it-1/2-i\gamma}\right) = \sum_{|t-\gamma|\ge1/2}\frac{\sigma-1/2}{(\sigma-1/2)^2+(t-\gamma)^2} \] \[
\le\sum_{|t-\gamma|\ge1/2}\frac{8}{1+(t-\gamma)^2}\le\sum_{\rho}\frac{8}{1+(t-\Im(\rho))^2}\ll\log(qt). \] Combining these gives $F(\sigma+it,\chi)=O\left(\frac{\log qT}{\delta}\right)$ as desired, so the middle term $-2\frac{V}{\log(qT)}F(\sigma+it,\chi)$ is $O(V/\delta)$, and we are done.
\end{proof}
If $\sigma$ is very close to $1/2$, we instead compute the following.
\begin{prop} \label{smallsglogbound} Let $\chi$ be a character $\pmod{q}$, $T$ be sufficiently large, $a(T,q)\le V\le b(T,q)$, and $t\in[T,2T]$ be $V$-typical of order $T$. Then we have for all $\frac12<\sigma\le\sigma_0=\frac12+\frac{V}{\log(qT)}$ that \[
\log|L(\sigma+it,\chi)|\ge\log|L(\sigma_0+it,\chi)|-V\log\frac{\sigma_0-1/2}{\sigma-1/2}-2(1+\delta)V\log\log V+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right). \] \end{prop}
\begin{proof} This is Proposition 16 of~\cite{halupsu}; we reproduce the proof here. First assume $\chi$ is primitive. Using Proposition~\ref{logderinF}, we have \[
\log|L(\sigma_0+it,\chi)|-\log|L(\sigma+it,\chi)|=\int_{\sigma}^{\sigma_0}\Re\frac{L'}{L}(u+it,\chi)du\le\int_{\sigma}^{\sigma_0}F(u+it,\chi)du \] \[ =\sum_{\gamma}\int_{\sigma}^{\sigma_0}\frac{u-1/2}{(u-1/2)^2+(t-\gamma)^2}du=\frac12\sum_{\gamma}\log\frac{(\sigma_0-1/2)^2+(t-\gamma)^2}{(\sigma-1/2)^2+(t-\gamma)^2}. \] Again we separate the sum according to the distance of $\gamma$ from $t$. By Property 3 of Definition~\ref{vtypical}, we have \begin{align*}
\frac12\sum_{|t-\gamma|<\pi V/(\log V\log(qT))}\log\frac{(\sigma_0-1/2)^2+(t-\gamma)^2}{(\sigma-1/2)^2+(t-\gamma)^2} &\le\frac12\sum_{|t-\gamma|<\pi V/(\log V\log(q_1T))}\log\frac{(\sigma_0-1/2)^2}{(\sigma-1/2)^2} \\ &\le V\log\frac{\sigma_0-1/2}{\sigma-1/2}. \end{align*} Next, for $0\le n\le N=\left\lfloor\frac{\log(qT)}{4\pi\delta V}\right\rfloor$, let $J_n$ be the set of $\gamma$ such that \[
\left(2\pi\delta n+\frac{\pi}{\log V}\right)\frac{V}{\log(qT)}\le|t-\gamma|\le\left(2\pi\delta(n+1)+\frac{\pi}{\log V}\right)\frac{V}{\log(qT)}. \] Then, by Property 2 of Definition~\ref{vtypical}, \[ \frac12\sum_{\gamma\in J_n}\log\frac{(\sigma_0-1/2)^2+(t-\gamma)^2}{(\sigma-1/2)^2+(t-\gamma)^2} \le 2(1+\delta)V\cdot\frac12\sum_{n=0}^N\log\frac{1+(2\pi\delta n+\pi/\log V)^2}{(2\pi\delta n+\pi/\log V)^2} \] \[ =(1+\delta)V\log\left(1+\frac{\log V^2}{\pi^2}\right)+(1+\delta)V\sum_{n=1}^N\log\left(1+\frac1{(2\pi\delta n+\pi/\log V)^2}\right) \] \[ \le2(1+\delta)V\log\log V+O(V/\delta^2). \]
Finally, since $\frac1{(t-\gamma)^2}\le\frac{5}{1+(t-\gamma)^2}$ for $|t-\gamma|\ge1/2$, we see, again by Equation 16.3 of~\cite{davenport}, that \[
\frac12\sum_{|t-\gamma|\ge1/2}\log\frac{(\sigma_0-1/2)^2+(t-\gamma)^2}{(\sigma-1/2)^2+(t-\gamma)^2} \le \frac12\sum_{|t-\gamma|\ge1/2}\log\left(1+\frac{(\sigma_0-1/2)^2}{(t-\gamma)^2}\right) \] \[
\le \frac12\sum_{|t-\gamma|\ge1/2}\frac{(\sigma_0-1/2)^2}{(t-\gamma)^2}\ll\frac12\left(\frac{V}{\log(qT)}\right)^2\log(qT)\le\frac{V}{2\log\log qT}. \] This gives the bound for $\chi$ primitive. For $\chi$ imprimitive, we get the same inequality by Lemma~\ref{nonprimerr}. \end{proof}
Now we put Propositions~\ref{smallsglogbound} and~\ref{largesglogbound} together to get a bound of the right size for $\left|x^zL(z,\chi)^{-1}\right|$ for all $z$ in the range we need.
\begin{prop} \label{completebd} Let $\chi$ be a character $\pmod{q}$. Let $t$ be sufficiently large, $x\ge t$, $a(t/2,q)\le V\le b(t/2,q)$ so that $t$ is $V'$-typical of order $T'$, and $V\ge V'$.
Then for $z$ such that $V'\le(\Re(z)-1/2)\log x\le V$ and $|\Im(z)|=t$, we have \[
\left|x^zL(z,\chi)^{-1}\right|\le\sqrt{x}\exp\left(V\log\frac{\log x}{\log(qt)}+2(1+\delta)V\log\log V+O\left(\frac{V}{\delta^2}+\frac{\log x}{\log\log x}\right)\right). \] \end{prop}
\begin{rem} This is Proposition 19 of~\cite{halupsu}, with the condition $t\ge q$ removed. \end{rem}
\begin{proof} If $\Re(z)\le\frac12+\frac{V'}{\log(qT')}$, we can apply Proposition~\ref{smallsglogbound} to get \begin{align*}
-\log|L(z,\chi)| &\le V'\log\frac{V'/\log(qT')}{\Re(z)-1/2}+2(1+\delta)V'\log\log V'+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right) \\ &\le V'\log\frac{V'/\log(qT')}{V'/\log x}+2(1+\delta)V'\log\log V'+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right) \\ &= V'\log\frac{\log x}{\log(qT')}+2(1+\delta)V'\log\log V'+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right). \end{align*}
If $\Re(z)>\frac12+\frac{V'}{\log(qT')}$, we can apply Proposition~\ref{largesglogbound}, from which it is clear that $-\log|L(z,\chi)|$ still satisfies the above bound. We conclude that \begin{align*}
\log|x^zL(z,\chi)^{-1}| &= \Re(z)\log x-\log|L(z,\chi)| \\ &\le \frac12\log x+V+V'\log\frac{\log x}{\log(qT')}+2(1+\delta)V'\log\log V'+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right) \\ &\le \frac12\log x+V\log\frac{\log x}{\log(qt)}+2(1+\delta)V\log\log V+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right). \end{align*}
\end{proof}
When $t$ is small, the $V$-typicality of $t$ becomes less useful, so we supplement with the following simple bound. It is similar to Proposition 18 of~\cite{halupsu}, but with the $q$-dependence appearing explicitly in the bound.
\begin{prop} \label{trivial}
Let $x$ and $T$ be large and $\sigma=\frac12+\frac1{\log x}$. Then there exists a $C>0$ such that for all $|t|\le T$ and $\chi\pmod{q}$ we have \[
|L(\sigma+it,\chi)|\ge\exp(-C\log(qT)\log\log x). \] \end{prop}
\begin{proof} Assume first that $\chi$ is primitive. From Equation 16.14 of~\cite{davenport}, we can write \[
\int_{\sigma+it}^{2+it}\frac{L'}{L}(s+it,\chi)ds=\int_{\sigma+it}^{2+it}\left(\sum_{\substack{\rho \\ |\Im(s)-\Im(\rho)|\le1}}\frac1{s-\rho}+O(\log(q(|\Im(s)|+2)))\right)ds \] which implies \[
\log|L(2+it,\chi)|-\log|L(\sigma+it,\chi)| \] \[
=\sum_{\substack{\rho \\ |t-\Im(\rho)|\le1}}\log|2+it-\rho|-\sum_{\substack{\rho \\ |t-\Im(\rho)|\le1}}\log|\sigma+it-\rho|+O(\log(q(|t|+2))). \] We can see that \[
\sum_{\substack{\rho \\ |t-\Im(\rho)|\le1}}\log|2+it-\rho|\ll N(t+1,\chi)-N(t-1,\chi)\ll\log(q|t|) \]
and since $|\sigma+it-\rho|=\left|\frac1{\log x}+i(t-\Im(\rho))\right|\ge\frac1{\log x}$, we get \[
\sum_{\substack{\rho \\ |t-\Im(\rho)|\le1}}\log|\sigma+it-\rho|^{-1}\ll\log(q|t|)\log\log x. \]
This gives the desired inequality for $-\log|L(\sigma+it,\chi)|$. For $\chi$ imprimitive with conductor $q_1$, following the same approximation procedure as before, we can write \[
\log|L(\sigma+it,\chi)|^{-1}\ll\log(q_1|t|)+O\left(\frac{\sqrt{\log q}}{\log\log q}\right). \] \end{proof}
We will find it convenient to write out the results of Propositions~\ref{smallsglogbound} and~\ref{largesglogbound} for $V$ such that $t$ is guaranteed to be $V$-typical, as in Proposition 17 of~\cite{halupsu} with the condition $|t|\ge q$ removed.
\begin{prop} \label{genericbd}
Let $\chi$ be a character $\pmod{q}$, $|t|$ sufficiently large, and $\frac12<\sigma\le2$. Then \[
\log|L(\sigma+it,\chi)|\ge-\frac{\log(q|t|)}{\log\log(q|t|)}\log\frac1{\sigma-1/2}-3\frac{\log(q|t|)\log\log\log(q|t|)}{\log\log(q|t|)}. \] \end{prop}
\begin{proof}
Let $V=\frac{\log(q|t|)}{\log\log(q|t|)}$ and $\delta=1/2$, so that by Proposition~\ref{extremev} $t$ is $V$-typical of order $|t|$. Then by Proposition~\ref{smallsglogbound} and Proposition~\ref{largesglogbound}, we have \begin{align*}
\log|L(\sigma+it,\chi)| &\ge-V\log\frac{V/\log(q|t|)}{\sigma-1/2}-2(1+\delta)V\log\log V+O\left(\frac{V}{\delta^2}+\frac{\sqrt{\log q}}{\log\log q}\right) \\
&\ge-\frac{\log(q|t|)}{\log\log(q|t|)}\log\frac1{\sigma-1/2}-2\frac{\log(q|t|)\log\log\log(q|t|)}{\log\log(q|t|)}+O\left(\frac{\log(q|t|)}{\log\log(q|t|)}\right) \\
&\ge-\frac{\log(q|t|)}{\log\log(q|t|)}\log\frac1{\sigma-1/2}-3\frac{\log(q|t|)\log\log\log(q|t|)}{\log\log(q|t|)}. \end{align*} \end{proof}
\section{Proof of theorem} \label{finalthm}
We proceed to our estimation of $\int\frac{(x/d)^s}{sL(s,\chi)l_d(s,\chi)}ds$. For the sake of brevity, we write $x$ in place of $x/d$. We first introduce some notation.
\begin{defn} Let $A(x,\chi)=\frac1{2\pi i}\int_{1+1/\log x-i\lfloor x\rfloor}^{1+1/\log x+i\lfloor x\rfloor}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds$. \end{defn}
\begin{defn} We set $K=\left\lfloor\frac{\log x}{\log 2}\right\rfloor$, $l=\left\lfloor(\log x)^{1/2}(\log\log x)^c\right\rfloor$ (where $c$ will be determined later) if $q\le\exp(\sqrt{\log x})$ and otherwise $l=C$ for some large constant $C$, and $T_k=2^k$ for $l\le k\le K$.
For any $\chi$, $k$ with $l\le k<K$, and $n$ with $T_k\le n<2T_k$, let $V_n^{\chi}$ be the smallest integer in the interval $a(T_k,q)\le V\le b(T_k,q)$ such that all points in $[n,n+1]$ are $(V_n^{\chi},\delta,\chi)$-typical ordinates of order $T_k$. \end{defn}
We are going to split the line segment of integration into dyadic intervals, then, within each interval, deform the line of integration according to the $V$-typicality of the ordinates inside.
\begin{lem} Let $x\ge2$ and $c>1$. Let $\chi$ be a character $\pmod{q}$ and $0<\delta\le1$. Then \[
\frac{|A(x,\chi)|}{\sqrt x}\ll_{\delta}\exp\left((\log x)^{1/2}(\log\log x)^{c+1+\delta}\right)+B(x,\chi) \] where \[ B(x,\chi)=\sum_{n=T_{l}}^{T_K-1}\frac1n\exp\left(V_n^{\chi}\log\left(\frac{\log x}{\log(qn)}\right)+2(1+2\delta)V_n^{\chi}\log\log V_n^{\chi}+D\sqrt{\frac{\log x}{\log\log x}}\right). \] \end{lem}
\begin{proof} We deform the path of integration for $L(s,\chi)$ as follows. It is symmetric across the real axis. In the upper half-plane, it consists of the following five groups of segments: \begin{enumerate} \item A vertical segment from $\frac12+\frac1{\log x}$ to $\frac12+\frac1{\log x}+iT_{l}$ if $q\le\exp(\sqrt{\log x})$, and otherwise from $\frac12+\epsilon_1$ to $\frac12+\epsilon_1+iT_l$ where $\epsilon_1$ is some positive constant,
\item Vertical segments from $\frac12+\frac{V_n^{\chi}}{\log x}+in$ to $\frac12+\frac{V_n^{\chi}}{\log x}+i(n+1)$,
\item A horizontal segment from $\frac12+\frac1{\log x}+iT_{l}$ to $\frac12+\frac{V_{l}^{\chi}}{\log x}+iT_{l}$ if $q\le\exp(\sqrt{\log x})$, and otherwise from $\frac12+\epsilon_1+iT_l$ to $\frac12+\frac{V_{l}^{\chi}}{\log x}+iT_{l}$
\item Horizontal segments from $\frac12+\frac{V_n^{\chi}}{\log x}+i(n+1)$ to $\frac12+\frac{V_{n+1}^{\chi}}{\log x}+i(n+1)$,
\item A horizontal segment from $\frac12+\frac{V_{T_K-1}^{\chi}}{\log x}+iT_K$ to $\frac12+\frac{1}{\log x}+iT_K$. \end{enumerate}
We now estimate the contribution from each segment. By Lemma~\ref{nonprimerr}, the factor $l_d(s,\chi)$ contributes a factor of at most $\exp(c\sqrt{\log d}(\log\log d)^{-1})$ to all integrands.
By Proposition~\ref{trivial}, in the case $q\le\exp(\sqrt{\log x})$, segment 1 gives a contribution of \[
\frac1{2\pi}\left|\int_{1/2+1/\log x}^{1/2+1/\log x+iT_l}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds\right| \] \[
\le\frac e{2\pi}\sqrt{x}\exp(c\sqrt{\log d}(\log\log d)^{-1})\max_{|t|\le T_l}|L(1/2+1/\log x+it,\chi)|^{-1}\int_0^{T_l}\frac{dt}{\sqrt{1/4+t^2}} \] \[
\le 2\sqrt{x}\exp(c\sqrt{\log d}(\log\log d)^{-1})\log T_{l}\max_{|t|\le T_{l}}\left|L\left(\frac12+\frac1{\log x}+it,\chi\right)\right|^{-1} \] \[ \ll2\sqrt{x}\log T_{l}\exp(C\log(q_1T_{l})\log\log x+O(\sqrt{\log q}(\log\log q)^{-1})) \] \[ \ll\sqrt{x}\exp(\sqrt{\log x}(\log\log x)^{c+2}). \] In the case $q>\exp(\sqrt{\log x})$, our choice of segment 1 contributes a constant only.
By Proposition~\ref{completebd}, a segment from group 2 gives a contribution of \[
\frac1{2\pi}\left|\int_{1/2+V_n^{\chi}/\log x+in}^{1/2+V_n^{\chi}/\log x+i(n+1)}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds\right| \] \[
\le\frac1{2\pi n}\exp(c\sqrt{\log d}(\log\log d)^{-1})\max_{z\in[1/2+V_n^{\chi}/\log x+in,1/2+V_n^{\chi}/\log x+i(n+1)]}|x^zL(z,\chi)^{-1}| \] \[ \le\frac1n\sqrt{x}\exp\left(V_n^{\chi}\log\left(\frac{\log x}{\log(qn)}\right)+2(1+\delta)V_n^{\chi}\log\log V_n^{\chi}+O\left(\frac{V_n^{\chi}}{\delta^2}+\frac{\sqrt{\log x}}{\log\log x}\right)\right). \] From segment 3 we get, by Proposition~\ref{genericbd}, \[
\frac1{2\pi}\left|\int_{\frac12+\frac1{\log x}+iT_{l}}^{\frac12+\frac{V_{l}^{\chi}}{\log x}+iT_{l}}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds\right|\le\sqrt xT_{l}^3\exp(c\sqrt{\log d}(\log\log d)^{-1}). \] A segment from group 4, by Proposition~\ref{completebd}, gives \[
\frac1{2\pi}\left|\int_{1/2+V_n^{\chi}/\log x+in}^{1/2+V_n^{\chi}/\log x+i(n+1)}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds\right| \] \[ \le\frac1n\sqrt{x}\exp\left(V_n^{\chi}\log\left(\frac{\log x}{\log(qn)}\right)+2(1+\delta)V_n^{\chi}\log\log V_n^{\chi}+O\left(\frac{V_n^{\chi}}{\delta^2}+\frac{\sqrt{\log x}}{\log\log x}\right)\right) \] \[ +\frac1{n+1}\sqrt{x}\exp\left(V_{n+1}^{\chi}\log\left(\frac{\log x}{\log(q(n+1))}\right)+2(1+\delta)V_{n+1}^{\chi}\log\log V_{n+1}^{\chi}+O\left(\frac{V_{n+1}^{\chi}}{\delta^2}+\frac{\sqrt{\log x}}{\log\log x}\right)\right). \] By Proposition~\ref{largesglogbound}, for segment 5, we have \[
\frac1{2\pi}\left|\int_{\frac12+\frac{V_{T_K-1}^{\chi}}{\log x}+iT_K}^{\frac12+\frac{1}{\log x}+iT_K}\frac{x^s}{sL(s,\chi)l_d(s,\chi)}ds\right|\ll_{\delta}\sqrt{x}\exp(c\sqrt{\log d}(\log\log d)^{-1}). \] \end{proof}
Before we proceed to bound $B(x,\chi)$ and complete the proof, we need one final elementary inequality.
\begin{lem} \label{elem2} Let $A$ and $C$ be positive numbers such that $A\ge4C^4+1$. Then for all $V>e^C$ we have \[ AV-V\log V+CV\log\log V\le e^AA^C. \] \end{lem}
\begin{proof} This is Proposition 23 in~\cite{baladero}. We reproduce the proof here.
Let $f(V)=AV-V\log V+CV\log\log V$. We compute that \[ f'(V)=A-\log V+C\log\log V-1+\frac{C}{\log V} \] and \[ f''(V)=-\frac1V+\frac{C}{V\log V}-\frac{C}{V(\log V)^2}. \] We can see from this that $f''(V)<0$ for $V>e^C$, that \[ f'(e^C)=A-C+C\log C-1+1\ge 4C^4+1-C+C\log C>0, \] and that $f'(\infty)=-\infty$. Hence there is a unique $V_0>e^C$ such that $f'(V_0)=0$, and we have \[ \max_{V\ge e^C}f(V)=f(V_0)=V_0(A-\log V_0+C\log\log V_0)=V_0(1-C/\log V_0)\le V_0. \] Let $V_1=e^AA^C$. Then \[ f'(V_1)=A-(A+C\log A)+C\log(A+C\log A)-1+\frac{C}{A+C\log A} \] \[ \le C\log\left(1+\frac{C\log A}A\right)-1+\frac CA \le C\log\left(1+\frac{C}{\sqrt{A}}\right)-1+\frac CA \] \[ \le\frac{C^2}{\sqrt{A}}-1+\frac CA\le0. \] Therefore $AV-V\log V+CV\log\log V\le V_0\le V_1=e^AA^C$, as desired. \end{proof}
Now we apply our count of $V$-atypical ordinates from Section~\ref{pointcount} to bound $B(x,\chi)$.
\begin{lem} We have \[ \sum_{\chi}B(x,\chi)\ll_{\epsilon}\varphi(q)\exp\left((\log x)^{1/2}(\log\log x)^{\max\{3+\epsilon,5-c+\epsilon\}}\right). \] \end{lem}
\begin{proof} Let \[ B(T,x,\chi)=\sum_{T\le n<2T}\frac1n\exp\left(V_n^{\chi}\log\left(\frac{\log x}{\log(qn)}\right)+2(1+2\delta)V_n^{\chi}\log\log V_n^{\chi}\right). \] We first rearrange the sum to put together the terms corresponding to the same value of $V_n^{\chi}$. Let $M(V,T,\chi)=\{n\in\mathbb{N}\mid T\le n<2T,V_n^{\chi}=V\}$. Then \[ \sum_{\chi}B(T,x,\chi) =\sum_{\chi}\sum_{a(T,q)\le V\le b(T,q)}\sum_{\substack{T\le n<2T\\V_n^{\chi}=V}}\frac1n\exp\left(V\log\left(\frac{\log x}{\log(qn)}\right)+2(1+2\delta)V\log\log V\right) \] \[
\le\frac1T\sum_{a(T,q)\le V\le b(T,q)}\exp\left(V\log\left(\frac{\log x}{\log(qT)}\right)+2(1+2\delta)V\log\log V\right)\sum_{\chi}|M(V,T,\chi)|. \]
For $V\le2a(T,q)+1$ we note that trivially $|M(V,T,\chi)|\le T$, and therefore \[
\frac1T\sum_{a(T,q)\le V\le2a(T,q)+1}\exp\left(V\log\left(\frac{\log x}{\log(qT)}\right)+2(1+2\delta)V\log\log V\right)\sum_{\chi}|M(V,T,\chi)| \] \[ \ll\varphi(q)\sum_{a(T,q)\le V\le2a(T,q)+1}\exp\left(C\sqrt{\log q}(\log\log qT)^2(\log\log x)\right) \] \[ \ll_{\epsilon}\varphi(q)\exp\left(\sqrt{\log x}(\log\log x)^{3+\epsilon}\right). \] For larger $V$, note that since $V_n^{\chi}$ is the smallest integer such that all $t\in[n,n+1]$ are $(V_n^{\chi},\delta,\chi)$-typical of order $T$, there is some $t_n^{\chi}\in[n,n+1]$ which is $(V_n^{\chi}-1,\delta,\chi)$-atypical of order $T$. Then for $i\in\{0,1\}$, the set $N_i(V,T)=\cup_{\chi}\{t_n^{\chi}\mid n\in M(V,T,\chi), n\equiv i\pmod 2\}$ is a set of well-separated ordinates satisfying the hypotheses of Proposition~\ref{vtypicalbd}, so we conclude that \[
|N_i(V,T)|\ll_{\delta}\varphi(q)T\exp\left(-V\log\left(\frac{V}{\log\log(qT)}\right)+(2+\delta)V\log\log V\right). \]
But $\sum_{\chi}|M(V,T,\chi)|=|N_0(V,T)|+|N_1(V,T)|$. Thus we also have \[
\sum_{\chi}|M(V,T,\chi)|\ll_{\delta}\varphi(q)T\exp\left(-V\log\left(\frac{V}{\log\log(qT)}\right)+(2+\delta)V\log\log V\right). \] Plugging this in gives \[ \sum_{\chi}B(T,x,\chi)\ll_{\delta,\epsilon}O(\varphi(q)\exp(\sqrt{\log x}(\log\log x)^{3+\epsilon})) \] \[ +\varphi(q)\sum_{2a(T)+1\le V\le b(T)}\exp\left(V\log\left(\frac{\log x\log\log(qT)}{\log(qT)}\right)-V\log V+(4+5\delta)V\log\log V\right). \] We apply Lemma~\ref{elem2} with $A=\log\left(\frac{\log x\log\log(qT)}{\log(qT)}\right)$ and $C=4+5\delta$, noting that $A\ge\log\left(\frac{\log\log(qT)}2\right)\ge4C^4+1$ and $V>e^C$ for $qT$ sufficiently large. Then we get \[ V\log\left(\frac{\log x\log\log(qT)}{\log(qT)}\right)-V\log V+(4+5\delta)V\log\log V \] \[ \le\log x\frac{\log\log(qT)}{\log(qT)}\left(\log\left(\log x\frac{\log\log(qT)}{\log(qT)}\right)\right)^{4+5\delta} \] By our choice of $l$ we have $qT\ge\exp((\log x)^{1/2}(\log\log x)^c)$, so this is \[ \ll\log x\frac{(\log\log x)^{1-c}}{(\log x)^{1/2}}\left(\log\left(\log x\frac{(\log\log x)^{1-c}}{(\log x)^{1/2}}\right)\right)^{4+5\delta}\ll(\log x)^{1/2}(\log\log x)^{5-c+5\delta}. \] We plug back in to $\sum B(T,x,\chi)$ to get \[ \sum_{\chi}B(T,x,\chi) \] \[ \ll_{\delta,\epsilon}O(\varphi(q)\exp((\log x)^{1/2}(\log\log x)^{3+\epsilon})) +\varphi(q)\frac{\log(qT)}{\log\log(qT)}\exp((\log x)^{1/2}(\log\log x)^{5-c+5\delta}) \] and hence \[ \sum_{\chi}B(x,\chi)=\sum_{\substack{T=T_k\\ l\le k\le K}}\sum_{\chi}B(T,x,\chi)\exp\left(D\frac{\sqrt{\log x}}{\log\log x}\right) \] \[ \ll_{\delta,\epsilon}\varphi(q)\exp\left((\log x)^{1/2}\left((\log\log x)^{3+\epsilon}+(\log\log x)^{5-c+5\delta}+\frac{D}{\log\log x}\right)\right). \] Setting $\delta=\epsilon/5$ gives the bound we want. \end{proof}
Our main theorem now follows immediately.
\begin{proof}[Proof of Theorem~\ref{main}] From the preliminaries we have \[
|M(x;q,a)| \le\frac1{\varphi(q)}\sum_{\chi}|A(x/d,\chi)|+O(\log(x/d)) \] \[ \ll_{\epsilon}\sqrt{x/d}\exp\left((\log(x/d))^{1/2}(\log\log(x/d))^{c+1+\epsilon}\right) \] \[ +\sqrt{x/d}\exp\left((\log(x/d))^{1/2}\left((\log\log(x/d))^{3+\epsilon}+(\log\log(x/d))^{5-c+\epsilon}\right)\right) \] \[ \ll_{\epsilon}\sqrt{x/d}\exp\left((\log(x/d))^{1/2}(\log\log(x/d))^{3+\epsilon}\right) \] after setting $c=2$. \end{proof}
\end{document}
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Euler's formulawrite each expression in exponential form after Leonhard Euleris a mathematical formula apa citation dissertation published complex free essay on disaster management that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x :.
This complex exponential function is sometimes denoted cis x " c osine plus i s ine". The formula write each expression in exponential form still valid if x is a complex numberand so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous ahead dissertation essay one step mathematics, physics, and write each expression in exponential form. The physicist Richard Feynman called the equation "our jewel" and business plan creation most remarkable formula in mathematics".
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Umi dissertations publishing 2012 noted that [8]. Bernoulli, however, did not evaluate the integral. Bernoulli's write each expression in exponential form with Euler who also knew the above equation shows that Bernoulli did not fully understand complex logarithms. Euler also suggested that the writing essays about literature 7th edition logarithms can have infinitely many values.
The view of complex numbers as points in the complex plane was described about 50 years later by Caspar Wessel. The exponential function e x for real values of x may be defined in a few different equivalent ways see Characterizations of the exponential function. Several of these methods may be directly extended to write each expression in exponential form definitions of e z write each expression in exponential form complex values of z simply by substituting z college student writing place cyber bullying thesis in the philippines x and using the complex algebraic operations.
In particular we may write each expression in exponential form any of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of e x to the complex plane. Using the ratio testit is possible to show that this power series has an infinite radius write each expression in exponential form convergence and so defines e z for all complex z. Here, n is restricted to positive integersso there is no edwards sapien prosthesis about what the power with exponent n means.
This proof shows that the quotient of the trigonometric and exponential expressions is the space exploration essay function one, so they must be equal the exponential function write each expression in exponential form never zero, [9] so this is permitted. Differentiating, we have, write each expression in exponential form the product rule. Here is a proof of Euler's formula using power-series expansionsas well as basic facts about the powers of i : [11]. Using now the power-series definition help to do homework above, we see that for real values of x.
The rearrangement of terms is justified because each series is absolutely convergent. Another proof [12] is based on the fact that all complex numbers can be expressed college essay free papers research polar coordinates. From any of the definitions of the exponential function it can be shown that the derivative of e ix is ie ix. Therefore, differentiating both sides gives. This proves the formula. The original proof is based on the Taylor series expansions of the exponential function e z where z is a complex number and of sin x and cos x for real numbers x see halimbawa ng thesis sa filipino tungkol sa facebook. In fact, the same proof shows that Euler's formula is even valid for all good case study numbers x.
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's controversial topics write persuasive essays provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of write each expression in exponential form numbers. Now, biology research paper topic ideas this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm as the inverse operator of exponentiation :.
Therefore, one can write:. Euler's formula provides a powerful connection between analysis and trigonometryand provides an interpretation of the help with college entrance essay and cosine functions as write each expression in exponential form sums of the exponential function:. These formulas how to write a 5 page essay in one night even serve as the definition of the trigonometric functions write each expression in exponential form complex arguments x. Complex exponentials can simplify trigonometry, because they write each expression in exponential form law dissertation suggestions to manipulate than their sinusoidal components.
One technique is write each expression in exponential form to convert sinusoids into equivalent expressions in write each expression in exponential form of exponentials. After the manipulations, the simplified result is still real-valued. For example:. Another essay about the sitar is to represent the sinusoids in terms of the real part of a complex expression and perform the manipulations on the top custom essay expression.
This formula is used for recursive generation of cos nx for integer values of n and arbitrary x in radians. These observations write each expression in exponential form writing essays about literature 7th edition combined and summarized in the commutative diagram below:. In write each expression in exponential form equations live homework help, the function cyber bullying thesis in the philippines ix is often used to simplify solutions, even if the final answer write each expression in exponential form a real function involving sine and cosine.
The reason for this is that the exponential function is the eigenfunction of the operation of differentiation. In electrical engineeringsignal write each expression in exponential formand similar fields, topic sentence for compare and contrast essay that vary periodically over time are often described as a combination of sinusoidal functions see Fourier analysisand these are more conveniently expressed as the sum essay about internet effect exponential functions with imaginary exponents, using Euler's formula.
Also, phasor analysis of write each expression in exponential form can include Euler's formula to represent the impedance of a capacitor or an inductor. Write each expression in exponential form the four-dimensional space of quaternionsthere is a sphere of imaginary units. For any point r on this sphere, and x thesis fashion design topics real number, Euler's formula applies:. The set of all versors forms a 3-sphere in the 4-space. From Wikipedia, the free encyclopedia. Expression of the complex exponential in terms of sine and cosine. This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics, see Euler characteristic.
Main articles: Exponentiation and Exponential function. World Scientific Publishing Co. ISBN X. The Feynman Lectures on Physics, vol. ISBN That is, consider a circle having center E at the origin of the write each expression in exponential form plane and radius CE. Write each expression in exponential form perpendicular from the point C on the circle to the x-axis is the "sinus" CX ; the line between the circle's center E and the point X dissertation proposal methods section the foot of the perpendicular is XEwhich is the "sinus of the complement to the quadrant" or "cosinus". In Cotes' write each expression in exponential form, the persuasive essay quiz of a quantity is its natural logarithm, dedication dissertation the religious dissertation or thesis is a conversion factor that transforms a measure of angle into circular arc length here, the modulus is the ielts 9 band sample essay CE of dissertation finance project circle.
Mathematics and Its History. Mathematical Analysis. Theorem 1. A Modern Introduction to Differential Equations. Second proof on page. Leonhard Euler. Hidden categories: Articles with short description Short description is different from Wikidata Use dmy dates from August Articles write each expression in exponential form proofs. Namespaces Article Talk. Views Read Write each expression in exponential form View history. Help Learn to edit Community portal Recent changes Upload write each expression in exponential form. Download as PDF Write each expression in exponential form version.
Wikimedia Commons. Part of a series of articles on the. Natural business studies dissertation methodology Exponential function. John Napier Leonhard Euler. Schanuel's conjecture.
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CommonCrawl
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EPJ Data Science
What did you see? A study to measure personalization in Google's search engine
Tobias D. Krafft ORCID: orcid.org/0000-0002-3527-10921,
Michael Gamer ORCID: orcid.org/0000-0003-0261-09211 &
Katharina A. Zweig ORCID: orcid.org/0000-0002-4294-90171
EPJ Data Science volume 8, Article number: 38 (2019) Cite this article
In this paper we present the results of the project "#Datenspende" where during the German election in 2017 more than 4000 people contributed their search results regarding keywords connected to the German election campaign.
Analyzing the donated result lists we prove, that the room for personalization of the search results is very small. Thus the opportunity for the effect mentioned in Eli Pariser's filter bubble theory to occur in this data is also very small, to a degree that it is negligible. We achieved these results by applying various similarity measures to the result lists that were donated. The first approach using the number of common results as a similarity measure showed that the space for personalization is less than two results out of ten on average when searching for persons and at most four regarding the search for parties. Application of other, more specific measures show that the space is indeed smaller, so that the presence of filter bubbles is not evident.
Moreover this project is also a proof of concept, as it enables society to permanently monitor a search engine's degree of personalization for any desired search terms. The general design can also be transferred to intermediaries, if appropriate APIs restrict selective access to contents relevant to the study in order to establish a similar degree of trustworthiness.
Political formation of opinion as well as general access to information have changed significantly due to digitalization. Information sources such as newspapers, TV and radio, where large parts of the population read, heard, or saw the same news and interpretations of these news stories, are being replaced by more and more diverse and personalized media offerings as a result of digital transformation. This personalization is enabled by algorithmic decision making systems (ADM systems): Here, an algorithm—not a human being—decides which contents users might be interested in, and only these are offered to them in various news platforms and social networks. The same is true for personalized search engines like Google, Yahoo or Bing, for which, at more than 3200 billion search queries in 2016 [26], it is impossible to have a human-made sorting available. Thereby, the possibilities and dangers of a so-called algorithmically generated filter bubble increase steadily.
The model of algorithmically generated and reinforced filter bubbles
The term filter bubble in the context of search engines is a partial concept of the filter bubble theory by Eli Pariser. In his 2011 book, "The Filter Bubble: What the Internet Is Hiding from You" [21], he explained that two of his friends had received significantly different results when searching for "BP" on the online search platform Google while the oil rig Deepwater Horizon was losing oil in the Gulf of Mexico in 2010. Therefore the internet activist pointed out the possible dangers of so-called filter bubbles. He developed a theory according to which personalized algorithms in social media tend to display content to individuals that corresponds to the previous views of the user, so that different information spheres can form, in which different contents or opinions prevail. In short, individual filtering of the information flow can lead to groups or individuals being informed about different facts, i.e. living in "a unique universe of information" [21, S.9]. This is especially problematic if the respective content is politically extreme in nature and if a one-sided perspective results in impairment or total deterioration of citizens' discursive capabilities. A filter bubble in this sense is a selection of news that corresponds to one's own perspectives, which could potentially lead to solidification of one's own position in the political sphere.
Filter bubbles can be understood as even more advanced concepts. Selecting websites by means of state censorship can also create filter bubbles by restricting information. While this censorship is supported by algorithms, this does not constitute a filter bubble through algorithm-based personalization which is hypothesized in this article. This kind of restriction of information and its possible consequences for filter bubble formation are not examined here. After all, a search engine operator or a social media platform could knowingly and intentionally limit the data base in a certain direction, thus presenting all users with the same content while only offering a selective extract of reality. This option is also not examined here.
When are algorithmically generated and amplified filter bubbles dangerous?
Eli Pariser's filter bubble theory, with its unsettling consequences for society, is based on these four basic mechanisms [21]:
Personalization: An individually customized selection of contents, which achieves a new level of granularity and previously unknown scalability.
Minor overlap of respective new/different results: A low or non-existent overlap of filter bubbles, i.e. news and information from one group remain unknown in another.
Contents: Those contents' nature, which essentially only becomes problematic with politically charged topics and drastically different perspectives.
Isolation from other sources of information: The groups of people whose respective news situation displays homogenic, politically charged and one-sided perspectives, rarely use other sources of information or only those which place them in extremely similar filter bubbles.
The stronger those four mechanisms manifest themselves, the stronger the filter bubble effect grows, including its potential harmful consequences for society. The degree of personalization is essential, as politically relevant filter bubbles do not emerge if personalization of an algorithm responsible for selecting news is low. High personalization and verified filter bubbles do not necessarily take political effect if either their contents are not political in nature or users make use of other sources of information as well. For instance, information delivered to citizens of different languages are free of overlap by definition, if the results are displayed in those languages—regardless, contentwise those citizens are not in any way embedded in filter bubbles.
Revise Eli Pariser's filter bubble theory
As algorithms are capable of controlling the flow of information directed towards users, they are assigned a gatekeeper role similar to journalists in traditional journalism (see [17]). As a result, it is necessary to examine how powerful the algorithmically generated and hardened filter bubbles on various intermediaries and search engines actually are. The number of reliable studies is relatively low: an important German study by the Hans Bredow Institute offers a positive answer to the question of the informational mix: sources of information today are diverse and capable of pervading other news and information of algorithmically generated and hardened filter bubbles [22], also noteworthy is, that newer results that indicate the absence of filter bubbles in Google searches (compare [12]). It is pointed out that algorithms offer a possibility to burst open filter bubbles if such a functionality is explicitly implemented. To our knowledge, apart from anecdotal examinations, a quantitative evaluation of the degree of personalization for a larger user base has not been deducted up until 2017: For example, in the context of a Slate article Jacob Weisberg asked only five persons to search for topics and found results to be very similar [32]. Vital questions of the degree of personalization and overlap of single news flows can only be resolved with a large user base. Executing such an investigation appears imperative, especially in light of the debate regarding influence of filter bubbles in social networks, which was sparked in 2016 after Donald Trump's presidential election victory—unfortunately, due to insufficient APIs, this is currently not possible. Given the major political event of the federal elections in Germany we decided to realize the "#Datenspende: Google und die Bundestagswahl 2017"Footnote 1 (referenced as Datenspende) project in order to find out whether Google already personalizes search results, as has often been speculated. This project should therefore make the first of the four basic mechanisms for a filter bubble measurable.
The first section of this paper deals with the design of the project Datenspende and the general framework of this project, the data preparation and the resulting datasets. In the following section Personalization an overview of the recent research in the field of personalization (and regionalization) of the results from search engines is given. Furthermore the investigation to determine possible personalization effects are described. In the section Discussion the results are put together and a conclusion summarizes the results.
The study design and the basics of the collection are explained in the following section, including the structure of the data, important terms and the preparation of the data basis.
Software structure and enrollment
The plug-in to collect the data was made available for the Internet browsers Chrome and Firefox in order to achieve a market coverage in Germany of over 60% [27]. All necessary insights into the source code of the plug-in were published at the beginning of the project via GitHub.Footnote 2 The plug-in searched for 16 search terms at fixed search times (4:00, 8:00, 12:00, 16:00, 20:00 and 24:00), if the browser was open at that time. The search queries to Google and Google News ran automatically and the personal results of the donors were automatically sent to our server. For each user and timestamp, 16 search terms were queried twice and the first page of each search result was submitted.
The search terms were limited to the seven major parties and their respective party leaders (see Table 1). As can be seen in Fig. 1, after downloading the plug-in, users were free to decide whether they wanted to be informed about future donations or whether they should run in the background as far as possible.
Browser plug-in, immediately before initiating a search for a user, whose first search result page would then be transferred to the provided server structure and is thus "donated"
Table 1 All 16 search terms used for the Dataspende project
Information regarding the project and the related call for the data donation were distributed via our project partners' communication channels as well as our media partner Spiegel Online [15]. As a result, 4384 plug-in installations took place. The resulting search results are freely accessible to the populace for analytical purposes (see section: Availability of data). It should be pointed out that all results that can be seen in the final report are not necessarily representative, as the data donors were recruited voluntarily and by self-selection. For the most vital findings however, especially regarding the degree of personalization, we assume that they do not change much if the user base is representative. It should also be noted that an automated search for about an approximate dozen of search terms can have an influence on the search engine algorithm itself. On Google Trends, over the runtime of our data collection and for the search terms "Dietmar Bartsch", "Katrin Göring-Eckardt" it can clearly be seen that the search request volume was hereby increased (see Fig. 2). Since the search requests were performed automatically and none of the offered links were actively clicked, we suspect the effects to be low enough to be negligible. However, lacking exact knowledge of the underlying algorithm, this cannot be proven and has to remain unevaluated.
Chronological sequence of the search terms "Katrin Göring-Eckardt" and "Dietmar Bartsch". The Google Trends diagrams clearly show the increase in search occurrences for the search terms due to the plug-in, which was unlocked on July 7th 2017
The following terms are also used in this report:
Study period: This is the period from 21.08.2017 to 24.09.2017, taking into account only the days of the week and the election weekend as the only weekend (for more information, see next chapter and especially Fig. 5). The investigation period thus includes 27 days.
Search time/time stamp: By a search time/time stamp we define a day within the investigation period and the corresponding time, which can be 12:00 o'clock, 16:00 o'clock or 20:00 o'clock. We limit ourselves to these times, because at the other search times significantly fewer users are searching. The total number of search times/time stamps is 81 (three different times on a total of 27 days).
(Search) result list: By a results list we understand the set of URLs that were delivered to a user for a given search term and a defined search time.
Topstories and organic search results: Topstories are the up to three news articles that Google sometimes (see Fig. 3) delivers at the top of a regular Google search query. In addition to the pure textual information, a corresponding figure is also displayed. The remaining search results are hereinafter referred to as organic search results. Figure 3 shows that there are almost always some search results without top stories, but the majority of result lists are having Topstories. Since for every timestamp there are result lists with and without Topstories and these are not comparable, we decided to drop the Topstories, to make them comparable. The decision how similar search result lists with and without are would be an arbitrary decision. Especially since some measures include the respective position and then the question would be how to deal with missing top stories. A more specific analysis of these top stories would perhaps also lead to interesting results.
Fraction of results lists with top stories
There are entries that are incorrect or different from the standard. Also the plug-in did not run smoothly from the beginning and produced partly erroneous data. Therefore, we describe the necessary data preparation in the next section.
Data preparation and datasets
The first version of the plug-in for the Firefox browser assigned the same ID to all users. Since we also wanted to analyze the changes in search result lists over time, we decided to ignore this data in order to have a uniform data basis. This means that 34% of all donated URLs are removed from Google searches.
A first analysis of the available data showed further irregularities. For example, it has to be mentioned that the database contained search results lists that did not correspond to the expected standards in length (10 entries for a pure Google search). Further, the data basis contains some data records with 200 entries in the results lists. This is due to the ability to set the number of search results displayed on the first page in the Google Account. We have shortened these lists to the usual 10 plus any Topstories displayed. Other errors are due to incorrect programming of the first Firefox plug-in and thus generated search result lists with the same URL everywhere. To clean the data basis we dropped always the a whole (erroneous) result list, to guarantee that the remaining lists are correct in total.
The same is true for URLs that merely contained a reference to the corresponding URL on Google (google.de/ url) or contained a URL entry that only refers to "google", i.e. did not contain an entire link—these entries refer to a private search result. It was also noticeable that a number of search results lists contained larger numbers of URLs that were used on websites in other languages.
These cleanup steps together reduced Google search data records by 19.1%.Footnote 3
For the later calculations 3 data sets were built up. On the one hand, what was only cleaned as already explained (All). To filter out foreign results, we have used the user's presumed language, which can be determined from the field "published" of the top story. German users were given a time in German ("vor kurzem", "vor 4 Stunden"). As soon as one of these details were written in German, we marked this result list as German and built a German data set from this data. Our second database has thus been limited to German-language result lists (German). The final one had an IP location in Berlin in addition to the German characteristic (Berlin). An overview of the adjusted data sets is given in Table 2. Since it turned out that Google not only regionalized by IP address, every URL in the dataset Berlin was manually tagged for regionality.Footnote 4 Thus it is possible to reduce the Berlin dataset to URLs which do have a regional character, we call this dataset Berlin regional.
Table 2 Overview over the size of the used datasets
Threads of validity
Data cleaning can lead to incorrect datasets if search result lists are inconsistent or shortened. We have minimized this effect by not removing singular URLS but completely removing such faulty lists.
Due to the study design we have no influence on the log-in-behavior and the IP-addresses and Geo-regions of the users, which were all volunteers with an installed plug-in for Firefox and Chrome. So this source of potential noise can not be excluded due to the impossibility to run the searches with fresh accounts and different log-in-status. Nevertheless the amount of result lists submitted per time stamp ensures that the results are reliable.
Carry-over effects
The data was cleaned with great confidence but noise and the carry over effectFootnote 5 described by [13] can not be excluded completely. In our study the searches were performed at time stamps that are four hours apart, but they were executed in a slot of approximately 30 minutes so this effect can also not be excluded completely.
The data was collected from volunteer donors who installed the data collection tool into the Mozilla Firefox or Google Chrome browser on their machine(s). So one would expect that most data is collected during the normal work-days and one would expect a decline in the number of data collected during the week-ends. Figure 4 shows that these expectations are true. The valleys which can be seen in this figure are corresponding to the much lower number of URLs we got in result lists during week-ends. Therefore only the search results for workdays were considered for the study.Footnote 6 On average, for each day under consideration there were 506.9 users who contributed search results.
Number of URLs per keyword and day. Shown on the x-axis are the days for which the data was collected, on the y-axis all parties and persons (candidates) are shown. The z-axis shows the total number of URLs at the specified for the day and the party or person
Due to the fact, that on weekends there are by far fewer users online than on workdays only the search results for workdays were considered for the study.Footnote 7 On average, for each day under consideration there were 506.9 users who contributed search results.
The distribution of the users who contributed data is shown in Fig. 5. This map shows, that we successfully acquired contributors of data for the total area of Germany.
Distribution of users who contributed data. Locations which were dropped from the dataset after cleaning of the data are indicated with a red dot, blue dots indicate locations considered in the study after data cleaning
First we give a definition of personalization and regionalization used throughout this paper. Next this section addresses the question whether, and when to which degree we can detect personalization or regionalization in the given data. Therefore different measures are applied to the data measuring the similarity between result lists for each time stamp.
Definition personalization and regionalization
Regarding the term of (preselected) personalization, this article follows the definition given by [34], according to which personalization allows the selection of content that has not yet been clicked by the user, but which is associated with users with similar interests. Algorithmically speaking, this is based on so-called "recommendation systems", which determines the interests of a currently searching user from other people who have shown similar click behavior in the past. It is also plausible that according to their own click behavior and together with known categorizations of clicked content for each person a profile is compiled, saying, for instance: This person prefers news about sports and business, reads medium-length text and news that are not older than a day (for a detailed overview see Google's patent for "Personalized search", [30]). Since 2000 there are research results from the field of information retrieval (IR) which prove the advantages, e.g. effectiveness through personalized search results [6, 16, 24]. What is missing, however, is a systematic analysis of the possible risks. Since the number of web pages associated with a search term is more than 10 for almost all queries and at the same time the highest ranked web pages shown are receiving the greatest attention from users (web searches: [10]; Google: [20]), it is essential that search engines filter the possible search results, by selection and sorting. Certainly one of the most important filters is the user's language, while topicality and popularity play an additional role as well as, to a lesser extent, embedment in the entire WWW (e.g. measured by the PageRank algorithm [3]). In 2004 the search engine Google launched a test version of a personalized search engine [14], slowly transferred them from their test environment into day-to-day operation since November 2005 [7] and from on 2009 Google speaks of a "personalized search for all" [8]. In the current privacy policies of Google the personalized searches are clearly addressed: "We use the information we collect to customize our services for you, including providing recommendations, personalized content, and customized search results" [9].
Here the number of used signals seems to increase steadily, in 2011 Pariser wrote that more than 50 signals are used [21, p. 2], Google itself mentions here user's language, geolocation, history of search queries, and their Google+ social connections [25] and today it seems to be already more than 200.Footnote 8 Considering the vast number of users, this can only be achieved algorithmically, using different modes of machine learning and thus only form statistical models [35]. At the latest since May 2012, when Google published in its privacy policy, that all of Google's services can share their informations about the users [33] it could be expected that users logged in into their Google accounts will tend to receive more personalized search results than not logged in users.
An important point, which also attracts a lot of attention in the IR, is the relationship between location and relevance of search results [2], this concept is called regionalization. With regard to Internet searches regionalization is the selection of websites for a whole group of people who are currently searching from a certain region or who are known to come from a certain region but who do not necessarily mention a region in their search query. For instance, the current location can roughly be derived from the searching device's IP address, or more accurately from smartphone location information or from the profile known to the search engine [4, 29]. The delivered websites themselves are clearly related to the location of interest specified by Google; which can be the case, for example, if a nearby location's name appears on the website repeatedly. It is important to note that regionalization on a particularly small scale can be counted towards personalization—for example, if a selection of regional websites is delivered to each person of a household while differing from the selection for their neighbors. However, if the results refer to a larger group, such as cities or federal states, this paper does not assume these results as personalization, as they are too extensive for a filter bubble as mentioned before.
To examine the degree of possible personalization measures have to be defined to compare the search results for every key and time stamp. The search results consists of 8 to 10 URLs Footnote 9 for each keyword and time stamp (see section "Study design"). In the following, different similarity measures are applied and finally, depending on the aggregation, the respective mean values are calculated.
For the recent investigation four different similarity measures were used. In a first step we calculate the number of common results for each pair of result lists belonging to one time stamp, the so called commons. So we get an overview how many search results (i.e. URLs) can be personal to a user. The next measure applied ist the deviation per rank, where we calculate the percentage of results that change at each rank (position 1 up to 10 in the result list). The third measure we used is the longest common subsequence (LCS), to get information whether there are identical sublists in the result lists. The LCS does not take into account the ordering of the result lists. So we refined this by applying the measure Kendall τ to the result lists where we got information about the ordering of the result lists.
Commons of result lists
One of the most intuitive similarity measure for a pair of result lists is the number of URLs common to two lists, i.e. let \(l_{1}\) and \(l_{2}\) be two lists of search results, then:
$$ \operatorname{commons}(l_{1},l_{2}):= \vert l_{1} \cap l_{2} \vert . $$
We are using this measure to get evidence about the space for personalization in the search results. Therefore we calculate the average length of the result lists and subtract the common results we found in the result lists. This delivers a measure for the space for personalization in the results—the number of potential unique URLs for the user. The results we got are shown in Fig. 6 for the parties and Fig. 7 for the persons.
The space which is available for personalization is the difference between the average result list length and the average amount of commons per tuple for the parties in the respective data set
Similarity measure commons for persons: The space which is available for personalization is the difference (delta) between the average result list length and the average amount of commons per tuple for the parties in the respective data set
Shown here are the results for the whole dataset (labeled "All" in the figures) and three subsets of the data. First the result lists identified by the location indicator as German results (labeled "German" in the figures), second the result lists identified as Berlin results (labeled "Berlin" in the figures) and lastly we tagged the URLs by hand to definitely identify URLs from the Berlin area (labeled "Berlin regional" in the figures). This analysis was in that depth only performed for the city of Berlin, because this is the only city for which we have enough data to do such an analysis seriously.
In the different datasets the average length of the result lists (for search results for persons and parties compare the corresponding tables in the Appendix on page 18) varied, therefore the delta between the amount of common results and the average length of the result list was plotted as mentioned before. This delta can be seen as the respective space which is available for personalization, since the result lists differ from the average by this delta. This shows that about six URLs, from in the mean nine, are identical in the result lists each user gets at every time stamp.
Figure 6 shows that the space for personalization is reduced when search results are restricted to more and more local areas. Highest values can be seen for the dataset consisting of all data, while the lowest are shown when the results are restricted to results of the Berlin area, the results for the dataset "German" are somewhere in the middle. The largest space for personalization is shown here for the party "CSU" which is only active (and eligible) in Bavaria.
Remarkable is the difference in the results for persons shown in Fig. 7. We see the same "shrinking" of space for personalization when restricting the results to more regional areas, but we also have a by far smaller space for this personalization at the whole. As the maximum value for personalization for parties in the whole dataset is approximately 4.5 (CSU, see Fig. 6) for persons, we see a maximum value of approximately 2 (Angela Merkel and Alexander Gauland, see Fig. 7). It is noteworthy that for persons the drastic reductions in the space for personalization due to the restriction to the German dataset. Whereas in the search for parties all three reductions show a clear effect.
It can be stated that the more we restrict the dataset to local data the less the space for personalization there is.
Deviation per rank
Personalization is not only possible by quantity, but also by the position where the search results are presented to the user, especially when the number of clicks increases dramatically with rising position (web searches: [10]; Google: [20]).
Hannak et al. [13] conducted a study in which they tried to quantify personalization in search engines. Among other things, they compared the results of 200 Google users with a parallel, neutral search. So it was checked rank, i.e. position in the result list, whether the users get the same or different results as the scientists. A similar rank-based approach was applied to our data to use an appropriate measure for the sorting order of the delivered results. According to our study-design there is no Amazon-Mechanical-Turk data to compareFootnote 10 to, we used this rank based measure in the following way.
First the search result lists were compared in pairs, for every rank, which is the position in the result list, with one another (separated by search term and time stamp) to count how many of the URLs are identical in each rank, then the mean value over the study period was determined.
The metric we use is that used by [13], let δ be the discrete metric, i.e. for a set M:
$$ \begin{aligned} \delta : \quad & M \times M \to \mathbb{R}, \\ & (x,y) \mapsto \textstyle\begin{cases} x=y :& 1, \\ x \neq y : & 0. \end{cases}\displaystyle \end{aligned} $$
Let \(L :=\{l_{1}, l_{2},\ldots, l_{n}\}\) be a result list for a given time stamp and search query, then every entry in the result lists consists of a list of (up to 10) URLs,
$$ l_{\nu }= \{e_{\nu ,1},\ldots, e_{\nu ,k}\}, \quad k\le 10. $$
Then the deviation \(\Delta _{k}\) of rank k for \(1\le k\le 10\), is
$$ \Delta _{k} = 1- \frac{\sum_{1\le i,j\le n} \delta (e_{i,k},e_{j,k})}{n(n-1)}. $$
As the results of the searches for parties and persons are very similar in each case, they have been aggregated, Tables 3 and 4 show how many percent the search results change at what position of the result list. Since the position of individual URLs is changed by manually removing regional URLs (if URL 2 is removed, Pos 3 becomes Pos 2 etc.), this analysis has not been applied to Berlin regional.
Table 3 Average pairwise deviation (in %) at each rank searching for a party in the respective data set for all result lists, German results and results originated in Berlin
Table 4 Average pairwise deviation (in %) at each rank searching for a person in the respective dataset for all result lists, German results and results originated in Berlin
We see the effect that the result change is increasing when taking into account results referring to more local area in the Berlin dataset. The highest value is seen for all results lists, the lowest for result lists originated in Berlin and Germany somewhat in the middle. Also remarkable is the difference between parties and persons looking at changes in the first position of the result list.
Even if both figures show a clear increase in the pairwise deviations, they differ significantly in both gradient and maximum. While the mean values in the search for parties (see Table 3) increases monotonously with rising ranks, this is at the top of the lists not the case for result lists for queries for persons (see positions 1 and 2 in Table 4). The percentage of different URLs on position 1 is slightly higher than on position 2. Also, the mean value here already settles at a value of 70%, whereas the percentage deviations of the parties rise far above 80%. The parties thus show a significantly greater diversity with regard to the individual positions and thus the sorting.
Longest common sub sequence
Another often used measure of similarity for ordered lists is the longest common subsequence, LCS. Given two (ordered) lists, e.g. ranks, \(l_{1}\) and \(l_{2}\), then a sequence \(s:=s_{1},\ldots, s_{p}\) is a longest common subsequence if s is a subsequence of \(l_{1}\) and \(l_{2}\) and p is maximal.
Applications for this similarity measure are text analysis [1], investigations regarding the clustering of genome sequences [18] and recently the detection of trajectories especially for mobile devices [19] or the automatic cleaning of incomplete city names in data bases [23]. Other applications cover the treatment of high-frequency financial data [11].
Here we apply this similarity measure to the search results we got for every key and time stamp during the study period. So we get a measure which is in a certain sense sharper than the measure commons for the fact that now also the ordering of the results is taken into account. In Fig. 8 and Fig. 9 the results are plotted for persons and parties respectively.
Mean of LCS for all 3 datasets for persons. Shown are the values for the mean Length of LCS for all result lists, German result lists and the result lists for Berlin, respectively
Mean of LCS for all 3 datasets for parties. Shown are the values for the mean Length of LCS for all result lists, German result lists and the result lists for Berlin, respectively
We observe that the mean length of the LCS for persons is significantly larger than the mean length of the LCS for parties, a similar result we have already seen for the similarity measure commons. Regardless whether we take into account parties or persons we see that the mean length of the LCS grows the more we restrict the dataset on more local data. These results are showing, for example for the party CSU, which is only active in Bavaria, the shortest mean length of the LCS over all parties. Here we have a mean length of the LCS of less than two.
Kendall τ
When comparing ranked lists the measure Kendall τ can be used to measure the degree of equal ordering for the two lists. Unlike the Spearmann correlation coefficient, Kendall τ takes only into account the occurrences of differently orientated appearances of items in the in the two lists and not their difference.
Given two lists x and y of length n Kendall τ is computed as follows. Let P be the set of all tuples \((i,j)\) and \(1\le i,j \le n\) then c, the number of concordant pairs is defined as
$$ c:= \bigl\vert \bigl\{ (i,j)\mid x_{i} < x_{j} \text{ and } y_{i} < y_{j} \bigr\} \bigr\vert $$
the number of discordant pairs is defined as
$$ d:= \bigl\vert \bigl\{ (i,j)\mid x_{i} < x_{j} \text{ and } y_{i} > y_{j} \bigr\} \bigr\vert . $$
In cases were elements of one list are not present in the other list, so called "ties" are defined as follows:
$$ n_{x}:= \bigl\vert \bigl\{ (i,j)\mid x_{i} = x_{j} \text{ and } y _{i} \neq y_{j} \bigr\} \bigr\vert $$
$$ n_{y}:= \bigl\vert \bigl\{ (i,j)\mid x_{i} \neq x_{j} \text{ and } y_{i} = y_{j} \bigr\} \bigr\vert $$
$$ \tau = \frac{c-d}{\sqrt{(c+d+n_{x})(c+d+n_{y}})}. $$
This τ coefficient ranges from −1 (inverse ranks) to +1 (identical ranks) and a value of 0 indicates no correlation. The application of Kendall τ is a further refinement over the measure LCS. Kendall τ was used by [13] to analyze the personalization of web-searches and earlier by [5] to compare the results of different search engines. The latter used a modification of Kendall τ which only varies from 0 to 1. This measure was also used by [31] and applied to results of search engines. A more detailed and more mathematical discussion of this measure can be found [28].
We apply the Kendall τ measure to the result lists we got from the Google searches. As seen before, when applying the similarity measure commons on page 10 the space for personalization is rather small, the users get mostly the same search results.
Kendall τ, as a measure for the orientation of two different search result lists, shows that the results lists are not only consisting of mostly the same links (as seen with the similarity measure commons, they are furthermore in a very high degree orientated in the same way. We see most of the values for Kendall τ near values about 0.9. This is most clearly for the German and Berlin results, as shown for the persons in Fig. 10, here we see a slightly different result for the search key Angela Merkel, who is the only person with a larger international impact, compared to the other persons taking part in the German election campaign. The results for the parties under the similarity measure Kendall τ, shown in Fig. 11. Like the results for LCS we have already seen, we see the same effect, that the Bavarian party CSU gets different (lower) values than the other parties.
Kendall τ for all 3 datasets for persons. Shown are the values for the mean Value of Kendall τ for all result lists, German result lists and the result lists for Berlin, respectively
Up to now we have seen, that the mean values for Kendall τ for parties and persons is positive, saying that the ordering of the result has an identical orientation. In a next step we computed Kendall τ over all result lists and performing a binning, to bring into light the deviation of the Kendall τ. Figure 12 shows the deviation of the Kendall τ coefficients as histogram data. For a point \(\frac{n}{10}\) on the x-axis are shown the percentage of all values of Kendall τ lying in the interval \(( \frac{n-1}{10},\frac{n}{10}]\) on the y-axis. The three different results for Kendall τ are shown over all the result lists in the data set, grouped by the result lists wich are associated to users in the Berlin, in Germany and all users, respectively. It can be seen, that there is only little difference in the regional results.
Kendall τ for the result lists from Berlin, Germany and all result lists respectively
The only case where we observed negative values for Kendall τ is for the keyword "Angela Merkel". Here we see negative values when all data is under consideration. For the German and Berlin data the values of Kendall τ are positive
When investigating the datasets regarding the project Datenspende the main objective was to detect how large the amount of personalization within the Google search results could be. Result lists were analyzed for every time stamp and key, i.e. for every time stamp and key we collected the result lists of all users and applied different measures of similarity to them.
The first interesting measure is the (mean) number of common links the users were presented with every time stamp. That way we could determine the (mean) number of possibly individual links users got, as could be seen in the previous sections this number was comparably small, for parties we got a maximum of about four URLs. For persons there are mostly less than two URLs which are potentially personalized (see Fig. 6 and Fig. 7).
This interesting difference when searching for persons in contrast to the search for parties shows that the space for personalization in these cases ranges between 0.5 and 4.5 (see Fig. 6). The largest space for personalization is seen for the party CSU which is only active in Bavaria.
Further for the analysis with similarity measure commons we divided the data in four different datasets with decreasing size:
All data,
German data,
Berlin data,
Berlin data regional.
We observe, that the space for personalization is decreasing, when we restrict the dataset to more local data. The largest amount for personalization is seen, if we take into account all data, which also covers search queries not originated in Germany.
When we investigate the same situation for searches regarding keys for persons (see Fig. 7), the space for personalization is even smaller and ranges from 0.5 to 2.
A possible personalization of search results is also the position at which a certain search result ist presented to the user. We addressed this problem by applying the measure deviation per rank which gives the percentage of results that change at each rank, usually one to ten, to the lists of search results, these results are shown in Table 4 and 3. Theses results show that regarding the search for parties we observe a relatively small amount of changes in the first two positions of the corresponding result lists, but when searching for persons the deviation per rank for the first four positions is less than 50%.
The next applied similarity measure for (ranked) lists is the longest common subsequence (LCS). For persons the results are shown in Fig. 8. It can be observed, that the lengths of the LCS for each person is increasing, when we restrict ourselves to more and more regional datasets. A similar result holds for result lists for parties, but with significantly lower values for LCS. As observed before the values for the Bavarian (regional) party CSU are smaller than for all the other parties.
With this measure, the LCS, the ordering of search result lists is not taken into account. The next measure we used is Kendall τ, a measure which also observes whether the lists of search results are ordered in the same way or not.
The similarity measure Kendall τ also takes into account the ordering of the URLs contained in the result lists. Therefore this measure is "sharper" than the measure commons. Our first observation is, that taking the mean of the measures Kendall τ over all time stamps for parties and persons we see that all the values are comparably large (mostly above 0.8 for persons 0.7 for parties) and the values are all positive, which means that the ordering of the result lists is overall consistent. Besides this the values are in a very small range. In Fig. 11 is shown that for the highly regionalized party CSU we see the smallest values for all parties.
For persons this measure shows that the results have a very similar ordering for all persons. An interesting observation in Fig. 10 is, that the values for all elements in the dataset, which also includes results obtained in countries other than Germany, shows that there is a significant deviation in the results for the keyword Angela Merkel. The result for Angela Merkel over all data is lowest in the field. We did indeed observe negative values for the keyword Angela Merkel when analyzing the whole dataset. But this was only the case for (non-German) result lists with two common elements in reverse order respectively.
Summarizing the results it can be claimed, that on the basis of the data we got, the space for a personalization of search results by Google is relatively small. Moreover we see the space for personalization decrease the more we restrict the data under investigation to a more regional basis. Thus we could show that one of the most important pillars of the filter bubble theory has been switched off by Google. Only regionalization seems to exist and without a fine-grained personalization, disjunctive information spheres hardly arise.
We can state that with such a "data donation", we have created a methodology for society as a whole to examine a black box like the Google search for important characteristics, without needing insight into the exact procedure of the algorithm.
https://datenspende.algorithmwatch.org/en/index.html
https://github.com/algorithmwatch/datenspende, published 07.06.2017.
From 4,416,585 to 3,707,302.
Since there were only 3793 unique URLs in the dataset Berlin, we could actually tag them manually.
Searches are influenced by each other when performed in a relatively small time slot. Hannak et al. [13] identified a slot size of less than 10 minutes were this effect van be seen.
The only weekend considered is the weekend of the election.
The only weekend considered is the election weekend.
John Mueller, a Google Webmaster Trends Analyst in an official video from Google, which explains how the Google search works: https://www.youtube.com/watch?v=ykfJUyD7y0A&t=170s.
Due to a disclaimer on "European data protection law provisions", some people get 9 results instead of 10 results, probably for space reasons.
The Amazon-Mechanical-Turk was used by [13].
ADM system:
algorithmic decision making system
IR:
LCS:
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We would like to take this opportunity to thank the State Media Authorities of Bavaria (BLM), Berlin-Brandenburg (mabb), Hesse (LPR Hessen), Rhineland-Palatinate (LMK), Saarland (LMS) and Saxony (SLM), who supported the project, and Spiegel Online as our media partner.
The data can be accessed via the following DOI https://doi.org/10.26204/DATA/1, only the user IDs are not included here. Due to data protection issues, these can only be shared upon personal request.
The project was financial supported by the State Media Authorities of Bavaria (BLM), Berlin-Brandenburg (mabb), Hesse (LPR Hessen), Rhineland-Palatinate (LMK), Saarland (LMS) and Saxony (SLM). As a media partner, we worked mainly with Spiegel Online.
Algorithm Accountability Lab, Department of Computer Science, Technische Universität Kaiserslautern, Kaiserslautern, Germany
Tobias D. Krafft
, Michael Gamer
& Katharina A. Zweig
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TDK, MG, KAZ analyzed and interpreted the data. TDK and MG have taken over the writing process of the manuscript. All authors read and approved the final manuscript.
Correspondence to Tobias D. Krafft.
A.1 Commons persons
Tables 5–8 are showing the common results for the result lists regarding persons for All, German, Berlin and Berlin without regional results.
Table 5 Mean of similarity commons over all result lists for persons
Table 6 Mean of similarity commons over all German result lists for persons
Table 7 Mean of similarity commons over all Berlin result lists for persons
Table 8 Mean of similarity commons over all Berlin result lists without regional urls for persons
A.2 Commons–parties
Tables 9–12 are showing the common results for the result lists regarding parties for All, German, Berlin and Berlin without regional results.
Table 9 Mean of similarity commons over all result lists for parties
Table 10 Mean of similarity commons over all German result lists for parties
Table 11 Mean of similarity commons over all Berlin result lists for parties
Table 12 Mean of similarity commons over all Berlin result lists without regional urls for parties
Table 13 Mean of similarity LCS over all result lists for persons
Table 14 Mean of similarity LCS over all German result lists for persons
Table 15 Mean of similarity LCS over all Berlin result lists for persons
Table 16 Mean of similarity LCS over all result lists for parties
Table 17 Mean of similarity LCS over all German result lists for parties
Table 18 Mean of similarity LCS over all Berlin result lists for parties
A.3 LCS person
A.4 LCS parties
A.5 Kendall τ
The only case where we observed negative values for Kendall τ is for the keyword "Angela Merkel". Here we see negative values when all data is under consideration. For the German and Berlin data the values of Kendall τ are positive.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Krafft, T.D., Gamer, M. & Zweig, K.A. What did you see? A study to measure personalization in Google's search engine. EPJ Data Sci. 8, 38 (2019) doi:10.1140/epjds/s13688-019-0217-5
DOI: https://doi.org/10.1140/epjds/s13688-019-0217-5
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What is the 17th odd positive integer?
Every odd positive integer can be expressed in the form $2x - 1$, for some integer $x$. Plugging in $x = 1$ gives $2 - 1 = 1$, which is the first odd positive integer. So the 17th odd positive integer is $2 \cdot 17 - 1 = \boxed{33}$.
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What is a state in physics?
What is a state in physics? While reading physics, I have heard many a times a "___" system is in "____" state but the definition of a state was never provided (and googling brings me totally unrelated topic of solid state physics), but was loosely told that it has every information of the system you desire to know. On reading further, I have found people talking of Thermodynamic state, Lagrangian, Hamiltonian, wave-function etc etc which I think are different from one another. So in general I want to know what do we mean by state in physics and is there a unique way to describe it?
terminology hilbert-space definition states-of-matter
Qmechanic♦
Manish Kumar SinghManish Kumar Singh
$\begingroup$ en.wikipedia.org/wiki/State $\endgroup$ – user83548 Dec 11 '15 at 18:51
$\begingroup$ exactly my question, there are so many definition for it, why shouldn't be there only one state. $\endgroup$ – Manish Kumar Singh Dec 11 '15 at 18:55
$\begingroup$ There are different notions of state because in physics, one often uses different mathematical models to describe systems in different contexts (e.g. different length scales), and in these different models, the state of a system is described by a different kind of mathematical object. $\endgroup$ – joshphysics Dec 11 '15 at 18:57
The definition of a state of a system, in physics, strongly depends on the area of physics one is dealing with and it comes as one of the initial definitions once such underlying theory has to be set up. In particular one has:
classical mechanics: a state of a system is a point $m\in TQ$ (or equivalently $T^*Q)$ in the tangent bundle of the configuration space (or the phase space, respectively). Such state is identified on a local chart with a set of coordinates $(q_i, \dot{q}_j)\in\mathbb{R}^N$ representing positions and velocities of all the particles at a given time $t$. Such description is equivalent to require the uniqueness of the solution of the Newton's equations once initial conditions are specified.
thermodynamics: a state is a set of extensive variables $(X_1,X_2,\ldots,X_N)$ that uniquely specify the value of the entropy function as $S(X_1,X_2,\ldots,X_N)\in\mathbb{R}$. Such variables represent the macroscopic extensive parameters (as volume, number of particles, total energy and so on and so forth) from which one can derive the corresponding associated intensive variables taking derivatives of the entropy as, for instance, $p=T(\partial S/\partial V)$ and similars.
quantum mechanics: a state is any element $|\psi\rangle\in\mathcal{H}$ of a Hilbert space together with a collection of self-adjoint operators $(A_1,\ldots,A_n, H)$. Special role is played by the Hamiltonian $H$, whose action mirrors classical mechanics giving the evolution in time of the state $|\psi(t)\rangle$. A collection of states (i. e. an ensamble) is instead described by a density matrix $\rho$ such that the expectation value of any operator on the ensamble can be defined as $\langle O \rangle = \textrm{tr}(\rho O)$.
field theories: very subtle as the definition of a state strongly depends on the theory at hand (quantum gravity, loop quantum gravity, string theory, QFT all have slightly different definitions of states).
EDIT: as per the suggestions in the comments below, more complex states and descriptions may and do arise, therefore the above is supposed to only be taken as general walkthrough.
gentedgented
$\begingroup$ I would argue, that your definition of thermodynamical state is a bit restrictive. One often works in ensembles with thermodynamic potentials, that do not necessarily only depend on extensive quantities (e.g. the canonical ensemble with $F(T, V, N)$). Plus there is a typo in an equation: $\partial_V S = p / T$. It also wouldn't hurt to mention the density matrix in quantum mechanics (as it is more general than a wave function). But this is nitpicking, so +1 anyway. $\endgroup$ – Sebastian Riese Dec 11 '15 at 19:51
$\begingroup$ Good answer. Just some remarks/questions: for point 2. I do not think you need to give such a specific role to the entropy function; it's a state function like the energy for instance. For point 3. is it really any element of Hilbert space? Shouldn't there be a set of observables associated to it somehow? I mean, if you take the Hilbert space of spin states of a spin half particle, good luck to get any information on the position of the particle. $\endgroup$ – gatsu Dec 11 '15 at 20:05
$\begingroup$ Yes, I agree with your remarks and I have edited my answer accordingly. $\endgroup$ – gented Dec 11 '15 at 20:31
$\begingroup$ For mechanical systems perhaps it would be better to take the coordinates and the impuls instead of coordinates and velocities? $\endgroup$ – HolgerFiedler Dec 12 '15 at 6:46
$\begingroup$ That is possible only when the Hessian of the Lagrangian is invertible. If so, yes. $\endgroup$ – gented Dec 12 '15 at 7:25
Our physics prof once put it informally that way:
A state is a set of variables describing a system which does not include anything about its history.
The set of variables (position, velocity vector) describes the state of a point mass in classical mechanics, while the path how the point mass got from point $A$ to point $B$ is not a state.
JensJens
$\begingroup$ That's nice and crisp. $\endgroup$ – Floris Dec 11 '15 at 20:58
Informally speaking, a complete description of a physical system is referred to as its state. Completeness of the state of a system means that it provides all the possible information about the system, i.e. everything that can be possibly known about the system has to be contained in the specification of its state.
Every physical theory is ultimately based on the following three fundamental postulates:
The postulate which defines the way we describe a state of a system.
The postulate which specify what kind of information about observables, i.e. measurable properties of the system, is contained in the description of its state.
And the postulate which provides us with a law that governs the time evolution of the system and allows us to predict its future state given the current one.
And in view of these fundamental postulates the meaning of completeness of the description provided by the state of a system is that all possible information about observables should be contained in the specification of the state and it should also be possible to use it to obtain all possible information about observables at any time in the future.
To make the definition of a state more formal and less vague we have to at least distinguish between classical and quantum theories because concrete manifestations of the above mentioned postulates for these two families of physical theories differ significantly. For instance, the meaning of the "all possible information about observables" phrase in quantum theories is quite unconventional from the classical point of view. And the rigorous definitions сan be given only for a particular physical theory since different mathematical objects are used to represent the state of a system in different theories as discussed in details in the answer given by Gennaro Tedesco.
WildcatWildcat
$\begingroup$ Good answer, but I would add that a state does not necessarily need to contain every bit of information that can possibly be known about the system. It just needs to contain all information that is relevant for the particular model being used. For example, a thermodynamic state contains only pressure, temperature, and number of particles, and completely ignores the individual motions of the particles, even though those could be known (in some thought-experimental system). $\endgroup$ – David Z Dec 12 '15 at 18:25
$\begingroup$ @DavidZ, true, but overall the question of what properties do we "attach" to a system is more about the system itself then about its state. And in that respect the notion of a system is even more metaphysical (at least when taken in general) than that of its state. Again, there will be at least a big difference between quantum and classical theories... $\endgroup$ – Wildcat Dec 12 '15 at 18:51
$\begingroup$ This is probably my favor answer, but I would like to ask about "physics laws give us time evolution of states", how about general relativity? $\endgroup$ – Shing Jan 5 '16 at 0:25
State in physics is a usefully ambiguous term, which is used in different ways in different fields; it's probably best understood in opposition to dynamics: state is static, and says nothing about motion; whereas dynamics tells you how one state evolves into another.
For example, in classical picture a state would be both the position and the momentum of a particle; knowing all the states of all the particles in the universe gives a snap-shot of the universe, or the state of the universe; but knowing all this does not tell you the state at some future moment - for this one also needs to know the dynamics - that is, the equations of motion; or simply how one state changes into another.
Another example, would be QM; there a state encodes the quantum system at hand, and (in the Schrodinger picture), are time-independent; the dynamics would then be given by Schrodingers equation which says how the state - the wave or potential - evolves.
(There is here, though the crucially complicating factor of observables, and acts of measurement).
However, it's also worth noting that there is another picture, the Heisenberg picture, where states do not evolve but observables do - this picture is more useful for the move into relativistic QM and/or QFT.
Mozibur UllahMozibur Ullah
Roughly, you describe state in physics as a series of particular values assigned to the different magnitudes that you can measure of the system, i.e. a value for the energy, pressure, temperature, ... or any magnitude that you're interested in.
So the state is a way of describing which properties has the system that you're going to study.
A. A.A. A.
$\begingroup$ I'm not sure if this really covers it, because e.g. a quantum state is not a measurable property. In quantum mechanics the state actually contains more information than what you can ever measure. $\endgroup$ – David Z Dec 12 '15 at 18:26
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ROBOMECH Journal
Quality visual landmark selection based on distinctiveness and repeatability
Masamichi Shimoda1 &
Kimotoshi Yamazaki1
ROBOMECH Journal volume 2, Article number: 16 (2015) Cite this article
In this study, a method for landmark selection from image streams captured by a camera mounted on a mobile robot is described. To select stable visual landmarks for mobile robots, two measures regarding landmark "visibility" are considered: distinctiveness and repeatability. In the proposed method, several neighboring feature points form a visual landmark and their distinctiveness is evaluated in each image. Then, under the assumption that a robot can actively seek a feasible landmark, the repeatability of the landmark is evaluated. Weighting techniques using feature-position relations are proposed, and landmark selection criteria using a variation coefficient are employed. These allow us to select high-visibility landmarks. Experimental results obtained using a real mobile robot demonstrate the effectiveness of the proposed method.
Mobile robots can have an extensive workspace in both indoor and outdoor environments. Thus, a reliable method for self-localization is very important. Several studies have examined the use of cameras and laser range finders to achieve environmental recognition and localization [1, 2].
The purpose of this study is to establish a framework for collecting visual landmarks from image streams. The image streams are assumed to be captured by a camera mounted on a mobile robot, and visual landmarks with high visibility are automatically extracted from the image streams. To understand "visibility" in the context of this study, we focus on distinctiveness and repeatability. Distinctiveness is represented by the uniqueness of a local image region in a robot's workspace, and repeatability is represented by the robustness of local image regions against possible viewpoint changes and occlusion. Both distinctiveness and repeatability are important for mobile robots because landmark detection might fail under various uncertain situations, e.g., accumulative positioning error and the kidnapping problem.
One conventional method to avoid such situations is the use of an image sequence [3, 4], which allows the determination of the current position with fast and light processing. However, it has a weakness: large occlusions or scene changing might cause a failure. Image features have also been used for reliable local information. For mobile robot navigation, hundreds of features detected from a single image have been used as landmarks [5–8]. Ogawa et al. [9] proposed a landmark selection method for robot navigation with a single camera. They extracted image features from each image and directly used them to describe a scene. Some studies employed an important theme relevant to feature point selection. Thompson et al. [10] proposed the use of landmarks selected automatically from panoramic images. "Turn Back and Look" behavior was used to evaluate potential landmarks. Normalized correlation enhanced a landmark's robustness against dramatic illumination change. Knopp et al. [11] proposed a method to suppress confusing features for increasing the success rate of localization. Hafez et al. [12] targeted a crowded urban environment and proposed a method to learn useful features through multiple experiences.
Some other studies have used local image regions as landmarks [13, 14]. Each image region includes some distinctive visual information, e.g., dozens of feature points. In contrast to the straightforward use of feature points, this setting allows easy viewpoint selection with a limited viewing field. In addition, if we annotate each landmark, they can be used for more semantic purposes. Both are suitable for autonomous mobile robots. In this study, we define a visual landmark using a local image region comprising dozens of neighboring image features. It is assumed that the robot travels multiple times on predefined courses, and useful landmarks are gradually selected during navigation.
We propose a method to select image regions with high distinctiveness and repeatability. Visual landmarks selected via the proposed method enable mobile robots to identify location using densely packed knowledge. However, well-designed evaluation criteria are required to select a quality landmark. One contribution of this study is to provide easily available criteria. Through experiments, we found that weighting each feature point in a local image region is important to describe a landmark with high distinctiveness and repeatability. The weight value is defined by the number of detections among input images. A high weight value is given to the feature point that is found in all images of a common scene from different observation points.
The remainder of this paper is organized as follows. "Visual landmarks" explains our representation of a visual landmark. "Landmark candidates collection" introduces landmark candidate selection, and "Landmark selection criteria" proposes landmark selection criteria. "Experiments" presents experimental results, and "Conclusion" concludes the paper.
Visual landmarks
Landmark availability
The quality of landmarks should be considered when extracting visual landmarks from image streams. In this study, we focus on the following four characteristics:
The landmark should be easy to distinguish from other parts of the scenes.
The landmark should be robust against occlusion.
There should be no significant difference of appearance even if the viewpoint changes.
The landmark should belong to a motionless object.
The above are based on conventional ideas for robust navigation. Item (1) conveys that distinguishable local image regions are easy to find from various viewpoints and capturing conditions. In addition, it suggests a way to eliminate confusing and redundant landmarks found in a scene. Item (2) is essentially achieved by using local image regions; however, it would be desirable to preemptively evaluate the possibility of occlusion. Item (3) is mostly applicable to mobile robots because a moving trajectory will not necessarily be the same for different navigations. Item (4) causes landmark deprivation, which negatively affects the reliability of self-localization.
Here, item (1) is associated with "distinctiveness," and items (2) to (4) are related to "repeatability." Landmarks that satisfy distinctiveness and repeatability are considered to have high "visibility." The proposed method selects quality visual landmarks in a step-by-step manner.
Distinctive feature region extraction based on feature point grouping
Image feature descriptions have been actively studied; therefore, we are now able to use high performance descriptors [15–17]. Since a tiny image region is required for many descriptors, using a group of image features affords good object detection performance that is robust against occlusion [18].
In this study, to generate a stable visual landmark, a rectangular region with dense image features is defined. The procedure to obtain a visual landmark is as follows. SIFT features are extracted from an input image. The detection criteria are the same as those described in [15]. Next, one feature is selected, and its neighboring features are searched. If the Euclidean distance between the selected feature and a neighboring feature in image coordinates is less than the predefined threshold D, they belong to the same group. Using the same procedure, another feature whose distance from the neighboring feature is less than the threshold is added to the group. This procedure enables the search for a cluster of image features. Finally, a circumscribed rectangular box that includes the cluster is generated as a local feature region of focus.
A local feature region is not necessarily required to have extremely dense feature points. If a landmark comprises highly distinctive features, it might have high visibility even if there are a less number of high-visibility features. However, a certain level of density is required; thus, parameter D is defined.
The abovementioned procedure might produce an uninformative image region comprising low distinctive features. Moreover, image regions without repeatability might be selected. To create a quality visual landmark, the feature region selection process is performed according to the procedure explained in "Landmark candidates collection" and "Landmark selection criteria".
Landmark candidates collection
Landmark selection procedure. Each landmark (red rectangular box) comprises dozens of image feature points. Through several phases, highly distinctive landmarks are selected, e.g., mutual consistency checks ensure the quality of the landmark
Landmark selection procedure
Figure 1 shows the landmark selection procedure. First, we outline the procedure. It consists of three phases:
Feature region detection:
SIFT features are extracted from an image, and rectangular regions that contain a feature cluster are selected ("Distinctive feature region extraction based on feature point grouping").
Landmark candidate selection:
Landmark candidate selection comprises two processes: small region elimination (explained below) and duplication avoidance (explained in "SIFT feature matching").
Landmark selection:
Note that only one image is considered in the above two phases. As we must select landmarks with high repeatability, robustness against viewpoint changes should be considered. Thus, the "Matching" process explained in "SIFT feature matching" is performed. Landmark repeatability is guaranteed by using dozens of images that capture the same scene from various viewpoints.
Here, we describe the small region elimination process. After the detection of local feature regions, the area size S of each region is calculated by image coordinates. Then, regions with S less than the predefined threshold \(S_s\) are eliminated. However, if a smaller region partly overlaps another larger region, the smaller region is considered over the larger region. The landmarks are also eliminated when the resulting region sizes are greater than the predefined threshold \(S_i\).
SIFT feature matching
The visual landmark used in this study comprises dozens of SIFT features. A SIFT feature is described by a 128-dimensional vector, and the representation is invariant to scale, translation, and rotation. In addition, its robustness against illumination is useful for robots in outdoor environments. Note that feature-to-feature matching is performed for both landmark detection and selection.
We apply two types of matching calculation. One is performed between two local feature regions cut from one input image to remove duplicate textures in the same scene. The other is applied for searching a local feature region from an input image to find one registered feature region from a present scene. We refer to the former matching as "Duplication Check" and the latter as "Matching."
Positional relationship between feature point and reference point. Using direction and scale information of SIFT features extracted in a training image, the position of a reference point is estimated in an input image
In Duplication Check, SIFT features are extracted from an image, and then, the local feature regions are generated. Let I be an image captured in a robot's workspace. Let \(\mathbf F_A = \{ \mathbf f^{(A)}_1, \mathbf f^{(A)}_2, \dots , \mathbf f^{(A)}_{N} \}\) be one local feature region extracted from I, where \(\mathbf f\) is a feature vector that corresponds to a feature point. Similarly, let \(\mathbf F_B = \{ \mathbf f^{(B)}_1, \mathbf f^{(B)}_2, \dots , \mathbf f^{(B)}_{M} \}\) be another local feature region, where \(N < M\). To calculate the similarity between \(\mathbf F_A\) and \(\mathbf F_B\), a feature vector \(\mathbf f^{(A)}_n\) is specified from \(\mathbf F_A\) and the Euclidean distances with all of feature vectors in \(\mathbf F_B\) are calculated. A feature vector \(\mathbf f^{(B)}_m\) with the minimum distance from \(\mathbf f^{(A)}_n\) is specified. If the distance is less than a pre-defined threshold, \(\mathbf f^{(A)}_n\) is considered to have correspondence. For all feature vectors in \(\mathbf F_A\), if the number of correspondences is greater than the pre-defined threshold, the two feature regions are eliminated because they are too similar to represent an independent region.
In Matching, the distance calculation is the same as that in Duplication Check. However, another distance threshold \(b_2\), which is looser than \(b_1\), is used. Then, a consistency check is performed against the resulting correspondences. First, the center of gravity of a local feature region is set as a reference point. As shown in the upper part of Fig. 2, a positional vector from each feature point to the reference point is calculated. The vector is transferred into a corresponding feature point extracted from an input image. Thus, the position of a reference point can be estimated in the input image. A SIFT feature contains information about intensity, direction, and scale; therefore, position (X, Y) is calculated using the following equations:
$$\begin{aligned} \begin{array}{lll} X = x_i - \displaystyle {\frac{\sigma _i}{\sigma _l}} \times \sqrt{\Delta x^2 + \Delta y^2} \times \cos (\theta + \theta _l - \theta _i) , \\ Y = y_i - \displaystyle {\frac{\sigma _i}{\sigma _l}} \times \sqrt{\Delta x^2 + \Delta y^2} \times \sin (\theta + \theta _l - \theta _i) , \\ \theta = \tan ^{-1} \displaystyle {\frac{\Delta y}{\Delta x}} , \end{array} \end{aligned}$$
where \(\sigma _l\) and \(\theta _l\) are the scale and angle of a feature point, respectively. In addition, \(\sigma _i\) and \(\theta _i\) are the same variables for a feature point in the input image; \(x_i\) and \(y_i\) are coordinates of the point in the image; and \((\Delta x, \Delta y)\) is a positional vector. If the number of estimated reference points, which are concentrated in a circle of radius d, is greater than the pre-defined threshold \(m_2\), the local feature region is considered to have correspondence.
The above idea is inspired by the implicit shape model [19], which is used for generic object recognition. Such positional relations are useful for eliminating mismatching when the similarity value becomes high with feature-to-feature correspondence [20].
Landmark selection criteria
Several local feature regions are selected through the procedure described in "SIFT feature matching", which considers distinctiveness. In other words, these feature regions satisfy item (1) ("Landmark availability"). Next, these regions are screened relative to repeatability based on items (2) to (4).
In this study, we have attempted to develop a visual function for an autonomous mobile robot. One assumption is that we can deploy an autonomous robot that moves in a workspace. Scene observation at various viewpoints enhances the quality of knowledge used for visual navigation. Based on the above discussion, landmark selection with multiple observations is performed. In other words, a camera is mounted on a robot, and n number of images are captured for one target scene while the robot moves.
Using these images, we employ the following four landmark selection methods.
Pairwise comparison of local feature regions [13].
Repeatability of local feature regions in input images.
Counting individual local feature correspondences in input images.
Using weight coefficient.
The details of these methods are described in the following order.
(a) Pairwise comparison of local feature regions
Here, "Duplication Check" techniques described in "SIFT feature matching" are used. First, one local feature region is selected and its similarity with another local feature region in another image is calculated. If the similarity value (i.e., the number of matched feature points) of the most similar region is greater than a predefined threshold, the two regions are associated (dark red line in Fig. 3). By applying this process to all local feature regions, a non-directed graph is obtained. Next, a set of local feature regions associated with each other is sought. The region with the greatest number of arcs \(l_c\) is selected as a visual landmark. Here, \(l_c\) is a criterion used to identify the visibility of a landmark.
Figure 3 shows three examples of landmark selection. Four sets of local feature regions are extracted from four different images. In the case of (A), the red-painted landmark candidate is selected by counting the number of arcs. When several landmark candidates have the same number of arcs, as shown in (B), a region with denser feature points is selected. Item (C) shows another case. When one region has the greatest number of arcs but large occlusion reduces the number of observable feature points, it is not selected as a landmark.
Mutual consistency check. Several local feature regions are extracted from four different images. a The red-painted landmark candidate is selected based on the number of arcs. b A landmark with dense feature points is selected if several landmark candidates have the same number of arcs. c When one region has the largest number of arcs but large occlusion reduces the number of observable feature points, it is not selected as a landmark
(b) Repeatability of local feature regions in input images
Here, "Matching" described in "SIFT feature matching" is used wherein a local feature region is selected in order and sought from each image. By applying the seeking process toward n images, the number of detections \(l_i\) is counted, where i indicates a serial number of a local feature region. If \(l_i\) is greater than a predefined threshold, then the ith local feature region is registered as a landmark.
In the processing explained in item (a), local feature region detection may fail when some feature points cannot be extracted from an input image. This means that the local feature regions extracted at different viewpoints might lose the correct correspondence. Meanwhile, the abovementioned process makes it possible to restore the situation.
(c) Counting individual local feature correspondences in input images
The abovementioned process considers landmark quality using the local feature region. However, better performance might be obtained if the number of correspondences between two image feature points is also considered. For example, if a feature point is extracted at the local region that captures two distant objects, its appearance is largely influenced by viewpoint changes. The local feature region having such a feature point should be assigned low reliability. Thus, we propose the following measure.
As with item (b), a local feature region is sought from images. In each of feature region seeking process, the number of feature correspondences is registered. This describes the frequency of finding respective feature points from several input images; thus, weight coefficient \(g_j\) is defined by the number of feature correspondences, where j denote the serial number of feature point. If \(g_j\) is greater than a pre-defined threshold, a parameter \(f_g\) is incremented. A landmark with large \(f_g\) has the potential to be a high repeatability landmark.
(d) Using weight coefficient
Using weight coefficient \(g_j\) described above, another weight coefficient G is calculated as follows:
$$\begin{aligned} G = \displaystyle {\frac{\Sigma g_j}{k}} \end{aligned}$$
When occlusion or appearance change occur by changes in viewpoint, G becomes small. In other words, large G are one criterion for selecting high repeatability landmarks.
\(f_g\) and G are similar criteria, where \(f_g\), which indicates the number of detection for each feature point, is binarized and G is a variable that directly considers the number of detections. The latter allows us to know the quality of a landmark in more detail. In addition, it allows us to represent additional information, e.g., the density of good features.
A mobile robot with a single mounted camera was used for our experiments. The mobile platform was "i-Cart mini" produced by the T-frog project [21], and the camera was a BSW32KM (Buffalo Americas Inc.). A laptop computer was mounted on the platform. It was used to capture VGA (\(640 \times 480\) pixels) images and control the platform. Image datasets were collected for both indoor (our experimental laboratory) and outdoor (ten different scenes on our university campus) environments.
Image capturing positions. Nine positions divided in a reticular pattern are given to the robot. are given to the robot
Quality landmark selection
Nine shooting locations were set in each of the target scenes, as shown in Fig. 4. The distance between neighboring locations was 0.2 m.
Landmark selection was performed by the four methods described in "Landmark selection criteria". The parameters used to select the local feature region were experimentally defined as follows:
Euclidean distance D to group two feature points ("Distinctive feature region extraction based on feature point grouping") was set to \(\sqrt{10}\).
Radius for investigating feature concentration ("SIFT feature matching") was set to \(d \le \sqrt{10}\)
\(b_1 \le 150\) and \(b_2 \le 250\) ("SIFT feature matching"),
Thresholds for the area of feature region \(S_s\) and \(S_l\) ("Landmark selection procedure") were set to 2500 and 40,000, respectively.
The parameters used to select a visual landmark determined by brute force. The results were as follows:
Number of high similarity regions: \(l_c \ge 3\).
Number of corresponding regions: \(l_i \ge 9\).
Number of detections of features: \(g_j \ge 9\)
These were the conditions used to select visual landmarks with respect to the criteria introduced in "Landmark selection criteria". Only the landmark candidates that satisfied each condition were selected as visual landmarks. These values were based on the assumption that nine images were used. If more images are to be used, these values should be increased linearly.
The abovementioned parameters were experimentally defined; therefore, one concern was their sensitivity. In our experience, it was not significantly high as long as we examined the proposed method using images captured in indoor and outdoor environments. When we slightly changed the parameters, the quality of the landmarks degraded in some scenes even though the changes improved the quality of landmarks in other scenes. The parameters given in this study might be rough estimates; however, they provided acceptable results.
Criterion for quality evaluation
In this study, it was assumed that a robot travels on a predefined course many times. While the robot moves along the course, the number of detections for each landmark was counted. The result was then used to evaluate the repeatability of the landmark.
A variation coefficient was used for this purpose. This calculation was performed by dividing the standard deviation (Std.) by the average (Ave.) with respect to the number of detections for each landmark. If the value is small, we consider the landmark to have high repeatability.
Landmark selection results. Numbers in columns 2 to 12 show the number of times of correct correspondence. (Ave. and Std. are calculated for each landmark)
Landmark examples
First, we present a landmark selection example from indoor environments. The visibility of these landmarks was confirmed through eleven automatic navigations. In each navigation, one hundred images were captured at 3 [fps]. Landmarks were then detected using these images.
The rightmost images in Fig. 5 are visual landmarks selected from the scene. The left columns in the table show the name of the landmark, and the top row shows the number of experiments. A to D show landmarks whose number of arcs was greater than 9 (\(l_i \ge 9\)). They were stably detected in the complete images with a small variation coefficient. On the other hand, E and F show \(l_i = 7\) and \(l_i = 8\), respectively. These were also relatively stable landmarks; however, the values of the variation coefficient were greater than those of the abovementioned case. These results indicate that \(l_i\) can be used to determine the landmark quality.
Visibility evaluation
The same procedure described in the previous subsection was performed using images obtained in ten outdoor locations. Methods (a)–(d) ("Landmark selection criteria") were used to determine whether they are suitable for selecting a quality landmark. Figure 6 shows a list of variation coefficients for all local feature regions. The blue and red points indicate landmarks and other local feature regions, respectively. It is not always true that landmarks with \(l_c\) greater than 3 have a smaller variation coefficient than the other local feature regions. The same holds true for Fig. 7, which shows the results for method (b). In addition, it is not always true that landmarks with \(l_i\) greater than 9 have a small variation coefficient.
The SIFT features included in the landmarks were examined to clarify the reason behind these observations. In some cases, the features were extracted from a spatial region where a large perspective change occurred. These features are not robust against viewpoint changes; thus, it is expected that they would not be included in the landmark. Another problem unique to method (a) is that a local feature region can differ according to the layout of the feature points. Figure 8 shows an example. One large region was extracted at one viewpoint; however, it was divided into two regions in another viewpoint. This caused a misdetection of the landmark.
Serial number of landmark/local feature region vs. variation coefficient by method (a). There are nearly no quality difference between the selected landmarks and the local feature region
Serial number of landmark/local feature region vs. variation coefficient by method (b). Compared with Fig. 6, there is no noteworthy difference
Landmark detection differences. One large region was extracted; however, it was divided into two regions in another viewpoint. This caused misdetection of the landmark
Method (c) considers the adequacy of SIFT features. Figure 9 shows the relation between the serial number of the landmark and the variation coefficient. Here, "all" means that all features were used for landmark detection and "only" means that only features with weight coefficient \(f_g\) were used for the detection. Obviously, using features with the weight coefficient resulted in small variation coefficients. This means that the criteria for defining the weight coefficient are useful for selecting high visibility landmarks.
Figure 10 shows the relation between the serial number of a landmark and the average number of correct correspondences. Using features with a weighted coefficient provided stable landmark detection. This means that the weighted features allow us to find correspondences with high repeatability because they were easy to find from a set of images captured at different viewpoints. In addition, the processing time to find landmarks with a weighted coefficient was 23.89 s for 100 frames. On the other hand, the same process using all feature points required 26.06 s, which is a reduction of 8.33 %.
Serial number of landmark vs. variation coefficient. Features with the weight coefficient show small variation coefficient. This means that the criterion for defining the weight coefficient is useful for selecting a high visibility landmark
Serial number of landmark vs. average number of correspondence. Features with a weighted coefficient yield stable landmark detection
As can be seen in Fig. 11, the average number of correspondences tended to be large when the feature points had large weight coefficient G. This graph shows that the proposed approach allows us to find quality landmarks using the weight coefficient, e.g., G greater than 5.0 statistically guarantees quality landmarks. This value can be predefined; thus, quality landmark selection can be automatically achieved. The same trend can be observed from the relation between the weight coefficient and the variation coefficient (Fig. 12). A greater weight coefficient results in a landmark with a lower variation coefficient.
Weight coefficient vs. average number of correspondence. For example, landmarks with G greater than 5.0 statistically guarantees its quality
Weight coefficient vs. variation coefficient
Other feature descriptors
A SIFT descriptor is robust against changes in scale, rotation, translation, and illumination because these characteristics are suitable for mobile robots. In addition, we can find other excellent descriptors with equivalent characteristics. A feature descriptor that provides scale and direction information is applicable to the proposed method; therefore, we attempted to replace SIFT with another feature descriptor.
Here, there are two primary steps to extract an image feature point: feature point detection and feature description. Speeded-Up Robust Features (SURF) [22] are well-known approximations of SIFT features. To detect SURF keypoints, a box filter was applied to calculate the scale-space extremum. Note that the density of the extremum tends to become sparse; thus, it may show poorer performance than SIFT because the proposed method requires densely extracted keypoints. This assumption was experimentally confirmed using the abovementioned images. We set a smaller \(l_c\), and this blurred the line between a landmark and other feature regions.
As another proof, feature description using FREAK [17] was also examined. Here, to generate a local feature region, the parameters were the same as those described in "Quality landmark selection". The FREAK descriptor was applied to each feature point extracted by the SIFT method. Figures 13 and 14 show the results obtained by method (c). The significance of the landmark quality was lesser than that of SIFT. Although the basic tendency was the same, i.e., a small coefficient variance was found by using feature regions with weight coefficient \(g_j\), the SIFT feature showed better performance compared with the proposed method.
Serial number of landmark vs. variation coefficient. Landmark distinctiveness is less than that of SIFT. However, the basic tendency was the same
Serial number of landmark vs. average number of correspondences
In this study, we have proposed a method for visual landmark selection from image streams captured by a camera mounted on a mobile robot. Using a visual landmark consisting of dozens of neighboring feature points, two evaluation criteria were considered: distinctiveness and repeatability. To evaluate visibility, distinctiveness was evaluated for each image. Then, under the assumption that robots can seek a feasible landmark actively, the repeatability of the landmark was evaluated. Experiments using real images demonstrated that weighting each feature point included in a local feature region is important to describe a landmark with high distinctiveness and repeatability.
In the future, we will examine automatic threshold determination. The existing method required a manually defined threshold; therefore, this burden should be reduced. Application to mobile robot is also important orientation.
Betke M, Gurvis L (1997) Mobile robot localization using landmarks. IEEE Trans Robot Autom 13(2):251–263
Thrun S, Fox D, Burgard W, Dellaert F (2000) Robust Monte Carlo localization for mobile robots. J Artif Intell 128(1–2):99–141
Matsumoto Y, Inaba M, Inoue H (1996) Visual navigation using view-sequenced route representation. In: Proceedings of International Conference on Robotics and Automation, pp 83–88
Kaneko Y, Miura J (2011) View Sequence Generation for View-Based Outdoor Navigation. In: Proceedings of 1st Asian Conference on Pattern Recognition, pp 139–144
Celaya E, Albarral J, Jimenez P, Torras C (2007) Natural landmark detection for visually-guided robot navigation. Artif Intell Hum Oriented Comput 4733–2007:555–566
Cummins M, Newman P (2008) FAB-MAP: Probabilistic localization and mapping in the space of appearance. Int J Robot Res 27(6):647–665
Sato T, Nishiumi Y, Susuki M, Nakagawa T, Yokoya N (2008) Camera position and posture estimation from a still image using feature landmark database. In: Proceedings of International Conference on Instrumentation, Control and Information Technology, pp 1514–1519
Se S, Lowe D, Little J (2001) Local and Global Localization for Mobile Robots using Visual Landmarks. In: Proceedings of International Conference on Intelligent Robots and Systems, pp 414–420
Ogawa Y, Shirai Y, Shimada N (2007) Environmental map-ping for mobile robot by tracking SIFT feature Points using trinocular vision. In: SICE, Annual Conference. IEEE, Takamatsu, pp 1996–2001
Thompson S, Matsui T, Zelinsky A (2000) Localisation using Automatically Selected Landmarks from Panoramic Images. In: Proceedings of Australian Conference on Robotics and Automation, pp 167–172
Knopp J, Sivic J, Pajadla T (2010) Avoiding confusing features in place recognition. In: Proceedings of 11th European Conference on Computer Vision, pp 671–748
Hafez A, Singh M, Krishna K, Jawahar C (2013) Visual Localization in Highly Crowded Urban Environments. In: Proceedings of IEEE/RSJ Conference on Intelligent Robots and Systems, pp 2778–2783
Hayet JB, Lerasle F, Devy M (2007) A visual landmark framework for mobile robot navigation. J Image Vis Comput 25(8):1341–1351
Mata M, Armingol JM, de la Escalera A, Salichs MA (2002) Learning visual landmarks for mobile robot navigation. In: 15th Triennial World Congress
Lowe D (2004) Distinctive image features from scale-invariant keypoints. Int J Comput Vis 60(2):91–110
Tuytelaars T, Gool LV (2004) Matching widely separated views based on affine invariant regions. Int J Comput Vis 50(1):61–85
Alahi RO, Vandergheynst P (2012) FREAK: Fast Retina Keypoint. In: IEEE Conference on Computer Vision and Pattern Recognition
Piccinini P, Prati A, Cucchiara R (2012) Real-time object detection and localization with SIFT-based clustering. J Image Vis Comput 30:573–587
Leibe B, Leonardis A, Schiele B (2006) An Implicit Shape Model for Combined Object Categorization and Segmentation. Toward Category-Level Object Recognition, Lecture Notes in Computer Science, vol 4170, pp 508–524
Ihara A, Fujiyoshi H, Takagi M, Kumon H, Tamatsu Y (2009) Improved Matching Accuracy in Traffic Sign Recognition by Using Different Feature Subspaces. In: Proceedings of International Conference on Machine Vision Applications, pp 130–133
http://t-frog.com/en/. (19/12/2014)
Bay H, Ess A, Tuytelaars T, Gool LV (2008) SURF: Speeded Up Robust Features. Comput Vis Image Underst 110(3):346–359
KY proposed the visual landmark method. MS improved the method, and carried out experiments. Both authors read and approved the final manuscript.
This work was partly funded by ImPACT Program of the Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).
Mechanical Systems Engineering, Faculty of Engineering, Shinshu University, 4-17-1, Wakasato, Nagano, Nagano, Japan
Masamichi Shimoda & Kimotoshi Yamazaki
Masamichi Shimoda
Kimotoshi Yamazaki
Correspondence to Kimotoshi Yamazaki.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Shimoda, M., Yamazaki, K. Quality visual landmark selection based on distinctiveness and repeatability. Robomech J 2, 16 (2015). https://doi.org/10.1186/s40648-015-0036-9
Landmark selection
Mobile robot
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How do I power Goldeneye?
I've been spending some time thinking about putting a satellite weapon into space. It would look something like this. Don't worry about those letters, they stand for something else.
This weapon can direct an electromagnetic pulse at the Earth of sufficient intensity to cripple a city. How do I provide enough power for this weapon?
The power system must be feasible with 1995 technology, able to survive in space for years to decades until called upon, and powerful enough to energize said pulse weapon.
Also, I may have other nefarious satellites in orbit, so this has to be a power source, not a bomb. I don't want to damage any of my other investments while I reduce London to ruin.
weapons engineering power-sources
kingledionkingledion
$\begingroup$ Your weapon would be much bigger than this. You'd need something like a gamma ray burst and from space, that would be a ton of energy. $\endgroup$ – The Anathema Sep 13 '18 at 19:03
$\begingroup$ "so this has to be a power source, not a bomb" But bombs are power sources! $\endgroup$ – RonJohn Sep 13 '18 at 19:04
$\begingroup$ @RonJohn Absolutely agree. They are like batteries. Limited charge capacity. It can be used until the charge is depleted (or exploded, same thing). $\endgroup$ – ArtificialSoul Sep 14 '18 at 9:46
$\begingroup$ To add a bit more detail to the people saying "it has to be a bomb": Nuclear EMPs result from gamma rays produced by the bomb interacting with diffuse gas in the upper atmosphere and Earth's magnetic field. At altitudes where satellites can orbit for decades on end without maintenance, there may not be enough gas to make an EMP. So, instead, your satellite could store several nuclear warheads that it can drop anywhere on the planet. If these nukes detonate in the upper atmosphere, they'll make EMPs. And if your other satellites are in high enough orbits, they shouldn't be affected. $\endgroup$ – Someone Else 37 Sep 15 '18 at 4:46
If you want to trigger an EMP with sufficient intensity to cripple a city from orbit, I'm sorry, but it's going to have to be a bomb.
EMPs are not efficient. You could use an directed energy weapon (probably microwaves), to disrupt specific targets - in fact, there's research into building such a weapon into a cruise missile. But your energy requirements increase non-linearly as the area you wish to affect increases. Additionally, the greater the distance from source to target, the greater the energy required. Eventually, if you're operating from orbit, you're going to need so much energy that it's more efficient just to make it a bomb.
(And, in fact, the Goldeneye weapons in the movie of the same name were single-shot bombs as well - from the James Bond wikia:
...the weapon consisted of two disposable satellites designated "Petya" and "Mischa", each one armed with a nuclear warhead. By detonating the device in the upper atmosphere, a pulse or a radiation surge, is generated; capable of destroying all electronic devices in a 30 mile radius.
jdunlopjdunlop
$\begingroup$ It seems that ability to focus the energy matters as well. My flashlight that uses 3 D-size batteries can light up the entire opposite wall of a room, but outside, I still can't see very far down a trail at night, for example. However, the laser pointer I annoyed my with uses a single button battery, can be seen reflecting on something hundreds of feet away, but only a coin-sized dot, no bigger. In other words, if I can restrict the "Goldeneye" beam very narrowly, I might not need as much energy at the source. I admit I'm not sure, but this answer feels incomplete without addressing it. $\endgroup$ – cobaltduck Sep 13 '18 at 18:53
$\begingroup$ It's true that you can restrict the beam (as in the directed energy weapon I linked in the second paragraph), but in that situation, you're not fulfilling the OP's desire to cripple a city. Collimation of an EMP is also somewhat impractical, as the chaotic nature of the field is what does the damage to electronics. You can do damage with focused VHF or microwave beams, but it has to be very targeted. $\endgroup$ – jdunlop Sep 13 '18 at 18:58
$\begingroup$ Can't believe I omitted the word "cat" as in "annoyed my cat with" Too late to edit, also. $\endgroup$ – cobaltduck Sep 13 '18 at 19:00
$\begingroup$ "energy requirements increase non-linearly as the area you wish to affect increases". But doesn't area increase non-linearly? $\endgroup$ – RonJohn Sep 13 '18 at 19:05
$\begingroup$ @RonJohn. It does, but EMP (even directed) affects a volume, rather than an area. So it's geometric in proportion to area, as well. $\endgroup$ – jdunlop Sep 13 '18 at 20:14
Since everybody loves math as much as I do
Here is some math
Geosynchronous orbit (GSO) is ca. 36,000km altitude.
Earth radius is 6,371km.
So distance to a satellite is about 30,000km.
The area of your target (London) is 1,572 km².
Imagine a nuclear bomb being set off at GSO. The blast would spread in all directions. So the portion that would be directed at London (if it was directly above London) would be $$R = \frac{A_{\text{London}}}{A_\text{Sphere;30000km}} = 1.39 \times 10^{-7}$$ or as a solid angle of $$\Omega = \frac{A_{\text{London}}}{(30,000km)²} = R \times 4 \pi = 1.75 \times 10^{-6}$$
That would be the focus a weapon would need. It's quite narrow, but a laser should manage to achieve that. (I am not a laser specialist, but I am certain that should work.)
What radiation passes through the atmosphere?
Microwaves pass through the atmosphere with only slight issues on some wavelengths. So we can assume they reach the surface just fine without accounting for further attenuation. (That is a simplification. There is some radiation blocked, but we're gonna ignore that. It's not that much.)
(High-Power Microwaves (HPM)) achieve a similar effect to EMPs, but are more difficult to harden against. Unfortunately, a big part of an EMP by nuclear explosion is caused by ionization of the air by gamma radiation. That radiation would mainly be absorbed by the atmosphere and would not reliable reach the target ground. At least not at high efficiency.
So we'll stick with an HPM concept for now.
The Non-bomb weapon
Information on how much energy is needed for a HPM to work is hard to come by. Bofors HPM Blackout is a weapon like that. Unfortunately, we don't know how much energy it needs. Just that it wighss less than 500kg.
Nuclear fission bombs range from under a ton to more than 500,000 kiltons of TNT. All of the sizes cause an EMP. - Obviously at different magnitudes. These sizes range from 4.2 GJ to 210 PJ energy output. In a nuclear explosion this energy is manifesting in several different ways. EMP, Air blast, Heat, light, etc.. Not all of those things would be useful for the desired result, so we would need less energy, if we manage to concentrate it.
But overall we would still need at minimum several GigaJoules of energy to reliably fry circuits in the target area.
How would you store this amount of energy electrically rechargable in a satellite with 1995 technology?
You couldn't. First of all until that would be charged using solar panels would probably take quite a while, unless your satellite is more than space station size at 230 W/m² for solar panels.
And how would you store that energy?
Well, considering there are no battery concepts now with more than 5 MJ/kg (Link in German because of a significantly more detailed list) you would need many tons of batteries.
And the additional problem of radiation hardening. To my knowledge batteries are not that fond of radiation, so having them out in space would require significant mass to protect them from it.
Even if the radiation wouldn't destroy them, it would probably lower the capacity over time as well as make it lose energy, making your recharging even more a problem.
I doubt it is possible. Maybe today with High-End secret military projects, but not with anything normal I could find.
But with 1995 technology it is even less likely.
ArtificialSoulArtificialSoul
$\begingroup$ "And how would you store that energy?" A bomb!!! :) $\endgroup$ – RonJohn Sep 14 '18 at 12:55
$\begingroup$ @RonJohn Bombs would certainly make that easier. $\endgroup$ – ArtificialSoul Sep 14 '18 at 12:57
The ionosphere will power it.
Lightning is a natural cause of electromagnetic pulses. These natural occurring EMP pulses can do significant damage to electronics on the ground. http://www.alphamarinesystems.com/lightning_and_emp_damage.htm
The ionosphere of the earth carries a high electrical charge. This charge builds up because the atmosphere is a good insulator, preventing the charge from going to ground.
http://www.metlink.org/wp-content/uploads/2013/06/PhysRev25_4_Nicoll.pdf
Global current The potential difference between the ionosphere and Earth's surface is approximately 250kV (2.5 × 105V). This very large potential difference means that the charged ions in the atmosphere will move and thus produce a vertical current.
Your Goldeneye capitalizes on this. It first emits a gas which diffuses out in a large radius, quickly becoming conductive plasma. Then it fires a "rod from god" orbital weapon from its onboard railgun (powered by a solar charged capacitor). As opposed to the typical role of these weapons to traverse the atmosphere intact and deliver a massive kinetic punch, this rod is made of silver and is intended to ablate on the way down, shedding particles and silver plasma en route. It will take it 20 seconds to make it to the ground.
But it will not reach the ground - the rod will ablate to nothing before making impact. It does not have to. Behind it, the immense charge coming down the conductive path of plasma will complete the process, arcing across the remaining atmosphere to ground and delivering an immense electromagnetic pulse.
Goldeneye produces a lightning bolt from the ionosphere. There is a lot of ionosphere. This trick will work more than once.
I have seen a video of lightning called down along a rocket-lifted copper wire. The portion along the wire was green, probably from the copper plasma. I am not sure what color silver plasma would be.
WillkWillk
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BMC Health Services Research
Active and adaptive case finding to estimate therapeutic program coverage for severe acute malnutrition: a capture-recapture study
Sheila Isanaka ORCID: orcid.org/0000-0002-4503-28611,2,
Bethany L. Hedt-Gauthier3,4,
Halidou Salou5,
Fatou Berthé6,
Rebecca F. Grais2 &
Ben G. S. Allen7
BMC Health Services Research volume 19, Article number: 967 (2019) Cite this article
Coverage is an important indicator to assess both the performance and effectiveness of public health programs. Recommended methods for coverage estimation for the treatment of severe acute malnutrition (SAM) can involve active and adaptive case finding (AACF), an informant-driven sampling procedure, for the identification of cases. However, as this procedure can yield a non-representative sample, exhaustive or near exhaustive case identification is needed for valid coverage estimation with AACF. Important uncertainty remains as to whether an adequate level of exhaustivity for valid coverage estimation can be ensured by AACF.
We assessed the sensitivity of AACF and a census method using a capture-recapture design in northwestern Nigeria. Program coverage was estimated for each case finding procedure.
The sensitivity of AACF was 69.5% (95% CI: 59.8, 79.2) and 91.9% (95% CI: 85.1, 98.8) with census case finding. Program coverage was estimated to be 40.3% (95% CI 28.6, 52.0) using AACF, compared to 34.9% (95% CI 24.7, 45.2) using the census. Depending on the distribution of coverage among missed cases, AACF sensitivity of at least ≥70% was generally required for coverage estimation to remain within ±10% of the census estimate.
Given the impact incomplete case finding and low sensitivity can have on coverage estimation in potentially non-representative samples, adequate attention and resources should be committed to ensure exhaustive or near exhaustive case finding.
ClinicalTrials.gov ID NCT03140904. Registered on May 3, 2017.
Program coverage is a measure of how many individuals in need are receiving treatment or an intervention. It is an important indicator to assess the performance of public health programs and is essential to inform program planning and prioritization of limited resources. Coverage, combined with program effectiveness, is critical to assess how many of those in need are accessing treatment or prevention activities and achieving the desired outcome.
In the management of severe acute malnutrition (SAM), several practical methods for treatment coverage estimation have been proposed [1] that identify cases using active and adaptive case finding (AACF). AACF is an informant-driven sampling method that yields a sample of individuals who possess specific characteristics and have been referred by others, starting with a "seed" or key informant(s) to begin the referral chain [1, 2]. Similar methods have commonly been used when sampling hard-to-reach populations such as injection drug users [3] or men who have sex with men [4]. When sampling children with SAM, AACF has two advantages: it is active and therefore does not rely on cases self-presenting as in central point sampling, thus avoiding cases not arriving due to stigma associated with the illness, distance or other factors [5]; and it is efficient as only houses of suspected cases, not all houses, in a sampling area are visited. AACF is particularly suitable for conditions with symptoms that can be visibly identified and that are rare and therefore require a larger sampling area in order to reach an adequate sample size. However, as this method can yield a non-representative sample, AACF should be exhaustive or nearly exhaustive to yield valid estimates of coverage [1].
Although practical guidelines have been proposed to indicate when sample exhaustion has been reached during AACF [1, 2], there is uncertainty around whether the method can ensure an adequate exhaustivity in operational settings [6]. Debate surrounding the practical validity of the case finding method thus remains. To inform the continued use of AACF in the estimation of SAM treatment coverage, we assessed the exhaustivity of AACF to identify SAM cases in northwestern Nigeria.
Study setting
This study was conducted in the Wamako Local Government Area in Sokoto State of northwestern Nigeria in 2017. The region is characteristic of the rural Sahel and has a stable population with a high burden of acute malnutrition (global acute malnutrition: 10.4, 95% CI: 7.5, 14.2% in 2015 [7]). From 2013 to 2017, International Medical Corps supported the Sokoto State Ministry of Health to deliver treatment of uncomplicated SAM at five outpatient centers, with community surveillance and outreach teams in approximately 430 villages (average village size: 483 people) [8].
For this study, we defined sensitivity as the probability of a sampling method to correctly identify a child in the community that has SAM or is recovering from SAM. We assessed the sensitivity of AACF using a capture-recapture design [9, 10]. Capture-recapture designs were first used in ecological studies to estimate animal populations [10] and have more recently been applied to assess the total case population of health conditions using two independent sources, such as two disease registers [10, 11]. In a capture-recapture study, two case finding methods are used to determine the size of the total case population, and with that information, the sensitivity of each case finding method can be estimated [9]. In this study, AACF was compared to a census method where all households were visited and all children 6–59 months screened on sequential days. While often considered to be a gold standard, the census method may miss cases, for example due to routine absence from the household on the day of recapture. The capture-recapture study design does not require that either case finding procedure find 100% of cases to estimate the total number of cases in the study population or method-specific sensitivity [9]. However to be valid, the capture-recapture study must adhere to five assumptions: closed population; ability to perfectly match cases captured in both methods; in both methods, perfect classification (perfect diagnosis of SAM and coverage status); within a method, any child with SAM has equal probability of capture; and independence of capture between methods [9,10,11] (Table 1).
Table 1 Descriptions of the assumptions underlying AACF and study procedures to reduce potential violations
Current operational guidance on the use of capture-recapture studies to validate SAM case finding recommends that the estimated number of cases found in both samples be greater than seven and the number of total cases found across both samples be greater than the estimated SAM population [9, 12]. A priori, we estimated AACF would capture 40% of SAM cases (sensitivityAACF = 40%) and that a census would capture 80% of SAM cases (sensitivitycensus = 80%). This would require 24 SAM cases to exist to meet the first condition. Assuming an average village size of 483 [8], a SAM prevalence of 2.7% [7] and the proportion of children aged 6–59 month to be 20% of the population [7], nine villages were estimated to be necessary to identify 24 SAM cases. Given the time and resources available, 15 villages were ultimately sampled to be sure that the minimum sample size would be reached for the first condition above.
Study procedures
AACF method
Prior to case finding, a SAM screening definition was developed using qualitative methods [2] [see Additional file 1]. Semi-structured interviews were first conducted in four villages. An interview guide was used to identify context-specific terms related to SAM, which were triangulated and used to devise a screening definition. This screening definition was then iteratively tested and revised with new information over three days until no new information was found. The resulting screening definition included terminology in two local languages (Hausa and Fulani) to describe the signs and symptoms of SAM as well as associated illnesses. Stigmatizing terms were identified to ensure they were avoided, and teams were aware if used by informants. Information on local beliefs about the etiology, health-seeking behaviors and the types of individuals with knowledge about children with SAM were also collected. This additional information was collected to allow enumerators to target individuals during case finding that would be more knowledgeable about the location of SAM cases.
During case finding, the context-specific screening definition iteratively developed for the study, as well as photos of malnourished children and packets of ready-to-use therapeutic food (RUTF) used for the treatment of SAM, were presented in each sampled village to help orient key informants towards suspected SAM or recovering cases. Key informants included traditional birth attendants, village leaders, caregivers, grandmothers, traditional healers, community nutrition volunteers, children and health center staff. The houses of all suspected cases were visited for individual evaluation. In each household, a brief household interview was conducted to ensure no child aged 6–59 months was sleeping or absent. All children present were assessed for SAM, defined as mid-upper arm circumference (MUAC) < 11.5 cm and/or bilateral pitting edema (Table 2). To identify recovering cases in the household, caregivers were asked if any child was undergoing treatment for SAM. RUTF sachets were presented to confirm enrollment. All identified SAM and recovering cases were confirmed to be resident in the village, and if so, name, age and sex were recorded to facilitate matching between case finding methods. Any identified SAM case not undergoing treatment was referred to the nearest outpatient center for treatment. In this study, AACF was considered exhaustive when teams were referred back to two cases already identified and all areas of the village had been visited.
Table 2 Case definitions used during case finding and coverage estimation [13]
Census method
During census case finding, the survey teams systematically visited each household in the village. Following the same household-level procedures as AACF, a household census was completed to identify all children 6–59 months of age, and all children present were evaluated using the standard case definition for SAM and recovering cases (Table 2). Census case finding was considered exhaustive when all households in the sampled village were visited.
Sensitivity was calculated as the proportion of all SAM and recovering cases that were correctly identified as such and estimated using the Chapman modification to the Lincoln-Petersen estimator [14]. The numerator was defined as the number of cases found using each method, and a denominator was defined as the total case population (N). The total case population (N) was estimated using Eq. 1 below [9, 14] with the observed number of cases identified using each method (a, b, and c in Table 3).
Table 3 2 × 2 table showing types of cases found in both samples
$$ N=\frac{\left(a+b+1\right)\times \left(a+c+1\right)}{a+1}-1\ \Big( $$
SAM treatment coverage was estimated using each case finding procedure according to current guidelines [1, 15]. To better understand the influence of the AACF sensitivity on coverage estimation, in a sensitivity analysis we calculated program coverage at varying levels of AACF sensitivity and distribution of coverage among missed cases and compared to coverage estimated using the census method.
In our study 59 SAM and recovering cases were found using AACF and 75 were found using the census method. Of those cases, 52 were found using both case finding methods, seven were found using only AACF, and 23 found only using the census method (Table 4). Three children were not found by either method. From this, we estimate the total SAM and recovering case population size across the 15 sampled villages to be 85. The sensitivity of our AACF method was 69.5% (95% CI: 59.8, 79.2) and for the census method was 91.9% (95% CI: 85.1, 98.8). The estimated SAM treatment coverage was 40.3% (95% CI 28.6, 52.0) using AACF and 34.9% (95% CI 24.7, 45.2) using the census method.
Table 4 Cases found during active and adaptive and census case finding
In sensitivity analyses, we found that AACF yielded coverage estimates very similar to that produced using the census method when either the AACF method had high sensitivity (e.g. 90–100%) or when the program coverage in the cases missed by AACF was approximately the same as the overall coverage of 34.9% (Table 5). In our study, six out of the 23 (26%) cases not found by AACF were covered by the program. This resulted in a non-significant over-estimate of coverage in this example (40.3% with AACF vs. 34.9% with census).
Table 5 Estimated coverage by sensitivity of AACF and corrected for the unobserved coverage of missed cases
AACF has been proposed as an efficient case finding method to estimate SAM treatment coverage. In this study, we estimated the sensitivity of AACF to be 69.5% (95% CI: 59.8, 79.2), or more specifically that AACF as applied in this study correctly identified approximately 7 out of 10 SAM and recovering cases.
Field-friendly approaches for obtaining coverage estimates are now available to help nutrition program managers directly measure treatment coverage [1]. These methodologies allow for routine assessment by program staff and support community engagement through participatory methods [16] . The current operational guidance on these methods for SAM coverage estimation introduce various case finding procedures. Selection of the most appropriate procedure is necessarily context-dependent, but in practice, AACF is often considered the default method. However, as AACF is an informant-driven procedure and may yield non-representative samples, AACF case finding should be exhaustive or nearly exhaustive to produce valid SAM coverage estimates. The operational guidance specific to AACF suggests 75% sensitivity to be adequate but offers limited guidance to know exactly when this has been achieved [9]. Notwithstanding implementation of a parallel capture-recapture study to measure sensitivity, guidance suggests simply that "sampling stops only when you are sure that you have found all SAM cases in the community" and "case-finding was considered to be exhaustive when no new leads to potential cases were forthcoming and when information given by different sources (e.g., key informants and carers) identified children that had already been seen by the team" [1] . In this study, we applied a stricter definition of exhaustivity, which required teams to be referred back to cases already identified at least two times, and significant resources were made to support exhaustivity, including iterative development of a sound case definition and appropriate training of enumerators to support complete case identification.
Despite these efforts, the AACF missed a total of 26 of a potential 85 cases (30.6%), including 23 cases found using census and an estimated three cases found by neither method. The missed cases lowered AACF sensitivity below the level suggested to be acceptable by operational guidance (75%) [9]. There is little published evidence that quantifies AACF sensitivity in the context of SAM treatment; however, a report of capture-recapture studies (2003–2011), including six comparing AACF to house-to-house case finding and 17 to a central location screening method, showed sensitivities of above 75% in 20/23 (87%) studies [17]. The authors of that report acknowledge that surveys analyzed were provided from early adopters of the coverage methodology, and that subsequent results using procedures locally adapted from these early studies in other settings may not replicate these findings.
The impact of incomplete case finding (e.g. low sensitivity) on coverage estimation is not well understood, and incomplete case-finding could result in bias in either direction depending on the distribution of coverage among missed cases. Sensitivity analyses suggested that case finding should generally have a sensitivity of ≥70% in order to avoid bias of more than 10% in coverage estimation, depending on the distribution of coverage among missed cases. Program managers using AACF should consider the resources and technical capacity needed to ensure such case finding sensitivity can be achieved for valid coverage estimation and consider alternative methods (e.g. census) if necessary.
This study has a number of strengths. First, the sample size ensured greater precision to estimate sensitivity and coverage estimates. Second, careful planning was made to ensure that the five assumptions underlying the capture-recapture design were adhered to (Table 1). For example, four individual-level identifiers were collected from confirmed cases in order to allow cases in both samples to be effectively matched. A well-developed and tested local case definition ensured key informants were able to orientate enumerators towards SAM and recovering cases. The same objective case definition was applied during both case finding methods and survey enumerators were trained, standardized and supervised in anthropometric assessment, assuring a correct and equal diagnosis in both methods. To maintain independence of capture between methods, the census method systematically assessed all households in a sampled village, irrespective of case finding results using AACF the previous day. Finally, efforts were made by teams to ensure each child had an equal risk of being captured, for example by finding the child if absent from the household but known to be in the village.
Despite using the same village boundaries, avoiding known market and treatment days and encouraging carers of cases to remain at home the following day, we were unable to guarantee a perfectly "closed population" to ensure the same population was present during both samples. Violation of the assumption of a closed population meant that seven cases were found during AACF and not during the census method the following day, and an additional seven cases were absent from the village during AACF. The direction of bias in the coverage point estimate due to such missed cases depend on the distribution of coverage among these children. In future use of AACF, absent cases could be reduced by informing village authorities and carers of children aged 6–59 months to stay at home between certain hours when the survey team were to visit.
With limited operational guidance on how to define and achieve exhaustivity, future coverage assessments that use AACF should take care to develop a strong screening definition and ensure exhaustivity by all reasonable measures. This may require dedicating additional personnel to each village during case finding, communicating with village leaders prior to arriving in the village and applying strict criteria to determine when exhaustivity has been reached, such as continuing case finding until re-directed to several cases already found that day. If there is any doubt in the sensitivity of AACF being adequate, a census method, such as door-to-door sampling, might be also considered as recommended in the operational guidance [1]. In this study, 15 days were needed to complete AACF and 14.5 days to complete for census case finding. As such, a census may not present substantially greater logistical or financial burden. We further note that AACF requires the development and testing a local screening definition, and in diverse study populations, this process may need to be repeated among different sub-groups that speak different languages or represent different socio-cultural contexts. In such settings, the census method which does not require context-specific adaptations may offer a comparative efficiency. In contrast, in a large homogenous population where the same screening definition could be reasonably used for case finding across many villages, AACF may prove to be a more efficient approach than a systematic census. These results may apply to assessing coverage of SAM treatment in other rural settings, though AACF is still not recommended for assessing coverage of moderate acute malnutrition treatment (where cases are less recognizable and may not be readily identifiable by key informants), or in urban or camp settings where community cohesion may be limited and key informants may not be aware of incident cases [18].
Given the impact incomplete case finding and low sensitivity can have on coverage estimation in potentially non-representative samples, adequate resources and capacity should be committed to ensure exhaustive or near exhaustive case finding.
The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.
AACF:
Active and adaptive case finding
MUAC:
Mid-upper arm circumference
Outpatient therapeutic program
RUTF:
Ready-to-use therapeutic food
Severe acute malnutrition
SQUEAC:
Semi-quantitative evaluation of access and coverage
Myatt M, Sadler K. Semi-Quantitative Evaluation of Access and Coverage (SQUEAC)/ Simplified Lot Quality Assurance Sampling Evaluation of Access and Coverage (SLEAC) Technical Reference. 2012;(October):1–241. Available from: www.fantaproject.org
Myatt M, Woodhead S. Developing an active and adaptive case-finding procedure for use in coverage assessments of therapeutic feeding programs [internet]. 2016. Available from: http://www.coverage-monitoring.org/wp-content/uploads/2016/01/Developing-an-active-and-adaptive-case-finding-procedure-for-use-in-coverage-assessments-of-therapeutic-feeding-programs.pdf
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Kendall C. An Emprical comparison of respondent-driven sampling, time location sampling, and snowball sampling for Behavioural surveillance in men who have sex with men, Fortaleza. Brazil AIDS Behav. 2008;12:S97–104.
Bliss JR, Njenga M, Stoltzfus RJ, Pelletier DL. Stigma as a barrier to treatment for child acute malnutrition in Marsabit County, Kenya. Matern Child Nutr. 2016;12(1):125–38.
Epicentre. Open Reivew of Coverage Methodologies: Questions, comments and ways forward. 2015; Available from: http://www.coverage-monitoring.org/wp-content/uploads/2015/03/Open-Review-of-Coverage-Methodologies-Questions-Comments-and-Way-Forwards.pdf
International Medical Corps. SMART Nutrition and Mortality Survey Report in Wamakko and Binji. 2015;
Sokoto State Ministry of Health. Nigeria National Population Commission Census. 2006.
Myatt M, Wegerdt J, Zanchettin M. Using capture-recapture studies to investigate the performance of case-finding procedures. 2016; Available from: http://www.coverage-monitoring.org/wp-content/uploads/2016/01/Developing-an-active-and-adaptive-case-finding-procedure-for-use-in-coverage-assessments-of-therapeutic-feeding-programs.pdf
Hook EB, Regal RR. Capture-recapture methods in epidemiology: methods and limitations. Epidemiol Rev. 1995;17(2):243–64.
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Nigerian Federal Ministry of Health, Family Health Department ND. National Guidelines for Community Management of Acute Malnutrition. 2011.
Chapman D. Some properties of the hypergeometric distribution with applications to zoological sample censuses. Berkeley: University of California Press; 1951. p. 131–59.
Balegamire BS, Siling K, Alvarez Moran J-L, Guevarra E, Woodhead S, Norris A, et al. A single coverage estimator for use in SQUEAC , SLEAC , and other CMAM coverage assessments. Field Exchange [Internet]. 2015;(49) Available from: https://www.ennonline.net/fex/49/singlecoverage.
Blanárová L, Rogers E, Magen C, Woodhead S. Taking severe acute malnutrition treatment Back to the community: practical experiences from nutrition coverage surveys. Front Public Heal. 2016;4(September):1–5.
Myatt M, Fieschi L, Ouma C, Guevarra E, Emary C. A review of historical data on the case-finding sensitivity of active and adaptive case-finding procedures for severe acute malnutrition; 2016.
Guerrero S, Kyalo K, Yishak Y, Kirichu S, Sebinwa U, Norris A. Debunking urban myths : access & coverage of SAM-treatment programmes in urban contexts. Field Exhange [Internet]. 2013;(46) Available from: https://www.ennonline.net/fex/46/debunking.
We sincerely thank Mark Myatt for his careful review of the methods and early version of the manuscript.
No specific funding was provided for preparation of this manuscript.
Department of Nutrition and Global Health and Population at Harvard School of Public Health, Boston, USA
Sheila Isanaka
Department of Research, Epicentre, Paris, France
& Rebecca F. Grais
Department of Global Health and Social Medicine (Harvard Medical School), Boston, USA
Bethany L. Hedt-Gauthier
Department of Biostatistics (Harvard School of Public Health), Boston, USA
Epicentre Niger, Maradi, Niger
Halidou Salou
Epicentre Nigeria, Sokoto, Nigeria
Fatou Berthé
Technical Rapid Response Team and International Medical Corps, Washington, DC, USA
Ben G. S. Allen
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SI and BGSA contributed to the conception and design of the study, analysis and interpretation of data, and drafted the manuscript. BHG contributed to the analysis and interpretation of data and critically reviewed the manuscript. RG contributed to the conception of the study, interpretation of data, and critically reviewed the manuscript for important intellectual content. HS and FB contributed to the design of the study and critically reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Sheila Isanaka.
Ethics approval was provided by the Harvard T.H. Chan School of Public Health and the Sokoto State Ministry of Health.
Additional file 1. Supplementary Methods Appendix 1
Isanaka, S., Hedt-Gauthier, B.L., Salou, H. et al. Active and adaptive case finding to estimate therapeutic program coverage for severe acute malnutrition: a capture-recapture study. BMC Health Serv Res 19, 967 (2019) doi:10.1186/s12913-019-4791-9
Active and adaptive
Case finding
Capture recapture
SQUEAC
Therapeutic feeding program
Community-based management of acute malnutrition
Quality, performance, safety and outcomes
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CommonCrawl
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\begin{document}
\title{Monotone Classes Beyond VNP}
\pagenumbering{gobble} \begin{abstract}
We study the natural monotone analogues of various equivalent definitions of $\mathsf{VPSPACE}$: a well studied class \cite{P08,KP09,M11,MR13} that is believed to be larger than $\mathsf{VNP}$.
We show an exponential separation between the monotone version of Poizat's definition of $\mathsf{VPSPACE}$ \cite{P08} and monotone $\mathsf{VNP}$.
We also show that unlike their non-monotone counterparts, these monotone analogues are not equivalent, with exponential separations in some cases.
The primary motivation behind our work is to understand the monotone complexity of \emph{transparent polynomials}, a concept that was recently introduced by \Hrubes{} and Yehudayoff \cite{HY21}.
In that context, we are able to show that \emph{transparent polynomials} of large sparsity are hard for the monotone analogues of all known definitions of $\mathsf{VPSPACE}$, except for the one due to Poizat. \end{abstract}
\pagenumbering{arabic} \setcounter{page}{1}
\section{Introduction} The aim of algebraic complexity is to classify polynomials in terms of how hard it is to compute them, and the most standard model for computing polynomials is that of an \emph{algebraic circuit}. An algebraic circuit is a rooted, directed acyclic graph where the leaves are labelled with variables or field constants and internal nodes are labelled with addition $(+)$ or multiplication $(\times)$. Every node therefore naturally computes a polynomial and the polynomial computed by the root is said to be the polynomial computed by the circuit. \autoref{defn:algebraic-circuits} is a formal definition.
The central question in the area is to show super-polynomial lower bounds against algebraic circuits for \emph{explicit} polynomials, or to show that $\mathsf{VP} \neq \mathsf{VNP}$, the algebraic analogue of the famed $\mathsf{P}$ vs. $\mathsf{NP}$ question. However, proving strong lower bounds against circuits has turned out to be a difficult problem. Much of the research therefore naturally focusses on various restricted algebraic models which compute correspondingly structured polynomials.
One such syntactic restriction is that of \emph{monotonicity}, where the models are not allowed to use any negative constants. Therefore, trivially, monotone circuits always compute polynomials with only non-negative coefficients. Such polynomials are called \emph{monotone polynomials}. We denote the class of all polynomials that are efficiently computable by monotone algebraic circuits by $\mathsf{mVP}$. Also note that any monomial computed during intermediate computation in a monotone circuit can never get cancelled out, making it a fairly weak model. As a result, several strong lower bounds are known against monotone circuits.
\paragraph*{Lower bounds in the monotone setting}
There has been a long line of classical works that prove lower bounds against monotone algebraic circuits \cite{Sch76, SS77, SS80, JS82, K85, KZ86, Gas87}. The most well-known among these, is the result of Jerrum and Snir \cite{JS82} where they showed exponential lower bounds against monotone circuits for many polynomial families, including the Permanent ($\operatorname{\sf Perm}_n$). In particular, they showed that every monotone algebraic circuit computing the $n^2$-variate $\operatorname{\sf Perm}_n$ must have size at least $2^{\Omega(n)}$. A few of the more recent works on monotone lower bounds include \cite{RY11, GS12, CKR20}.
Additionally, many separations that are believed to be true in the general setting have actually been proved to be true in the monotone setting \cite{SS77, HY16, Y19, S19}. Most remarkably, Yehudayoff \cite{Y19} showed an exponential separation between the computational powers of the the monotone analogues of $\mathsf{VP}$ and $\mathsf{VNP}$ (denoted by $\mathsf{mVP}$ and $\mathsf{mVNP}$ respectively).
Another line of work in this setting tries to understand the power of non-monotone computational models while computing monotone polynomials. Valiant \cite{V80}, in his seminal paper, showed that there is a family of monotone polynomials which can be computed by polynomial sized non-monotone algebraic circuits such that any monotone algebraic circuit computing them must have exponential size. More recent works \cite{HY13, CDM21, CDGM22, CGM22} have shown even stronger separations between the relative powers of monotone and non-monotone models while computing monotone polynomials.
\paragraph*{Newton polytopes, transparency and monotone complexity} Returning briefly to the general setting, an interesting conjecture relating the algebraic complexity of a bivariate polynomial to its geometric property, is the `Tau-conjecture' (also written as $\tau$ conjecture). The Newton polytope of an $n$-variate polynomial $f$, denoted by $\mathsf{Newt}(f)$, is the convex hull in $\R^n$ of the \emph{exponent vectors} of the monomials in the support of $f$. Recently, \Hrubes{} and Yehudayoff \cite{HY21} proposed the notion of \emph{Shadows of Newton polytopes} (the maximum number of vertices in any linear projection of the polytope to a plane) as an approach to refute the $\tau$-conjecture for Newton polygons made by Koiran, Portier, Tavenas and Thomass\'{e} \cite{KPTT15}.
Informally, the $\tau$-conjecture for Newton polygons \cite{KPTT15} states that if $f$ is a bivariate polynomial that can be written as an $s$-sum of $r$-products of $p$-sparse polynomials, then its Newton polygon has at most $\poly(s,r,p)$ vertices. \autoref{defn:newton-polytopes} and \autoref{conj:tau-conjecture-newton-polygons} are the formal definition of Newton polytopes and the formal statement of the $\tau$-conjecture for Newton polygons respectively.
This is a fairly strong conjecture and it implies, among other things, that $\mathsf{VP} \neq \mathsf{VNP}$. However, observe that the Newton polygon retains no information about the coefficients of the polynomial. Since the algebraic complexity of polynomials is believed to be heavily dependent on coefficients (for example the determinant ($\operatorname{\sf Det}_n$) is efficiently computable by algebraic circuits and this is expected to not be the case for $\operatorname{\sf Perm}_n$ even though they have the same set of monomials), the $\tau$-conjecture for Newton polygons is believed to be false.
The approach suggested by \Hrubes{} and Yehudayoff~\cite{HY21} used shadows of Newton polytopes as a means to move from the multivariate setting to the bivariate setting, and use polynomials like determinant ($\operatorname{\sf Det}_n$) to refute the conjecture.
The difficulty in this strategy however, is to find a polynomial in $\mathsf{VP}$ that exhibits high \emph{shadow complexity}, since even when a candidate polynomial is fixed, say $\operatorname{\sf Det}_n$, it is not easy to design a suitable bivariate projection.
As a means to tackle this issue, \Hrubes{} and Yehudayoff introduced the notion of \emph{transparent polynomials} --- polynomials that can be projected to bivariates in such a way that all of their monomials become vertices of the resulting Newton polygon. Further, they also gave examples of polynomials with exponentially large sets of monomials that are provably transparent. Therefore a proof of any one of these polynomials being in $\mathsf{VP}$ would directly refute the $\tau$-conjecture for Newton polytopes.
Even though \Hrubes{} and Yehudayoff~\cite{HY21} were not able to actually use this approach to refute the conjecture, they used the notions of shadows and transparency to come up with yet another method for proving lower bounds against monotone algebraic circuits. They showed that the monotone circuit complexity of a polynomial is lower bounded by its shadow complexity when the polynomial is transparent.
\begin{theorem}[{\cite[Theorem 2]{HY21}}]
If $f$ is transparent then every monotone circuit computing $f$ has size at least $\Omega(\abs{\operatorname{supp}(f)})$. \end{theorem}
As a corollary, they present an $n$-variate polynomial such that any monotone algebraic circuit computing it must have size $\Omega(2^{n/3})$.
\subsection{Our Contribution} Here we state our contributions informally; the formal statements can be found in \autoref{sec:contributions-formal}. The goal of this work is two-fold.
The first goal is to understand how restrictive the notion of transparency is.
Our search begins with an observation by Yehudayoff~\cite{Y19}, that any lower bound against $\mathsf{mVP}$ depending solely on the support of the hard polynomial, automatically ``lifts'' to $\mathsf{mVNP}$ with the same parameters\footnote{\cite{Y19}: ``If a monotone circuit-size lower bound for $q(\vecx)$ holds also for all polynomials that are equivalent to $q(\vecx)$ then it also holds for every $\mathsf{mVNP}$ circuit computing $q(\vecx)$.''}. Since transparency is a property solely of the Newton polytope, and hence of the support of the polynomial, the above observation shows that any transparent polynomial that is non-sparse (has super-polynomially large support) is hard to compute even for $\mathsf{mVNP}$. However we believe that transparency is a very strong property and a natural question for us therefore, is whether there are even larger classes of monotone polynomials that do not contain non-sparse, transparent polynomials.
This brings us to the second goal of this work --- studying monotone models of computation that can possibly compute polynomials outside even $\mathsf{mVNP}$. Classes larger than $\mathsf{VNP}$ had not been defined in the monotone world prior to this work, and we therefore turn to the literature in the non-monotone setting. Here, $\mathsf{VPSPACE}$ is a well studied class \cite{P08,KP09,M11,MR13} that is believed to be strictly larger than $\mathsf{VNP}$. Interestingly there are multiple definitions of $\mathsf{VPSPACE}$, resulting from varied motivations, which are all known to be essentially equivalent \cite{M11, MR13}.
We study the natural monotone analogues of these definitions and show that unlike the non-monotone setting, the powers of the different resulting models varies greatly. This allows us to then analyse the technique of \Hrubes{} and Yehudayoff against monotone classes that are possibly larger than $\mathsf{mVNP}$. The following figure succinctly describes our main results.
\begin{figure}
\caption{Nodes represent classes of polynomial families; $A\dashrightarrow B$ denotes $A\subseteq B$ and $A\longrightarrow B$ denotes $A\subsetneq B$.}
\label{fig:classes_inclusion}
\end{figure}
In \autoref{fig:classes_inclusion}, the node labels refer to the following classes of polynomial families that have $\poly(n)$ complexity under various models, as follows: \begin{itemize}
\item \textsf{msuccABP} - monotone succinct ABPs (\autoref{defn:monotone-succinct-ABPs}),
\item $\mathsf{mVP}_{\mathsf{quant}}$ - quantified monotone circuits (\autoref{defn:quantified-monotone-algebraic-circuits}),
\item $\mathsf{mVP}_{\mathtt{sum},\mathtt{prod}}$ - monotone circuits with summation and production gates (\autoref{defn:monotone-circuits-summation-production}),
\item $\mathsf{mVP}_{\mathsf{proj}}$ - monotone circuits with projection gates (\autoref{defn:monotone-algebraic-circuits-with-projection}). \end{itemize} The orange, rectangular nodes denote the classes in which sparsity of transparent polynomials in it is bounded by a constant factor of the size of the smallest $\mathcal{M}$ computing it, if $\mathcal{M}$ is the computational model corresponding to the class (\autoref{thm:transparency-hard-for-QuantMonVP}). An interesting point to note here is that there is an exponential separation between $\mathsf{mVP}_{\mathsf{quant}}$ and $\mathsf{mVP}_{\mathsf{proj}}$, which means that at least one of the inclusions: $\mathsf{mVP}_{\mathsf{quant}}$ to $\mathsf{mVP}_{\mathsf{sum,prod}}$, and $\mathsf{mVP}_{\mathsf{sum,prod}}$ to $\mathsf{mVP}_{\mathsf{proj}}$ is strict with an exponential separation.
\subsection{Organisation of the paper}
We begin in \autoref{sec:prelims} with formal definitions for all the models of computation that we will be using. Next, we define the monotone analogues of the various definitions of $\mathsf{VPSPACE}$, and outline our results about them in \autoref{sec:contributions-formal}. The proofs of our results are discussed in \autoref{sec:succinct-monotone-abp}, \autoref{sec:quantified-circuits}, \autoref{sec:monotone-circuits-summation-production} and \autoref{sec:monotone-circuits-projection}. We conclude with \autoref{sec:conclusion}, where we discuss some of the important open threads from our work.
\section{Preliminaries}\label{sec:prelims}
We shall use the following notation for the rest of the paper. \begin{itemize}
\item We use the standard shorthand $[n] = \set{1,2,\ldots,n}$.
\item We use boldface letters like $\vecx,\vecz,\vece$ to denote tuples/sets of variables or constants, individual members are expressed using indexed version of the usual symbols: $\vece = (e_1,e_2,\ldots,e_n)$, $\vecx=\set{x_1,\ldots,x_n}$.
We also use $\abs{\vecy}$ to denote the size/length of a vector $\vecy$.
For vectors $\vecx$ and $\vece$ of the same length $n$, we use the shorthand $\vecx^{\vece} $ to denote the monomial $x_1^{e_1} x_2^{e_2} \cdots x_n^{e_n} $.
\item For a polynomial $f(\vecx)$ and a monomial $m = \vecx^{\vece} $, we refer to the coefficient of $m$ in $f$ by $\operatorname{coeff}_f(m) $.
The support $\operatorname{supp}(f)$ of a polynomial $f$ is given by $\set{m : \operatorname{coeff}_f(m) \neq 0}$, and the \emph{sparsity} of a polynomial is the size of its support, $\abs{\operatorname{supp}(f)}$. \end{itemize}
\paragraph*{Algebraic Circuits and basic monotone classes}
We now formally define algebraic circuits. \begin{definition}[Algebraic circuits]\label{defn:algebraic-circuits}
An algebraic circuit is a directed acyclic graph with leaves (nodes with in-degree zero) labelled by formal variables and constants from the field, and other nodes labelled by addition $(+)$ and multiplication $(\times)$.
The leaves compute their labels, and every other node computes the operation it is labelled by, on the polynomials along its incoming edges.
There is a unique node of out-degree zero called the root, and the circuit is said to compute the polynomial computed at the root.
The \emph{size} of a circuit, $\mathcal{C}$, denoted by $\operatorname{size}(\mathcal{C})$, is the number of nodes in the graph.
An algebraic circuit over $\Q$ or $\R$ is said to be \emph{monotone}, if all the constants appearing in it are non-negative. \end{definition}
Next, we formally define the relevant classes of monotone polynomials that are already present in literature, namely the monotone analogues of $\mathsf{VP}$ and $\mathsf{VNP}$.
\begin{definition}[Monotone $\mathsf{VP}$ ($\mathsf{mVP}$)]\label{defn:mVP}
A family $\set{f_n}$ of monotone polynomials is said to be in $\mathsf{mVP}$, if there exists a constant $c \in \N$ such that for all large $n$, $f_n$ depends on at most $n^c $ variables, has degree at most $n^c $, and is computable by a monotone algebraic circuit of size at most $n^c $. \end{definition}
\begin{definition}[Monotone $\mathsf{VNP}$ ($\mathsf{mVNP}$)]\label{defn:mVNP}
A family $\set{f_n}$ of monotone polynomials is said to be in $\mathsf{mVNP}$, if there exists a constant $c \in \N$, and a family $\set{g_m} \in \mathsf{mVP}$, such that for all large enough $n$, and $m \leq n^c$, $f_n $ satisfies the following.
\[
f_n(\vecx) = \sum_{\veca \in \set{0,1}^{\abs{\vecy}}} g_{m}(\vecx,\vecy = \veca) \qedhere
\] \end{definition}
\paragraph*{Newton Polytopes and the tau conjecture for Newton polygons}
\begin{definition}[Newton polytopes]\label{defn:newton-polytopes}
For a polynomial $f(\vecx)$, its Newton polytope $\mathsf{Newt}(f) \subseteq \R^n$, is defined as the convex hull of the \emph{exponent vectors} of the monomials in its support.
\[
\mathsf{Newt}(f) := \operatorname{conv}\inparen{\set{ \vece : \vecx^{\vece} \in \operatorname{supp}(f)}}
\]
A point $\vece \in \mathsf{Newt}(f)$ is said to be a \emph{vertex}, if it cannot be written as a convex combination of \emph{other} points in $\mathsf{Newt}(f)$.
We denote the set of all vertices of a polytope $\mathcal{P}$ using $\operatorname{vert}(\mathcal{P})$. \end{definition}
\begin{conjecture}[\texorpdfstring{$\tau$}{Tau} conjecture for Newton polytopes \cite{KPTT15}]\label{conj:tau-conjecture-newton-polygons}
Suppose $f(x,y)$ is a bivariate polynomial that can be written as $\sum_{i \in [s]} \prod_{j \in [r]} T_{i,j}(x,y)$, where each $T_{i,j}$ has sparsity at most $p$.
Then the Newton polygon of $f$ has $\poly(s,r,p)$ vertices. \end{conjecture}
\noindent We now move on to formally stating the various definitions of $\mathsf{VPSPACE}$. This will allow us to then define their monotone analogues.
\subsection{Various definitions of \textsf{VPSPACE}}
Koiran and Perifel~\cite{KP09, KP07} were the first to define $\mathsf{VPSPACE}$ as the class of polynomials (of degree that is potentially exponential in the number of underlying variables) whose coefficients can be computed in $\mathsf{PSPACE}/\poly$ and $\mathsf{VPSPACE}_\text{b}$ to be the polynomials in $\mathsf{VPSPACE}$ that have degree bounded by a polynomial in the number of underlying variables. They showed that if $\mathsf{VP} \neq \mathsf{VPSPACE}_\text{b}$ then either $\mathsf{VP} \neq \mathsf{VNP}$ or $\mathsf{P}/\poly \neq \mathsf{PSPACE}/\poly$.
Later, Poizat~\cite{P08} gave an alternate definition that does not rely on any boolean machinery, but instead uses a new type of gate called a \emph{projection gate}.
\begin{definition}[Projection gates \cite{P08}]\label{defn:projection}
A \emph{projection} gate is a \emph{unary} gate that is labelled by a variable $z$ and a constant $b \in \set{0,1}$, denoted by $\project{z}{b}$.
It returns the partial evaluation of its input polynomial, at $z=b$, that is, $\project{z}{b}(f(z,\vecx)) = f(b,\vecx)$. \end{definition}
Poizat defined algebraic circuits with projection gates and then defined $\mathsf{VPSPACE}$ to be the class of polynomial families that are efficiently computable by this model. Poizat showed\footnote{The work of Poizat is written in French, Malod~\cite{M11} provides an alternate exposition of some of the main results in English.} that this definition is equivalent to that of Koiran and Perifel.
\begin{definition}[Algebraic circuits with projection gates \cite{P08}]\label{defn:algebraic-circuits-with-projection}
An \emph{algebraic circuit with projection gates} is a directed acyclic graph with leaves (nodes with in-degree zero) labelled by formal variables and constants from the field, and other nodes labelled by addition $(+)$, multiplication $(\times)$ or projection $(\project{z}{b})$.
The leaves compute their labels, the nodes labelled by addition and multiplication compute the operation they are labelled by, on the polynomials along its incoming edges, and nodes labelled by projection gates compute the polynomial described in \autoref{defn:projection}.
There is a unique node of out-degree zero called the root, and the circuit is said to compute the polynomial computed at the root.
The \emph{size} of an algebraic circuit with projection gates is the number of nodes in the graph. \end{definition}
Adding to Poizat's work, Malod~\cite{M11} characterised $\mathsf{VPSPACE}$ using exponentially large \emph{algebraic branching programs (ABPs)} that are \emph{succinct}.
Malod's work defines the \emph{complexity} of an ABP as the size of the smallest algebraic circuit that encodes its graph --- outputs the corresponding edge label when given the two endpoints as input. An $n$-variate ABP is then said to be \emph{succinct}, if its complexity is $\poly(n)$.
\begin{definition}[Succinct ABPs {\cite{M11}}]\label{defn:succinct-ABPs}
A \emph{succinct ABP} over the $n$ variables $\vecx = \set{x_1,\ldots,x_n}$ is a three tuple $(B,\vecs,\vect)$ with $\abs{\vecs} = \abs{\vect} = r$, where
\begin{itemize}
\item $\vecs$ is the label of the source vertex, and $\vect$ is the label of the sink(target) vertex.
\item $B(\vecu,\vecv,\vecx)$ is an algebraic circuit that describes a directed acyclic graph $G_B$ on the vertex set $\set{0,1}^r$ in the following way.
For any two vertices $\veca,\vecb \in \set{0,1}^r $, the output $B(\vecu = \veca,\vecv = \vecb,\vecx)$ is the label of the edge from $\veca$ to $\vecb$ in the ABP.
\end{itemize}
The polynomial computed by the ABP is the sum of polynomials computed along all $\vecs$ to $\vect$ paths in $G_B$; where each path computes the product of the labels of the constituent edges.
The size of the circuit $B$ is said to be the \emph{complexity} of the succinct ABP.
The number of vertices $2^r$ is the \emph{size} of the succinct ABP, and the length of the longest $\vecs$ to $\vect$ path is called the \emph{length} of the ABP. \end{definition}
In the same work~\cite{M11}, Malod alternatively characterised $\mathsf{VPSPACE}$ using an interesting algebraic model that resembles \emph{(totally) quantified boolean formulas} that are known to characterise $\mathsf{PSPACE}$. This model, which we refer to as ``quantified algebraic circuits'', is defined using special types of projection gates called \emph{summation} and \emph{production} gates.
\begin{definition}[Summation and Production gates \cite{M11}]\label{defn:summation-production}
\emph{Summation} and \emph{production} gates are unary gates that are labelled by a variable $z$, and are denoted by $\mathtt{sum}_z $ and $\mathtt{prod}_z$ respectively.
A summation gate returns the sum of the $(z=0)$ and $(z=1)$ evaluations of its input, and a production gate returns the product of those evaluations.
That is, $\mathtt{sum}_z(f(z,\vecx)) = f(0,\vecx) + f(1,\vecx) $, and $\mathtt{prod}_z(f(z,\vecx)) = f(0,\vecx) \cdot f(1,\vecx) $.
We sometimes use $\mathtt{sum}_{\set{z_1,\ldots,z_k}}$ to refer to the nested expression $\mathtt{sum}_{z_1}\cdots\mathtt{sum}_{z_k}$ (similarly for $\mathtt{prod}$); it can be checked that the order does not matter here. \end{definition}
A quantified algebraic circuit has the form $\mathtt{Q}^{1}_{z_1} \mathtt{Q}^{2}_{z_2} \cdots \mathtt{Q}^{m}_{z_m} \mathcal{C}(\vecx,\vecz) $, where each $\mathtt{Q}^i $ is a summation or a production, and $\mathcal{C}(\vecx,\vecz)$ is a usual algebraic circuit. \begin{definition}[Quantified Algebraic Circuits \cite{M11}]\label{defn:quantified-algebraic-circuits}
A quantified algebraic circuit is an algebraic circuit has the form
\[
\mathtt{Q}^{(1)}_{z_1} \mathtt{Q}^{(2)}_{z_2} \cdots \mathtt{Q}^{(m)}_{z_m} \mathcal{C}(\vecx,\vecz)
\]
where $\abs{\vecz} = m$, $\mathtt{Q}^{(i)} \in \set{\mathtt{sum},\mathtt{prod}}$ for each $i \in [m]$, and $\mathcal{C}$ is an algebraic circuit.
The size of a quantified algebraic circuit is $m + \operatorname{size}(\mathcal{C})$. \end{definition}
Finally, Mahajan and Rao~\cite{MR13} defined algebraic analogues of small space computation (e.g. $\textsf{L}$, $\textsf{NL}$) using the notion of \emph{width} of an algebraic circuit. They use their definitions to import some of the relationships from the boolean world to the algebraic world (e.g, they show $\textsf{VL} \subseteq \mathsf{VP}$). They further show that their definition of uniform polynomially-bounded-space computation coincides with that of \textsf{uniform}-$\mathsf{VPSPACE}$ as defined by Koiran and Perifel \cite{KP09}.
We now narrow our focus to the definitions due to Poizat~\cite{P08} and Malod~\cite{M11}. We choose these definitions because they are algebraic in nature, and have fairly natural monotone analogues. We elaborate a bit more about this decision in \autoref{sec:boolean-definitions}.
\begin{remark*} It should be noted that all the above mentioned definitions of $\mathsf{VPSPACE}$ allow for the polynomial families to have large degree --- as high as $\exp(\poly(n))$. The main focus of our work, however, is to compare the monotone analogues of these models with $\mathsf{mVP}$ and $\mathsf{mVNP}$. Since the latter classes only contain low-degree polynomials, we will only work with polynomials of degree $\poly(n)$, or $\mathsf{VPSPACE}_\text{b}$ as defined in \cite{KP09}, for the rest of this paper. \end{remark*}
\section{Monotone analogues of \textsf{VPSPACE}, and our contributions}\label{sec:contributions-formal}
We now define monotone analogues for the various definitions of $\mathsf{VPSPACE}$ outlined in the previous section, and compare the powers of the resulting monotone models/classes.
\paragraph*{Monotone succinct ABPs.} We first consider the natural monotone analogue of the definition due to Malod \cite{M11} which uses succinct algebraic branching programs (\autoref{defn:succinct-ABPs}).
Malod showed that every family $\set{f_n}$ in $\mathsf{VPSPACE}$ can be computed by $2^{\poly(n)}$ sized ABPs that have \emph{complexity} $\poly(n)$. Recall that the complexity of a succinct ABP is the size of the smallest algebraic circuit that encodes its graph.
We therefore define monotone succinct ABPs as ABPs that can be succinctly described by \emph{monotone} algebraic circuits of size $\poly(n)$. However this restriction forces that if the monomial $\vecx^{\vece}$ appears in any edge-label $(\veca,\vecb)$, then it also appears in the label of $(\bar{1},\bar{1})$. Therefore self-loops are inevitably present in succinct ABPs in the monotone setting. To handle this, we additionally allow the \emph{length} of the ABP, say $\ell$, to be predefined\footnote{It is not hard to see that the analogous definition in the non-monotone setting is equivalent to Malod's definition (\autoref{defn:succinct-ABPs}). This is essentially because of the connection to Iterated Matrix Multiplication.} so that now the polynomial computed by the ABP can be defined to be the sum of polynomials computed by all $\vecs$ -- $\vect$ paths of length at most $\ell$.
\begin{definition}[Monotone Succinct ABPs]\label{defn:monotone-succinct-ABPs}
A \emph{monotone succinct ABP} over $\vecx = \set{x_1,\ldots,x_n}$ is a four tuple $(B,\vecs,\vect,\ell)$ with $\abs{\vecs} = \abs{\vect} = r$, where
\begin{itemize}
\item $\ell$ is the \emph{length} of the ABP.
\item $\vecs$ is the label of the source vertex, and $\vect$ is the label of the sink(target) vertex.
\item $B(\vecu,\vecv,\vecx)$ is a \emph{monotone} algebraic circuit that describes a directed graph $G_B$ on the vertex set $\set{0,1}^r$ in the following way.
For any two vertices $\veca,\vecb \in \set{0,1}^r $, the output $B(\vecu = \veca,\vecv = \vecb,\vecx)$ is the label of the edge from $\veca$ to $\vecb$ in the ABP.
\end{itemize}
The polynomial computed by the ABP is the sum of polynomials computed along all $\vecs$ to $\vect$ paths in $G_B$ of length at most $\ell$; where each path computes the product of the labels of the constituent edges.
The size of the circuit $B$ is said to be the \emph{complexity} of the monotone succinct ABP.
The number of vertices $2^r $ is the \emph{size} of the succinct ABP. \end{definition} Note that since $B$ is a monotone algebraic circuit, all the edge-labels in the ABP are monotone polynomials over $\vecx$. It is also not hard to see that any polynomial $f \in \mathsf{mVP}$ is computable by this model. If $\mathcal{C}$ is the monotone circuit computing $f$, then the monotone succinct ABP computing $f$ is $(\mathcal{C}', 0, 1, 1)$ where $\mathcal{C}'(u,v,\vecx) = v \cdot \mathcal{C}(\vecx)$.
However, surprisingly, we show that the computational power of monotone succinct ABPs when computing polynomials of \emph{bounded degree} does not go beyond $\mathsf{mVNP}$.
\begin{restatable}{theorem}{MonSuccinctABPisMonVNP}\label{thm:fullMonABP-inside-mVNP}
If an $n$-variate polynomial $f(\vecx)$ of degree $\poly(n)$ is computable by a monotone succinct ABP of complexity $\poly(n)$, then $f(\vecx) \in \mathsf{mVNP}$. \end{restatable}
In contrast, Malod~\cite{M11} showed that every family in $\mathsf{VPSPACE}$ admits succinct ABPs of polynomial complexity and we expect $\mathsf{VPSPACE}_b$ to be a much bigger class than $\mathsf{VNP}$.
\paragraph*{Quantified monotone circuits.} As mentioned earlier, Malod \cite{M11} had also characterised the class $\mathsf{VPSPACE}$ using the notion of quantified algebraic circuits (\autoref{defn:quantified-algebraic-circuits}). We now consider its natural monotone analogue, which we call quantified monotone circuits.
\begin{definition}[Quantified Monotone Algebraic Circuits]\label{defn:quantified-monotone-algebraic-circuits}
A quantified monotone algebraic circuit has the form
\[
\mathtt{Q}^{(1)}_{z_1} \mathtt{Q}^{(2)}_{z_2} \cdots \mathtt{Q}^{(m)}_{z_m} \mathcal{C}(\vecx,\vecz)
\]
where $\abs{\vecz} = m$, $\mathtt{Q}^{(i)} \in \set{\mathtt{sum},\mathtt{prod}}$ for each $i \in [m]$, and $\mathcal{C}$ is a monotone algebraic circuit.
The size of the quantified monotone algebraic circuit above is $m + \operatorname{size}(\mathcal{C})$. \end{definition}
Quantified monotone circuits of polynomial size can clearly compute any polynomial in $\mathsf{mVNP}$. It is therefore interesting to check if there is any polynomial of \emph{bounded degree} that is outside $\mathsf{mVNP}$. This turns out to be a tricky question. A reason for that is as follows.
\begin{restatable}{lemma}{SupportSummationProduction} \label{lem:support-summation-production}
Let $f(\vecx)$ be a monotone polynomial whose support cannot be written as a non-trivial product of two sets.
Further for some monotone polynomial $g(\vecx,\vecz)$, suppose $f(\vecx) = \mathtt{Q}^{(1)}_{z_1} \mathtt{Q}^{(2)}_{z_2} \cdots \mathtt{Q}^{(m)}_{z_m} g(\vecx,\vecz) $ with $\mathtt{Q}^{(i)} \in \set{\mathtt{sum},\mathtt{prod}}$ for each $i \in [m]$.
Then $\operatorname{supp}(f(\vecx)) = \operatorname{supp}(g(\vecx,\bar{1}))$. \end{restatable}
This is an extension of the observation due to Yehudayoff \cite{Y19} and essentially shows that any lower bound proof against $\mathsf{mVNP}$ for a polynomial computable by quantified monotone circuits will require an argument that relies on some structure in the coefficients.
While there are instances of such arguments in the literature \cite{Y19, CDGM22,CDM21}, extending those ideas to work against $\mathsf{mVNP}$ does not seem like an easy task. The following theorem sheds some light on the cause of this difficulty.
\begin{restatable}{theorem}{ExpSumForQuantifiedMVP}
\label{thm:exp-sum-for-quantified-mvp}
Suppose $f(\vecx)$ is an $n$-variate, degree-$d$ polynomial computed by a quantified monotone circuit of size $s$, which uses $\ell$ summation gates.
Then for a set of variables $\vecw$ of size at most $d \cdot \ell$, there is a monotone circuit $h(\vecx,\vecw)$ of size at most $d \cdot s$, and a polynomial $A(\vecw)$ such that,
\begin{equation}\label{eq:almost-vnp-expression}
f(\vecx) = \sum_{\vecb \in \set{0,1}^{\abs{\vecw}}} A(\vecw) \cdot h(\vecx,\vecw),
\end{equation}
where $A(\vecw)$ potentially has size and degree that is exponential in $n$ and $\ell$. \end{restatable}
We now discuss how \autoref{thm:exp-sum-for-quantified-mvp} helps us understand the main barriers towards separating quantified monotone $\mathsf{VP}$ from $\mathsf{mVNP}$.
\begin{enumerate}
\item If the polynomial $A(\vecw)$ from \autoref{thm:exp-sum-for-quantified-mvp} were to have degree and size that is polynomial in $n$, then quantified monotone $\mathsf{VP}$ would collapse to $\mathsf{mVNP}$.
Further since $A$ is free of $\vecx$, its exponential degree and size can be leveraged only for designing coefficients of $f$.
Moreover, the monotone nature of $A$ and $h$ ensures that $A(\mathbf{1})$ is the largest value, and contributes \emph{equally} to all monomials in the support of $f$, since $\operatorname{supp}(f) = \operatorname{supp}(h(\vecx,\vecw=\mathbf{1}))$.
\item Another consequence that is quite interesting is the following.
Suppose there is a different monotone polynomial $B(\vecw)$ of small degree and size that agrees with $A(\vecw)$ on all $\set{0,1}$-inputs, then $f(\vecx) = \sum_{\vecb} B(\vecw) h(\vecx,\vecw)$.
That is, we can replace $A$ by $B$ in our expression and then $f$ clearly has an efficient `$\mathsf{mVNP}$-expression'.
Thus, any separation between $\mathsf{mVNP}$ and quantified monotone $\mathsf{VP}$ will provide a polynomial $A(w)$ which is hard to compute for $\mathsf{mVNP}$, even as a function over the boolean hypercube; a result that perhaps stands on its own. \end{enumerate}
\paragraph*{Monotone circuits with summation and production gates}
Next we consider a model that further generalises quantified monotone circuits. Here summation and production gates are allowed to appear anywhere in the circuit.
\begin{definition}[Algebraic circuits with summation and production gates]\label{defn:monotone-circuits-summation-production}
An algebraic circuit with summation, production gates is a directed acyclic graph with leaves (nodes with in-degree zero) labelled by formal variables and constants from the field, and other nodes labelled by addition $(+)$, multiplication $(\times)$, summation $(\mathtt{sum}_z)$ or production $(\mathtt{prod}_z)$.
The leaves compute their labels, and addition, multiplication nodes compute the operation they are labelled by, on the polynomials along its incoming edges.
The nodes labelled by summation or production computes the polynomial described in \autoref{defn:summation-production}.
There is a unique node of out-degree zero called the root, and the circuit is said to compute the polynomial computed at the root.
The \emph{size} of a circuit, $\mathcal{C}$, denoted by $\operatorname{size}(\mathcal{C})$, is the number of nodes in the graph.
An algebraic circuit with summation, production gates is said to be \emph{monotone}, if all the constants appearing in it are non-negative. \end{definition}
Note that even in the non-monotone setting this model is clearly as powerful as quantified circuits and less powerful than circuits with projection gates. Therefore since Malod \cite{M11} showed that quantified circuits and circuits with projection gates are equivalent in power, the class of polynomials efficiently computable by this model is again $\mathsf{VPSPACE}$.
In the monotone setting, however, it is not clear if the power of quantified monotone circuits is the same as that of this model. In particular, it is unclear if a version of \autoref{lem:support-summation-production} is true for this model. However, we show that even this additional power does not help much in monotone computation of transparent polynomials.
\begin{restatable}{theorem}{TransparencyLBforQuantCircuits}\label{thm:transparency-hard-for-QuantMonVP}
Any monotone algebraic circuit with summation and production gates that computes a transparent polynomial $f$, has size at least $\abs{\operatorname{supp}(f)}/4$. \end{restatable}
This shows that transparent polynomials with large support are hard even for this model.
Recall that one way to refute the $\tau$-conjecture for Newton polygons is to show a transparent polynomial in (non-monotone) $\mathsf{VP}$. \autoref{thm:transparency-hard-for-QuantMonVP} shows that any transparent polynomial from $\mathsf{VP}$ that refutes the conjecture would also witness a separation between $\mathsf{VP}$ and a class potentially much bigger than $\mathsf{mVNP}$\footnote{That is, the class of bounded degree polynomials computable by monotone algebraic circuits with summation and production gates.}.
Even though stark separations between monotone and non-monotone models are not unheard of \cite{HY13, CDM21}, such a result would be very interesting and would further highlight the power of subtractions.
\paragraph*{Monotone circuits with projection gates.} Finally, adapting the definition of $\mathsf{VPSPACE}$ due to Poizat (\autoref{defn:algebraic-circuits-with-projection}) \cite{P08}, we define monotone circuits with projection gates.
\begin{definition}[Monotone algebraic circuits with projection gates]\label{defn:monotone-algebraic-circuits-with-projection}
A \emph{monotone algebraic circuit with projection gates} is an algebraic circuit with projection (as defined in \autoref{defn:algebraic-circuits-with-projection}) in which only non-negative constants from the field are allowed to appear as labels of leaves.
The \emph{size} of a monotone algebraic circuit with projection gates is the number of nodes in the underlying graph. \end{definition}
This model is clearly at least as powerful as monotone circuits with summation and production gates, since $\mathtt{sum}_z = \project{z}{0} + \project{z}{1}$ and $\mathtt{prod}_z = \project{z}{0} \times \project{z}{1}$. It would therefore be interesting to show a separation between the power of the two models.
Even though we are unable to do that, we show that monotone circuits with projection gates are indeed more powerful than quantified monotone circuits, with a $2^{\Omega(\sqrt{n})}$ separation. We do so by first showing that the Permanent family is efficiently computable by the first model (\autoref{thm:Perm-mvpspace-ub}), and then using \autoref{lem:support-summation-production} against the second model.
\begin{restatable}{theorem}{QuantCircuitsVsProjectionCircuits}\label{thm:QuantMonVP-neq-VP-proj} The polynomial family $\set{\operatorname{\sf Perm}_n}$ can be computed by a monotone circuit with projection gates of size $O(n^3)$, but any quantified monotone circuit computing it must have size $2^{\Omega(\sqrt{n})} $. \end{restatable}
We end this section with a conjecture. We believe that transparency is a highly restrictive property, especially for monotone computation. Therefore we conjecture that if $f$ is a transparent polynomial being computed by a monotone circuit with projection gates of size $s$, then $\abs{\operatorname{supp}(f)} \leq 2^{\operatorname{polylog}(s)}.$
\section{Monotone succinct algebraic branching programs}\label{sec:succinct-monotone-abp}
In this section we prove \autoref{thm:fullMonABP-inside-mVNP}.
\MonSuccinctABPisMonVNP*
\begin{proof}
Let $\mathcal{A} = (B,\vecs,\vect, \ell)$ be the monotone succinct ABP computing $f$, with $\abs{\vecs} = \abs{\vect} = r$.
\begin{claim}
If $\ell > 1$, then $\ell \leq \deg(f)+2$.
\end{claim}
\begin{proof}
Let $b(\vecu,\vecv,\vecx)$ be the \emph{monotone} $(2r+n)$-variate polynomial computed by the circuit $B$.
Due to the monotonicity of $B$, for any $\vece \in \N^n$ we have that if the monomial $\vecx^{\vece}$ appears in any edge-label $(\veca,\vecb)$, then it also appears in the label of $(\bar{1},\bar{1})$.
Therefore $\deg_{\vecx}(B(\veca,\vecb,\vecx)) \leq \deg_{\vecx}(B(\bar{1},\bar{1},\vecx)) $ for all $\veca,\vecb$.
Similarly, $\deg_{\vecx}(B(\vecs,\vecb,\vecx)) \leq \deg_{\vecx}(B(\vecs,\bar{1},\vecx))$ and $\deg_{\vecx}(B(\veca,\vect,\vecx)) \leq \deg_{\vecx}(B(\bar{1},\vect,\vecx))$ for all $\veca,\vecb$.
This shows that if $\ell > 1$, then
\[
\deg(f) = \deg(B(\vecs,\bar{1},\vecx) \cdot B(\bar{1},\bar{1},\vecx)^{\ell - 2} \cdot B(\bar{1},\vect,\vecx)) \geq \ell - 2. \qedhere
\]
\end{proof}
As a result of the above claim, for $d = \deg(f)$, we have the following.
\begin{align*}
f(\vecx)
&= B(\vecs,\vect,\vecx) + \sum_{j = 1}^{d - 1} \inparen{\text{sum of $\vecs$--$\vect$ paths through $j$ intermediate vertices}}\\
&= B(\vecs,\vect,\vecx) + \sum_{j = 1}^{d - 1} \inparen{\sum_{\veca_1,\ldots,\veca_j \in \set{0,1}^r} B(\vecs,\veca_1,\vecx) \cdot \inparen{\prod_{k = 1}^{j-1} B(\veca_k,\veca_{k+1},\vecx) } \cdot B(\veca_{j},\vect,\vecx)}\\
&= B(\vecs,\vect,\vecx) + \sum_{\veca_1,\ldots,\veca_{d-1} \in \set{0,1}^r} \sum_{j = 1}^{d-1} 2^{-r(d-1-j)} \inparen{B(\vecs,\veca_1,\vecx) \cdot \inparen{\prod_{k = 1}^{j-1} B(\veca_k,\veca_{k+1},\vecx) } \cdot B(\veca_{j},\vect,\vecx)}\\
&=\sum_{\veca_1,\ldots,\veca_{d-1}} \inparen{2^{-r(d-1)}B(\vecs,\vect,\vecx) + \sum_{j = 1}^{d-1} 2^{-r(d-1-j)} B(\vecs,\veca_1,\vecx) \inparen{\prod_{k = 1}^{j-1} B(\veca_k,\veca_{k+1},\vecx) } B(\veca_{j},\vect,\vecx)}
\end{align*}
which is clearly a monotone $\mathsf{VNP}$ expression, since $d = \poly(n)$ and $B$ is a monotone circuit of size $\poly(n)$.
\end{proof}
\section{Quantified monotone circuits}\label{sec:quantified-circuits}
In this section we first prove \autoref{lem:support-summation-production}, which is an extension of the following observation due to Yehudayoff~\cite{Y19}.
\begin{observation}[{\cite{Y19}}]\label{obs:support-monotone-projection}
Let $g(\vecx,z)$ be a monotone polynomial and let $c > 0$.
Then for any monomial $m = \vecx^\vece z^j $ in the support of $g$, $\vecx^\vece \in \operatorname{supp}(g,z=c)$. \end{observation}
We now restate \autoref{lem:support-summation-production} and complete its proof.
\SupportSummationProduction*
\begin{proof}
Observe that it is enough to show the statement of the lemma for $m = 1$.
Therefore, suppose $f(\vecx) = \mathtt{sum}_z g(\vecx,z)$, then $f(\vecx) = g(\vecx,0) + g(\vecx,1)$, and hence $\operatorname{supp}(f) = \operatorname{supp}(g(\vecx,1))$, since $g$ is monotone.
Next, $f(\vecx) = \prod_z g(\vecx,z)$ means that $f(\vecx) = g(\vecx,0) \cdot g(\vecx,1)$.
As $\operatorname{supp}(f)$ cannot be written as a non-trivial product of two sets, and since $g$ is monotone, this must mean that $g(\vecx,0)$ is a constant and $\operatorname{supp}(f(\vecx)) = \operatorname{supp}(g(\vecx,1))$ as claimed. \end{proof}
Let us now move on to the proof of \autoref{thm:exp-sum-for-quantified-mvp}, which we first restate.
\ExpSumForQuantifiedMVP*
The proof requires us to repeatedly use the following simple observation. It is easy to verify and therefore we omit its proof. \begin{observation}[Product of exponential sums] \label{obs:product-of-exp-sums}
\[
\mathtt{prod}_z \mathtt{sum}_{\vecy} g(\vecx,\vecy,z) = \mathtt{sum}_{\vecy_0,\vecy_1} \inparen{g(\vecx,\vecy_0,0) \cdot g(\vecx,\vecy_1,1)} \qedhere
\] \end{observation}
We now use a toy example to exhibit the trivial way of moving from a quantified expression to an exponential sum, using \autoref{obs:product-of-exp-sums}. \begin{align*}
f(x) &= \mathtt{sum}_{y_1} \mathtt{prod}_{z_1} \mathtt{sum}_{y_2} \mathtt{prod}_{z_2,z_3} \mathtt{sum}_{y_3} g(x,y_1,y_2,y_3,z_1,z_2,z_3)\\
&= \mathtt{sum}_{y_1} \mathtt{prod}_{z_1} \mathtt{sum}_{y_2} \mathtt{prod}_{z_2} \mathtt{sum}_{y_{3,0},y_{3,1}} \inparen{\prod_{a_3 \in \set{0,1}} g(x,y_1,y_2,y_{3,a_3},z_1,z_2,a_3)}\\
&= \mathtt{sum}_{y_1} \mathtt{prod}_{z_1} \mathtt{sum}_{y_2, y_{3,(00)},y_{3,(01)},y_{3,(10)},y_{3,(11)}} \inparen{\prod_{a_2,a_3 \in \set{0,1}} g(\ldots,y_{3,(a_2 a_3)},z_1,a_2,a_3)}\\
&= \mathtt{sum}_{y_1} \mathtt{sum}_{y_{2,\ast},y_{3,\ast\ast\ast}}\inparen{\prod_{a_1,a_2,a_3 \in \set{0,1}} g(x,y_1,y_{2,a_1},y_{3,(a_1 a_2 a_3)},a_1,a_2,a_3)} \end{align*} In the last line, $\ast$ runs over $\set{0,1}$, so there are $1 + 2+ 8 = 11$ auxiliary variables in total. Note that $y_3$ has $8$ copies, which is due to the $3$ production gates `above' the summation gate labelled by it. Similarly $y_2$ has just $2$ copies, while $y_1$ has just one. In particular, it should be noted that the number of such copies is independent of the number of \emph{alternations}. Also if instead of single auxiliary variables $y_2$ and $y_3$ we had sets of auxiliary variables $\vecy_2 $ and $\vecy_3$, nothing much would change. That is, we would have had $8$ copies of the set $\vecy_3$ and $2$ copies of $\vecy_2$, irrespective of their sizes.
In general, what this shows is that we can trivially move from a quantified expression to an expression which has the form \[
f(\vecx) = \mathtt{sum}_{\mathbf{Y}} \prod_{\veca \in \set{0,1}^r} g_\veca(\vecx, \vecy_\veca) \] where $\mathbf{Y} = \cup_{\veca} \set{\vecy_\veca}$, $r$ is the number of production gates in the quantified expression, $\abs{\mathbf{Y}}$ is potentially exponential (since the number of copies of some auxiliary variable might be exponential) but $g_\veca(\vecx, \vecy_\veca) = g(\vecx, \vecy = \vecy_\veca, \vecz = \veca)$ for a poly-sized circuit $g(\vecx, \vecy, \vecz)$.
The key observation allowing us to prove \autoref{thm:exp-sum-for-quantified-mvp} is that if $f$ has degree $d$, then the number of copies of each auxiliary variable needed in the outer summation gate is at most $d$. This is because, due to monotonicity, $\deg_{\vecx}(g_\veca(\vecx, \vecy_\veca)) \neq 0$ for only $d$ many $\veca \in \set{0,1}^r$.
Moving on to a formal proof, we introduce a new shorthand for the remainder of this section. For a vector $\veca = \set{a_1,a_2,\ldots,a_{\ell}}$ and a number $k \leq \ell$, we use $\veca[:k]$ to denote the \emph{prefix} vector $\set{a_1,a_2,\ldots,a_k}$. With this new notation, we can express the last line of our toy example as follows. \[
f(x) = \mathtt{sum}_{y_1} \mathtt{sum}_{y_{2,\ast},y_{3,\ast\ast\ast}}\inparen{\prod_{\veca \in \set{0,1}^3} g(x,y_1,y_{2,\veca[:1]},y_{3,\veca[:3]},a_1,a_2,a_3)} \] We are now ready to prove \autoref{thm:exp-sum-for-quantified-mvp}. We start by recalling the statement of the theorem.
\ExpSumForQuantifiedMVP*
\begin{proof}
The first step is to obtain a trivial exponential sum for the quantified expression, as in the discussion above.
\begin{claim}\label{claim:trivial-exp-sum}
Suppose $f(\vecx)$ can be expressed as the following quantified circuit.
\[
f(\vecx) = \mathtt{sum}_{\vecy_1} \mathtt{prod}_{\vecz_1} \mathtt{sum}_{\vecy_2} \mathtt{prod}_{\vecz_2} \cdots \mathtt{prod}_{\vecz_{k}} \mathtt{sum}_{\vecy_{k+1}} g(\vecx,\vecy_1,\ldots,\vecy_{k+1},\vecz_1,\ldots,\vecz_{k})
\]
Let $m_i = \abs{\vecz_i}$, and further let $M_i = m_1 + m_2 + \cdots + m_i$, for each $i \in [k]$.
Also, let $\vecy = \vecy_1 \cup \vecy_2 \cup \cdots \cup \vecy_{k+1}$, and $\vecz = \vecz_1 \cup \vecz_2 \cup \cdots \cup \vecz_{k}$
Then $f(\vecx)$ can also be expressed as the following exponential sum.
\[
f(\vecx) = \mathtt{sum}_{\mathbf{Y}} \inparen{\prod_{\veca \in \set{0,1}^{M_{k}}} g(\vecx,\vecy_{1},\vecy_{2,\veca[:M_1]},\vecy_{3,\veca[:M_2]},\ldots,\vecy_{k+1,\veca[:M_k]},\vecz = \veca) }
\]
Here $\mathbf{Y}$ is a set of all $y$-variables, of size $\inparen{1 + \sum_i 2^{M_i}}$ that is defined as follows.
\[
\mathbf{Y} = \bigcup_{\veca \in \set{0,1}^{M_k}} ( \vecy_1 \cup \vecy_{2,\veca[:M1]} \cup \cdots \cup \vecy_{k+1,\veca[:M_k]}) \qedhere
\]
\end{claim}
Even though the claim is fairly verbose, it is easy to verify given the discussion before the lemma, so we will not explicitly prove it.
As the next step, we shall use the fact that the `inner circuit' $g$ is monotone, to lower bound the degree of $f$.
\begin{align*}
\deg(f) &= \deg_{\vecx}\inparen{ \mathtt{sum}_{\mathbf{Y}} \inparen{\prod_{\veca \in \set{0,1}^{M_{k}}} g(\vecx,\vecy_{1},\vecy_{2,\veca[:M_1]},\ldots,\vecy_{k+1,\veca[:M_k]},\vecz = \veca) } }\\
\text{($g$ is monotone)} &= \deg_{\vecx}\inparen{\prod_{\veca \in \set{0,1}^{M_{k}}} g(\vecx,\mathbf{1},\vecz = \veca) }\\
&\geq \sum_{\veca \in \set{0,1}^{M_k}} \deg(g(\vecx,\mathbf{1},\vecz = \veca))
\end{align*}
Therefore, since $f$ has degree $dg(\vecx,\vecy,\veca)$, it must be the case that for all but $d$ fixings $\veca$ of $\vecz$, $g(\vecx,\vecy,\veca)$ is a constant in terms of $\vecx$ for any $\set{0,1}$-assignment\footnote{Note that $g(\vecx,\vecy,\veca)$ can be a non-constant polynomial in $\vecx$ and still have this property: e.g. $y + x(y^2 - y)$. It can be checked that the proof goes through despite this.} to the variables in $\vecy$.
Let $\mathcal{A} := \set{\veca \in \set{0,1}^{M_k} : \deg_{\vecx}\inparen{g(\vecx,\vecb,\veca)} > 0 \text{ for some } \vecb \in \set{0,1}^{\abs{\vecy}} }$, and let $\mathcal{A}_0 := \set{0,1}^{M_k} \setminus \mathcal{A}$.
We therefore have that $\abs{\mathcal{A}} \leq d$.
Further, let $\mathbf{Y}_1 := \bigcup_{\veca \in \mathcal{A}} ( \vecy_1 \cup \vecy_{2,\veca[:M1]} \cup \cdots \cup \vecy_{k+1,\veca[:M_k]})$, and let $\mathbf{Y}_0 := \mathbf{Y} \setminus \mathbf{Y}_1 $.
Note that now $\abs{\mathbf{Y}_1} \leq \abs{\mathcal{A}} \cdot \abs{\vecy} \leq d \cdot m$.
We can now simplify the exponential sum in \autoref{claim:trivial-exp-sum} and finish the proof as follows, where $\vecy_{\veca}$ refers to $(\vecy_1,\vecy_{2,\veca[:M_1]},\cdots,\vecy_{k+1,\veca[:M_k]})$.
\begin{align*}
f(\vecx) &= \mathtt{sum}_{\mathbf{Y}} \inparen{\prod_{\veca \in \set{0,1}^{M_{k}}} g(\vecx,\vecy_{\veca},\vecz = \veca) }\\
\text{(for appropriate $\vecy_{\veca}$)} &= \mathtt{sum}_{\mathbf{Y}} \inparen{ \inparen{\prod_{\veca \in \mathcal{A}_0} g(\vecx,\vecy_{\veca},\vecz = \veca)} \cdot \inparen{\prod_{\veca \in \mathcal{A}} g(\vecx,\vecy_{\veca},\vecz = \veca)} }\\
\text{(first term ``$\vecx$-free'')} &= \mathtt{sum}_{\mathbf{Y}} \inparen{ \inparen{\prod_{\veca \in \mathcal{A}_0} g(\mathbf{0},\vecy_{\veca},\vecz = \veca)} \cdot \inparen{\prod_{\veca \in \mathcal{A}} g(\vecx,\vecy_{\veca},\vecz = \veca)} }\\
&= \mathtt{sum}_{\mathbf{Y}_1,\mathbf{Y}_0} \inparen{ \inparen{\prod_{\veca \in \mathcal{A}_0} g(\mathbf{0},\vecy_{\veca},\vecz = \veca)} \cdot \inparen{\prod_{\veca \in \mathcal{A}} g(\vecx,\vecy_{\veca},\vecz = \veca)} }\\
\text{(regroup terms)} &= \mathtt{sum}_{\mathbf{Y}_1} \inparen{ \mathtt{sum}_{\mathbf{Y}_0} \inparen{\prod_{\veca \in \mathcal{A}_0} g(\mathbf{0},\vecy_{\veca},\vecz = \veca)}} \cdot \inparen{\prod_{\veca \in \mathcal{A}} g(\vecx,\vecy_{\veca},\vecz = \veca)}\\
\text{(simplify)}&= \mathtt{sum}_{\mathbf{Y}_1} A(\mathbf{Y}_1) \cdot h(\vecx,\mathbf{Y}_1)
\end{align*}
As claimed, the size of $h$ is at most $\abs{\mathcal{A}} \cdot \operatorname{size}(g) \leq d \cdot s$, while $A(\mathbf{Y}_1)$ is a fairly structured polynomial despite its exponential size and degree. \end{proof}
\section{Monotone circuits with summation and production gates}\label{sec:monotone-circuits-summation-production}
In this section, we prove \autoref{thm:transparency-hard-for-QuantMonVP}. We start by recalling the theorem. \TransparencyLBforQuantCircuits*
This result is an extension of the ideas in the work of \Hrubes{} and Yehudayoff~\cite{HY21}. Their argument shows that any bivariate monotone circuit of size $s$ that computes a polynomial with \emph{convexly independent support} outputs a polynomial with support at most $4s$. They achieve this by keeping track of the largest polygon (having most vertices) that one can build using the polynomials computed at all the gates in the circuit. They then inductively show that no gate (leaf, addition, multiplication) can increase the number of vertices by $4$. We are able to show the same bound for production and summation gates, by working with a monotone bivariate circuit over $y_1,y_2 $ that is allowed some auxiliary variables $z$ for summations and productions.
An important component of the proof in \cite{HY21} is that if the sum or product of two monotone polynomials in convexly independent, then so are each of the two inputs. However, the allowing for summations and productions means that some monomials that are computed internally could get ``zeroed out''. In fact, summation and production gates do not quite ``preserve convex dependencies''. For example, the convexly dependent support $\set{y_1 y_2, y_1 y_2 z, y_1 y_2 z^2} $ when passed through $\mathtt{sum}_z $ produces just $\set{y_1 y_2}$, which is convexly independent.
In order to prove \autoref{thm:transparency-hard-for-QuantMonVP}, we get around this by working directly with the support projected down to the ``true'' variables, which we call $\vecy$-support in our arguments. It turns out that summations and productions indeed preserve convex dependencies that are in the $\vecy$ support of the input polynomial. We now show a complete proof.
We start by recalling the concepts of \emph{shadow complexity} and \emph{transparent polynomials}.
\begin{definition}[Shadow complexity {\cite{HY21}}]\label{defn:shadow-complexity}
For a polynomial $f(x_1,\ldots,x_n)$, its shadow complexity $\sigma(f)$ is defined as follows.
\begin{equation*}
\sigma(f) := \max_{L : \R^n \rightarrow \R^2} \abs{\operatorname{vert}(L(\mathsf{Newt}(f)))} \qedhere
\end{equation*} \end{definition}
For any $n$, a set of points in $\R^n $ is said to be \emph{convexly independent} if no point in the set can be written as a convex combination of other points from the set. Note that if a polynomial has \emph{convexly independent support}, then all the monomials in its support correspond to vertices of its Newton polytope. The following definition is an even stronger condition. \begin{definition}[Transparent polynomials {\cite{HY21}}]\label{defn:transparent-polynomials}
A polynomial $f$ is said to be transparent, if $\sigma(f) = \abs{\operatorname{supp}(f)}$. \end{definition}
The following lemma states that the linear map that witnesses the shadow complexity of a polynomial over the reals, can be assumed to be ``integral'' without loss of generality. \begin{lemma}[Consequence of {\cite[Lemma 4.2]{HY21}}]\label{lem:real-polynomial-integral-projections}
Let $f(\vecx) \in \R[\vecx]$ be an $n$-variate polynomial.
Then there is an $M \in \Z^{2 \times n} $, such that for $L(\vece) := M \cdot \vece$, $\abs{\operatorname{vert}(L(\mathsf{Newt}(f)))} = \sigma(f)$. \end{lemma}
We also require the following concepts from the work of \Hrubes and Yehudayoff \cite{HY21}.
\begin{definition}[Laurent polynomials and high powered circuits]\label{defn:hp-circuits-laurent}
A \emph{Laurent polynomial} over the variables $\set{x_1,\ldots,x_n}$ and a field $\F$, is a finite $\F$-linear combination of terms of the form $x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n} $, where $p_1,p_2,\ldots,p_n \in \Z$.
A high powered circuit over the variables $\set{x_1,\ldots,x_n}$ and a field $\F$, is an algebraic circuit whose leaves can compute terms like $\alpha x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n} $ for any $\alpha \in \F$ and $\vecp \in \Z^n $.
In other words, a high powered circuit can compute an arbitrary Laurent monomial with size $1$; the size of the high powered circuit is the total number of nodes as usual. \end{definition}
Using the above definition, we can easily infer the following by replacing each leaf with the corresponding Laurent monomial. \begin{observation}\label{obs:shadows-hp-circuits}
Let $f(\vecx)$ be computable by a monotone circuit of size $s$, and suppose $\sigma(f) = k$.
Then there exists a bivariate Laurent polynomial $P(y_1,y_2)$ that is computable by a high powered circuit of size $s$, whose Newton polygon has $k$ vertices. \end{observation}
We now have all the concepts required to prove the main theorem of this section, \autoref{thm:transparency-hard-for-QuantMonVP}. The following results and their proofs closely follow those in \cite{HY21}. We reproduce the overlapping parts for the sake of completeness and ease of exposition.
\begin{lemma}[{\cite[Lemma 5.8]{HY21}}]\label{lem:mink-sums-2d}
Let $A, B \subset R^2 $ be finite sets, such that $A + B$ is convexly independent.
Then if $\abs{A} \geq \abs{B}$, then either $\abs{A},\abs{B} \leq 2$ or $\abs{B} = 1$. \end{lemma}
\begin{theorem}[Extension of {\cite[Theorem 5.9]{HY21}}]\label{thm:transparent-summation-production}
Let $f(y_1,y_2)$ be a monotone Laurent polynomial with convexly independent support, and let $C(y_1,y_2,\vecz)$ be a monotone high-powered circuit with summation and production gates\footnote{All auxiliary variables only appear with non-negative powers in the circuit.}, that computes $f$.
Then $\operatorname{size}(C) \geq \abs{\operatorname{supp}(f)}/4$. \end{theorem} \begin{proof}
For a multi-set\footnote{We assume that copies of the same set $A \in \mathcal{A}$ can be referred distinctly.} $\mathcal{A}$ that contains sets of points in $\R^2$, we define a measure $\mu$ that relates to the ``largest'' convexly independent set that can be constructed using it.
For a sub-collection $\mathcal{B} \subseteq \mathcal{A}$ and a map $v : \mathcal{B} \rightarrow \R^2 $, the resulting set $\mathcal{B}(v)$ is defined as follows.
\begin{equation*}
\mathcal{B}(v) := \bigcup_{A \in \mathcal{B}} (\set{v(A)} + A)
\end{equation*}
The measure $\mu$ is then defined as follows.
\begin{equation}\label{eqn:definition-mu}
\mu(\mathcal{A}) := \max_{\mathcal{B},v} \set{ \abs{\mathcal{B}(v)} : \mathcal{B}(v)\text{ is convexly independent}}
\end{equation}
For a Laurent polynomial $g(y_1,y_2,\vecz)$, let $\operatorname{supp}_{\vecy}(g) := \set{(a,b) : \exists \vece, y_1^a y_2^b \vecz^{\vece} \in \operatorname{supp}(g)}$ be its $\vecy$-support.
Corresponding to the circuit $C(y_1,y_2,\vecz)$ of size $s$, we will consider the collection $\mathcal{A}$ of $s$ sets, which will be the $\vecy$-supports of the polynomials computed by the $s$ gates.
The following claim will help us prove the theorem by induction.
\begin{claim}
For $\mathcal{A}' = \mathcal{A} \cup \set{B}$, and $A_1,A_2 \in \mathcal{A} $,
\begin{align}
\mu(\mathcal{A}') &\leq \mu(\mathcal{A}) + \abs{B}, &\label{eq-case:1}\\
\mu(\mathcal{A}') &\leq \mu(\mathcal{A}) + 2 &\text{if $B = u + A_1 $},\label{eq-case:2}\\
\mu(\mathcal{A}') &\leq \mu(\mathcal{A}) + 4 &\text{if $B = A_1 \cup A_2 $},\label{eq-case:3}\\
\mu(\mathcal{A}') &\leq \mu(\mathcal{A}) + 4 &\text{if $B = A_1 + A_2 $},\label{eq-case:4}\\
\mu(\mathcal{A}') &\leq \mu(\mathcal{A}) + 4 &\text{if $B = A_1 + A' $ for $A' \subseteq A_1$ }\label{eq-case:5}.
\end{align}
\end{claim}
\begin{proof}
It is trivial to see that \eqref{eq-case:1} holds.
For \eqref{eq-case:2}, suppose $\mathcal{B}$ is the subset that achieves $\mu(\mathcal{A}') > \mu(\mathcal{A})$.
Then $A_1,B \in \mathcal{B} $ as otherwise one can mimic the contribution of $B$ using $A_1 $; further $v(A_1) \neq v(B) + u$ because otherwise the translates of $A_1 $ and $B$ overlap.
Now note that $(\set{v(A_1)}+A_1) \cup (\set{v(B)}+B) $ is a convexly independent set of points, and also that $(\set{v(A_1)}+A_1) \cup (\set{v(B)}+B) = \set{v(A_1),v(B)+u} + A_1$.
Therefore by \autoref{lem:mink-sums-2d}, we see that $\abs{B} = \abs{A_1} \leq 2$, which finises the proof using \eqref{eq-case:1}.
For \eqref{eq-case:3}, observe that $\mu(\mathcal{A}) \leq \mu(\mathcal{A} \cup {A_1,A_2})$.
The required bound then follows by two applications of \eqref{eq-case:2}.
In \eqref{eq-case:4}, if $B$ is convexly \emph{dependent}, then it cannot contribute to $\mu(\mathcal{A}')$, so suppose it is.
Assuming $\abs{A_1} \geq \abs{A_2}$ without loss of generality, by \autoref{lem:mink-sums-2d}, either $\abs{B} \leq \abs{A_1} \cdot \abs{A_2} \leq 4$, or $B = u + A_1 $ for some $u$, and \eqref{eq-case:2} finishes the proof.
Clearly \eqref{eq-case:4} implies \eqref{eq-case:5}, as its proof does not depend on whether $A_2 \in \mathcal{A}$, or $A_2 \not\subseteq A_1$. \end{proof}
We now argue that the polynomial computed at every gate in $C(y_1,y_2,\vecz)$ has convexly independent $\vecy$-support. Since the $\vecy$-supports of addition and multiplication gates are unions and Minkowski sums of their children respectively, if any of their input is convexly dependent, then so is the output. For a summation gate $g = \mathtt{sum}_z g'$, $\operatorname{supp}_{\vecy}(g) = \operatorname{supp}_{\vecy}(g')$ using \autoref{lem:support-summation-production}. For a production gate $g = \mathtt{prod}_z g' $, $\operatorname{supp}_{\vecy}(g) = S' + \operatorname{supp}_{\vecy}(g')$ for some $S' \subseteq \operatorname{supp}_{\vecy}(g')$, so any convex dependency in $\operatorname{supp}_{\vecy}(g')$ would transfer to $\operatorname{supp}_{\vecy}(g)$. Since the output of $C(x,y,\vecz)$ is convexly independent, the above observations imply that each gate $g \in C$ has convexly independent $\operatorname{supp}_{\vecy}(g)$.
Let us now prove the theorem by inductively building the collection $\mathcal{A}$ with respect to the circuit $C$: a gate is added only after adding all of its children. When the gate being added is a leaf, then $\mu$ increases by at most $1$ due to \eqref{eq-case:1}. For an addition gate computing $g$, $\operatorname{supp}_{\vecy}(g) $ is the union of the $(x,y)$-supports of its children; so we can apply \eqref{eq-case:3}. For an multiplication gate computing $g$, $\operatorname{supp}_{\vecy}(g) $ is the Minkowski sum of the $(x,y)$-supports of its children; so we can use \eqref{eq-case:4}. For a summation gate that computes $g$, note that its $(x,y)$-support is exactly the same as that of its child (\eqref{obs:support-monotone-projection}); therefore \eqref{eq-case:2} applies. Finally for a production gate, we can use \eqref{eq-case:5}, as $\operatorname{supp}_{\vecy}(\mathtt{prod}_z g) = \operatorname{supp}_{\vecy}(g\vert_{z=0}) + \operatorname{supp}_{\vecy}(g\vert_{z=1})$, and $\operatorname{supp}_{\vecy}(g\vert_{z=0}) \subseteq \operatorname{supp}_{\vecy}(g\vert_{z=1}) = \operatorname{supp}_{\vecy}(g)$.
Since the measure $\mu$ increases by at most $4$ in each of the $s$ steps, we have that $\abs{\operatorname{supp}(f)} \leq \mu(\mathcal{A}) \leq 4s$, as required. \end{proof}
\noindent The above result then lets us prove \autoref{thm:transparency-hard-for-QuantMonVP}, which we first restate. \TransparencyLBforQuantCircuits* \begin{proof} Let $C$ be a quantified monotone circuit computing $f_n$, of size $s$. Since $f_n(\vecx) \in \R[\vecx] $ is transparent, there exists a matrix $M \in \Z^{2 \times n} $, such that the linear map $L(\vece) = M\vece $, satisfies $\abs{\operatorname{vert}({L(\mathsf{Newt}(f))})} = \abs{\operatorname{supp}(f)}$. Further using \autoref{obs:shadows-hp-circuits}, there exists a size-$s$ high powered monotone circuit with summation and production gates, that computes a Laurent polynomial $P(y_1,y_2)$ which has $\abs{\operatorname{supp}(f)}$ vertices in its Newton polytope. The bound then easily follows from \autoref{thm:transparent-summation-production}. \end{proof}
\section{Monotone circuits with projection gates}\label{sec:monotone-circuits-projection}
In this section, we prove \autoref{thm:QuantMonVP-neq-VP-proj}. First we describe an efficient monotone circuit with projection gates that computes $\operatorname{\sf Perm}_n$.
\begin{theorem}\label{thm:Perm-mvpspace-ub}
There is a monotone circuit with projection gates of size $O(n^3)$ that computes $\operatorname{\sf Perm}_n$. \end{theorem}
\begin{proof}
We first define a polynomial $P_0$ such that all its monomials contain at most one $\vecx$-variable from each row.
\[
\text{Let } P_0(\vecx,\vecy) := \inparen{\sum_{j=1}^{n} y_{1,j} x_{1,j}} \inparen{\sum_{j=1}^{n} y_{2,j} x_{2,j}} \cdots \inparen{\sum_{j=1}^{n} y_{n,j} x_{n,j}}.
\]
Note that $P_0$ has $n^2 $-many auxiliary variables $\vecy$, one attached to each `true' variable $x_{i,j}$.
We now want to use these to progressively prune the monomials that pick up multiple variables from the $j$th column by projecting the $n$ variables $y_{1,j},\ldots,y_{n,j}$.
Let $e_1,\ldots,e_n \in \set{0,1}^n $ such that $e_i(k) = 1 \Leftrightarrow i=k$, and define for each $j \in [n]$,
\begin{equation}
P_j := \sum_{i \in [n]} \project{y_{1,j}}{e_i(1)}\inparen{\project{y_{2,j}}{e_i(2)}\inparen{\cdots\inparen{\project{y_{n,j}}{e_i(n)}\inparen{P_{j-1}}}}}.
\end{equation}
The following claim is now easy to verify.
\begin{claim}
For all $j \in [n]$, $P_{j} $ contains all the monomials from $P_{j-1}$ that are supported on exactly one $\vecx$-variable from the $j$th column.
\end{claim}
As a result, the monomials in $P_{n} $ are exactly those of the monomials in $\operatorname{\sf Perm}_n $.
Additionally for each $j$, the auxiliary variables in $P_j $ are only from the columns $j+1,\ldots,n$; thus $P_n = \operatorname{\sf Perm}_n $.
The size of our circuit is $O(n^3)$, since $\operatorname{size}(P_0) = O(n^2)$ and $\operatorname{size}(P_j) = \operatorname{size}(P_{j-1}) + O(n^2)$.
This proves \autoref{thm:Perm-mvpspace-ub}. \end{proof}
\begin{remark}
Our upper bound above also implies that any polynomial (family) that can be expressed as the permanent of a monotone matrix of size $\poly(n)$ (called monotone $p$-projection of $\operatorname{\sf Perm}_n$) can also be computed by efficient monotone circuits with projection gates.
Although $\operatorname{\sf Perm}_n$ is complete for non-monotone $\mathsf{VNP}$, it is \emph{not} the case that all monotone polynomials in $\mathsf{VNP}$ are monotone $p$-projections of $\operatorname{\sf Perm}_n$, as shown by Grochow~\cite{G17}. \end{remark}
Finally, we complete the proof of \autoref{thm:QuantMonVP-neq-VP-proj}.
\QuantCircuitsVsProjectionCircuits*
\begin{proof}
Trivially follows from \autoref{thm:Perm-mvpspace-ub} and \autoref{lem:support-summation-production}, since $\operatorname{\sf Perm}_n$ is irreducible. \end{proof}
\section{Conclusion}\label{sec:conclusion}
Our work is an attempt at understanding the hardness of transparent polynomials for monotone algebraic models. We observe that the lower bound of \Hrubes{} and Yehudayoff~\cite{HY21} extends beyond monotone $\mathsf{VNP}$, and therefore turn to exploring the class $\mathsf{VPSPACE}$ from the non-monotone world. This exploration reveals that the natural monotone analogues of the multiple equivalent definitions of $\mathsf{VPSPACE}$ have contrasting powers. Additionally, transparent polynomials turn out to be as hard for some of these analogues as they are for usual monotone circuits. Following are some interesting open threads from our work.
\begin{itemize} \item The first and most natural question related to the motivation behind our work is to prove (or refute) our conjecture that a transparent polynomial computed by a size-$s$ monotone circuit with projection gates, has sparsity at most $\exp(\poly\log{s})$.
An immediate hurdle in extending \autoref{thm:transparency-hard-for-QuantMonVP} to handle arbitrary projection gates is that unlike summations and productions, $0$-projections do not preserve convex dependencies, that is, the $0$-projection of a convexly dependent polynomial could be convexly independent.
\item Along similar lines, a possibly simpler goal is to show a non-monotone circuit upper bound for a transparent polynomial.
Note that transparency only restricts the support of the polynomial, so one is free to choose any real coefficients that do not affect the transparency.
It could therefore be possible to compute transparent polynomials that have positive and negative coefficients (\autoref{lem:real-polynomial-integral-projections} works for all real polynomials) in $\mathsf{VP}$ using fundamentally non-monotone tricks like interpolation.
Among other things, such a result would refute the notoriously open $\tau$-conjecture for Newton polygons.
\item Another question we would like to highlight is separating $\mathsf{mVNP}$ and quantified monotone circuits. As mentioned in the discussion following \autoref{thm:exp-sum-for-quantified-mvp}, such a separation would yield a (high degree) polynomial that is hard for $\mathsf{mVNP}$ even as a function over the boolean hypercube. Such a polynomial might be of interest, perhaps, even in the non-monotone setting.
\end{itemize}
\appendix
\section{Definitions of \textsf{VPSPACE} relying on boolean computation}\label{sec:boolean-definitions} In this section we briefly address why we did not study monotone analogues of the definitions due to Koiran and Perifel~\cite{KP07, KP09}, and Mahajan and Rao~\cite{MR13}.
Koiran and Perifel define uniform $\mathsf{VPSPACE}$ as the class of families $\set{f_n}$ of $\poly(n)$-variate polynomials of degree at most $2^{\poly(n)} $, such that there is a $\mathsf{PSPACE}$ machine that computes the \emph{coefficient function} of $\set{f_n}$. Here, the coefficient function of $\set{f_n}$ can be seen to map a pair $(1^n,\vece)$ to the coefficient of $\vecx^{\vece} $ in $f_n $.
Non-uniform $\mathsf{VPSPACE}$ is then defined by replacing $\mathsf{PSPACE}$ by its non-uniform analogue, $\mathsf{PSPACE}/\poly$. Since there are no monotone analogues of Turing machines, perhaps the only possible monotone analogue of this definition is to insist on the coefficient function being monotone, which results in an absurdly weak class (the ``largest'' monomial will always be present).
Mahajan and Rao~\cite{MR13} look at the notion of \emph{width} of a circuit --- all gates are assigned heights, such that the height of any gate is \emph{exactly} one larger than the height of its highest child. The width of the circuit is the maximum number of nodes that have the same height. They then define $\mathsf{VSPACE}(S(n))$, as the class of families that are computable by circuits of width $S(n)$ and size at most $\max\set{2^{S(n)},\poly(n)} $.
The class uniform $\mathsf{VSPACE}(S(n))$ further requires that the circuits be $\mathsf{DSPACE(S(n))}$-uniform. Although their non-uniform definition is purely algebraic, it is a bit unnatural for space $S(n) \gg \log{n}$ (as also pointed out in their paper), since such circuits may not even have a $\poly(n)$-sized description. We therefore do not analyse a monotone analogue for their definition.
\end{document}
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arXiv
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December 2017 , Volume 7, Issue 8, pp 4207–4218 | Cite as
Contemplating the feasibility of vermiculate blended chitosan for heavy metal removal from simulated industrial wastewater
N. Prakash
M. Soundarrajan
S. Arungalai Vendan
P. N. Sudha
N. G. Renganathan
Wastewater contaminated by heavy metals pose great challenges as they are non biodegradable, toxic and carcinogenic to the soil and aquifers. Vermiculite blended with chitosan have been used to remove Cr(VI) and Cd(II) from the industrial wastewater. The results indicate that the vermiculite blended with chitosan adsorb Cr(VI) and Cd(II) from industrial waste water. Batch adsorption experiments were performed as a function of pH 5.0 and 5.5 respectively for chromium and cadmium. The adsorption rate was observed to be 72 and 71 % of chromium and cadmium respectively. The initial optimum contact time for Cr(VI) was 300 min with 59.2 % adsorption and 300 min for Cd(II) with 71.5 % adsorption. Whereas, at 4–6 there is saturation, increasing the solid to liquid ratio for chitosan biopolymers increases the number of active sites available for adsorption. The optimum pH required for maximum adsorption was found to be 5.0 and 5.5 for chromium and cadmium respectively. The experimental equilibrium adsorption data were fitted using Langmuir and Freundlich equations. It was observed that adsorption kinetics of both the metal ions on vermiculite blended chitosan is well be analyzed with pseudo-second-order model. The negative free energy change of adsorption indicates that the process was spontaneous and vermiculite blended chitosan was a favourable adsorbent for both the metals.
Chromium(VI) Cadmium(II) Chitosan mixed vermiculite Adsorption isotherms Kinetics
Chromium and cadmium are highly toxic heavy metals and these have to be removed from the water sources. These metals are from various industrial effluents such as tanneries, electroplating and paints. The chromium toxicity is mainly induced from its hexavalent form, Cr(VI) which is readily soluble in water. Its concentration should not exceed 0.05 mg l−1 in drinking water and become more toxic with potential carcinogenic effects (Baird and Cann 2005). Cadmium belongs to the hazardous metal group. It is fairly mobile in soil and is primarily present as an organically bound, exchangeable and water soluble species (Holm et al. 1995; Chlopecka 1996). The removal of cadmium from the wastewater by various techniques such as chemical precipitation, electro deposition, electro coagulation process, ion exchange and emulsion liquid membrane (Vasudevan et al. 2011; Elkady et al. 2011; Ahmad et al. 2012). These techniques are expensive and ineffective at low concentration of metal ions. Adsorption method is reported to be a favourable method owing to its low cost and high efficiency for high and low concentration of metal ions (Ngah and Hanafiah 2008).
Novel adsorbents prepared from orange peel and Fe2O3 nano particles have been used (Gupta et al. 2012) to remove cadmium from aqueous solutions. The removal of cadmium from the electroplating industry effluent is reported and is shown to have desorption and reusability without loss of efficiency. Material from the low cost fertilizer industry viz., carbon slurry, has been chemically treated, activated and used as adsorbent to remove hexavalent chromium from aqueous solutions (Gupta et al. 2010) and the kinetics of adsorption follows pseudo second order rate equation based on batch experiments. The removal of lead and chromium from aqueous solutions, has been reported using inexpensive red mud, an aluminium industry waste and bagasse fly ash. The blast furnace waste generated in steel plants has been used for the removal of lead and chromium (Srivastava et al. 1997) and the authors concluded that the efficiency is higher for lead than that of chromium.
Low cost adsorbents have been investigated from agricultural and fishery wastes such as rice hull and saw dust (Asadi et al. 2008), peanut husk (Li et al. 2007), pumpkin waste (Horsfall et al. 2006), Nipah palm (Nypa fruticans Wurmb), shoot biomass (Wankasi et al. 2006), prawn shell (Chu 2002), crab shell (Dahiya et al. 2008), chitin (Jayakumar et al. 2009) and chitosan (Tao et al. 2009) to remove metal ions from wastewater. Chitosan biopolymer (Kumari et al. 2009; ShahinHydari et al. 2012; Prakash et al. 2011) and vermiculite clay (Malandrino et al. 2006; Maria Rosaria Panuccio et al. 2009) are being explored recently for the removal of heavy metals from wastewater.
Chitosan, poly (β(1-4)-2-amino-2-deoxy-d-glucose is obtained from partial N-deacetylation of chitin with a strong alkali solution such as sodium hydroxide. It is characterized by a high content of nitrogen, present as amine groups capable of binding the metal ion through several mechanisms, including chemical interactions, such as chelation, electrostatic interactions (or) ion-exchange. The interaction depends (Ng et al. 2003; Dzul Erosa et al. 2001) on the metal ions and initial pH of the medium. The chemical stability of chitosan is enhanced by several techniques such as cross linking (Du et al. 2009) carboxymethylation (Xu et al. 2009) grafting (Morimoto et al. 2002) blending (Wang and Kuo 2008; Ngah et al. 2004) coating (Popuri et al. 2009) and sulphonation (Holme and Perlin 1997). Among all the techniques cross linking is mainly focused because of its simple procedure and there are enormous opportunities to form macromolecular super structures for various specific applications (Crini 2005). Several studies have emphasized that chitosan and its modified form (cross linking coating and blending) is a very efficient adsorbent to remove various toxic and strategic metals (Gerente et al. 2007; Chang et al. 2007).
Vermiculite refers to a class of hydrated ferro-magnesium–aluminum silicates that contains various mineral groups. It looks like mica and has a lattice layer of Mg2+ (or) K+ with water molecules (McBride 1994). Vermiculite can adsorb metals via two mechanisms: (1) cation exchange at the planar sites, resulting from the interaction between metal ions and negative permanent charge (outer-sphere complexes) and (2) formation of inner-sphere complexes through Si–O- and Al–O-groups at the clay particle edges. Removal of toxic metals such as Cd(II), Pb(II) and Cu(II) by vermiculite (Allan et al. 2007; Prakash et al. 2012) has also been reported.
Multiwall carbon nanotubes with the magnetic properties of iron oxide have been used as a composite adsorbent for the adsorption of Cr(III) and it has been shown (Gupta et al. 2011) that the adsorption of Cr(III) is strongly dependent on contact time, agitation speed and pH in the batch mode and on flow rate and the bed thickness in the fixed bed mode. Chemical treatment technologies for wastewater recycling has been reviewed in (Gupta et al. 2012) and a brief guideline for the selection of the appropriate technologies for specific applications has been neatly evaluated. The carbon occurs adsorbent prepared from carbon slurry is reported (Jain et al. 2003) to be efficient and can be used for the removal of dyes from solution.
Bottom ash and despoiled soya have been used as adsorbents for the removal of a hazardous azodye and reported in Mittal et al. (2008). A hazardous textile dye, safranin-T, has been successfully photo degraded using Tio2 as catalyst and reported in Gupta et al. (2007). The adsorption performance of the materials namely bottom ash, a power plant water and de-oiled soya waste has been appreciable while employing batch and column methods as reported (Mittal et al. 2010). The hexavalent chromium, Cr(VI) has been successfully bioadsorbed on raw and acid treated Oedogonium hatei from aqueous solutions (Gupta et al. 2010c).
From the available literature reports, it is observed that the investigations on ternary blends with the presence of cross linking agents likely to enhance the thermal and chemical stability for the removal of heavy metals are scanty and inadequate. Hence, in the present study an attempt has been made to investigate the possibility of the vermiculite blended chitosan in the presence of glutaraldehyde as the cross linking agent for the adsorption of Cr(VI) and Cd(II) ions from the aqueous solution. Vermiculite is mixed with chitosan was prepared in ratios of (1:1), (3:1) in the presence and absence of cross linking agent glutaraldehyde. The blend was characterized by using FT-IR, XRD, TGA and DSC techniques. The kinetic adsorption studies of these blends with chromium and cadmium in aqueous solution were evaluated using different isotherms.
FT-IR studies
Fourier Transform Infra Red spectra of all the adsorbents of various combinations were recorded in the frequency range of 400–4000 cm−1 using Thermo Nicolet AVATAR 330 spectrophotometer. The samples were pressed into pellets with KBr.
X-ray diffraction studies
X-ray diffractograms of powdered samples were obtained using a X-ray powder diffractometer (XRD-Shimadzu XD-D1) with Ni-filter, Cu Kα, radiation source. The relative intensity was recorded in the scattering range 2θ of 10°–90°.
Thermogravimetric analysis
Thermo gravimetric analysis of all the adsorbents with various combinations was conducted by SOT Q600. V8. This analysis is to find out the thermal stability of the synthesized polymer and the heating range is 20–50 °C and the heating rate is 20 °C per min.
Differential scanning calorimetric analysis
The thermal behaviour of the adsorbents was studied using NET 2 SCH DSC thermal analyzer. The sample was inserted into the Al pan and DSC scan was made from 30 to 300 °C in a nitrogen atmosphere at a heating rate of 20 °C min−1. The results were recorded and analyzed.
Stock solution
Stock solutions of chromium(VI) ions and cadmium ions were prepared (Dean 1995). All other concentration values, varying between 40 and 670 mg l−1 of Cd(II) were prepared by dilution from the stock solution. The initial pH of the working solution was obtained by adding small volumes of 0.5 N HNO3 (for pH 2.0 and 3.0), or acetate buffer (CH3COOH/CH3COONa), for 3.0 < pH < 6.0.
Preparation of adsorbent: chitosan mixed vermiculite
The chitosan polymer solution was prepared by dissolving 1 g of chitosan in 2 % acetic acid and mixed with a known weight of the vermiculite. The mixture was then stirred thoroughly in the presence of glutaraldehyde (10 ml) as a cross linking agent at room temperature for 1 h. Then the above prepared mixture were poured into a petri dish and dried in vacuum for 10 h to remove the solvent completely.
Methodology for batch process of adsorption studies
For the study of equilibrium isotherms, the adsorption studies were done at varying concentrations of the metal, i.e., 30, 60, 90, 120, 150, 180, and 210 mg l−1 of chromium and cadmium and keeping the amount of chitosan (2.0 g l−1) constant at pH 5.0. Polymer blended solutions (30 ml) were added to the reactor vessel. Solutions were stirred magnetically throughout the experiments. Samples (100 μl) were taken from the feed cell into which metal was adsorbed and the solution is estimated for chromium and cadmium using the usual EDTA method (Prakash et al. 2012). Chromium and cadmium (30 mg l−1) were added to the reactor vessel. The reactor vessel is stirred magnetically throughout the experiments. Following the chromium and cadmium concentrations were estimated, the equilibrium isotherm was prepared by plotting uptake in mg g−1 versus the equilibrium concentration of chromium and cadmium in mg l−1.
Removal of chromium and cadmium
Synthetic solutions of Cr(VI) ions and Cd(II) ions were taken in stopper bottles and agitated with the prepared adsorbents separately at 30 °C in orbit shaker at fixed speed of 160 rpm. The extent of heavy metal removal was investigated separately by changing the amount of adsorbent, contact time and different pH of the solution. After attaining the equilibrium the adsorbent was separated by treating it with suitable reagents after leaching the metals from the adsorbents. Analysis of the concentration of the metals has been done using atomic adsorption spectrophotometer (Varian AAA 220 FS). Chromium has been determined from the analysis by fixing the wave length at 357.9 nm and cadmium has been obtained by fixing the wavelength at 228.8 nm.
Vermiculite blended chitosan were characterized by IR spectroscopy, X-ray diffraction, thermal gravimetric analysis (TGA/DTA), Differential Scanning Calorimetry (DSC). Further, kinetic analysis have also been reported and discussed.
FT-IR spectra analysis
The FT-IR spectrum of chitosan gave a characteristic band at 3450 cm−1 which is attributed to –NH2 and –OH stretching vibration and the band for amide I at 1652 cm−1 is observed in the infrared spectrum of chitosan. The characteristic carbonyl stretching of chitosan observed at 1733 cm−1 and the broader peak at 1635 cm−1 corresponding to the chitosan NH2 band. In 3:1 ratio the peaks observed at 3421 cm−1 are due to OH stretching, –NH stretching and intermolecular hydrogen bonding. The peaks at 2935, 1640 and 1579 cm−1 indicate asymmetric C–H stretching, C=O stretching and N–H bending respectively. The peaks at 1386, 1254, 1029 and 452 cm−1 arises due to C–H bending, O–H bending, Si–O stretching, Al–O stretching vibrations respectively. The IR spectrum of chitosan:vermiculite (1:1) shows the peak at 3430 cm−1 corresponding to the –OH stretching, –NH stretching, which may be due to intermolecular hydrogen bonding and polymeric association. The peaks at 2927, 1605 cm−1 correspond to the asymmetric C–H stretching and N–H bending. The peaks at 1383, 1021 and 460 cm−1 are due to the C–H bending, C–OH was stretching, Si–O and Al–O stretching vibrations.
The spectrum of chitosan:vermiculite+Glu (1:1), blend has a peak at 3000–3600 cm−1, which is due to OH stretching, NH stretching and intermolecular hydrogen bonding. The peak positions are slightly modified in the presence of cross linker. The peaks at 2938, 1713, 1583 and 1381 cm−1 correspond to the asymmetric C–H stretching, C=O stretching, N–H bending and C–H bending vibrations. The peak at 1020, 681, 499 and 450 cm−1 indicate the C–O stretching, N–H deformation and Al–O & Si–O stretching vibration respectively. The IR spectrum for the ratio of chitosan:vermiculite+Glu (3:1) has stretching and due to intermolecular hydrogen bonding. The peaks at 2927 cm−1 to aldehydic asymmetric and symmetric CH stretching, the peaks at 1716, 1383 cm−1 the corresponds to C=O stretching, N–H bending and C–H bending vibrations respectively. The peaks at 1022 and 449 cm−1 correspond to C–OH stretching and Al–O, Si–O stretching vibrations respectively.
The sharp peak observed at 1610 cm−1 may be due to C=N (amide I band) of the Schiff's base formed between the free amino groups from the chitosan and aldehyde groups from glutaraldehyde which acts as a cross linking agent. In addition, the observed decrease in the intensity of the broad peak around may be attributed to the cross linking via acetalization reaction the glucosamine units of chitosan and aldehyde groups of glutaraldehyde. Similar observations were reported by Ruiz et al. (2000). An intensity of the sharp peak was observed at 1624 cm−1 which may be due to the N–H bending (amide II band). The increase is the chitosan composition in blend leads to more number of –NH functional groups. When chitosan was mixed with vermiculite the absorption peaks were observed at lower frequencies, due to the intermolecular hydrogen bonding of chitosan with vermiculite (Xie and Wang 2009).
When chitosan was blended with vermiculite in the presence of glutaraldehyde as cross linking agent an intensity of the sharp peak was observed at 1610 cm−1 which may be due to the C=N (amide I band) of the formed Schiff's base between the free amino groups that are present in the chitosan and aldehyde groups of the cross-linker glutaraldehyde (Ruiz et al. 2000).
Figure 1a, b show the XRD spectrum of chitosan:vermiculite (1:1) and (3:1). From the XRD spectrum, the appearance of short sharp peaks at 2θ = 19°, 33° and 36° for 1:1 and three sharp peaks at 2θ = 11°, 15° and 25° for 3:1 indicates high crystalline nature of blends.
a XRD diffractogram of chitosan mixed vermiculite (1:1), b XRD diffractogram of chitosan mixed vermiculite (3:1), c XRD diffractogram of chitosan mixed with vermiculite (1:1)+Glu, d XRD diffractogram of chitosan mixed with vermiculite (3:1)+Glu
In the presence of glutaraldehyde, the XRD spectrum of chitosan:vermiculite for ratio (1:1) and (3:1) are shown in Fig. 1c, d. The XRD peaks are observed at 2θ = 19°, 26°, 34° for 1:1 and at 2θ = 25°, 39° for 3:1. From the XRD data it is clear that the nature of the sample is semi crystalline in nature for 1:1 ratio, where as in the case of 3:1, the observation of two long sharp peaks at 2θ = 25° indicate crystalline nature of the blend.
Thermo gravimetric analysis (TGA)
Figure 2a, b represents the thermogram of chitosan:vermiculite mixture for (1:1) and (3:1) ratios. The maximum weight loss takes place at the temperature range from 250 to 500 °C. Around 54.57 % of the sample had disintegrated at the temperature of approximately 731.25 °C. At the end of the experiment, 45.43 % of the sample was obtained as a residue. 12 % of the sample had disintegrated at the temperature range of 510–860 °C. It may be due to loss of water. In the case of 3:1 ratio, the maximum weight loss takes place at the temperature range from 200 to 663 °C. Around 63.10 % of the sample had disintegrated at the temperature of about 657.12 °C. At the end of the experiment 36.90 % of the sample remained as a residue. 15 % of the sample had disintegrated at a temperature range of 450–200 °C.
a TGA of chitosan mixed with vermiculite (1:1), b TGA of chitosan mixed with vermiculite (3:1), c TGA of chitosan mixed with vermiculite (1:1)+Glu, d TGA of chitosan mixed with vermiculite (3:1)+Glu
Figure 2c, d represents the thermogram chitosan:vermiculite (1:1)+Glu (10 ml) ratio. The maximum weight loss takes place at the temperature range from 150 to 375 °C. Around 75.311 % of the sample had disintegrated at the end of the experiment. 24.68 % of the sample remained as a residue. 5 % of the sample had disintegrated at the temperature range of 500–150 °C. The weight loss between 50 and 150 °C may be due to water loss. The maximum weight loss occurs at 150 and 375 °C. The weight loss may be due to the decomposition of the –CH2-linkages, H-bonding and other interaction between the two polymers. After 400 °C there was a sudden rise in the decomposition of the pyranose ring of the polysaccharides. When we compare the thermal stability with or without cross linking agent, the cross linked chitosan was thermally more stable.
Based on Fig. 2d, the thermogram of chitosan:vermiculite+Glu (3:1), the maximum weight loss takes place at the temperature range from 200 to 400 °C. Around 63.552 % of the sample had disintegrated at the temperature range of 700 °C. At the end of the experiment, 36.448 % of the sample remained as a residue. The maximum weight loss that occurred at 200–400 °C may be due to the decomposition of the –CH2-linkage, H-bonding and other interaction between the polymers.
Adsorption experiments
Experimental procedure
Series of experiments were performed to evaluate the influence of the following variables: initial pH value of the solution, contact time, and adsorbent dosage. The batch experiments were performed by adding chitosan mixed vermiculite to a 25 ml solution containing Cr(VI) ions and Cd(II) ions under intermittent stirring, at room temperature (25 ± 0.5 °C) and, with the exception of only one series, at a constant contact time of 24 h. The influence of the initial solution pH on Cr(VI) and Cd(II) ions capacity of adsorption onto chitosan mixed vermiculite was investigated over the 4.0–8.0 pH range, at a constant dosage of chitosan mixed vermiculite. To evaluate the effect of adsorbent dosage on Cr(VI) and Cd(II) adsorption, the metal ion solutions of a constant concentration (71.3 and 74.3 mg l−1) was treated with variable dosages of chitosan mixed vermiculite. (Resulting in adsorbent concentrations in the mixture from 5 to 40 g l−1, at pH 5.0 of the initial solution). The influence of contact time of both the metal ions concentration on chitosan mixed vermiculite adsorption capacity was studied over the 30–360 min range, at pH 5.0 of the initial solution and constant dosage of chitosan mixed vermiculite (5 g l−1 concentration in the reaction volume). It is seen from Fig. 3, that cadmium recovery is higher when compared to chromium recovery at all times. It is seen from the graph that the complexes of Cr6+ and Cd2+ behave in the same manner and cadmium and chromium complexes are stable during the studies. This may also be due to the stability of the complexes, cadmium complex is more stable compared to chromium complex (Prakash et al. 2012). The percentage of removal of cadmium is always higher compared to chromium at all contact time Fig. 3. Since cadmium is easily polarizable and charge to size ratio is less for cadmium compared to chromium the recovery of cadmium is more at all time compared to chromium. (Cr6+ = 11.6 × 10−24 and Cd2+ = 7.2 × 10−24) Fig. 3 also shows the effect of contact time on the removal of chromium and cadmium metals in the artificial wastewater.
Percentage of metal removal as a function of time
Percentage of metal removal as a function of pH
The adsorption process is strongly affected by the pH of the solution. The effect of pH change on adsorption was studied for both the metals by changing the pH of the contents from 4 to 8, using dilute solutions of HCl and NaOH. The observation shows that the maximum adsorption of both the metals took place in acidic media and this is evident from Fig. 4. The maximum adsorption for Cr(VI) was observed at pH 5 (72 % removal) and for cadmium at pH 5.5 (71 % removal) (Prakash et al. 2012). In general, at very high and very low pH values, the surface of the adsorption is surrounded mainly by H+ and OH− ions respectively. These positively and negatively charged ions may compete with the metal ions and as a result, adsorption decreases. That's why metal ions show low adsorption at very high and low pH values (Saeed and Iqbal 2003). Influence of pH on adsorption phenomenon is also related with the functional groups present on the bio-adsorbent.
Percentage of Cr(VI) and Cd(II) recovery by chitosan mixed with vermiculite at different pH
The percentage removal of chromium and cadmium using chitosan mixed with vermiculite at different adsorbent dosage
Various operational parameters like solid/liquid ratio, pH, contact time, metal ion concentration and temperature were employed to determine the adsorption characteristics of chitosan biopolymers for heavy metal ion adsorption from aqueous solutions. The effect of the solid/liquid ratio for chromium adsorption on chitosan biopolymer was examined. In these experiments, operational parameters were kept constant T = 298 K, C 0 = 200 mg l−1 and the pH was varied initially for various time intervals. Figure 5 indicates that the percentage of adsorption of chromium and cadmium increases steadily with, solid to liquid ratio of 1–3 %. Whereas, at 4–6 there is saturation, increasing the solid to liquid ratio for chitosan biopolymers increases the number of active sites available for adsorption.
Percentage removal of chromium and cadmium using chitosan mixed with vermiculite at different adsorbent dosage
Effect of contact time
It has been observed that at a constant concentration of metal ions and fixed amount of adsorbent, the adsorption efficiency increases with increase in the contact time up to a certain level. Figure 5 shows that the adsorption rate first increased rapidly, but after reaching the optimum time value, the removal efficiency slightly decreased with an increase in the contact time. The effect may be due to the saturation of adsorption sites with metal ions on the solid particle. The initial optimum contact time for Cr(VI) was 300 min with 59.2 % adsorption and 300 min for Cd(II) with 71.5 % adsorption. The slight decrease in adsorption after the optimum contact time may be due to the breakage of newly formed weak adsorption bonds due to the constant shaking.
Kinetics of adsorption
Pseudo first and second order kinetics of adsorption
The sorption data of Cr(VI) and Cd(II) uptake by chitosan:vermiculite adsorbent was fitted using Lagergren pseudo first and second order model and these are given in Figs. 6 and 7. The linearized form of the first order Lagergren equation is given as Eq. 1.
$$ \log \left( {q_{\text{e}} - q_{\text{t}} } \right) = \log q_{\text{e }} - \frac{{K_{{1 ,{\text{ads}} }} }}{2.303}t $$
Plot of ln (q e/q e − q) versus time in chromium and cadmium
Plot of t/q versus time in chromium and cadmium
The pseudo-second-order Eq. 2
$$ \frac{t}{q} = \frac{1}{{k_{{2,{\text{ads}}}} q_{\text{e}}^{2} }} + \frac{t}{{q_{\text{t}} }} $$
where q e is the mass of metal adsorbed at equilibrium (mg g−1), q t the mass of metal at time (min), k 1, ads the pseudo first order reaction rate of adsorption (per min), k 2, ads the pseudo second order rate constant of adsorption mg g−1 min−1. A comparison between the two kinetic models suggested is presented in Table 1. These facts suggest that the obtained Cr(VI) and Cd(II) adsorption onto chitosan mixed vermiculite kinetic data followed the pseudo second order kinetic model which describes the biosorption as the rate limiting step (Hanif et al. 2007).
Comparison between Lagergren pseudo first and second order kinetic of adsorption
Metal ion
Pseudo first order kinetic model
Experimental value
Pseudo second order kinetic model
q e (mg g−1)
k 1 (min−1)
k 2 (g mg−1 min−1)
Cr(VI)
Cd(II)
Adsorption isotherm models
Effect of adsorbent mass
Adsorption to a maximum of 5 g/100 ml (using 50 ppm metal solution) for Cr(VI) and 5 g/100 ml (10 ppm metal solution) for Cd(II). By increasing the quality of adsorption beyond the optimal mass, there is no significant change in the removal efficiency. These results indicate that the removal efficiency is directly related to the number of available adsorption sites. Once equilibrium is attained there is no effect on adsorption efficiency. The optimum adsorbent mass is 5 g/100 ml for both metals with 71.3 % removal of Cr(VI) and 74.3 % removal of Cd(II).
Adsorption isotherms
Langmuir adsorption isotherms
The Langmuir is used to describe the experimental adsorption isotherm. The Langmuir isotherm equation may be formulated as
$$ \frac{{C_{\text{eq}} }}{{C_{\text{ads}} }} = \left( {\frac{1}{Qb}} \right) + \left( {\frac{{C_{\text{eq}} }}{Q}} \right) $$
where non linearity is indicated in Figs. 8 and 9. From this it is not possible to correlate to the isotherm with the experimental data and it is clear that the experimental data is not following Langmuir isotherm. Another essential dimensionless equilibrium constant of Langmuir isotherm is R L which is applied in many operations and is expressed as:
$$ R_{\text{L}} = \frac{1}{{1 + bC_{\text{i}} }} $$
where, C i the initial concentration is the analyst in the solution. If R L > 1, then it is considered unfavourable. R = 1 corresponds to linear and 0 < R < 1 is favourable (Table 2).
Langmuir isotherm plot for chromium
Langmuir isotherm plot for cadmium
Adsorption isotherm constant, C max and correlation coefficients
Langmuir constants
KL (dm3 g−1)
b (dm3 mg)−1
C max (mg g−1)
Freundlich adsorption isotherms
The Freundlich model is used in essential of adsorption concentration of the adsorbate on the adsorbent surface (Freundlich 1926) and expressed as:
$$ C_{\text{ads}} = KC_{\text{eq}}^{{\frac{1}{n}}} $$
The linearized (Figs. 10, 11) form of this equation is expressed as
$$ \log C_{\text{ads}} = \frac{1}{n}\log C_{\text{eq}} + \log K $$
whereas, 1/n is Freundlich characteristic constant related to energy, and K is the maximum adsorption capacity. Both n and K are temperature dependant. By plotting log C ads against log C eq, if a straight line is obtained, then the value of 1/n and K can be calculated from the slope and intercept. At the regions of moderate coverage Freundlich isotherm is similar to the Langmuir isotherm. Values of 1/n indicate the steepness or flatness of the slopes. 1/n close to 1 indicates the steep slope and high adsorption capacity and equilibrium concentrations which diminishes at low equilibrium concentrations. On the other hand 1/n ≪ 1 indicates a flat slope and adsorption capacity is slightly reduced at low equilibrium concentration (Faust and Aly 1987).
Freundlich isotherm plot for chromium
Freundlich isotherm plot for cadmium
Adsorption isotherms results Langmuir and Freundlich
The adsorption isotherm is fundamental in describing the interactive behaviour between solutes and adsorbent (Chiou and Li 2003; Ofomoja and Ho 2006). According to Ofomaja and Ho (2006), the isotherm yields certain constant values, which expresses the surface properties and affinity of the adsorbent. It also plays an important role in the design of an adsorption system.
The parameter indicates the effect of separation factor on isotherm shape according to Table 3. The values of R L calculated for different initial metal ion concentration are given in Table 4. If the R L values are in the range of 0 < R L < 1, it indicates that the adsorption of Cr(IV) and Cd(II) onto chitosan beads are favourable. Thus, chitosan beads are favourable adsorbents. The mechanism of ion adsorption on porous adsorbents may involve three steps: (1) diffusion of the ions to the external surface of adsorbent; (2) diffusion of ions into the pores of adsorbents; (3) adsorption of the ions on the internal surface of adsorbent.
Adsorption isotherm constant Freundlich
Freundlich constants
R L values based on Langmuir adsorption
Initial concentration C 0 (mg dm−3)
Final concentration C f (mg dm−3)
R L values
Cr(VI) ion
Cd(II) ion
Chitosan mixed with vermiculite has been successfully used as the adsorbent for the removal of Cr(VI) and Cd(II) from wastewater. Cadmium recovery is slightly higher at all times than chromium. The maximum adsorption for Cr(VI) has been obtained at pH 5.0 and for Cd(II) at pH 5.5. The percentage of adsorption of chromium and cadmium increases steadily with solid to liquid ratio of 1–3 % whereas at 4–6 saturation is attained. The adsorption of Cr(VI) and Cd(II) is found to be 59.2 and 71.5 % respectively with the contact time of 300 min. Kinetic data of adsorption studies indicate that adsorption follows pseudo second order model and bio adsorption is found to have rate limiting step. The optimum adsorption mass is 5 g/100 ml for both metals with 71.3 % removal of Cr(VI) and 74.3 % removal of Cd(II). The cross linker being glutaraldehyde enhances the thermal and chemical stability. Further, cross linking increases intra molecular hydrogen bonding enabling greater adsorption. The feasibility of using chitosan with vermiculite added with glutaraldehyde in the ratio 3:1 is established and this composition yields greater capacity for adsorption
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.Department of ChemistryThangavelu Engineering CollegeChennaiIndia
2.Department of ChemistryValliammai Engineering CollegeKattankulathurIndia
3.School of Electrical EngineeringVIT UniversityVelloreIndia
4.Department of ChemistryD.K.M. College for WomenVelloreIndia
5.Department of ChemistryVel Tech UniversityAvadi, ChennaiIndia
Prakash, N., Soundarrajan, M., Arungalai Vendan, S. et al. Appl Water Sci (2017) 7: 4207. https://doi.org/10.1007/s13201-015-0366-z
Accepted 24 November 2015
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# Theory behind Shanks' square forms factorization
Shanks' square forms factorization is a method for factoring large integers. It is based on the idea that if a number can be factored into two smaller factors, then its square form will have a certain structure. This structure can be used to find the factors of the original number.
To understand Shanks' method, let's first discuss square forms. A square form is a representation of a number in the form of a quadratic equation:
$$ax^2 + bxy + cy^2 = N$$
The variables $x$ and $y$ are integers, and $a$, $b$, and $c$ are integers such that $4ac - b^2 \neq 0$.
Shanks' method involves finding square forms for a given number $N$ and checking if they can be factored into two smaller factors. If a square form can be factored, then it is considered a proper square form. Otherwise, it is considered an improper square form.
The algorithm for Shanks' square forms factorization involves the following steps:
1. Start with an initial square form.
2. Advance to the next square form using the formulas (2.2), (2.3), and (2.4) from the research notes.
3. Check if the current square form is a proper square form.
4. If it is, then the factors of $N$ can be found.
5. If it's not, then continue to the next square form.
## Exercise
Consider the following square form:
$$x^2 - 2xy + 2y^2 = 14$$
Determine if it is a proper square form and find the factors of $14$ if it is.
# Implementing Shanks' square forms factorization in Python
To implement Shanks' square forms factorization in Python, you'll need to follow the algorithm described in the previous section. Here's a high-level outline of the steps:
1. Define a function to advance to the next square form.
2. Define a function to check if a square form is a proper square form.
3. Define a function to factor $N$ using Shanks' method.
4. Call the factoring function with the given number $N$.
Here's a sample implementation in Python:
```python
def advance_square_form(a, b, c):
# Implement the formulas (2.2), (2.3), and (2.4) here
pass
def is_proper_square_form(a, b, c):
# Implement the logic to check if the square form is proper
pass
def factor_using_shanks(N):
# Implement the Shanks' method for factoring N
pass
# Example usage
N = 14
factors = factor_using_shanks(N)
print(factors)
```
# Comparing the efficiency of Shanks' square forms factorization to other prime factorization algorithms
Shanks' square forms factorization is a powerful method for factoring large integers. However, it's not the only prime factorization algorithm available. There are other algorithms like the trial division method, the Pollard's rho algorithm, and the Elliptic Curve Method.
The efficiency of Shanks' method can be compared to these other algorithms based on factors such as time complexity, memory usage, and the types of numbers it can handle. In general, Shanks' method is more efficient for large numbers, but it may not be the best choice for all types of numbers.
# Applications of prime factorization in Python programming
Prime factorization is a fundamental concept in number theory and has numerous applications in computer science and cryptography. Some of the applications include:
- Factoring large numbers for cryptographic purposes.
- Solving Diophantine equations.
- Generating prime numbers for cryptographic keys.
- Optimizing algorithms and data structures.
In Python programming, prime factorization can be used to implement various algorithms and data structures. For example, it can be used to implement efficient algorithms for primality testing, modular exponentiation, and integer factorization.
# Real-world examples of prime factorization
Prime factorization is used in various fields, including cryptography, number theory, and computer science. Here are some real-world examples of prime factorization:
- RSA cryptosystem: RSA relies on the difficulty of factoring large numbers to secure data transmission.
- Primality testing: Prime numbers are used in cryptography and number theory.
- Integer factorization: Factoring large numbers is a fundamental problem in number theory.
# Handling large numbers and optimizing performance
When working with large numbers in Python, it's important to handle them efficiently to avoid memory and performance issues. Some strategies for handling large numbers include:
- Using the `math` library for mathematical operations.
- Using the `numpy` library for numerical computations.
- Implementing efficient algorithms for prime factorization and other number theory operations.
To optimize performance, you can use techniques like memoization, dynamic programming, and parallel computing. These techniques can help reduce the time complexity of your algorithms and improve the overall efficiency of your code.
# Debugging and testing your implementation
Debugging and testing your implementation of Shanks' square forms factorization is crucial to ensure its correctness and reliability. Here are some steps you can follow:
1. Write unit tests for each function in your implementation.
2. Use a debugger to step through your code and identify any issues or errors.
3. Verify the correctness of your implementation by comparing the results with known values or other algorithms.
For example, you can use the `pytest` library to write unit tests for your functions and the `pdb` module to debug your code.
# Conclusion and future developments
In conclusion, Shanks' square forms factorization is a powerful method for factoring large integers. It has numerous applications in computer science and cryptography.
Future developments in this area may involve improving the efficiency of the algorithm, extending it to handle other types of numbers, and exploring new applications of prime factorization in computer science and cryptography.
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Find the Lagrange Interpolation Formula given below, Solved Examples Question: Find the value of y at x = 0 given some set of values (-2, 5), (1, 7), (3, 11), (7, 34)? However, as n gets smaller, this approximation At one step they say something like "and obviously we can use the Stirling formula to show that ..." and show the equation in … We have step-by-step solutions for your textbooks written by Bartleby experts! ., x n with step length h.In many problems one may be interested to know the behaviour of f(x) in the neighbourhood of x r (x 0 + rh).If we take the transformation X = (x - (x 0 + rh)) / h, the data points for X and f(X) can be written as (n−k)!, and since each path has probability 1/2n, the total probability of paths with k right steps are: 5 To evaluatex 2 p(x)dx z ∞ =s, we proceed as before, integrating on only the positive x-axis and doubling the value.Substituting what we know of p(x), we have 2 2 2 0 2 2 k 2 x e dx k x p s ∞ z − = . 3 Stirlings approximation is n n n e n 8 In order for find the P i we use the from PHYS 346 at University of Texas, Rio Grande Valley k! Stirling's interpolation formula looks like: (5) where, as before,. The binomial coe cient can often be used to compute multiplicities - you just have to nd a way to formulate the counting problem as choosing mobjects from nobjects. eq. The person has definitely birthday on one day in the year, so we can say the probability p 1 = 1 = 365 365 The Stirling engine efficiency formula you have derived is correct except that number of moles (n) should have canceled out. Formula (5) is deduced with use of Gauss's first and second interpolation formulas [1]. Using Stirling's formula [cf. . To prove Stirling's formula, we begin with Euler's integral for n!. Stirling's approximation for approximating factorials is given by the following equation. The Boltzmann distribution is a central concept in chemistry and its derivation is usually a key component of introductory statistical mechanics courses. k R N Nk S k N g g D = - ln2 ln 2 ln BBoollttzzmmaannnn''ss ccoonnssttaanntt In the Joule expansion above, Proof of … ~ (n/e) n There are a couple ways of deriving this result. DERIVATION OF THE IMPROVED STIRLING FORMULA FOR N! Stirling's interpolation formula. or the gamma function Gamma(n) for n>>1. $\begingroup$ @JohnDonne In the proof I wrote above (you can find more details in Griffiths) there is no explicit mention of entropy and the logarithm only serves to break production in summation and to exploit Stirling approximation (even if the maximization of entropy is certainly a possible angle from which see this problem). At first glance, the binomial distribution and the Poisson distribution seem unrelated. Normal approximation to the binomial distribution . \[ \ln(n! 264-267), but it also offers several different approaches to deriving the deep and powerful Euler-Maclaurin summation formula, of which Stirling's formula is … James Stirling, (born 1692, Garden, Stirling, Scotland—died December 5, 1770, Edinburgh), Scottish mathematician who contributed important advances to the theory of infinite series and infinitesimal calculus.. No absolutely reliable information about Stirling's undergraduate education in Scotland is known. The formula is: n! f '(x) = 0. For using this formula we should have – ½ < p< ½. Sometimes this takes some ingenuity. x = μ. which says that the bell shaped curve peaks out above the mean, which we suspected to be true to begin with. Not only does the book include the very derivation of Stirling's formula that Professor Gowers has presented here (on pp. It turns out the Poisson distribution is just a… Mean and variance of the binomial distribution; Normal approximation to the binimial distribution assumption that jf00(x)j K in the Trapezoid Rule formula. A random variable has a standard Student's t distribution with degrees of freedom if it can be written as a ratio between a standard normal random variable and the square root of a Gamma random variable with parameters and , independent of . In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (). A useful step on the way to understanding the specific heats of solids was Einstein's proposal in 1907 that a solid could be considered to be a large number of identical oscillators. 12:48. Another formula is the evaluation of the Gaussian integral from probability theory: (3.1) Z 1 1 e 2x =2 dx= p 2ˇ: This integral will be how p 2ˇenters the proof of Stirling's formula here, and another idea from probability theory will also be used in the proof. Which is zero if and only if. (−)!.For example, the fourth power of 1 + x is The approximation can most simply be derived for n an integer by approximating the sum over the terms of the factorial with an integral, so that lnn! Stirling S Approximation To N Derivation For Info. The efficiency of the Stirling engine is lower than Carnot and that is fine. We can get very good estimates if - ¼ < p < ¼. ... My textbook is deriving a certain formula and I'm trying to follow the derivation. Stirling's approximation gives an approximate value for the factorial function n! = ln1+ln2+...+lnn (1) = sum_(k=1)^(n)lnk (2) approx int_1^nlnxdx (3) = [xlnx-x]_1^n (4) = nlnn-n+1 (5) approx nlnn-n. If not, and I know this is a rather vague question, what is the simplest but still sufficiently rigorous way of deriving it? From the standpoint of a number theorist, Stirling's formula is a significantly inaccurate estimate of the factorial function (n! Title: ch2_05g.PDF Author: Administrator Created Date: 1/12/2004 10:58:48 PM CENTRAL DIFFERENCE FORMULA Consider a function f(x) tabulated for equally spaced points x 0, x 1, x 2, . = nne−n √ 2πn 1+O 1 n , we have f(x) = nne−n √ 2πn xxe−x √ 2πx(n− x)n−xe−(n−x) p 2π(n− x) pxqn−x 1+O 1 n = (p/x) x(q/(n− x))n− nn r n 2πx(n− x) 1+O 1 n = np x x nq n −x n−x r n 2πx(n− x) 1+O 1 n . (Note that this formula passes some simple sanity checks: When m= n, we have n n = 1; when m= 1 we get n 1 = n. Try some other simple examples.) Consider: i) ( ), ( ) ln( ( )) ( ) ( ) ( ) b b b pr x dx a x R a a f x e f x pr x dx f x pr x dx Î Õ = £ò ò We have shown in class, by use of the Laplace method, that for large n, the factorial equals approximately nn!e≅−2πnn xp(n)]dt u This is referred to as the standard Stirling's approximation and is quite accurate for n=10 or greater. Stirlings central difference Formula - Duration: 12:48. This formula gives the average of the values obtained by Gauss forward and backward interpolation formulae. To find maxima and minima, solve. It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n, and is given by the formula =!! However, this is not true! Here, with only a little more effort than what is needed for the dV E dN dV dE dU d W g f F i i i i + = b (ln) + ∑ (2.5.18) Comparing this to the thermodynamic identity: Derivation of Gaussian Distribution from Binomial The number of paths that take k steps to the right amongst n total steps is: n! What this is stating is that the magnitude of the second derivative must always be less than a number K. For example, suppose that the second derivative of a function took all of the values in the set [ 9;8] over a closed interval. Textbook solution for Calculus (MindTap Course List) 11th Edition Ron Larson Chapter 5.4 Problem 89E. I had a look at Stirling's formula: proof? The quantum approach to the harmonic oscillator gives a series of equally spaced quantized states for each oscillator, the separation being hf where h is Planck's constant and f is the frequency of the oscillator. But a closer look reveals a pretty interesting relationship. Improvement on Stirling's Formula for n! February 05 Lecture 2 3 Proof of "k ln g" guess. = 1*2*3*...*(n-1)*(n)). x - μ = 0. or. However, the derivation, as outlined in most standard physical chemistry textbooks, can be a particularly daunting task for undergraduate students because of the mathematical and conceptual difficulties involved in its presentation. (11.1) and (11.5) on p. 552 of Boas], n! by Marco Taboga, PhD. Stirling numbers of the second kind, S(n, r), denote the number of partitions of a finite set of size n into r disjoint nonempty subsets. Stirlings approximation does not become "exact" as ##N \rightarrow \infty ##. (2) To recapture (1), just state (2) with x= nand multiply by n. One might expect the proof of (2) to require a lot more work than the proof of (1). NPTEL provides E-learning through online Web and Video courses various streams. Student's t distribution. ∑dU d W g f dE EF dN i = b (ln) + i i i + (2.5.17) Any variation of the energies, E i, can only be caused by a change in volume, so that the middle term can be linked to a volume variation dV. According to one source, he was educated at the University of Glasgow, while … The will solve it step by step before deriving the general formula. n = 1: There is only one person in the group. Now that we have the formula, we can locate the critical points in the bell shaped curve. using Product Integrals (The following is inspired by Tyler Neylon's use of Product Integrals for deriving Stirling's Formula-like expressions). Wikipedia was not particularly helpful either since I have not learned about Laplace's method, Bernoulli numbers or … (1) Study Buddy 21,779 views. So the formula becomes. but the comments seems quite messy. The formula is: There are also Gauss's, Bessel's, Lagrange's and others interpolation formulas. The integral on the left is evaluated by parts withu=x and dv xe k x = − 2 Let"s say the number of people in the group is denoted by n. We also assume that a year has 365 days, thus ignoring leap years. formula duly extends to the gamma function, in the form Γ(x) ∼ Cxx−12 e−x as x→ ∞.
deriving stirling's formula
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deriving stirling's formula 2020
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Song Sun
Song Sun (Chinese: 孙崧; pinyin: Sūn Sōng, born in 1987) is a Chinese mathematician whose research concerns geometry and topology. A Sloan Research Fellow, he is a professor at the Department of Mathematics of the University of California, Berkeley, where he has been since 2018. In 2019, he was awarded the Oswald Veblen Prize in Geometry.
Biography
Sun attended Huaining High School in Huaining County, Anhui, China, before being admitted to the Special Class for the Gifted Young at the University of Science and Technology of China in 2002.[1] After graduating from the program with a B.S. in 2006, he moved to the United States to pursue graduate studies at the University of Wisconsin, obtaining his Ph.D in mathematics (differential geometry) in 2010.[1][2] His doctoral advisor was Xiuxiong Chen, and his dissertation was titled "Kempf–Ness theorem and uniqueness of extremal metrics".[3]
Sun worked as a research associate at Imperial College London before becoming an assistant professor at Stony Brook University in 2013.[2] He was awarded the Sloan Research Fellowship in 2014.[2] In 2018, he was appointed an associate professor at the Department of Mathematics of the University of California, Berkeley.[4]
He was an invited speaker at the 2018 International Congress of Mathematicians, in Rio de Janeiro.[5] For 2021 he received the Breakthrough Prize in Mathematics – New Horizons in Mathematics.[6]
Conjecture on Fano manifolds and Veblen Prize
In 2019, Sun was awarded the prestigious Oswald Veblen Prize in Geometry, together with his former advisor Xiuxiong Chen and Simon Donaldson, for proving a long-standing conjecture on Fano manifolds, which states that "a Fano manifold admits a Kähler–Einstein metric if and only if it is K-stable". It had been one of the most actively investigated topics in geometry since a rough version of it was conjectured in the 1980s by Shing-Tung Yau, who had previously proved the Calabi conjecture.[7] The conjecture was later given a precise formulation by Donaldson, based in part on earlier work of Gang Tian. The solution by Chen, Donaldson and Sun was published in the Journal of the American Mathematical Society in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III".[7][8]
Major publications
• Donaldson, Simon; Sun, Song (2014). "Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry". Acta Math. 213 (1): 63–106. doi:10.1007/s11511-014-0116-3. S2CID 120450769.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015). "Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities". J. Amer. Math. Soc. 28 (1): 183–197. arXiv:1211.4566. doi:10.1090/S0894-0347-2014-00799-2. S2CID 119641827.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015). "Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π". J. Amer. Math. Soc. 28 (1): 199–234. arXiv:1212.4714. doi:10.1090/S0894-0347-2014-00800-6. S2CID 119140033.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015). "Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof". J. Amer. Math. Soc. 28 (1): 235–278. arXiv:1302.0282. doi:10.1090/S0894-0347-2014-00801-8. S2CID 119575364.
References
1. "陈秀雄孙崧荣获维布伦奖". University of Science and Technology of China Initiative Foundation. 2018-11-20. Retrieved 2019-04-09.
2. "Song Sun Awarded Prestigious Sloan Fellowship for Mathematics". Stony Brook University. 2014-02-18. Retrieved 2019-04-03.
3. "Song Sun". The Mathematics Genealogy Project. Retrieved 2019-04-03.
4. "Song Sun". Department of Mathematics, University of California Berkeley. Retrieved 2019-04-03.
5. "Invited Section Lectures Speakers". ICM 2018. Retrieved 2019-04-03.
6. Breakthrough Prize in Mathematics 2021
7. "2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun". American Mathematical Society. 2018-11-19. Retrieved 2019-04-09.
8. "Song Sun to receive the 2019 Oswald Veblen Prize in Geometry. Congratulations!". Department of Mathematics, University of California Berkeley. Retrieved 2019-04-03.
Recipients of the Oswald Veblen Prize in Geometry
• 1964 Christos Papakyriakopoulos
• 1964 Raoul Bott
• 1966 Stephen Smale
• 1966 Morton Brown and Barry Mazur
• 1971 Robion Kirby
• 1971 Dennis Sullivan
• 1976 William Thurston
• 1976 James Harris Simons
• 1981 Mikhail Gromov
• 1981 Shing-Tung Yau
• 1986 Michael Freedman
• 1991 Andrew Casson and Clifford Taubes
• 1996 Richard S. Hamilton and Gang Tian
• 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins
• 2004 David Gabai
• 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó
• 2010 Tobias Colding and William Minicozzi; Paul Seidel
• 2013 Ian Agol and Daniel Wise
• 2016 Fernando Codá Marques and André Neves
• 2019 Xiuxiong Chen, Simon Donaldson and Song Sun
Authority control: Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Wikipedia
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Combinatorial commutative algebra
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role.
One of the milestones in the development of the subject was Richard Stanley's 1975 proof of the Upper Bound Conjecture for simplicial spheres, which was based on earlier work of Melvin Hochster and Gerald Reisner. While the problem can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques.
A signature theorem in combinatorial commutative algebra is the characterization of h-vectors of simplicial polytopes conjectured in 1970 by Peter McMullen. Known as the g-theorem, it was proved in 1979 by Stanley (necessity of the conditions, algebraic argument) and by Louis Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question was the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim Adiprasito.
Important notions of combinatorial commutative algebra
• Square-free monomial ideal in a polynomial ring and Stanley–Reisner ring of a simplicial complex.
• Cohen–Macaulay ring.
• Monomial ring, closely related to an affine semigroup ring and to the coordinate ring of an affine toric variety.
• Algebra with a straightening law. There are several versions of those, including Hodge algebras of Corrado de Concini, David Eisenbud, and Claudio Procesi.
See also
• Algebraic combinatorics
• Polyhedral combinatorics
• Zero-divisor graph
References
A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory:
• Hochster, Melvin (1977). "Cohen–Macaulay rings, combinatorics, and simplicial complexes". Ring Theory II: Proceedings of the Second Oklahoma Conference. Lecture Notes in Pure and Applied Mathematics. Vol. 26. Dekker. pp. 171–223. ISBN 0-8247-6575-3. OCLC 610144046. Zbl 0351.13009.
The first book is a classic (first edition published in 1983):
• Stanley, Richard (1996). Combinatorics and commutative algebra. Progress in Mathematics. Vol. 41 (2nd ed.). Birkhäuser. ISBN 0-8176-3836-9. Zbl 0838.13008.
Very influential, and well written, textbook-monograph:
• Bruns, Winfried; Herzog, Jürgen (1993). Cohen–Macaulay rings. Vol. 39. Cambridge Studies in Advanced Mathematics: Cambridge University Press. ISBN 0-521-41068-1. OCLC 802912314. Zbl 0788.13005.
Additional reading:
• Villarreal, Rafael H. (2001). Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 238. Marcel Dekker. ISBN 0-8247-0524-6. Zbl 1002.13010.
• Hibi, Takayuki (1992). Algebraic combinatorics on convex polytopes. Glebe, Australia: Carslaw Publications. ISBN 1875399046. OCLC 29023080.
• Sturmfels, Bernd (1996). Gröbner bases and convex polytopes. University Lecture Series. Vol. 8. American Mathematical Society. ISBN 0-8218-0487-1. OCLC 907364245. Zbl 0856.13020.
• Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics. Springer. doi:10.1007/b105283. ISBN 978-0-387-76355-2. Zbl 1168.13001.
A recent addition to the growing literature in the field, contains exposition of current research topics:
• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. Springer. ISBN 0-387-22356-8. Zbl 1066.13001.
• Herzog, Jürgen; Hibi, Takayuki (2011). Monomial Ideals. Graduate Texts in Mathematics. Vol. 260. Springer. ISBN 978-0-85729-106-6. Zbl 1206.13001.
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Wikipedia
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\begin{document}
\title[Uniqueness theorems]{Uniqueness theorems for weighted harmonic functions\\ in the upper half-plane}
\date{\today}
\author{Anders Olofsson} \address[Anders Olofsson]{Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden} \email{[email protected]}
\author{Jens Wittsten} \address[Jens Wittsten]{Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden, and Department of Engineering, University of Bor{\aa}s, SE-501 90 Bor{\aa}s, Sweden} \email{[email protected]}
\subjclass[2010]{31A05, 35A02 (primary), 31A20, 33C05 (secondary)}
\keywords{Harmonic function, uniqueness problem, open upper half-plane, hyper\-geometric function}
\begin{abstract} We consider a class of weighted harmonic functions in the open upper half-plane known as $\alpha$-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case ($\alpha\neq0$) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case ($\alpha=0$) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series.
Our findings shed new light on the theory of harmonic functions, for which we provide uniqueness results under vanishing conditions at infinity along a) geodesics, and b) rays emanating from the origin. The geodesic uniqueness results require vanishing on two distinct geodesics which is best possible. The ray uniqueness results involves an arithmetic condition which we analyze by introducing the concept of an admissible function of angles. We show that the arithmetic condition is to the point and that the set of admissible functions of angles is minimal with respect to a natural partial order. \end{abstract}
\maketitle
\section{Introduction}
Let $\mathbb H$ be the open upper half-plane in the complex plane $\mathbb C$ and consider the weighted Laplace differential operator \begin{equation}\label{HalphaLaplacian} \Delta_{\mathbb H;\alpha,z}=\partial_z (\operatorname{Im} z)^{-\alpha}\bar\partial_z,\quad z\in\mathbb H, \end{equation} where $\partial$ and $\bar\partial$ are the usual complex partial derivatives and $\alpha>-1$. Here $\operatorname{Im} z$ is the imaginary part of the complex number $z\in\mathbb C$ and the function $w_{\mathbb H;\alpha}(z)=(\operatorname{Im} z)^{\alpha}$ for $z\in\mathbb H$ has an interpretation of a standard weight function for $\mathbb H$. The study of differential operators of the form \eqref{HalphaLaplacian} is suggested by a classical paper of Paul Garabedian \cite{Garabedian}.
We refer to the differential operator $\Delta_{\mathbb H;\alpha}$ in \eqref{HalphaLaplacian} as the $\alpha$-Laplacian for $\mathbb H$. An $\alpha$-harmonic function $u$ in $\mathbb H$ is a twice continuously differentiable function $u$ in $\mathbb H$ (in symbols: $u\in C^2(\mathbb H)$) such that $$ \Delta_{\mathbb H;\alpha}u=0\quad \text{in}\ \mathbb H. $$ Notice that $\Delta_{\mathbb H;0}=\partial\bar\partial$ is the usual Laplacian and that a $0$-harmonic function in $\mathbb H$ is a harmonic function in $\mathbb H$ in the usual sense. The restriction $\alpha>-1$ on the weight parameter ensures good supply of $\alpha$-harmonic functions with well-behaved boundary values.
In this paper we address the uniqueness problem for $\alpha$-harmonic functions in $\mathbb H$, that is, we wish to characterize the identically zero function $u\equiv 0$ within the class of $\alpha$-harmonic functions in $\mathbb H$. A natural condition is that of a vanishing boundary value \begin{equation}\label{vanishingDirichletbdrycondition} \lim_{\mathbb H\ni z\to x}u(z)=0,\quad x\in\mathbb R, \end{equation} on the real line $\mathbb R$. A condition of this type is often referred to as a vanishing Dirichlet boundary value. There are plenty of non-trivial $\alpha$-harmonic functions in $\mathbb H$ satisfying \eqref{vanishingDirichletbdrycondition}. A simple such example is the function $$ u(z)=(\operatorname{Im} z)^{\alpha+1},\quad z\in\mathbb H, $$ which is $\alpha$-harmonic in $\mathbb H$ and satisfies \eqref{vanishingDirichletbdrycondition}. In order to obtain satisfactory results on the uniqueness problem above it is thus natural to complement \eqref{vanishingDirichletbdrycondition} with some condition(s) taking into account the behavior of the $\alpha$-harmonic function $u$ at infinity. Here we think of the point at infinity $\infty$ as a boundary point of $\mathbb H$ in the extended complex plane $\mathbb C_\infty=\mathbb C\cup\{\infty\}$. In the case of usual harmonic functions in $\mathbb H$ ($\alpha=0$) this problem setup is classical. We mention here a recent contribution by Carlsson and Wittsten \cite{carlsson2016dirichlet} concerned with a uniqueness result in this flavor for $\alpha$-harmonic functions in $\mathbb H$ with $\alpha>-1$. This result of Carlsson and Wittsten has served as a guidance for the present investigations.
We say that a function $u$ in $\mathbb H$ is of temperate growth at infinity if it satisfies an estimate of the form $$ \lvert u(z)\rvert \leq C(\lvert z\rvert^2/\operatorname{Im}(z))^N $$ for $z\in\mathbb H$ with $\lvert z\rvert>R$, where $C$, $R$ and $N$ are positive constants. We refer to the parameter $N$ as an order of growth at infinity for the function $u$.
A first main result concerns $\alpha$-harmonic functions $u$ in $\mathbb H$ that are of temperate growth at infinity. We prove that such a function $u$ satisfies \eqref{vanishingDirichletbdrycondition} if and only if it has the form \begin{equation}\label{alphaobstructionfunction} u(z)=\sum_{k=0}^n c_k(\operatorname{Im} z)^{\alpha+1} p_{k,\alpha}(z),\quad z\in\mathbb H, \end{equation} for some $n\in\mathbb N=\{0,1,2,\dots\}$ and $c_0,\dots,c_n\in\mathbb C$, where \begin{equation*} p_{k,\alpha}(z)=\sum_{j=0}^k\frac{(\alpha+1)_j}{j!}z^{k-j}\bar z^j \end{equation*} for $k=0,1,\dots$ (see Corollary \ref{Valphacharacterization}). We point out that the polynomials $p_{k,\alpha}$ are naturally derived from the classical binomial series. We also establish \eqref{alphaobstructionfunction} under a weaker distributional version of \eqref{vanishingDirichletbdrycondition} (see Theorem \ref{alphaobstructionclassdistributionalversion}).
We denote by $\mathcal{V}_\alpha$ the set of all functions $u$ of the form \eqref{alphaobstructionfunction} for some $n\in\mathbb N$ and $c_0,\dots,c_n\in\mathbb C$. The set $\mathcal{V}_\alpha$ is naturally filtered in the sense that $$ \mathcal{V}_\alpha= \cup_{n=0}^\infty\mathcal{V}_{\alpha,n}, $$ where $\mathcal{V}_{\alpha,n}$ is the set of all functions of the form \eqref{alphaobstructionfunction} for some $c_0,\dots,c_n\in\mathbb C$. The set $\mathcal{V}_{\alpha,n}$ has a natural structure of a complex vector space of finite dimension $n+1$. The space $\mathcal{V}_{\alpha,n}$ admits a natural description within the class $\mathcal{V}_{\alpha}$ using order of growth at infinity (see Lemma \ref{coefficientlemma} and Proposition \ref{Vanfromrelaxedgrowth}). The finer study of behaviors at infinity of functions in the class $\mathcal{V}_{\alpha}$ depends on the parameter $\alpha>-1$ and divides naturally into cases whether $\alpha\neq0$ or $\alpha=0$. The reason behind this dichotomy is a different geometry of zero sets for the polynomials $p_{k,\alpha}$.
We consider next the problem of characterizing the null function in the class $\mathcal{V}_\alpha$ using a vanishing condition at infinity. For parameters $\alpha>-1$ with $\alpha\neq0$, we prove that if $u\in\mathcal{V}_\alpha$ is such that \begin{equation}\label{sequencevanishingalphaneq0} \lim_{j\to\infty} \frac{u(z_j)}{(\operatorname{Im}(z_j))^{\alpha+1}}=0 \end{equation} for some sequence $\{z_j\}$ in $\mathbb H$ with $z_j \to\infty$ in $\mathbb C_\infty$ as $j\to\infty$, then $u(z)=0$ for all $z\in\mathbb H$ (see Theorem \ref{Vauniquenessaneq0}). We emphasize the big freedom allowed in the choice of sequence $\{z_j\}$ in \eqref{sequencevanishingalphaneq0} above. This analysis leads to the following highly flexible uniqueness result for $\alpha$-harmonic functions in the case $\alpha\neq0$ as well as a distributional version thereof (see Theorem \ref{thm:distributionaluniquenessaneq0}).
\begin{theorem}\label{introuniquenessthm} Let $\alpha>-1$ and $\alpha\ne0$. Let $u$ be an $\alpha$-harmonic function in $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item\label{classicalvanishingrealline} Assume that \eqref{vanishingDirichletbdrycondition} holds. \item Assume that there exists a sequence $\{z_j\}$ in $\mathbb H$ with $z_j \to\infty$ in $\mathbb C_\infty$ as $j\to\infty$ such that \eqref{sequencevanishingalphaneq0} holds. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
We now turn our attention to the case $\alpha=0$ of usual harmonic functions in $\mathbb H$. A function $u$ belongs to the class $\mathcal{V}_0$ if and only if it has the form $$ u(z)=\sum_{k=1}^n c_k\operatorname{Im}(z^k),\quad z\in\mathbb H, $$ for some $n\in \mathbb Z^+=\{1,2,3,\dots\}$ and $c_1,\dots,c_n\in\mathbb C$ (see Proposition \ref{harmonicobstructionclass}). Observe that the harmonic polynomial $$ u_k(z)= \operatorname{Im}(z^k),\quad z\in\mathbb C, $$ vanishes on a union of $k$ lines passing through the origin. Notice also that the zero set of $u_k$ intersects the unit circle at the $2k$-th roots of unity. These examples make evident that the flexible uniqueness results for $\alpha$-harmonic functions in $\mathbb H$ with $\alpha\neq0$ are no longer true when $\alpha=0$ (compare with Theorem \ref{introuniquenessthm} above).
In order to obtain satisfactory uniqueness results for usual harmonic functions in $\mathbb H$ we shall restrict condition \eqref{sequencevanishingalphaneq0} to suitable classes of curves. We consider two such classes of curves, namely, geo\-desics in $\mathbb H$ and rays in $\mathbb H$ emanating from the origin.
By a geodesic in $\mathbb H$ we understand a ray in $\mathbb H$ which is parallel to the imaginary axis. We prove that if $u\in\mathcal{V}_0$ is such that \begin{equation*} \lim_{y\to+\infty}u(x+iy)/y=0 \end{equation*} for $x=x_j\in\mathbb R$ ($j=1,2$) with $x_1\neq x_2$, then $u(z)=0$ for all $z\in\mathbb H$ (see Theorem \ref{V0uniquenessgeodesic}). This analysis leads to corresponding uniqueness results for usual harmonic functions in $\mathbb H$ (see Theorems \ref{classicaluniquenessthm2geodesics} and \ref{distributionaluniquenessthm2geodesics}). We emphasize that those geodesic uniqueness results require vanishing on two ($2$) distinct geodesics which is best possible. Together with the success results for $\alpha\neq 0$, this marks a significant advancement from an earlier uniqueness result of Carlsson and Wittsten \cite[Corollary 1.9]{carlsson2016dirichlet} for $\alpha$-harmonic functions which required vanishing on an interval of geodesics.
By a ray in $\mathbb H$ emanating from a point $a\in\mathbb R$ we understand a set of the form $\{a+te^{i\theta}:\ t>0\}$, where $0<\theta<\pi$. We shall restrict our attention to rays emanating from the origin ($a=0$). In view of translation invariance of the class of harmonic functions this restriction is minor. In order to discuss vanishing of functions along rays we introduce a notion of admissible function of angles which is a function element $(E,\eta)$ with $E\subset(0,\pi)$ and $\eta:E\to\mathbb Z^+$ having the property that for every $k\in\mathbb Z^+$ there exists $\theta\in E$ such that $\sin(k\theta)\neq0$ and $k\geq \eta(\theta)$ (see Definition \ref{dfnadmissiblefcnangles}). We prove that if $u\in\mathcal{V}_0$ and there exists an admissible function of angles $(E,\eta)$ such that $$ \lim_{t\to+\infty}u(te^{i\theta})/t^{\eta(\theta)}=0 $$ for every $\theta\in E$, then $u(z)=0$ for all $z\in\mathbb H$ (see Theorem \ref{V0uniquenessray}). This analysis leads to corresponding uniqueness results for usual harmonic functions in $\mathbb H$ (see Theorems \ref{classicalfcnofanglesvanishing} and \ref{fcnofanglesvanishing}).
The notion of admissible function of angles involves an arithmetic element. In order to illuminate this fact we mention the following result.
\begin{theorem}\label{classicalgenericrayuniquenessresult} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item Assume that \eqref{vanishingDirichletbdrycondition} holds. \item Assume that \begin{equation}\label{rayvanishingcondition} \lim_{t\to+\infty}u(te^{i\theta})/t=0 \end{equation} for some $0<\theta<\pi$ which is not a rational multiple of $\pi$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
The examples $u_k$ above show that the arithmetic condition on $\theta$ in \eqref{rayvanishingcondition} is to the point. Behind Theorem \ref{classicalgenericrayuniquenessresult} lies a construction of a one point admissible function of angles $(E,\eta)$, where $E=\{\theta\}$ and $\eta(\theta)=1$. The arithmetic condition on $\theta$ in \eqref{rayvanishingcondition} ensures that this latter function element $(E,\eta)$ is an admissible function of angles. More elaborate constructions of admissible functions of angles lead to corresponding uniqueness results for usual harmonic functions in $\mathbb H$ (see Corollaries \ref{genericrayvanishing}, \ref{rationalrayvanishing}, \ref{testthm} and \ref{testthm2}).
In the final section we provide constructions of admissible functions of angles (see Theorems \ref{constructionfoainfinite} and \ref{constructionfoafinite}). The set $\mathcal{A}$ of admissible functions of angles is structured by a natural partial order. We show that every element in $\mathcal{A}$ has a lower bound which is minimal (see Theorem \ref{lowerboundfoa} and Lemma \ref{cfoaminimal}). Our constructions of admissible functions of angles yield precisely the minimal elements in $\mathcal{A}$ (see Theorem \ref{minimalelementsA}).
There is a substantial literature on boundary uniqueness problems for (sub-)harmonic functions with contributions of Wolf \cite{Wolf}, Shapiro \cite{ShapiroVL}, Dahlberg \cite{Dahlberg}, Berman and Cohn \cite{BermanCohn} and Borichev with collaborators \cite{BorichevCT,BorichevT} to name a few. As far as usual harmonic functions in $\mathbb H$ are concerned our uniqueness results supersede an earlier uniqueness result of Siegel and Talvila \cite[Corollary 3.1]{SiegelTalvila}.
The $\alpha$-Laplacian (or rather its symmetric part) is also related to the Laplace-Beltrami equation in the Riemannian space defined by the metric $$ ds^2=x_n^{-\alpha/(n-2)}\sum_1^n dx_i^2,\quad n>2, $$ as studied by Weinstein \cite{Weinstein} and Huber \cite{Huber}, among others. For historic reasons, solutions of said Laplace-Beltrami equation are referred to as generalized axially symmetric potentials. For a discussion on this connection and more recent applications in this direction we refer to Wittsten \cite{Wittsten}. Another related area of interest is the recent study of higher order Laplacians initiated by Borichev and Hedenmalm \cite{BH}.
The present paper is rooted in previous investigations. In an earlier paper \cite{olofsson2013poisson} we initiated a theory of $\alpha$-harmonic functions in the open unit disc $\mathbb D$ in the complex plane. A function $u\in C^2(\mathbb D)$ is called $\alpha$-harmonic in $\mathbb D$ if $\Delta_{\mathbb D;\alpha}u=0$ in $\mathbb D$, where $$ \Delta_{\mathbb D;\alpha,z}= \partial_z(1-\lvert z\rvert^2)^{-\alpha}\bar\partial _z,\quad z\in\mathbb D, $$ is the $\alpha$-Laplacian for $\mathbb D$ and $\alpha\in\mathbb R$. A main concern in this theory is the representation of an $\alpha$-harmonic function in $\mathbb D$ as a Poisson integral $u=P_\alpha[f]$ in $\mathbb D$ with respect to the kernel \begin{equation} \label{aPoissonkernel} P_\alpha(z)=\frac{(1-\lvert z\rvert^2)^{\alpha+1}}{(1-z)(1-\bar z)^{\alpha+1}}, \quad z\in\mathbb D. \end{equation} We refer to the function $P_\alpha$ as the $\alpha$-harmonic Poisson kernel for $\mathbb D$.
We prove that a Poisson integral representation $u=P_\alpha[f]$ with $f$ a distribution on the unit circle $\mathbb T=\partial\mathbb D$ exists if and only if $u$ is $\alpha$-harmonic in $\mathbb D$, $u$ has temperate growth in $\mathbb D$ and a certain spectral condition is satisfied (see Theorem \ref{characterizationPIrepresentation}). This latter spectral condition is automatically satisfied when $\alpha$ is not a negative integer (see Corollary \ref{genericcharacterizationPIrepresentation}). For $\alpha>-1$, the distribution $f\in\mathscr{D}'(\mathbb T)$ has a natural interpretation as a (distributional) boundary limit of the function $u$. On the other hand, for $\alpha\leq-1$, existence of a distributional boundary limit of an $\alpha$-harmonic function $u$ in $\mathbb D$, forces $u$ to be analytic in $\mathbb D$ (see Theorem \ref{boundarybehavioralphaleq-1}). We thus establish Poisson integral representations $u=P_\alpha[f]$ with $f\in\mathscr{D}'(\mathbb T)$ in situations where a distributional boundary value of $u$ is non-existent. To overcome those difficulties we resort to a study of related hyper\-geometric functions, found in Section \ref{sectionhypergeometricfunction}.
A link between the settings of the upper half-plane $\mathbb H$ and the unit disc $\mathbb D$ is provided by a certain conformal invariance property of $\alpha$-harmonic functions which was recently studied by the first author \cite{olofsson2017on}. Let $\varphi:\mathbb D\to\mathbb H$ be a bi\-holomorphic map. For a function $u$ in $\mathbb H$ we consider the weighted pull-back \begin{equation}\label{weightedpullback} v(z)=u_{\varphi,\alpha}(z)=\varphi'(z)^{-\alpha/2} u(\varphi(z)),\quad z\in\mathbb D, \end{equation} of $u$ by $\varphi$ with respect to the parameter $\alpha$. Here the power in \eqref{weightedpullback} is defined in the usual way using a logarithm of $\varphi'$ in $\mathbb D$. We shall use the fact that the function $v$ is $\alpha$-harmonic in $\mathbb D$ if and only if the function $u$ is $\alpha$-harmonic in $\mathbb H$. This latter fact follows easily from \cite[Theorem 1.1]{olofsson2017on}. We refer to Geller \cite{Geller} or Ahern et al. \cite{ABC,AC} for earlier results.
Let us return to an $\alpha$-harmonic function $u$ in $\mathbb H$ satisfying some appropriate conditions, notably \eqref{vanishingDirichletbdrycondition} and temperate growth at infinity. We consider a weighted pull-back $v$ of $u$ of the form \eqref{weightedpullback}. The function $v$ is $\alpha$-harmonic in $\mathbb D$ and we propose to study this function by means of its Poisson integral representation $v=P_\alpha[f]$ in $\mathbb D$. An ambiguity lies in the choice of bi\-holomorphic map $\varphi:\mathbb D\to\mathbb H$ which we chose as the M\"obius transformation \begin{equation*} \varphi(z)=i\frac{1+z}{1-z}. \end{equation*} Notice that $\varphi(0)=i$ and $\varphi(1)=\infty$.
From \eqref{vanishingDirichletbdrycondition} we have that the boundary value $f$ for $v$ vanishes on the set $\mathbb T\setminus\{1\}$. Standard distribution theory then dictates that the distribution $f\in\mathscr{D}'(\mathbb T)$ is a finite linear combination of derivatives of a Dirac mass $\delta_1$ located at the point $1$ on $\mathbb T$: $f=\sum_{k=0}^n c_k\delta_1^{(k)}$ in $\mathscr{D}'(\mathbb T)$ (see H\"ormander \cite[Theorem 2.3.4]{Hormander}). The Poisson integral $P_\alpha[\delta_1']$ is naturally interpreted as an angular derivative $iAP_\alpha$ of the Poisson kernel $P_\alpha$, and by iteration we find that $v=\sum_{k=0}^n c_k (iA)^kP_\alpha$ (see Corollary \ref{angularderivativePa}). In Section \ref{sectionangularderivatives}, those angular derivatives $(iA)^k P_\alpha$ of $P_\alpha$ are carefully investigated using the M\"obius transformation $\varphi$ above, and in Section \ref{representationthms} this part of the analysis culminates in the proof of the representation formula \eqref{alphaobstructionfunction} (see Theorem \ref{alphaobstructionclass}).
\section{Series expansion of \texorpdfstring{$\alpha$}{alpha}-harmonic functions in \texorpdfstring{$\mathbb D$}{D}}\label{sectionhypergeometricfunction}
In this section we revisit the series expansion of $\alpha$-harmonic functions in $\mathbb D$. Of particular concern is a characterization of temperate growth of an $\alpha$-harmonic function in $\mathbb D$ in terms of polynomial growth of coefficients (see Theorem \ref{polynomialgrowthcoefficients}). The proof of this latter result depends on properties of related hyper\-geometric functions.
For $a\in\mathbb C$, we set $(a)_0=1$ and $$ (a)_k=\prod_{j=0}^{k-1}(a+j) $$ for $k=1,2,\dots$. The numbers $(a)_k$ are known as Pochhammer symbols. Notice that $(a)_k=\Gamma(a+k)/\Gamma(a)$ for $k\in\mathbb N$, where $\Gamma$ is the standard Gamma function.
The hyper\-geometric function is the function defined by the power series expansion \begin{equation}\label{hypergeometricfunction} F(a,b;c;z)= \sum_{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k} \frac{z^k}{k!}, \quad z\in\mathbb D, \end{equation} for parameters $a,b,c\in\mathbb C$ with $c\neq 0,-1,-2,\dots$. Convergence in \eqref{hypergeometricfunction} follows by the standard ratio test.
Recall also the classical binomial series \begin{equation}\label{binomialseries} (1-z)^{-a}=\sum_{k=0}^\infty \frac{(a)_k}{k!} z^k, \quad z\in\mathbb D, \end{equation} for $a\in\mathbb C$. Notice that $$ F(a,b;b;z)=(1-z)^{-a}, $$ which follows from \eqref{hypergeometricfunction} and \eqref{binomialseries}.
We take as our starting point a certain series expansion of $\alpha$-harmonic functions in $\mathbb D$. It is known that a function $u$ is $\alpha$-harmonic in $\mathbb D$ if and only if it has the form \begin{equation}\label{generalizedpowerseries} u(z)=\sum_{k=0}^\infty c_{k} z^k + \sum_{k=1}^\infty c_{-k}F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k,\quad z\in\mathbb D, \end{equation} for some sequence $\{c_k\}_{k=-\infty}^\infty$ of complex numbers such that \begin{equation}\label{coefficientgrowth} \limsup_{\lvert k\rvert\to\infty}\lvert c_k\rvert^{1/\lvert k\rvert}\leq 1, \end{equation} where $F$ is the hyper\-geometric function \eqref{hypergeometricfunction} (see \cite[Theorem 1.2]{olofsson2013poisson}). Condition \eqref{coefficientgrowth} ensures that the series expansion \eqref{generalizedpowerseries} is absolutely convergent in the space $C^\infty(\mathbb D)$ of in\-definitely differentiable functions in $\mathbb D$. As a consequence we have that $u\in C^\infty(\mathbb D)$. We refer to Klintborg and Olofsson \cite{KO} for an updated account on these matters.
The property $F(a,b;c;0)=1$ of the hyper\-geometric function \eqref{hypergeometricfunction} yields a normalization of the expansion \eqref{generalizedpowerseries}. As a consequence we have the coefficient formulas $$ c_k=\partial^k u(0)/k! \quad \text{and}\quad c_{-k}=\bar\partial^k u(0)/k! $$ for $k\in\mathbb N$, where $u$ is as in \eqref{generalizedpowerseries} (see \cite[Theorem 5.3]{KO}). In particular, an $\alpha$-harmonic function in $\mathbb D$ is uniquely determined by its germ at the origin.
As an example of a series expansion of the form \eqref{generalizedpowerseries} we mention that of the $\alpha$-harmonic Poisson kernel \begin{equation}\label{Paseriesexpansion} P_\alpha(z)= \sum_{k=0}^\infty z^k + \sum_{k=1}^\infty \frac{(\alpha+1)_k}{k!} F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k \end{equation} for $z\in\mathbb D$ (see \cite[Theorem 6.3]{KO}).
We shall make use of a classical result known as Euler's integral formula for the hyper\-geometric function. This result says that \begin{equation}\label{EulerintegralF} F(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a} \, dt \end{equation} for $z\in\mathbb D$ when $\operatorname{Re} c>\operatorname{Re} b>0$, where $\Gamma$ is the standard Gamma function (see \cite[Theorem 2.2.1]{AAR}).
In particular, from \eqref{EulerintegralF} we have that \begin{equation}\label{Ffactorintegralformula} F(-\alpha,k;k+1;x)= k\int_0^1t^{k-1}(1-xt)^\alpha\, dt,\quad 0\leq x<1, \end{equation} for $k\in \mathbb Z^+$ and $\alpha\in\mathbb R$. From this latter formula we see that the function $F(-\alpha,k;k+1;\cdot)$ is non\-negative on the interval $[0,1)$. Furthermore, the function $F(-\alpha,k;k+1;\cdot)$ is de\-creasing on the interval $[0,1)$ if $\alpha\geq0$ and, similarly, the function $F(-\alpha,k;k+1;\cdot)$ is in\-creasing on the interval $[0,1)$ if $\alpha\leq0$.
A passage to the limit in \eqref{Ffactorintegralformula} shows that \begin{equation}\label{Gausssummationformula} \lim_{x\to1} F(-\alpha,k;k+1;x)= \frac{\Gamma(k+1)\Gamma(\alpha+1)}{\Gamma(k+\alpha+1)} \end{equation} for $\alpha>-1$, where we have used a standard formula for the Beta function. When $\alpha\leq-1$, the quantity $F(-\alpha,k;k+1;x)$ diverges to $+\infty$ as $x\to1$.
We shall need some more detailed estimates of the hyper\-geometric functions appearing in \eqref{generalizedpowerseries}.
\begin{lemma}\label{Festimationparameter-1} Let $k\in\mathbb Z^+$. Then $$ F(1,k;k+1;x)\leq\frac{k}{x}\log\Big(\frac{1}{1-x}\Big) $$ for $0<x<1$. \end{lemma}
\begin{proof} From \eqref{Ffactorintegralformula} we have that \begin{equation*} F(1,k;k+1;x)= k\int_0^1 \frac{t^{k-1}}{1-xt}\, dt \end{equation*} for $0\leq x<1$. An integration by parts shows that $$ F(1,k;k+1;x)=\frac{k}{x}\log\Big(\frac{1}{1-x}\Big) +\frac{k(k-1)}{x}\int_0^1 t^{k-2}\log(1-xt)\, dt $$ for $0< x<1$. Observe that the logarithm in the right\-most integral is negative. This yields the conclusion of the lemma. \end{proof}
We shall use also another result of Euler which says that \begin{equation}\label{Eulerformula} F(a,b;c;z)=(1-z)^{c-a-b}F(c-a,c-b;c;z) \end{equation} for $z\in\mathbb D$ (see \cite[Theorem 2.2.5]{AAR}).
\begin{lemma}\label{Festimationparameter<-1} Let $\alpha<-1$ and $k\in\mathbb Z^+$. Then $$ F(-\alpha,k;k+1;x)\leq\max\Big(1,-\frac{k}{\alpha+1}\Big)(1-x)^{\alpha+1} $$ for $0\leq x<1$. \end{lemma}
\begin{proof} We first apply \eqref{Eulerformula} to see that \begin{equation}\label{factorizationformula} F(-\alpha,k;k+1;x)=(1-x)^{\alpha+1}F(k+\alpha+1,1;k+1;x) \end{equation} for $0\leq x<1$. We shall divide into cases depending on whether $k+\alpha+1>0$ or $k+\alpha+1\leq0$.
Assume first that $k+\alpha+1\leq0$. By \eqref{EulerintegralF} we have that $$ F(k+\alpha+1,1;k+1;x) =k\int_0^1 (1-t)^{k-1} (1-xt)^{-(k+\alpha+1)}\, dt $$ for $0\leq x<1$. Since $k+\alpha+1\leq0$, we have that this latter hyper\-geometric function $F(k+\alpha+1,1;k+1;\cdot)$ is decreasing on $[0,1)$. Therefore $F(k+\alpha+1,1;k+1;x)\leq1$ for $0\leq x<1$. In view of \eqref{factorizationformula} this yields the conclusion of the lemma for $k+\alpha+1\leq0$.
We next assume that $k+\alpha+1>0$. By symmetry and \eqref{EulerintegralF} we have that \begin{align*} F(k+\alpha+1,1;k+1;x) &=F(1,k+\alpha+1;k+1;x)\\ & =\frac{\Gamma(k+1)}{\Gamma(k+\alpha+1)\Gamma(-\alpha)} \int_0^1 t^{k+\alpha}(1-t)^{-(\alpha+1)}\frac{1}{1-xt}\, dt \end{align*} for $0\leq x<1$. By monotonicity we have that \begin{align*} F(k+\alpha+1,1;k+1;x)&\leq \frac{\Gamma(k+1)}{\Gamma(k+\alpha+1)\Gamma(-\alpha)} \int_0^1 t^{k+\alpha}(1-t)^{-(\alpha+1)-1}\, dt\\ &= \frac{\Gamma(k+1) \Gamma(-(\alpha+1))}{\Gamma(-\alpha)\Gamma(k)} =-\frac{k}{\alpha+1} \end{align*} for $0\leq x<1$, where the last two equalities follows by standard formulas for the Beta and Gamma functions. In view of \eqref{factorizationformula} this yields the conclusion of the lemma for $k+\alpha+1>0$. \end{proof}
We say that a function $u$ in $\mathbb D$ is of temperate growth in $\mathbb D$ if it satisfies an estimate of the form \begin{equation}\label{utemperategrowthdisc} \lvert u(z)\rvert \leq C(1-\lvert z\rvert^2)^{-N}, \quad z\in\mathbb D, \end{equation} for some positive constants $C$ and $N$.
\begin{theorem}\label{polynomialgrowthcoefficients} Let $\alpha\in\mathbb R$. Let $u$ be an $\alpha$-harmonic function in $\mathbb D$ and consider the expansion \eqref{generalizedpowerseries}. Then the function $u$ is of temperate growth in $\mathbb D$ if and only if the sequence of coefficients $\{c_k\}_{k=-\infty}^\infty$ in \eqref{generalizedpowerseries} has at most polynomial growth. \end{theorem}
\begin{proof} Assume that $u$ has temperate growth in $\mathbb D$. We first show that the coefficient $c_k$ has at most polynomial growth as $k\to+\infty$. Let $k\in\mathbb N$ and $0<r<1$. From \eqref{generalizedpowerseries} we have that $$ c_k r^k=\frac{1}{2\pi}\int_\mathbb T u(re^{i\theta})e^{-ik\theta}\, d\theta. $$ From the triangle inequality and \eqref{utemperategrowthdisc} we have that $$ \lvert c_k\rvert r^k\leq C/(1-r^2)^N\leq C/(1-r)^N $$ for $k\in\mathbb N$ and $0<r<1$, where $C$ is as in \eqref{utemperategrowthdisc}. Choosing $r=1-1/k$ with $k$ big in this latter in\-equality, we see that $\lvert c_k\rvert\leq C'k^N$ as $k\to +\infty$.
We next show that the coefficient $c_k$ has at most polynomial growth as $k\to-\infty$. Let $k\in\mathbb Z^-=\mathbb Z\setminus\mathbb N$ and $0<r<1$. From \eqref{generalizedpowerseries} we have that \begin{equation}\label{coefficientformula} c_k F(-\alpha,\lvert k\rvert; \lvert k\rvert+1 ;r^2) r^{\lvert k\rvert} =\frac{1}{2\pi}\int_\mathbb T u(re^{i\theta})e^{-ik\theta}\, d\theta. \end{equation} Let us first consider the case $\alpha\leq0$. Since the function $F(-\alpha,\lvert k\rvert; \lvert k\rvert+1 ;\cdot)$ is increasing on $[0,1)$, we have from \eqref{coefficientformula} and the triangle inequality that $$ \lvert c_k\rvert r^{\lvert k\rvert}\leq C/(1-r)^N, $$ where $C$ is as in \eqref{utemperategrowthdisc}. Choosing $r=1-1/\lvert k\rvert$ in this latter inequality we see that $\lvert c_k\rvert\leq C'\lvert k\rvert^N$ as $k\to -\infty$.
Let us now consider the case $\alpha\geq0$. Since the function $F(-\alpha,\lvert k\rvert; \lvert k\rvert+1 ;\cdot)$ is decreasing on $[0,1)$, we have from \eqref{coefficientformula} and the triangle inequality that \begin{equation}\label{prelcoefficientcontrol} \lvert c_k\rvert \frac{\Gamma(\lvert k\rvert+1)\Gamma(\alpha+1)}{\Gamma(\lvert k\rvert+\alpha+1)}r^{\lvert k\rvert} \leq C/(1-r)^N, \end{equation} where we have used \eqref{Gausssummationformula} and $C$ is as in \eqref{utemperategrowthdisc}. Stirling's formula ensures that a quotient of Gamma functions $\Gamma(x)/\Gamma(x+\alpha)$ behaves asymptotically as $1/x^\alpha$ when $x\to+\infty$ (\cite[Section 1.4]{AAR}). Choosing $r=1-1/\lvert k\rvert$ in \eqref{prelcoefficientcontrol} we see that $\lvert c_k\rvert\leq C'\lvert k\rvert^{N+\alpha}$ as $k\to -\infty$.
Assume now that the sequence of coefficients $\{c_k\}_{k=-\infty}^\infty$ in \eqref{generalizedpowerseries} has at most polynomial polynomial growth, that is, $\lvert c_k\rvert\leq C(1+\lvert k\rvert)^N$ for $k\in\mathbb Z$, where $N\geq0$. We shall show that the function $u$ has temperate growth in $\mathbb D$. We consider first the left\-most sum $$ f(z)= \sum_{k=0}^\infty c_{k} z^k, \quad z\in\mathbb D, $$ appearing in \eqref{generalizedpowerseries}. Observe that $$ (1+k)^N\leq \frac{(k+N)!}{k!}= N! \frac{1}{k!} \prod_{j=1}^k(N+j) = N!\frac{(N+1)_k}{k!} $$ for $k\in\mathbb N$. By the triangle inequality we have that $$ \lvert f(z)\rvert\leq C \sum_{k=0}^\infty (1+k)^N \lvert z\rvert^k \leq C N! \sum_{k=0}^\infty \frac{(N+1)_k}{k!} \lvert z\rvert^k = C N! (1-\lvert z\rvert)^{-(N+1)} $$ for $z\in\mathbb D$, where in the last equality we have used \eqref{binomialseries}. This proves that the function $f$ above has temperate growth in $\mathbb D$.
We consider next the function $$ g(z)=\sum_{k=1}^\infty c_{-k}F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k, \quad z\in\mathbb D, $$ appearing in \eqref{generalizedpowerseries}. Assume first that $\alpha\geq0$. Then the function $F(-\alpha,k;k+1;\cdot)$ is decreasing on $[0,1)$. From the triangle inequality we have that $$ \lvert g(z)\rvert\leq \sum_{k=1}^\infty \lvert c_{-k}\rvert \lvert z\rvert^k \leq C \sum_{k=1}^\infty (1+k)^N \lvert z\rvert^k \leq C N! (1-\lvert z\rvert)^{-(N+1)} $$ for $z\in\mathbb D$, where the last two in\-equalities follows as in the previous paragraph. This proves that the function $g$ above has temperate growth in $\mathbb D$ if $\alpha\geq0$.
Assume next that $-1<\alpha\leq0$. In this case the function $F(-\alpha,k;k+1;\cdot)$ is increasing on $[0,1)$. By the triangle inequality and \eqref{Gausssummationformula} we have that $$ \lvert g(z)\rvert\leq \sum_{k=1}^\infty \lvert c_{-k}\rvert \frac{\Gamma(k+1)\Gamma(\alpha+1)}{\Gamma(k+\alpha+1)} \lvert z\rvert^k $$ for $z\in\mathbb D$. Since $\Gamma(x)/\Gamma(x+\alpha)$ behaves asymptotically as $1/x^\alpha$ when $x\to+\infty$ we deduce as above that the function $g$ has temperate growth in $\mathbb D$ if $-1<\alpha\leq0$.
Assume next that $\alpha=-1$. By the triangle inequality and Lemma \ref{Festimationparameter-1} we have that $$ \lvert g(z)\rvert\leq \sum_{k=1}^\infty \lvert c_{-k}\rvert \frac{k}{\lvert z\rvert^2}\log\Big(\frac{1}{1-\lvert z\rvert^2}\Big) \lvert z\rvert^k\leq \frac{C}{\lvert z\rvert^2}\log\Big(\frac{1}{1-\lvert z\rvert^2}\Big) \sum_{k=1}^\infty (1+k)^{N+1} \lvert z\rvert^k $$ for $z\in\mathbb D$. We can now deduce as above that the function $g$ has temperate growth in $\mathbb D$ if $\alpha=-1$.
Assume finally that $\alpha<-1$. By the triangle inequality and Lemma \ref{Festimationparameter<-1} we have that $$ \lvert g(z)\rvert\leq \sum_{k=1}^\infty \lvert c_{-k}\rvert kC(\alpha) (1-\lvert z\rvert^2)^{\alpha+1} \lvert z\rvert^k\leq C C(\alpha) (1-\lvert z\rvert^2)^{\alpha+1} \sum_{k=1}^\infty (1+k)^{N+1} \lvert z\rvert^k $$ for $z\in\mathbb D$, where $C(\alpha)=\max(1,-1/(\alpha+1))$. We can then once again deduce as above that the function $g$ has temperate growth in $\mathbb D$ if $\alpha<-1$. \end{proof}
The case division in the proof of Theorem \ref{polynomialgrowthcoefficients} depends on different behaviors of the hyper\-geometric functions appearing in \eqref{generalizedpowerseries}. One can notice that the quantity $F(-\alpha,k;k+1;x)$ is decreasing in $\alpha\in\mathbb R$. This observation leads to a shorter proof of Theorem \ref{polynomialgrowthcoefficients} with less precision. An earlier result can be found in Olofsson \cite[Proposition 3.2]{O14}.
\section{Poisson integrals of distributions}\label{sectionpoissonintegrals}
A purpose of this section is to further develop a theory of Poisson integral representations of $\alpha$-harmonic functions in $\mathbb D$. Of particular interest is a characterization of Poisson integrals of distributions on $\mathbb T$ (see Theorem \ref{characterizationPIrepresentation} and Corollary \ref{genericcharacterizationPIrepresentation}). Along the way we introduce some notation needed later.
We denote by $\mathscr{D}'(\mathbb T)$ the space of distributions on $\mathbb T$. An integrable function $f\in L^1(\mathbb T)$ on $\mathbb T$ is identified with the distribution $$ \langle f,\varphi\rangle =\frac{1}{2\pi}\int_\mathbb T f(e^{i\theta})\varphi(e^{i\theta})\, d\theta, \quad \varphi\in C^\infty(\mathbb T), $$ where $C^\infty(\mathbb T)$ is the space of indefinitely differentiable test functions on $\mathbb T$. The space $\mathscr{D}'(\mathbb T)$ is topologized in the usual way using the semi-norms $$ \mathscr{D}'(\mathbb T)\ni f\mapsto \lvert \langle f,\varphi\rangle \rvert $$ for $\varphi\in C^\infty(\mathbb T)$. Notice that $f_k\to f$ in $\mathscr{D}'(\mathbb T)$ as $k\to\infty$ means that the limit $\lim_{k\to\infty} \langle f_k,\varphi\rangle =\langle f,\varphi\rangle$ holds for every $\varphi\in C^\infty(\mathbb T)$.
Let $$ \phi_k(e^{i\theta})=e^{ik\theta},\quad e^{i\theta}\in\mathbb T, $$ for $k\in\mathbb Z$ be the exponential monomials on $\mathbb T$. The Fourier coefficients of a distribution $f\in \mathscr{D}'(\mathbb T)$ are defined by $$ \hat{f}(k)=\langle f,\phi_{-k} \rangle,\quad k\in\mathbb Z. $$ A distribution on $\mathbb T$ is uniquely determined by its sequence of Fourier coefficients. It is well-known that a sequence of complex numbers $\{c_k\}_{k=-\infty}^\infty$ is of at most polynomial growth if and only if it is the sequence of Fourier coefficients for some distribution on $\mathbb T$, that is, there exists $f\in \mathscr{D}'(\mathbb T)$ such that $\hat{f}(k)=c_k$ for $k\in\mathbb Z$.
The derivative of a distribution $f\in \mathscr{D}'(\mathbb T)$ is the distribution $f'$ in $\mathscr{D}'(\mathbb T)$ determined by $(f')\hat{\ }(k)=ik\hat f(k)$ for $k\in\mathbb Z$.
The convolution $h=f*g$ of the distributions $f\in \mathscr{D}'(\mathbb T)$ and $g\in \mathscr{D}'(\mathbb T)$ is the distribution $h\in \mathscr{D}'(\mathbb T)$ determined by $\hat{h}(k)=\hat{f}(k)\hat{g}(k)$ for $k\in\mathbb Z$. Notice that $f*g\in C^\infty(\mathbb T)$ if $f\in \mathscr{D}'(\mathbb T)$ and $g\in C^\infty(\mathbb T)$.
For a suitably smooth function $u$ in $\mathbb D$ we set \begin{equation}\label{urfcn} u_r(e^{i\theta})=u(re^{i\theta}),\quad e^{i\theta}\in\mathbb T, \end{equation} for $0\leq r <1$. Notice that the function $u_r$ is essentially the restriction of $u$ to the circle $\{z\in\mathbb C:\ \lvert z\rvert=r\}$.
Recall formula \eqref{aPoissonkernel}. The $\alpha$-harmonic Poisson integral of a distribution $f\in\mathscr{D}'(\mathbb T)$ is defined by $$ P_\alpha[f](z)=(P_{\alpha,r}\ast f)(e^{i\theta}),\quad z=re^{i\theta}\in\mathbb D, $$ where $\ast$ denotes convolution, $P_{\alpha,r}$ is defined in accordance with \eqref{urfcn}, $r\geq0$ and $e^{i\theta}\in\mathbb T$. Observe that $$ P_\alpha[f](z)=\frac{1}{2\pi}\int_\mathbb T P_{\alpha}(ze^{-i\tau}) f(e^{i\tau})\, d\tau, \quad z\in\mathbb D, $$ if $f\in L^1(\mathbb T)$.
We next calculate the series expansion of the Poisson integral.
\begin{prop}\label{PIseriesexpansion} Let $\alpha\in\mathbb R$ and $f\in\mathscr{D}'(\mathbb T)$. Then \begin{equation*} P_\alpha[f](z)= \sum_{k=0}^\infty \hat{f}(k) z^k + \sum_{k=1}^\infty \hat{f}(-k) \frac{(\alpha+1)_k}{k!} F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k \end{equation*} for $z\in\mathbb D$, where $\hat{f}(k)$ is the $k$-th Fourier coefficient of $f$. \end{prop}
\begin{proof} Let $0\leq r<1$. From \eqref{Paseriesexpansion} we have that $$ P_{\alpha,r}(e^{i\theta})= \sum_{k=0}^\infty r^k e^{ik\theta} + \sum_{k=1}^\infty \frac{(\alpha+1)_k}{k!} F(-\alpha,k;k+1;r^2) r^k e^{-ik\theta} $$ for $e^{i\theta}\in\mathbb T$. Passing to the convolution we have that \begin{align*} P_\alpha[f](z)&=(P_{\alpha,r}\ast f)(e^{i\theta})= \sum_{k=0}^\infty r^k\hat{f}(k) e^{ik\theta}\\ & \ + \sum_{k=1}^\infty \frac{(\alpha+1)_k}{k!} F(-\alpha,k;k+1;r^2) r^k \hat{f}(-k) e^{-ik\theta} \end{align*} for $z=re^{i\theta}\in\mathbb D$ with $r\geq0$ and $e^{i\theta}\in\mathbb T$. This yields the conclusion of the proposition. \end{proof}
Notice that the expansion in Proposition \ref{PIseriesexpansion} is a series expansion of the form \eqref{generalizedpowerseries}. As a consequence, we have that the function $P_\alpha[f]$ is $\alpha$-harmonic in $\mathbb D$. We record also that \begin{equation}\label{coefficientbinomialseries} \frac{(\alpha+1)_k}{k!}=\frac{\Gamma(\alpha+k+1)}{\Gamma(\alpha+1)\Gamma(k+1)}\sim \frac{1}{\Gamma(\alpha+1)}(1+k)^\alpha \end{equation} as $k\to+\infty$ which follows by Stirling's formula.
Recall the notion of Fourier spectrum of a distribution on $\mathbb T$ defined by $$ \operatorname{Spec}(f)=\{k\in\mathbb Z:\ \hat{f}(k)\neq 0\} $$ for $f\in\mathscr{D}'(\mathbb T)$. We shall need a similar notion of spectrum of an $\alpha$-harmonic function in $\mathbb D$. Recall that an $\alpha$-harmonic function in $\mathbb D$ is uniquely determined by its sequence of coefficients in \eqref{generalizedpowerseries}. For such a function $u$ we set $$ \operatorname{Spec}(u)=\{k\in\mathbb Z:\ c_k\neq 0\}, $$ where the $c_k$'s are as in \eqref{generalizedpowerseries}. Observe that $\operatorname{Spec}(u)=\emptyset$ if and only if $u=0$.
Let us denote by $\mathbb Z^-$ the set of negative integers. In view of the series expansion \eqref{Paseriesexpansion} of the function $P_\alpha$ we have that $\operatorname{Spec}(P_\alpha)=\mathbb Z$ if $\alpha\in \mathbb R\setminus \mathbb Z^-$ whereas $$ \operatorname{Spec}(P_\alpha)=\{\alpha+1,\ \alpha+2,\dots \} $$ if $\alpha\in\mathbb Z^-$.
\begin{cor}\label{spectraofPI} Let $\alpha\in\mathbb R$ and $f\in\mathscr{D}'(\mathbb T)$. Then $$ \operatorname{Spec}(P_\alpha[f])=\operatorname{Spec}(P_\alpha)\cap\operatorname{Spec}(f). $$ In particular, $P_\alpha[f]=0$ if $\alpha\in\mathbb Z^-$ and $\operatorname{Spec}(f)\subset\{k\in\mathbb Z:\ k\leq\alpha\}$. \end{cor}
\begin{proof} The result is evident from Proposition \ref{PIseriesexpansion}. When $\alpha\in\mathbb Z^-$ the inclusion $\operatorname{Spec}(f)\subset\{k\in\mathbb Z:\ k\leq\alpha\}$ ensures that $\operatorname{Spec}(P_\alpha)\cap\operatorname{Spec}(f)=\emptyset$. \end{proof}
We next turn our attention to a characterization of Poisson integrals of distributions.
\begin{theorem}\label{characterizationPIrepresentation} Let $\alpha\in \mathbb R$. Then a function $u$ in $\mathbb D$ has the form of a Poisson integral $u=P_\alpha[f]$ in $\mathbb D$ for some $f\in\mathscr{D}'(\mathbb T)$ if and only if $u$ is $\alpha$-harmonic in $\mathbb D$, $u$ has temperate growth in $\mathbb D$, and $\operatorname{Spec}(u)\subset\operatorname{Spec}(P_\alpha)$. \end{theorem}
\begin{proof} Assume first that $u=P_\alpha[f]$ in $\mathbb D$ for some $f\in\mathscr{D}'(\mathbb T)$. From a well-known characterization of Fourier coefficients of distributions on $\mathbb T$ we know that the sequence of Fourier coefficients $\{\hat{f}(k)\}_{k=-\infty}^\infty$ has at most polynomial growth. Proposition \ref{PIseriesexpansion} supplies us with the series expansion of $u$. Clearly $u$ is $\alpha$-harmonic in $\mathbb D$. From Corollary \ref{spectraofPI} we have that $\operatorname{Spec}(u)\subset\operatorname{Spec}(P_\alpha)$. By Theorem \ref{polynomialgrowthcoefficients} we conclude that $u$ has at most temperate growth in $\mathbb D$.
Assume next that $u$ is $\alpha$-harmonic in $\mathbb D$, $u$ has temperate growth in $\mathbb D$, and $\operatorname{Spec}(u)\subset\operatorname{Spec}(P_\alpha)$. Consider the series expansion \eqref{generalizedpowerseries}. By Theorem \ref{polynomialgrowthcoefficients} we conclude that the sequence of coefficients $\{c_k\}_{k=-\infty}^\infty$ in \eqref{generalizedpowerseries} has at most polynomial growth. We set $a_k=c_k$ for $k\in\mathbb N$, $$ a_k=\frac{(-k)!}{(\alpha+1)_{-k}}c_{k} $$ for $k\in\mathbb Z^-$ and $k\in \operatorname{Spec}(P_\alpha)$, and $a_k=0$ for $k\in\mathbb Z^-$ and $k\not\in \operatorname{Spec}(P_\alpha)$. From \eqref{coefficientbinomialseries} we have that the sequence $\{a_k\}_{k=-\infty}^\infty$ is of polynomial growth. From a well-known characterization of Fourier coefficients of distributions on $\mathbb T$ there exists $f\in\mathscr{D}'(\mathbb T)$ such that $\hat{f}(k)=a_k$ for $k\in\mathbb Z$. Since $\operatorname{Spec}(u)\subset\operatorname{Spec}(P_\alpha)$, we have by Proposition \ref{PIseriesexpansion} that $u=P_\alpha[f]$ in $\mathbb D$. \end{proof}
The spectral condition in Theorem \ref{characterizationPIrepresentation} is redundant when $\alpha$ is not a negative integer.
\begin{cor}\label{genericcharacterizationPIrepresentation} Let $\alpha\in \mathbb R\setminus\mathbb Z^-$. Then a function $u$ in $\mathbb D$ has the form of a Poisson integral $u=P_\alpha[f]$ in $\mathbb D$ for some $f\in\mathscr{D}'(\mathbb T)$ if and only if it is $\alpha$-harmonic in $\mathbb D$ and of temperate growth there. \end{cor}
\begin{proof} Recall that $\operatorname{Spec}(P_\alpha)=\mathbb Z$ in the present situation. The result follows from Theorem \ref{characterizationPIrepresentation}. \end{proof}
The $\alpha$-harmonic Poisson kernel $P_\alpha$ has bounded $L^1$-means when $\alpha>-1$ (see Olofsson \cite[Theorem 3.1]{O14}). As a consequence, for $\alpha>-1$, one has that $u_r\to f$ in $\mathscr{D}'(\mathbb T)$ as $r\to1$ if $f\in\mathscr{D}'(\mathbb T)$ and $u=P_\alpha[f]$, where $u_r$ is as in \eqref{urfcn}.
The boundary behavior of $\alpha$-harmonic functions in $\mathbb D$ is conceptually different when $\alpha\leq-1$. The next result exemplifies this fact.
\begin{theorem}\label{boundarybehavioralphaleq-1} Let $\alpha\leq-1$ and let $u$ be an $\alpha$-harmonic function in $\mathbb D$. Assume that the limit $f=\lim_{r\to1}u_r$ in $\mathscr{D}'(\mathbb T)$ exists, where $u_r$ is as in \eqref{urfcn}. Then $u$ is analytic in $\mathbb D$. \end{theorem}
\begin{proof} Consider the series expansion \eqref{generalizedpowerseries}. We shall prove that $c_k=0$ for $k<0$. Let $k\in\mathbb Z^+$. From \eqref{generalizedpowerseries} we have that \begin{equation}\label{ccoefficientformula} c_{-k} F(-\alpha,k; k+1 ;r^2) r^{k} =\frac{1}{2\pi}\int_\mathbb T u(re^{i\theta})e^{ik\theta}\, d\theta \end{equation} for $0<r<1$. By assumption the right hand side in \eqref{ccoefficientformula} has a limit as $r\to 1$. Recall from the discussion following formulas \eqref{Ffactorintegralformula}-\eqref{Gausssummationformula} that the quantity $F(-\alpha,k; k+1 ;x)$ increases to $+\infty$ as $x\to1$. Passing to the limit in \eqref{ccoefficientformula} as $r\to1$ we conclude that $c_{-k}=0$. This yields the conclusion of the theorem. \end{proof}
We refer to Olofsson \cite[Theorem 2.3]{O14} for a result analogous to Theorem \ref{boundarybehavioralphaleq-1} phrased in another setting of generalized harmonic functions in $\mathbb D$. Under an assumption of pointwise boundary limit, a similar result also exists for the class of generalized axially symmetric potentials mentioned in the introduction, see Huber \cite[Theorem 2]{Huber}.
A traditional approach to the Poisson integral representation \begin{equation}\label{PIrepresentation} u(z)=P_\alpha[f](z), \quad z\in\mathbb D, \end{equation} is to exhibit the distribution $f\in\mathscr{D}'(\mathbb T)$ as a limit point of the $u_r$'s as $r\to1$ and then deduce the Poisson integral representation \eqref{PIrepresentation} from a uniqueness argument. See for instance Olofsson and Wittsten \cite[Theorem 5.5]{olofsson2013poisson} for an elaboration on this theme. An interesting point of Theorem \ref{characterizationPIrepresentation} is that this result establishes Poisson integral representations \eqref{PIrepresentation} in cases where boundary limits are non-existent.
Structure in \eqref{generalizedpowerseries} suggests use of the differential operator $A=z\partial-\bar z\bar\partial$. Observe that \begin{equation}\label{Aaction} Au(z)= \sum_{k=1}^\infty kc_{k} z^k -\sum_{k=1}^\infty k c_{-k}F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k ,\quad z\in\mathbb D, \end{equation} if $u$ has the form \eqref{generalizedpowerseries}. In particular, the function $Au$ is $\alpha$-harmonic in $\mathbb D$ if $u$ is.
\begin{prop}\label{A'intertwining} Let $\alpha\in\mathbb R$ and $f\in\mathscr{D}'(\mathbb T)$. Then $$ iA P_\alpha[f](z)=P_\alpha[f'](z) $$ for $z\in\mathbb D$, where $f'$ is the distributional derivative of $f$. \end{prop}
\begin{proof} Recall that $(f')\hat{\ }(k)=ik\hat f(k)$ for $k\in\mathbb Z$. By Proposition \ref{PIseriesexpansion} we have that \begin{align*} P_\alpha[f'](z)&=\sum_{k=0}^\infty ik\hat{f}(k) z^k + \sum_{k=1}^\infty(-ik) \hat{f}(-k) \frac{(\alpha+1)_k}{k!} F(-\alpha,k;k+1;\lvert z\rvert^2)\bar z^k\\ &=iAP_\alpha[f](z) \end{align*} for $z\in\mathbb D$, where in the last equality we have used \eqref{Aaction}. \end{proof}
We refer to the differential operator $iA=i(z\partial-\bar z\bar\partial)$ as the angular derivative. Our interest in this operator arose in connection to the paper Olofsson \cite{olofsson2018lipschitz}.
Let $\delta_1\in\mathscr{D}'(\mathbb T)$ be the unit Dirac mass located at the point $1\in\mathbb T$, that is, $\langle\delta_1,\varphi\rangle=\varphi(1)$ for $\varphi\in C^\infty(\mathbb T)$.
\begin{cor}\label{angularderivativePa} Let $\alpha\in\mathbb R$ and $k\in\mathbb N$. Then $$ P_\alpha[\delta_1^{(k)}](z)=(iA)^k P_\alpha(z) $$ for $z\in\mathbb D$, where $\delta_1^{(k)}$ denotes the $k$-th distributional derivative of $\delta_1$. \end{cor}
\begin{proof} It is straight\-forward to check that $(\delta_1)\hat{\ }(k)=1$ for $k\in\mathbb Z$. By Proposition \ref{PIseriesexpansion} and formula \eqref{Paseriesexpansion} we have that $P_\alpha[\delta_1](z)=P_\alpha$. The result now follows by Proposition \ref{A'intertwining}. \end{proof}
\section{Angular derivatives of Poisson kernels} \label{sectionangularderivatives}
This section is devoted to a careful analysis of angular derivatives of Poisson kernels. We shall make good use of the M\"obius transformation \begin{equation}\label{mobiusmap} \varphi(z)=i\frac{1+z}{1-z} \end{equation} in our calculations. Notice that $\varphi$ maps $\mathbb D$ one-to-one onto $\mathbb H$, $\varphi(0)=i$ and $\varphi(1)=\infty$. From standard theory we have that the M\"obius transformation $\varphi$ is uniquely determined by these three properties. For the sake of easy reference we record also the formulas \begin{equation}\label{mobiusmapderivative} \operatorname{Im} \varphi(z)=\frac{1-\lvert z\rvert^2}{\lvert 1-z\rvert^2} \quad\text{and}\quad \varphi'(z)=\frac{2i}{(1-z)^2}, \end{equation} which are straight\-forward to check.
A most natural $\alpha$-harmonic function in $\mathbb H$ is the $(\alpha+1)$-th power of the imaginary part: $$ u(z)=(\operatorname{Im} z)^{\alpha+1},\quad z\in\mathbb H. $$ We first calculate the weighted pull-back $u_{\varphi,\alpha}$ from \eqref{weightedpullback} of this function by $\varphi$.
\begin{theorem}\label{Paaswpullback} Let $\alpha\in\mathbb R$. Then \begin{equation*} P_\alpha(z)= c\varphi'(z)^{-\alpha/2} (\operatorname{Im} \varphi(z))^{\alpha+1}, \quad z\in\mathbb D, \end{equation*} where $\varphi$ is as in \eqref{mobiusmap} and $c\varphi'(0)^{-\alpha/2}=1$. \end{theorem}
\begin{proof} Recall formula \eqref{aPoissonkernel}. The two formulas in \eqref{mobiusmapderivative} make evident that $$ P_\alpha(z)=c\varphi'(z)^{-\alpha/2} (\operatorname{Im} \varphi(z))^{\alpha+1}, \quad z\in\mathbb D, $$ where $c\in\mathbb C$. From $P_\alpha(0)=1$ we see that $c\varphi'(0)^{-\alpha/2}=1$. \end{proof}
Notice that Theorem \ref{Paaswpullback} with $\alpha=0$ yields the well-known formula $$ P_0(z)=\operatorname{Re}\Big( \frac{1+z}{1-z} \Big),\quad z\in\mathbb D, $$ for the usual Poisson kernel for $\mathbb D$.
Our next task is to calculate angular derivatives of Poisson kernels. We begin with a preparatory lemma.
\begin{lemma}\label{iAmobiusmap} Let $\varphi$ be as in \eqref{mobiusmap}. Then $iA\varphi=\frac{1}{2}(\varphi^2+1)$. \end{lemma}
\begin{proof} From formula \eqref{mobiusmapderivative} we have that $$ iA\varphi(z)=iz\varphi'(z)=-2z/(1-z)^2. $$ Using some elementary algebra we now calculate that $$ iA\varphi(z)=\frac{1}{2}\frac{(1-z)^2-(1+z)^2}{(1-z)^2} =\frac{1}{2}\Big(1- \Big(\frac{1+z}{1-z}\Big)^2 \Big) =\frac{1}{2}(1+\varphi(z)^2), $$ where the last equality follows by \eqref{mobiusmap}. \end{proof}
Let us record some properties of the angular derivative. The differential operator $iA$ satisfies the product rule for differen\-tiation: $iA(fg)=giA(f)+fiA(g)$ for suitable $f$ and $g$. Denote by $\bar f$ the point\-wise complex conjugate of a complex-valued function $f$. We shall use the chain rule in the form \begin{equation}\label{chainrule} iA(h\circ f)=((\partial h)\circ f)iAf +((\bar\partial h)\circ f)\overline{iAf} \end{equation} for suitable $f$ and $h$. Formula \eqref{chainrule} is straight\-forward to check. As a consequence of \eqref{chainrule} we have that the differential operator $iA$ commutes with the action of complex conjugation of functions: $iA(\bar f)=\overline{iAf}$ for suitable $f$.
We now turn to differentiation of powers.
\begin{lemma}\label{iAimmobiusmappower} Let $\varphi$ be as in \eqref{mobiusmap} and $\alpha\in\mathbb R$. Then $iA((\operatorname{Im} \varphi)^{\alpha+1}) =(\alpha+1)(\operatorname{Re} \varphi) (\operatorname{Im} \varphi)^{\alpha+1}$ in $\mathbb D$. \end{lemma}
\begin{proof} Let $u=(\operatorname{Im} \varphi)^{\alpha+1}$ in $\mathbb D$. By a standard rule for differentiation we have that $$ iAu=(\alpha+1)(\operatorname{Im} \varphi)^{\alpha} iA(\frac{1}{2i}(\varphi-\bar\varphi)) $$ in $\mathbb D$, compare with \eqref{chainrule}. Since the differential operator $iA$ commutes with the action of complex conjugation of functions we have that $$ iAu=(\alpha+1)(\operatorname{Im} \varphi)^{\alpha} \frac{1}{2i}(iA\varphi-\overline{iA\varphi}) $$ in $\mathbb D$. We now use Lemma \ref{iAmobiusmap} and calculate that $$ iAu=(\alpha+1)(\operatorname{Im} \varphi)^{\alpha} \frac{1}{2i} (\frac{1}{2}(\varphi^2+1)-\frac{1}{2}(\bar\varphi^2+1))= (\alpha+1)(\operatorname{Im} \varphi)^{\alpha} \frac{1}{2i}\frac{1}{2}(\varphi^2-\bar\varphi^2) $$ in $\mathbb D$, where the last equality follows by cancellation. By some elementary algebra we now have that $$ iAu=(\alpha+1)(\operatorname{Im} \varphi)^{\alpha} \frac{1}{2i}\frac{1}{2} (\varphi-\bar\varphi)(\varphi+\bar\varphi)= (\alpha+1)(\operatorname{Im} \varphi)^{\alpha}(\operatorname{Im} \varphi)(\operatorname{Re} \varphi) $$ in $\mathbb D$. This yields the conclusion of the lemma. \end{proof}
\begin{lemma}\label{iAmobiusmappower} Let $\varphi$ be as in \eqref{mobiusmap} and $\alpha\in\mathbb R$. Then $iA((\varphi')^{-\alpha/2}) =\frac{\alpha}{2} (i-\varphi)(\varphi')^{-\alpha/2}$ in $\mathbb D$. \end{lemma}
\begin{proof} We first show that $iA\varphi'=(\varphi-i)\varphi'$. From \eqref{mobiusmapderivative} we have that $$ iA\varphi'(z)=iz\varphi''(z)=(2i)^2\frac{z}{(1-z)^3} =i\frac{2z}{1-z}\varphi'(z), $$ where the last equality again follows by \eqref{mobiusmapderivative}. By some elementary algebra we now have that $$ iA\varphi'(z)= i\frac{(1+z)-(1-z)}{1-z}\varphi'(z) =i\Big(\frac{1+z}{1-z}-1\Big)\varphi'(z)=(\varphi(z)-i)\varphi'(z), $$ where the last equality follows by \eqref{mobiusmap}.
We now consider the general case. By a well-known differentiation formula we have that $$ iA((\varphi')^{-\alpha/2})=-\frac{\alpha}{2}(\varphi')^{-\alpha/2-1}iA\varphi' $$ in $\mathbb D$, compare with \eqref{chainrule}. We now use the result of the previous paragraph to conclude that $$ iA((\varphi')^{-\alpha/2})=-\frac{\alpha}{2}(\varphi')^{-\alpha/2-1} (\varphi-i)\varphi'=\frac{\alpha}{2}(i-\varphi)(\varphi')^{-\alpha/2} $$ in $\mathbb D$. \end{proof}
We next calculate the angular derivative of the Poisson kernel.
\begin{theorem}\label{iAderivativePa} Let $\alpha\in\mathbb R$ and let $\varphi$ be as in \eqref{mobiusmap}. Then $$ iA P_\alpha(z)=\frac{1}{2}(\varphi(z)+(\alpha+1)\overline{\varphi(z)} +i\alpha )P_\alpha(z) $$ for $z\in\mathbb D$. \end{theorem}
\begin{proof} Let $u=(\varphi')^{-\alpha/2} (\operatorname{Im} \varphi)^{\alpha+1}$ in $\mathbb D$. In view of Theorem \ref{Paaswpullback} it suffices to prove the conclusion of the theorem with $P_\alpha$ replaced by the function $u$. By the product rule for differentiation we have that $$ iAu= (\operatorname{Im} \varphi)^{\alpha+1} iA((\varphi')^{-\alpha/2}) + (\varphi')^{-\alpha/2} iA((\operatorname{Im} \varphi)^{\alpha+1}) $$ in $\mathbb D$. We now use Lemmas \ref{iAimmobiusmappower} and \ref{iAmobiusmappower} to conclude that \begin{align*} iAu&=(\operatorname{Im} \varphi)^{\alpha+1} \frac{\alpha}{2} (i-\varphi)(\varphi')^{-\alpha/2} + (\varphi')^{-\alpha/2} (\alpha+1)(\operatorname{Re} \varphi) (\operatorname{Im} \varphi)^{\alpha+1}\\ &= \Big(\frac{\alpha}{2}(i-\varphi) + (\alpha+1) \frac{1}{2}(\varphi+\bar\varphi) \Big) u \end{align*} in $\mathbb D$, where the last equality is straight\-forward to check. This yields the conclusion of the theorem. \end{proof}
An earlier version of Theorem \ref{iAderivativePa} appears in Olofsson \cite[Theorem 1.11]{olofsson2018lipschitz}; see also Klintborg and Olofsson \cite[Corollary 1.3]{KO} for a generalization along similar lines. Here our focus is on applications to the setting of the upper half-plane and we express matters using the M\"obius transformation $\varphi$.
We denote by $\mathbb C[z,\bar z]$ the algebra of polynomials in $z$ and $\bar z$ with complex coefficients. Define polynomials $\{h_{k,\alpha}\}_{k=0}^\infty$ in $\mathbb C[z,\bar z]$ by $h_{0,\alpha}=1$ and \begin{align}\label{hkpolynomials} h_{k+1,\alpha}(z)&=\frac{1}{2}(z^2+1)\partial h_{k,\alpha}(z)+ \frac{1}{2}(\bar z^2+1)\bar\partial h_{k,\alpha}(z)\\ & \quad +\frac{1}{2}(z+(\alpha+1)\bar z+i\alpha) h_{k,\alpha}(z) \notag \end{align} for $k\geq0$. It is straight\-forward to check that $h_{k,\alpha}$ has degree at most $k$ for $k=0,1,\dots$.
\begin{theorem}\label{iApowerPa} Let $\alpha\in\mathbb R$ and let $\varphi$ be as in \eqref{mobiusmap}. Then $$ (iA)^k P_\alpha(z)=h_{k,\alpha}(\varphi(z))P_\alpha(z),\quad z\in\mathbb D, $$ for $k=0,1,2,\dots$, where the $h_{k,\alpha}$'s are as in \eqref{hkpolynomials}. \end{theorem}
\begin{proof} For $k=0$ the result is evident. For $k=1$ the result follows by Theorem \ref{iAderivativePa}. We proceed by induction and assume that $(iA)^k P_\alpha=(h_{k,\alpha}\circ\varphi)P_\alpha$ for some $k\geq0$. Applying the operator $iA$ we have that $$ (iA)^{k+1} P_\alpha= P_\alpha iA(h_{k,\alpha}\circ\varphi) +(h_{k,\alpha}\circ\varphi) iAP_\alpha, $$ where we have used the product rule. From the chain rule \eqref{chainrule} and Lemma \ref{iAmobiusmap} we have that $$ iA(h_{k,\alpha}\circ\varphi) =((\partial h_{k,\alpha})\circ\varphi) \frac{1}{2}(\varphi^2+1) +((\bar\partial h_{k,\alpha})\circ\varphi)\frac{1}{2}(\bar\varphi^2+1). $$ We now return to the function $(iA)^{k+1} P_\alpha$ and use Theorem \ref{iAderivativePa} to conclude that \begin{align*} (iA)^{k+1} P_\alpha&= P_\alpha\Big( ((\partial h_{k,\alpha})\circ\varphi) \frac{1}{2}(\varphi^2+1) +((\bar\partial h_{k,\alpha})\circ\varphi)\frac{1}{2}(\bar\varphi^2+1)\Big)\\ & \ + (h_{k,\alpha}\circ\varphi) \frac{1}{2}(\varphi+(\alpha+1)\bar\varphi+i\alpha )P_\alpha = (h_{k+1,\alpha}\circ\varphi) P_\alpha, \end{align*} where in the last equality we have used \eqref{hkpolynomials}. The result now follows by invoking the induction principle. \end{proof}
We shall study the polynomials $h_{k,\alpha}$ in some more detail.
\begin{lemma}\label{dehka} Let $h_{k,\alpha}\in\mathbb C[z,\bar z]$ be as in \eqref{hkpolynomials} for some $\alpha\in\mathbb R$ and $k\geq0$. Then the function $\mathbb H\ni z\mapsto (\operatorname{Im} z)^{\alpha+1}h_{k,\alpha}(z)$ is $\alpha$-harmonic in $\mathbb H$. \end{lemma}
\begin{proof} Recall that the function $iAu$ is $\alpha$-harmonic in $\mathbb D$ if $u$ is, see \eqref{Aaction}. From Theorem \ref{iApowerPa} we thus have that the function $(h_{k,\alpha}\circ\varphi)P_\alpha$ is $\alpha$-harmonic in $\mathbb D$. Theorem \ref{Paaswpullback} displays the Poisson kernel $P_\alpha$ as a weighed pull-back by the function $\varphi$. From Olofsson \cite[Theorem 1.1]{olofsson2017on} we now conclude that the function $z\mapsto (\operatorname{Im} z)^{\alpha+1}h_{k,\alpha}(z)$ is $\alpha$-harmonic in $\mathbb H$, see discussion following formula \eqref{weightedpullback}. \end{proof}
Let $\alpha\in\mathbb R$ and let us denote by $y$ the imaginary part. A calculation shows that $$ \Delta_{\alpha;\mathbb H}y^{\alpha+1}=\frac{1}{2i}D_\alpha $$ in the sense of differential operators, where \begin{equation}\label{Dalphaoperator} D_{\alpha,z}=(z-\bar z)\partial_z\bar\partial_z +\bar\partial_z-(\alpha+1)\partial_z. \end{equation} As a consequence, we have that a product $y^{\alpha+1}h$ is $\alpha$-harmonic in $\mathbb H$ if and only if $D_\alpha h=0$ in $\mathbb H$, where $D_\alpha$ is as in \eqref{Dalphaoperator}.
Notice that if $p\in\mathbb C[z,\bar z]$ is homogeneous of degree $m$, then the polynomial $D_\alpha p$ is homo\-geneous of degree $m-1$.
Consider the partial sums \begin{equation}\label{eq:skallealfa} s_{k,\alpha}(z)=\sum_{j=0}^k \frac{(\alpha+1)_j}{j!} z^j \end{equation} for $k=0,1,\dots$ of a binomial series \eqref{binomialseries} with $a=\alpha+1$. We shall need the associated homo\-geneous polynomials \begin{equation}\label{eq:pkallealfa} p_{k,\alpha}(z)=z^k s_{k,\alpha}(\bar z /z) =\sum_{j=0}^k\frac{(\alpha+1)_j}{j!}z^{k-j}\bar z^j \end{equation} for $k=0,1,\dots$, where $\alpha\in\mathbb R$. Notice that $p_{k,\alpha}\in\mathbb C[z,\bar z]$ is homo\-geneous of degree $k$.
We now return to the differential operator $D_\alpha$ in \eqref{Dalphaoperator}.
\begin{theorem}\label{significancepkapolynomial} Let $\alpha\in\mathbb R$. Let $p$ in $\mathbb C[z,\bar z]$ be homo\-geneous of degree $k$. Then $D_\alpha p=0$ if and only if $p$ is a constant multiple of $p_{k,\alpha}$, where $p_{k,\alpha}$ is as in \eqref{eq:pkallealfa}. \end{theorem}
\begin{proof} We shall evaluate the differential operator $D_\alpha$ on a polynomial $p$ of the form \begin{equation}\label{homogeneouspolynomial} p(z)=\sum_{j=0}^k a_j z^{k-j}\bar z^j \end{equation} for some $a_0,\dots,a_k\in\mathbb C$. Calculations show that $$ \bar\partial p(z)=\sum_{j=0}^{k-1} (j+1)a_{j+1} z^{k-1-j}\bar z^j,\quad \partial p(z)=\sum_{j=0}^{k-1} (k-j)a_{j} z^{k-1-j}\bar z^j, $$ and $$ (z-\bar z)\partial\bar\partial p(z)= \sum_{j=0}^{k-2} (j+1)(k-1-j)a_{j+1} z^{k-1-j}\bar z^j- \sum_{j=1}^{k-1} j(k-j)a_{j} z^{k-1-j}\bar z^j. $$ From these formulas we have that $$ D_\alpha p(z)= \sum_{j=0}^{k-1} (k-j)((j+1)a_{j+1}-(j+\alpha+1)a_{j}) z^{k-1-j}\bar z^j $$ for $p$ of the form \eqref{homogeneouspolynomial}.
From the result of the previous paragraph we have that $D_\alpha p=0$ if and only if $$ (j+1)a_{j+1}-(j+\alpha+1)a_{j}=0 $$ for $j=0,\dots, k-1$. We conclude that $D_\alpha p=0$ if and only if $p=a_0 p_{k,\alpha}$. \end{proof}
\begin{cor}\label{reprtermharmonic} Let $\alpha\in\mathbb R$ and $k\in\mathbb N$. Let $p_{k,\alpha}$ be as in \eqref{eq:pkallealfa}. Then the function $\mathbb H\ni z\mapsto (\operatorname{Im} z)^{\alpha+1}p_{k,\alpha}(z)$ is $\alpha$-harmonic in $\mathbb H$. \end{cor}
\begin{proof} By Theorem \ref{significancepkapolynomial} we have that $D_\alpha p_{k,\alpha}=0$. Therefore $\Delta_\alpha y^{\alpha+1}p_{k,\alpha}=0$ in $\mathbb H$ by the factorization formula preceding \eqref{Dalphaoperator}. \end{proof}
We now return to the $h_{k,\alpha}$'s.
\begin{theorem}\label{hkalincombofpka} Let $\alpha\in\mathbb R$ and $k\in\mathbb N$. Let $h_{k,\alpha}$ be as in \eqref{hkpolynomials}. Then $h_{k,\alpha}$ is a linear combination of $p_{0,\alpha},\dots,p_{k,\alpha}$, where the $p_{j,\alpha}$'s are as in \eqref{eq:pkallealfa}. \end{theorem}
\begin{proof} From Lemma \ref{dehka} we have that the product $y^{\alpha+1}h_{k,\alpha}$ is $\alpha$-harmonic in $\mathbb H$. Therefore $D_\alpha h_{k,\alpha}=0$ in $\mathbb C[z,\bar z]$, where $D_\alpha$ is as in \eqref{Dalphaoperator}. We next write the polynomial $h_{k,\alpha}$ as a sum of homo\-geneous polynomials: $h_{k,\alpha}=\sum_{j=0}^k p_j$, where $p_j$ is homogeneous of degree $j$. Clearly $p_0$ is a constant multiple of $p_{0,\alpha}$. Applying the differential operator $D_\alpha$ we see that $$ 0=D_\alpha h_{k,\alpha}=\sum_{j=1}^k D_\alpha p_j $$ in $\mathbb C[z,\bar z]$. Since $D_\alpha p_j$ is homogeneous of degree $j-1$, we have that $D_\alpha p_j=0$ for $j=1,\dots k$. From Theorem \ref{significancepkapolynomial} we have that $p_j$ is a constant multiple of $p_{j,\alpha}$ for $j=1,\dots k$. We conclude that $h_{k,\alpha}$ is a linear combination of $p_{0,\alpha},\dots,p_{k,\alpha}$. \end{proof}
\section{A general representation theorem} \label{representationthms}
In this section we consider $\alpha$-harmonic functions in $\mathbb H$ which are of temperate growth at infinity and vanish on the real line. A main task is to establish a re\-presentation of such functions. The analysis uses results from Section \ref{sectionangularderivatives}.
\begin{theorem}\label{alphaobstructionclass} Let $\alpha>-1$. Let $u$ be an $\alpha$-harmonic function in $\mathbb H$ which is of temperate growth at infinity. Assume that $u(z)\to0$ as $\mathbb H\ni z\to x$ for every $x\in\mathbb R$. Then \begin{equation}\label{alphaobstructionfcn} u(z)=\sum_{k=0}^n c_k(\operatorname{Im} z)^{\alpha+1} p_{k,\alpha}(z),\quad z\in\mathbb H, \end{equation} for some $n\in\mathbb N$ and $c_0,\dots,c_n\in\mathbb C$, where the $p_{k,\alpha}$'s are as in \eqref{eq:pkallealfa}. \end{theorem}
\begin{proof} We consider the weighted pull-back $$ v(z)=\varphi'(z)^{-\alpha/2}u(\varphi(z)),\quad z\in\mathbb D, $$ where $\varphi$ is as in \eqref{mobiusmap}. From \cite[Theorem 1.1]{olofsson2017on} it follows that $v$ is $\alpha$-harmonic in $\mathbb D$. From temperate growth of $u$ at infinity and vanishing of $u$ on $\mathbb R$ we have by a compactness argument that $v$ is of temperate growth in $\mathbb D$. By Theorem \ref{characterizationPIrepresentation} we conclude that $v=P_\alpha[g]$ for some distribution $g\in\mathscr{D}'(\mathbb T)$. Since $\alpha>-1$, we have from general theory that $v_r\to g$ in $\mathscr{D}'(\mathbb T)$ as $r\to1$, where the $v_r$'s are defined as in \eqref{urfcn} (see for instance \cite[Theorem 5.4]{olofsson2013poisson}). Since $u(z)\to0$ as $\mathbb H\ni z\to x$ for $x\in\mathbb R$, we see that $g$ vanishes on $\mathbb T\setminus\{1\}$ in a distributional sense. Now since $\operatorname{supp}(g)\subset\{1\}$, standard distribution theory dictates that $g=\sum_{k=0}^na_k\delta_1^{(k)}$ in $\mathscr{D}'(\mathbb T)$ for some $n\in\mathbb N$ and complex numbers $a_0,\dots, a_n\in\mathbb C$ (see H\"ormander \cite[Theorem 2.3.4]{Hormander}). By Corollary \ref{angularderivativePa} we have that $P_\alpha[\delta_1^{(k)}]=(iA)^kP_\alpha$ for $k\ge0$. Passing to the Poisson integral we have that \begin{equation}\label{vprelformula} v(z)= \sum_{k=0}^n a_k (iA)^kP_\alpha(z) =\Big(\sum_{k=0}^n a_k h_{k,\alpha}(\varphi(z)) \Big) P_\alpha(z) \end{equation} for $z\in\mathbb D$, where the last equality follows by Theorem \ref{iApowerPa}. By Theorem \ref{hkalincombofpka} the function $h_{k,\alpha}$ is a linear combination of $p_{0,\alpha},\dots,p_{k,\alpha}$. From \eqref{vprelformula} we conclude that \begin{equation*} v(z)=\Big(\sum_{k=0}^n b_k p_{k,\alpha}(\varphi(z)) \Big) P_\alpha(z),\quad z\in\mathbb D, \end{equation*} for some $b_0,\dots, b_n\in\mathbb C$. Invoking Theorem \ref{Paaswpullback} we see that \begin{equation*} v(z)=\varphi'(z)^{-\alpha/2} \Big(\sum_{k=0}^n cb_k (\operatorname{Im}\varphi(z))^{\alpha+1} p_{k,\alpha}(\varphi(z)) \Big) \end{equation*} for $z\in\mathbb D$, where $c\varphi'(0)^{-\alpha/2}=1$. A passage back to the function $u$ yields \eqref{alphaobstructionfcn} with $c_k=cb_k$ for $k=0,\dots,n$. This completes the proof of the theorem. \end{proof}
For $\alpha>-1$ and $n\in\mathbb N$, we denote by $\mathcal{V}_{\alpha,n}$ the set of all functions $u$ of the form \eqref{alphaobstructionfcn} for some $c_0,\dots,c_n\in\mathbb C$, where the $p_{k,\alpha}$'s are as in \eqref{eq:pkallealfa}. We also set $$ \mathcal{V}_{\alpha}=\cup_{n=0}^\infty\mathcal{V}_{\alpha,n}. $$ The space $\mathcal{V}_{\alpha,n}$ is a complex vector space of finite dimension $n+1$. Notice that the set $\mathcal{V}_{\alpha}$ is a complex vector space which is naturally filtered by the sets $\mathcal{V}_{\alpha,n}$ for $n=0,1,\dots$.
Observe that the conclusion of Theorem \ref{alphaobstructionclass} says that $u\in\mathcal{V}_{\alpha}$. We next describe the class $\mathcal{V}_{0}$ in some more detail.
\begin{prop}\label{harmonicobstructionclass} Let $n\in\mathbb N$. Then a function $u$ belongs to the space $\mathcal{V}_{0,n}$ if and only if it has the form \begin{equation}\label{harmonicobstructionfcn} u(z)=\sum_{k=1}^{n+1} c_k \operatorname{Im}(z^k),\quad z\in\mathbb H, \end{equation} for some $c_1,\dots,c_{n+1}\in\mathbb C$. \end{prop}
\begin{proof} A calculation using the formula for a finite geometric sum shows that $$ p_{k,0}(z)=\sum_{j=0}^k z^{k-j}\bar z^j=\frac{z^{k+1}-\bar z^{k+1} }{z-\bar z}. $$ Therefore $$ (\operatorname{Im} z)p_{k,0}(z)=\frac{z-\bar z}{2i} \frac{z^{k+1}-\bar z^{k+1}}{z-\bar z} =\operatorname{Im}(z^{k+1}). $$ The result is now evident from definition of the space $\mathcal{V}_{0,n}$. \end{proof}
We shall next investigate the order of growth of a function of the form \begin{equation}\label{ukafunction} u_{k,\alpha}(z)= (\operatorname{Im} z)^{\alpha+1} p_{k,\alpha}(z),\quad z\in\mathbb H, \end{equation} for some $k\in\mathbb N$ and $\alpha>-1$, where $p_{k,\alpha}$ is as in \eqref{eq:pkallealfa}. From the proof of Proposition \ref{harmonicobstructionclass} we have that $$ u_{k,0}(z)= \operatorname{Im}(z^{k+1}) ,\quad z\in\mathbb C, $$ for $k\in\mathbb N$.
\begin{prop}\label{tempgrowthprop} Let $u_{k,\alpha}$ be as in \eqref{ukafunction} for some $k\in\mathbb N$ and $\alpha>-1$. Then \begin{equation*} \lvert u_{k,\alpha}(z)\rvert \leq \max_{0<\theta<\pi} \sin^{k+2\alpha+2}(\theta) \lvert p_{k,\alpha}(e^{i\theta})\rvert \Big(\lvert z\rvert^2/\operatorname{Im}(z)\Big)^{k+\alpha+1} \end{equation*} for $z\in \mathbb H$. In particular, if $u\in\mathcal{V}_{\alpha,n}$ for some $n\in\mathbb N$, then $u$ has order of growth at most $n+\alpha+1$ at infinity. \end{prop}
\begin{proof} Let $z\in\mathbb H$ and write $z=te^{i\theta}$ with $t>0$ and $0<\theta<\pi$. By homogeneity we have that $$ u_{k,\alpha}(z) =\sin^{\alpha+1}(\theta) p_{k,\alpha}(e^{i\theta})t^{k+\alpha+1}. $$ Notice also that $\lvert z\rvert^2/\operatorname{Im}(z)=t/\sin(\theta)$. From these two facts we have that \begin{equation*} u_{k,\alpha}(z)= \sin^{k+2\alpha+2}(\theta) p_{k,\alpha}(e^{i\theta}) \Big(\lvert z\rvert^2/\operatorname{Im}(z)\Big)^{k+\alpha+1} \end{equation*} for $z\in\mathbb H$, where $\theta=\arg z$. This yields the conclusion of the proposition. \end{proof}
The next result points out the significance of the class $\mathcal{V}_\alpha$.
\begin{cor}\label{Valphacharacterization} Let $\alpha>-1$. Then a function $u$ belongs to the class $\mathcal{V}_\alpha$ if and only if it is $\alpha$-harmonic in $\mathbb H$, has temperate growth at infinity and vanishes on the real line in the sense that $u(z)\to0$ as $\mathbb H\ni z\to x$ for every $x\in\mathbb R$. \end{cor}
\begin{proof} The if part is a re\-statement of Theorem \ref{alphaobstructionclass}. From Corollary \ref{reprtermharmonic} we have that every function $u\in \mathcal{V}_\alpha$ is $\alpha$-harmonic in $\mathbb H$. From Proposition \ref{tempgrowthprop} we have that every function $u\in\mathcal{V}_\alpha$ is of temperate growth at infinity. It is evident that every function $u\in \mathcal{V}_\alpha$ vanishes on the real line. \end{proof}
Corollary \ref{Valphacharacterization} explains how the space $\mathcal{V}_\alpha$ can be thought of as the class of obstructions for the uniqueness problem for $\alpha$-harmonic functions in $\mathbb H$ with respect to a vanishing Dirichlet boundary value on the real line and temperate growth at infinity.
\begin{lemma}\label{ulimitlemma} Let $u\in\mathcal{V}_{\alpha,n}$ be of the form \eqref{alphaobstructionfcn} for some $\alpha>-1$ and $n\in\mathbb N$. Then \begin{equation*} \lim_{t\to+\infty}u(te^{i\theta})/t^{n+\alpha+1}=c_n \sin^{\alpha+1}(\theta) p_{n,\alpha}(e^{i\theta}) \end{equation*} for $0<\theta<\pi$. \end{lemma}
\begin{proof} Let $t>0$ and $0<\theta<\pi$. By homogeneity we have that $$ u(te^{i\theta})= \sum_{k=0}^n c_k(\sin \theta)^{\alpha+1} p_{k,\alpha}(e^{i\theta}) t^{k+\alpha+1}. $$ The result now follows by a passage to the limit. \end{proof}
We shall dissect the space $\mathcal{V}_\alpha$ using orders of growth at infinity. We say that a function $u$ in $\mathbb H$ has order of growth $n+\alpha+1$ at infinity if it satisfies an estimate of the form \begin{equation}\label{utempgrowth} \lvert u(z)\rvert\leq C(\lvert z\rvert^2/\operatorname{Im}(z))^{n+\alpha+1} \end{equation} for $z\in\mathbb H$ with $\lvert z\rvert>R$, where $C\geq0$ and $R>1$ are finite constants.
\begin{lemma}\label{coefficientlemma} Let $u\in \mathcal{V}_\alpha$ for some $\alpha>-1$. Let $n\in\mathbb N$. Then $u\in \mathcal{V}_{\alpha,n}$ if and only if \eqref{utempgrowth} holds. Moreover, there exists a constant $C'=C'_{\alpha,n,R}$ such that $\lvert c_k\rvert\leq C'C$ for $k=0,1,\dots,n$ whenever $u\in\mathcal{V}_{\alpha,n}$ has the form \eqref{alphaobstructionfcn} and $C$ is as in \eqref{utempgrowth}. \end{lemma}
\begin{proof} From Proposition \ref{tempgrowthprop} we know that \eqref{utempgrowth} holds if $u\in\mathcal{V}_{\alpha,n}$. Assume next that $u\in\mathcal{V}_{\alpha}$ satisfies \eqref{utempgrowth}. We shall prove that $u\in\mathcal{V}_{\alpha,n}$. If $u\in\mathcal{V}_{\alpha,0}$ there is nothing to prove. Assume therefore that $u\in\mathcal{V}_{\alpha,m}$ and $u\not\in\mathcal{V}_{\alpha,m-1}$ for some $m\in\mathbb Z^+$. An application of Lemma \ref{ulimitlemma} with $0<\theta<\pi$ chosen such that $p_{m,\alpha}(e^{i\theta})\neq0$ shows that $u$ has order of growth at least $m+\alpha+1$. Therefore $m\leq n$, so that $u\in\mathcal{V}_{\alpha,n}$.
We equip the space $\mathcal{V}_{\alpha,n}$ with the semi-norm given by the best constant $C$ in \eqref{utempgrowth}. Another application of Lemma \ref{ulimitlemma} shows that this latter semi-norm is in fact a norm. The space $\mathcal{V}_{\alpha,n}$ is thus a normed complex vector space of finite dimension $n+1$. Since any two norms on such a space are equivalent, we conclude that $\lvert c_k\rvert\leq C'C$ for $k=0,1,\dots,n$ whenever $u\in \mathcal{V}_{\alpha,n}$. \end{proof}
We denote by $\mathscr{D}'(\mathbb R)$ the space of distributions on the real line $\mathbb R$. An integrable function $f\in L^1(\mathbb R)$ is identified with the distribution $$ \langle f,\varphi\rangle=\int_{-\infty}^\infty f(x)\varphi(x)\, dx, \quad \varphi \in C_0^\infty(\mathbb R), $$ where $C_0^\infty(\mathbb R)$ is the space of indefinitely differentiable test functions on $\mathbb R$ with compact support. By $u_j\to u$ in $\mathscr{D}'(\mathbb R)$ as $j\to\infty$ we understand that $\lim_{j\to\infty}\langle u_j,\varphi\rangle = \langle u,\varphi\rangle$ for every $\varphi \in C_0^\infty(\mathbb R)$. A standard reference for distribution theory is H\"ormander \cite{Hormander}.
We shall next extend the validity of Theorem \ref{alphaobstructionclass} to allow for a distributional boundary value.
\begin{theorem}\label{alphaobstructionclassdistributionalversion} Let $\alpha>-1$. Let $u$ be an $\alpha$-harmonic function in $\mathbb H$ which is of temperate growth at infinity. Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. Then $u\in \mathcal{V}_\alpha$. \end{theorem}
\begin{proof} The theorem is proved by a regularization argument. Let $\psi\in C_0^\infty(\mathbb R)$ be a non\-negative compactly supported test function with $\int_{-\infty}^\infty\psi(x)\, dx=1$ and set $\psi_\varepsilon(x)=\psi(x/\varepsilon)/\varepsilon$ for $x\in\mathbb R$ and $\varepsilon>0$. We consider the regularizations $$ u_\varepsilon(z)=\int_{-\infty}^\infty u(z-t)\psi_\varepsilon(t)\, dt,\quad z\in\mathbb H, $$ for $\varepsilon>0$. A differentiation under the integral shows that the function $u_\varepsilon$ is $\alpha$-harmonic in $\mathbb H$. It is straight\-forward to check that $u_\varepsilon(z)\to0$ as $\mathbb H\ni z\to x$ for every $x\in\mathbb R$. Using that $u$ is of temperate growth at infinity it is straight\-forward to check that \begin{equation}\label{tempgrowthbound} \lvert u_\varepsilon(z)\rvert\leq C(\lvert z\rvert^2/\operatorname{Im}(z))^{n+\alpha+1} \end{equation} for $z\in\mathbb H$ with $\lvert z\rvert>R$ and $0<\varepsilon<1$, where $C\geq0$ and $R>1$ are finite constants and $n\in\mathbb N$. Notice also that $u=\lim_{\varepsilon\to0}u_\varepsilon$ in $\mathbb H$ in the sense of normal convergence.
We now proceed to details. By Theorem \ref{alphaobstructionclass} we have that $u_\varepsilon\in \mathcal{V}_\alpha$. From \eqref{tempgrowthbound} and Lemma \ref{coefficientlemma} we have that $u_\varepsilon\in \mathcal{V}_{\alpha,n}$. Therefore \begin{equation}\label{alphaobstructionfcnregularization} u_\varepsilon(z)=\sum_{k=0}^{n} c_k(\varepsilon) (\operatorname{Im} z)^{\alpha+1} p_{k,\alpha}(z),\quad z\in\mathbb H, \end{equation} for some $c_0(\varepsilon),\dots,c_{n}(\varepsilon)\in\mathbb C$. Another application of Lemma \ref{coefficientlemma} gives that $\lvert c_k(\varepsilon)\rvert\leq C'C$ for $k=0,1,\dots,n$, where $C$ is as in \eqref{tempgrowthbound} and $C'=C'_{\alpha,n,R}$ is a positive constant. By a compactness argument we can extract a subsequence $\varepsilon=\varepsilon_j\to 0$ of positive real numbers such that $c_k(\varepsilon_j)\to c_k$ as $j\to\infty$ for $k=0,1,\dots,n$. The conclusion of the theorem now follows by setting $\varepsilon=\varepsilon_j$ in \eqref{alphaobstructionfcnregularization} and letting $j\to\infty$. \end{proof}
We point out that the assumption in Theorem \ref{alphaobstructionclassdistributionalversion} is that $$ \lim_{y\to 0}\int_{-\infty}^\infty u(x+iy)\varphi(x)\, dx=0 $$ for every $\varphi \in C_0^\infty(\mathbb R)$.
The case of usual harmonic functions deserves special mention.
\begin{cor}\label{harmonicobstructionclassdistributionalversion} Let $u$ be a harmonic function in $\mathbb H$ which is of temperate growth at infinity. Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. Then $u\in \mathcal{V}_0$. \end{cor}
We record also the following slight improvement of the if part of Lemma \ref{coefficientlemma}.
\begin{prop}\label{Vanfromrelaxedgrowth} Let $u\in \mathcal{V}_\alpha$ for some $\alpha>-1$. Let $n\in\mathbb N$. Assume that \begin{equation*} u(z)=o((\lvert z\rvert^2/\operatorname{Im}(z))^{n+\alpha+2}) \end{equation*} as $\mathbb H\ni z\to\infty$ in the extended complex plane. Then $u\in \mathcal{V}_{\alpha,n}$. \end{prop}
\begin{proof} If $u\in\mathcal{V}_{\alpha,0}$ there is nothing to prove. Assume therefore that $u\in\mathcal{V}_{\alpha,m}$ and $u\not\in\mathcal{V}_{\alpha,m-1}$ for some $m\in\mathbb Z^+$. An application of Lemma \ref{ulimitlemma} with $0<\theta<\pi$ chosen such that $p_{m,\alpha}(e^{i\theta})\neq0$ shows that $u$ has order of growth at least $m+\alpha+1$ at infinity. Therefore $m< n+1$, so that $u\in\mathcal{V}_{\alpha,n}$. \end{proof}
Proposition \ref{Vanfromrelaxedgrowth} is included for the sake of convenience. We shall prove more refined versions of Proposition \ref{Vanfromrelaxedgrowth} in later sections.
\section{A uniqueness result for the case \texorpdfstring{$\alpha\neq0$}{nonzero alpha}} \label{sectionuniguenessalphaneq0}
We now turn our attention to uniqueness results for $\alpha$-harmonic functions in $\mathbb H$. In this section we study the case of parameters $\alpha>-1$ such that $\alpha\neq 0$. The special case of usual harmonic functions is investigated in later sections. We begin our analysis with a study of the zeros of the polynomials $p_{k,\alpha}$.
A classical result of Enestr\"om-Kakeya says that if \begin{equation}\label{analyticpolynomial} p(z)=\sum_{k=0}^n a_kz^k \end{equation} is an analytic polynomial of degree $n\geq0$ such that $a_k\leq a_{k+1}$ for $0\leq k<n$, then the zeroes of $p$ are all located in the closed unit disc $\bar\mathbb D$. An easy proof of this result can be found in Gardner and Govil \cite[Theorem 1.3]{gardner2014enestrom}.
\begin{cor}\label{EKcor} Let $p$ be an analytic polynomial of the form \eqref{analyticpolynomial} with positive coefficients: $a_k>0$ for $0\leq k\leq n$. Set $$ r=\min_{0\leq k<n}a_k/a_{k+1}\quad \text{and} \quad R=\max_{0\leq k<n}a_k/a_{k+1}. $$ Then the zeroes of $p$ are all located in the annulus $\{z\in\mathbb C:\ r\leq\lvert z\rvert\leq R\}$. \end{cor}
\begin{proof} We first show that the zeroes of $p$ are all located in the disc $\{z\in\mathbb C:\ \lvert z\rvert\leq R\}$. Consider the polynomial $g(z)=p(Rz)$. By assumption the polynomial $g$ satisfies the assumptions of the Enestr\"om-Kakeya theorem quoted above. We conclude that the zeroes of $g$ are all located in $\bar\mathbb D$. This yields that the zeroes of $p$ are all located in the disc $\{z\in\mathbb C:\ \lvert z\rvert\leq R\}$.
We next show that the zeroes of $p$ are all located in the exterior disc $\{z\in\mathbb C:\ \lvert z\rvert\geq r\}$. Consider the polynomial $h(z)=z^np(1/z)$. Notice that $$ h(z)=\sum_{k=0}^n a_{n-k}z^k. $$ By the result of the previous paragraph, the zeroes of $h$ are all located in the disc $\{z\in\mathbb C:\ \lvert z\rvert\leq 1/r\}$. Therefore the zeroes of $p$ are all located in the exterior disc $\{z\in\mathbb C:\ \lvert z\rvert\geq r\}$. \end{proof}
Our interest in zeroes of polynomials stems from the following result.
\begin{theorem}\label{pkakzerofree} Let $\alpha>-1$ and $\alpha\neq0$. Let $p_{k,\alpha}$ be as in \eqref{eq:pkallealfa} for some $k\in\mathbb N$. Then $p_{k,\alpha}(e^{i\theta})\neq0$ for $e^{i\theta}\in\mathbb T$. \end{theorem}
\begin{proof} By \eqref{eq:pkallealfa} it suffices to show that $s_{k,\alpha}(e^{i\theta})\neq0$ for $e^{i\theta}\in\mathbb T$, where $s_{k,\alpha}$ is as in \eqref{eq:skallealfa}. Set $a_{j}(\alpha)=\frac{1}{j!}(\alpha+1)_j$. Observe that \begin{equation}\label{coefficientquotient} \frac{a_{j}(\alpha)}{a_{j+1}(\alpha)}=\frac{j+1}{\alpha+j+1} =1-\frac{\alpha}{\alpha+j+1} \end{equation} for $0\leq j<k$.
Assume next that $\alpha>0$. In this case the quotients in \eqref{coefficientquotient} increase in $j$ and we have that $a_{j}(\alpha)/a_{j+1}(\alpha)\leq k/(\alpha+k)$ for $0\leq j<k$. By Corollary \ref{EKcor} the zeroes of $s_{k,\alpha}$ are all located in the disc $\{z\in\mathbb C:\ \lvert z\rvert\leq k/(\alpha+k)\}$ which is compactly contained in $\mathbb D$.
Assume next that $-1<\alpha<0$. In this case the quotients in \eqref{coefficientquotient} decrease in $j$ and we have that $a_{j}(\alpha)/a_{j+1}(\alpha)\geq 1/(\alpha+1)$ for $0\leq j<k$. By Corollary \ref{EKcor} the zeroes of $s_{k,\alpha}$ are all located in the exterior disc $\{z\in\mathbb C:\ \lvert z\rvert\geq 1/(\alpha+1)\}$ which is disjoint from $\bar\mathbb D$. \end{proof}
\begin{remark} We point out that the zero set of $p_{k,0}$ intersects the unit circle if $k\geq1$. In fact, from \eqref{eq:skallealfa} we have that $$ s_{k,0}(z)=\sum_{j=0}^kz^j=(1-z^{k+1})/(1-z) $$ for $k\in\mathbb N$. \end{remark}
We now consider the class $\mathcal{V}_{\alpha}$.
\begin{theorem}\label{Vauniquenessaneq0} Let $u\in\mathcal{V}_{\alpha}$ for some $\alpha>-1$ with $\alpha\ne0$. Assume that there is a sequence $\{z_j\}$ in $\mathbb H$ with $\lvert z_j\rvert \to\infty$ as $j\to\infty$ such that \begin{equation}\label{sequencevanishing} \lim_{j\to\infty} \frac{u(z_j)}{(\operatorname{Im}(z_j))^{\alpha+1}}=0. \end{equation} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} We put $u\in\mathcal{V}_{\alpha}$ on the form \eqref{alphaobstructionfcn} for some $n\in\mathbb N$ and constants $c_0,\dots,c_n\in\mathbb C$, where the $p_{k,\alpha}$'s are as in \eqref{eq:pkallealfa}. Recall that the polynomial $p_{k,\alpha}$ is homogeneous of degree $k\in\mathbb N$. By Theorem \ref{pkakzerofree} the polynomial $p_{k,\alpha}$ has no zeroes on the unit circle. From these two properties of $p_{k,\alpha}$ we have that the quotient $\lvert p_{k,\alpha}(z)\rvert/\lvert z\rvert^k$ is bounded from above and below by finite positive constants uniformly in the punctured plane. From \eqref{sequencevanishing} we now have that $$ 0=\lim_{j\to\infty} \frac{u(z_j)}{p_{n,\alpha}(z_j)(\operatorname{Im}(z_j))^{\alpha+1}}=c_n. $$ Repeating the argument we conclude that $c_k=0$ for $0\leq k\leq n$. Therefore $u(z)=0$ for all $z\in\mathbb H$. \end{proof}
We emphasize that condition \eqref{sequencevanishing} can be checked along any sequence $\{z_j\}$ in $\mathbb H$ such that $z_j\to\infty$ in the extended complex plane $\mathbb C_\infty$ as $j\to\infty$.
We now turn to uniqueness theorems for $\alpha$-harmonic functions with $\alpha\neq0$. We first prove Theorem \ref{introuniquenessthm} stated in the introduction.
\begin{proof}[Proof of Theorem \ref{introuniquenessthm}] By assumption \eqref{classicalvanishingrealline} (formula \eqref{vanishingDirichletbdrycondition}) we can apply Theorem \ref{alphaobstructionclass} to conclude that $u\in\mathcal{V}_\alpha$. We next apply Theorem \ref{Vauniquenessaneq0} to conclude that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
We next extend the validity of Theorem \ref{introuniquenessthm} to allow for a distributional boundary value instead of \eqref{vanishingDirichletbdrycondition}.
\begin{theorem}\label{thm:distributionaluniquenessaneq0} Let $\alpha>-1$ and $\alpha\ne0$. Let $u$ be an $\alpha$-harmonic function in $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that there is a sequence $\{z_j\}$ in $\mathbb H$ with $z_j\to\infty$ in $\mathbb C_\infty$ as $j\to\infty$ such that \eqref{sequencevanishing} holds. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} In view of assumption \eqref{classicalvanishingrealline}' we can apply Theorem \ref{alphaobstructionclassdistributionalversion} to conclude that $u\in\mathcal{V}_\alpha$. We next apply Theorem \ref{Vauniquenessaneq0} to conclude that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
We close this section with a relaxed version of Theorem \ref{Vauniquenessaneq0}.
\begin{theorem}\label{Vancharacterizationaneq0} Let $u\in\mathcal{V}_{\alpha}$ for some $\alpha>-1$ with $\alpha\neq 0$. Let $n\in\mathbb N$. Assume that there is a sequence $\{z_j\}$ in $\mathbb H$ with $z_j\to\infty$ in $\mathbb C_\infty$ as $j\to\infty$ such that \begin{equation*} \lim_{j\to\infty} \frac{u(z_j)}{\lvert z_j\rvert^{n+1}(\operatorname{Im}(z_j))^{\alpha+1}}=0. \end{equation*} Then $u\in\mathcal{V}_{\alpha,n}$. \end{theorem}
The proof of Theorem \ref{Vancharacterizationaneq0} follows along the same lines as the proof of Theorem \ref{Vauniquenessaneq0}. We omit the details.
\section{Harmonic functions vanishing along geodesics}
We shall now turn to uniqueness results for classical harmonic functions in $\mathbb H$ ($\alpha=0$). Recall the harmonic polynomials $$ u_{k,0}(z)=\operatorname{Im}(z^{k+1}),\quad z\in\mathbb C, $$ for $k\in\mathbb N$ which all vanish on the real line $\mathbb R$. Observe that the zero set of $u_{k,0}$ is a union of $k+1$ lines passing through the origin. A treatment of the uniqueness problem for classical harmonic functions along the lines of what we did in Section \ref{sectionuniguenessalphaneq0} thus calls for a more demanding vanishing condition at infinity. In this section we shall consider vanishing conditions along geodesics in $\mathbb H$, that is, along rays in $\mathbb H$ that are parallel to the positive imaginary axis.
Let us first consider the class $\mathcal{V}_{0}$.
\begin{theorem}\label{V0uniquenessgeodesic} Let $u\in\mathcal{V}_{0}$. Assume that \begin{equation}\label{geodesicvanishing} \lim_{y\to+\infty}u(x+iy)/y=0 \end{equation} for $x=x_j\in\mathbb R$ ($j=1,2$) with $x_1\neq x_2$. Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} We put $u\in\mathcal{V}_{0}$ on the form \eqref{harmonicobstructionfcn} for some $c_1,\dots, c_{n+1}\in\mathbb C$ and $n\in\mathbb N$. By adding an extra term in \eqref{harmonicobstructionfcn} if necessary, we can arrange that $n\geq1$ is odd. We shall prove that $c_n=c_{n+1}=0$. This will complete the proof of the theorem.
We now proceed to details. Let $x\in\mathbb R$. From the binomial theorem we have that \begin{align*} (x+iy)^{n+1}&= (iy)^{n+1}+(n+1)x(iy)^{n} +O(y^{n-1})\\ &=(-1)^{(n+1)/2}y^{n+1}+i(n+1)(-1)^{(n-1)/2}xy^{n}+O(y^{n-1}) \end{align*} as $y\to +\infty$. Passing to the imaginary part we have that \begin{equation}\label{(n+1)thtermasymptotics} \operatorname{Im}((x+iy)^{n+1})=(n+1)(-1)^{(n-1)/2}xy^{n}+O(y^{n-1}) \end{equation} as $y\to +\infty$. A similar consideration shows that \begin{equation}\label{nthtermasymptotics} \operatorname{Im}((x+iy)^{n})=(-1)^{(n-1)/2}y^{n}+O(y^{n-1}) \end{equation} as $y\to +\infty$.
We now return to the function $u$. From \eqref{(n+1)thtermasymptotics} and \eqref{nthtermasymptotics} we have that $$ \lim_{y\to+\infty}u(x+iy)/y^{n}=c_{n+1} (n+1)(-1)^{(n-1)/2}x+c_{n}(-1)^{(n-1)/2} $$ for $x\in\mathbb R$. From \eqref{geodesicvanishing} we have that $$ c_{n+1} (n+1)(-1)^{(n-1)/2}x+c_{n}(-1)^{(n-1)/2}=0 $$ for $x=x_j$ ($j=1,2$). Since $x_1\neq x_2$, we conclude that $c_{n+1}=c_{n}=0$. \end{proof}
We next turn to uniqueness theorems for harmonic functions.
\begin{theorem}\label{classicaluniquenessthm2geodesics} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item \label{classicalvanishingonrealboundary} Assume that $u(z)\to 0$ as $\mathbb H\ni z\to x$ for every $x\in\mathbb R$. \item\label{vanishingalonggeodesics} Assume that $$ \lim_{y\to+\infty}u(x+iy)/y=0 $$ for $x=x_j\in\mathbb R$ ($j=1,2$) with $x_1\neq x_2$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} From Theorem \ref{alphaobstructionclass} we have that $u\in\mathcal{V}_0$. An application of Theorem \ref{V0uniquenessgeodesic} yields that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
Recall the definition of $u_{k,\alpha}$ in \eqref{ukafunction} and let $a\in\mathbb R$. We point out that the function $$ u(z)=u_{1,0}(z-a)=\operatorname{Im}((z-a)^2)=2(x-a)y,\quad z=x+iy\in\mathbb C, $$ is of quadratic growth, harmonic, and vanishes on the lines $x=a$ and $y=0$. Thus, the vanishing assumption \eqref{vanishingalonggeodesics} in Theorem \ref{classicaluniquenessthm2geodesics} can not be relaxed to vanishing along one geodesic $x=a$ only.
We next extend the validity of Theorem \ref{classicaluniquenessthm2geodesics} to allow for a distributional boundary value in \eqref{classicalvanishingonrealboundary}.
\begin{theorem}\label{distributionaluniquenessthm2geodesics} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that $$ \lim_{y\to+\infty}u(x+iy)/y=0 $$ for $x=x_j\in\mathbb R$ ($j=1,2$) with $x_1\neq x_2$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} From Theorem \ref{alphaobstructionclassdistributionalversion} we have that $u\in\mathcal{V}_0$. An application of Theorem \ref{V0uniquenessgeodesic} yields that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
The results of this section together with corresponding results from Section \ref{sectionuniguenessalphaneq0} yield considerable advancement from a recent result by Carlsson and Wittsten \cite[Corollary 1.9 ]{carlsson2016dirichlet}. In fact, instead of \eqref{sequencevanishing} or \eqref{geodesicvanishing} those authors considered functions such that $$ \lim_{y\to\infty}u(x+iy)/y^{\alpha+1}=0 $$ for all $\lvert x\rvert<\delta$, where $\delta>0$.
The importance of condition \eqref{geodesicvanishing} for two distinct geodesics brings to mind an analogous situation in the study non-tangential limits. If $u$ is a bounded harmonic function in $\mathbb H$ such that $$ \lim_{r\to0}u(re^{i\theta_1})=L=\lim_{r\to0}u(re^{i\theta_2}) $$ for some $0<\theta_1<\theta_2<\pi$, then $u$ has non-tangential limit $L$ at the origin (see Axler et al. \cite[Theorem 2.10]{ABR}). This latter result can be thought of as a harmonic function version of a classical result of Lindel\"of (see Rudin \cite[Theorem 12.10]{Rudin}).
\section{Harmonic functions vanishing along rays}\label{section:rays}
In this section we continue our study of uniqueness results for classical harmonic functions in $\mathbb H$ ($\alpha=0$). We now consider functions vanishing along rays in $\mathbb H$ emanating from the origin, that is, along rays of the form $\{te^{i\theta}:\ t>0\}$, where $0<\theta<\pi$.
We shall use a concept of admissible function of angles that we proceed to define.
\begin{dfn}\label{dfnadmissiblefcnangles} Let $E\subset(0,\pi)$ be a set and $\eta:E\to\mathbb Z^+$ a positive integer valued function on $E$. We say that the function element $(E,\eta)$ is an {\it admissible function of angles} if it has the property that for every $k\in\mathbb Z^+$ there exists $\theta\in E$ such that $\sin(k\theta)\neq0$ and $k\geq\eta(\theta)$. \end{dfn}
Our interest in admissible functions of angles stems from the following theorem.
\begin{theorem}\label{V0uniquenessray} Let $u\in\mathcal{V}_{0}$. Assume that there is an admissible function of angles $(E,\eta)$ such that $$ \lim_{t\to+\infty}u(te^{i\theta})/t^{\eta(\theta)}=0 $$ for every $\theta\in E$. Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} We put $u\in\mathcal{V}_{0}$ on the form $$ u(z)=\sum_{k=1}^n c_k\operatorname{Im}(z^k),\quad z\in\mathbb H, $$ for some $c_1,\dots,c_n\in\mathbb C$ and $n\in \mathbb Z^+$. It suffices to show that $c_n=0$. For such functions, $$ \lim_{t\to+\infty}u(e^{i\theta}t)/t^{n} =\lim_{t\to+\infty}\sum_{k=1}^nc_k\operatorname{Im}(e^{ik\theta}t^k)/t^n=c_n \sin(n\theta). $$ Since $(E,\eta)$ is an admissible function of angles we may choose $\theta\in E $ such that $\sin(n\theta)\ne0$ and $\eta(\theta)\le n$. It follows that $$ c_n\sin(n\theta)=\lim_{t\to+\infty}\frac{u(te^{i\theta})}{t^{\eta(\theta)}}\frac{t^{\eta(\theta)}}{t^{n}}=0. $$ Since $\sin(n\theta)\ne0$ we conclude that $c_n=0$. \end{proof}
We next turn to uniqueness results.
\begin{theorem}\label{classicalfcnofanglesvanishing} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item Assume that $u(z)\to 0$ as $\mathbb H\ni z\to x$ for every $x\in\mathbb R$. \item\label{vanishingatfcnofangles} Assume that there is an admissible function of angles $(E,\eta)$ such that $$ \lim_{t\to+\infty}u(te^{i\theta})/t^{\eta(\theta)}=0 $$ for every $\theta\in E$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} From Theorem \ref{alphaobstructionclass} we have that $u\in\mathcal{V}_0$. An application of Theorem \ref{V0uniquenessray} yields that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
We next extend the validity of Theorem \ref{classicalfcnofanglesvanishing} to allow for a distributional boundary value in \eqref{classicalvanishingonrealboundary}.
\begin{theorem}\label{fcnofanglesvanishing} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that there is an admissible function of angles $(E,\eta)$ such that $$ \lim_{t\to+\infty}u(te^{i\theta})/t^{\eta(\theta)}=0 $$ for every $\theta\in E$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{theorem}
\begin{proof} From Theorem \ref{alphaobstructionclassdistributionalversion} we have that $u\in\mathcal{V}_0$. An application of Theorem \ref{V0uniquenessray} yields that $u(z)=0$ for $z\in\mathbb H$. \end{proof}
Let us comment on the generic situation.
\begin{cor}\label{genericrayvanishing} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that $u$ satisfies the vanishing condition that $\lim_{t\to+\infty}u(te^{i\theta})/t=0$ for some $0<\theta<\pi$ which is not a rational multiple of $\pi$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{cor}
\begin{proof} Let $E=\{\theta\}$ and $\eta(\theta)=1$. Then $(E,\eta)$ is an admissible function of angles for which condition (2) of Theorem \ref{fcnofanglesvanishing} holds. Thus, an application of the mentioned theorem gives the result. \end{proof}
Theorem \ref{classicalgenericrayuniquenessresult} in the introduction is proved in the same way as Corollary \ref{genericrayvanishing} by using Theorem \ref{classicalfcnofanglesvanishing} instead of Theorem \ref{fcnofanglesvanishing}.
The examples $$ u_k(z)=\operatorname{Im}(z^k) $$ for $k=1,2,\dots$, show that the arithmetic condition on $\theta\in(0,\pi)$ in Corollary \ref{genericrayvanishing} is to the point. Clearly, the polynomial $u_k$ vanishes on $\mathbb R$. Moreover, if $\theta=\frac{m}{n}\pi$ for some integers $m$ and $n$ with $n\geq1$, then the function $u_n$ vanishes on the line $e^{i\theta}\mathbb R$.
\begin{cor}\label{rationalrayvanishing} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. Let $\tau=\frac{m}{n}\pi$ with $m,n\in\mathbb Z^+$ relatively prime and $m<n$. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that $\lim_{t\to+\infty}u(te^{i\tau})/t=0$. \item[(3)] Assume that $\lim_{t\to+\infty}u(te^{i\theta})/t^n=0$ for some $0<\theta<\pi$ which is not a rational multiple of $\pi$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{cor}
\begin{proof} Let $E=\{\tau,\theta\}$ and set $\eta(\tau)=1$ and $\eta(\theta)=n$. Then $(E,\eta)$ is an admissible function of angles such that condition (2) of Theorem \ref{fcnofanglesvanishing} is satisfied. Applying the mentioned theorem thus gives the result. \end{proof}
It might be worthwhile to state the special case of Corollary \ref{rationalrayvanishing} obtained when $\tau=\pi/2$:
\begin{cor}\label{testthm} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that $u$ satisfies the vanishing condition that $\lim_{y\to+\infty}u(iy)/y=0$. \item[(3)] Assume that $u$ satisfies the vanishing condition that $\lim_{t\to+\infty}u(te^{i\theta})/t^2=0$ for some $0<\theta<\pi$ which is not a rational multiple of $\pi$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{cor}
Let us elaborate some more on the theme of Corollary \ref{testthm}.
\begin{cor}\label{testthm2} Let $u$ be a harmonic function in the open upper half-plane $\mathbb H$ which is of temperate growth at infinity. \begin{enumerate} \item[(1)'] Assume that $\lim_{y\to0+} u(\cdot+iy)=0$ in $\mathscr{D}'(\mathbb R)$. \item[(2)] Assume that $$ \lim_{t\to+\infty}u(te^{i2^{-k}\pi})/t^{2^{k-1}}=0 $$ for $k=1,2,\dots$. \end{enumerate} Then $u(z)=0$ for all $z\in\mathbb H$. \end{cor}
\begin{proof} Let $E=\{2^{-k}\pi:\ k=1,2,\dots\}$ and set $\eta(2^{-k}\pi)=2^{k-1}$ for $k=1,2,\dots$. From Theorem \ref{constructionfoainfinite} in the next section we have that $(E,\eta)$ is an admissible function of angles. Clearly condition (2) of Theorem \ref{fcnofanglesvanishing} is satisfied. Applying the mentioned theorem thus gives the result. \end{proof}
\section{Admissible functions of angles}\label{section:construction}
In this section we shall provide some constructions of admissible functions of angles (see Theorems \ref{constructionfoainfinite} and \ref{constructionfoafinite}). Examples of such function elements are needed for successful applications of Theorems \ref{classicalfcnofanglesvanishing} or \ref{fcnofanglesvanishing}. The set of admissible functions of angles is equipped with a natural partial order. We show that the admissible functions of angles constructed are in fact the minimal elements in the partial order (see Theorem \ref{minimalelementsA}). We also show that every admissible function of angles has a lower bound which is minimal (see Theorem \ref{lowerboundfoa}). The construction uses some notions from ideal theory for the ring of integers $\mathbb Z$.
Let $\theta\in\mathbb R$ be a real number and consider the set of integers $$ \mathcal{I}(\theta)=\{m\in \mathbb Z: \ \sin(m\theta)=0 \}. $$ Using standard properties of the sine function, it is straight\-forward to check that the set $\mathcal{I}(\theta)$ is an ideal in $\mathbb Z$. Since the ring $\mathbb Z$ is a principal ideal domain, we have that $$ \mathcal{I}(\theta)=d(\theta)\mathbb Z $$ for some integer $d(\theta)$. Since the units in $\mathbb Z$ are $\pm1$, this integer $d(\theta)$ is uniquely determined by the condition that $d(\theta)\geq0$. This defines a function $d:\theta\mapsto d(\theta)$ from $\mathbb R$ to $\mathbb N$.
We observe that $d(\theta)=0$ if and only if $\theta\in\mathbb R$ is not a rational multiple of $\pi$. Furthermore, if $\theta=\frac{r}{s}\pi $ with $r\in\mathbb Z$ and $s\in \mathbb Z^+$ relatively prime, then $d(\theta)=s$. These assertions are straight\-forward to check.
We denote by $\operatorname{lcm}(a_1,\dots,a_n)$ the least common multiple of integers $a_1,\dots,a_n$. Recall that $\operatorname{lcm}(a_1,\dots,a_n)$ is the non-negative generator of the ideal $\cap_{k=1}^na_k\mathbb Z$: $$ \operatorname{lcm}(a_1,\dots,a_n)\mathbb Z=\cap_{k=1}^na_k\mathbb Z. $$ The symbol $a\vert b$ means that the integer $b$ is divisible by the integer $a$ in the usual sense that $b=ac$ for some integer $c$.
We are now ready for the construction of admissible functions of angles.
\begin{theorem}\label{constructionfoainfinite} Let $\{\theta_k\}_{k=1}^\infty$ be an infinite sequence of rational multiples of $\pi$ from the interval $(0,\pi)$ such that $$ d(\theta_k)\nmid \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ for $k=2,3,\dots$. Set $E=\{\theta_k:\ k=1,2,\dots \}$ and define $\eta:E\to\mathbb Z^+$ by $\eta(\theta_1)=1$ and $$ \eta(\theta_k)= \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ for $k=2,3,\dots$. Then $(E,\eta)$ is an admissible function of angles. \end{theorem}
\begin{proof} Observe first that the requirements on the sequence $\{\theta_k\}_{k=1}^\infty$ guarantee that $$ 2\leq \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{j})) <\operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k})) $$ for $1\leq j<k$. Thus $\theta_j\neq\theta_k$ if $j\neq k$ and $\cap_{k=1}^\infty d(\theta_k)\mathbb Z=\{0\}$.
The function $\eta:E\to\mathbb Z^+$ is well-defined since the $\theta_k$'s are distinct. We proceed to prove that $(E,\eta)$ is an admissible function of angles. Let $m\in\mathbb Z^+$ be a positive integer. Since $\cap_{k=1}^\infty d(\theta_k)\mathbb Z=\{0\}$, there exists $k\in\mathbb Z^+$ such that $m\not\in d(\theta_k)\mathbb Z$. Furthermore, by the well-ordering of positive integers we can also arrange that $m\in d(\theta_j)\mathbb Z$ for $1\leq j<k$. Since $m\not\in d(\theta_k)\mathbb Z$, we have that $\sin(m\theta_k)\neq0$. If $k=1$, we clearly have that $m\geq1=\eta(\theta_1)$. Assume next that $k\geq2$. Since $m\in d(\theta_j)\mathbb Z$ for $1\leq j<k$, we have $$ m\in \cap_{j=1}^{k-1}d(\theta_j)\mathbb Z = \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1}))\mathbb Z, $$ which allows us to conclude that $m\geq \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1}))=\eta(\theta_k)$. This completes the proof of the theorem. \end{proof}
Theorem \ref{constructionfoainfinite} provides a multitude of examples of admissible functions of angles. For example, let $\theta_k=\frac{n_k}{2^k}\pi$ with $1\leq n_k<2^k$ odd for $k=1,2,\dots$. Now $d(\theta_k)=2^{k}$ for $k=1,2,\dots$, which makes evident that the assumption of Theorem \ref{constructionfoainfinite} is satisfied. In this case $\eta(\theta_k)=2^{k-1}$ for $k=1,2,\dots$.
We next turn to the construction of admissible functions of angles $(E,\eta)$ with $E$ finite.
\begin{theorem}\label{constructionfoafinite} Let $\{\theta_k\}_{k=1}^n$ be a finite sequence of rational multiples of $\pi$ from the interval $(0,\pi)$ such that $$ d(\theta_k)\nmid \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ for $2\leq k\leq n$, where $n\in\mathbb N$. Let $\theta_{n+1}\in(0,\pi)$ be an irrational multiple of $\pi$. Set $E=\{\theta_k:\ 1\leq k\leq n+1 \}$ and define $\eta:E\to\mathbb Z^+$ by $\eta(\theta_1)=1$ and $$ \eta(\theta_k)= \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ for $2\leq k\leq n+1$. Then $(E,\eta)$ is an admissible function of angles. \end{theorem}
\begin{proof} The function $\eta:E\to\mathbb Z^+$ is well-defined since the $\theta_k$'s are distinct. We proceed to prove that $(E,\eta)$ is an admissible function of angles. Let $m\in\mathbb Z^+$ be a positive integer. If $m\geq \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))$, then it is straight\-forward to check that $\sin(m\theta_{n+1})\neq0$ and $m\geq\eta(\theta_{n+1})$.
It remains to consider the case $1\leq m< \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))$. Since \begin{equation*} \cap_{k=1}^n d(\theta_k)\mathbb Z=\operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))\mathbb Z, \end{equation*} there exists $k\in\mathbb Z^+$ with $1\leq k\leq n$ such that $m\not\in d(\theta_k)\mathbb Z$ and $m\in d(\theta_j)\mathbb Z$ for $1\leq j<k$. Since $m\not\in d(\theta_k)\mathbb Z$, we have that $\sin(m\theta_k)\neq0$. If $k=1$, we clearly have that $m\geq1=\eta(\theta_1)$. Assume next that $k\geq2$. Since $m\in d(\theta_j)\mathbb Z$ for $1\leq j<k$, we have $$ m\in \cap_{j=1}^{k-1}d(\theta_j)\mathbb Z = \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1}))\mathbb Z, $$ which allows us to conclude that $m\geq \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1}))=\eta(\theta_k)$. This yields the conclusion of the theorem. \end{proof}
We point out that the case of an empty sequence $\{\theta_k\}_{k=1}^n$ ($n=0$) in Theorem \ref{constructionfoafinite} yields the admissible function of angles used in the proof of Corollary \ref{genericrayvanishing}.
Let $E_j\subset(0,\pi)$ and $\eta_j:E_j\to\mathbb Z^+$ ($j=1,2$) be such that $E_1\subset E_2$ and $\eta_1(\theta)\geq \eta_2(\theta)$ for $\theta\in E_1$. If $(E_1,\eta_1)$ is an admissible function of angles, then so is $(E_2,\eta_2)$. For applications of Theorems \ref{classicalfcnofanglesvanishing} or \ref{fcnofanglesvanishing} it is of interest to have an admissible function of angles $(E,\eta)$ which is as economical as possible in this respect.
Let us denote by $\mathcal{A}$ the set of all admissible functions of angles. We equip the set $\mathcal{A}$ with a relation ($\leq$) defined by $(E_1,\eta_1)\leq (E_2,\eta_2)$ if $E_1\subset E_2$ and $\eta_1(\theta)\geq\eta_2(\theta)$ for $\theta\in E_1$. It is straight\-forward to check that this relation gives the set $\mathcal{A}$ a structure of a partial order.
We denote by $\mathcal{A}_0$ the sub\-set of $\mathcal{A}$ consisting of those admissible functions of angles $(E,\eta)$ that are constructed as in Theorem \ref{constructionfoainfinite} or \ref{constructionfoafinite}.
We next prove that every element in $\mathcal{A}$ has a lower bound from the set $\mathcal{A}_0$.
\begin{theorem}\label{lowerboundfoa} Let $(E_1,\eta_1)\in\mathcal{A}$ be arbitrary. Then there exists $(E,\eta)\in\mathcal{A}_0$ such that $(E,\eta)\leq (E_1,\eta_1)$ in $\mathcal{A}$. \end{theorem}
\begin{proof} The proof relies on a construction of a suitable sequence $\{\theta_k\}$ from the set $E_1\subset (0,\pi)$. Since $(E_1,\eta_1)$ is an admissible function of angles there exists $\theta\in E_1$ such that $\eta_1(\theta)=1$. We set $\theta_1=\theta$. If $\theta_1$ is an ir\-rational multiple of $\pi$, then $(E,\eta)\leq (E_1,\eta_1)$ in $\mathcal{A}$, where $E=\{\theta_1\}$ and $\eta(\theta_1)=1$ are as in Theorem \ref{constructionfoafinite} with $n=0$ there. If $\theta_1$ is a rational multiple of $\pi$, then we proceed to construct an element $\theta_2$ as described in the following paragraph.
Assume that $\theta_1,\dots,\theta_{n}\in E_1$ are rational multiples of $\pi$ such that $$ d(\theta_k)\nmid \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ and $$ \eta_1(\theta_k)\leq \operatorname{lcm}(d(\theta_1),\dots,d(\theta_{k-1})) $$ for $2\leq k\leq n$, where $n\geq1$. Let $m=\operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))$. Clearly $m\in\mathbb Z^+$. Since $(E_1,\eta_1)$ is an admissible function of angles there exists $\theta\in E_1$ such that $\sin(m\theta)\neq 0$ and $m\geq \eta_1(\theta)$. We set $\theta_{n+1}=\theta$. If $\theta_{n+1}$ is an irrational multiple of $\pi$, then the construction from Theorem \ref{constructionfoafinite} provides us with a function of angles $(E,\eta)\in\mathcal{A}_0$ such that $(E,\eta)\leq (E_1,\eta_1)$ in $\mathcal{A}$.
Assume next that $\theta_{n+1}$ is a rational multiple of $\pi$. Since $\sin(m\theta_{n+1})\neq0$, we have $$ m\not\in \mathcal{I}(\theta_{n+1})=d(\theta_{n+1})\mathbb Z, $$ which gives that $d(\theta_{n+1})\nmid m=\operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))$. Notice also that the in\-equality $\eta_1(\theta_{n+1})\leq\operatorname{lcm}(d(\theta_1),\dots,d(\theta_{n}))$ is evident from construction. We are now in position to repeat the procedure from the previous paragraph.
Unless the above procedure terminates in a finite number of steps, the principle of induction provides us with with an in\-finite sequence $\{\theta_k\}_{k=1}^\infty$ of rational multiples of $\pi$ from the set $E_1$ satisfying the assumptions of Theorem \ref{constructionfoainfinite}. The construction from Theorem \ref{constructionfoainfinite} now provides us with an admissible function of angles $(E,\eta)\in\mathcal{A}_0$ such that $(E,\eta)\leq (E_1,\eta_1)$ in $\mathcal{A}$. \end{proof}
Recall that an element $m$ in a partial order $\mathcal{M}$ is called minimal (in $\mathcal{M}$) if it has the property that $x\in \mathcal{M}$ and $x\leq m$ implies that $x=m$.
We next prove that the admissible functions of angles $(E,\eta)$ from the set $\mathcal{A}_0$ are minimal elements in $\mathcal{A}$.
\begin{lemma}\label{cfoaminimal} Let $(E,\eta)\in \mathcal{A}_0$. Let $(E_1,\eta_1)\in\mathcal{A}$ be such that $(E_1,\eta_1)\leq (E,\eta)$ in $\mathcal{A}$. Then $(E_1,\eta_1)=(E,\eta)$. \end{lemma}
\begin{proof} We consider the case when $(E,\eta)$ is as in Theorem \ref{constructionfoainfinite}. The remaining case when $(E,\eta)$ is as in Theorem \ref{constructionfoafinite} is similar and therefore omitted. By construction we have that $\eta(\theta_1)=1$, $\eta(\theta_j)<\eta(\theta_k)$ for $1\leq j<k$ and $\sin(\eta(\theta_k)\theta_j)=0$ for $1\leq j<k$. We need to prove that $\theta_k\in E_1$ and $\eta_1(\theta_k)=\eta(\theta_k)$ for $k=1,2,\dots$. We proceed by induction.
Since $(E_1,\eta_1)$ is an admissible function of angles there exists $\theta\in E_1$ such that $\eta_1(\theta)=1$. Since $(E_1,\eta_1)\leq (E,\eta)$ we have that $\theta=\theta_j$ for some $j\in\mathbb Z^+$ and $\eta(\theta_j)=1$. The condition $\eta(\theta_j)=1$ forces $j=1$. We have shown that $\theta _1\in E_1$ and $\eta_1(\theta_1)=\eta(\theta_1)=1$.
Assume next that $\theta_j\in E_1$ and $\eta_1(\theta_j)=\eta(\theta_j)$ for $1\leq j<k$, where $k\geq2$. We shall prove that $\theta_k\in E_1$ and $\eta_1(\theta_k)=\eta(\theta_k)$. Let $m=\eta(\theta_k)\in\mathbb Z^+$. Since $(E_1,\eta_1)$ is an admissible function of angles, there exists $\theta\in E_1$ such that $\sin(m\theta)\neq0$ and $m\geq \eta_1(\theta)$. Since $(E_1,\eta_1)\leq (E,\eta)$ we have that $\theta=\theta_j$ for some $j\in\mathbb Z^+$ with $\eta_1(\theta_j)\geq\eta(\theta_j)$. The inequality $\eta(\theta_j)\leq\eta(\theta_k)$ gives that $1\leq j\leq k$. The condition $\sin(m\theta_j)\neq0$ now forces $j=k$. We have shown that $\theta _k\in E_1$ and $\eta_1(\theta_k)=\eta(\theta_k)$. The conclusion of the lemma now follows by induction. \end{proof}
We are now in position to calculate the minimal elements in $\mathcal{A}$.
\begin{theorem}\label{minimalelementsA} The minimal elements in $\mathcal{A}$ are exactly those that belong to the set $\mathcal{A}_0$. \end{theorem}
\begin{proof} By Lemma \ref{cfoaminimal} we know that the admissible functions of angles $(E,\eta)$ from the set $\mathcal{A}_0$ are minimal elements in $\mathcal{A}$. We proceed to show that every minimal element in $\mathcal{A}$ has such form.
Let $(E_1,\eta_1)\in\mathcal{A}$ be minimal in $\mathcal{A}$. By Theorem \ref{lowerboundfoa} there exists $(E,\eta)\in\mathcal{A}_0$ such that $(E,\eta)\leq (E_1,\eta_1)$ in $\mathcal{A}$. Since $(E_1,\eta_1)$ is minimal in $\mathcal{A}$, we conclude that $(E,\eta)=(E_1,\eta_1)$. \end{proof}
\end{document}
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arXiv
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NTS ABSTRACTSpring2021
Revision as of 16:39, 29 April 2021 by Ashankar22 (talk | contribs)
2 Feb 4
15 May 6
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Padmavathi Srinivasan
Towards a unified theory of canonical heights on abelian varieties
p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for rational points. This is joint work in progress with Amnon Besser and Steffen Mueller.
Retrieved from "https://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2021&oldid=21193"
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CommonCrawl
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Breakthrough discovery in astronomy: first ever image of a black hole
European Commission · Youtube · 2195 HN points · 6 HN comments
HN Theater has aggregated all Hacker News stories and comments that mention European Commission's video "Breakthrough discovery in astronomy: first ever image of a black hole".
On 10 April 2019 at 15:00 CEST (Brussels time) the European Commission presented a ground-breaking discovery by Event Horizon Telescope - an international scientific collaboration aiming to capture the first image of a black hole by creating a virtual Earth-sized telescope. EU-funded researchers play a key role in the project.
Black holes are extremely compressed cosmic objects, containing incredible amounts of mass within a tiny region. Their presence affects their surroundings in extreme ways, by warping spacetime and super-heating any material falling into it. The captured image reveals the black hole at the centre of Messier 87, a massive galaxy in the constellation of Virgo. This black hole is located 55 million light-years from Earth and has a mass 6.5-billion times larger than our sun.
Six press conferences around the world took place simultaneously. In Europe, Commissioner Moedas and lead scientists funded by the European Research Council held a press conference in Brussels to unveil the discovery.
- Carlos Moedas, European Commissioner for Research, Science and Innovation
- Prof. Anton Zensus, Director at Max-Planck-Institut für Radioastronomie, Bonn, Germany (Chair of the EHT Collaboration Board)
- Prof. Heino Falcke, Radboud University, Nijmegen, The Netherlands (Chair of the EHT Science Council)
- Dr Monika Mościbrodzka, Radboud University, Nijmegen, The Netherlands (EHT Working Group Coordinator)
- Prof. Luciano Rezzolla, Goethe Universität, Frankfurt, Germany (EHT Board Member)
- Prof. Eduardo Ros, Max-Planck-Institut für Radioastronomie, Bonn, Germany, (EHT Board Secretary)
EC press release: https://europa.eu/!nr97rr
Scientific press release by EHT: https://blackholecam.org/erc_bhc_pr-2/
More insights by Jonathan O'Callaghan's article: https://europa.eu/!mb77wB
Behind the scene with the black hole scientists: https://europa.eu/!xx46nf
#EHTBlackHole #BlackHoleDay #RealBlackHole #BlackHoles
Ranked #2 all time · view
Unveiling the first-ever image of a black hole [video]
Apr 10, 2019 · 2164 points, 488 comments · submitted by doktorn
⬐ piker
Higher resolution official release seems to be: https://www.nsf.gov/news/special_reports/blackholes/download...
⬐ lisper
And a really great explanation of why it looks the way it does:
https://www.youtube.com/watch?v=zUyH3XhpLTo
⬐ localhostdotdev
editing that image reveals all kinds of details: https://imgur.com/a/TvQsgbA (and I don't even know what I'm doing)
⬐ Obi_Juan_Kenobi
Those 'details' aren't real. The image is the result of a complex interpolation algorithm that takes very noisy and incomplete data as its input. The resolution is quite limited.
⬐ PavlikPaja
I do (I think...) the bands are atifacts:
https://imgur.com/a/dhN9Pf9
⬐ vanderZwan
Mind you, that's a JPG you are editing, so in the 8 bit sRGB color space. That means manipulations can lead to errors and illusions due to the color space being non-linear and clipped[0].
The original data was likely linear and at a much higher precision. If the source was a 16 bit linear grayscale PNG for example you could be much more assured you're not seeing the effects of JPG compression and things that were actually measured.
EDIT: Found better sources:
16-bit sRGB PNG: https://eventhorizontelescope.org/files/eht/files/20190410-7...
180 MiB original TIFF: https://www.eso.org/public/images/eso1907a/
[0] https://www.youtube.com/watch?v=LKnqECcg6Gw
Thanks a lot for the explanation and the better sources, the TIFF image should probably be a torrent because it's not downloading very fast, the PNG image does give much nicer renders: https://i.imgur.com/zZcD5Na.jpg
⬐ aepiepaey
The article linking to that image: https://nsf.gov/news/news_summ.jsp?cntn_id=298276
⬐ aubricus
Thanks for the source article! I had heard about this but not seen any material on it until now.
⬐ hliyan
The additional pixel density probably doesn't add any details though. The original image looks like a smoothed-out version of a low-res image anyway...
⬐ nabnob
Is this link down for anyone else?
⬐ yeukhon
Probably because of HN effect again. Yes it is down.
Slashdot effect
⬐ bspammer
Can someone stick it on IFPS?
⬐ apo
It really needs a scale marker. How many light years across?
⬐ nprz
whoever took this picture should be fired, it's way out of focus!
⬐ asdfasdfasdfa
"High res"
⬐ hathym
So how much this picture cost ?
⬐ derpherpsson
Whenever this type of question comes up, how much science cost, think about the following:
It costs more to do a sciency Hollywood movie about than it costs to actually do the science. Sending an actual probe to the orbit of Mars is in general cheaper than making a sci-fi movie.
⬐ fabricexpert
So basically nothing in the grand scheme of anything.
⬐ acqq
That's half the cost of a single F-35 fighter plane. Just US alone will pay for around 2500 of them (2010 estimate), and plans to pay around 4 times more than that to maintain them. So just F-35 total costs for US are equivalent to 25000 scientific projects like this one. To compare, it would be one new project like this every day for 70 years!
This specific project is more European or even a world project than just US, if I understood correctly.
⬐ imtringued
It makes sense to compare the costs to a single plane but to the whole fleet is just stupid. You can't defend a country with scientific projects.
The thing is every time someone proposes the idea to slash the military budget to fund something else there are at least a hundred other people with a different idea on what to use the funds on. If the funds were spread over so many different projects you end up with an insignificant sum in each of them. Spending the money on military might be actually be advantageous because of large investments in new military technology end up benefiting the civilian sector. (Isn't that the point of the F-35?)
It's the opposite: most of the military money goes to the stuff which, if actually used, ends the human civilisation as we know it.
The same goals that US has could be achieved with orders of magnitude less military spending, while also reducing the risks for the whole humanity.
So every alternative to the current practices is infinitely better.
⬐ coldpie
I've taken to measuring moderately large amounts of money in "juiceros". Forty million dollars is one third of a juicero.
⬐ LoSboccacc
the original tiff is 7416x4320
it's still too many pixel, it can be a fifth of that while conveying the same amount of information.
I'm puzzled by this, if each of those pixel is actually captured by the lenses, why is it all this much uniform? was it smoothed or does this suggest that it's actually a gigantic uniform cloud of gas?
haven't seen the whole video of the release, I'll have to catch up later in the evening, but this really seems to have captured way too much compared to the actual lens resolution and I wonder what would be the "confidence interval" or astrophysical equivalent on each of those pixels.
⬐ petschge
The picture is not captured by lenses, or an optical telescope at all. It is created by inverting the data received at eight radio telescopes (or eighty individual dishes) around the world. And the smoothing is just due to the inherent limit in the resolving power of the telescope array. The impressive bit is that we see more than a single bright dot.
⬐ theWheez
From the press conference, they said that the raw captured data took up 6 cubic meters of hard drives
> cubic meters of hard drives
Astronomy and high energy physics are pretty much the only science I know where this is an applicable unit of measurement
⬐ princekolt
There is no lens. This was done using eight radio telescopes (or arrays) around the globe to create a "virtual" radio telescope which is effectively the size of the earth. After that the data of the individual telescopes was processed to produce an image.
There is some more info in the wikipedia article for the Event Horizon Telescope (EHT): https://en.wikipedia.org/wiki/Event_Horizon_Telescope
⬐ joering2
Thanks for the link. More reading reveals the amount of data produced goes into Petabytes. You can't just upload it to drop box or push it via FTP; hence correct icon of an airplane because those drives with data are indeed delivered by regular means of transportation.
https://en.wikipedia.org/wiki/Event_Horizon_Telescope#/media...
⬐ hrdwdmrbl
Does that image have a weird pulsing optical illusion for anyone else?
⬐ bparsons
I think my eye is trying to bring it into focus. It is like when the camera on your phone can't quite figure out what it is looking it and keeps adjusting.
⬐ simias
I suspect this is because it's extremely blurry, if you watch it on a large screen it might trick your eye into trying to refocus, obviously without success.
⬐ dreamcompiler
Interesting that so many people see this but for me it's absolutely static.
⬐ sametmax
If you are focused and calm, or have bad sight, you won't see it naturally. To force it, move your focus from 1cm to the left, then right, again and again, quickly.
⬐ pcmaffey
Yes, reminds me of the Eye of Sauron.
⬐ EForEndeavour
The supreme astronomical example of the Eye of Sauron is this Hubble shot of the star Fomalhaut and its debris disk: https://en.wikipedia.org/wiki/Fomalhaut#/media/File:Fomalhau...
⬐ the_arun
Reminds me of Indian food - Vada - https://en.wikipedia.org/wiki/Vada_(food) :)
⬐ manaskarekar
Funnily, it reminds me of Soundgarden's album, Superunknown, which had the famous song 'Black Hole Sun.'
https://www.amazon.com/Superunknown-Soundgarden/dp/B00IXLQJ8...
⬐ chki
Yes, fascinating! If you shift the focus of your eyes from the outside to the inside of the hole, it seems to "pulsate"
⬐ banehoodie
looks like the image data was cleaned up with bicubic interpolation, or something of the like.
https://en.wikipedia.org/wiki/Bicubic_interpolation
⬐ Tor3
I don't even have to move my eyes, I stare at it and it's constantly moving away from me. I had to check that it's not actually a movie. Strangest thing.
⬐ kyllo
Yes, and I also see a spot in the shape of it after looking away, though that may just mean my monitor is too bright
⬐ tacone
Some time ago I've seen a picture over the Internet where the description said the more you see it "moving" the more likely you're stressed.
As for the black hole, looking it at full screen I see it pulsating a little.
⬐ anonytrary
Yes if you dance your focus around the image, it appears to pulsate. Probably your pupils constricting and dilating due to the high contrast in the image.
⬐ Insanity
It does look like it pulsates to me! That's an interesting illusion :)
⬐ neom
Stare at it till there are two, then have fun snapping them in and out.
⬐ luizfzs
Relieved it isn't just me
⬐ geoah
I get it if I move a bit when on an angle, I assume it's just my crappy monitor's viewing angles causing it.
moving the picture on my screen caused some eye bug
⬐ mrhappyunhappy
You mean to tell me this is not a gif?!
I literally did double-check the file extension
⬐ thanatropism
It's the color scheme. I think in matplotlib it's called "hot". I do a lot of 2D kernel density plots that by having a black zero level and dozens of contour levels produce a smooth look with aesthetics similarities to this. I used to use that scheme because "heatmaps" but stopped because of the pulsing illusion.
I'm not sure if this image is real color or just lightness value and they used a color scheme for drama.
⬐ 317070
It's the latter. This is an image in radio-frequency brightness of the object, so not visible frequencies of light. But yeah, it also looks like the 'hot' colormap to me.
⬐ Mizza
Grrr, why aren't scientists all using perceptually uniforms color schemes! This would still look awesome in `plasma`. Is there any where I can get the data and do it myself?
One of my favorite talks ever is on this subject: https://www.youtube.com/watch?v=xAoljeRJ3lU
⬐ kardos
Can you run it through the colormap in reverse to (approximately) recover the data?
Technically yes -- colormaps are 1-dimensional non-self intersecting curves in RGBA space. You just grab them by the... I mean just straighten them.
I don't know if an explicit formula for the colormap is given, but you can always do "xs = np.linspace(0,1,100); ys = { cm.hot(x):x for x in xs}" and recover an approximate inverse. Then apply this function to each pixel of the image.
⬐ yehar
Based on my analysis the published image is too processed to be able to invert the colormap to get to the brightness temperature data. I had better luck with Paper IV Fig. 15. See: https://physics.stackexchange.com/questions/472641/what-woul...
⬐ knolan
You can easily remap it between different colour maps if you assume that is is a standard hot map.
⬐ Osmium
'Inferno' would be nice for this, also perceptually uniform, and has the benefit that space is still black.
⬐ turshija
Related video, made by Veritasium yesterday, is one of my favorite videos in a long time. He explained how the prediction of this image was made (before the image got released) and the video is great and fun to watch.
⬐ ragebol
Note that the prediction of the light being brighter on one side did come out.
⬐ jessriedel
What are you basing that on? (Edit:) From one of the papers released today:
> The ring is brighter in the south than the north. This can be explained by a combination of motion in the source and Doppler beaming. As a simple example we consider a luminous, optically thin ring... Then the approaching side of the ring is Doppler boosted, and the receding side is Doppler dimmed...This sense of rotation is consistent with the sense of rotation in ionized gas at arcsecond scales ..Notice that the asymmetry of the ring is consistent with the asymmetry inferred from 43 GHz observations of the brightness ratio between the north and south sides of the jet and counter-jet
https://iopscience.iop.org/article/10.3847/2041-8213/ab0f43
(Edit 2:) Ahh, I see your comment now says "did come out". I initially read it as "did not come out", which was either a misreading on my part (likely) or an earlier edit by you.
Are north and south in astronomy defined relative to Earth's poles? What about "lateral" directions, since east and west are relative (no poles, ie. no east of earth)?
I don't know for sure how that's defined (I ctrl-f'd and it's not explained in the paper), but this says the "North" is to the right of the image, and from context it sounds like it's the north pole of the accretion disk, i.e., the direction of the rotation axis with the right-hand rule.
> The approaching side of the large-scale jet in M87 is oriented west–northwest (position angle $\mathrm{PA}\approx 288^\circ ;$ in Paper VI this is called ${\mathrm{PA}}_{\mathrm{FJ}}$), or to the right and slightly up in the image.
⬐ thro_away_n
In paper I, Figure 3, it says North is to the top of the image and East is to the left.
Whoop. You're right. I misread again.
⬐ pi-squared
Using the right-hand rule: knowing the direction of spinning, if you point your thumb up and wrap the other four fingers in the direction of rotation, the thumb will be pointing North. Oposite of that is South. East can then be defined along the direction of spinning (eastward or counterclockwise looked from North, the way Earth is spinning) and West - opposite to that, clockwise looked from North, opposite the direction of rotation.
It's confirmed in the press conference by the scientists. They said its the Doppler beam effect
You mean the asymmetry was confirmed, contra ragebol's comment? (See the added quotes in my comment.)
⬐ a_d
Previous work on this was done for the movie Interstellar. The resolution of the rendering software was so high that team members were able to examine the black hole very closely - Because Gargantua was spinning at almost the speed of light, the rendering showed that spacetime warped into shapes never seen before. This led to the publication of —> https://arxiv.org/abs/1502.03808
Kip Thorne describes his work not this in a book called the science of interstellar.
Kip's description of black holes here is also fascinating: https://youtu.be/oj1AfkPQa6M — first time I learnt what "warped" space-time means :)
⬐ rubicon33
> Kip's description of black holes here is also fascinating: https://youtu.be/oj1AfkPQa6M — first time I learnt what "warped" space-time means :)
In this video he makes a comment which I struggle to fully understand:
He says that ALL of the matter which belonged to the cooled-off star is DESTROYED in the process of creating a black hole.
That concept of complete destruction eludes me. I assume what he means is, the matter was converted entirely to energy. Right?
But if that's true - where is all of that energy? Is it stored (somehow?) in the Black Hole? Is it dispersed throughout the galaxy? What HAPPENED to the mass (energy)?
⬐ roywiggins
It's all squished into the black hole and from the perspective of everyone outside, converted into... more mass of the black hole. The mass of the black hole comes from the mass that created it. As you feed it more stuff, it gets more massive.
As to what physically happens to the stuff once it's inside, I don't know if we know for certain. It gets dragged towards the center. From the point of view of the rest of the universe, it might never actually reach the singularity: GR would make it look like it's going slower and slower and slower.
Speculation about what is actually inside the event horizon is at most mathematical extrapolation, since we can't actually crack one open and look.
⬐ pavanred
Sean Caroll has a great podcast, mindscape [0]. One of the recent episode featured Kip Thorne as a guest and had some great discussions about Gravitational Waves, Time Travel, and Interstellar [1]. It's a very informative and entertaining podcast, I recommend it.
[0] https://www.preposterousuniverse.com/podcast/
[1] https://www.preposterousuniverse.com/podcast/2018/11/26/epis...
⬐ jperras
Wow, that brings me back. I studied GR under Robert Brandenberger, and we used Caroll's book. What a wonderful text.
Definitely going to listen to his podcast!
Agree with the recommendation.
Kip has studied black holes all his life — this podcast goes into the work on LIGO that finally got Kip (and collaborators) the Nobel Prize. I found it amusing that there is some "Nobel guilt" for scientists that comes with the prize, because the size of them teams that usually collaborate and make a large project like LIGO happen (over 20 years) is incredibly large.
I also find it inspiring that Kip speaks with so much... love ... about warped space time :)
There is a video that I cannot find where Christopher Nolan describes the process of rendering the black hole for his movie - they used Kip's equations to render Gargantua and when the first images were seen, he realized that Kip has never actually seen a black hole before - even though he has spent his entire life studying it.
⬐ yaseer
International collaboration on scientific projects (International space station, CERN) always fills me with hope and optimism for humanity.
It's a nice contrast to opening the papers and reading the regular news, dominated by politics, with all the pessimism that creates.
Hooray for science.
⬐ blablabla123
So true, I was also recently thinking I should read more science/engineering news because that's actually positive stuff happening.
⬐ hinkley
If I understand it, this class of "telescope" is made up of arrays of telescopes spread as widely over the hemisphere as possible. We can only get data like this by collaborating with as many different sites as possible. It literally can't exist without broad support from many countries.
As one of the scientists said in the interview, the next step is a telescope bigger than the earth. Hopefully we can collaborate on those too but if that involves a lot of satellites in a heliocentric orbit that may limit contributions considerably.
⬐ aaco
I know this comment will sound a bit unrelated to the main topic but reading this made this thought pop up in my mind and I thought it would make sense to share it. This comment is, however, related to the sub topic of the quoted sentence:
> It literally can't exist without broad support from many countries.
This is the same constraint for the, let's call it, "peace on Earth" problem, or just "peace".
If only more of us could realize this is what it takes to solve that problem... which itself is part of the puzzle, i.e. how to increase awareness about the need to solve this.
While there are a number of people and organizations trying to do this, I see that there seems to be more possible fronts that could be used to tackle this and accelerating the reach for stability and sustainability of the state of peace.
One example of a possible front (and I honestly don't know if those already exist) is: through marketing it would be possible to influence people enough to be interested in the outcome of "peace on Earth" and pay some money for that, in a way that it doesn't feel like a donation, and more like an investment or maybe acquiring a service that would be hopefully realized in the coming years (hence the importance of the marketing capacity of that entity, as this mindset needs to be set in the consumers in order to make them buy the good).
Of course, the reporting to the consumers on the use of the invested money toward that effort has to be as transparent and honest as possible, as those approaches are arguably required for a sustainable state of peace. And hopefully it would make enough sense for an entity to operate in the way of the outcome it is seeking. Even though we are moving from a state of no-peace, which is hopefully unsustainable. In other words the effort could be be defined as "safely and confidently accelerating the maximum point of unsustainability of the no-peace state such that it inevitably transitions to a sustainable state of peace".
That's then possibly a private endeavor (not that it could not be a public one as well, but you need to raise enough money to pay for the possibly expensive marketing and then pay all of its employees), because there is now an identifiable market willing to pay some amount in exchange for obtaining the "product" of peace, which in other words just mean the modulation of humanity and its mindset in order for it to operate in such a way that it is always aligned to its own common good, or maximum known state of well-being, sustainably.
We already know that groups and individuals are not great at doing that, on average. So if an individual is not always able to operate towards its own good, or maybe some are but don't have access to the resources that would allow them to do so, how could then a group of individuals be able to do so? Unlikely.
And yes, exploring the universe and finding more about its mysteries and teaching humanity about them is a valid and great approach and a subset of all the possible approaches.
It is a subset because in order for an individual to be interested in knowing more about the mysteries of the universe, or consciousness and other topics, they have to have this mindset, well, set in the mind.
Therefore there are many more fronts that could be, and to many extents currently are, covered. So all I'm arguing here is we are not doing enough to reach the tipping point before possible big catastrophes happen, therefore we should do a lot more than what we're currently doing. There are many entity/company/organization models to explore that could benefit us in a spectrum of possibilities ranging from private to public.
⬐ aurailious
Constructing a telescope on Mars would be a valuable investment when a permanent presence is based there.
⬐ TeMPOraL
I hope one day we'll construct a telescope that uses the Sun for gravitational lensing. I've seen this paper once that claimed you could use it to image surface of exoplanets directly with pretty high (for our current astronomy standards) resolution. I think it talked about this: https://en.wikipedia.org/wiki/FOCAL_(spacecraft).
⬐ bashinator
Mars is smaller than Earth, so planet-wide radio interferometry would be a smaller "aperture" than possible on Earth. If you're talking about extending the telescope to include both Earth and Mars, I imagine that doing the interferometry over changing speeds and distances would be challenging to say the least.
The speeds and distances of the planets are well known at this point and easily predicted. By the time of establishing a permanent presence on Mars the requirements of communications would already put in place the information needed if the DSN isn't already capable of it now. The compute needed would be greater, but so would the availability of it in the future too.
⬐ okket
The papers with the scientific details are here (open access):
https://iopscience.iop.org/journal/2041-8205/page/Focus_on_E...
Article in physics world with comparisons to simulations:
https://physicsworld.com/a/first-images-of-a-black-hole-unve...
"AskScience" AMA on Reddit about the breakthrough:
https://www.reddit.com/r/askscience/comments/bbknik/askscien...
Article in physicsworld with comparison to simulations:
⬐ nonbel
From reddit:
>"Hi, regarding the image itself: What I don't understand is why does it look like a donut and not a bright sphere? Assuming the black hole is actually spherical and not disc shaped, I would expect the Halo to be spherical and surrounding the black hole? so all we would see would be the ball of bright gas, even though there is a black hole in the middle?"
This is also what I would have expected.
⬐ Sharlin
The same reason planetary rings, solar systems, and (spiral) galaxies are flat: friction and conservation of angular momentum. Internal drag forces will eventually cancel out velocity components perpendicular to the plane of rotation, turning a cloud into a flat plane.
Things don't simply form a halo around a black hole. Instead you get a relatively flat accretion disk of things that orbit around the black hole.
⬐ VikingCoder
This was a great explanation:
⬐ db48x
Material can come in towards the black hole from any angle. However, because matter can't just pass through other matter, matter travelling in the wrong directions will collide more frequently than matter travelling in the right direction. Because of the conservation of angular momentum, the "right" direction will depend on the average angular momentum of everything in the cloud that is collapsing into the disk.
You might be interested to know that this is the same reason all the planets in our solar system orbit in the same disk: all the matter that is now in our solar system was originally a very thin cloud of gas with a small amount of overall angular momentum. As gravity drew it together it flattended out into a disk and eventually the clumps became planets (and the sun in the middle.)
⬐ jjeaff
I am not trying to throw cold water on this, but I have some questions.
This ted talk has a very basic explanation of how they constructed this image. I was curious if anyone with image interpolation experience could weigh in on the method. https://www.youtube.com/watch?v=P7n2rYt9wfU
When she first starts explaining their method around 8:00m in, I was initially very skeptical of this result because she said that they feed images of what we "think" a black hole should look like and use algorithms to compare the captured data with those images.
She then goes into explaining the measure they take to keep the resulting image from being biased by passing environmental images and images of other astronomical anomaly to make sure that those images return similar results.
But I can't for the life of me figure out how passing non-stellar imagery could return something similar. And if it does, why do we need to feed it an example of what we think it should look like at all?
Yes, at ~6:53 in the video she shows how they are selecting images to include that look like a black hole. If you plug in enough noisy images you will eventually get a few that look like a bright donut.
⬐ neals
Was just wondering this myself. How opinionated is this photo?
⬐ ImaniBlack
Probably very opinionated. I can't help but take this with a grain of salt. relevant example .How long did it take for people to evolve our view of dinosaurs as information was reevaluated.
⬐ RedOrGreen
(I haven't watched the video, but I do have professional expertise on this topic.)
With interferometry, you're getting an incomplete sampling of the Fourier transform of the sky image, and if you just invert the samples, you get what we call a "dirty" image.
But you know your sampling of the Fourier plane exactly, since that's just a function of the projected baselines between every pair of telescopes during the observation, so you can create a "dirty beam" - now all you have to do is remove the effects of the dirty beam from the dirty image. Of course, that's a deconvolution problem, and given that you don't have all the information - you sampled it - it can never be exact.
But it can be very good! There are very sophisticated radio synthesis image deconvolution algorithms, including CLEAN and Maximum Entropy. For Maximum Entropy methods, you can apply a Bayesian prior on your images - most of the time, the prior we apply is a blank sky (seriously!) but if we have other constraints that we can use (e.g., the approximate size of the region with extended emission), Bayes tells us that we would be remiss not to use it.
If you look at this image [1] from Paper IV [2], we show the image results from different techniques on different observing days. Those are the inputs to what is the "consensus image" - you can check how close they all are to each other.
Does that make sense...?
[1] https://iopscience-event-horizon.s3.amazonaws.com/2041-8205/... [2] https://iopscience.iop.org/article/10.3847/2041-8213/ab0e85
⬐ itake
This 9 min video [0] does an excellent job explaining what we are looking at.
[0] - https://www.youtube.com/watch?v=zUyH3XhpLTo&feature=youtu.be
⬐ kristofferR
The US unveiling is WAY better than the EU unveiling that is linked to above. They have images, animations and graphs that's easily understandable by the layperson, and it's scientists instead of politicians speaking.
https://www.youtube.com/watch?v=2DxjuE7WDlk
They talk about how the image was produced, and how they made such a small image out of the 5 PetaByte of data they gathered from stations all over the world.
I really really liked this press conference, I highly recommend it, well worth the watch. It's fairly short (~30m for the main part) and they explain the process step by step: how it was captured -> how it was processed -> what it means.
It's very well communicated in a way most can understand, it's concise and it has great accompanying graphics.
⬐ Karto
Thanks. Commissioners aren't even proper politicians : they get to power without election, without ever needing to convince the public, most often without even being known from the public. Nevertheless the decisions they make must be implemented in national laws by elected MPs. Hence you can imagine how despised and hated they are.
⬐ drilldrive
Yeah, no kidding. The presentation in the American conference is much more insightful and gives a stronger perspective. And the EU conference didn't even show the image until 8 minutes in.
⬐ AnimalMuppet
One of the cool things about this was that the data was too large to ship over the internet (in a reasonable amount of time). They actually shipped physical disks full of data.
Even today, never underestimate the bandwidth of a station wagon full of disks...
⬐ r721
"This is more realistic of the uncertainties involved in this high-end image reconstruction. Still amazing though! Fig. 4 in https://iopscience.iop.org/article/10.3847/2041-8213/ab0e85 "
https://twitter.com/karlglazebrook/status/111598136971105894...
⬐ SiempreViernes
Yeah, apparently radio interferometry is still very manual and so involves many relatively subjective decisions to produce an image.
That's probably why they had several analysers that they then combined into progressively larger teams until they could produce this Consensus-A picture.
There is non-manual ways to do it (called self-calibrating), but those need many more antennas. (What you really need is good coverage of the (u,v) plane AND many more different baselines between pairs of telescopes than the number of individual antennas.) If you do not have that self-calibration fails and leads to horrible image artifact or does not converge at all. They limited the influence of the manual calibration by observing a Quasar which is basically a point source between subsequent observations of M87* to get an independent amplitude calibration.
Some quasar other than M87 I suppose :P
(It is a quasar, simbad says so and that's good enough for this setting!)
Yes. They used 3C 279 which is more than 100 times further away than M87.
⬐ symlock
For those wondering how the image was constructed: https://www.youtube.com/watch?v=hMsNd1W_lmE
Basically, the image has been constructed by calculations on massive measurement data-sets from multiple synchronized telescopes around the world.
So this isn't a "photo" in the normal sense. It's a reconstruction of many, many radio waves.
It doesn't sound like they just snapped a picture. The one guy says they used "supercomputers" for 6 months to get the image.
Sunspots look black relative to the rest of the sun but are actually very bright. Could this be the same thing? How did they set the black level? Is there a description of the procedure somewhere?
Found the paper describing the data processing: https://iopscience.iop.org/article/10.3847/2041-8213/ab0c57
⬐ julienchastang
Indeed, from what I gather from this thread and external links this "image" seems to be more of a plot than an image. In fact, are we looking at a matplotlib plot of the data with a "hot" colormap? The iopscience paper even references matplotlib. I'm just making some educated guesses here, but it's still fun to think about. BTW: Python for the win! iopscience paper makes several references to Python tools (e.g., numpy, Jupyter etc.)
Another interesting thing about that iopscience paper is that you can turn MathJax on/off with a link at the top. Mathjax is off by default, but when you turn it on, you get a nicely rendered equation instead of a gif image.
⬐ ants_a
Every image is a plot... This one just has had a bit more processing gone into it than your average demosaiced and noise reduced vacation photo.
Summary of their summary:
"arising from the rapid atmospheric phase fluctuations, wide recording bandwidth, and highly heterogeneous array"
(Filtering out bad data)
⬐ LolWolf
As far as I know K. Bouman [0] was the scientist leading the charge on the image processing/reconstruction. A few of her later papers probably have hints [1, 2] about how this is done, but I haven't seen the official release.
[0] http://people.csail.mit.edu/klbouman/
[1] https://arxiv.org/pdf/1903.08832.pdf
⬐ MagicPropmaker
The project's director was Shep Doeleman
https://news.harvard.edu/gazette/story/2019/04/harvard-scien...
Thanks, I think I added the link to the paper while you were writing this. From skimming it I can't tell the answer to my question though.
How do they know what appears "black" in the image is really black vs. relatively black?
If you observe a very wide band of light (e.g., EM radiation) and there is nothing received from those spots, then, for all intents and purposes, the region of that image is black.
Now, if you're asking if, perhaps, the region isn't really black, but rather it's emitting some sort of small radiation relative to the bright region, it would be essentially impossible to know without much higher resolving powers (since it might even be indistinguishable from the background noise generated by the surrounding region). There is no way to really know if it's "perfectly black" vs. "orders of magnitude darker than the surrounding regions."
Makes sense. So I guess they could probably tell us an upper bound on how bright it is.
Indeed! That's for sure: it's probably not hard to extract a bound on the magnitude of a part of the spectrum from this analysis.
Any idea how bright it might be? Eg, could it be as bright as the sun? The moon?
⬐ _void
I don't get the whole "oh it's too blurry and nothing is visible" comments. It's a black hole, what did you expect to see? Interstellar CGI?
⬐ crimsonalucard
It is a letdown. Although I wasn't expecting Jupiter level detail this pic doesn't blow me away because it's just a blurry ring. Jupiter level detail will blow me away.
⬐ lanewinfield
Consider yourself lucky that a black hole isn't Jupiter level distance from us.
At a jupiter level distance we'd be well within the event horizon. At that distance I expect to see four dimensional book cases and my daughter.
⬐ HenryBemis
Jupiter level quality would mean that we are so close that we would be instantly torn apart.
Jupiter level quality is what I want. Not jupiter level distance.
⬐ C4stor
The significance of the image is not in its quality, but in its mere existence.
It's enough to confirm various predictions, and should give a new baseline for truth about black holes.
⬐ KnightOfWords
Yes, we're using a telescope the size of the Earth to look at an object the size of our solar system in a galaxy about 55 million light years away. It's an infinitesimally small patch of sky.
On the other hand, the jet from the M87 black hole is quite large and we have good images of it. It's even resolvable by amateurs, I hope to take a picture of it over Easter with a small telescope.
https://en.wikipedia.org/wiki/Messier_87#Jet
⬐ O_H_E
> 55 million light years
didn't he correct himself and said kilometers
⬐ MR4D
He corrected himself to say 500 billion billion kilometers after mistakingly saying light years.
I haven't done the math, but I believe that roughly equates to 55 million light years.
Yup, My bad.
55 million light years is about correct. Wikipedia still has 53.5 ± 1.63 Mly from older observations, but appendix I in paper https://iopscience.iop.org/article/10.3847/2041-8213/ab1141 has a bit more on the distance measurements they used. They arrive at 16.76 ± 0.75 Mpc, which translates into 54.7 ± 2.4 Mly.
⬐ montenegrohugo
It's just a stupid collection of small pixels but somehow it feels very overwhelming looking at it for the first time.
⬐ int_19h
I mean, in the end, everything that you're looking at is just a collection of cones and rods in your retina. It's the information that it conveys that really matters.
⬐ bytematic
Because that collection of pixels is really out there
⬐ novaRom
Excellent example of successful international collaboration, with distributed team. Great results and promising future work: they say Sagitarius-A* is their next target!
⬐ jcims
Todays' Veritasium video had a shot of SgtA. Didn't know they did that.
https://www.youtube.com/watch?v=VnsZj9RvhFU
Did some more digging and cant find the image in that video anywhere, lol. The M87 shot looks better, but still.
⬐ niklasrde
The telescope's announcement is up, too: https://eventhorizontelescope.org/
⬐ ckugblenu
The EU is finally learning from the US when it comes making a lot of fanfare for discoveries and other inventions.
⬐ chicob
Why is the European Commission doing this? When I recognized Carlos Moedas I immediately skipped to what mattered. The reason why he was ever appointed as Commissioner for Research, Science and Innovation is, to me, more remote than the black hole about which he knows absolutely nothing.
⬐ blibble
because it's one Commissioner per member state, and apparently nationality is the most important criteria for appointing ministers
⬐ mlindner
Well this was mostly a US-run effort. It's headed by Harvard University and the key South Pole Telescope is North American.
⬐ Gravityloss
Too bad they don't present it well. I just see self congratulating politicians and officials on some youtube stream.
Telling that it's a US organization that hosts the actual picture.
esa.int and eso.org seem to be down actually.
Both ESO and the NFS hosts the images, don't know what gave you any other impression.
And while the EC stream was pretty bad, at least they let you skip back in the stream and left it up after the presentation ended. The NFS stream wouldn't allow you to go back (useful if you joined late) while it was up.
Personally I liked the ALMA stream best, but it's down now :(
https://eventhorizontelescope.org/ is hosted in the US and the linked image links to a Harvard.edu web server.
⬐ taksintikk
The European broadcast presentation is not great.
The speakers are referencing images and diagrams that are not visible on screen.
The accomplishment speaks for itself. The delivery can be improved.
⬐ huffmsa
So is a black hole 3 dimensional? Is it a sphere? Or does it only work in certain directions? Does everything "fall" the same direction?
I ask because even in a brief history of time, the diagrams are very "single plane of space-time, pulled infinitely deep by the black hole"
⬐ oppositelock
1. Yes, but it's also up for debate, since it's also known that the information content of a black hole is proportional to its surface area, so 2D.
2. The event horizon is spherical. Mathematically we treat it as a point. Nobody knows.
3. It's a gravity well, so everything falls towards it like it does towards the earth. Since it's the center of a galaxy, and galaxies form a 2D spin plane, most matter will be circling it, like our solar system circling the sun, so most matter will come from that planar distribution.
We will never see anything fall into a black hole, as in our space-time, this would happen at infinite time. As things get closer, they red shift from our frame of reference due to time dilation, until the light frequencies are so low that we can no longer detect them
Hmm, interesting. I'm curious why this is getting down voted.
⬐ beat
Is an image of a sphere printed on paper three dimensional? Think about that for a moment, and you'll see the problem.
I am not a physicist, but from my lay understanding, a black hole (and all other forms of matter) exists in a four-dimensional space-time form that we can experience directly, plus a number of higher dimensions that we can only extrapolate, and observe indirectly via experiment (kind of how a picture of a sphere is not a sphere due to a missing dimension, but you can still tell it's a sphere based on other characteristics). The exact number of higher dimensions is the realm of string theory, and could be 10, 11, 26, or something else.
The singularity of the black hole is one-dimensional, I think (I could be wrong, I'm often wrong). The event horizon is... funny. The outside of the event horizon is in four-dimensional space-time and is more or less "spherical". The inside is a land of theory and debate, because by definition we can't directly observe it from our space-time. Do space and time even exist inside the event horizon? I dunno, ask someone who knows what they're talking about. The best I can do is an analogy. Imagine it's the surface of the ocean. Above the surface, you're in air. Below the surface, you're in water.
⬐ rtkwe
The stretchy sheet of space time illustration is just a simplified visualization space is 3 dimensional so the warping is 3D. The stretchy sheet visual is used a lot because showing a warped 3D grid to be more accurate is a lot harder and a lot less clear and illustrative. Trying to show a 3D grid doesn't even fully capture the situation because it doesn't show how the passage of time gets distorted as well.
⬐ JimBrimble35
Let's start a movement similar to flat Earth, but instead it's Spherical Black Hole. I refuse to believe that a black hole is anything but a sphere no matter what evidence I'm shown.
⬐ JumpCrisscross
> is a black hole 3 dimensional? Is it a sphere?
The event horizon is, to our understanding, roughly spherical when viewed from the outside. We don't know what happens inside.
⬐ SpeakMouthWords
1. Yes 2. Basically 3. No 4. No
⬐ PeanutNore
A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point. The event horizon around it is roughly a sphere. The diagrams that you are talking about are a visual metaphor that represent 3D space as a 2D plane and the 3rd dimension standing in for the influence of gravity. IRL, spacetime has 3 spatial dimensions, not 2, and gravity is not a dimension but a force. It's hard to visually represent gravitational distortion of 3 dimensional space without a 4th spatial dimension to do it with, so textbook diagrams use a 2D plane.
⬐ pflats
I think you've got a typo here: points should be 0-dimensional.
⬐ curlypaul924
Are you sure it's a typo?
https://physics.stackexchange.com/a/194947
(I'm not a physicist.)
⬐ codethief
Not sure why you're getting downvoted, this is a legitimate question. However, in this particular case, I think the final sentence of the accepted answer on Physics.SE, namely that
> In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime.
is wrong or at least very misleading – the answer does (correctly) say that asking for the "dimensionality of a singularity […] is a meaningless question because the spacetime geometry is undefined at a singularity".
⬐ checkyoursudo
I believe the previous poster is referring to a typo following from this:
>A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point.
In Euclidean geometry,
A cube is 3 dimensions.
A plane is 2 dimensions.
A line is 1 dimension.
A point is ...
⬐ cuspycode
A point is 0-dimensional, but in a space-time diagram the time dimension is added, which makes it 1-dimensional. Penrose diagrams for black holes assume spherical symmetry, so all of space is represented by a single radial coordinate, which makes it possible to display such diagrams in 1+1=2 dimensions.
A singularity doesn't have a dimension. It is a portion of spacetime that is missing, not a point or set of points. We can't define its dimensionality, either.(×) What we can say is that the singularity in Schwarzschild black holes is spacelike.
×) Counterexample: Consider the manifold M := R³\B, where B is the closed unit ball, equipped with the standard Euclidean metric. This manifold is certainly not Cauchy-complete and we can reach the singularity at r=1 in finite time. Now, if we had to define the dimension of the singularity, what dimension n should it have? n=2 (a sphere)? Maybe. At least we could extend M by the unit sphere to make it complete. But could the singularity also be a point (i.e. n=1)? Yes, certainly. By diffeomorphism invariance, we could simply find new coordinates and map R³\B to R³\{0}, so the singularity would suddenly become a point. So, as you can see, interpreting the singularity as a point or set of points that have a topological dimension doesn't work.
⬐ antidesitter
> A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point.
A point is 0-dimensional. A line is 1-dimensional.
The singularity is 1-dimensional (more precisely, a ring) if the black hole is rotating. [1]
[1] https://en.wikipedia.org/wiki/Ring_singularity
⬐ pavelrub
The singularity is likely a mathematical artifact of the fact that GR is insufficient to describe black holes. In reality (quantum gravity) they probably do not exist.
⬐ lostmsu
I would not be so sure. Its presence might simply indicate, that black hole's inner volume is infinite.
Could you elaborate on your definition of volume here (are you talking about spatial volume or spacetime volume?) and how the curvature going to infinity at the singularity should imply its infiniteness?
My thought process here is the following: The inside of an (eternal) black hole carries four (Schwarzschild) coordinates t, r, theta, phi – r now being timelike and confined to the interval (0, 2M) and t now being spacelike and being any real number. That is, depending on when (at what time t) you cross the event horizon, you end up at a different point in space. The singularity at r=0 is then a point in your future which, like your own death, you cannot actually see but which you will nevertheless hit in finite proper time.
So in this sense I'd say the volume is very finite (if we disregard the (trivially unbounded) spacelike coordinate t which, as mentioned before, simply corresponds to the time of entering the BH).
I should add:
1. In the interior of the BH, the determinant of the metric is bounded since the Schwarzschild factors in the metric cancel out. So the volume measure doesn't do anything crazy as one gets closer to the singularity and boundedness of coordinates implies boundedness of the volume. Again, I'm disregarding the spacelike t coordinate because to me the relevant fact is that all matter reaches the singularity in finite proper time, so while we could theoretically stack lots of (actually, an infinite amount of) (massless) cubes inside a black hole, they would soon all get crushed.
2. Of course the situation is slightly different if we're talking about a growing black hole whose mass (and, therefore, radius) increases as we throw matter into it.
Unfortunately, I am not strong on mathematical part of GR, so feel free to disregard the rest of this idea, if math's the sole thing you are looking for.
A thought experiment: imagine we have a clock, falling into Schwarzschild black hole. Obviously, any real clock would have some non-zero size in all space dimensions. Here we will be concerned with just two: r and any orthogonal one. So for simplicity let the clock be a simple rubber-like oscillating ring with a fixed k and infinite resistance to tearing. (You could also take infinite k, but I'd argue that would not be physically meaningful in this setup)
As the ring is closing to the r = 0, its oscillations will slow down and come to a halt due to physical stretching along the r dimension. What I am trying to say is that maybe these oscillations make more sense as the measure of time for the ring people, than what a numerical value of proper time tells us. In a similar way the time singularity at the horizon is nothing special for a freely falling observer.
I am unsure how to interpret the fact, that the number of oscillations per proper time unit is going down though. Seems to be quite the opposite of my original note about the volume, yet something is ringing.
Unfortunately can't reply to the original question anymore. But here's another hypothesis I just read elsewhere (RU: https://don-beaver.livejournal.com/212422.html): the matter-energy falling towards "center" inevitably has some non-zero momentum, orthogonal to the direction along r. That means its fall is going to be (sort of) spiral, causing it to produce gravitational waves. The argument here is that the whole mass will be converted into gravitational waves before reaching the central point. Now what happens to those waves is a question still.
⬐ alkonaut
So much higher res than I thought (I was expecting a 3x3px black and white).
Does anyone know if this is aggregated over a long time so it's unlikely to improve with more observation? And what is limiting the resolution at this point?
⬐ stargazer-3
One thing missing from @petschge's reply: adding more telescopes! This will not improve the imaging the smallest possible features, but will make features larger (disk and / or jet components) more prominent. The more telescope pairs with different distance between them are added, the more complete the picture will get - up to a limit of having a single antenna roughly the size of Earth.
Furthermore, adding more telescopes will help better image the fainter features in the image (due to larger total collecting surface).
Very good point. I was too much focused on resolution. Adding more telescopes would indeed give a better picture of large structures (most are cropped out of the current view because they are not well imaged and you would mostly see artifacts from the image reconstruction) and improve dynamic range, i.e. allow to see things much fainter than the brightest spot in the image.
The limit in some sense is the size of the Earth. This picture is from an interferometer observing at very high radio frequencies (wavelength of 1.3 mm) using telescope around the Earth. To improve resolution you would either have to go to even higher frequencies (we have a very hard time doing interferometry there) or find baselines larger than what fits on Earth. And you don't just need a long baseline, but need to keep the baseline constant to within a fraction of the wavelength. So if you want to use satellites to get longer baselines you would have to know their orbit to within a tenth of a millimeter.
As an aside: that image is basically black and white. The intensity is just mapped to black -> orange -> white instead of black -> gray -> white.
I wonder how important time is in collecting these data? Because if 6 months isn't an issue, then the array size could be expanded in one axis to the diameter of Earth's orbit around the sun, no?
The collecting time is important, but mainly to 1) sample the rotation of the Earth in the Fourier space for better angular resolution and 2) for raising your overall signal-to-noise ratio is the image (but the one released is already pretty good, so not much improvement can be done there unless you're trying to go for the faint features in the image).
Unfortunately, you can only do interferometry with simultaneous measurements (we need information about the difference in the phase of light hitting the receiving antennas), so the motion of Earth around the Sun is largely irrelevant, unless you can park another antenna at a trailing orbit (see space VLBI for that).
What you're thinking of is probably parallax measurements of distance - that's how missions like Gaia can pinpoint distances to stars in Milky Way (and some in its satellites as well).
We seem to need a big radiotelescope on the moon then? That should give simultaneous measurement on a much larger baseline?
I doesn't even need to be that big. A 60 meter dish would do nicely.
Problem is, you would want to have several Gigahertz of (radio) bandwidth. You are not going to down link that raw, but rather as a number of channels, integrated over a number of microseconds and digitized at something between 4 and 64 samples per bit, but we are still talking a down link data rate of gigabits per second.
⬐ veryworried
Is this what we would see with the naked eye if we were close enough to the black hole? Or would we see nothing, because at this distance we would be dead (or the universe ended all around us)?
⬐ kiliantics
This image is a representation of radio observations. What we would see in the visible part of the spectrum would be different but the structure would likely be similar.
⬐ azernik
You'd probably be dead from the radiation (that light peaks in the x-ray wavelengths) long before any of the spatial distortion effects got to you.
⬐ m_mueller
regular orbits around black holes are possible. Depending on activity if it's far enough away the radiation should be survivable, it just depends on how much matter it's eating up and whether the orbit is planar enough to its spin (otherwise its jets can be deadly).
⬐ andygates
These obs are in radio. We could replace your eyes with radio eyes... ..but it's friction heating so it ought to be broadly blackbody, so I would say yes.
⬐ e0m
"This is like viewing a mustard seed in Washington DC from Brussels"
Pedantic/fun comment: I will assume that one of the presenters said that (I will watch the full video later tonight). One would imagine that that person knows that you cannot see a mustard seed from Brussels because the planet's crust is blocking your line of sight ;) unless the scientist who said that is a flat-earth-believer!
A statement like "it is like viewing a mustard seed X kilometres/miles away would be more appropriate.
And in all seriousness, I have started watching again everything-Star-Trek again (for the 5th time in my life), and news like that make me look up to the sky and think that as a species we do have a chance to move out of here and to a better future.
⬐ zild3d
for what it's worth, with an active sensor (not a camera) you can see over the horizon.
https://en.wikipedia.org/wiki/Over-the-horizon_radar
⬐ the8472
Such a photon-centric perspective. Why don't you use your neutrino perceptors to look at the seed?
⬐ dugluak
Why did they choose this very distant galaxy? why not something that's close to us like the andromeda galaxy or even the center of our own galaxy?
⬐ bbrks
They did. M87* was the widely publicised "first" photo, but they also got images of Sagittarius A* (the supermassive black hole at the center of our galaxy).
The difficulties with Sgr A* are twofold:
- The black hole moves too much during an observation (because it's much closer, and parallax is a thing)
- Sgr A* is much smaller than M87* by mass, so even though it's much closer, the angular diameter is almost the same, and the accretion disk is dimmer.
⬐ t3hz0r
As I understand it, by being more massive this black hole has roughly the same angular size in the sky as Sag A* at the center of our galaxy, but none of the dusty foreground in the way so we get a clearer vantage point.
⬐ negamax
Does anyone remember Interstellar. Their black hole simulation was based on hard science. Looks fairly similar
⬐ ezekg
Interstellar is one of my favorite movies in recent memory. If anybody else is interested in reading more, the book "The Science of Interstellar" is fascinating and very easily digestible even if you're not big on physics.
⬐ vxNsr
just had my wife watch it and yea, that was the first thing I noticed, they really did a good job on the science of interstellar.
When interstellar came out wired did a full magazine spread on the movie, covering its science, how they came up with the story and what the director did to make sure it was believable. idk what it looks like online but I've kept the magazine.
⬐ nikofeyn
this presentation by kip thorne is very detailed about the science behind the movie. it's one of the best presentations i have seen, and i have watched it multiple times.
https://www.youtube.com/watch?v=lM-N0tbwBB4
⬐ etatoby
The visualization I liked the best in Interstellar was that of the wormhole, seen from the outside. (The "travel" through it was rubbish.)
The thing would look like a mirror sphere, floating in the middle of space, reflecting the space around you. Except it would be the space on the other side of the wormhole, not that behind you, and the motion of the "mirrored" image would be opposite of that on a real mirror. (The apparent image on the wormhole's spherical surface would appear to move in your same direction, as you're moving around it, not in the opposite direction as it does on a spherical mirror.)
It took me a while to reason about it and figure out that it was indeed what a wormhole would look like, if we could find or create one.
https://youtu.be/f3ptQ0CPMmU
Relevant paper: https://aapt.scitation.org/doi/pdf/10.1119/1.4916949?class=p...
⬐ metalliqaz
It doesn't look anything like Gargantua in Interstellar, but it's not really supposed to, except for the most prominent features. Interstellar showed what a black hole might look like in visible light while in orbit around it. This is a radio telescope image from a far away galaxy.
Interstellar also turned off the doppler effect, maybe to get a boring symmetric ring instead of an awesome asymmetric one as nature intended it to be!
⬐ CharlesColeman
IIRC, the black hole in Interstellar was also rapidly rotating, which I think was significant to how it looked:
https://www.wired.com/2014/10/astrophysics-interstellar-blac...
> Nolan's story relied on time dilation: time passing at different rates for different characters. To make this scientifically plausible, Thorne told him, he'd need a massive black hole—in the movie it's called Gargantua—spinning at nearly the speed of light.
> Von Tunzelmann tried a tricky demo. She generated a flat, multicolored ring—a stand-in for the accretion disk—and positioned it around their spinning black hole. Something very, very weird happened. "We found that warping space around the black hole also warps the accretion disk," Franklin says. "So rather than looking like Saturn's rings around a black sphere, the light creates this extraordinary halo."
Sure, the BH spin affects the photon orbits somewhat. But the effect of the rotation of the accretion disk itself is completely neglected despite being one of the mayor effects.
I'm still wondering why the accretion disk from the released images isn't warped in a way we saw in interstellar. Are we looking at it top-down rather than from its orbital disk? would a differently oriented black hole look more like interstellar? Veritasium also has an explanation and I don't think it has anything to do with its spin.
Intersterllar chose to view the disk edge on, when the distortions got the most funky at all.
I think the rotation from M87 is 15 degrees off from our perspective, so yes we are looking mostly "down" at it.
Also, the following post on reddit might help you visualize this:
https://www.reddit.com/r/space/comments/bc343r/for_those_con...
just for people's reference, here is the more realistic picture: https://cdn.iopscience.com/images/0264-9381/32/6/065001/Full...
i really feel they should have keep the more realistic one and dialed it up a bit. the more whispy distortions are much more awe-inspiring than the symmetrical, oversaturated version.
the source paper: https://iopscience.iop.org/article/10.1088/0264-9381/32/6/06...
⬐ zakki
Is black hole a 3D object like a sun? If so, I assume the light in the event horizon cover the whole object, i mean 3D shaped as well. How the picture taken from the telescope shown the dark area where the black hole is located? I mean, shouldn't the whole black hole covered by the light thus we can't see the black hole?
⬐ kvartz
I think the 'glow' around the black hole you are thinking about is a light that was bent by gravity but hasn't gotten to event horizon.
⬐ usaphp
It almost looks like a sun eclipse.
⬐ tempestn
The black hole itself is spherical, but the light we're seeing around it isn't projected by the event horizon; rather, it's light from behind/around the black hole that isn't blocked. This video explains in much more detail: https://www.youtube.com/watch?v=zUyH3XhpLTo
But if it's spherical, should not the light be all around it?
⬐ Neeek
The light is emitted from an accretion disk, which does not envelope the entire sphere.
Actually, this particular black hole is (probably) not exactly spherically symmetric but only axisymmetric since it is rotating (i.e. it is a so-called Kerr black hole). But disregard the lack of spherical symmetry for a moment (it can still be approximated quite well by a sphere for our purposes, see below), the crucial points are the following:
1. Along with its rotation comes the fact that the black hole drags the surrounding spacetime along with it (whatever this means), including matter. So matter near such a Kerr black hole will start orbiting it automatically. Closely related(×) to this is the fact that, in the close vicinity of a black hole, you typically find a so-called accretion disk of matter that is orbiting the black hole and slowly being eaten by it, while also emitting light because the infalling matter is heating up in the process. Now, the important point is that the disk is really a disk, though(!), meaning that it doesn't completely surround the black hole in all directions, so there are (lots of) angles from which you could actually "look at" the black hole and your view would not be (entirely) blocked by the matter (and the light it emits). I hope this answers your question as to whether the light "should not […] be all around it".
2. In the case of M87 it seems like the axis of rotation is pretty much parallel to our line of sight, meaning that we're actually looking at the black hole "from above" and that our line of sight is pretty much perpendicular to the accretion disk surrounding the hole. In particular, this means we get to see the accretion disk and the black hole's "bald head" in their full glory. Moreover, since we're looking at the black hole "from above", its slight deviation from spherical symmetry doesn't matter and it still looks like a disk to us due to its rotational symmetry in the direction in which it rotates. (Think of how a cylinder looks like a disk/sphere from above.)
(×) To be precise, infalling matter often carries angular momentum (as measured with respect to the black hole's location), i.e. it doesn't fall into the black hole exactly radially but rather sideways, possibly after having orbited the black hole multiple times. This means that when it finally gets absorbed by the black hole, the latter will absorb the matter's angular momentum, too, and start spinning.(××) So the rotation of the black hole, on the one hand, and of the matter outside, on the other hand, are tightly coupled phenomena and disentangling what came first is a "chicken or egg" kind of problem.
(××) Side note: Infalling matter transferring angular momentum to a black hole is the reason why we expect most, if not all black holes in nature to carry angular momentum, i.e. to be of the (axisymmetric) Kerr type instead of the simpler (non-rotating and perfectly spherically symmetric) Schwarzschild type.
If it were emitting the light, yes, but it's not. It's more complicated than this, but imagine holding a black sphere in front of a light bulb. You'll see a ring of light around the sphere.
Somewhat separately, there's the accretion disk, which again is a disk not a sphere, much like other orbiting systems like solar systems or galaxies−the gravity between bodies orbiting the same central gravity source causes them to arrange roughly into a plane, rather than all having their own unrelated orbits. We're not seeing the accretion disk directly though, but rather the light from it, and from other sources, that is able to pass around the black hole. (ie the black sphere in front of the light bulb.)
Watch that video; it explains it in a very approachable way.
⬐ zeristor
How big a VLBI baseline would they need to see much more detail? Are there any plans for a space based VLBI; not easy when you consider the utterly huge amounts of information they have to transfer, a radio telescope in a L5 would be a start.
Also I'd be interested to read how they got around scintallation of the Interstellar Medium.
⬐ thatcherc
Space VLBI is a thing already! Seems to have been around for a while [0][1]. I imagine that the challenge with during EHT-style VLBI in space is downlink bandwidth... there's just no way to get the petabytes of data they talk about down from a satellite. It's already so much data that it's easier to fly hard drives around the world than to send it over the internet, so the transfer would probably take ages over a microwave link. Maybe with laser communication in the near future but even then... it's a lot of data.
Resolution-wise, the angular resolution is inversely proportional to the farthest baseline, as given by the Rayleigh criterion. [2] To get twice the resolution, you "just" need to double the size of your baseline (and do that in both dimensions, otherwise the angular resolution will be different in x & y). We've maxed out the Earth's baseline, so it seems like orbital radio telescopes are the only way to better resolution. Pretty exciting!
[0] - https://www.jpl.nasa.gov/missions/space-very-long-baseline-i...
[1] - https://asd.gsfc.nasa.gov/blueshift/index.php/2016/07/25/thi...
[2] - https://en.wikipedia.org/wiki/Angular_resolution
⬐ privong
> We've maxed out the Earth's baseline, so it seems like orbital radio telescopes are the only way to better resolution.
Going to higher frequency gives you higher resolution for a given physical baseline length.
⬐ JoeAltmaier
For some observations, they've used the earth's moving position in space to create a synthetic-radar style image. That gives an 'aperture' of 180M miles.
> For some observations, they've used the earth's moving position in space to create a synthetic-radar style image. That gives an 'aperture' of 180M miles.
That may not work for imaging the immediate surroundings of black holes – one can only combine the data for that SAR-style imaging if the source you're observing doesn't vary significantly (e.g., in brightness or flux distribution) between observations.
⬐ richk449
> I imagine that the challenge with during EHT-style VLBI in space is downlink bandwidth... there's just no way to get the petabytes of data they talk about down from a satellite.
Why do you say that? The latest High Throughput commercial satellites do 500Gpbs downlink throughput, so shouldn't the transfer time for a petabyte of data be reasonable?
Of course those satellites are configured to transmit data all over the earth, so the technology would have to be used differently for this application.
>> Also I'd be interested to read how they got around scintallation of the Interstellar Medium.
One of the reasons they've imaged M87* rather than a black hole in our own galaxy was to avoid dealing with ISM scattering - we don't have to look through our edge-on disk, so it's easier to image another galaxy's center somehow. But the Sgr A* image might be in the works already, it was mentioned in both the press-conference and one of the paper's future work section.
How does one even deal with ISM scattering?
The optical equivalent can be dealt with by active optics, or picking frames with best seeing, but for ISM I imagine it is a fair bit different.
⬐ pps
You can ask them directly: https://www.reddit.com/r/askscience/comments/bbknik/askscien...
> Are there any plans for a space based VLBI; not easy when you consider the utterly huge amounts of information they have to transfer, a radio telescope in a L5 would be a start.
There is space-based VLBI at lower radio frequencies, with the "Radio Astron" project[0]. That effort works at frequencies which are roughly a factor of ~10 lower than that of the EHT. I'm not aware of any plans for millimeter Space VLBI, but the higher frequency would require higher data rates.
[0] http://www.asc.rssi.ru/radioastron/
⬐ mythz
Katie Bouman herself did a fascinating Ted Talk behind the effort involved for creating the first image of a black hole:
https://www.ted.com/talks/katie_bouman_what_does_a_black_hol...
⬐ Aardwolf
So we got an image of one in a far away galaxy before the one in the center of our own galaxy!
⬐ freejak
Is this due to occlusion?
A lot of the problems with imaging the black hole at the center of the milkway is that we would have to look along the midplane of the galaxy towards the center. That is where a lot of gas and dust is to be found and the radio waves interact with that stuff. They did take data and are working on it, but the image of M87* was easier to reconstruct and went out first.
Event horizon telescope has been taking data of the Sag A* black hole but the data is still being processed and yet to be released.
⬐ ArtWomb
Breakthrough discovery in astronomy: press conference
https://www.youtube.com/watch?v=Dr20f19czeE
And the image itself:
https://pbs.twimg.com/media/D3y037OW0AQmpAf.jpg
Is this image lossy compressed by Twitter? Is there an official release?
Not really, that's just the level of detail available.
⬐ nathanm412
https://www.nsf.gov/news/special_reports/blackholes/formedia...
⬐ fsakura
I always thought we have photographed black holes. Any idea/link to article where it explains why this took so long and why it was difficult?
Well, we have in the same sense that you take a picture of bacteria every time you take a selfie: there certainly are lots of pictures which contain black holes that are too small to be seen in them.
In line with that analogy, the basic difficulty is simply that they are tremendously small compared to the sizes of galaxies.
⬐ greeneggs
Here's an NY Times article about the project:
https://www.nytimes.com/2018/10/04/magazine/how-do-you-take-...
The author, Seth Fletcher, also wrote a book about it, "Einstein's Shadow: A Black Hole, a Band of Astronomers, and the Quest to See the Unseeable", if you want more details.
https://www.goodreads.com/book/show/36300597-einstein-s-shad...
There have been plenty of indirect observations such as their jets, stars in very tight orbits around Sagittarius A*, xray binaries etc.
This is a direct observation of the immediate environment of a black hole, i.e. its accretion disk and other light bent around it.
⬐ yk
This is as far as I understand the first picture of a black hole shadow, that is the dark disc that indicates that there is actually light missing. Before that, there were astrophysical observations that strongly indicated that there is a black hole, for example QSOs, that is a bright combination of accretion disc and jet, that is powered by a black hole, or one paper I really liked tried to argue that there has to be a black hole in Cygni X1 because otherwise we would see the missing matter crashing into something. (Cygni X1 is a binary of a star and something very bright and the star is loosing mass.)
Because they're very small, very dark, and very distant, compared to visible objects like planets in our solar system, stars in our galaxy, and other galaxies.
⬐ lovehashbrowns
Their website talks about some of the challenges of getting a picture of a black hole: https://eventhorizontelescope.org/home
All of the data used to generate the image was gathered by around a dozen different telescopes around the world. The black hole itself also needs to be in a specific configuration in order for us to be able to see it. It needs to have an accretion disk that's generating light. It needs to be sufficiently large or close. And it can't be obfuscated by other astronomical objects like stars or nebulae. This black hole itself is hugeeeeee and far. It's about the size of our solar system, but it's ~52 million light years away.
⬐ harshvladha
Does anybody noticed that there are two brighter areas (not just one) in bottom part of the image.
One at approx 8PM and one at approx 5-6PM (if seen as clock)
⬐ pulse7
The image of the black hole is not sharp. Isn't it possible, that the optics at that distance may make "optical mistakes" and this is not a black hole after all but maybe just some circular lightning or some darker, but not black object before some star - like in solar eclipse?
⬐ pontifier
The thought struck me that the interferometry technique used must have gathered data from a much wider field of view than of just the black hole.
Burried in that 5 petabytes of data is likely a scan of a much larger field of view at a similar resolution.
Yes. But since you have a bright spot in the center of the larger image and you only have a limited number of baseline between pairs of telescopes there will be all kinds of artefacts in the outer portion of that large image. So it get's cropped away. (Most of it is actually never even computed).
I still think it would be worth pursuing. I hope the raw data will be made available at some point... Not that I have 5 petabytes free space available right now...
⬐ mkurz
Here it is: https://image.futurezone.at/images/cfs_932w/3398269/bild-ist...
⬐ angrygoat
Here's the image via twitter: https://twitter.com/ehtelescope/status/1115964692802019328
Here's the image via a URL:
https://pbs.twimg.com/media/D3yzi3dX4AEnoEp.png
> Scientists have obtained the first image of a black hole, using Event Horizon Telescope observations of the center of the galaxy M87. The image shows a bright ring formed as light bends in the intense gravity around a black hole that is 6.5 billion times more massive than the Sun
⬐ vertline3
So how come it's a ring of plasma that forms and not a sphere? Like we can see the hole unobstructed? Is it because gravity tends to clump things together? I guess Saturn's rings are the same?
⬐ kerbalspacepro
Conservation of Angular momentum collapses shapely sort of things into a disk
Saturns ring, the flattening of the accretion disk, the fact that planet systems will have most planets more or less in one plane and that spiral galaxies are flat (much thinner than they are wide at least) all have the same reason, yes. Basically most orbits align due to initial orbital momentum and the stuff that orbits in other directions sooner or later experiences enough friction to be forced into the common orbital plane.
⬐ sus_007
Here's an animated explanation of the process by the Event Horizon Telescope. https://youtu.be/hMsNd1W_lmE
⬐ ken
Misspeak? This video says it's 4 million times as massive as the sun, but today's press release and Wikipedia article about it both say it's 6.5 billion times as massive.
⬐ ProAm
The black hole at the center of the milky way is 4 million times the mass of our sun, the one in the photograph is much more massive. Im guessing that is what they were referring to.
⬐ jschrf
Let's say I were to jump into this black hole for fun. What happens to me? What are the chances I go somewhere cool, like another universe, versus getting turned into a spaghetti noodle?
⬐ doctorstupid
Don't it's no fun you'll get space sickness trust me.
⬐ shireboy
I understand this was done by coordinating multiple telescopes to create a virtual earth size scope. Why is this not done more? It seems like cloud services would make this relatively easy to share bits technically. Are the issues mostly political or are there tech issues making this harder to do more often? Tangential question: if we can do this with an earth-size virtual scope, what could we do with a larger one? Scopes on earth/moon/Lagrange points synced together.
⬐ ceejayoz
It's been done since the 1970s.
https://en.wikipedia.org/wiki/Astronomical_interferometer
There are some proposals to do it in space.
https://www.universetoday.com/139566/instead-of-building-sin...
https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Ant...
⬐ jhallenworld
So how can we make a planet sized synthetic aperture optical telescope?
I'm wondering if there is a way to do it with holograms (since they preserve phase information): take holograms of the object from opposite sides of the earth and then combine them offline. There are some papers in this direction:
https://hal.archives-ouvertes.fr/hal-00654840/document
(spy satellites probably already do it...)
Can we see nearest stars and planets using this method? Or is it only for radio sources?
One day we may even connect millions of smartphones to observe various interesting space phenomena.
⬐ myth_buster
On a tangent, interesting closing remarks from the commissioner on the importance of courage, dreams and science. Targeted towards the present anti-science climate.
⬐ gigatexal
Any word if Einstein's predictions are correct or refuted? There was word that if the shadow looked a certain way it could mean that relativity is incomplete.
⬐ mercwear
USA Today is live streaming a presentation about this from the National Science Foundation and they seem to be taking the point of view that Einstein's theory of general relativity has held up SO FAR. There is still quite a bit of research to do on the image / data which means the theory could be further proven or disproven.
⬐ hacker_9
General relativity predicted the event horizon, but it breaks down when trying to predict what is going on in the center, as all equations go to infinity, which is likely to be more a problem with our current understanding than what is actually happening.
⬐ SketchySeaBeast
I wonder if this is a call for another level of math. Are we just not able to take the derivative of something in respect to space time rather than just time?
⬐ JanSt
Pretty much exactly what was predicted as to the livestream :-)
⬐ adamson
A Bloomberg journalist at the press conference asked this question and the researchers didn't really commit to an answer.
⬐ misnome
In the discussions they said that it appeared entirely consistent with GR. Presumably more specific analysis is in papers coming afterwards.
⬐ zaph0d_
Are there any papers already out which focus on the technical aspect of reconstructing the image? I heard that they analyzed 3.5 Petabytes of data for this.
5 PB. The American unveiling focus on just what you talk about: https://www.youtube.com/watch?v=2DxjuE7WDlk
Have a look at the papers listed at the bottom of https://iopscience.iop.org/journal/2041-8205/page/Focus_on_E... .
Keep in mind that a lot of the details will NOT be in those papers as they have used CASA and AIPS, two standard software tools that have been developed over more than a decade. Details are consequently scattered over many papers. Radio interferometry is not new and there is entire textbooks on the subject. The exiting bit here is not that we go a first image from interferometry but that we have a first image of the region just around a black hole.
⬐ brink
Here's an article (and image) from the guardian: https://www.theguardian.com/science/2019/apr/10/black-hole-p...
Why there are two live events right now? One is from Europe, another from US:
EU: https://www.youtube.com/watch?v=Dr20f19czeE
US: https://www.youtube.com/watch?v=re_o0uckG-M
⬐ bionsystem
There are 6. It's an international accomplishment.
⬐ docker_up
If there's a God, He must be so incredibly proud of us. What we have accomplished as a species is just mind-blowing.
I wonder how it compares to the one from Interstellar?
https://www.newscientist.com/article/dn26966-interstellars-t...
This blog post describes the stuff Interstellar mostly gets right, and the big thing it's lacking (a brightness asymmetry from the rotation of the accretion disk).
http://blogs.nature.com/aviewfromthebridge/2017/03/28/imagin...
Side note - there's a really neat write-up by someone who realistically rendered a rotating black hole in 1979, partly by computer and partly by hand [0]. He goes through all the important visual effects, like GR ray tracing, disk brightness, and Doppler shift. The final image is pretty cool.
[0] - https://blogs.futura-sciences.com/e-luminet/2015/02/18/black...
⬐ java-man
If I recall this correctly, the movie intentionally simplified the rendering of the black hole (by eliminating the Doppler shift) to make it more visually appealing.
Pity, really. Then again, there were other things that made no sense in the movie...
NASA hq launching a rocket from the next door room, was my warning.
Danny Boyle was confident he had "done SciFi" with Sunshine, perhaps Christopher Nolan just wanted to tick that box too.
Where Kubrick led others followed; Ridley Scott was doing great to Bladerunner...
⬐ TheHypnotist
Since we're on it, there's actually a book about the science of interstellar.
https://www.amazon.com/Science-Interstellar-Kip-Thorne/dp/03...
The author (Thorne) was actually a producer for the movie, as well as a being a physics professor.
Google Cache because site got stomped with traffic
http://webcache.googleusercontent.com/search?q=cache:https:/...
Even that page is not opening for me ...
⬐ basicplus2
I have a real problem with this..
they have not taken a picture of a black hole because that is not possible..
they have at best constructed an image of some effects of a black hole.
This is where top scientists do damage to science for ordinary people when they make fundamental errors in public statements.
⬐ sigmaprimus
Anyone else getting sick of the media calling this the first "PHOTOGRAPH" of a black hole? It's a spectrograph at best and more realistically a rendering. Don't get me wrong it's really cool but it's not a photo.
⬐ qlk1123
You are wrong, this is definitely a photo, shot by multiple telescopes global-wise.
Multiple radio telescopes right? Then the data was compiled and rendered into an image, you really think it's that bright? How about orange? I wonder why it's not symmetrical? I suppose you can take multiple images combine them into one, run it through a bunch of filters and call it a photo but I still think it's more of a rendering than a photo.
>an array of relatively large sensors with very high exposure looking at information from something very far away, and combining that data with a 'brightness' scale, instead of a 'colour' scale I like the sound of that far more than photograph, but I will relent, not worth getting into a slap fight over what constitutes a photograph...face palm
⬐ brandonjm
> Then the data was compiled and rendered into an image
A digital camera is just an array of tiny sensors that detect how much light hit them and a computer takes those values and renders an image from them.
> you really think it's that bright?
If you take a photo in a dark room with increased exposure then the resulting image is brighter than what you see with your own eyes. This was taken over a long period of time, effectively a long exposure. If you were close enough to see it at the scale you are seeing it on your computer screen it would probably be far brighter.
> How about orange?
As mentioned in another comment on this thread, it's just the colour scheme used in the output to show the brightness differences. The accretion disk is not necessarily orange, this photo is essentially greyscale mapped to a black -> orange -> white scale.
This is just an array of relatively large sensors with very high exposure gathering information from something very far away and combining that data in greyscale.
> I wonder why it's not symmetrical?
See Veritasium's video [1] on why it looks like it does, in short, the effects of the black hole and our angle to the accretion disk.
[1] https://www.youtube.com/watch?v=zUyH3XhpLTo
⬐ gianpaj
Also relevant: How to take a picture of a black hole | Katie Bouman | TEDxBeaconStreet https://www.youtube.com/watch?v=P7n2rYt9wfU
⬐ it
If you want to jump directly to the image, here it is: https://www.youtube.com/watch?v=Dr20f19czeE&t=8m28s
Can the equipment used from several places on earth for this be deployed in space (may be to a geostationary orbit)? Not thinking about cost for a moment is this a far-fetched thought or such a thing is technically possible.
⬐ Rebelgecko
Yeah. There were a few NASA projects attempting to do this that got cancelled circa 2010 (programs called TPF-I and SIM).
Something that has been done for a while is using a single telescope in space in conjunction with some on Earth. Russia has a radio telescope in orbit called Spektr-R. I think it died recently, but Spektr-R did interferometry with other telescopes on Earth. When it was operational, I believe it was the widest VLBI array— its orbit was higher than GEO (and even intersected the moon's!), so it got pretty good distance from the ground.
⬐ amy12xx
Katie Bouman's 2017 TED video explaining how the image was taken https://www.youtube.com/watch?v=BIvezCVcsYs
⬐ skanderbm
Reminds me of this project (building a realistic blackhole raytracer) http://rantonels.github.io/starless
⬐ zyngaro
This is fantastic. But why did it take so long to prove the existence of blackholes ? are there any scientific breakthroughs that made it possible or it's more a technological achievement ?
⬐ return0
To be clear, they detected the black hole because it looked like the way we expect a supermassive black hole to be? Or were there other hints that led to it being discovered where it was?
⬐ omarchowdhury
The Eye was rimmed with fire, but was itself glazed, yellow as a cat's, watchful and intent, and the black slit of its pupil opened on a pit, a window into nothing.
⬐ acl777
There are people that can't believe the Earth is round... I can't imagine what "alternative explanation" will be made to explain a black hole...
From whatever videos I have watched I don't believe so called "flat earthers" really believe that the earth is flat. At most they seem to be trolls and just like to oppose whatever scientists say. So IMO nobody should waste time explaining anything to them.
⬐ 781
? As you can obviously see in the image, the black hole is also flat.
That its a black circle?
Photoshop!
⬐ nukeop
There is still zero evidence that there exists such a thing as a black hole with all its fantastic properties. We have a picture of a red-yellow accretion disk and that's it. Black holes remain a mathematical artifact of general relativity and there is as much evidence of their existence as of crystal healing and astral travel. There is at best some evidence of very massive objects at certain points.
⬐ posix_me_less
How do you explain observed gravitational wave observed by LIGO? It is almost the same as the one calculated for coalescence of two black holes in GR.
It's still no proof of existence of black holes. It's like saying that a lightning storm is proof of Thor, because the legends say he wields the very same lightning. The cause could be something entirely different (spontaneous spacetime ripples, alien generators, whatever really).
⬐ Retra
General Relativity is based on familiar observations, not mythical beings. It's not nearly that complex. Crystal healing and astral travel likewise don't even have a mathematical basis that is compatible with observations about reality, so there's no comparison to draw.
⬐ ncallaway
> It's still no proof of existence of black holes
You've shifted the goal posts in this sentence. Previously, you said there was no evidence of black holes. Now you're upped the scale to demanding proof.
The previous commenter wasn't claiming that LIGO detections were proof of black holes. They said that they were evidence of black holes.
The LIGO detections, by all reasonable metrics, would certainly qualify as evidence of black holes. Of course, I agree with your assertion that taken alone, LIGO detections are not proof of the existence of black holes. Taken with the significant body of other evidence we have, though, I would say the sum total of evidence that we have is a strong indicator that black holes exist.
The fact that scientists were able to accurately predict what this black whole should look like using the math at our disposal is about as good a piece of evidence as one can get.
I think it's on par with the LIGO detections (which I think also qualify as really strong evidence) given the specificity of the predictions and how closely they aligned with the observed data.
Taken as a whole, the body of evidence is really strong and it's amazing how much stronger it's become in just the last few years!
This is truly an amazing image.
⬐ Ono-Sendai
I think you're right, that they probably are a mathematical artifact of GR. Regardless, what they are in the real world will share quite a few properties with black holes.
⬐ swamp40
This is what convinced me: https://www.youtube.com/watch?v=B0QRpid5_QU
Likewise, we have zero evidence that there exists such a thing as atoms... if you only use direct evidence of the unaided sense, rather than indirect evidence.
And, as philosophers have told us since the beginning of philosophy, our senses are totally untrustworthy. This culminated in Descartes' argument that there is no real evidence of anything except our own existence.
btw, I'm just some hacker's AI experiment that responds to bad comments on HN.
⬐ sam0x17
FML I just took this literally and was breaking down the post to see how this could have been an ML-based response from a bot. Then I looked at the poster's other comments and realized I'm an idiot. :(
We're all idiots, my friend. Remembering that is the hardest and most important work we can do.
⬐ izzydata
Potentially none. They can just deny its existence and pretend this is a fabrication.
⬐ hashfunktion
Pardon my naïveté, I never took advance physics, but if light can't escape from a blackhole, then how can we "take a picture" of one?
Same way we can take a picture of a shadow - by taking a picture of the stuff surrounding it. This black hole is surrounded by a bright accretion disk.
⬐ ValleyOfTheMtns
It's a photo of the absence of light (in the middle) and light/particles swirling around the black hole that haven't passed the event horizon (yet).
⬐ m3kw9
How come it so happens we are looking at the black part while the sides are bright? Does it look like a O from any angle of the sphere?
How jaded have I become that when I see four old men and one young person I automatically assume the younger one did all the work?
Now I'm really curious if Quantum Entanglement will work if you send one of the entangled particles into a black hole.
⬐ dangban
Is this a 2D slice of the actual spherical black hole?
Because if a black hole is a sphere, then shouldn't the whole thing be golden?
⬐ scottie_m
The black hole is warping spacetime around it in a severe fashion. The innermost (unstable) orbit is called the photon sphere, and light can make several orbits around the hole before either falling in, or going off to infinity. The result is that (depending on the hole's rotation relative to your point of view) you're going to see roughly the same image we have today. You're actually going to see multiple images of the entire hole, stretched around the outer edge, and those images multiplied and distorted again. If you could somehow "stand" by the photon spehere you'd see the back of your own head many many times.
The other factor is that while the event horizon itself is a spheroid, the accretion disk of bright infalling material is not. The horizon is totally black, so it will always look roughly like seeing a disk face-on no matter where you look at it. The accretion disk is a "hoop" that's stretched and the image is multiplied and distorted by the strong warping of spacetime in the region. As a result you get smeared and repeated views of the entire disk including portion behind the hole in a kind of bent band. Plus the whole thing is subject to strong Doppler beaming hence the bright and dark regions.
⬐ codesternews
It's really amazing that human mind can predict and tell the things without even seeing it.
How come we even know and predicted the radius and things far away in galaxy without even seeing it. It's just amazing. Wow it just amazes you that scientist even have predicted the radius of thing and how it work etc.[1]
⬐ stunt
Numbers about measures and dimensions are mind blowing. You can find articles with more details.
⬐ zframroze
This is so phenomenal. What an amazing and tremendous effort from that team!
⬐ malikNF
The reveal happens at -27:06
This is one of the most beautiful photos I've ever seen. Fascinating.
⬐ csomar
Does this confirm or disapprove Steven Hawking Black-hole radiation?
⬐ symmetricsaurus
The Hawking radiation for a black hole with this mass would be really small. It actually decreases with added mass. So, this observation does not say anything about Hawking radiation.
⬐ ajuc
Shouldn't it increase, just with square of the radius, so it's dwarfed by all the effects that scale with cube of the radius?
⬐ cyberfart
Here is an answer from SE https://physics.stackexchange.com/a/318734
The temperature is actually inversely proportional to black hole mass and the power that is radiated away even falls like the square of the mass. (A rough but illustrative derivation can be found in https://en.wikipedia.org/wiki/Hawking_radiation#A_crude_anal... )
⬐ GolDDranks
Neither. Hawking radiation - if it exists – is far too weak to be detected this way.
⬐ quadcore
100 billion kilometres wide, is that correct?
one time he also said 100 billion billion kilometers.
⬐ wallace_f
Can someone explain in layman terms: If graviton particles or gravity waves travel at light speed, how does this information escape the event horizon?
It doesn't. It escapes from outside the event horizon. The event horizon is the dark area in the middle.
A rephrasing: How does the gravity-information about what's inside the event horizon get out?
That's a really interesting question. (And you did say "graviton particles or gravity waves" in your first post, and I missed that.)
At first glance, I think that the gravity information can't escape from inside the event horizon, just like light can't. That means that the event horizon describes a frozen version of the mass inside it, not a current "live" version.
And that seems to work, if you think about gravity waves. There aren't any changes to the gravitational field coming out from inside the event horizon. But it doesn't work so well if you think about gravitons. "There aren't any gravitons coming out" should be equivalent to "flat worldlines", which is very much not true just outside the event horizon.
It also doesn't seem to work for a situation like a black hole merger. The spacetime outside the event horizon is this frozen snapshot, but it can still do this spiral around this other black hole? That doesn't seem to make a ton of sense.
So I'm not sure my answer is very good. But it's a fascinating question. If anyone has a real answer, I'd love to hear it.
⬐ Pharmakon
It's not that interesting unfortunately, although I liked your approach. A graviton would be just another boson like a photon, and like a photon would be unable to escape. All of the worldlines of a graviton within the event horizon would lead to a collision with th singularity. It's just another aspect of "No Hair" on the hole.
Remember that this applies everything where r≤1. As far as "flat" worldlines I think you might be thinking of a geodesic approaching r=1 in terms of a null geodesic, which isn't necessarily true unless we're dealing with a photon or graviton. Regions I,II of the Classic Kruskal-Szekeres extension illustrates this pretty clearly.
https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates...
The total mass of the black hole (and all other information possible about it) can be described in terms of the boundary at r=1, so there's no problem with mergers or accretion. To answer Wallace's original question, we see no information escaping the black hole. What we're seeing is sort of like shining a light on an absence of information, and observing the shadow cast. That's not quite right, but it's close.
But there pretty clearly is a gravitational field at r > 1. If that field is made up of gravitons, and a graviton can't escape from the mass to outside r = 1, then what is the source of the gravitons that compose the field at r > 1? If they don't originate at the mass, then... what?
I think the downvote brigaiding on this thread is coming from a particular set of users.
I noticed my karma is jumping down in waves. Unrelated comments are all being downvoted at the same time, suggesting unlikely coincidence or that some users are clicking on my profile and going through downvoting all the comments.
The manifold is well-behaved and continuous at r=1, and mass is one of the few characteristics a black hole has other than spin and charge. Gravitons from within the event horizon won't escape, but the event horizon itself can be thought of as the entire black hole (for everything outside of the black hole).
This discussion might help where my ability to answer your excellent question is failing: https://physics.stackexchange.com/questions/937/how-does-gra...
I also saw that stackx discussion when I asked the question and Google'd it. But I was surprised by the fact that while I'd seen in social media, QA, etc this question had been asked before, I was looking for something of a longer or more authoritative source that was accessible to people outside of academic study.
Thank you for that link - it was very helpful. Summarizing:
The same problem would exist for the electric field from a charged black hole. However, static fields don't need propagating photons to establish them, so you don't have to get photons from inside the black hole in order for the electric field to be established outside.
The same would be true of gravitons. But one respondent indicated that general relativity can't do a second quantization like electromagnetism, and therefore gravitons are... suspect? Impossible? Not proven? It wasn't clear to me how strongly to take that statement.
⬐ n00bdude
What do you reckon is in the black hole?
⬐ sbhn
Why are they all wearing suits?
⬐ laythea
Is this as real as the computer edited NASA images (eg of earth) or is the actually real?
⬐ briarpatch
Here's today's xkcd comparing it to the size of our solar system: https://xkcd.com/2135/
⬐ dandigangi
What a time to be alive! I wish Hawking got to see it.
Lol, why would someone downvote this?
⬐ jahrule
black ties and revelations...
⬐ modzu
why is it oblong???
⬐ delecti
If you mean the brightness asymmetry, it's because the accretion disc is spinning. This video predicts pretty well all of the features we expected to find, and it lines up with the actual image.
Live Stream: https://www.youtube.com/watch?v=Dr20f19czeE
⬐ i_cant_speel
Have they already shown it?
Thanks! Over 150,000 viewers right now.
⬐ steve76
I remember reading about measuring stars with something other than light.
What would that look like? Knowing all particles between us and your target, and then popping a mass into existence in such a way you get a measurement?
⬐ samzevo
https://coincircle.com/l/sfX5us6vcQ
⬐ stewartjarod
At what time do they show the black hole? i hate links to videos that aren't at the time that the title is about...
Will it be possible to use this data to study black hole radiation or pair production?
https://i.redd.it/p55wgfxsrfr21.jpg
even the black hole photo is sucking up it's own press conference! :-)
⬐ tsukurimashou
Screenshot from press conference: https://imgur.com/UBsPgzw
Edit: Better posted above: https://pbs.twimg.com/media/D3y037OW0AQmpAf.jpg
⬐ OnlyRepliesToBS
more masturbation to the fanfare than the actual research picture
⬐ comiskey1905
thanks, i didn't watch it.
⬐ colinwilyb
INT. EUROPEAN COMMISSION – PHONE RINGS
Seven days...
⬐ tomc1985
Why does the ESA always couch their discoveries in these stupid press conference panels? Every time I try to watch something of theirs it's a bunch of old people blabbing. Show the pictures and stop talking!
This isn't ESA, it's unrelated EU-funded research.
⬐ crocal
My God, it's full of stars!
⬐ ape4
A small thing. It could have been nice if the scientists didn't have commercial bottled water on stage. As an example to the world. Reusable personal containers would have been nice.
Most scientist I know have a reusable water bottle on their desk. And a coffee cup. But it will likely have a cat or a joke of the "astronomers do it in the dark" flavour on it. Not something you typically want to show on TV. Also they traveled from their home institutes to the location of the press conference and got handed that plastic bottle by some well-meaning PR person.
For this event they could have purchased multiple use containers for each person. Wasteful but a better image. So not really their personal containers.
⬐ patagonia
One difference between a Tesla or Apple style event vs a staid, scientific one such as this. Starting in the evening vs 9am...
⬐ legatus
Isn't that related to the fact that it has been announced across the world at the same time? In europe it was announced at 3 pm for example. I'm sorry, I don't understand your point
That showmanship counts. I don't believe HN would argue with me if I said "a strong sales group can make or break a company". Why is it unfashionable to suggest that, in the marketplace of ideas, with an announcement of this magnitude, this exciting, this "sexy", the presentation of the announcement would benefit from being delivered in a likewise suitable atmosphere. At night, as an event, in a cool warehouse with cool music. I'm not trying to take anything away from the inherent excitement and importance of the news. Just suggesting that the cause could be furthered even more if there was a little hype, a little showmanship. Rather than a well lit lecture hall populated by scruffy reporters, early in the morning, morning as defined in the locale the event is being held.
I would argue that "hype", or as you say "showmanship", is actually damaging when it comes to science. Maybe it's just me, but I try to stay away from pop-science news sites because of all the hype they're filled with. As others have said, I like scientific conferences to provide the evidence, that is enough for me. As for the "early in the morning", as I already said, is a consequence of trying to make it possible for as many people as possible to follow the conference, I don't think anyone would be against another conference in the evening for american viewers. There were 5 "locales": Brussels, Santiago, Taipei, Tokyo, Washington. Unless I missed something, I still disagree with you.
⬐ kryptiskt
Apple always have their unveilings in the morning pacific time.
Ha. My bad. Thanks for correcting. I always streamed after the fact. Prob because they were early.
Appropriate types and degrees of showmanship/salesmanship depend on the target customer. And scientists, by and large, don't seem to be concerned with impressing the general public - or at least, that is far secondary to the importance of impressing other scientists.
One could argue that making an impression on other scientists is the basis of the scientific method. And we impress with the quality of our evidence and the repeatability of our experiments.
> And scientists, by and large, don't seem to be concerned with impressing the public...
Some quotes from European Research Commissioner Moedas, answering the first question.
"...which is that this is linking between the citizens and science, how important is that?..."
"Because we want European citizens to feel connected."
"I've never seen this room so full."
"It's so refreshing to come here, to see so many people, to see people clap. I mean it's very rare in a press room to have people clapping."
During his introductory comments also, he is clearly excited and wants to engage people, and not simply through the scientific method. He talks about watching sci-fi movies as a kid and books on science.
People. It's ok to throw Science a party.
⬐ chirau
MASSSIVELY UNDERWHELMING
As a person who has no interest in these intergalactic shenanigans, it looks just like another ball on fire. I wanted to be excited, I really was, but this is just another picture.
A giant leap for mankind, and i fully recognize that, but the awe... nada. It's just another picture really. There is nothing fascinating about it.
Hats off to the people who brought this to us though. I know gravity of the matter and how daunting a task it was. Keep on!
⬐ jazzyjackson
If it fails to pique your interest, oh well, but the more you read about black holes and what is observable about them, the stranger it feels to be looking at it face to face.
⬐ ttul
Perhaps consider this: "VLBI allows the EHT to achieve an angular resolution of 20 micro-arcseconds — enough to read a newspaper in New York from a sidewalk café in Paris [6]."
What's surprising about this image is how utterly microscopic the thing is that was observed - from our location. The width of the event horizon is 40 billion kilometers. That's only 267x the diameter of our orbit around the Sun. But the M87 black hole is 26,000 light years away. 26,000 light years is 2.45979e+17 km, or 639,903,746,098 times farther away than the moon.
⬐ phonypc
53 million light years. 26 thousand would still be in our own galaxy.
Quite right. Thank you for the correction.
I am fully aware of the theory and excitement over all this. I just expected something... different. I thought i'd be wowed by something new. It literally looks just like a regular photo.
Is there a lot of context behind it? Sure. Is it a HUGE leap in our advancement? Definitely. Is it an eye-turning picture? NO. That is all.
Black holes are fascinating, their picture simply isn't, for me at least.
Good video that correctly predicted the image and describes why it looks the way it does [1].
TL; DR The dark area is the entire surface of the event horizon, including the side facing away from us, plus some more due to photons missing the event horizon "directly" being drawn in. One side is brighter due to its being Doppler boosted.
⬐ mrandish
Wow, the video you posted is even more informative and clear than the actual press conference, and it was created by someone who hadn't even seen the image yet based purely on the mathematical predictions of what we would see.
Kind of sad that after all the amazing effort and resources that have gone into the creating the image that the international team couldn't have featured an explanation as clear as this in their actual press conference.
⬐ teej
Different target audiences for a science YouTube channel and a scientific findings press conference.
It'd be extremely shocking if he hadn't been able to. The math has been known for a very long time the largest differences would be based on the orientation of disk relative to us but that has been mostly known since the original Hubble picture. If he'd been significantly wrong that'd mean our understanding of the physics was wrong or something unknown was happening at a pretty large scale.
Everyone knew what the image was going to look like, so it's not any harder to prepare in advance.
⬐ arriu
The video by Veritasium is by a guy who literally got a PHD on the subject of making physics more approachable through videos. He is exactly the person I would expect to provide a more clear and understandable explanation.
https://www.youtube.com/watch?v=S1tFT4smd6E&feature=youtu.be...
⬐ rexpop
Why doesn't he work with the actual astronomers, then?
⬐ cicloid
Just by sharing the news he is working with them helping the global effort.
⬐ code_duck
Because he's busy making physics more approachable through videos.
Perhaps I should have phrased it as "why don't astronomers work with _him_?"
⬐ mgalgs
You don't have to watch his channel for very long to learn that he often does (see his video about the recent gravitational wave detection [1], plus a bunch more).
[1] https://youtu.be/iphcyNWFD10
⬐ everdev
Here's the follow up video after the image was released: https://www.youtube.com/watch?v=S_GVbuddri8
So amazing that scientists were able to predict what something would look like that we have never seen before.
⬐ forgot-my-pw
Einstein predicted the existence of black hole back in early 1915. Pretty amazing.
⬐ ben_w
I thought (relativistic) black holes were first predicted by Schwarzschild in 1916?
(As an aside, I have found a whole extra level to nominative determinism since starting to learn German — Schwarzschild = Black shield)
⬐ dmix
It's a little of both, but yes Schwarzschild technically first predicted them.
> In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months later, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass.
https://en.wikipedia.org/wiki/Black_hole#History
⬐ limbicsystem
Or was it the Revd. John Michell? https://en.wikipedia.org/wiki/John_Michell
⬐ TheOtherHobbes
Michell. And maybe Laplace.
Of course they "invented" Newtonian black holes, not relativistic black holes.
Even so - well ahead of the rest.
⬐ jcoffland
This is also called an aptronym.
https://en.m.wikipedia.org/wiki/Aptronym
⬐ WhitneyLand
Einstein first developed the theory and the equations that allowed for them to be discovered.
You can imagine that space-time equations have many solutions and properties that can't be contemplated all at once even having them right in front of you.
Schwarzschild took the equations and obsessed over them for countless hours and eventually discovered that one solution to them implied this phenomenon and therefore he discovered black holes by discovering a specific solution to Einstein's equations.
Of course no one knew at the time if the mathematical solution represented real physical objects that exist in the universe, because it doesn't always happen that way. Occasionally some obscure corner of the math predicts something that's a dead end or anomaly that doesn't have any meaning of value as far as it is known.
They had no way to know one possibility from the other.
⬐ noir_lord
It's good science.
Been able to make testable predictions and then confirming them or disproving them is the entire (awesome) point.
⬐ trickstra
ad Good science, I just randomly bumped into this video today that very nicely explains the differences between good science and bad science. https://www.youtube.com/watch?v=umo6pMCkcXs (safely skip the first 3 minutes)
After having said that, I did a bit more research into how the image was made. I am of course reserving judgement as I don't fully understand the underlying technology. But it sounds like they used an interpolation algorithm to come up with the image based on renderings of what we "think" a black hole should look like. This high level overview from a ted talk goes into how they 'unbias' the data. But it is obviously on a very basic overview: https://www.youtube.com/watch?v=P7n2rYt9wfU
Not sure why you're getting downvoted because (as a physicist) I'd say that especially in the face of a high-level discovery like today's, a healthy amount of scepticism is a good thing.
That being said, it seems your concerns are being addressed in the TED talk you linked to from 8:45 onward?
Moreover, in the NSF press conference today it was said that they had four different teams in four different locations across the globe last year, working on interpolating the data and generating the images and they basically asked the teams to lock themselves in, i.e. to not communicate with each other at all, and use (more or less) whatever interpolation algorithm they thought would fit the data best. And at the end, when the four teams met up last year, they had supposedly arrived at very similar-looking images.
I briefly(!) looked at the papers that were published today ("First M87 Event Horizon Telescope Results" I-VI) and while I'm anything but an expert when it comes to radioastronomy and imaging technology (I'm more a theoretical physics/mathematical general relativity kind of guy), I came across the following statements which, to me, all suggest that they've at least evaluated the data with due diligence (emphases all mine):
"IV. Imaging the Central Supermassive Black Hole" (https://iopscience.iop.org/article/10.3847/2041-8213/ab0e85):
Section 5.2 confirms the statements from the press conference today:
> The imaging teams worked on the data independently, without communication, for seven weeks, after which teams submitted images to the image comparison website using LCP data (because the JCMT recorded LCP on April 11). After ensuring image consistency through a variety of blind metrics (including normalized cross-correlation, Equation (15)), we compared the independently reconstructed images from the four teams.
> Figure 4 shows these first four images of M87. All four images show an asymmetric ring structure. For both RML teams and both CLEAN teams, the ring has a diameter of approximately 40 μas, with brighter emission in the south. In contrast, the ring azimuthual profile, thickness, and brightness varies substantially among the images. Some of these differences are attributable to different assumptions about the total compact flux density and systematic uncertainties (see Table 2).
Section 6, in turn, confirms the statements from the TED talk:
From the introduction to section 6:
> To explore the dependence of the reconstructed images on imaging assumptions and impartially determine a combination of fiducial imaging parameters, we introduced a second stage of image production and analysis: performing scripted parameter surveys for three imaging pipelines. To objectively evaluate the fidelity of the images reconstructed by our surveys—i.e., to select imaging parameters that were independent of expert judgment—we performed these surveys on synthetic data from a suite of model images as well as on the M87 data. The synthetic data sets were designed to have properties that are similar to the EHT M87 visibility amplitudes (e.g., prominent amplitude nulls). This suite of synthetic data allowed us to test the scripted reconstructions with knowledge of the corresponding ground truth images and, thereby, select fiducial imaging parameters for each method. These fiducial parameters were selected to perform well across a variety of source structures, including sources without the prominent ring observed in our images of M87.
From section 6.2:
> We then reconstructed images from all M87 and synthetic data sets using all possible parameter combinations on a coarse grid in the space of these parameters. We chose large ranges for each parameter, deliberately including values that we expected to produce poor reconstructions.
Finally, in the caption of figure 4 of "I. The Shadow of the Supermassive Black Hole" (https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7) they write:
> Note that although the fit to the observations is equally good in the three cases, they refer to radically different physical scenarios; this highlights that a single good fit does not imply that a model is preferred over others
…which, assuming that I'm understanding this correctly, means that the bias in the fits towards one model over another is low.
Again, I cannot stress enough that I've only skimmed the papers but from what I did read, I see no good reason not to trust their results.
⬐ salty_biscuits
This sort of reconstruction problem from VLBI measurements is under-determined so you need to insert priors/regularization to get anything at all. The priors in this case are pretty weak (from a quick read of the CHIRP paper). https://arxiv.org/pdf/1512.01413.pdf
⬐ sandworm101
Indeed. Whenever you go looking for something you think is already exists, as opposed to stumbling across an object, there is a danger that the parameters of your search will favor your preconceived notions. One will also tend to describe observed objects in terms that tends to fit your theory. I'm not saying that happened here just that it is a danger.
Astronomy/cosmology is one of those strange disciplines where rather than discover new objects in situ, one discovers their possibility in the mathematics and then goes out to find them. So I and many others were hoping that this image was radically different than the math, potentially opening the door to some new theories. Confirmation just isn't as much fun as raw discovery of the unknown. Example: the recent "cannonball star" observations. We are going to need some new science to explain how that is a thing.
⬐ vbuwivbiu
The Event Horizon Telescope site has had predicted images up for ages https://eventhorizontelescope.org/science
⬐ z3t4
Side observation: This video, and the video you linked to got two million views in just a few hours. I didn't know black holes where this popular. (market opportunity here)
It's a very well made pop-sci video, which probably substantially increases the likelihood of it being reshared.
⬐ eu
If one needs an intro, this video should be watched before watching the press release..
⬐ Abishek_Muthian
In the video he talks about the Schwarzchild radius but doesn't go into details. It is the distance from the center of the black hole to the event horizon.
Anything which is not in that radius or not already in a path towards it should be safe from not getting sucked by the black hole.
E.g. If our sun becomes a black hole, Schwarzchild radius would be 2.954Km i.e. anything outside ~3Km would be safe.
This was explained in the scishow video on that topic[1].
[1]:https://youtu.be/Mm_ks1ce3C4
⬐ Retric
That "not already on a path towards it" is very misleading. A random object passing anywhere near the Schwarzchild radius is will be eaten by the black hole. The only way for something to escape 'at 3km' is for an object falling into a back hole to emit light which just happens to be pointing in the opposite direction from the black hole.
Even light can't orbit at that distance.
That's an oversimplification. The Schwarzschild radius is where to find an uncharged & non-rotating black hole's event horizon, and the event horizon is the surface at which newly generated photos (and all causal influences) can no longer escape.
The innermost stable circular orbit is further out than the event horizon, 3 times the Schwarzschild radius IIRC. Anything closer to that has an unstable orbit.
⬐ mattfrommars
If the general theory of relatively was tested again and it proved to provide the next result, what does it say about actually is an black hole?
Thus far, from all the experiment and result observed, the theory has been proven to be correct.
Hence, it can be said with 99% certainty whatever it predicts must be correct. I hope it does mention about possibility of creating a worm hole.
⬐ r_c_a_d
No, that is not how science works. We can say with 100% certainty that the theory has not been falsified by any test to date.
A very good (technical) talk about the EHT project and the physics they try to do here: https://www.youtube.com/watch?v=JiS1OJNBrvk
It is a bit old (2012), but comprehensive and with both good audio and readable* slides.
*: In the sense that you can see the letters on them
⬐ jtr_47
Another link describing what a black hole is:
Part I: https://www.youtube.com/watch?v=VnJYo6LKzgA
Part II: https://www.youtube.com/watch?v=Nlry6LqWwJ0
⬐ kaycebasques
I just stumbled upon Veritasium a week ago while learning about the double slit experiment in quantum theory and trying to see some actual evidence [1] of the experiment.
[1]: https://youtu.be/Iuv6hY6zsd0
Not sure if I missed it, but can we tell which way the accretion disk is from our view of it?
apparently from the top. i hope we get to find another one sideways because they look cool ;)
perpendicular - we see it almost exactly from the top. Mentioned during the Q&A (41:33 to be exact)
⬐ pjungwir
The video JumpCrisscross shared above says this is the black hole at the center of our galaxy, so why isn't its accretion disk oriented the same as the rest of the galaxy? Isn't that weird?
⬐ alpaca128
It's actually in a galaxy called Messier 87 which is 55 million lightyears away.
⬐ manigandham
The observed both supermassive black holes at the center of M87 and our Milky Way galaxy Sagittarius A
And FWIW, we don't orbit in the galactic plane. Per Wikipedia, "the galactic plane is inclined by about 60° to the ecliptic (the plane of Earth's orbit)." https://en.wikipedia.org/wiki/Milky_Way And "the Sun is currently 5–30 parsecs (16–98 ly) from the central plane of the Galactic disk." Id.
Not according to Veritasium's comment on his own video as well as the RelAstro group who produced the material[1]: "As there seems to be some general confusion, please note that the image shown here is a simulated one and not an actual image. So far we only have an image of M87. Kind regards, the RelAstro group. "
[1] https://www.youtube.com/watch?v=VnsZj9RvhFU
Ah you're right. Looks like they all should've skipped Sagittarius A instead of adding confusion.
Oh, interesting! Wouldn't it be easier to photograph our own? :-)
⬐ jug
During the press conference, they said photographing the M87 black hole was like shooting a hibernating bear, and photographing the Sag A* black hole of our galaxy like photographing a quickly moving toddler. Something about the speed making it much harder. It is also much smaller, but that's less of a problem because it's much closer. All in all apparently making it about the same angular size. But it sounds like they'll get to it, it's just harder.
⬐ SamBam
That's a... weird analogy.
⬐ salthound
I think the actual analogy was lost in translation: the point is that, unlike its older brother, Sgr A* is not going to "pose" when you point your camera at it.
Ah, I thought it was about actually shooting bears and toddlers.
M87's black hole is currently eating something big, which makes it brighter. The black hole at the center of the milky way doesn't seem to have eaten anything lately, so it's accretion disk may be small or nonexistent.
⬐ schlowmo
> is currently eating something big
You mean: was eating something big 55 million years ago ;)
⬐ losteric
Nope, for the same reason we had photos of the Moon before we had photos of Earth.
⬐ nazgul17
Can you expand on this?
⬐ CydeWeys
What they're saying is that we had to send a camera away from the Earth (in a rocket) in order to photograph it properly, and similarly you'd need to send a camera away from our galaxy (in a biiiig rocket) in order to be able to photograph it properly.
Our solar system orbits at a 60 degree inclination from the galactic plane. See https://web.archive.org/web/20160809194418im_/http://sob.nao...
There's a great general description here: https://medium.com/starts-with-a-bang/ask-ethan-37-the-earth...
I'm not an astronomer nor is geometry my strong suit, so I don't quite know how to interpret and convey the descriptions of our relative motion. But AFAIU while quite low we're nonetheless currently outside the galactic plane. How well positioned we are to see our black hole free of obstruction would, I imagine, depend on the average inclination of everything else. But it seems like our inclination is relatively extreme and for the next 50 million years or so our view should become increasingly more clear.
Nope. The view of our own galaxy's supermassive black hole is completely obstructed by matter within our own galaxy. You can't see to the core of our galaxy; it's too dense. You'd have to send a rocket quite some distance outside of the galactic plane to get a good view of it.
⬐ nwallin
The images are made with radio telescopes, which cuts through the dust quite easily. We have many other radio observations of Sagittarius A* [1], albeit at much lower resolutions. There are also numerous observations of Sagittarius A* in X-ray wavelengths, which is also fine because they are so energetic they simply punch through. All the dust and gas in the galaxy is transparent at most wavelengths except the visible one.
The Event Horizon Telescope is interesting because it is, in essence, a radio telescope that uses a "sensor" that is the size of the entire Earth. As such, it is able to make much higher resolution observations.
[1] https://en.wikipedia.org/wiki/Sagittarius_A*#/media/File:Clo...
[2] https://en.wikipedia.org/wiki/Sagittarius_A*#/media/File:X-R...
Actually not. M87 is a 1000 times farther away than Sgr A* but also a 1000 heavier and thus a 1000 times bigger in diameter. (Diameter/radius scale proportionally with the black hole's mass.) Therefore, the actual angular size on our night sky is the same for both black holes and, from this point of view, both would be equally difficult to observe.
However, as they mention in the press conference, Sgr A* moves a lot faster relative to us than M87, so it's much harder to take a still image. (In the press conference they used the example of trying to take a photo of a toddler with an exposure time of 8 hours.)
⬐ muterad_murilax
If we see it almost exactly from the top, then why is one half of the ring so much brighter than the other?
⬐ zenzen
Relativistic beaming! https://youtu.be/zUyH3XhpLTo?t=490
Already watched the video, thanks. However in his example the disk is not perpendicular to the viewer so the beaming makes more sense there I think.
On the other hand, "almost exactly from the top" is not the same as "exactly from the top".
⬐ hypothete
So that would mean that the right side is tipped slightly away from us, right? Because the matter in the accretion disk starts approaching us at about halfway down the ring on the right side?
Yes, from paper 1:
"Third, adopting an inclination of 17° between the approaching jet and the line of sight (Walker et al. 2018), the west orientation of the jet, and a corotating disk model, matter in the bottom part of the image is moving toward the observer (clockwise rotation as seen from Earth). "
⬐ fspeech
He appeared to have released a new video that incorporates the actual image: https://m.youtube.com/watch?v=S_GVbuddri8
⬐ meko
That was a great video, never grasped why interstellar and such showed black holes with the rings on top and bottom.
⬐ nsxwolf
So is that hazy diagonal line in the image the accretion disc, viewed edge-on from the Earth?
Great video but he really doesn't do himself any favors by flapping his arms like a rabid goose. Distracting as heck.
⬐ m87
It doesn't seem to be that accurate, as expectation was for it to have smoother photo sphere, but it has irregular bulges (5 of them). I think they discussed it briefly in the press conference as well. It would be interesting to see if this would change understanding of general relativity and may be give a hint for a theory of quantum gravity.
Look at the expectation examples in the scientific paper from 2013 that is, from ca. 6 years ago:
https://arxiv.org/abs/1309.3519
It matches quite good, I'd say. Page 4.
What bothered me more about the video is that of you watch the whole thing, he ends up talking about some theory which leads to another type of image prediction. He never actually explains why he chose to go with the former prediction rather than the latter.
⬐ albedoa
The whole video was him explaining why the image would look the way it ended up looking. darkpuma was overly patient when explaining this to you.
You were a user acting in bad faith that very literally jumped in at the end just to write a troll bait comment in that thread. Your comment had nothing substantive to even do with any discussion, or even any relation to anything I said.
There is nothing complex about my original statement there. The EHT website itself has a gallery of simulated images and I'd like to know why he chose that one specifically.
In the video he says "just trust me."
This is a perfectly reasonable criticism, I don't care how many downvotes or personal attacks I get.
It has nothing to do with anyone "being patient." That thread was 90% bullying, which you are taking part of.
Can you link to the comment of yours that I replied to? I can't see it for some reason.
I'm not sure how much we can truly draw from small irregularities in the image. This isn't actually a low resolution image. It is an algorithmically interpolated image created by comparing possible interpretations of the spotty and noisy data gathered from multiple points at different times processed against images of what we think a black hole should look like.
⬐ amelius
> Good video that correctly predicted the image and describes why it looks the way it does
This is of course a bummer, since this means that the acquired image does not give us any new clues of where our understanding of physics is wrong.
It's probably a bit too early to say because e.g the magnetic field data that was also collected hasn't even been scrutinized yet. This will also almost for sure lend a better understanding of the relativistic jets, in order to hopefully one day tell why they are this way or another depending on the particular black hole rather than just "they are somehow often there".
It's still very early and like the detection of gravitational waves, I think it feels like more of a symbolic step into a new era of space science. It's easy to forget that yesterday, black holes were a result of mathematics and only indirectly shown that they "ought to exist".
So first, I think we need to cut them some slack! Second, I think that if we at all WANT to shatter the Standard Model, I think we first need to be able to do science at the extremes of it! The LHC is one way, probing into the details of black hole mechanics might end up being another
We've furthered our understanding of the truth of the universe. That's not a bummer.
I expect that at the resolution that this picture is taken it would be surprising if new radically physics was found, since it would require our current models to be very different from reality to see significantly different results
⬐ mxwsn
I think this is unhelpful parroting of comments on actual cutting edge physics experiments like the LHC. The degree matters substantially - there, we have good guesses of what we might see, but there's uncertainty and new data is immensely valuable for narrowing hypotheses. It's the research frontier.
Reasoning about an image of a black hole is very much within the realm of standard science. Veritasium was able to explain the prediction using essentially ideas that are so basic they're at the high school level. If our basic understanding of physics down to the high school level was wrong (e.g., very far from the research frontier), there would be very very serious issues.
Ok, wait a second. I liked a lot of this video, but there are some aspects which are kind of ridiculous. He says his reason for his confidence in this prediction is "I think it's going to look like a fuzzy coffee mug stain." He doesn't give an actual reason.
He does talk a lot about theory, a lot of it interesting and novel to me, but by the end of the video, most of this theory suggests a different-looking image!
⬐ bdamm
The entire video is his description of the reasons for why it'll look like a fuzzy coffee mug stain. It's "fuzzy" because of the low resolution, not because the black hole itself is fuzzy. Undoubtedly there will be work to improve the best quality photograph of a black hole, now that we have one at all.
That's not correct. Around 25% of the video is discussion around the concept of the radius of photon sphere to the event horizon, and what constitutes the light surrounding the photon sphere (one explanation he gives is that there are infinite reflections). Then he spends the last 30% of the video talking about reflections of the acretion disk. This is the theory that he never included in his original prediction, but doesn't explain why he made that call.
Uh.. It's the reason that the "shadow" is larger than the Schwarzschild radius. He's describing the ratio of the "rim" to the "hole". So while the picture may be fuzzy, there's information in it anyway about what the pictures means relative to how large/spin we think the black hole is. In fact I wish he'd said more about the Doppler effect.
Ok I mean, now what you're writing, that's not even wrong. The black part of the image being larger than the actual radius of the blackhole is a discussion about the relative size of the black area you see in the image, to the actual black hole itself. I think that should make sense to you? It's not an argument about why he predicts it will look like a "fuzzy coffee stain," as opposed to other simulated images. Are you able to see the difference between those two ideas?
I think you have some other discussion you were having confused with this one. To remind you about your earlier comment, you stated disagreement with me, saying this video was him only talking about why he predicted the image to look like a "fuzzy coffee stain" as opposed to some other simulated and theorized predicgions. I think the above paragraph and reply should obviously show you why this is actually not true. Are you still with me?
I do think most of the video is interesting, but he never states an argument about why he chose that prediction. That was what I thought was ridiculous. Science is about reason and evidence, not just saying "believe me."
Maybe you watched a different video? You should watch this video, the last 30% talks about simulations done in which the acretion disk reflects around the black hole, he uses example images which look a bit different!
⬐ darkpuma
The reason for the fuzziness should be obvious. It's really far away and it's really hard to see. Expecting a jump from "never seen before" to "seen in amazing visual clarity" is unrealistic.
That's also not even wrong. You can't see why?
Let's say we're imagining what Uranus would look like. I draw a picture, and say it should look like a "fuzzy Jupter." But you ask, why should it look that way? Do you have a reason? And I say you should be confident, but don't provide an argument. By the end, I start talking about how it might look like Saturn.
Then I come along and say, "it should be obvious the fuzziness is because it is far away."
That final statement is not even wrong. It misses the point.
Are you upset that the Veritasium video didn't explicitly spell out for you that this black hole is very far away, so the first image of it ever is basically certain to be fuzzier than IMAX fidelity computer simulations used in a Hollywood movie?
I think you're being very silly.
What? You obviously didn't read what I wrote.
> "What? You obviously didn't read what I wrote. Are you ok?"
You're ranting about Saturn and Jupiter for some reason. Why don't you calm down and look at page eight of the paper that Veritasium video was based on: https://ve42.co/luminet "Image of a spherical black hole with thin accretion disk Astronomy and Astrophysics, vol. 75, no. 1-2, May 1979, p. 228-235"
Look at that last image, and squint at it until it gets fuzzy. Lo and behold, a fuzzy coffee mug stain!
Perhaps it was a mistake for him to make the focus of the video on the physics of black holes, rather than the limitations of state of the art radio interferometry... but I don't think so.
It's really strange that you are unable to understand such a simple complaint I made, even after going to extra effort to spell it out. You even have even become aggressive insulting. I understand that some people become that way when they get confused, but how is it possible you are confused here?
I think you are literally not even reading my comments. Either that, or you are going through some personal issues right now.
I think it might help you if you tried to first understand that I am not complaining about the image being fuzzy. I have no idea why you keep going back to that. I spelled out an analogy to explain this to you and you got angry and insulting.
Are you really sure you are ok? You clearly have something going on.
Mate, you insulted me first, and have just repeated that insult again. I've tried to be charitable with you.
How did you come to the conclusion that I "insulted you first?"
Everyone present can clearly see which person in this conversation is getting their jimmies rustled.
You truly believe that all of the downvotes on all of your comments are the work of one account?
Really awesome link, thanks for that! It's always interesting to see theory that is decades out in front of experimental confirmation, and then proves to be dead right.
Yeah, I did a double-take when I saw that was published in 1979. I think that's really cool.
⬐ khamoud
He isn't explaining why the blackhole would look like a fuzzy coffee mug stain. He's explaining why the _picture_ of the blackhole will look like a fuzzy coffee mug stain.
To your point, it's like taking a picture of Uranus with film and waiting for it to develop. People familiar with the matter can guess what the _image_ will look like not what Uranus actually looks like.
This is all very clear in the first 25 seconds of the video if you actually listen to what he's saying.
The first 25 seconds of this video actually does not have sound. The 25 seconds after he begins talking is him talking about Einstein and history.
Unless I'm misunderstanding your intentions and you just meant that as a condescending insult?
I think what you're hoping for is a more exhaustive survey of what black holes are theorized to look like, with different possibilities. Is that right?
The problem here is your expectation does not match the product. It's like you went to a car dealer and are upset they didn't sell you an airplane.
It isn't surprising to me that Derek focuses on one form, since Veritassium is providing content for the armchair consumer, that he chooses what he believes to be the best model and presents that. This isn't a PhD defense, after all, it's just a timely video so that folks can appreciate the image that the EHT group has released (is going to release, at the time of Veritassium's video.)
Did you have some other models in mind?
Interesting to me is that Veritassium's presented model doesn't explain the corona-like features, nor any attempt at explaining the "blobs" although he does say that blobs would be exiting to see. And there they are! What fun.
Yea sort of. I mean I said I liked the episode, I just think this part is bad form (I'm paraphrasing):
>why can you expect it to look like this? well because it's just going to look like this.
Now, I'm not trying to say his prediction was unwise. I just think it's first of all bad form to say something like "the reason is just trust me," in scientific discussions (even if you are correct)... but second I actually do want to know at least some explanation to that question. Granted, I'm not saying the video does not explain anything about the image. I wanted to know: why can we be sure it will look like this, and not other simulated images?
That's all.
You can see on the EHT's own website a gallery of other simulated models of what could be expected from a radio image.
And the second part of my criticism, was that by the end of the video, he was using images inspired by other models, and particularly of one where the accretion disk dominated the image.
I dont care how many downvotes I get, I know the difference between right and wrong and this is a perfectly reasonable criticism.
Anyways, since my posts lost 60 karma in 1 hour somehow (almost uniformly coming from posts in other threads, wtf?) some other opportunistic types see it as a chance for bullying. Even writing stuff like 'you've obviously got your jimmies rustled mate!' or other extremely bad faith assumptions like 'if you cant understand why it's fuzzy, it's far away!' The best is the troll bait comment, 'do you really think one person is downvoting you?' Groupthink, bullying, and authority define right and wrong for some people--they can't even write something that even addresses an actual comment or argument. People are crazy. Eventually they just start addressing the negativity itself, abandoning any substantive argument, and focusing on the negativity itself. The next step is using the negativity as its own justification (you deserve negativity because otherwise you wouldnt be receiving negativity kind of assumptions implicig in the above troll bait comment).
But still all that I dont think explains why comments in unrelated posts (even ones that were being complimented) got the same time-unform mass-downvotes?? I think I haven't experienced anything like this before on this site until recently. I lost 60 karma in the a few hours.
Well, for the record, it wasn't me. I did however read through your comments just now, and it seems there is a pattern of not really engaging in dialogue but just blasting your opinion over and over. Looking at your comments on Julian Assange, for example, it seems clear that you do not think there is any difference between what a NYTimes reporter does and what Assange did. I can't speak to everyone else but to me the difference is quite obvious, and probably that has something to do with why you're being downvoted. You're repeatedly stating that they're the same, without describing why you believe this, and then kind of insulting other people for their belief that in fact they are different. So anyway, I've spent way, way too much time responding to you. Good luck to you sir.
Hmm there is some groupthink, lack of reason, or bullying going on here.
Look at this image for yourself. The article is titled: "Here is what scientists think a black hole. Looks like:" https://www.sciencemag.org/news/2019/04/here-s-what-scientis...
There are not massive differences in the images, but the Vetiasium prediction was (and this is very plain) much more accurate.
This isn't controversial at all.
I also thought the particular statement, "why is it like this, because it is just going to look like this" was bad form.
This is very plainly reasonable.
I think the fact that you and a few other users turned this into an opportunity to go through the effort of writing belittling comments and even put downs and troll bait over something as plain and ordinary as this is indicative of some bad qualities of humanity expressing themselves here.
Of course, your only response will be further negativity as bad people dont possess the ability to admit when they were wrong.
And the statement about Assange is very widely expressed. See here(1). There are articles all over the media, just like that one, echoing the same exact view. They are literally everywhere.
Finally, I noticed over the last 30 minutes all my recent comments went down by -1 each. That really makes it look like I'm engaging with some quite petty and insecure people.
Edit: see, right after I wrote this comment, all my recent comments each, in perfect synchronization, down by -1 again haha.
⬐ coldtea
>He says his reason for his confidence in this prediction is "I think it's going to look like a fuzzy coffee mug stain." He doesn't give an actual reason.
He doesn't need to give a reason. The reasons why it would look like a "fuzzy coffee mug stain" are well known since Hawkings...
Did you see the movie Interstellar? He included it in his video. That's what the black hole looks like, except with the relativistic beaming make one side brighter than the other.
Here's the image from the movie: https://www.wired.com/wp-content/uploads/2014/10/ut_interste...
Now imagine that image being taken far away by several ground-based telescopes put together at the edge of their capabilities and using math to error correct and stitch together the final result. What you get is what we saw.
Ah, sure. Also, if you go to the EHT's own website, they have a gallery of predicted, simulated images.
Here is another prediction: https://www.sciencemag.org/news/2019/04/here-s-what-scientis...
I just wanted to know why he went with that one because his prediction was really accurate. And I thought him saying 'just trust me' was bad form.
He did talk some about this, but he didn't really say anything about why he thought his illustration would be so accurate compared to a lot of other stuff seen in the press.
I dont care (or have any idea why) how many downvotes or insults I get. It is a perfectly reasonable question and criticism.
I don't see the difference, the predictions are the same.
The only changes in the image depend on what angle the black hole is being viewed at which would influence whether we see a band across the middle and the slimmer inner ring.
Lol the image I linked is obviously very different.
There is some groupthink going on here affecting people like you and others. The above is obviously plain.
There are 4 images on that page, 1 of which is the Interstellar movie rendition that I already linked to. They are also all the same model only in better detail as graphics technology has improved.
Perhaps you should post exactly what image you're talking about and what you think is different.
In the images I linked to, the black hole images don't have any "blobbiness," and seem to have these perfect gradiants.
In the veritasium video the "coffee stain" was not really as blobby as the real image, but it seemed a lot closer than the smooth-gradiant, no blobbiness and no irregularities predictions.
I dont just mean the fuzziness from low resolution.
This isn't really a big deal, but it's also obvious that I am just stating plain facts about what is in these images.
At the time I saw this video when he said "you can be confident, because... (no reason given)" was really the thing that I thought was annoying.
I think you can see the differences in the images. They're not huge but the smooth gradiants vs irregularities/coffee stain/blobbiness is plain to see I think.
Edit: from the horses mouth himself, one of the lead researchers says he didn't expect the image to look like it did: https://youtu.be/ZrDhHDBHkQY
Some of the people here in other parts of this thread have been really offensive for this. It's honestly pretty ridiculous.
The general model of what a black hole looks like is well understood. The highest definition rendering is that in Interstellar, except for one side being brighter than the other due to the relativistic beaming. They kept that part out of the movie to just make it look nicer.
All the models are the same, and the real picture is "blobby" only because of the process in how it was taken. I think you are refusing to accept that but there's nothing else to say about it. It wasn't a direct photo, it was a complex assembly of several different radio telescopes around the world stitching data together. If we were actually next to it, it would very much look like the one from interstellar.
The video you linked isn't about the prediction being wrong, more that he just didn't expect to really see a black hole at all. Even though black holes are generally understood for decades, there's a certain shock and awe to seeing it real for the first time.
Lol ok then we can agree to disagree. Even the YT video I just linked to opens with an intro containintlf a simulation, which once again, has subtle but pretty obvious and appreciable differences.
Also, my complaint was in fact that I didnt know why Veritasium was confident in their prediction. This complaint is for a matter of fact completely consistent with one of the lead researchers outright saying they didn't know what to expect. I never said I was exclusively complaining about there being simulated models which have some differences. You and others criticized me after I said he should have substantiated why he was confident in his prediction. I gave what I believed was my the foremost reasoning for saying that.
I had little idea what the picture would look like...
I have no idea why you're so intent in disagreeing with me. I'm substantiating my ideas with facts. And saying 'just bbelieve me' I think is also bad form.
At this point I feek like your disagreement has to do with psychological or social biases unless you are able to address the factual content of my comment.
But the one thing you said that was interesting was about the blobbiness. I think what you are trying to saya is that it is fully expected by the researchers to be error. Do you have a good interview or other source on this?
How about forgetting that video for a moment and trying to consider the following picture:
https://static.projects.iq.harvard.edu/files/styles/os_files...
Taken from here: https://eventhorizontelescope.org/science (the official site of the project).
On the left is how it would look like if we weren't so far -- we are 55 million light years far from that. You know the distance from us to our Sun, which you see on the sky but can cover with your own thumb? That object is 3,500,000,000,000 times farther than the Sun is far from us.
On the right is what we can reconstruct from the signals measured because we are so far and we have "only" the telescope the size of the Earth. More details would be visible (the picture would look more like the one on the left) either if we had even much bigger telescope than the Earth, or if the black hole of the same size were much closer to us, which it is not.
⬐ GuB-42
Smart move by Veritasium, making a video commenting the news just before the news actually happens. Time works in weird ways around black holes.
Anyways, it is a good one. So is that channel in general.
It is also a good video if you just watched Interstellar, because it also explains why the black hole looks the way it does in the movie. Note that the movie black hole rendering is slightly incorrect for artistic reasons, the video shows the more scientifically accurate version.
⬐ koheripbal
It's too bad we cannot see the accretion disk edge-on as in the video. That would have made it a perfect prediction. Maybe it's so thin that it's overwhelmed by the projections of the top and bottom of the back side of the disk.
⬐ darkhorn
May be we can see higher resolution if we build an additional telescope.
⬐ dredmorbius
More distant (longer baseline), not merely additional sensors.
Resolving power is proportional to the (virtual) aperture size, not the total sensor area (that gives more signal strength).
⬐ exelius
So put some telescopes in orbit around the Moon and Mars.
⬐ Cthulhu_
A guy on Reddit actually asked this in the AMA. While that would increase the resolution, it would also be extremely difficult. The algorithms used to combine the data from the dishes relies on the exact position of the dish being known at the time of measuring, to a precision of fractions of millimeters. It's already hard to do on the earth's surface, but imagine doing it with a sattelite zipping around the earth at 20K km/s, or the moon at >1 km/s around the earth, or Mars at 24K km/s around the sun.
⬐ ataturk
I predict we'll do it, though.
⬐ wool_gather
Not to mention getting the data back down here. For the analysis of M87, there were multiple petabytes generated: they had to use good old sneakernet and ship hard drives.
Given the observation period has been multiple years, does that virtual size include the orbit of the Earth? Or is there something that limits it to still being Earth-sized?
⬐ tqkxzugoaupvwqr
As far as I know, Veritasium and others occasionally join efforts and coordinate around soon to be published scientific discoveries in a goal to increase exposure. Don't know if this was the case with the black hole image.
It probably wasn't coordinated, because he got the black hole that was being imaged wrong.
He said we would see a picture of Sagittarius A*, but we actually got the black hole at the center of M87.
⬐ spectre256
I thought we got both, which is the best possible outcome: no one is wrong and there are more black hole photos.
We didn't get both, we only got M87. Sgr A is dimmer, so it needs more number crunching to get a good image
They observed both, watch the follow-up: https://www.youtube.com/watch?v=S_GVbuddri8
"The Event Horizon Telescope Collaboration observed the supermassive black holes at the center of M87 and our Milky Way galaxy (SgrA*) finding the dark central shadow in accordance with General Relativity, further demonstrating the power of this 100 year-old theory."
But in the press conference they specifically said they weren't releasing SgrA* yes because they hadn't completed their analysis. They released pictures after that?
You're right, it looks like they are still observing Sagittarius A but what they released was only a simulation for that one.
Apr 10, 2019 · mino on Event Horizon Telescope Live Press Conference
Direct link to youtube streaminng: https://www.youtube.com/watch?v=Dr20f19czeE
Breakthrough discovery in astronomy press conference [video]
Apr 10, 2019 · 29 points, 6 comments · submitted by matco11
⬐ teilo
First picture of a black hole from the EHT. No decent picture released yet. Just what's on the projection screen at the press conference.
https://imgur.com/qub7OVD https://imgur.com/atwFELN
https://www.nytimes.com/interactive/2019/04/10/science/event...
Scroll down to see the released image.
⬐ ycombonator
I thought last year they were working on getting the picture of Sagittarius A. Not sure why they changed the plan.
⬐ melling
Einstein's Shadow?
https://www.sciencenews.org/article/einstein-shadow-explores...
⬐ aasasd
Duplicate of the post from yesterday: https://news.ycombinator.com/item?id=19624226
⬐ WhuzzupDomal
It wasn't Einstein who first conceived the idea of a black hole, one of the earliest records of someone speculating what would happen if a star got so massive that it's gravity would pull back light was John Michell, who termed them 'dark stars'. If that name doesn't ring a bell, think of the Cavendish experiment to weigh the Earth. That apparatus was conceived by the same man.
⬐ lelf
The image https://www.nsf.gov/news/special_reports/blackholes/download...
Apr 09, 2019 · 2 points, 0 comments · submitted by friede
Apr 03, 2019 · jacobedawson on Astronomers Worldwide Are About to Make a Groundbreaking Black Hole Announcement
OP here - The URL is admittedly off-putting & the date of the announcement is odd, but it appears legit:
https://www.usatoday.com/story/news/nation/2019/04/02/black-... https://interestingengineering.com/will-we-actually-see-a-bl... http://www.newser.com/story/273418/no-ones-ever-seen-a-black... https://www.eso.org/public/announcements/ann19018/?lang https://www.youtube.com/watch?reload=9&v=Dr20f19czeE (live stream)
Could be an elaborate joke but from what I understand it is a set of images from the Event Horizon Telescope https://eventhorizontelescope.org/ capturing Sagittarius A (the supermassive black hole in the center of the Milky Way)
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CommonCrawl
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A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $17$ complete rotations. The value of $h/r$ can be written in the form $m\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
The path is a circle with radius equal to the slant height of the cone, which is $\sqrt {r^{2} + h^{2}}$. Thus, the length of the path is $2\pi\sqrt {r^{2} + h^{2}}$.
Also, the length of the path is 17 times the circumference of the base, which is $34r\pi$. Setting these equal gives $\sqrt {r^{2} + h^{2}} = 17r$, or $h^{2} = 288r^{2}$. Thus, $\dfrac{h^{2}}{r^{2}} = 288$, and $\dfrac{h}{r} = 12\sqrt {2}$, giving an answer of $12 + 2 = \boxed{14}$.
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Math Dataset
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What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)?
This is a confused part ever since I started learning electricity. What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)? All of them have the same SI unit of Volt, right? I would appreciate an answer.
electricity electric-circuits terminology potential voltage
Qmechanic♦
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new hernew her
EDIT: Put simply, potential difference is the work done by electrostatic force on a unit charge, while EMF is the work done by anything other than electrostatic force on a unit charge.
I don't like the term "voltage". It seems to mean anything measured in volts. I'd rather say electric potential and electromotive force.
And the two are fundamentally different.
Electrostatic field is conservative, that is, over any loop $l$ we have $\oint_l \vec{E}\cdot\mathrm{d}\vec{l}=0$. In other words, the line integral of electrostatic field does not depend on the path, but only on end points. So we can define point by point a scalar value electrostatic potential $\varphi$, such that $$\varphi_A-\varphi_B=\int_A^B \vec{E}\cdot\mathrm{d}\vec{l},$$
$$q \left( \varphi_A-\varphi_B \right)=\int_A^B q\vec{E}\cdot\mathrm{d}\vec{l},$$
so $q\Delta\varphi$ equals the work done by electrostatic force.
In pratical application, electrons (and other carriers) flow in circuits. Since electrostatic field is conservative, it alone cannot move electrons in circles; it can only move them from lower potential to higher potential. You need another kind of force to move them from higher potential to lower ones in order to complete a cycle. This other force could be chemical, magnetic or even electric (vortex electric field, different from electrostatic field), and their equivalent contribution is called electromotive force. $$\mathrm{E.M.F.}=\int_\text{Circuit} \frac{\vec{F}}{q}\cdot\mathrm{d}\vec{l}$$
Siyuan RenSiyuan Ren
$\begingroup$ your explanation (which repeats what I said regarding useful work) is confusing because it doesn't account for the difference in potentials when the circuit is not closed in a loop and which is called alternatively emf or voltage. $\endgroup$ – ganzewoort Oct 6 '11 at 14:16
$\begingroup$ Also, observing electrons traveling spontaneously from lower potential to higher potential, as in your reply, is conterintuitive. Since it's a matter of conventions it would be preferable to choose a convention in a reverse sense. $\endgroup$ – ganzewoort Oct 6 '11 at 14:23
$\begingroup$ @ganzewoort: Well, my explanation may be confusing, but potential and emf are fundamentally different. Even when the circuit is not closed, potential difference is not the same as emf. $\endgroup$ – Siyuan Ren Oct 6 '11 at 14:52
$\begingroup$ first it should be understood that emf doesn't apply only for a closed loop, as you have inferred. As for whether or not it is a potential difference, it is, in the sense which I already explained. $\endgroup$ – ganzewoort Oct 6 '11 at 15:01
$\begingroup$ @ganzewoort: I concede I was wrong about closed loop. But your explanation is not an explanation at all. You just describe how you measure the two, but does not address the conceptual difference. And your explanation is wrong. EMF cannot be directly measured. For example, the EMF of an inductor with non-zero resistance is different from the potential difference, and the only thing you can directly measure is that difference. $\endgroup$ – Siyuan Ren Oct 6 '11 at 15:13
Anyway the simple answer is e.m.f. is not a force in the mechanical sense. It measures the amount of work to be done for a unit charge to travel in a closed loop of a conducting material.
Let's make it more clear. In static case (ignoring time variation of any magnetic field), electric field at a point can be derived solely from a scalar as the negative of the gradient of this scalar. This scalar at any point is called the "electric potential" at that point. If two points are at different potentials then we say there exists a potential difference. Obviously it is the difference in the potentials that matters and not their absolute values. One can therefore arbitrarily assign a value zero for some fixed point who's potential may be considered constant and compare the potentials of other points with respect to it. In this way one need not have to always speak of potential difference but simply potentials.
Now, often this "electric potential" at some point in a conductor or a dielectric is called "voltage" at that point assigning the value of the voltage to be zero for earth since the potential of earth is constant for all practical purposes.
If there is no variation of magnetic field then the work done by an unit charge in a closed loop will be $0$. But if the magnetic field varies then it will be nonzero. Recall the formula: $$\nabla \times {E} = -\frac {\partial {B}}{\partial {t}}.$$
What it really implies is, it is impossible for an electric field, derived solely from a scalar potential, to maintain an electric current in a closed circuit. So an e.m.f. implies presence of some source other then a source which can only produce a scalar potential.
The following equation tells the whole story:
$$E = -\nabla \phi - \frac{\partial A}{\partial t},$$ where $\phi$ is the scalar potential and $A$ is the vector potential.
$\begingroup$ People downvote sometimes not because you are wrong, but because you are repeating other people's answers without adding anything new. $\endgroup$ – Ron Maimon Oct 6 '11 at 16:53
$\begingroup$ sb1, your explanation again fails to explain the open-circuit emf. Even more interestingly, I'm curious to hear your explanation as to how the Faraday's law you're mentioning accounts for the voltage drop measured across a unipolar generator. This, perhaps, is for a separate question to be asked in stackexchange. $\endgroup$ – ganzewoort Oct 6 '11 at 17:01
$\begingroup$ @ganzewoort: "e.m.f. implies presence of some source other then a source which can only produce a scalar potential." That means the electric field is not conservative any more. That's all. In open circuit condition, a voltage will be generated between the ends which is not just the difference of scalar potential at the two ends. As for unipolar generator, yes, it will be a good idea to ask as a separate question. $\endgroup$ – user1355 Oct 6 '11 at 17:52
$\begingroup$ I agree about the scalar potential (I think you've explained it very well and doesn't repeat what's been said so far). However, scalar potential is only a mathematical construct, created for convenience, which isn't inherent in the phenomena. I'm adding a separate question regarding the unipolar generator. $\endgroup$ – ganzewoort Oct 6 '11 at 18:43
Electromotive force (Note; not a force) is simply the source of voltage in a circuit.
Martin BeckettMartin Beckett
A very short answer:
Voltage is a potential difference, due to the energy dissipation. Emf is a potential difference, due to the energy generation.
Martin GalesMartin Gales
$\begingroup$ "Voltage is a potential difference, due to the energy dissipation" are you sure? What about the voltage across an Inductor? $\endgroup$ – user1355 Oct 7 '11 at 7:27
$\begingroup$ @sb1 Ok, for non-DC replace "dissipation" with "dissipation and consumption". $\endgroup$ – Martin Gales Oct 10 '11 at 7:44
$\begingroup$ Sorry still wrong. A.C. or D.C., energy is always conserved in a pure inductor and never "consumed" or "dissipated". If you apply a dc source across an inductor through a resistance then energy will be dissipated but again by the resister ($I^2r$ loss)and not by the inductor. In practice, an inductor will always have some resistance and a portion of energy will be dissipated and but again that's because of the resistance. $\endgroup$ – user1355 Oct 11 '11 at 15:41
$\begingroup$ @sb1 At a time when the energy is converted into magnetic energy of the inductor, there is no difference, is the energy conserved or not. At this time the conductor draw energy from the circuit like a resistor. At a time when the inductor returns the stored energy into the circuit it works like a generator(a source of emf). $\endgroup$ – Martin Gales Oct 12 '11 at 7:56
$\begingroup$ No point in arguing :( You don't even know the meaning of consumption, dissipation, even energy conservation. $\endgroup$ – user1355 Oct 12 '11 at 8:54
EMF is used as a more general term to also include those situations where the integral of the electric field around a closed curve is not zero, so that the E field doesn't come from a pure potential. Usually, when people say potential, they mean that the potential is a function of the position, and when they say EMF, they mean it is a function of the loop.
You have nonintegrable E fields when you have changing magnetic fields, an inductance. Since the "voltage" is usually used for the pure electrical potential, people call the voltage produced by an inductance an "EMF". Outside the circuitry, the fields are negligible usually, and the EMF at any point is the electrostatic potential at that point. But inside the circuitry, in inductors, there's a difference.
Ron MaimonRon Maimon
Actually these are are same thing but usage is at different places.
Whenever we talk about batteries or a DC system, we use the Potential difference, as there is potential difference of 3.7 Volt.
The phrase "electro-motive force" (EMF) is used when a conductor cuts the flux inside the machine (Transformer, Generator, etc)
Voltage is used as Output from an electrical machine.
McGarnagle
SADAM HUSSAIN GAADSADAM HUSSAIN GAAD
To help you understand the difference, think of EMF as a measurement of Work being done and think of Electric Potential Energy as energy that has the "potential" to perform Work. As an analogy, EMF could be thought of (in the Mechanical realm) as one pushing a wheel barrel up a hill. (Or better yet, a car, with gasoline prices as they are today lol.). And think of Electric Potential Energy as the wheel barrel being at the top of the hill. If the wheel barrel was released, its' potential energy would be transformed into several different forms of energy in rolling down the hill (Frictional-Heat, Work done on air resistance; and if it collided with a wall at the bottom and came to rest, its' original Potential Energy would all have been transformed into different forms of energy upon coming to rest at the bottom. Now to get a bit more technical...
A. EMF (Electromotive Force)
work that has been done is by definition, the Work done within the EMF "seat" (the battery in this case) in raising the charges (Chemically) from the negative (-) terminal up to the positive (+) terminal thus maintain the ability to still provide the circuit with current.
B. ELECTRIC POTENTIAL ENERGY
As an analogy (I'll get a little funny on this one), imagine that a Woman and a Man see each other; say from six feet apart. They instantly have an attraction (overbearing) for one-another; enter into a trance and begin walking towards each other. The energy other folks would have to apply to hold both from continuing to walk towards each other, is analogically, the Electric Potential Energy. the folks would be holding the Man and Woman still while they retain their trance - maintain the force to come together. And in a direct definitional realm, it is the potential energy two separated oppositely charged (for positive potential) particles posses in the attraction to come together. As well, such can think of this from the perspective of the energy required to hold the two charges at rest (in a Static State) not allowing them to move towards one-another.
Kyle Kanos
aleenaaleena
The amount of work done by unit charge between any two nodes of current carrying circuit is called the potential difference between those nodes.
The amount of work done against the electric field by displacing (without acceleration) a unit test charge from one terminal to other terminal in an open circuit is called the electromotive force.
Obviously when we deal in static electricity the potential difference between two points in electric field is amount of work done against the electric field by displacing (without acceleration) a unit test charge from one point to another point, off course it doesn't depends on path because the electric field is conservative field.
Same is happen when current is flowing in a circuit, in this case the electric field is confined in physical boundaries of circuit components, but still it is conservative in nature. Hence the potential difference in a current carrying circuit will also the amount of work done by moving a unit test charge from one node to another node. In other scenario we can observe that the charge is already moving in the current carrying circuit, so amount of work done by these moving charges in the current carrying circuit is converted in heat, light, mechanical work etc.
In case of emf, when any circuit is open, the open terminals do have charge density difference, this difference in charge density create an electric field, the work done against this electric field in moving a unit test charge without acceleration from one terminal to another is called the electromotive force..
Pushpkant YadavPushpkant Yadav
potential difference and e.m.f has same unit because of voltage. firstly,potential difference is is define as the work done upon charge , while e.m.f(electro motive force) is the potential differnce maintain across the battery.we are normally cosidering the external cicuit there is also an inner circuit. V=IR
and E.M.F=Ir+IR sice E.M.F=I(r+R) therefore E.M.F=Ir+IR
AS WE KNOW V=IR
E.M.F=V+Ir
hasnainhasnain
Potential differance is the electrical pressure between two points but voltage is the electrical pressure between any two live wires or one live wire and earth.
Potential, voltage and emf are practically the same thing. Potential is the value of volts of a given electrode you measure with respect to some standard electrode whose potential is considered zero (Normal Hydrogen Electrode (NHE), saturated calomel electrode (SCE) etc.) Voltage is the difference between two thus measured potentials of two electrodes. So, you see, potential is same as voltage but one of the electrodes is considered conditionally of potential zero. The term electromotive force you'd use in the stead of voltage if you intend to talk about the change of the Gibbs free energy which would amount to the useful work you can get from the given Galvanic element, say. In any event, that's just splitting hairs in my opinion, so you can use those terms interchangeably as long as it is clear what the reference electrode is.
ganzewoortganzewoort
protected by Qmechanic♦ Jul 13 '15 at 8:49
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CommonCrawl
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Polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.
More specifically, if we denote by $D(z,r)$ the open disc of center z and radius r in the complex plane, then an open polydisc is a set of the form
$D(z_{1},r_{1})\times \dots \times D(z_{n},r_{n}).$
It can be equivalently written as
$\{w=(w_{1},w_{2},\dots ,w_{n})\in {\mathbf {C} }^{n}:\vert z_{k}-w_{k}\vert <r_{k},{\mbox{ for all }}k=1,\dots ,n\}.$
One should not confuse the polydisc with the open ball in Cn, which is defined as
$\{w\in \mathbf {C} ^{n}:\lVert z-w\rVert <r\}.$
Here, the norm is the Euclidean distance in Cn.
When $n>1$, open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.[1]
When $n=2$ the term bidisc is sometimes used.
A polydisc is an example of logarithmically convex Reinhardt domain.
References
1. Poincare, H,Les fonctions analytiques de deux variables et la r?epresentation conforme, Rend. Circ. Mat. Palermo23 (1907), 185-220
• Steven G Krantz (Jan 1, 2002). Function Theory of Several Complex Variables. American Mathematical Society. ISBN 0-8218-2724-3.
• John P D'Angelo, D'Angelo P D'Angelo (Jan 6, 1993). Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press. ISBN 0-8493-8272-6.
This article incorporates material from polydisc on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia
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A prism has 15 edges. How many faces does the prism have?
If a prism has 2 bases and $L$ lateral faces, then each base is an $L$-gon, so the two bases collectively have $2L$ edges. Also, there are $L$ edges connecting corresponding vertices of the two bases, for a total of $3L$ edges. Solving $3L=15$, we find that the prism has 5 lateral faces and hence $5+2=\boxed{7}$ faces in total.
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Math Dataset
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\begin{document}
\title[Complex equiangular lines from mutually unbiased bases]{Constructions of complex equiangular lines from mutually unbiased bases}
\author{Jonathan Jedwab and Amy Wiebe} \address{Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada, V5A 1S6} \date{21 August 2014 (revised 4 March 2015)} \email{[email protected], [email protected]} \thanks{J. Jedwab is supported by an NSERC Discovery Grant. A. Wiebe was supported by an NSERC Canada Graduate Scholarship.}
\begin{abstract} A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: \begin{enumerate} \item{adapting a set of $d$ MUBs in $\mathbb{C}^d$ to obtain $d^2$ equiangular lines in $\mathbb{C}^d$,} \item{using a set of $d$ MUBs in $\mathbb{C}^d$ to build $(2d)^2$ equiangular lines in $\mathbb{C}^{2d}$,} \item{combining two copies of a set of $d$ MUBs in $\mathbb{C}^d$ to build $(2d)^2$ equiangular lines in $\mathbb{C}^{2d}$.} \end{enumerate} For each construction, we give the dimensions $d$ for which we currently know that the construction produces a maximum-sized set of equiangular lines. \end{abstract}
\maketitle
\section{Introduction} \label{S:Intro}
\allowdisplaybreaks
Equiangular lines have been studied for over 65 years \cite{Haantjes}, and their construction remains ``[o]ne of the most challenging problems in algebraic combinatorics'' \cite{Khatirinejad}. In particular, the study of equiangular lines in complex space has intensified recently, as its importance in quantum information theory has become apparent \cite{Appleby-Clifford, Grassl-tomography, Renes, Scott-Grassl}. It is well-known that the maximum number of equiangular lines in $\mathbb{C}^d$ is $d^2$ \cite{DGS-bounds, Godsil-Roy}. Zauner \cite{Zauner-thesis} conjectured 15 years ago that this upper bound can be attained for all $d\geq2$. This conjecture is supported by exact examples in dimensions 2, 3 \cite{DGS-bounds, Renes}, 4, 5 \cite{Zauner-thesis}, 6 \cite{Grassl-dim6}, 7 \cite{Appleby-Clifford, Khatirinejad}, 8 \cite{Appleby-imprimitivity, Grassl-tomography, Hoggar-two, Scott-Grassl}, 9--15 \cite{Grassl-tomography, Grassl-slides, Grassl-computing}, 16 \cite{Appleby-monomial}, 19 \cite{Appleby-Clifford, Khatirinejad}, 24 \cite{Scott-Grassl}, 28 \cite{Appleby-imprimitivity}, 35 and 48 \cite{Scott-Grassl}, and by examples with high numerical precision in all dimensions $d\leq 67$ \cite{Renes, Scott-Grassl}. However, Scott and Grassl \cite{Scott-Grassl} note that ``[a]lthough our confidence in its truth has grown considerably, we seem no closer to a proof of Zauner's conjecture than Gerhard Zauner was at the time of his doctoral dissertation.''
M.~Appleby \cite{Appleby-abstract} observed in 2011: ``In spite of strenuous attempts by numerous investigators over a period of more than 10 years we still have essentially zero insight into the structural features of the equations [governing the existence of a set of $d^2$ equiangular lines in $\mathbb{C}^d$] which causes them to be soluble. Yet one feels that there must surely be such a structural feature \dots (one of the frustrating features of the problem as it is currently formulated is that the properties of an individual [set of $d^2$ equiangular lines in $\mathbb{C}^d$] seem to be highly sensitive to the dimension).'' In light of this difficulty, one of the aims of this paper is to illuminate structural features of sets of equiangular lines that are common across several dimensions.
There are many papers addressing both the topic of maximum-sized sets of equiangular lines and that of mutually unbiased bases \cite{Appleby-prime, Appleby-monomial, Beneduci-operational, Bengtsson-Eddington, Bengtsson-KS, Greaves, Kibler, Wootters}. In 2005, Appleby \cite{Appleby-prime} even stated: ``There appear to be some intimate connections [between the study of complex equiangular lines and] the theory of mutually unbiased bases \dots''. Nonetheless, in this paper we show that there appear to be still deeper connections between these two objects than previously recognized.
The remainder of this paper is organized as follows. In Section~\ref{S:defns} we define the major objects that we use in the rest of the paper; in Section~\ref{S:Zauner} we give an overview of the standard method of construction of equiangular lines; in Sections~\ref{S:LinesfromMUBs}--\ref{S:Lblocks} we describe three new construction methods of equiangular lines from MUBs, including examples from the dimensions $d$ for which we currently know they succeed; and in Section~\ref{S:conclusion} we give some concluding remarks.
\section{Definitions} \label{S:defns}
We now introduce the main objects of study.
A line through the origin in $\mathbb{C}^d$ can be represented by a nonzero vector $\mbox{\boldmath $x$}\in\mathbb{C}^d$ which spans it. A set of $m\geq 2$ distinct lines in $\mathbb{C}^d$, represented by vectors $\mbox{\boldmath $x$}_1,\ldots,\mbox{\boldmath $x$}_m$, is {\em equiangular} if there is some real constant $c$ such that
\begin{equation*} \frac{{|\langle \mbox{\boldmath $x$}_j,\mbox{\boldmath $x$}_k\rangle|}}{||\mbox{\boldmath $x$}_j||\cdot||\mbox{\boldmath $x$}_k||} = c \hspace{10pt}\mbox{ for all } j\neq k, \end{equation*}
where $\langle \mbox{\boldmath $x$},\mbox{\boldmath $y$}\rangle$ is the standard Hermitian inner product in $\mathbb{C}^d$ and $||\mbox{\boldmath $x$}|| = \sqrt{|\langle \mbox{\boldmath $x$}, \mbox{\boldmath $x$}\rangle|}$ is the norm of $\mbox{\boldmath $x$}$. We simplify notation by always taking $\mbox{\boldmath $x$}_1,\ldots,\mbox{\boldmath $x$}_m$ to have equal norm, and then it suffices that there is a constant $a$ such that
\begin{equation} |\langle \mbox{\boldmath $x$}_j,\mbox{\boldmath $x$}_k\rangle| = a \hspace{10pt}\mbox{ for all } j\neq k. \label{EQ:equiang2} \end{equation}
Furthermore, if each vector has unit norm, then we will refer to $|\langle \mbox{\boldmath $x$}_j,\mbox{\boldmath $x$}_k\rangle|$ as the {\em angle} between $\mbox{\boldmath $x$}_j$ and $\mbox{\boldmath $x$}_k$ (although this value is strictly the cosine of the angle).
It is known that there can be at most $d^2$ equiangular lines in $\mathbb{C}^d$~\cite{DGS-bounds}. This is a specific instance of more general results obtained by Delsarte, Goethals and Seidel~\cite{DGS-bounds} using Jacobi polynomials. They found special bounds on the number of lines with a small set of angles that can exist in $\mathbb{C}^d$ when the angle values are specified (see~\cite[Table~I]{DGS-bounds}), as well as absolute bounds on the number of lines with a small set of angles that can exist in $\mathbb{C}^d$ without specifying angle values (see~\cite[Table~II]{DGS-bounds}). They also noted that if $\{\mbox{\boldmath $x$}_1,\ldots, \mbox{\boldmath $x$}_{d^2}\}$ is a set of unit vectors representing a maximum-sized set of complex equiangular lines, then the value of $a$ in (\ref{EQ:equiang2}) is determined.
\begin{prop} Let $\{\mbox{\boldmath $x$}_1,\ldots, \mbox{\boldmath $x$}_{d^2}\}$ be a set of unit vectors representing equiangular lines in $\mathbb{C}^d$. Then
$$|\langle \mbox{\boldmath $x$}_j,\mbox{\boldmath $x$}_k\rangle| = \frac{1}{\sqrt{d+1}}$$ for all $j\neq k$. \label{PROP:angle} \end{prop}
The value $1/\sqrt{d+1}$ given in Proposition~\ref{PROP:angle} can be determined by taking $s=1, \varepsilon=0$ over~$\mathbb{C}$ in \cite[Table II]{DGS-bounds}. An alternative self-contained proof using linear algebra is given in \cite[Proposition 9]{Wiebe-thesis}, following the method described by Godsil \cite{Godsil-slides}.
A basis for $\mathbb{C}^d$ is called {\em orthogonal} if the inner product of any two distinct basis elements is 0. Let $\{\mbox{\boldmath $x$}_1,\ldots, \mbox{\boldmath $x$}_d\},\{\mbox{\boldmath $y$}_1,\ldots,\mbox{\boldmath $y$}_d\}$ be two distinct orthogonal bases for $\mathbb{C}^d$. They are called {\em unbiased bases} if
\begin{equation} \frac{|\langle \mbox{\boldmath $x$}_j,\mbox{\boldmath $y$}_k\rangle|}
{||\mbox{\boldmath $x$}_j||\cdot||\mbox{\boldmath $y$}_k||} = \frac{1}{\sqrt{d}} \label{EQ:MUBdef} \quad \mbox{for all $j,k$}. \end{equation} A set of orthogonal bases is a set of {\em mutually unbiased bases (MUBs)} if all pairs of distinct bases are unbiased.
\begin{ex} Consider the following orthogonal bases for $\mathbb{C}^2$: \begin{eqnarray*} B_1 = \left\{ \begin{array}{@{\; (}cc@{)\;}} 1 & 0 \\ 0 & 1 \end{array}\right\} & B_2 = \left\{ \begin{array}{@{\; (}cc@{)\;}} 1 & 1 \\ 1 & -1 \end{array}\right\} & B_3 = \left\{ \begin{array}{@{\; (}cc@{)\;}} 1 & i \\ 1 & -i \end{array}\right\}. \end{eqnarray*} Then for $\mbox{\boldmath $x$},\mbox{\boldmath $y$}$ in distinct bases we have \begin{eqnarray*}
\frac{|\langle \mbox{\boldmath $x$},\mbox{\boldmath $y$}\rangle|}{||\mbox{\boldmath $x$}||\cdot||\mbox{\boldmath $y$}||}
& = & \begin{cases} \frac{1}{1\cdot\sqrt{2}} & \text{ for one of } \mbox{\boldmath $x$},\mbox{\boldmath $y$}\in B_1\\[3pt]
\frac{\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}} & \text{ for }\mbox{\boldmath $x$},\mbox{\boldmath $y$}\notin B_1 \end{cases} \\
& = & \frac{1}{\sqrt{2}}, \end{eqnarray*} satisfying $(\ref{EQ:MUBdef})$, so $\{B_1,B_2,B_3\}$ is a set of $3$ MUBs in $\mathbb{C}^2$. \label{EX:MUB} \end{ex}
An upper bound on the number of MUBs in $\mathbb{C}^d$ is $d+1$ \cite[Table I]{DGS-bounds} (using $\alpha=1/d,\beta=0$ over $\mathbb{C}$). An alternative proof of this bound is given in \cite[Proposition 16]{Wiebe-thesis} using linear algebra, following the method described by Bandyopadhyay {\em et al.} \cite{Bandyopadhyay}. As with equiangular lines, the central question concerning MUBs is whether this bound can be attained in all dimensions. In contrast to the situation for equiangular lines, there seems to be more doubt that this is possible. Bengtsson \cite{Bengtsson-Eddington} in 2011 observed: ``The belief in the community is that a complete set of $[d+1]$ MUB[s] does not exist for general $[d]$, while the [maximum-sized sets of equiangular lines] do.'' However, it is known that this upper bound for MUBs is attainable in prime power dimensions $d$ using the method of Godsil and Roy \cite{Godsil-Roy} which we follow here.
Let $G$ be a group of order $mn$, containing a normal subgroup $N$ of order~$n$. A {\em $(m,n,k,\lambda)$-relative difference set (RDS)}\index{relative difference set}\index{RDS} in $G$ relative to $N$ is a subset $R\subset G$ of size $k$, such that the multiset $$\{r_1r_2^{-1} : r_1,r_2\in R, r_1\neq r_2\}$$ contains each element of $G\backslash N$ exactly $\lambda$ times and does not contain any elements of $N$.
\begin{ex} Let $G$ be the abelian group of order $16$ given by $\langle x\rangle\times\langle y\rangle$, with $x^4=y^4=1$. Let $N$ be the subgroup $\langle x^2\rangle\times\langle y^2\rangle$ of order $4$. Then $R =\{1,x,y,x^3y^3\}$ is a $(4,4,4,1)$-RDS in $G$ relative to $N$. \label{EX:RDS4} \end{ex}
A {\em character} of a finite abelian group $G$ is a map $\chi:G\to\mathbb{C}$ which is a group homomorphism. If $G$ has order $v$, then there are $v$ characters, each of which maps the elements of $G$ to roots of unity. These characters form a group $G^*$ which is isomorphic to $G$. (See Pott \cite{Pott}, for example, for background on the use of characters to study relative difference sets.)
\begin{thm}[Theorem 4.1, \cite{Godsil-Roy}] Let $R=\{r_1,\ldots, r_d\}$ be a $(d,d,d,1)$-RDS in an abelian group $G$ relative to some subgroup of order $d$. Then the set of vectors $$\{(\chi(r_1),\ldots, \chi(r_d)) : \chi\in G^*\}$$ comprises a set of $d$ MUBs in $\mathbb{C}^d$. \label{THM:RDStoMUBs} \end{thm}
\begin{ex} Take $R$ to be the RDS of Example~$\ref{EX:RDS4}$. The group $G$ has $16$ characters given by $\{\chi_{j,k}: j,k\in\mathbb{Z}_4\}$, where $\chi_{j,k}(x) = i^j, \chi_{j,k}(y)=i^k$. Using the construction of Theorem~$\ref{THM:RDStoMUBs}$ we get the following $4$ MUBs $B_1, B_2, B_3, B_4$ in $\mathbb{C}^4$: \begin{displaymath} {\renewcommand{1.1}{1.1}
\begin{array}{c|c@{(}cccc@{)\hspace{5pt}}c}
&& 1 & x & y & x^3y^3 \\ \hline
\chi_{0,0} && 1 & 1 & 1 & 1 & \multirow{4}{*}{$\left. \rule{0pt}{32pt}\right\} B_1 $} \\
\chi_{0,2} && 1 & 1 & -1 & -1 \\
\chi_{2,0} && 1 & -1 & 1 & -1 \\
\chi_{2,2} && 1 & -1 & -1 & 1 \\[2pt] \hdashline[2pt/6pt]
\chi_{0,1} && 1 & 1 & i & -i & \multirow{4}{*}{$\left. \rule{0pt}{32pt}\right\} B_2 $} \\
\chi_{0,3} && 1 & 1 & -i & i \\
\chi_{2,1} && 1 & -1 & i & i \\
\chi_{2,3} && 1 & -1 & -i & -i \\[2pt] \hdashline[2pt/6pt]
\chi_{1,0} && 1 & i & 1 & -i & \multirow{4}{*}{$\left. \rule{0pt}{32pt}\right\} B_3 $} \\
\chi_{1,2} && 1 & i & -1 & i \\
\chi_{3,0} && 1 & -i & 1 & i \\
\chi_{3,2} && 1 & -i & -1 & -i \\[2pt] \hdashline[2pt/6pt]
\chi_{1,1} && 1 & i & i & -1 & \multirow{4}{*}{$\left. \rule{0pt}{32pt}\right\} B_4. $} \\
\chi_{1,3} && 1 & i & -i & 1 \\
\chi_{3,1} && 1 & -i & i & 1 \\
\chi_{3,3} && 1 & -i & -i & -1 \end{array} } \end{displaymath} \label{EX:MUBs4} \end{ex}
We note that to attain the upper bound of $d+1$ MUBs in $\mathbb{C}^d$ when $d$ is a prime power, we can include the standard basis with the $d$ bases obtained from Theorem~\ref{THM:RDStoMUBs}.
When $d$ is not a prime power, the smallest dimension for which a set of $d+1$ MUBs in $\mathbb{C}^d$ is not known is $d=6$. No one has even found 4 MUBs in $\mathbb{C}^6$; furthermore, the existence of sets of 3 MUBs which are provably not extendable to 4 leads some researchers to suspect that it may not be possible to find more than 3 MUBs in $\mathbb{C}^6$ \cite{Bengtsson-three-ways}.
\section{Zauner's Construction} \label{S:Zauner}
Zauner \cite{Zauner-thesis} was the first to conjecture that maximum-sized sets of equiangular lines exist in all dimensions. Along with this conjecture, he presented a construction for such sets of lines which has become the standard construction in the area. We give a brief overview of this construction here.
In most of the literature regarding complex equiangular lines, maximum-sized sets of equiangular lines are constructed as the orbit of a {\em fiducial vector} under the action of a group of matrices. Zauner's thesis \cite{Zauner-thesis} describes both the group to use and where to find an appropriate fiducial vector.
Let $\omega = e^{2\pi i/d}$ and $$U = \left[\begin{array}{cccccc}
1 & 0 & 0 & \cdots & 0 \\
0 & \omega & 0 & \cdots & 0 \\
0 & 0 & \omega^2 & \cdots & 0 \\
\vdots & & & \ddots & \vdots
\\
0 & 0 & 0 & \cdots & \omega^{d-1}
\end{array}\right]
\mbox{\hspace{10pt} and \hspace{10pt}} V = \left[\begin{array}{cccccc}
0 & 1 & 0 & 0 & \cdots & 0 \\
0 & 0 & 1 & 0 & \cdots & 0 \\
\vdots & & & \ddots & & \vdots \\
\\
0 & 0 & 0 & 0 & \cdots & 1 \\
1 & 0 & 0 & 0 & \cdots & 0
\end{array}\right].$$ Matrices $U,V$ generate a group of matrices known as the Weyl-Heisenberg group \cite{Weyl} (see \cite{Bengtsson-Eddington, Scott-Grassl}, for example, for an overview). Modulo its center, the group is isomorphic to $\mathbb{Z}_d\times\mathbb{Z}_d$, and the elements $U^jV^k$ for $j,k\in\mathbb{Z}_d$ form a set of coset representatives for the center in this group.
Next define a $d\times d$ matrix $Z_u=(z_{jk})$, for $j,k\in\{0,\ldots,d-1\},$\index{$Z_u$} by \begin{equation*} z_{jk} = \frac{e^{\pi i(d-1)/12}}{\sqrt{d}}e^{\pi i(2jk+(d+1)k^2)/d}. \end{equation*} Then $Z_u$ is a unitary matrix (often referred to as {\em Zauner's unitary}\index{Zauner's unitary}) which satisfies $$Z_u^3 = I_d $$ and normalizes the Weyl-Heisenberg group.
Zauner's full conjecture is the following: \begin{conj} For each $d\geq 2$, there exists a set of $d^2$ equiangular lines in $\mathbb{C}^d$ that is constructed as $$\{A\mbox{\boldmath $x$}^T:A\in G\},$$ where $G$ is the Weyl-Heisenberg group and $\mbox{\boldmath $x$}^T$ is some eigenvector of $Z_u$ having eigenvalue $1$. \label{CONJ:ZaunerFull} \end{conj}
In all dimensions where a maximum-sized set of equiangular lines is known, there is a set constructed as in Conjecture~\ref{CONJ:ZaunerFull}. Furthermore, almost all known maximum-sized sets of equiangular lines can be constructed in this way.
Notice that Conjecture~\ref{CONJ:ZaunerFull} does not state that every eigenvector of eigenvalue 1 will produce a maximum-sized set of equiangular lines. For further details on the computational methods by which appropriate eigenvectors are found and the difficulty of doing so, we refer the reader to \cite{Grassl-tomography, Grassl-computing, Scott-Grassl}.
\begin{ex} $d=4$ \cite[p.~62]{Zauner-thesis}: In dimension $4$, the generators of the Weyl-Heisenberg group and Zauner's unitary are as follows: $$U = \left[\begin{array}{@{\,}cccc@{\,}}
1 & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -i
\end{array}\right],\, V = \left[\begin{array}{@{\,}cccc@{\,}}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right],\, {\renewcommand{1.1}{1.5} Z_u = \frac{1}{2}\left[\begin{array}{@{\,}cccc@{\,}}
\frac{1+i}{\sqrt{2}} & -i & -\frac{1+i}{\sqrt{2}} & -i \\
\frac{1+i}{\sqrt{2}} & 1 & \frac{1+i}{\sqrt{2}} & -1 \\
\frac{1+i}{\sqrt{2}} & i & -\frac{1+i}{\sqrt{2}} & i \\
\frac{1+i}{\sqrt{2}} & -1 & \frac{1+i}{\sqrt{2}} & 1
\end{array}\right].
}$$ A fiducial vector is given by the eigenvector \[ \left( \begin{array}{c} x_0 \\ x_1 \\ x_2 \\ x_3 \end{array} \right) = \frac{1}{2\sqrt{6}}\sqrt{3-\frac{3}{\sqrt{5}}} \left ( \begin{array}{c} \omega+1 \\ i \\ \omega-1 \\ i \end{array} \right) + \frac{1}{2\sqrt{2}}\sqrt{1+\frac{3}{\sqrt{5}}} \left ( \begin{array}{c} 0 \\ \omega \\ 0 \\ -\omega \end{array} \right) \] of $Z_u$, where $\omega=e^{i\pi/4}$. Then $16$ equiangular lines in $\mathbb{C}^4$ are given by \begin{displaymath} {\renewcommand{1.1}{1.1} \begin{array}{@{}rrrr@{}} (x_0 & x_1 & x_2 & x_3)\; \\ (x_0 & ix_1 & -x_2 & -ix_3)\; \\ (x_0 & -x_1 & x_2 & -x_3)\; \\ (x_0 & -ix_1 & -x_2 & ix_3)\; \\ (x_1 & x_2 & x_3 & x_0)\; \\ (x_1 & ix_2 & -x_3 & -ix_0)\; \\ (x_1 & -x_2 & x_3 & -x_0)\; \\ (x_1 & -ix_2 & -x_3 & ix_0)\; \\ (x_2 & x_3 & x_0 & x_1)\; \\ (x_2 & ix_3 & -x_0 & -ix_1)\; \\ (x_2 & -x_3 & x_0 & -x_1)\; \\ (x_2 & -ix_3 & -x_0 & ix_1)\; \\ (x_3 & x_0 & x_1 & x_2)\; \\ (x_3 & ix_0 & -x_1 & -ix_2)\; \\ (x_3 & -x_0 & x_1 & -x_2)\; \\ (x_3 & -ix_0 & -x_1 & ix_2). \end{array} } \end{displaymath} \label{EX:WH4} \end{ex}
Now we describe three new constructions of equiangular lines in $\mathbb{C}^d$, each of which involves a set of MUBs.
\section{Construction 1} \label{S:LinesfromMUBs}
The underlying structure of the 16 equiangular lines in $\mathbb{C}^4$ of Example~\ref{EX:WH4} seems strictly tied to the Weyl-Heisenberg group and requires complicated constants. However, Appleby {\em et al}. \cite{Appleby-monomial} recently reinterpreted Zauner's construction (as described in \S\ref{S:Zauner}), leading to a new example in dimension 4 with simpler constants. We will show how the resulting simplified set of lines, given in the following example, has additional underlying combinatorial structure.
\begin{ex} A set of $16$ vectors representing equiangular lines in $\mathbb{C}^4$, as constructed in \cite{Appleby-monomial}, is \begin{displaymath} \begin{array}{r@{\;}cccc@{\;}l} (&\sqrt{2+\sqrt{5}} & 1 & 1 & 1&) \\ (&\sqrt{2+\sqrt{5}} & 1 & -1 & -1&) \\ (&\sqrt{2+\sqrt{5}} & -1 & 1 & -1&) \\ (&\sqrt{2+\sqrt{5}} & -1 & -1 & 1&) \\[8pt] (&1 & 1 & i\sqrt{2+\sqrt{5}} & -i&) \\ (&1 & 1 & -i\sqrt{2+\sqrt{5}} & i&) \\ (&1 & -1 & i\sqrt{2+\sqrt{5}} & i&) \\ (&1 & -1 & -i\sqrt{2+\sqrt{5}} & -i&)\\[8pt] (&1 & i & 1 & -i\sqrt{2+\sqrt{5}}&) \\ (&1 & i & -1 & i\sqrt{2+\sqrt{5}}&) \\ (&1 & -i & 1 & i\sqrt{2+\sqrt{5}}&) \\ (&1 & -i & -1 & -i\sqrt{2+\sqrt{5}} &)\\[8pt] (&1 & i\sqrt{2+\sqrt{5}} & i & -1&) \\ (&1 & i\sqrt{2+\sqrt{5}} & -i & 1&) \\ (&1 & -i\sqrt{2+\sqrt{5}} & i & 1&) \\ (&1 & -i\sqrt{2+\sqrt{5}} & -i & -1 &). \end{array} \end{displaymath} \label{EX:d=4} \end{ex}
This set of vectors was also found by Belovs \cite{Belovs-thesis} in 2008 via another method and was known even earlier by Zauner (unpublished notes, 2005, referenced in \cite{Appleby-monomial}). However, here we describe another construction of this set of equiangular lines which demonstrates that the underlying structure of this set can be interpreted as a set of 4 MUBs in $\mathbb{C}^4$ coming from a $(4,4,4,1)$-RDS as in Theorem~\ref{THM:RDStoMUBs}. The general construction is as follows.
We exploit the already constrained angles between vectors in a set of MUBs to produce sets of equiangular lines by allowing the multiplication of a single entry in each of the $d^2$ vectors by a constant. Specifically, let $B^R_1,\ldots, B^R_d$ be $d$ MUBs in $\mathbb{C}^d$ formed from a $(d,d,d,1)$-RDS $R$ according to Theorem~\ref{THM:RDStoMUBs}. Let $\pi\in S_d$ be a permutation of $\{1,\ldots,d\}$ (which we represent as an ordered list of images). Let $B^R_j(\pi,v)$ denote the set of vectors formed by multiplying entry $\pi(j)$ of each vector in $B_j^R$ by $v\in\mathbb{C}$. Let \begin{equation} L^R_d(\pi,v) = \bigcup_{j=1}^d B^R_j(\pi,v).\index{$L^R_d(\pi,v)$} \label{EQ:Lblock} \end{equation}
\begin{ex} Take $B_1^R, \ldots, B_4^R$ to be the $4$ MUBs of Example~$\ref{EX:MUBs4}$. Let $\pi = [1,3,4,2]$ and let $v\in\mathbb{C}$. Then $L^R_4(\pi,v)$ consists of vectors \begin{displaymath} {\renewcommand{1.1}{1.1} \begin{array}{@{(}cccc@{)\hspace{5pt}}c}
v & 1 & 1 & 1 & \multirow{4}{*}{$\left. \rule{0pt}{30pt}\right\} B^R_1(\pi,v) $} \\
v & 1 & -1 & -1 \\
v & -1 & 1 & -1 \\
v & -1 & -1 & 1 \\[5pt]
1 & 1 & iv & -i & \multirow{4}{*}{$\left. \rule{0pt}{30pt}\right\} B^R_2(\pi,v) $} \\
1 & 1 & -iv & i \\
1 & -1 & iv & i \\
1 & -1 & -iv & -i \\[5pt]
1 & i & 1 & -iv & \multirow{4}{*}{$\left. \rule{0pt}{30pt}\right\} B^R_3(\pi,v) $} \\
1 & i & -1 & iv \\
1 & -i & 1 & iv \\
1 & -i & -1 & -iv \\[5pt]
1 & iv & i & -1& \multirow{4}{*}{$\left. \rule{0pt}{30pt}\right\} B^R_4(\pi,v) $} \\
1 & iv & -i & 1 \\
1 & -iv & i & 1 \\
1 & -iv & -i & -1 \end{array} } \end{displaymath} and is a set of equiangular lines when $v\in\{ \pm \sqrt{2+\sqrt{5}}$, $\pm i\sqrt{2+\sqrt{5}}\}$. In fact, for $v = \sqrt{2+\sqrt{5}}$ it is the same set of lines as given in Example~$\ref{EX:d=4}$. \label{EX:MUBLines4} \end{ex}
\begin{thm} For $d=2,3,4$ there exists a $(d,d,d,1)$-RDS $R$, a permutation $\pi\in S_d$ and a constant $v(d)\in\mathbb{C}$ such that $L^R_d(\pi,v(d))$ is a set of $d^2$ equiangular lines in $\mathbb{C}^d$. \label{THM:1constantconstruct} \end{thm}
Example~\ref{EX:MUBLines4} together with the following two examples prove the theorem:
\begin{ex} $d=2:$ Take the $(2,2,2,1)$-RDS $R=\{1,x\}$ in $\langle x\rangle\cong \mathbb{Z}_4$ relative to $\langle x^2\rangle\cong\mathbb{Z}_2$, the permutation $\pi=[1,2]$ and the constant $v$. This gives a set of $2$ MUBs in $\mathbb{C}^2$ ($B_2,B_3$ from Example~$\ref{EX:MUB}$). Then $L_2^R(\pi,v)$ consists of vectors \begin{displaymath}\begin{array}{cc} \begin{array}{@{\;(}cc@{)\hspace{3pt}}c} v & 1 & \multirow{2}{*}{$\left. \rule{0pt}{12pt}\right\} B^R_1(\pi,v) $} \\ v & -1 \\[8pt] 1 & iv & \multirow{2}{*}{$\left. \rule{0pt}{12pt}\right\} B^R_2(\pi,v) $} \\ 1 & -iv \end{array} \end{array}\end{displaymath} which are equiangular for $v\in\left\{\frac{1}{2}(\sqrt{2}\pm\sqrt{6})\right.$, $-\frac{1}{2}(\sqrt{2}\pm\sqrt{6})$, $\frac{1}{2}i(\sqrt{2}\pm\sqrt{6})$, $\left.-\frac{1}{2}i(\sqrt{2}\pm\sqrt{6})\right\}$. \label{EX:MUBLines2} \end{ex}
\begin{ex} $d=3:$ Take the $(3,3,3,1)$-RDS $R = \{1,y,xy^2\}$ in $\langle x\rangle\times\langle y\rangle \cong\mathbb{Z}_3\times\mathbb{Z}_3$ relative to $\langle x\rangle \times \langle 1\rangle \cong \mathbb{Z}_3$. The resulting $3$ MUBs are as follows: {\renewcommand{1.1}{1.1} \begin{displaymath}
\begin{array}{c|c@{(}ccc@{)\;}l}
&& 1 & y & xy^2 \\ \hline
\chi_{0,0} && 1 & 1 & 1 & \multirow{3}{*}{$\left.\rule{0pt}{22pt}\right\} B^R_1$} \\
\chi_{0,1} && 1 & \omega & \omega^2 \\
\chi_{0,2} && 1 & \omega^2 & \omega \\[2pt] \hdashline[2pt/6pt]
\chi_{1,0} && 1 & 1 & \omega & \multirow{3}{*}{$\left.\rule{0pt}{22pt}\right\} B^R_2$} \\
\chi_{1,1} && 1 & \omega & 1 \\
\chi_{1,2} && 1 & \omega^2 & \omega^2 \\[2pt] \hdashline[2pt/6pt]
\chi_{2,0} && 1 & 1 & \omega^2 & \multirow{3}{*}{$\left.\rule{0pt}{22pt}\right\} B^R_3$} \\
\chi_{2,1} && 1 & \omega & \omega \\
\chi_{2,2} && 1 & \omega^2 & 1 \end{array} \end{displaymath}} where $\omega=e^{2\pi i/3}$. Take the permutation $\pi = [1,2,3]$ and the constant $v$. Then $L_3^R(\pi,v)$ consists of vectors \begin{displaymath} \begin{array}{@{(}ccc@{)\;}l}
v & 1 & 1 & \multirow{3}{*}{$\left.\rule{0pt}{20pt}\right\} B^R_1(\pi,v)$} \\
v & \omega & \omega^2 \\
v & \omega^2 & \omega \\[8pt]
1 & v & \omega & \multirow{3}{*}{$\left.\rule{0pt}{20pt}\right\} B^R_2(\pi,v)$} \\
1 & v\omega & 1 \\
1 & v\omega^2 & \omega^2 \\[8pt]
1 & 1 & v\omega^2 & \multirow{3}{*}{$\left.\rule{0pt}{20pt}\right\} B^R_3(\pi,v)$} \\
1 & \omega & v\omega \\
1 & \omega^2 & v \end{array} \end{displaymath} which are equiangular for $v=0$. \label{EX:MUBLines3} \end{ex}
It turns out that in Examples~\ref{EX:MUBLines2} ($d=2$) and \ref{EX:MUBLines3} ($d=3$), every choice of permutation $\pi\in S_d$ will produce a maximum-sized set of equiangular lines with the given constant(s) $v$. However, in Example~\ref{EX:MUBLines4} ($d=4$), the choice of permutation becomes important, as only 8 of the 24 possible permutations, namely, $[1,3,4,2]$, $[1,4,2,3]$, $[2,3,1,4]$, $[2,4,3,1]$, $[3,1,2,4]$, $[3,2,4,1]$, $[4,1,3,2]$, and $[4,2,1,3]$, admit a constant $v$ which results in a set of equiangular lines. The occurrence of these permutations can be explained by the following theorem, which we state here without proof.
\begin{thm}[Theorem 46, \cite{Wiebe-thesis}] Fix a $(4,4,4,1)$-RDS $R$ in $\mathbb{Z}_4\times\mathbb{Z}_4$. The vectors of $L^R_4(\pi,\sqrt{2+\sqrt{5}})$ comprise $16$ equiangular lines in $\mathbb{C}^4$ if $\pi$ is such that the inner product between each pair of distinct vectors in $L^R_4(\pi,0)$ has magnitude~$\sqrt{2}$. \label{THM:d=4_1constant} \end{thm}
Theorem~\ref{THM:d=4_1constant} suggests that as we increase the dimension $d$ there may be additional restrictions on when this construction produces maximum-sized sets of equiangular lines, which leads us to ask the following question:
\begin{question} Does the construction used in Theorem~$\ref{THM:1constantconstruct}$ produce maximum-sized sets of equiangular lines for some $d>4$? \label{Q:1constantconstruct} \end{question}
And more generally,
\begin{question} Can we transform a set of $d$ MUBs in $\mathbb{C}^d$ into $d^2$ equiangular lines in $\mathbb{C}^d$ through multiplication by constants? \label{Q:MUBstoLines} \end{question}
Conversely, we wonder if there is a complementary construction to the one we have described, which could be used to extract MUBs from maximum-sized sets of equiangular lines other than those given in Examples~\ref{EX:MUBLines4}, \ref{EX:MUBLines2} and \ref{EX:MUBLines3}.
\begin{question} Can we transform a set of $d^2$ equiangular lines in $\mathbb{C}^d$ into $d$ MUBs in $\mathbb{C}^d$ through multiplication by constants? \label{Q:LinestoMUBs} \end{question}
We now derive a necessary condition that could assist in answering Question~\ref{Q:1constantconstruct}. We observe that the construction used in Theorem~$\ref{THM:1constantconstruct}$ will always produce {\em almost flat} vectors; that is, vectors having all but one entry of equal magnitude. Furthermore, if this set of vectors represents a set of equiangular lines, then the following lemma determines the single exceptional magnitude:
\begin{lemma} Let $R$ be a $(d,d,d,1)$-RDS in an abelian group. Let $\pi\in S_d$ and $v\in\mathbb{C}$. Suppose that $L^R_d(\pi,v)$ is a set of $d^2$ equiangular lines. Then the magnitude of $v$ is $\sqrt{2\pm\sqrt{d+1}}$. \end{lemma}
\begin{proof}
Let $\mbox{\boldmath $x$},\mbox{\boldmath $y$}$ be distinct lines of $L^R_d(\pi,v)$ originating from the same basis. Since the original vectors are orthogonal, the inner product of $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$ is $\xi(|v|^2-1)$ for some root of unity $\xi$. Each of $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$ has norm $\sqrt{d-1+|v|^2}$. By Proposition~\ref{PROP:angle}, we must have $$
\frac{1}{\sqrt{d+1}} = \frac{|\langle \mbox{\boldmath $x$},\mbox{\boldmath $y$}\rangle|}{||\mbox{\boldmath $x$}||\cdot||\mbox{\boldmath $y$}||} = \frac{\big||v^2|-1\big|}{{d-1+|v|^2}}, $$
which is easily solved to find $|v|^2 = 2\pm\sqrt{d+1}$. \end{proof}
Though this construction is notably different than Zauner's, we observe that having an exceptional magnitude of $\sqrt{2\pm\sqrt{d+1}}$ is equivalent to the necessary condition given in \cite[6.4.1.\ Lemma]{Roy-thesis} for constructing maximum-sized sets of almost flat equiangular lines using Zauner's construction.
As a final observation about this construction, notice that MUBs formed as in Theorem~\ref{THM:RDStoMUBs} have elements all of whose entries lie on the complex unit circle. Thus if we take a single basis from the set and write its elements as the rows of a matrix $H$, then this matrix will satisfy the equation \begin{equation} HH^\dagger = dI_d \end{equation} (where $H^\dagger$ is the conjugate transpose of $H$), which means $H$ is a complex Hadamard matrix of order $d$. From this observation, we can consider the single Hadamard matrix construction of \cite{Jedwab-Wiebe-simple} for 64 equiangular lines in $\mathbb{C}^8$ as an example of a construction relying on MUBs. This now links the construction of maximum-sized sets of equiangular lines in dimensions $2,3,4$ and $8$ via MUBs.
\section{Construction 2}\label{S:d8}
We now examine another example of a maximum-sized set of equiangular lines whose connection to MUBs has not previously been recognized.
One of the first dimensions for which a set of $d^2$ equiangular lines in $\mathbb{C}^d$ was discovered was $d=8$. In 1981, Hoggar gave a construction for 64 lines as the (complexified vectors associated to the) diameters of a quaternionic polytope~\cite{Hoggar-two}. (See \cite{Hoggar-t-designs, Hoggar-64} for more details on this construction.)
This set of lines was reinterpreted by Zauner~\cite{Zauner-thesis}. It was also reinterpreted by Godsil and Roy~\cite{Godsil-Roy} using the following variation of Zauner's construction method of \S\ref{S:Zauner}.
Recall that in dimension 2, the Weyl-Heisenberg group has generators $$U = \left[\begin{array}{cc} 1& 0 \\ 0 & -1\end{array}\right], \hspace{20pt} V = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right] $$ and a set of coset representatives for its center is $\{I_4,U,V,UV\}$. Now consider the 3-fold tensor product $G$ of this set: it has 64 elements given by $G = \{A\otimes B\otimes C: A,B,C\in \{I_4,U,V,UV\}\}$ with each element being an $8\times 8$ matrix. Let $$\mbox{\boldmath $x$} = \left(0,0, \frac{1+i}{\sqrt{2}}, \frac{1-i}{\sqrt{2}}, \frac{1+i}{\sqrt{2}},- \frac{1+i}{\sqrt{2}},0,\sqrt{2}\right).$$ Then 64 equiangular lines in $\mathbb{C}^8$ are given by $\{A\mbox{\boldmath $x$}^T : A\in G\}$.
There are several operations that map a set of equiangular lines to an equivalent set of equiangular lines, including: \begin{enumerate} \item{permuting the entries of each vector according to the same permutation,} \item{multiplying all entries of a single vector by a complex constant of magnitude 1,} \item{multiplying the same entry of each vector by a complex constant of magnitude 1.} \end{enumerate} Under a suitable combination of these operations, we can transform Hoggar's 64 lines (as interpreted above) into a form that exposes a new link with MUBs.
\begin{ex} Let $B_1,B_2,B_3,B_4$ be the $4$ MUBs of Example~$\ref{EX:MUBs4}$. Let $C_j$ be the $4\times 4$ matrix with $-1+i$ in the $j$-th column and zeros elsewhere. Then Hoggar's $64$ lines in $\mathbb{C}^8$ are equivalent to \begin{displaymath} \begin{array}{@{[}cc@{]\hspace{15pt}[}cc@{]\hspace{15pt}[}cc@{]\hspace{15pt}[}cc@{]}} B_1 & C_1 & B_1 & -C_1 & -C_1 & B_1 & C_1 & B_1 \\ B_2 & C_2 & B_2 & -C_2 & -C_2 & B_2 & C_2 & B_2 \\ B_3 & C_3 & B_3 & -C_3 & -C_3 & B_3 & C_3 & B_3 \\ B_4 & C_4 & B_4 & -C_4 & -C_4 & B_4 & C_4 & B_4 \\ \end{array} \end{displaymath} where $[B\;\,C]$ represents the set of vectors which are the concatenation of corresponding vectors in $B$ and $C$. \label{EX:Hoggartwist} \end{ex}
In Example~\ref{EX:Hoggartwist} we can see how the 4 MUBs of Example~\ref{EX:MUBs4} are embedded in a set of equiangular lines equivalent to Hoggar's. In this way, we can view Example~\ref{EX:Hoggartwist} as constructing 64 equiangular lines in $\mathbb{C}^8$ from 4 MUBs in $\mathbb{C}^4$. In fact, it is just one instance of the following construction of a one-parameter family of sets of 64 equiangular lines in $\mathbb{C}^8$.
Let $a$ be a real parameter. Let $C_j(a)$ be the $4\times 4$ matrix with $\frac{a-1+i(a+1)}{\sqrt{1+a^2}}$ in column $j$ and zeros elsewhere and $D_j(a)$ be the $4\times 4$ matrix with $\frac{a+1+i(a-1)}{\sqrt{1+a^2}}$ in column $j$ and zeros elsewhere. Then one can verify using a computer algebra system, or even by hand, that the following vectors are a set of 64 equiangular lines in $\mathbb{C}^8$ for all values of $a\in\mathbb{R}$: \begin{displaymath} \begin{array}{@{[}cc@{]\hspace{15pt}[}cc@{]\hspace{15pt}[}cc@{]\hspace{15pt}[}cc@{}l} B_1 & C_1(a) & B_1 & -C_1(a) & D_1(a) & B_1 & -D_1(a) & B_1&] \\ B_2 & C_2(a) & B_2 & -C_2(a) & D_2(a) & B_2 & -D_2(a) & B_2&] \\ B_3 & C_3(a) & B_3 & -C_3(a) & D_3(a) & B_3 & -D_3(a) & B_3&] \\ B_4 & C_4(a) & B_4 & -C_4(a) & D_4(a) & B_4 & -D_4(a) & B_4&]. \\ \end{array} \end{displaymath} The lines of Example~\ref{EX:Hoggartwist} can be obtained by setting $a=0$.
Notice that in contrast to the construction of \S\ref{S:LinesfromMUBs}, this construction involves a change of dimension, namely using MUBs in $\mathbb{C}^4$ to construct equiangular lines in $\mathbb{C}^8$. It is then natural to ask the following question: \begin{question} Can we construct lines having similar form to those in Example~$\ref{EX:Hoggartwist}$ in dimensions other than $8$? \label{Q:Hoggar} \end{question} And more generally, \begin{question} Can we construct $D^2$ equiangular lines in $\mathbb{C}^D$ from $d$ MUBs in $\mathbb{C}^d$ for $D\neq d$? \label{Q:MUBstoLines2}\end{question}
\section{Construction 3}\label{S:Lblocks}
In our final construction, we suggest another approach to answering Question~\ref{Q:MUBstoLines2}. We will show how to join together blocks of the form $L_d^R(\pi,v)$ as constructed in \S\ref{S:LinesfromMUBs} to give a set of $d^2$ equiangular lines in $\mathbb{C}^{2d}$ for infinitely many values of $d$. Furthermore, we will show how in dimension 8 we can extend this set of 16 lines to a maximum-sized set of 64 equiangular lines.
As described in \S\ref{S:LinesfromMUBs}, the vectors in $L^R_d(\pi,v)$ are derived from a set of $d$ MUBs constructed from a $(d,d,d,1)$-RDS in an abelian group. Write $[L^R_d(\pi,v) \hspace{5pt} L^R_d(\pi,v')]$ for the set of vectors in which each vector is the concatenation of corresponding vectors in $L^R_d(\pi,v)$ and $L^R_d(\pi,v')$.
\begin{ex} $d=4:$\! Take $R$ to be the RDS of Example~$\ref{EX:RDS4}$. Let $\pi = [1,3,4,2]$, and let $v,v'\in\mathbb{C}$. Construct $L^R_4(\pi,v)$ and $L^R_4(\pi,v')$ as in $(\ref{EQ:Lblock})$ (see Example~$\ref{EX:MUBLines4}$). Then \begin{displaymath} [L^R_4(\pi,v)\hspace{5pt} L^R_4(\pi,v')] = \left\{ \begin{array}{@{\;(}cccccccc@{)\;}} v & 1 & 1 & 1 & v' & 1 & 1 & 1 \\ v & 1 & -1 & -1 & v' & 1 & -1 & -1 \\ v & -1 & 1 & -1 & v' & -1 & 1 & -1 \\ v & -1 & -1 & 1 & v' & -1 & -1 & 1 \\[8pt] 1 & 1 & iv & -i & 1 & 1 & iv' & -i \\ 1 & 1 & -iv & i & 1 & 1 & -iv' & i \\ 1 & -1 & iv & i & 1 & -1 & iv' & i \\ 1 & -1 & -iv & -i & 1 & -1 & -iv' & -i\\[8pt] 1 & i & 1 & -iv & 1 & i & 1 & -iv' \\ 1 & i & -1 & iv & 1 & i & -1 & iv' \\ 1 & -i & 1 & iv & 1 & -i & 1 & iv' \\ 1 & -i & -1 & -iv & 1 & -i & -1 & -iv' \\[8pt] 1 & iv & i & -1 & 1 & iv' & i & -1 \\ 1 & iv & -i & 1 & 1 & iv' & -i & 1 \\ 1 & -iv & i & 1 & 1 & -iv' & i & 1 \\ 1 & -iv & -i & -1 & 1 & -iv' & -i & -1 \end{array} \right\}. \end{displaymath} \label{EX:2L4blocks} \end{ex}
\begin{lemma} Let $R$ be a $(d,d,d,1)$-RDS in an abelian group, let $\pi\in S_d$ and let $a,b\in\mathbb{R}$. Then all inner products between distinct vectors of the $d^2$ vectors of $[L^R_{d}(\pi,a+ib) \hspace{5pt} L^R_d(\pi,2-a-ib)]$ in $\mathbb{C}^{2d}$ have magnitude ${2}({b^2+(a-1)^2})$ or $2\sqrt{d}$. \label{LEM:2Lblockangles} \end{lemma}
\begin{proof} We consider two cases, according to whether distinct vectors of $L_d^R(\pi,v)$ originate from the same basis or from distinct bases.
In the first case, consider the inner product of distinct vectors of $L^R_d(\pi,v)$ constructed from vectors in the same basis $B^R_j$. Since the original vectors are orthogonal, this inner product is $\xi(|v|^2-1)$ for some root of unity~$\xi$. When $v=a+ib$, the inner product becomes $\xi(a^2+b^2-1)$ and when $v=2-a-ib$ it becomes $\xi((2-a)^2+b^2-1)$. Thus the corresponding concatenated vectors have inner product $\xi(a^2+b^2-1+(2-a)^2+b^2-1)$, which has magnitude ${2}({b^2+(a-1)^2})$.
In the second case, consider vectors of $L^R_d(\pi,v)$ constructed from vectors in distinct bases $B^R_j, B^R_k$ . Let these constructed vectors be given by \begin{equation} \begin{array}{rcccccccc} \mbox{\boldmath $x$} & = & (x_{1} & x_2 & \ldots & \ldots & vx_{\pi(j)} & \ldots & x_d)\\ \mbox{\boldmath $y$} & = & (y_{1} & y_2 & \ldots & vy_{\pi(k)} & \ldots & \ldots & y_d). \end{array} \end{equation} When $v=1$, by construction all of the entries $x_\ell$, $y_\ell$ are roots of unity (see Theorem~\ref{THM:RDStoMUBs}) and so each vector has norm $\sqrt{d}$. Therefore the inner product $\sum_{\ell=1}^d x_{\ell}\overline{y_\ell}$ of these vectors when $v=1$ has magnitude $\sqrt{d}$, by (\ref{EQ:MUBdef}). Now the inner product of $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$ in $L^R_d(\pi,v)$ is \begin{equation*} \begin{split} x_1\overline{y_1} + \cdots + vx_{\pi(j)}\overline{y_{\pi(j)}}+\cdots+\overline{v}x_{\pi(k)}\overline{y_{\pi(k)}}+\cdots+x_d\overline{y_d} \\ = \sum_{\ell=1}^d x_{\ell}\overline{y_\ell} + (v-1)x_{\pi(j)}\overline{y_{\pi(j)}}+(\overline{v}-1)x_{\pi(k)}\overline{y_{\pi(k)}}. \end{split} \end{equation*} This means that the corresponding concatenated vectors have inner product \begin{equation*}\begin{split} \sum_{\ell=1}^d x_{\ell}\overline{y_\ell} + (a+ib-1)x_{\pi(j)}\overline{y_{\pi(j)}}+(a-ib-1)x_{\pi(k)}\overline{y_{\pi(k)}}\hspace{60pt} \\ +\;\sum_{\ell=1}^d x_{\ell}\overline{y_\ell} + (2-a-ib-1)x_{\pi(j)}\overline{y_{\pi(j)}}+(2-a+ib-1)x_{\pi(k)}\overline{y_{\pi(k)}} \\ =2\sum_{\ell=1}^d x_{\ell}\overline{y_\ell}. \end{split} \end{equation*}
Since $\left|\sum_{\ell=1}^d x_{\ell}\overline{y_\ell}\right|=\sqrt{d}$, the concatenated vectors have inner product of magnitude $2\sqrt{d}$. \end{proof}
\begin{cor} Let $R$ be a $(d,d,d,1)$-RDS in an abelian group, and let $\pi\in S_d$. Then there are infinitely many choices of $a,b \in \mathbb{R}$ so that the vectors of $[L^R_{d}(\pi,a+ib) \hspace{5pt} L^R_d(\pi,2-a-ib)]$ form a set of $d^2$ equiangular lines in $\mathbb{C}^{2d}$. \label{COR:Lblocks} \end{cor}
Corollary~\ref{COR:Lblocks} follows from Lemma~\ref{LEM:2Lblockangles}, since for every dimension $d$ there are infinitely many choices of $a,b$ such that ${2}(b^2+(a-1)^2)=2\sqrt{d}$. Thus Corollary~\ref{COR:Lblocks} gives $d^2$ equiangular lines in $\mathbb{C}^{2d}$ whenever $d=p^r$ for some prime $p$, as we can construct blocks of the form $L_d^R(\pi,v)$ in these dimensions. So the list of dimensions for which one can construct $\Theta(d^2)$ equiangular lines in $\mathbb{C}^d$, which was previously known to include $d=3\cdot 2^{2t-1}-1$ \cite{deCaen}, and $d=p^r+1$ for $p$ prime \cite{Godsil-Roy, Konig}, can now be extended to include $d=2p^r$ for $p$ prime\footnote{A function $f$ from $\mathbb{N}$ to $\mathbb{R}^+$ is $\Theta(d^2)$ if there are positive constants $c$ and $C$, independent of $d$, for which $c d^2 \le f(d) \le C d^2$ for all sufficiently large~$d$.}.
Notice that the angle between each pair of distinct lines in Corollary~\ref{COR:Lblocks} is $\frac{1}{1+\sqrt{d}}$ (as is easily checked by normalizing these vectors). Using the special bounds calculated by Delsarte, Goethals and Seidel (\cite[Table I]{DGS-bounds} with $\alpha=\beta=\frac{1}{1+\sqrt{d}}$ and $n=2d$ over $\mathbb{C}$), we find that the function \begin{equation*} f(d) = \frac{d(2d+1)(2\sqrt{d}+d)^2}{d^2+4d+2\sqrt{d}} \end{equation*} is an upper bound on the number of vectors in $\mathbb{C}^{2d}$ having angle $\frac{1}{1+\sqrt{d}}$ between each pair of distinct vectors. In the range $d \ge 1$ we have $2d^2 < f(d) \le 4d^2$, and the larger value $4d^2$ is attained exactly at $d=4$ and the smaller value $2d^2$ is the asymptotic value of $f(d)$. Since Corollary~\ref{COR:Lblocks} gives only $d^2$ equiangular lines with the specified angle $\frac{1}{1+\sqrt{d}}$, we ask the following questions:
\begin{question} Can we extend the set of $d^2$ equiangular lines in $\mathbb{C}^{2d}$ given in Corollary~$\ref{COR:Lblocks}$ by adding some or all of the vectors of additional blocks of the form $L_d^R(\pi,v)$ for suitable $\pi$ and $v$, and if so what is the largest possible number of resulting equiangular lines? \label{Q:LblockExtend} \end{question}
\begin{question} For large $d$, can we achieve the asymptotic bound of $2d^2$ equiangular lines having angle $\frac{1}{1+\sqrt{d}}$ in $\mathbb{C}^{2d}$? \label{Q:LblockAsymp} \end{question}
The value $d=4$ in Question~\ref{Q:LblockExtend} is of special interest, because it is the only value of $d \ge 1$ for which $f(d) = 4d^2$ (so that there is a possibility of extending the $d^2$ equiangular lines in $\mathbb{C}^{2d}$ to a maximum-sized set of size $4d^2$. We can alternatively identify the candidate value $d=4$ by equating the specified angle $\frac{1}{1+\sqrt{d}}$ with the angle $\frac{1}{\sqrt{2d+1}}$ given by Proposition~\ref{PROP:angle}.) It turns out that when $d=4$ we can indeed combine several blocks of the form $L_4^d(\pi,v)$ to form a set of 64 equiangular lines in $\mathbb{C}^8$.
\begin{ex} Let $R$ be the RDS of Example~$\ref{EX:RDS4}$ and let $\pi=[1,3,4,2]$. The following is a set of $64$ equiangular lines in $\mathbb{C}^8$: $${\renewcommand{1.1}{1.1} \begin{array}{@{[}r@{(\pi,\;}r@{\,)\, \hspace{5pt}}r@{(\pi,\; }r@{\,)\,]\hspace{10pt}[}r@{(\pi,\;}r@{\,)\, \hspace{5pt}}r@{(\pi,\; }r@{\,)\,]}} L^R_4 & 2+i & L^R_4 & -i & L^R_4 & -i & L^R_4 & 2+i \\[2pt] L^R_4 & -1+2i & -L^R_4 & 1 & L^R_4& 1 & -L^R_4 &-1+2i \end{array} }$$ The lines are given explicitly by the following vectors:
$$ \fontsize{9}{9.75}\selectfont \begin {array}{@{(}*{4}{@{\!}c@{}}*{4}{@{\,}c@{\;}}@{\!)}} 2+i&1&1&1&-i&1&1&1 \\ \noalign{
}2+i&1&-1&-1&-i&1&-1&-1\\ \noalign{
}2+i&-1 &1&-1&-i&-1&1&-1\\ \noalign{
}2+i&-1&-1&1&-i&-1&-1&1 \\ \noalign{
}1&1&-1+2\,i&-i&1&1&1&-i\\ \noalign{
}1&1&1 -2\,i&i&1&1&-1&i\\ \noalign{
}1&-1&-1+2\,i&i&1&-1&1&i \\ \noalign{
}1&-1&1-2\,i&-i&1&-1&-1&-i\\ \noalign{
}1&i &1&1-2\,i&1&i&1&-1\\ \noalign{
}1&i&-1&-1+2\,i&1&i&-1&1 \\ \noalign{
}1&-i&1&-1+2\,i&1&-i&1&1\\ \noalign{
}1&-i& -1&1-2\,i&1&-i&-1&-1\\ \noalign{
}1&-1+2\,i&i&-1&1&1&i&-1 \\ \noalign{
}1&-1+2\,i&-i&1&1&1&-i&1\\ \noalign{
}1&1-2 \,i&i&1&1&-1&i&1\\ \noalign{
}1&1-2\,i&-i&-1&1&-1&-i&-1 \\[15pt] -1+2\,i&1&1&1&-1&-1&-1&-1 \\ \noalign{
}-1+2\,i&1&-1&-1&-1&-1&1&1\\ \noalign{
}-1+ 2\,i&-1&1&-1&-1&1&-1&1\\ \noalign{
}-1+2\,i&-1&-1&1&-1&1&1&-1 \\ \noalign{
}1&1&-2-i&-i&-1&-1&-i&i\\ \noalign{
}1&1&2+ i&i&-1&-1&i&-i\\ \noalign{
}1&-1&-2-i&i&-1&1&-i&-i \\ \noalign{
}1&-1&2+i&-i&-1&1&i&i\\ \noalign{
}1&i&1&2+ i&-1&-i&-1&i\\ \noalign{
}1&i&-1&-2-i&-1&-i&1&-i \\ \noalign{
}1&-i&1&-2-i&-1&i&-1&-i\\ \noalign{
}1&-i&- 1&2+i&-1&i&1&i\\ \noalign{
}1&-2-i&i&-1&-1&-i&-i&1 \\ \noalign{
}1&-2-i&-i&1&-1&-i&i&-1\\ \noalign{
}1&2+i& i&1&-1&i&-i&-1\\ \noalign{
}1&2+i&-i&-1&-1&i&i&1 \end {array} \hspace{12pt} \begin {array}{@{(}*{4}{@{\,}c@{\;\,}}*{4}{@{\!}c@{}}@{)}} -i&1&1&1&2+i&1&1&1 \\ \noalign{
}-i&1&-1&-1&2+i&1&-1&-1\\ \noalign{
}-i&-1& 1&-1&2+i&-1&1&-1\\ \noalign{
}-i&-1&-1&1&2+i&-1&-1&1 \\ \noalign{
}1&1&1&-i&1&1&-1+2\,i&-i\\ \noalign{
}1&1&- 1&i&1&1&1-2\,i&i\\ \noalign{
}1&-1&1&i&1&-1&-1+2\,i&i \\ \noalign{
}1&-1&-1&-i&1&-1&1-2\,i&-i\\ \noalign{
}1&i &1&-1&1&i&1&1-2\,i\\ \noalign{
}1&i&-1&1&1&i&-1&-1+2\,i \\ \noalign{
}1&-i&1&1&1&-i&1&-1+2\,i\\ \noalign{
}1&-i& -1&-1&1&-i&-1&1-2\,i\\ \noalign{
}1&1&i&-1&1&-1+2\,i&i&-1 \\ \noalign{
}1&1&-i&1&1&-1+2\,i&-i&1\\ \noalign{
}1&-1& i&1&1&1-2\,i&i&1\\ \noalign{
}1&-1&-i&-1&1&1-2\,i&-i&-1 \\[15pt] 1&1&1&1&1-2\,i&-1&-1&-1 \\ \noalign{
}1&1&-1&-1&1-2\,i&-1&1&1\\ \noalign{
}1&-1& 1&-1&1-2\,i&1&-1&1\\ \noalign{
}1&-1&-1&1&1-2\,i&1&1&-1 \\ \noalign{
}1&1&i&-i&-1&-1&2+i&i\\ \noalign{
}1&1&-i&i &-1&-1&-2-i&-i\\ \noalign{
}1&-1&i&i&-1&1&2+i&-i \\ \noalign{
}1&-1&-i&-i&-1&1&-2-i&i\\ \noalign{
}1&i&1& -i&-1&-i&-1&-2-i\\ \noalign{
}1&i&-1&i&-1&-i&1&2+i \\ \noalign{
}1&-i&1&i&-1&i&-1&2+i\\ \noalign{
}1&-i&-1& -i&-1&i&1&-2-i\\ \noalign{
}1&i&i&-1&-1&2+i&-i&1 \\ \noalign{
}1&i&-i&1&-1&2+i&i&-1\\ \noalign{
}1&-i&i&1 &-1&-2-i&-i&-1\\ \noalign{
}1&-i&-i&-1&-1&-2-i&i&1 \end {array} $$ Notice that this is also an example of almost flat equiangular lines. \label{EX:LblockLines} \end{ex}
It is not the case that for every choice of permutation $\pi$, the set $[L_4^R(\pi,a+ib)\;L_4^R(\pi,2-a-ib)]$ can be extended to 64 equiangular lines in $\mathbb{C}^8$. In fact, the ability to extend the set of Lemma~\ref{LEM:2Lblockangles} to a set of maximum size is highly sensitive to the choice of additional blocks. Notice also that in Example~\ref{EX:LblockLines}, the lower blocks of vectors do not follow the exact structure of Lemma~\ref{LEM:2Lblockangles}, but instead are of the form $[L_4^R(\pi,i(a+ib))\;-L_4^R(\pi,i(2-a-ib))]$. These subtle differences indicate that answering Question~\ref{Q:LblockExtend} (for $d > 4$) and Question~\ref{Q:LblockAsymp} might involve careful parameter choices as well as small variations in the form of the blocks. This discussion also motivates a final question:
\begin{question} Is there a variant of the block construction of Lemma~$\ref{LEM:2Lblockangles}$ in $\mathbb{C}^d$ from which we can construct $d^2$ equiangular lines in $\mathbb{C}^D$ with angle $\frac{1}{\sqrt{D+1}}$ for some $D > d$ (so that the resulting lines have the correct angle required for a maximum-sized set of size $D^2$)? \label{Q:AlterLblocks} \end{question}
\section{Conclusion} \label{S:conclusion}
We have seen that MUBs and sets of equiangular lines are more deeply intertwined than previously recognized. We believe that the new constructions presented here, and the questions posed, open new avenues for exploring the existence of maximum-sized sets of equiangular lines.
\end{document}
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arXiv
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Preliminary study of the production of metabolites from in vitro cultures of C. ensiformis
Juan F. Saldarriaga1,
Yuby Cruz1 &
Julián E. López2
Canavalia ensiformis is a legume native to Central and South America that has historically been a source of protein. Its main proteins, urease, and lectin have been extensively studied and are examples of bioactive compounds. In this work, the effect of pH and light effects on the growth of C. ensiformis were analyzed. Also, the bioactive compounds such as phenols, carotenoids, chlorophyll a/b, and the growth of callus biomass of C. ensiformis from the effect of different types of light treatments (red, blue and mixture) were evaluated. Likewise, the antioxidative activity of C. ensiformis extracts were studied and related to the production of bioactive compounds. For this, a culture of calluses obtained from seeds were carried out. For the light experiments, polypropylene boxes with red, blue, combination (1/3, 3/1 and 1/1 R-B, respectively) lights and white LED were used as control. In each treatment, three glass containers with 25 ml of MS salts containing 0.25 g of fresh callus were seeded.
The results have shown that the pH of the culture medium notably affects the increase in callogenic biomass. It shows that the pH of 5.5 shows better results in the callogenic growth of C. ensiformis with an average increase of 1.3051 g (198.04%), regarding the initial weight. It was found that the pH 5.5 and the 1/3 R-B LED combination had higher production of bioactive compounds and better antioxidant activity. At the same time, the red-light treatment was least effective.
It was possible to find the ideal conditions of important growth under conditions of pH and light of C. ensiformis. Likewise, it is evaluated whether the production of compounds of interest, such as phenolic compounds and carotenoids, occurs under these conditions. The highest production of calluses occurs in the 1/3 R-B LED combined light treatment, which showed a significant increase in biomass, followed by B. From this study, it could be demonstrated that C. ensiformis produces compounds such as phenols and carotenoids in vitro culture that are essential for the antioxidant activity of the plant.
In biochemical research, Canavalia ensiformis has been historically a promising source of protein [1]. Urease and lectin from C. ensiformis, are widely studied proteins and notable examples of the importance of bioactive compounds of this plant species [2,3,4]. In plants, the amount of these compounds depends mostly on the plants growing conditions (in vivo or in vitro), on the photoperiod to which it is exposed and on other important factors such as planting density, nutrient supply, temperature, etc. [5]. Therefore, the absorption of nutrients in plants grown in vitro are affected by several factors, such as gelling agents [6], the breakdown of carbohydrates and chelating agents [7]. Furthermore, it has been reported that light is a factor that can affect the production of these compounds [8,9,10], thereby inducing physiological changes in plants [11].
Extracellular pH can affect ion absorption and, at the same time, creates ionic competitions [12]. Studies have shown that low pH levels are associated with inhibition of cation absorption, while anion uptake may be slightly influenced or not influenced [12]. In particular, attention has focused on the effect of pH on nitrogen uptake by roots and on how the predominant form of nitrogen (i.e., NH4+ or NO3) in the nutrient solution influences the absorption of the other ions [12,13,14].
Light is the origin of the direct source of energy for many plants as IT controls the growth rate in processes such as phototropism, photosynthesis, photomorphogenesis, among others that affect the metabolism related to pigments [15,16,17]. Four of the most important pigments are chlorophyll a (Chla), chlorophyll b (Chlb), phytochrome PR (red light) and PFR (far-red light), this according to Zhou et al. [16] and Wright [18]. These mainly absorb blue (400–500 nm), red (580–680), and far-red (690–800 nm) light. According to Carvalho et al. [19], the quality of these lights affects the accumulation of photosynthetic pigments in the leaves, which can increase absorption of light in low light conditions or act as screening pigments, and free radical scavengers in high light conditions. These effects have been extensively studied due these lengths are absorbed by photosynthetic pigments, and their important impact occurs in the development of plants [20, 21]. Studies have found that the wavelengths of R and B always coexist in natural light environments and that the optimal mixture of these differs from the plant species. For example, for the strawberry, a 7/3 ratio has been found [22]. In contrast, for rapeseed, a ratio 1/3 [23] of the RB mix has been reported.
Chen et al. [24], argue that blue light is necessary during plant growth for normal photosynthesis and that the regulated responses of this plant quantitatively resemble those of irradiation intensity. Also, it was shown that plants cultivated in B have a higher proportion of Chl a/b, more significant activity of Rubisco, and higher activity of transport of photosynthetic electrons than plants grown in R [25,26,27,28]. Also, B can trigger photomorphogenesis processes in plants and provides enough energy through photosynthesis to maintain normal growth and development [20]. Costa et al. [29] indicated that B is indispensable for photo-acclimatization and protection of diatoms at high light intensities.
On the other hand, plants that were grown with R exhibit a significantly lower Chl a/b ratio, lower rates of photosynthetic CO2 fixation, and total plant biomass than plants grown with white light or a combination of R and B [28, 30, 31]. According to Wang et al. [28], plants are sensitive to their light environment not only because the light is the sole source of energy but also due to its effect on growth and development. An example of the above takes place in autumn, spring, and winter, where the shortening of the light time and the considerable fluctuation of irradiation are quite serious problems for the development of plants [28]. Plants that grow under low light intensity are more sensitive to photoinhibition caused by exposure to increase light irradiation [32].
In vitro tissue culture has been widely used for rapid plant propagation and obtaining bioactive compounds from cell culture [33]. There are previous studies of a successful induction and propagation of calluses of C. ensiformis under in vitro conditions [34]. However, the production of bioactive compounds such as photosynthetic pigments, phenols, and carotenoids from cell culture has not been evidenced for C. ensiformis. The main objective of this work was to evaluate different types of pH of the culture medium to determine the optimum in which the planting of C. ensiformis achieves adequate growth, and the production of metabolites of interest that can be used in its crops.
Furthermore, starting from the optimal pH, the effect of different light treatments (red, blue, and combination), the production of chlorophyll, phenols, and carotenoids on the growth of callus biomass of C. ensiformis were evaluated. Also, the antioxidant activity of the extracts of C. ensiformis callus were analyzed, and these have related to the production of bioactive compounds. In this way, it was possible to determine under which light conditions, R, B, or combination present the best production of bioactive compounds and antioxidant activity.
The results have shown that the pH of the culture medium notably affects the increase in callogenic biomass. Table 1 shows that the pH of 5.5 presents better results in the callogenic growth of C. ensiformis, with an average increase of 1.3051 g (198.04%) after 30 days. While the pH of 6.0 evidences the least results of an increase in callogenic biomass. And finally, the pH of 5.0 and 5.7 show very similar results of biomass increase with approximately 104% on average for both treatments.
Table 1 Callogenic biomass variation of C. ensiformis at different cultivation pH
Table 2 shows the analysis of variance components for the effect of pH on callus growth, showing that pH is the most important factor in the effect of growth (p-value 0.005). It is observed as in the data in Table 1, that it is the pH of 5.5 that presents a positive effect on callus development. ANOVA shows that pH is the most significant factor and that it is important in callus growth. Of these, pH 5.5 is the one with the highest percentage increase since planting after 30 days of planting. While the pH of 6.0 and 4.5 are the treatments that show lower results of biomass increase. With these results, it is shown that the optimal growth for a significant increase of biomass in vitro cultures of C. ensiformis, is with a pH of 5.5.
Table 2 Analysis of variance for the effect of pH on growth
Df: degrees of freedom.
From these tests, the effect of the growth of calluses of C. ensiformis were evaluated in a medium with pH 5.5 but with variation in the intensity of light.
Effect of light on callus growth
Figure 1 shows the effect of light on the growth of the C. ensiformis callus at 30 days, where it is observed that treatments with B have a significant effect on the increase of weight of callus in this plant, having the highest production with the 1/3 R-B combination. While the treatments in which the R is the main one, the increase in callus weight is smaller, being less than 0.35 g in all treatments.
Effect of lights on weight increase of callus
Table 3 shows the statistical analysis regarding the increase of weight in calluses. According to the Tukey test, the combination of 1/3 R-B LED represents an average increase of 1.08 g. In contrast, white light and a combination of 3/1 R and B LED with 0.28, and 0.30 g respectively are below average. These results are engaging in the cultivation of C. ensiformis, because the rapid growth of calluses under these conditions helps to have enough biomass for the effects of a scaled process, or even on industrial levels.
Table 3 Statistical analysis of the effect on increasing callus weight
Total phenols content
In Fig. 2, the effect of light on the production of phenolic/polyphenolic compounds is shown. It is observed that the production of phenolic compounds increased in all treatments with B, compared to W (white light). In contrast, treatment with R resulted in a much lower mean than that of W. These results may be related to the production of calluses, because the combination of 1/3 R-B LED produced more calluses, results in higher accumulation of total phenolics.
Effect of light on the production of phenolic/polyphenolic compounds
After applying statistical analysis and a Tukey test to the effect of light on the production of phenolic compounds, it was observed that there is a higher production of phenolic compounds when treatment B is in equal or more significant proportion than R. The principal effect of the light occurs in the combined light treatments 1/3 R-B LED and 1/1 R-B LED with an average phenolic production of 11.52 and 10.75 mg GAE/g FW, respectively.
Chlorophyll a and b content
The effect of light on the production of both Chla and Chlb is shown in Fig. 3. It is observed that in both treatments, the white light has a production above the average of all the Chlb treatments, while the Chla is given below the average of all the treatments. However, the highest production is of Chla, which reaches an average of 0.32 mg g− 1 biomass while the Chlb is 0.24 mg g− 1 biomass.
Effect of light on chlorophyll production, (a) Chla and (b) Chlb
The combination of 1/3 R-B LED and the treatment B has higher concentrations of Chla and Chlb (Fig. 3a and b), this is because it has been widely proven that B favors photosynthesis processes. An interesting effect is observed in the production of Chla compared to Chlb in the 1/3 R-B and B treatment in which the highest chlorophyll productions occur because Chla has a higher production with 0.5 mg g− 1 biomass compared to Chlb of 0.3 mg g− 1 biomass. In the case of Chlb, there is a higher standard deviation and a more extensive range. Figure 3 shows that the three best treatments to produce Chla and Chlb are the combination 1/3 R-B LED, B LED, and W LED. Of the above, the treatment of 1/3 R-B LED is the one with the highest concentration of both chlorophyll a and b with an average of 0.49 and 0.31 mg g− 1 biomass, respectively. Moreover, treatment with R, has a lower concentration of chlorophyll both Chla and Chlb, demonstrating that exposing C. ensiformis only to R to produce Chla and Chlb is not adequate.
Carotenoids content
Regarding the production of carotenoids, observed in Fig. 4, the R is the one that presents the least number of carotenoids and, again, as in the previous results, it is the light with the lowest production response of bioactive compounds. In the case of the 1/3 R-B combination, it is the best response in the production of carotenoids with an average of 0.25 mg g− 1 biomass. This combination is the one that had the best response to the production of compounds of interest and also in the callogenic production, even much greater than that of the control with W. Similarly, only B presented good results, all of them above average in each of the parameters evaluated.
Effect of lights on carotenoid production
Antioxidant activity capacity
According to the evaluation carried out on the antioxidant capacity of C. ensiformis callus extracts from different light treatments, it could be evidenced that B is the one that has the best effect on this parameter evaluated. Conversely, the R is the one with the lowest activity, with an average of 30.67 μmol TE g− 1 FW. The same behavior is observed in the production of phenols and carotenoids concerning the higher antioxidant activity, in which the B in equal or greater quantity than R favors a better production of bioactive compounds (Table 4). Similarly, Table 4 shows that the standard deviation of all treatments is very low compared to the production of Chla and Chlb.
Table 4 Statistical analysis of the effect on antioxidant activity
Principal component analysis PCA
The PCA analysis offers a graphical representation that simplifies the visualization and understanding of data and variables. The analysis shows a relationship between the factors and the different parameters measured. An interaction between Chla and Chlb is observed, in the same way, there is a correlation between the content of phenols and carotenoids (Fig. 5).
Principal component analysis for the physiological response of C. ensiformis
From the PCA analysis, it was found that the first two components explained 92.37% of the overall variation. A high positive correlation was observed between the antioxidant activity and the content of phenols and carotenoids. Callus weight had also shown a positive correlation with the chlorophyll content. The treatments mainly influence this with B and the combination of R and B. Likewise, Table 5 shows the analysis of six principal components showing the percentage of each one.
Table 5 Principal components analysis (PCA)
It could be verified that the pH is an important factor in the increase of the biomass of C. ensiformis, this may be because it affects the intake of minerals, as well as the activity or metabolism of the phytohormones supplied in the medium [35, 36]. In the literature, there is little or no work on the effect of pH on in vitro cultures of C. ensiformis, considering that this parameter is essential and that depending on it, the plant will take nutrients better. Its development will also be optimum. Similarly, it has been proven that pH has a profound effect on crop productivity as primary metabolism and biosynthetic pathway enzymes are affected by culture media pH [37,38,39,40].
Studies have evaluated on other plants the pH effect on the growth of plants such as tomatoes, Catharanthus roseus, Withania somnifera, carrot, Bacopa monnieri, ectomycorrhizal fungi, among others [37, 41,42,43,44,45,46,47]. For this reason, the optimal pH for the in vitro culture of C. ensiformis is 5.5. From this, the preliminary production of metabolites such as phenolic compounds and carotenoids were evaluated, as well as antioxidant activity.
In the case of C. ensiformis to date, there are few studies on the effect of light on callogenic growth, the production of phenols, chlorophyll a/b, carotenoids, and antioxidant activity. In this study, it was found that B positively affects callus weight, and these results again demonstrate that the light with this wavelength is necessary during plant growth for normal photosynthesis to occur. Similarly, the data accord with other authors who evaluated this light in other plant species [24]. The results of this study show that the considerable increase occurs in a combination of 1/3 R-B LED, and this agrees with other authors who argue that these wavelengths are mostly absorbed by photosynthetic pigments giving an important impact on development [21, 48]. While other authors have found different R-B ratios, in the case of this study (the best 1/3 R-B ratio), which agrees with the results of Li et al. [23] for rapeseed, regarding the biomass increased. Likewise, this behavior is similar to what was argued by Ahmed et al. [49], which says that this type of spectra promotes plant growth, photosynthetic velocity, and biomass accumulation.
Concerning the production of chlorophyll, the same results were exposed as in other studies where it was found that with R the Chl a/b production is significantly low compared to B and W [28, 31]. In the case of B, this work found that there is a production equal or higher than the 1/3 R-B LED combination. These results coincide with other studies in which it was found that this favors the production of Chl a/b [25,26,27]. With these results, those argued by other authors are proven that B is essential for the development of chloroplasts, stoma opening, and photomorphogenesis, as well as regulating the biosynthesis of chlorophyll and anthocyanin [50,51,52,53]. Also, it was proven with the results that a mixture of lights is necessary for the normal growth of the plant since it favors normal photosynthesis, and besides, the response of the plant can quantitatively resemble those found in the intensity of the radiation [24].
C. ensiformis shows a significant association between antioxidant capacity, phenols, and carotenoids. These results are similar to those found by Hoffmann et al. [54], which have shown that B illumination favors the potential accumulation of carotenoids. Figure 6 is observed as the correlation between phenolic compounds and the antioxidant capacity, which is directly proportional and accords with other authors who report within the chemical compounds with antioxidant capacity, are the phenolic compounds [55]. Kapoor et al. [10] observed a higher production of phenolic compounds in in vitro cultured corns of the Rhodiola imbricata species exposed to B, as well as an increase in the antioxidant capacity of callus extracts. According to the results, these phenolic compounds, that in C. ensiformis reach average values of up to 11.52 mg eq. AG/g biomass, intensely contribute to the defense mechanisms against biotic and abiotic stress [56].
Correlations between antioxidant activity and phenolic and carotenoid compounds (n = 18)
Some authors have linked the phenolic content with stress tolerance, either contributing to indirect light protection or participating directly as antioxidants acting as free radical scavengers due to their reducing properties [57, 58]. Also, it has proven that phenolic compounds can act as hydrogen donor agents or singlet oxygen extinguishing electrons and metal chelators [59, 60].
Figure 6 shows the direct connection between the carotenoid and phenolic contents with the antioxidant activity. According to some authors, this happens because compounds have a positive effect on the antioxidants activity [61, 62]. Furthermore, antioxidants are species that can protect organisms from damage caused by oxidative stress induced by free radicals and in this work, it was found that the combination 1/3 R-B LED is the best response to the production of the compounds evaluated [63]. The presence of these antioxidant compounds is important in C. ensiformis because it favors the production of vital proteins such as the urease and lectin of the plant. These results demonstrate that in vitro culture with a combination of R and B in 1/3 proportion favors the production of different compounds such as chlorophyll a/b, phenolic contents, and carotenoids that help in antioxidant activity of the plant.
With this work, it was possible to verify that the initial pH in the culture medium is a fundamental factor in giving the proper nutrients to calluses and in the production of compounds of interest such as phenols, carotenoids, and antioxidant activity. Due to that, it has been proven that these are fundamental in the growth of plants and that it affects different structures of it [64, 65]. pH and light are important because they are a significant step for the in vitro production of compounds of importance from this plant, such as urease and lectin. Therefore, it is necessary to carry out new works to investigate how these factors can improve the production of these proteins and how their products could be increased through in vitro cultures.
It was possible to find the ideal conditions of highest growth under conditions of pH and light of C. ensiformis, to evaluate whether under these conditions the production of compounds of interest such as phenolics and carotenoids occurs. In addition to observing the antioxidant activity as a primary factor in the response of the plant to external factors such as the culture media in the callogenic multiplication.
Determining the in vitro culture conditions of C. ensiformis represents a valuable instrument for the study and production of important metabolites such as phenols and carotenoids. Its independence from environmental conditions allows for the continuous supply of materials, as well as the use of several strategies to stimulate the specific production of these metabolites. The results of this study show that the pH of the culture medium influences the growth of C. ensiformis. In the statistical analysis, it can be seen how the pH of 5.5 shows a better response to callus growth concerning the other four pH evaluated.
The highest production of calluses occurs in the 1/3 R-B LED combined light treatment, which showed a significant increase in biomass, followed by B, while the least effective was with W. However, the R treatment, combination 1/1, and 3/1 R-B LED, show similar behaviors. These results are because the 1/3 RB LED combination is repeated in all the bioactive compounds analyzed in this study, which makes this combination the best treatment for obtaining a large amount of biomass and compounds of interest such as phenols, carotenoids, and chlorophyll a/b.
From this study, it could be demonstrated that C. ensiformis not only has a high production of compounds such as urease and lectin but also of compounds such as phenols and carotenoids that are essential for the antioxidant activity of the plant. The in vitro culture of this could be promising to obtain the compounds produced by C. ensiformis and it could be of interest at an industrial level as a source of protein.
With these results, the efficiency of B in carotenoid production and photosynthetic activity in C. ensiformis was verified. These contrast with the results found by other authors and show once again that the combination of R and B with a higher proportion of B favors a long ratio between oxidizing capacity, phenols, and carotenoids produced in C. ensiformis.
Culture conditions
C. ensiformis was grown in glass jars (55 cm in diameter by 70 cm high) in the Research Center in Environmental Engineering, Universidad de los Andes, from seeds (Semillas-Camposeeds, Bogotá, Colombia). Callogenic induction and callus propagation were achieved using MS salts and vitamins [66] as a basal medium (Thermo Fisher, New Jersey, USA), and supplemented with 3.0% (w/v) sucrose (Merck, New Jersey, USA)) and 5 g l− 1 agar-agar (Merck, New Jersey, USA) callus propagation medium [34].
pH experiments
To determine the effect of pH on callus growth, these were sowed in MS salts (described Section 2.1). The initial pH of the medium was adjusted with 1 N NaOH (Merck, New Jersey, USA) and 0.1 N HCl (Merck, New Jersey, USA), then they were autoclaved (121 °C, 18 PSI, 20 min). In this work, the pH between 4.5 and 6.0 was used. For the initial weight, a single range was taken, and it is higher than 0.2500 g because it is where the important growth effect was observed for the initial mass. All experiments were performed in quadruplicate. Two hundred eighty-eight glass containers were taken, sterilized at 121 °C, 18 PSI (lbf in− 2) for 20 min, weighed empty, and then with the explant, they were immediately cultured at 23 ± 1 °C, with continuous white light for 30 days. After this time, the calluses were weighed again, and the biomass growth over time was obtained by weight difference.
LED experiments
For LED (light-emitting diode) experiments, black painted polypropylene boxes (33 X 52 X 31 cm) were used with red (R; peak at 657 nm), and blue (B; peak at 455 nm) LEDs (ILUMAX, Shenzhen, China). Six light treatments were applied, which were: 100% white light (control) (LED W) and several red (R) to blue (B) ratios, as follows: 100% B, 100% R, 25% R and 75% B (1/3 R-B), 75% R and 25% B (3/1 R-B), and 50% R and 50% B (1/1 R-B), and LED W. In each treatment, three glass containers with 25 ml of MS salts containing 0.25 g of fresh callus were seeded. Calluses were grown for 60 days, until the callogenic mass increased and exposed to light treatments for 30 days. The laboratory conditions were a relative humidity of 52%, with an average temperature of 20/16 °C (light/day), the plants were exposed to a 12 h/12 h photoperiod. Then, callus biomass was analyzed and used for the determination of total phenolics, chlorophyll, carotenoids, and antioxidant activity.
Total phenolics
The content of total phenolic in callus extract was determined by the Folin-Ciocalteu method [67]. Fresh calluses (200 mg) were extracted with methanol using a Soxhlet apparatus. 1 mL of methanolic callus extract was mixed with 5 ml of Folin-Ciocalteu reagent (Sigma-Aldrich, St. Louis, United States) (diluted 10-fold) and 4 ml of sodium carbonate solution (7500 mg l− 1). Then, the absorbance at 765 nm was measured after 1 h. The calibration curve was prepared with methanolic gallic acid solutions, which were mixed with the same reagents described above, and after 1 h the absorbance was measured. Total phenolic content in callus methanolic extracts was expressed in Gallic Acid Equivalents (GAE) by the equation:
$$ TP=\frac{\left(c\ast V\right)}{FW} $$
Where, TP is the total content of phenolic compounds expressed like mg GAE g− 1 FW, c is the concentration of gallic acid deduced from the calibration curve (mg ml− 1), V is extract volume (ml), and FW is the weight of the fresh callus (g).
Chlorophyll and carotenoids determination
The content of chlorophyll and carotenoids was quantified in acetone extract. In other words, 100 mg of fresh callus were crushed in 5 ml of chilled acetone (80% v/v). Then, the extract was centrifuged at 2500 rpm for 5 min, and absorbance of the supernatant was read at 660, 645, and 470 nm. The content of Chla, Chlb, and carotenoids was calculated in mg g− 1 FW biomass using the following equations, according to Wellburn [68].
$$ CHla=\left\{\frac{\left[12.21\left({A}_{660}\right)-2.81\left({A}_{645}\right)\right]V}{\left(1000\ast FW\right)}\right\} $$
$$ CHlb=\left\{\frac{\left[20.13\left({A}_{645}\right)-5.03\left({A}_{660}\right)\right]V}{\left(1000\ast FW\right)}\right\} $$
$$ Carotenoid=\left\{\frac{\left[1000\left({A}_{470}\right)-3.27(CHla)-104\left(\mathrm{C} Hlb\right)\right]V}{\left(1000\ast FW\right)}\right\} $$
Where A660, A645, and A470 are the value of absorbance in nm, V is the extract volume, and FW is the weight of the fresh callus.
The antioxidant activity of callus methanolic extract was determined by 2,7′- dichlorodihydrofluorescein diacetate (DCFH) probe, which reacts indiscriminately with reactive oxygen species (ROS) and reactive nitrogen species (RNS) generated by the compound azo, 2,2′-diazobis (2-amidinopropane dihydrochloride) (AAPH) in an aqueous medium and forms the fluorescent compound 2,7-dichlorofluorescein (DCF). The antioxidants in the samples capture ROS and RNS and reduce the fluorescence emitted by DCF. 50 μl of a 0.3 M AAPH solution, 50 μl of a 2.4 mM DCFH ethanolic solution, 2850 μl of 75 mM phosphate buffer (pH 7.4), and 50 μl of the methanolic callus extract (described Section 2.4), which was obtained by mixing 0.3 g of macerated fresh callus using liquid nitrogen and mortar, with 2 ml of 10 mM phosphate buffer (pH 7.0). The intensity of fluorescence emitted during the first 10 min was read and compared with the intensity emitted in the absence of the sample (λ excitation: 326 nm, a λ emission: 432 nm and 10 nm slit). The results are expressed as mg μmol of Trolox Equivalents (TE) per g of fresh callus biomass by constructing a standard curve using different concentrations of TROLOX® [69].
All statistical analysis were performed using two softwares. Statgraphics 18 centurion software was used for the analysis of the effect of pH on callus growth. For the analysis of the effect of light on the growth of calluses and the effect of the production of bioactive substances, the software JMP-Pro version 13.1.0 was used. All experiments were performed in quadruplicate.
All data generated and analyzed during this study are included in the published article.
A660, A645, A470 :
Value of absorbance (nm)
AAPH:
Azo, 2,2′-diazobis (2-amidinopropane dihydrochloride)
ANOVA:
Concentration of gallic acid (mg ml− 1)
Chla :
Chlb :
DCF:
2,7-dichlorofluorescein
DCFH:
2,7′- dichlorodihydrofluorescein diacetate
df:
FW:
Weight of the fresh callus (g)
GAE:
Gallic Acid Equivalents
PCA:
PFR:
Far-red light
RNS:
Reactive nitrogen species
TE:
Trolox Equivalents
TP:
Total content of phenolic compounds expressed like mg (GAE g− 1 FW)
V:
Extract volume (ml)
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Caprioli G, Maggi F, Bendif H, Miara MD, Cinque B, Lizzi AR, et al. Thymus lanceolatus ethanolic extract protects human cells from t-BHP induced oxidative damage. Food Funct. 2018;9(7):3665–72.
Jabri Karoui I, Msaada K, Abderrabba M, Marzouk B. Bioactive compounds and antioxidant activities of ThymeEnriched refined corn oil. J Agr Sci Tech. 2016;18:79–91.
Čanadanović-Brunet JM, Djilas SM, Ćetković GS, Tumbas VT, Mandić AI, Čanadanović VM. Antioxidant activities of different Teucrium montanum L. extracts. Int J Food Sci Technol. 2006;41(6):667–73.
Buchanan BB, Gruissem W, Jones RL. Editores. Biochemistry & Molecular Biology of plants. 1st ed. Hoboken: Wiley; 2002. p. 1408.
Pallardy SG. Physiology of Woody plants. Edición: 3. Amsterdam: Academic Press; 2007. p. 464.
Murashige T, Skoog F. A revised medium for rapid growth and bio assays with tobacco tissue cultures. Physiol Plant. 1962;15(3):473–97.
Misra P, Toppo DD, Gupta N, Chakrabarty D, Tuli R. Effect of antioxidants and associate changes in antioxidant enzymes in controlling browning and necrosis of proliferating shoots of elite {Jatropha} curcas {L}. Biomass Bioenergy. 2010;34(12):1861–9.
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Zapata K, Cortes FB, Rojano BA. Polifenoles y Actividad Antioxidante del Fruto de Guayaba Agria (Psidium araca). Información tecnológica. 2013;24(5):103–12.
This work was carried out with financial support from the Department of Civil and Environmental Engineering of the Universidad de los Andes. This publication was partially made possible by the call for proposals CI-0120: "Publish your new knowledge or expose your new creations" from the Office Vice President for Research and Creation at Universidad de los Andes.
This research received no external funding.
Dept. of Civil and Environmental Engineering, Universidad de los Andes, Carrera 1Este #19A-40, Bogotá, Colombia, 111711
Juan F. Saldarriaga & Yuby Cruz
Dept. of Environmental Engineering, Universidad de Medellín, Carrera 87 #30-65, Medellín, Colombia, 050026
Julián E. López
Juan F. Saldarriaga
Yuby Cruz
JFS, YC, and JEL planned and designed the experiments, JFS and JEL carried out the experiments, JFS, YC, and JEL wrote the manuscript. All authors have read and approved the entire manuscript.
Correspondence to Juan F. Saldarriaga.
12896_2020_642_MOESM1_ESM.docx
12896_2020_642_MOESM2_ESM.xlsx
Saldarriaga, J.F., Cruz, Y. & López, J.E. Preliminary study of the production of metabolites from in vitro cultures of C. ensiformis. BMC Biotechnol 20, 49 (2020). https://doi.org/10.1186/s12896-020-00642-x
DOI: https://doi.org/10.1186/s12896-020-00642-x
C. ensiformis
LED red
Bioactive compounds
In vitro culture
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CommonCrawl
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How to convert from power signal received by Antenna to voltage signal?
I am considering wireless transmission. Let us define a cosine signal with frequency 2.4 GHz, i.e. $$s(t) = A\cos(2\pi ft),$$ where $$f=2.4\times10^9.$$
Let us assume that when TX transmits the signal s(t) of power 50 W, RX receives the signal of power 0.5 W (that is, 49.5 W is attenuated in the air.) Saying more detail, transmitted signal and received signal are, respectively, $$s_{TX}(t)=10\cos(2\pi ft)$$ and $$s_{RX}(t) = \cos(2\pi ft).$$
Does the received signal represent voltage for load? I mean, if load is 1 Ohm resistor, is the power for resistor $$P_{Load}=\int_{T}\frac{s_{RX}(t)^2}{1}dt=\frac{1}{2}=0.5 W\quad?$$
If so, if I use 0.1 ohm, the power will be 5 W? (Received power is just 0.5...)
Please someone teach me where I have misunderstanding
rf antenna wireless wireless-charging
God DannyGod Danny
If your transmitting antenna has low losses and is driven at the right frequency it will emit power efficiently. A straightforward simple antenna like a dipole will emit power evenly in all directions in one plane and, it will emit zero power in other directions. I mention this to set the scene.
The electromagnetic power emitted is made from two fields and those two fields are an electric field and a magnetic field measured in volts per metre and amps per metre respectively. The power of these two fields is simply volt.amps per square metre i.e. we talk about power in watts per square metre and the square metre part does represent the watts that flow through the air per square metre.
However, as you get further from the transmit antenna the power per square metre drops with distance squared. Think of a light bulb emitting light in all directions - your eyes have a certain area with which to capture that power and if you move away from the light the power per square metre reduces.
So a transmit antenna transmits a fixed amount of power and the further you are away, you receive less of that power because it is spreading out. It isn't attenuating, it is spreading out.
Now think about your receive antenna and importantly, don't think of it as a wire, think of it as a collector of power with a certain area - this is called the aperture of the antenna and is measured in square metres. Up close to the transmit antenna the energy density is greater and more watts are packed into a given area. Further away there are fewer watts per square metre so your receiver (with a fixed aperture) cannot receive the same power.
If your receive antenna's aperture "collects" 0.5 watts at position A it will collect one quarter of that power at position 2A.
There is no trade off with load resistance - you get the power that is collected by the receive antennas aperture. If of course your receive antenna is up really close to the transmit antenna you are in, what is called, the "near field" and this is much less predictable - you get induction and electric coupling effects and you can, under the right situation "load" the transmit antenna and produce much more difficult to understand relationships.
So, if you are just talking about EM radiation from an antenna you are in, what is called, the far-field. This nominally begins about 1 wavelength from the transmit antenna so, at 2.4 GHz this is round about 10 cm.
So, what I've largely said above, is about far-field power transfer and not near-field stuff. In the far-field the power you get is immutable by impedances - the receive antenna can only "collect" what is delivered and that is nearly the end of the story however, if you don't provide the correct impedance to the antenna you won't maximize the power potential. Any antenna has an impedance and this is vastly frequency dependent so, you are limited even more.
In short, in the far-field you get what power you are given and, you have to match impedances to maximize that power.
Andy akaAndy aka
\$\begingroup\$ Very good explanation, you give me very good intuition. Thank you so much. \$\endgroup\$ – God Danny Sep 19 '17 at 4:02
The antenna has a design output impedance, usually 50 ohms, it is NOT a voltage source, but appears to be a voltage source in series with a resistor equal to its designed output impedance.... All RF work is better thought of as power transfer rather then voltage transfer as impedance is so easy to change.
For maximum power transfer you need to have a load that matches the output impedance of the antenna (Graph the power in a load for a voltage source in series with a resistor driving a variable load and you will find that maximum power transfer occurs when the source and load impedance are equal, seriously worth taking the time to do this). For sources having a complex source impedance the maximum power transfer occurs when the load is the complex conjugate of the source impedance.
Now one probably could design an antenna to match into 1 ohm, or 0.1 ohm, or 0.01 ohm (But things are getting VERY lossy because of secondary effects), but in all cases the power delivered will be equal (apart from the losses due to poor antenna efficiency), you will see much less voltage and more current as the system impedance drops.
Consider an antenna with a design output impedance of 1 ohm (Unlikely for all sorts of reasons, but go with it), model it as say a 1.414V voltage source in series with a 1 ohm resistor. Into a matched load (1 ohm) you get 0.707V (Voltage divider) and 0.707A, multiplying these gives you your 0.5W. If you load this up into 0.1 ohms you get ~0.13V @ 1.3A = 0.169W, hardly an improvement.
Now a free space path loss of only 20dB seems unlikely to me, with a 1m separation and isotropic antennas for example you have ~40dB of path loss at 2.4GHz, and you can add 20dB for every decade of path length, so you might want to look at your numbers.
Dan MillsDan Mills
\$\begingroup\$ Very kind explanation, nice, thank you. I actually want to grasp of how to change power signal to voltage signal, ignoring the impedance matching, i.e. ignoring reflecting effects and so on. \$\endgroup\$ – God Danny Sep 15 '17 at 1:53
\$\begingroup\$ Just use a very high impedance input (Good luck with that at 2.4GHz!), you get zero power transferred but can measure the voltage. \$\endgroup\$ – Dan Mills Sep 19 '17 at 10:08
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The Journal of Mathematical Neuroscience
The geometry of rest–spike bistability
Giuseppe Ilario Cirillo ORCID: orcid.org/0000-0002-1162-32251 &
Rodolphe Sepulchre1
The Journal of Mathematical Neuroscience volume 10, Article number: 13 (2020) Cite this article
Morris–Lecar model is arguably the simplest dynamical model that retains both the slow–fast geometry of excitable phase portraits and the physiological interpretation of a conductance-based model. We augment this model with one slow inward current to capture the additional property of bistability between a resting state and a spiking limit cycle for a range of input current. The resulting dynamical system is a core structure for many dynamical phenomena such as slow spiking and bursting. We show how the proposed model combines physiological interpretation and mathematical tractability and we discuss the benefits of the proposed approach with respect to alternative models in the literature.
Conductance-based models are by now well established as a fundamental modeling framework to connect the physiology and the dynamics of excitable cells. Ever since the seminal work of Hodgkin and Huxley [1], there has been a continuing effort in the literature to develop models that combine mathematical tractability and physiological interpretation. An interesting example is the two-dimensional model published by Morris and Lecar in 1981 [2]. Like Hodgkin–Huxley model, it captures the essential physiology of excitability: a spike results from the fast activation of an inward current followed by the slow activation of an outward current. The former provides positive feedback in the fast time-scale whereas the latter provides negative feedback in the slow time-scale. Because it is only two-dimensional, the model is also amenable to phase-portrait analysis without any reduction. Its geometry is similar to the one of FitzHugh–Nagumo model [3], the first mathematical model proposed to understand the core dynamics of the Hodgkin–Huxley model. In that sense, Morris–Lecar model combines the physiological interpretation of Hodgkin–Huxley model and the mathematical tractability of FitzHugh–Nagumo circuit.
In the present paper, we aim at capturing in a similar way the essence of rest–spike bistability, that is, the coexistence of a stable spiking attractor and a stable fixed point in a slow–fast model. The importance of this phenomenon is well acknowledged in the neurodynamics literature due to its role as a building block of neuronal patterns such as bursting [4, 5]. We obtain rest–spike bistability by adding one extra current in the Morris–Lecar model: an inward current with slow activation. The resulting model combines the three following features:
For a range of input currents, the model is rest–spike bistable, that is, a stable equilibrium coexists with a stable limit cycle. The geometry of the two attractors is robust to the time-scale separation in the sense that it persist in the limit of infinite time-scale separation.
The model has the direct physiological interpretation of dynamics and attractors being shaped by three distinct currents (fast positive feedback (e.g. sodium activation), slow negative feedback (e.g. potassium activation), and slow positive feedback (e.g. calcium activation)).
The model is amenable to a mathematical analysis by geometric singular perturbation theory.
We are not aware of other single-cell models in the literature combining those three features. Mathematical models of rest–spike bistability often lack the first feature above. For instance, a homoclinic bifurcation in the Morris–Lecar model only exists for a specific time-scale separation (see e.g. Table 3.1 in Sect. 3.2 of [4]). Limitations of such models with respect to the geometry of the attractors and the robustness of bursting are discussed in [6, 7]. We are only aware of two published models in which rest–spike bistability persists in the singular limit of infinite time-scale separation. The first one is the model proposed by Hindmarsh and Rose in 1982 [8] as a mathematical model aimed at capturing low-frequency spiking. The second one is the transcritical model proposed in 2011 as a two-dimensional reduction of a physiological model combining the currents of the Hodgkin–Huxley model with a slow inward (calcium) current [9]. Both models are planar and lack the second feature, that is, they can only be regarded as a mathematical reduction of a physiological conductance-based model.
This paper aims to contribute to the idea that balancing positive and negative feedback in the slow time scale is a key mechanism to generate rest–spike bistability. This viewpoint is at the core of the planar model in [6] and its importance from a physiological viewpoint is highlighted by [7]. Here we complement that work by studying how this mechanism can be naturally implemented in a physiological context: using two distinct slow currents, one providing negative feedback to restore the membrane potential, the other providing positive feedback to obtain two attractors separated by the stable manifold of a saddle.
The remainder of the paper is organized as follows. Section 2 presents the model and recalls the notions of geometric singular perturbation theory needed for its analysis. In Sect. 3 we study numerically the dynamics on the critical manifold, highlighting its persistence properties. Section 4 builds on this picture to derive conditions for multistability and monostability, we focus on the singular case and mention what hypotheses guarantee persistence. In Sect. 5 we discuss some variations of the same geometric picture, while in Sect. 6 we relate the Hindmarsh–Rose and the transcritical models to the one we are studying. We draw some conclusions in Sect. 7. Two appendices report additional details.
A model of rest–spike bistability
We consider a three-dimensional slow–fast conductance-based model defined by
$$ \begin{gathered} \varepsilon \dot{v} = i - i_{\mathrm{ion}}(v, n, p), \\ \dot{n} = -n + S_{n}(v), \\ \tau \dot{p} = - p + S_{p}(v), \end{gathered} $$
where ε is a small parameter. The total ionic current is the sum of a leak current and three voltage-gated currents:
$$ \begin{aligned}[b] i_{\mathrm{ion}} &= g_{l}(v-v_{l}) + S_{m}(v) (v - 1) + n(v + 1) + p(v - 1) \\ &= c(v) + n(v + 1) + p(v - 1). \end{aligned} $$
The parameters \(-1 < v_{l} < +1\) that appear in the equation can be thought of as reversal potentials. In the absence of an external current i, the voltage range \([-1,1]\) is positively invariant. The currents \(S_{m}(v)(v - 1)\) and \(p(v - 1)\) are then negative (inward currents) whereas the current \(n(v + 1)\) is positive (outward current). The variables n and p are gating (positive) variables that model the slow activation of the inward current \(p(v - 1)\) and of the outward current \(n(v + 1)\). The inward current \(S_{m}(v)(v - 1)\) has instantaneous activation, a standard simplification for currents that activate in the fast time-scale. The functions \(S_{x}(v)\) correspond to activation functions that we assume of the form
$$ S_{x} = \frac{g_{x}}{2} \biggl(\tanh \biggl( \frac{v - a_{x}}{b_{x}} \biggr) + 1 \biggr). $$
Here the multiplicative factor \(g_{x}\) corresponds to the maximal conductance associated to the current x. We find it convenient to write the equations in this form, rather than including maximal conductances in the voltage equation, because this allows us to change maximal conductances of slow currents without modifying the critical manifold of the system. A consequence of this is that the dynamics of p and n lies between zero and the corresponding maximal conductance, i.e. \(p \in [0, g_{p}]\) and \(n \in [0, g_{n}]\).
The key property of the model is the presence of the slow inward current \(p(v - 1)\). In the absence of this current, the model is two-dimensional and has a phase portrait similar to the classical FitzHugh–Nagumo model. With this additional slow inward current, both continuous spiking and rest coexist for the same value of applied current, as shown by the simulation in Fig. 1 (see Appendix B for numerical values of the parameters).
Rest–spike bistability in the model (1)
We note that similar phenomena can be obtained with a current of the type \(p(v + 1)\) where p inactivates, i.e. decreases as v increases. Physiologically this corresponds to an outward current that inactivates slowly, rather than an inward current that activates slowly. Both types of currents model a source of positive feedback in the slow time-scale [10]. A classical example of slowly inactivating outward current is the A-type potassium current [11].
We use geometric singular perturbation theory [12] to study the slow–fast system (1) as ε tends to zero. The singular limit of this model is the differential-algebraic system
$$ \begin{gathered} 0 = i - i_{\mathrm{ion}}(v, n, p), \\ \dot{n} = -n + S_{n}(v), \\ \tau \dot{p} = - p + S_{p}(v), \end{gathered} $$
which we call slow dynamics or reduced system. After rescaling time, the same limit leads to the layer dynamics
$$ \begin{gathered} v' = i - i_{\mathrm{ion}}(v, n, p), \\ n' = 0, \\ p' = 0, \end{gathered} $$
where ′ refers to differentiation with respect to the fast time \(\tau = t / \varepsilon \).
The reduced system (4) is constrained to the critical manifold \(\mathcal{C}_{0}\), defined by
$$ i_{\mathrm{ion}}(v, n, p) = i $$
and corresponding to fixed points of the layer dynamics (5). Normally-hyperbolic compact subsets of \(\mathcal{C}_{0}\) persist as invariant manifolds of (1) for ε small enough. This manifolds are not necessarily unique, but we assume one family of perturbation \(\mathcal{C}_{\varepsilon }\) has been fixed and call them slow manifolds.
Perturbations of subsets of \(\mathcal{C}_{0}\) maintain their type of stability with corresponding (local) stable and unstable manifolds. These admit invariant foliations, with each point on the critical manifold acting as base for a fiber. Invariance of the foliation can be interpreted as points on each fiber "shadowing" the corresponding base point, in forward time for the stable manifold and backward for the unstable. Points on \(C_{\varepsilon }\) follow a dynamics that is a regular perturbation of the reduced system (4); in the following we refer to this perturbation as slow dynamics.
A point x on the critical manifold is normally hyperbolic if it is a hyperbolic fixed point of the layer dynamics (5). If this is the case, as \(\varepsilon \to 0\) the fibers based at x tend to its stable and unstable manifolds in the layer dynamics (5). For (1) the layer dynamics is one dimensional, so that hyperbolic fixed points are either attractive or repulsive, with their invariant manifolds corresponding to lines with n and p constant.
We consider parameter ranges for which the critical manifold can be divided in three normally-hyperbolic branches. These are separated by two lines of folds that we call \(F_{l}\) and \(F_{h}\), and verify
$$ \frac{\partial i_{\mathrm{ion}}}{\partial v} = \frac{dc}{dv}(v) + n + p = 0. $$
The two lines of folds are connected by an unstable branch M. The other branches, \(S_{l}\) and \(S_{h}\), are both stable. Figure 2 shows the typical shape of the critical manifold for fixed i.
Reduced dynamics (4) on the critical manifold (6), and its projection onto the v–p plane, together with the lines of folds \(F_{l}\), \(F_{h}\) and their projections \(P_{l}\), \(P_{h}\). A saddle point and its stable (blue) and unstable (red) manifolds in the reduced system are shown on the critical manifold. The black trajectory is a singular relaxation oscillation composed of two slow parts (single arrow) connected by two trajectories along fast fibers (double arrow, dotted in the projection)
Around any point away from the lines of folds, the critical manifold admits a parametrization in the slow variables n and p. However, this local parametrization cannot be made global due to the presence of folds. Following [13–15], we use v and p to obtain a parametrization valid in the interval \(v\in (-1, 1)\). This is achieved by solving (6) for \(n(v, p, i)\). The corresponding projection is shown in Fig. 2.
The reduced dynamics in these coordinates is obtained differentiating (6):
$$ \begin{gathered} \frac{\partial i_{\mathrm{ion}}}{\partial v} \dot{v} = - \frac{\partial i_{\mathrm{ion}}}{\partial n} \dot{n} - \frac{\partial i_{\mathrm{ion}}}{\partial p} \dot{p}, \\ \tau \dot{p} = -p + S_{p}(v). \end{gathered} $$
The first equation becomes singular on the lines of folds \(\frac{\partial i_{\mathrm{ion}}}{\partial v} = 0\). Multiplication by \(\frac{\partial i_{\mathrm{ion}}}{\partial v}\) recovers a regular differential equation:
$$ \begin{gathered} \dot{v} = -\frac{\partial i_{\mathrm{ion}}}{\partial n} \dot{n} - \frac{\partial i_{\mathrm{ion}}}{\partial p} \dot{p}, \\ \tau \dot{p} = \frac{\partial i_{\mathrm{ion}}}{\partial v}\bigl(-p + S_{p}(v)\bigr). \end{gathered} $$
The two systems (8) and (9) share the same trajectories with different time parametrizations. Moreover, in (9) time is reversed on the unstable branch \(\frac{\partial i_{\mathrm{ion}}}{\partial v} < 0\) and new fixed points can appear on the lines of folds. These verify
$$ \frac{\partial i_{\mathrm{ion}}}{\partial v} = 0, \qquad \frac{\partial i_{\mathrm{ion}}}{\partial n} \dot{n} + \frac{\partial i_{\mathrm{ion}}}{\partial p} \dot{p} = 0. $$
They are called folded singularities [13].
Away from the lines of folds the two systems (8) and (9) are largely equivalent, but important differences occur in the neighborhood of \(F_{l}\) and \(F_{h}\). Moreover, near these lines the perturbed dynamics is no longer constrained by normal hyperbolicity, in particular it cannot be obtained as a regular perturbation of the reduced system (8). Different phenomena are possible. The least degenerate situation occurs when the desingularized vector field is never zero along these lines:
$$ \frac{\partial i_{\mathrm{ion}}}{\partial n}\dot{n} + \frac{\partial i_{\mathrm{ion}}}{\partial p} \dot{p} \neq0. $$
Under this assumption the desingularized vector field (9) can point either to the unstable branch or the stable one. Assuming the additional nondegeneracy condition
$$ \frac{\partial ^{2} i_{\mathrm{ion}}}{\partial v^{2}} \neq 0, $$
the first case corresponds to jump points, at which the reduced system (8) admits two solutions backwards in time but none in forward time. For \(\varepsilon >0\) a stable branch of \(\mathcal{C}_{\varepsilon }\) near these points can be continued using the flow [14]. Doing so shows that trajectories on the slow manifold pass the folds and reach a fiber contained in the stable manifold of the other stable branch of \(C_{\varepsilon }\), with the flow contracting the direction transverse to the manifold.
Condition (11) corresponds to the vector field being transverse to the critical manifold, a condition which is violated at folded singularities. These are fixed points of the desingularized system (9), but not necessarily fixed points of the reduced dynamics (8). As a consequence, they can be reached in finite time. Depending on the type of fixed point they can correspond to the singular limit of canard trajectories, i.e. intersections between stable and unstable branches of the slow manifold [13]. Generically, the desingularized flow changes direction at these points. Hence, a folded singularity delimits the set of jump points on a line of folds [15].
Reduced dynamics
We will now study the reduced system (8), often with the aid of its desingularized version (9). Fixed points can be parametrized by v through the steady-state i–v curve
$$ i_{s}(v) := i_{\mathrm{ion}}\bigl(v, S_{n}(v), S_{p}(v)\bigr). $$
This is shown in Fig. 6 and is an S-shaped curve, with two folds separating three families of fixed points \(\mathcal{X}_{l}\), \(\mathcal{X}_{m}\) and \(\mathcal{X}_{h}\); \(\mathcal{X}_{l}\) corresponds to low voltages, \(\mathcal{X}_{m}\) to intermediate voltages and \(\mathcal{X}_{h}\) to high voltages. For fixed i, we denote points in each family with corresponding lower-case letters \(x_{l}\), \(x_{m}\) and \(x_{h}\). In addition to these three fixed points, the desingularized dynamics (9) has a folded singularity \(x_{f}\in F_{l}\). For parameter values reported in Appendix B, and i in the range of interest in this section, this point is a focus and does not lead to canard trajectories [13]; it only delimits jump points on \(F_{l}\).
Figure 3 shows the typical phase portrait of the reduced system (8). The fixed point \(x_{l}\) is a stable node, while \(x_{m}\) and \(x_{h}\) are both saddle points. Their stable and unstable manifolds do not extend beyond \(F_{l}\) and \(F_{h}\) due to loss of existence and uniqueness along these lines. In particular, unstable manifolds terminate at jump points.
A typical phase portrait of the reduced system (8). Fixed points of the desingularized system (9) are denoted by crosses, \(x_{l}\) is a stable node, \(x_{m}\) and \(x_{h}\) are saddle points and \(x_{f}\) is a folded focus (unstable). Stable and unstable manifolds of the saddle points are shown in blue and red, respectively. Along the two lines of folds \(F_{l}\) and \(F_{h}\) the system is singular: trajectory at those points are defined only in forward or backward time; the first of these two cases corresponds to jump points. The stable manifold of \(x_{m}\) separates initial conditions in \(S_{l}\) (left of \(F_{l}\)) that reach a jump point from those that converge to \(x_{l}\)
For \(\varepsilon >0\) hyperbolic fixed points persist in the slow dynamics with their stable and unstable manifolds [12, 16]. In the perturbed system (1) these fixed points are still hyperbolic. In particular, saddle points remain saddle, with their invariant manifolds being obtained as a combination of trajectories in the slow dynamics and fast fibers. The unstable manifold of \(x_{m}\) is completely contained in the slow manifold. Its stable manifold, instead, is two dimensional; it includes the stable manifold in \(\mathcal{C}_{\varepsilon }\) and all fast fibers based on that curve. In the singular limit this surface tends to the stable manifold of \(x_{m}\) in the reduced system (8) and all nearby segment with constant n and p that intersect it. Similarly, \(x_{h}\) perturbs to a saddle with a one-dimensional stable manifold and a two-dimensional unstable manifold.
Adding a trivial equation for i to (1), the same is true for the family of fixed points \(x_{m}(i)\). At least for i in a small interval, this family persists together with its two-dimensional unstable manifold and three-dimensional stable one. Sections of these manifolds for fixed i coincide with the invariant manifolds of the corresponding fixed point.
When i varies on larger domains, the outlined phase portrait can undergo two distinct qualitative changes: increasing i leads to a fold of the i–v curve at which \(x_{l}\) and \(x_{m}\) merge in a saddle-node bifurcation, leaving only one fixed point \(x_{h}\in M\). Likewise, decreasing i, \(x_{m}\) and \(x_{h}\) reach a similar fate, leaving \(x_{l}\in S_{l}\) as the only fixed point.
To obtain this second bifurcation it is necessary that one of the two fixed points crosses the line \(F_{l}\) and changes branch.Footnote 1 In our case \(x_{m}\) crosses \(F_{l}\). This passage corresponds to an exchange of stability with the folded singularity through a folded saddle-node [17]. Beyond this crossing, the folded singularity is a saddle, while \(x_{m}\) is a node of the reduced system. In a similar fashion increasing the applied current leads to \(x_{h}\) crossing \(F_{h}\), which happens once \(x_{h}\) is the only fixed point left. After this crossing, \(x_{h}\) is a stable fixed point on an attractive branch.
Finally, varying i can lead to changes in the type of folded singularity. As already mentioned \(x_{m}\in F_{l}\) corresponds to a folded saddle-node, thus varying i and moving \(x_{m}\) between branches leads to different types of folded singularity: it is a saddle when \(x_{m} \in M\) and a node when \(x_{m} \in S_{l}\). Both situations lead to canard trajectories [13]. Moreover, since \(x_{f}\) is a focus in the phase portrait described above, it has to change to a node before becoming a folded saddle-node.
Rest–spike bistability
Returning to the phase portrait in Fig. 3, we now analyze the global return mechanism that leads to rest–spike bistability.
In the singular limit \(\varepsilon = 0\), trajectories on stable branches of the critical manifold \(\mathcal{C}_{0}\) stay on it until they reach a line of fold in correspondence of a jump point. Once one of these points is reached the singular trajectory is continued along a fast fiber with constant n and p, reaching the opposite branch as shown in Fig. 2. The points at which these singular trajectories arrive correspond to the projections of \(F_{l}\) and \(F_{h}\) along fast fibers. We call these projections \(P_{l} \subset S_{l}\) and \(P_{h} \subset S_{h}\).
Based on this property, we can analyze the singular system referring only to the v–p plane and the reduced dynamics: when a trajectory reaches a jump point it is transported to the corresponding projection keeping p fixed, as shown in Fig. 2 for a limit cycle.
Rest–spike bistability follows from how the stable and unstable manifold of \(x_{m}\) constrain trajectories. The role of the stable manifold is simple, it separates initial conditions on \(S_{l}\) that reach a jump point on \(F_{l}\) from those that remain on the critical manifold and tend to \(x_{l}\). The unstable manifold, instead, determines if the system is multistable. This is the case if the unstable manifold stays away from \(x_{l}\). Otherwise almost all trajectories converge to \(x_{l}\). We treat these two situations separately in the next sections.
Bistability
In the following we denote by \(x_{1}\) the intersection of the unstable manifold of \(x_{m}\) with \(F_{l}\), and by \(x_{-1}\) the intersection of the stable manifold of \(x_{m}\) with \(P_{l}\). Following the singular flow from \(x_{1}\) leads to \(x_{2}\in P_{h}\), then to \(x_{3}\in F_{h}\) and back to \(P_{l}\) at \(x_{4}\) (see Fig. 4). We recall that given the dynamics (1) we can assume that p lies in the interval \([0, g_{p}]\), where \(g_{p}\) is the maximal conductance appearing in (3).
Reduced dynamics (8) in the multistable case. The stable manifold of \(x_{m}\) (blue) separates initial conditions that reach a jump point on \(F_{l}\) from those that converge to \(x_{l}\). Jump points are mapped to their projections (e.g. \(x_{1}\) to \(x_{2}\) and \(x_{3}\) to \(x_{4}\)). The unstable manifold of \(x_{m}\) (red) delimits an invariant set for the dynamics
Assume that the trajectory starting at \(x_{4}\) reaches a jump point on \(F_{l}\) (\(x_{5}\)), as shown in Fig. 4. Consider the segment \(I_{l}\subset P_{l}\) between \(x_{4}\) and \(p=g_{p}\). The reduced dynamics maps this segment to \(F_{l}\) in finite time, defining a map \(\varPi _{l}:I_{l} \to F_{l}\). Clearly the same map can be defined using the desingularized reduced system (9), thus as long as this vector field is transverse to \(F_{l}\) at all points in \(\varPi _{l}(I_{l})\) the map is smooth. We note that this is equivalent to \(\varPi _{l}(I_{l})\) not containing folded singularities. Similarly, on \(S_{h}\) we define the segment \(I_{h}\subset P_{h}\) between \(x_{2}\) and \(p=g_{p}\), and a corresponding map \(\varPi _{h}:I_{h} \to F_{h}\). We denote the projection along fast fibers by \(\varPi _{f}\) (from \(F_{l}\) to \(P_{h}\) and from \(F_{h}\) to \(P_{l}\)). Since the dynamics is bounded by the line \(p=g_{p}\), by construction we have
$$ \varPi _{f} \circ \varPi _{l} (I_{l}) \subset I_{h}, \qquad \varPi _{f} \circ \varPi _{h}(I_{h}) \subset I_{l}, $$
which allows us to define the singular Poincaré map
$$ \varPi = \varPi _{f} \circ \varPi _{h} \circ \varPi _{f} \circ \varPi _{l}: I_{l} \to I_{l}. $$
This construction shows that the stable manifold of \(x_{m}\) divides the state space into two invariant sets. One is the basin of attraction of \(x_{l}\), while the other one has dynamics characterized by the Poincaré map (15). Since this is a smooth map of an interval into itself it admits at least one fixed point, which corresponds to a singular relaxation oscillation. As shown in [14], if this fixed point is hyperbolic, under the additional hypothesis that the singular trajectory intersects \(P_{l}\) and \(P_{h}\) transversally, it perturbs to a hyperbolic limit cycle for \(\varepsilon >0\). In fact the Poincaré map (15) is (up to conjugacy) a global version of the one used in that reference.
We remark that this construction only guarantees multistability. Further analysis of the map (15) is required to obtain a more accurate picture. While this is beyond the scope of this work, numerical simulations confirm that this map has a unique attracting fixed point.
Monostability
Constructing the Poincaré map (15) requires that \(x_{4}\) falls inside the interval defined by \(x_{-1}\) and \(p=g_{p}\) on \(P_{l}\). The situation in which this assumption fails is illustrated in Fig. 5. In this case most trajectories on \(S_{l}\) and \(S_{h}\) are attracted by the stable fixed point \(x_{l}\), the only exception being the stable manifold of \(x_{m}\).
Reduced dynamics in the monostable case. The stable manifold of \(x_{m}\) separates initial condition that arrive at a jump point on \(F_{l}\) from those that converge to \(x_{l}\) (not shown). The unstable manifold of \(x_{m}\) (red) converges to \(x_{l}\) after one jump. Similarly, almost all initial conditions on stable branches converge to it, the only exception being the ones that form the stable manifold of \(x_{m}\)
To see this, we start from \(x_{-1}\) and consider its anti-image through \(\varPi _{f}\) on \(F_{h}\). Continuing to follow the singular flow "backwards", as shown in Fig. 5, leads back to \(P_{l}\) at a point that we call \(x_{-2}\). Any compact segment in \(P_{l}\) that lies between these points is mapped by the singular flow strictly inside the segment delimited by \(x_{4}\) and \(x_{-1}\). Since any point strictly inside this second segment converges to \(x_{l}\), the same conclusion extends to all points in the original segment.
The same argument shows that points in the portion of \(S_{l}\) delimited by the trajectories starting at \(x_{-1}\) and \(x_{-2}\) tend to \(x_{l}\). The only exceptions are these boundary trajectories that reach \(x_{m}\) and belong to its stable manifold. As long as the stable manifold of \(x_{m}\) is unbounded in the p coordinate, the same argument can be iterated on all \(S_{l}\) and adapted to \(S_{h}\), leading to the conclusion that almost all points on \(S_{l}\) and \(S_{h}\) are in the basin of attraction of \(x_{l}\). This situation persists for small enough \(\varepsilon >0\), and since most points are attracted to stable branches of the slow manifold we see that for almost all initial conditions the perturbed dynamics converges to \(x_{l}\).
Homoclinic trajectory and bifurcation diagram
Transitions between monostability and bistability in system (1) are controlled by the applied current i. The phase portraits in Figs. 4 and 5 suggest the presence of a homoclinic trajectory, which can be obtained by decreasing the applied current from the bistable case. In the singular limit this trajectory corresponds to the condition \(x_{4} = x_{-1}\) and delimits the boundary of bistability. We denote by \(i_{H}\) the value of current at which this happens. While we cannot expect this homoclinic trajectory to persist for \(\varepsilon >0\) with i fixed, it is natural to ask whether for \(\varepsilon >0\), fixed and small, we can find an \(i_{H}(\varepsilon )\), close to \(i_{H}\), at which a homoclinic trajectory exists. There is a natural transversality condition that guarantees this property. The family of fixed points \(x_{m}(i)\) admits a three-dimensional stable manifold and a two-dimensional unstable one. Their intersection is a homoclinic trajectory. In the singular limit, following the unstable manifold of \(x_{m}(i)\) leads back to \(S_{l}\) after two jumps. Extending \(\mathcal{C}_{0}\) to include i, \(x_{m}(i)\) is a (normally hyperbolic) invariant set in it, with two-dimensional invariant manifolds. The continuation of the unstable one using the singular flow, after two jumps intersects the stable manifold in the plane \(i=i_{H}\). If this intersection is transverse then it persists for small ε and i close to \(i_{H}\). We show this in Appendix A adapting the arguments used in [14] to prove existence of relaxation oscillations.
To conclude this section, Fig. 6 shows the bifurcation diagram of the whole system (1) computed with AUTO-07p [18] for parameter values reported in Appendix B. The numerics confirms the presence of a family of limit cycle (red curves) and its coexistence with a family of fixed points (blue curve). The family of periodic solutions terminates in a homoclinic trajectory for low values of i (the numerical continuation was stopped at period \(T=10^{4}\)).
Bifurcation diagram of (1). Solid lines denote stable solutions, dotted correspond to unstable ones; blue lines correspond to fixed points, red lines to limit cycle, in the latter case both maximum and minimum are shown
A common geometric picture
The bifurcation diagram illustrated in the previous section is understandably only one among many possible scenarios compatible with the three-dimensional geometry of Fig. 2. While a detailed study of all possible cases is beyond the scope of this work, we wish to highlight how different types of bistability could have the same geometric structure. To do this we use ideas and techniques from [19]. As in Sect. 3 we identify fixed points with the i–v curve
$$ i_{s}(v) = i_{\mathrm{ion}}\bigl(v, S_{n}(v), S_{p}(v)\bigr) $$
and divide them in three families \(\mathcal{X}_{l}\), \(\mathcal{X}_{m}\) and \(\mathcal{X}_{h}\), separated by two folds. As noted in Sect. 3 there is a value of current \(i_{c}\), between the two folds, at which \(x_{m}\) crosses \(F_{l}\) to enter the unstable branch M. The scenario studied in Sect. 4 assumes \(i_{c} < i_{H}\) since the homoclinic bifurcation occurs when \(x_{m} \in S_{l}\).
As a first variation we consider what happens when the bistable range extends to current values for which \(x_{m}\in M\). The bifurcation \(x_{m}\in F_{l}\) corresponds to a folded saddle-node. Beyond this bifurcation \(x_{m}\in M\) is a node of the reduced dynamics while \(x_{f}\) is a saddle. In this case the analysis is easily adapted from Sect. 4. One must simply substitute the stable manifold of \(x_{m}\) with the one of \(x_{f}\), and use \(\varPi _{f}(x_{f})\) in place of \(x_{2} = \varPi _{f}(x_{1})\). Figure 7 shows the corresponding geometric construction. A classical example where this scenario occurs is the Hodgkin–Huxley model with the reversal potential of potassium increased. This situation of bistability has been studied in the early work [20]. Its planar reduction leads to the transcritical model [10]. Also in this case the boundary of bistability is a singular homoclinic trajectory. This trajectory, however, has to go through the folded singularity \(x_{f}\) to reach \(x_{m}\) on the unstable branch M.
Alternative scenarios that lead to bistability. Top: geometric construction when a folded saddle (\(x_{f}\)) takes the place of \(x_{m}\). Bottom: bistability between two fixed points (\(x_{l}\) and \(x_{h}\))
Both cases discussed so far assume that \(x_{m}\) and \(x_{h}\) collide in a fold on M. Yet another scenario corresponds to this fold occurring on \(S_{l}\), after \(x_{h}\) crosses \(F_{l}\). Also this crossing leads to a folded saddle-node, after which \(x_{h}\in S_{l}\) can perturb to a stable fixed point. Local analysis around folded saddle-node shows the possibility of Hopf bifurcations [17], which are indeed found numerically. After this the system presents two stable fixed points. The relevant part of the reduced dynamics in this case is shown in Fig. 7: the stable manifold of \(x_{m}\) acts as separatrix between the basins of attraction of the two stable fixed points, while the one of \(x_{f}\) (a folded saddle) separates initial conditions that reach a jump point on \(F_{l}\) from those that remain on the critical manifold.
The examples above suggest that many possible variants for transitions between monostability and bistability are possible. We also note that many of the geometric constructions used in [6, 7, 19] have an analog in our setting, allowing, for example, non-plateau oscillations, contrary to the case showed in Fig. 1. This flexibility is interesting in the perspective of connecting the present approach to the classification of bursting types according to the transitions that occur from rest to spike and vice versa (see e.g. [21]).
Connections with phase-portrait analysis
We close this paper by clarifying the connection between the proposed three-dimensional model and two published slow–fast phase portraits of rest–spike bistability.
The first phase portrait goes back to the seminal work of Hindmarsh and Rose [8, 22]. In one of the earliest attempts to model slow spiking and bursting, Hindmarsh and Rose proposed to modify the FitzHugh–Nagumo model with a recovery variable that has a nonmonotonic activation function. Geometrically, this situation corresponds to a degenerate case of the planar pictures described in Sect. 3 and Sect. 4, in which all essential elements are contained on a line. As a result, the main elements of the three dimensional dynamics can be captured by constraining it to a plane, resulting in a simplified two-dimensional model of rest–spike bistability. This is characterized by the classical N-shaped critical manifold, as shown in Fig. 8. The price paid for this simplification is that the flexibility of the two-dimensional slow dynamics described in Sect. 5 is lost. For instance, bistability is only possible if \(x_{l}\) lies out of the stripe delimited by \(P_{l}\) and \(F_{l}\), ruling out patterns in which the voltage of the resting state is between maximum and minimum of the spike. We note that the nonmonotonicity of the activation function in Hindmarsh–Rose model has the natural interpretation of summarising in one variable the distinct roles of an inward and an outward slow current.
Bistable slow–fast phase portraits as reduction of a larger dimensional model. Left: critical manifolds obtained as the intersection of a higher-dimensional one (green) with a surface (gray). Right: corresponding phase plane with the critical manifold obtained (green) and a possible nullcline for the slow variable (dashed) that completes the dynamics. Top: Hindmarsh–Rose model can be obtained constraining the dynamics to a plane, the critical manifold in the phase plane is the classical N-shaped one, but presents nontrivial dynamics leading to rest–spike bistability. Bottom: the transcritical model obtained constraining the dynamics to a surface. The transcritical bifurcation is obtained when this surface is tangent to a line of folds at a point. This bifurcation is responsible for a singular homoclinic trajectory in the planar reduction
The second rest–spike bistable phase portrait is the transcritical model of [6]. This model was obtained as a two-dimensional reduction of a conductance-based model that adds a slow calcium current to the Hodgkin–Huxley model [9]. The analysis of [6] rests on the presence of a transcritical bifurcation of the critical manifold. This bifurcation also directly relates to the mixed role of the slow variable as a source of both positive and negative feedback in the slow time-scale. A main motivation of the present paper was to understand the geometric picture generated by this motif in conductance-base models, where these two roles are often played by distinct variables.
To connect the transcritical bifurcation of the planar model [6] to the three-dimensional geometry of the present paper we consider how this planar reduction can be obtained. Referring to our model (1) for simplicity, a planar reduction is typically obtained imposing an algebraic constraint between n and p, which can be interpreted as a path \(n(s)\), \(p(s)\) [20]. After obtaining a dynamic equation for s from a combination of ṅ and ṗ, the system becomes
$$ \begin{gathered} \varepsilon \dot{v} = i - i_{\mathrm{ion}}\bigl(v, n(s), p(s)\bigr), \\ \dot{s} = g(v, s), \end{gathered} $$
which is a slow–fast planar model. Its critical manifold is given by
$$ i = i_{\mathrm{ion}}\bigl(v, n(s), p(s)\bigr) $$
It corresponds to the intersection of the critical manifold of the larger system with the surface
$$ n = n(s), \qquad p = p(s). $$
A transcritical bifurcation is obtained when
$$ \begin{gathered} i_{\mathrm{ion}} = i, \\ \frac{\partial i_{\mathrm{ion}}}{\partial v} = 0, \\ \frac{\partial }{\partial s}\bigl(i_{\mathrm{ion}}\bigl(v, n(s), p(s)\bigr)\bigr) = \frac{\partial i_{\mathrm{ion}}}{\partial n} \frac{d n}{d s} + \frac{\partial i_{\mathrm{ion}}}{\partial p} \frac{d p}{d s}=0. \end{gathered} $$
Geometrically this corresponds to a point at which the surface (19) is tangent to the line of folds of the critical manifold, as shown in Fig. 8. Similar geometric constructions lead to the presence of a transcritical bifurcation when reducing the Hodgkin–Huxley model with increased potassium reversal potential, as well as when reducing the same model augmented with a calcium current, as done in [6].
An equivalent interpretation of how the transcritical bifurcation arises is that the path \((n(s), p(s))\) defining the surface (19) is tangent to the line of folds
$$ i = i_{\mathrm{ion}}(v, n, p), \qquad \frac{\partial i_{\mathrm{ion}}}{\partial v}(v, n, p) = 0, $$
projected onto the n–p plane. This is the simplest example of how singularities in the sense of [23] can be generated from elementary catastrophes, the core idea in the path formulation of [23, Ch.3 §12]. This is particularly interesting in view of [24], where singularity theory is used to obtain a global description of the critical manifolds of slow–fast planar systems relevant to neuronal dynamics. Two singularities play a prominent role: hysteresis, in connection with spiking, and winged cusp, for rest–spike bistability. Both these singularities can be realized as paths in the unfolding of the cusp catastrophe [23]. Interestingly, this bifurcation is often found in the fast subsystem of neuronal models (an early example being [25]), and it is typically related to the appearance and disappearance of bistability. For example, decreasing the sodium conductance in the Hodgkin–Huxley model leads to the appearance of this bifurcation, and the same is achieved by reducing \(g_{m}\) in (1). The presence of this type of bifurcation in these models suggests that those singularities can arise from model reduction similarly to what happens in the transcritical case.
We studied a simplified slow–fast model of neuronal activity that exhibits rest–spike bistability. The simplest physiological models of excitability include a fast-activating inward current and a slowly-activating outward current. Our model adds a slowly-activating inward current to this basic motif. We think of this model as a core structure for the generation of multistability in more general and realistic conductance-based models. We speculate that similar results are possible using a slowly inactivating outward current, which would have the same functional role of a slow positive feedback.
Through geometric singular perturbation theory we could analyze the geometry of this three-dimensional model. This geometry is rather simple, with the slow dynamics taking place on a classical N-shaped critical manifold. The saddle point on the critical manifold is a key feature of the proposed model. Its stable manifold acts as separatrix, while its unstable manifold determines whether multiple attractors are present. Moreover, a same geometric picture captures different types of bistability, suggesting a common framework to study different phenomena important to neuronal dynamics.
This is by no means the first study of a slow–fast systems with one fast and two slow variables, nor the first single-cell model of bistability. The value of this model is in that it explains how bistability can arise in a physiologically relevant context using a mechanism that is generic but not widely acknowledged. Our hope is that it contributes to the view that a combination of positive and negative feedback in the slow time-scale is a core element in the generation of neuronal patterns.
This assumes that the bifurcation does not happen exactly on \(F_{l}\).
We need a relaxed version of the implicit function theorem since the dependence on ε is not smooth but only continuous; this is proven in [26].
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GIC has received funding from EPSRC (RG80792) and Qualcomm Inc. The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645.
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, UK
Giuseppe Ilario Cirillo & Rodolphe Sepulchre
Giuseppe Ilario Cirillo
Rodolphe Sepulchre
All authors formulated the problems. GIC carried out the details of the theoretical analysis and performed the simulations. All authors wrote, read and approved the manuscript.
Correspondence to Giuseppe Ilario Cirillo.
Appendix A: Existence of a homoclinic trajectory
In this section we show that under the assumption of transversality, the intersection of stable and unstable manifolds that leads to a singular homoclinic trajectory persists for \(\varepsilon >0\). We do this using the setting of [14] and in particular their results on maps defined by the flow of (1). We recast these results in the notation of Sect. 4 and refer the reader to the original work for details.
As in Sect. 4, \(i_{H}\) is the value of i at which a singular homoclinic trajectory exists. We consider the reduced dynamics for this value of i, and fix a point \(x_{u}\) on the unstable manifold of \(x_{m}\) between \(x_{m}\) and \(F_{l}\). Similarly, we fix a point \(x_{s}\) on the stable manifold between \(P_{l}\) and \(x_{m}\).
After a local change of coordinates we can find two neighborhoods of these points, \(N_{s}\) and \(N_{u}\), such that the critical manifold \(\mathcal{C}_{0}\) corresponds to the plane \(v=0\). The intersections of these neighborhoods with the planes \(n=n_{s}\) and \(n=n_{u}\) determine two surfaces \(\varSigma _{s}\) and \(\varSigma _{u}\). Rotating n and p if necessary, we can assume that \(\varSigma _{u} \cap \mathcal{C}_{0}\) intersects the unstable manifold of \(x_{m}\) transversally and only at \(x_{u}\), and similarly for \(\varSigma _{s} \cap \mathcal{C}_{0}\). For fixed δ we let \(N_{\delta } = (i_{H} - \delta , i_{H} + \delta )\) and consider \(\varSigma _{s} \times N_{\delta }\) and \(\varSigma _{u} \times N_{\delta }\). If δ is small enough, stable and unstable manifolds of \(x_{m}(i)\) intersect transversally these extended neighborhoods (in the critical manifold extended to include i). In the following we assume that \(N_{s}\), \(N_{u}\), \(N_{\delta }\) are shrunk whenever necessary.
In Sect. 3, we have characterized the stable manifold of \(x_{m}\) for small \(\varepsilon >0\), this is composed of a line on \(\mathcal{C}_{\varepsilon }\) and the fibers based on it. In the limit \(\varepsilon \to 0\), the singular stable manifold intersect \(\varSigma _{s}\) transversally along one of these fibers. Thus if ε and δ are small enough the same will be true for the stable manifold of \(x_{m}(i)\) for fixed i and ε. Moreover, since at \(\varepsilon =0\), \(i=i_{H}\) this intersection is a line of constant p, we can find a parametrization of it that has the form \(p = p_{s}(v, i, \varepsilon )\). Similarly, the intersection of \(\varSigma _{u}\) with the unstable manifold of \(x_{m}(i, \varepsilon )\) defines two functions \(v_{u}(i, \varepsilon )\) and \(p_{u}(i, \varepsilon )\).
Notice that in this section we use v and p to parametrize the two slices \(\varSigma _{s}\) and \(\varSigma _{u}\), so that v preserves its nature of fast variable. This differs from the use of v and p to parametrize the critical manifold as done in Sect. 3 and Sect. 4.
We can now use the same construction of [14] to obtain a map \(\varPi :\varSigma _{u} \to \varSigma _{s}\) corresponding to the action of the flow. This has the form
$$ \varPi \begin{pmatrix} v \\ p \end{pmatrix} = \begin{pmatrix} R(v, p, i, \varepsilon ) \\ G(v, p, i, \varepsilon ) \end{pmatrix} ;$$
R is exponentially small in ε (\(\lvert R \rvert + \lVert \nabla R \rVert < \exp (-c/\varepsilon )\)) and in particular verifies
$$ R(v, p, i, 0) = 0. $$
G has the form
$$ G = G_{0}(p) + \mathcal{O}\bigl(\varepsilon \ln ( \varepsilon )\bigr) $$
where \(G_{0}:\varSigma _{u} \cap \mathcal{C}_{0} \to \varSigma _{s} \cap \mathcal{C}_{0}\) is the map defined by the singular flow. Smooth dependence on i follows from standard results.
The only difference between this map and the Poincaré map defined in [14] is that we consider two different sections \(\varSigma _{s}\) and \(\varSigma _{u}\) rather than one.
Applying this map to \((v_{u}, p_{u})\) we obtain the intersection of the unstable manifold of \(x_{m}\) with \(\varSigma _{s}\)
$$ \varPi \begin{pmatrix} v_{u} \\ p_{u} \end{pmatrix} = \begin{pmatrix} R(v_{u}, p_{u}, i, \varepsilon ) \\ G(v_{u}, p_{u}, i, \varepsilon ) \end{pmatrix} . $$
In this setting an intersection of stable and unstable manifolds corresponds to a solutions of
$$ G(v_{u}, p_{u}, i, \varepsilon ) = p_{s}\bigl(R(v_{u}, p_{u}, i, \varepsilon ), i, \varepsilon \bigr) $$
where \(v_{u}=v_{u}(i, \varepsilon )\) and \(p_{u}=p_{u}(i, \varepsilon )\). Thus, we can define
$$ P_{u}(i, \varepsilon ) = G(v_{u}, p_{u}, i, \varepsilon ), \qquad P_{s}(i, \varepsilon ) = p_{s}\bigl(R(v_{u}, p_{u}, i, \varepsilon ), i, \varepsilon \bigr) $$
and a homoclinic trajectory corresponds to \(P_{u} - P_{s}=0\).
At \(\varepsilon =0\)
$$ \begin{gathered} P_{u}(i, 0) = G_{0}\bigl(p_{u}(i, 0)\bigr), \\ P_{s}(i, 0) = p_{s}\bigl(R\bigl(0, p_{u}(i, 0), i, 0\bigr), i, 0\bigr) = p_{s}(0, i, 0), \end{gathered} $$
and the existence of the singular homoclinic trajectory at \(i = i_{H}\) means that
$$ P_{u}(i_{H}, 0) = G_{0} \bigl(p_{u}(i_{H}, 0)\bigr) = p_{s}(0, i_{H}, 0) = P_{s}(i_{H}, 0). $$
Assuming that
$$ \frac{\partial P_{u}}{\partial i}(i_{H}, 0) - \frac{\partial P_{s}}{\partial i}(i_{H}, 0) \neq 0, $$
an application of the implicit function theoremFootnote 2 guarantees the existence of a continuous functions \(i_{H}(\varepsilon )\) such that \(P_{u}(i_{H}(\varepsilon ), \varepsilon ) = P_{s}(i_{H}(\varepsilon ), \varepsilon )\).
At \(\varepsilon =0\), \(P_{u}(i, 0)\) is the intersection of the singular unstable manifold (after two jumps) with \(\varSigma _{s} \cap \mathcal{C}_{0}\), while \(P_{s}(i, 0)\) corresponds to the intersection of the stable manifold of the reduced flow and \(\varSigma _{s} \cap \mathcal{C}_{0}\). Condition (30) corresponds to transversality of the intersection between the manifolds \(p=P_{s}(i,0)\) and \(p=P_{u}(i,0)\) in the extended neighborhood \(\varSigma _{s} \times N_{\delta }\). Since the invariant manifolds of \(x_{m}(i)\) can be obtained applying the singular flow to these two sections, we see that condition (30) is equivalent to transversality of the intersection between the invariant manifolds of \(x_{m}(i)\) on the critical manifold (where the unstable manifold has been continued past two jumps using the singular flow).
Appendix B: Parameters
The analysis in Sects. 3 and 4 uses the following numerical values for the parameters of (1)
$$ \begin{gathered} \varepsilon = 0.05,\qquad v_{l} = -0.8, \quad gl = 2, \\ g_{m} = 4.4, \quad a_{m} = -0.19, \quad b_{m} = 0.18, \\ g_{n} = 8.0, \quad a_{n} = -0.16, \quad b_{n} = 0.29, \\ g_{p} = 2.0, \quad a_{p} = -0.5, \quad b_{p} = 0.3, \\ \tau = 1.5. \end{gathered} $$
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Cirillo, G.I., Sepulchre, R. The geometry of rest–spike bistability. J. Math. Neurosc. 10, 13 (2020). https://doi.org/10.1186/s13408-020-00090-z
Neuronal models
Slow–fast models
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CommonCrawl
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Mathematical Biosciences & Engineering
2008 , Volume 5 , Issue 4
A special issue on
Tribute to Thomas G. Hallam's Contributions to Mathematical Ecology,
Ecotoxicology, and the Academic Community
Azmy S. Ackleh, Linda J. S. Allen, Graciela Canziani, Shandelle M. Henson, Jia Li and Zhien Ma
2008, 5(4): i-iii doi: 10.3934/mbe.2008.5.4i +[Abstract](2281) +[PDF](713.7KB)
Thomas Guy Hallam began his career as a faculty member in the Department of Mathematics at Florida State University, working in the area of comparison theorems for ordinary differential equations. While at Florida State he organized a mathematical modeling course and thus became interested in mathematical biology. He began to wonder how he, as a mathematician, might address the mounting environmental problems. He took courses in oceanography and ecology and delved deeply into the literature. During the summer of 1974, he gave a full series of lectures on mathematical biology at the University of São Carlos in São Paulo, Brazil. In 1976, he took a year's leave at the University of Georgia, Athens, in the Departments of Mathematics, Zoology, and the Institute of Ecology, where he met Tom Gard and Ray Lassiter, with whom he has had career-long interactions. In Athens, he became interested in ecotoxicology and partial differential equation models of physiologically structured populations.
For more information please click the "Full Text" above.
Azmy S. Ackleh, Linda J. S. Allen, Graciela Canziani, Shandelle M. Henson, Jia Li, Zhien Ma. Preface. Mathematical Biosciences & Engineering, 2008, 5(4): i-iii. doi: 10.3934/mbe.2008.5.4i.
Parameter estimation in a structured erythropoiesis model
Azmy S. Ackleh and Jeremy J. Thibodeaux
2008, 5(4): 601-616 doi: 10.3934/mbe.2008.5.601 +[Abstract](2606) +[PDF](253.3KB)
We develop a numerical method for estimating parameters in a structured erythropoiesis model consisting of a nonlinear system of partial differential equations. Convergence theory for the computed parameters is provided. Numerical results for estimating the growth rate of precursor cells as a function of the erythropoietin concentration and the decay rate of erythropoietin as a function of the total number of precursor cells from computationally generated data are provided. Standard errors for such parameters are also given.
Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5(4): 601-616. doi: 10.3934/mbe.2008.5.601.
Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission
Chandrani Banerjee, Linda J. S. Allen and Jorge Salazar-Bravo
Arenaviruses are associated with rodent-transmitted diseases in humans. Five arenaviruses are known to cause human illness: Lassa virus, Junin virus, Machupo virus, Guanarito virus and Sabia virus. In this investigation, we model the spread of Machupo virus in its rodent host Calomys callosus. Machupo virus infection in humans is known as Bolivian hemorrhagic fever (BHF) which has a mortality rate of approximately 5-30% [31].
Machupo virus is transmitted among rodents through horizontal (direct contact), vertical (infected mother to offspring) and sexual transmission. The immune response differs among rodents infected with Machupo virus. Either rodents develop immunity and recover (immunocompetent) or they do not develop immunity and remain infected (immunotolerant). We formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. The four states represent susceptible, immunocompetent, immunotolerant and recovered rodents, denoted as $S$, $I^t$, $I^c$ and $R$, respectively. A unique disease-free equilibrium (DFE) is shown to exist and a basic reproduction number $\mathcal R_0 $ is computed using the next generation matrix approach. The DFE is shown to be locally asymptotically stable if $\mathcal R_0<1$ and unstable if $\mathcal R_0>1$.
Special cases of the general model are studied, where there is only one immune stage, either $I^t$ or $I^c$. In the first model, $SI^cR^c$, it is assumed that all infected rodents are immunocompetent and recover. In the second model, $SI^t$, it is assumed that all infected rodents are immunotolerant. For each of these models, the basic reproduction numbers are computed and their relationship to the basic reproduction number of the general model determined. For the $SI^t$ model, it is shown that bistability may occur, the DFE and an enzootic equilibrium, with all rodents infectious, are locally asymptotically stable for the same set of parameter values. A simplification of the $SI^t $ model yields a third model, where the sexes are not differentiated, and therefore, there is no sexual transmission. For this third simplified model, the dynamics are completely analyzed. It is shown that there exists a DFE and possibly two additional equilibria, one of which is globally asymptotically stable for any given set of parameter values; bistability does not occur. Numerical examples illustrate the dynamics of the models. The biological implications of the results and future research goals are discussed in the conclusion.
Chandrani Banerjee, Linda J. S. Allen, Jorge Salazar-Bravo. Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission. Mathematical Biosciences & Engineering, 2008, 5(4): 617-645. doi: 10.3934/mbe.2008.5.617.
Quantifying uncertainty in the estimation of probability distributions
H.T. Banks and Jimena L. Davis
We consider ordinary least squares parameter estimation problems where the unknown parameters to be estimated are probability distributions. A computational framework for quantification of uncertainty (e.g., standard errors) associated with the estimated parameters is given and sample numerical findings are presented.
H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5(4): 647-667. doi: 10.3934/mbe.2008.5.647.
Optimal control applied to a model for species augmentation
Erin N. Bodine, Louis J. Gross and Suzanne Lenhart
Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this paper, we develop a differential equations model and optimal control formulation for a continuous time augmentation of a general declining population. We find a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. The numerical results provide considerably more detail about the exact dynamics of optimal augmentation than can be readily intuited. The work and results presented in this paper are a first step toward building a general theory of population augmentation, which accounts for the complexities inherent in many conservation biology applications.
Erin N. Bodine, Louis J. Gross, Suzanne Lenhart. Optimal control applied to a model for species augmentation. Mathematical Biosciences & Engineering, 2008, 5(4): 669-680. doi: 10.3934/mbe.2008.5.669.
Age-of-infection and the final size relation
Fred Brauer
We establish the final size equation for a general age-of-infection epidemic model in a new simpler form if there are no disease deaths (total population size remains constant). If there are disease deaths, the final size relation is an inequality but we obtain an estimate for the final epidemic size.
Fred Brauer. Age-of-infection and the final size relation. Mathematical Biosciences & Engineering, 2008, 5(4): 681-690. doi: 10.3934/mbe.2008.5.681.
Artificial neural networks and remote sensing in the analysis of the highly variable Pampean shallow lakes
Graciela Canziani, Rosana Ferrati, Claudia Marinelli and Federico Dukatz
2008, 5(4): 691-711 doi: 10.3934/mbe.2008.5.691 +[Abstract](2484) +[PDF](2927.5KB)
Suspended organic and inorganic particles, resulting from the interactions among biological, physical, and chemical variables, modify the optical properties of water bodies and condition the trophic chain. The analysis of their optic properties through the spectral signatures obtained from satellite images allows us to infer the trophic state of the shallow lakes and generate a real time tool for studying the dynamics of shallow lakes. Field data (chlorophyll-a, total solids, and Secchi disk depth) allow us to define levels of turbidity and to characterize the shallow lakes under study. Using bands 2 and 4 of LandSat 5 TM and LandSat 7 ETM+ images and constructing adequate artificial neural network models (ANN), a classification of shallow lakes according to their turbidity is obtained. ANN models are also used to determine chlorophyll-a and total suspended solids concentrations from satellite image data. The results are statistically significant. The integration of field and remote sensors data makes it possible to retrieve information on shallow lake systems at broad spatial and temporal scales. This is necessary to understanding the mechanisms that affect the trophic structure of these ecosystems.
Graciela Canziani, Rosana Ferrati, Claudia Marinelli, Federico Dukatz. Artificial neural networks and remote sensing in the analysis of the highly variable Pampean shallow lakes. Mathematical Biosciences & Engineering, 2008, 5(4): 691-711. doi: 10.3934/mbe.2008.5.691.
Spatial spread of sexually transmitted diseases within susceptible populations at demographic steady state
Carlos Castillo-Chavez and Bingtuan Li
In this study, we expand on the susceptible-infected-susceptible (SIS) heterosexual mixing setting by including the movement of individuals of both genders in a spatial domain in order to more comprehensively address the transmission dynamics of competing strains of sexually-transmitted pathogens. In prior models, these transmission dynamics have only been studied in the context of nonexplicitly mobile heterosexually active populations at the demographic steady state, or, explicitly in the simplest context of SIS frameworks whose limiting systems are order preserving. We introduce reaction-diffusion equations to study the dynamics of sexually-transmitted diseases (STDs) in spatially mobile heterosexually active populations. To accomplish this, we study a single-strain STD model, and discuss in what forms and at what speed the disease spreads to noninfected regions as it expands its spatial range. The dynamics of two competing distinct strains of the same pathogen on this population are then considered. The focus is on the investigation of the spatial transition dynamics between the two endemic equilibria supported by the nonspatial corresponding model. We establish conditions for the successful invasion of a population living in endemic conditions by introducing a strain with higher fitness. It is shown that there exists a unique spreading speed (where the spreading speed is characterized as the slowest speed of a class of traveling waves connecting two endemic equilibria) at which the infectious population carrying the invading stronger strain spreads into the space where an equilibrium distribution has been established by the population with the weaker strain. Finally, we give sufficient conditions under which an explicit formula for the spreading speed can be found.
Carlos Castillo-Chavez, Bingtuan Li. Spatial spread of sexually transmitted diseases within susceptible populations at demographic steady state. Mathematical Biosciences & Engineering, 2008, 5(4): 713-727. doi: 10.3934/mbe.2008.5.713.
Modeling evolution and persistence of neurological viral diseases in wild populations
Dobromir T. Dimitrov and Aaron A. King
Viral infections are one of the leading source of mortality worldwide. The great majority of them circulate and persist in wild reservoirs and periodically spill over into humans or domestic animals. In the wild reservoirs, the progression of disease is frequently quite different from that in spillover hosts. We propose a mathematical treatment of the dynamics of viral infections in wild mammals using models with alternative outcomes. We develop and analyze compartmental epizootic models assuming permanent or temporary immunity of the individuals surviving infections and apply them to rabies in bats. We identify parameter relations that support the existing patterns in the viral ecology and estimate those parameters that are unattainable through direct measurement. We also investigate how the duration of the acquired immunity affects the disease and population dynamics.
Dobromir T. Dimitrov, Aaron A. King. Modeling evolution and persistence of neurological viral diseases in wild populations. Mathematical Biosciences & Engineering, 2008, 5(4): 729-741. doi: 10.3934/mbe.2008.5.729.
Bat population dynamics: multilevel model based on individuals' energetics
Paula Federico, Dobromir T. Dimitrov and Gary F. McCracken
Temperate-zone bats are subject to serious energetic constraints due to their high surface area to volume relations, the cost of temperature regulation, the high metabolic cost of flight, and the seasonality of their resources. We present a novel, multilevel theoretical approach that integrates information on bat biology collected at a lower level of organization, the individual with its physiological characteristics, into a modeling framework at a higher level, the population. Our individual component describes the growth of an individual female bat by modeling the dynamics of the main body compartments (lipids, proteins, and carbohydrates). A structured population model based on extended McKendrick-von Foerster partial differential equations integrates those individual dynamics and provides insight into possible regulatory mechanisms of population size as well as conditions of population survival and extinction. Though parameterized for a specific bat species, all modeling components can be modified to investigate other bats with similar life histories. A better understanding of population dynamics in bats can assist in the development of management techniques and conservation strategies, and to investigate stress effects. Studying population dynamics of bats presents particular challenges, but bats are essential in some areas of concern in conservation and disease ecology that demand immediate investigation.
Paula Federico, Dobromir T. Dimitrov, Gary F. McCracken. Bat population dynamics: multilevel model based on individuals\' energetics. Mathematical Biosciences & Engineering, 2008, 5(4): 743-756. doi: 10.3934/mbe.2008.5.743.
Modeling the effect of information campaigns on the HIV epidemic in Uganda
Hem Joshi, Suzanne Lenhart, Kendra Albright and Kevin Gipson
The increasing prevalence of HIV/AIDS in Africa over the past twenty-five years continues to erode the continent's health care and overall welfare. There have been various responses to the pandemic, led by Uganda, which has had the greatest success in combating the disease. Part of Uganda's success has been attributed to a formalized information, education, and communication (IEC) strategy, lowering estimated HIV/AIDS infection rates from 18.5% in 1995 to 4.1% in 2003. We formulate a model to investigate the effects of information and education campaigns on the HIV epidemic in Uganda. These campaigns affect people's behavior and can divide the susceptibles class into subclasses with different infectivity rates. Our model is a system of ordinary differential equations and we use data about the epidemics and the number of organizations involved in the campaigns to estimate the model parameters. We compare our model with three types of susceptibles to a standard SIR model.
Hem Joshi, Suzanne Lenhart, Kendra Albright, Kevin Gipson. Modeling the effect of information campaigns on the HIV epidemic in Uganda. Mathematical Biosciences & Engineering, 2008, 5(4): 757-770. doi: 10.3934/mbe.2008.5.757.
Model analysis of a simple aquatic ecosystems with sublethal toxic effects
B. W. Kooi, D. Bontje and M. Liebig
The dynamic behaviour of simple aquatic ecosystems with nutrient recycling in a chemostat, stressed by limited food availability and a toxicant, is analysed. The aim is to find effects of toxicants on the structure and functioning of the ecosystem. The starting point is an unstressed ecosystem model for nutrients, populations, detritus and their intra- and interspecific interactions, as well as the interaction with the physical environment. The fate of the toxicant includes transport and exchange between the water and the populations via two routes, directly from water via diffusion over the outer membrane of the organism and via consumption of contaminated food. These processes are modelled using mass-balance formulations and diffusion equations. At the population level the toxicant affects different biotic processes such as assimilation, growth, maintenance, reproduction, and survival, thereby changing their biological functioning. This is modelled by taking the parameters that described these processes to be dependent on the internal toxicant concentration. As a consequence, the structure of the ecosystem, that is its species composition, persistence, extinction or invasion of species and dynamics behaviour, steady state oscillatory and chaotic, can change. To analyse the long-term dynamics we use the bifurcation analysis approach. In ecotoxicological studies the concentration of the toxicant in the environment can be taken as the bifurcation parameter. The value of the concentration at a bifurcation point marks a structural change of the ecosystem. This indicates that chemical stressors are analysed mathematically in the same way as environmental (e.g. temperature) and ecological (e.g. predation) stressors. Hence, this allows an integrated approach where different type of stressors are analysed simultaneously. Environmental regimes and toxic stress levels at which no toxic effects occur and where the ecosystem is resistant will be derived. A numerical continuation technique to calculate the boundaries of these regions will be given.
B. W. Kooi, D. Bontje, M. Liebig. Model analysis of a simple aquatic ecosystems with sublethal toxic effects. Mathematical Biosciences & Engineering, 2008, 5(4): 771-787. doi: 10.3934/mbe.2008.5.771.
A malaria model with partial immunity in humans
Jia Li
In this paper, we formulate a mathematical model for malaria transmission that includes incubation periods for both infected human hosts and mosquitoes. We assume humans gain partial immunity after infection and divide the infected human population into subgroups based on their infection history. We derive an explicit formula for the reproductive number of infection, $R_0$, to determine threshold conditions whether the disease spreads or dies out. We show that there exists an endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that models having the same reproductive number but different numbers of progression stages can exhibit different transient transmission dynamics.
Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5(4): 789-801. doi: 10.3934/mbe.2008.5.789.
The existence of positive periodic solutions of a generalized
Meili Li, Maoan Han and Chunhai Kou
In this paper, the existence of positive periodic solutions of a class of periodic $n$-species Gilpin-Ayala impulsive competition systems is studied. By using the continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions is obtained. Our results are general enough to include some known results in this area.
Meili Li, Maoan Han, Chunhai Kou. The existence of positive periodic solutions of a generalized. Mathematical Biosciences & Engineering, 2008, 5(4): 803-812. doi: 10.3934/mbe.2008.5.803.
Local resource competition and the skewness of the sex ratio: a demographic model
Lorenzo Mari, Marino Gatto and Renato Casagrandi
Most animal populations are characterized by balanced sex ratios, but there exist several exceptions in which the sex ratio at birth is skewed. An interesting hypothesis proposed by Clark (1978) to explain male-biased sex ratios is the local resource competition theory: the bias may be expected in those species in which males disperse more than females, which are thus more prone to local competition for resources. Here we discuss some of the ideas underlying Clark's theory using a spatially explicit approach. In particular, we focus on the role of spatiotemporal heterogeneity as a possible determinant of biased sex ratios. We model spatially structured semelparous populations where either Ricker density dependence or environmental stochasticity can generate irregular spatiotemporal patterns. The proposed discrete-time model describes both genetic and complex population dynamics assuming that (1) sex ratio is genetically determined, (2) only young males can disperse, and (3) individuals locally compete for resources. The analysis of the model shows that no skewed sex ratios can arise in homogeneous habitats. Temporal asynchronized fluctuations between two distinct patches coupled with dispersal of young males is the minimum requirement for obtaining skewed sex ratios of demographic nature in local adult populations. However, the establishment of a male-biased sex ratio at birth in the long run is possible if dispersal is genetically determined and there is genetic linkage between sex ratio determination and dispersal.
Lorenzo Mari, Marino Gatto, Renato Casagrandi. Local resource competition and the skewness of the sex ratio: a demographic model. Mathematical Biosciences & Engineering, 2008, 5(4): 813-830. doi: 10.3934/mbe.2008.5.813.
Modeling the daily activities of breeding colonial seabirds: Dynamic occupancy patterns in multiple habitat patches
Andrea L. Moore, Smruti P. Damania, Shandelle M. Henson and James L. Hayward
We constructed differential equation models for the diurnal abundance and distribution of breeding glaucous-winged gulls (Larus glaucescens) as they moved among nesting and non-nesting habitat patches. We used time scale techniques to reduce the differential equations to algebraic equations and connected the models to field data. The models explained the data as a function of abiotic environmental variables with $R^{2}=0.57$. A primary goal of this study is to demonstrate the utility of a methodology that can be used by ecologists and wildlife managers to understand and predict daily activity patterns in breeding seabirds.
Andrea L. Moore, Smruti P. Damania, Shandelle M. Henson, James L. Hayward. Modeling the daily activities of breeding colonial seabirds: Dynamic occupancy patterns in multiple habitat patches. Mathematical Biosciences & Engineering, 2008, 5(4): 831-842. doi: 10.3934/mbe.2008.5.831.
Density-dependent dispersal in multiple species metapopulations
Jacques A. L. Silva and Flávia T. Giordani
A multiple species metapopulations model with density-dependent dispersal is presented. Assuming the network configuration matrix to be diagonizable we obtain a decoupling of the associated perturbed system from the homogeneous state. It was possible to analyze in detail the instability induced by the density-dependent dispersal in two classes of $k$-species interaction models: a hierarchically organized competitive system and an age-structured model.
Jacques A. L. Silva, Fl\u00E1via T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5(4): 843-857. doi: 10.3934/mbe.2008.5.843.
Variation in risk in single-species discrete-time models
Abhyudai Singh and Roger M. Nisbet
Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, $R$, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of $R$. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.
Abhyudai Singh, Roger M. Nisbet. Variation in risk in single-species discrete-time models. Mathematical Biosciences & Engineering, 2008, 5(4): 859-875. doi: 10.3934/mbe.2008.5.859.
Food web dynamics in a seasonally varying wetland
Donald L. DeAngelis, Joel C. Trexler and Douglas D. Donalson
A spatially explicit model is developed to simulate the small fish community and its underlying food web, in the freshwater marshes of the Everglades. The community is simplified to a few small fish species feeding on periphyton and invertebrates. Other compartments are detritus, crayfish, and a piscivorous fish species. This unit food web model is applied to each of the 10,000 spatial cells on a 100 x 100 pixel landscape. Seasonal variation in water level is assumed and rules are assigned for fish movement in response to rising and falling water levels, which can cause many spatial cells to alternate between flooded and dry conditions. It is shown that temporal variations of water level on a spatially heterogeneous landscape can maintain at least three competing fish species. In addition, these environmental factors can strongly affect the temporal variation of the food web caused by top-down control from the piscivorous fish.
Donald L. DeAngelis, Joel C. Trexler, Douglas D. Donalson. Food web dynamics in a seasonally varying wetland. Mathematical Biosciences & Engineering, 2008, 5(4): 877-887. doi: 10.3934/mbe.2008.5.877.
Modeling frequency-dependent selection with an application to cichlid fish
Sheree L. Arpin and J. M. Cushing
Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis exhibit lateral asymmetry with respect to jaw morphology: the mouth either opens to the right or left side of the body. Field data illustrate a temporal phenotypic oscillation in the mouth-handedness, and this oscillation is maintained by frequency-dependent selection. Since both genetic and population dynamics occur on the same time scale in this case, we develop a (discrete time) model for P. microlepis populations that accounts for both dynamic processes. We establish conditions on model parameters under which the model predicts extinction and conditions under which there exists a unique positive (survival) equilibrium. We show that at the positive equilibrium there is a 1:1 phenotypic ratio. Using a local stability and bifurcation analysis, we give further conditions under which the positive equilibrium is stable and conditions under which it is unstable. Destabilization results in a bifurcation to a periodic oscillation and occurs when frequency-dependent selection is sufficiently strong. This bifurcation is offered as an explanation of the phenotypic frequency oscillations observed in P. microlepis. An analysis of the bifurcating periodic cycle results in some interesting and unexpected predictions.
Sheree L. Arpin, J. M. Cushing. Modeling frequency-dependent selection with an application to cichlid fish. Mathematical Biosciences & Engineering, 2008, 5(4): 889-903. doi: 10.3934/mbe.2008.5.889.
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CommonCrawl
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Nonlinear Differential Equations and Applications NoDEA
February 2020 , 27:10 | Cite as
Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions
Peter S. Morfe
We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical \(\epsilon ^{\frac{1}{2}}\) rate, while we obtain an \(\epsilon ^{\frac{1}{7}}\) rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.
Hamilton–Jacobi equations Junction problems Stratification problems Vanishing viscosity limit Monotone finite difference schemes
35F20 65N12 65M12
It is a pleasure to acknowledge P.E. Souganidis for suggesting this problem and for enlightening discussions. Credit is due as well to the anonymous reviewers for their sage advice and for pointing out a number of typos, and to M. Sardarli for helpful comments. The author was partially supported by the National Science Foundation Research Training Group Grant DMS-1246999.
Appendix A: Reformulated Kirchoff condition
We present an equivalent definition of viscosity solutions of Kirchoff problems for the general problem:
$$\begin{aligned} \left\{ \begin{array}{l l} F_{i}(x,u,u_{x_{i}}) = 0 &{} \text {in} \, \, I_{i} \\ {\sum }_{i = 1}^{K} u_{x_{i}} = B &{} \text {on} \, \, \{0\} \end{array} \right. \end{aligned}$$
where \(F_{i} : \overline{I_{i}} \times {\mathbb {R}} \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is continuous for each i. We start with sub-solutions:
\(u \in USC({\mathcal {I}})\) is a sub-solution of (82) if and only if for each \(\varphi \in C^{2}({\mathcal {I}})\) and any local maximum \(x_{0}\) of \(u - \varphi \), the following inequalities are satisfied:
$$\begin{aligned} \left\{ \begin{array}{l l} F_{i}(x_{0},u(x_{0}),\varphi _{x_{i}}(x_{0})) \le 0 &{} \text {if} \, \, x_{0} \in I_{i} \\ \min _{i, {\tilde{\theta }} \in [0,1]} F_{i}\left( 0,u(0),\varphi _{x_{i}}(0) + {\tilde{\theta }} \left( {\sum }_{j = 1}^{K} \varphi _{x_{i}}(0) - B\right) ^{-}\right) \le 0 &{} \text {if} \, \, x_{0} = 0 \end{array} \right. \end{aligned}$$
In fact, the proposition only requires \(\varphi \in C^{1}({\mathcal {I}})\), but we will not expand on that here.
Before proceeding, we need to recall the definitions of first-order differential sub-jets and super-jets. Given an upper semi-continuous function \(u : {\mathcal {I}} \rightarrow {\mathbb {R}}\) and an \(x \in \overline{I_{i}}\), we say that \(p \in J^{+}_{i}u(x)\) if and only if
$$\begin{aligned} u(y) \le u(x) + p(y - x) + o(|y - x|) \quad \text {if} \, \, y \in \overline{I_{i}}, \end{aligned}$$
where \(\lim _{y \rightarrow x} \frac{o(|y - x|)}{|y - x|} = 0\). Similarly, given a lower semi-continuous function \(v : {\mathcal {I}} \rightarrow {\mathbb {R}}\) and an \(x \in \overline{I_{i}}\), we say that \(q \in J^{-}_{i}v(x)\) if and only if
$$\begin{aligned} v(y) \ge v(x) + q(y - x) + o(|y-x|) \quad \text {if} \, \, y \in \overline{I_{i}}. \end{aligned}$$
Note that there is a \(\varphi \in C^{2}({\mathcal {I}})\) such that \(u - \varphi \) has a local maximum at 0 if and only if \(\xi _{i} := \varphi _{x_{i}}(0) \in J_{i}^{+}u(0)\) for each i. Therefore, in what follows, we often work with K-tuples \((\xi _{1},\dots ,\xi _{K})\) instead of test functions.
As in the Neumann problem, Proposition 10 rests on the next lemma.
Lemma 9
Fix \(i \in \{1,\dots ,K\}\) and assume that u is an upper semi-continuous sub-solution of
$$\begin{aligned} F_{i}(x,u,u_{x_{i}}) = 0 \quad \text {in} \, \, I_{i}. \end{aligned}$$
Let \(\xi _{i} \in J_{i}^{+}u(0)\) and set \(\lambda _{i,0} = \sup \{\lambda \ge 0 \, \mid \, \xi _{i} + \lambda \in J^{+}_{i}u(0) \}\). If \(\lambda _{i,0} < \infty \), then
$$\begin{aligned} F_{i}(0,u(0),\xi _{i} + \lambda _{i,0}) \le 0. \end{aligned}$$
The proof is the same as the one in [18, Lemma 3]. For the sake of completeness, we reproduce it here. Since \(J_{i}^{+}u(0)\) is closed, we can pick \(\psi \in C^{2}(\overline{I_{i}})\) such that \(u - \psi \) has a strict local maximum at 0 in \(\overline{I_{i}}\), \(u(0) = \psi (0)\), and \(\psi _{x_{i}}(0) = \xi _{i} + \lambda _{i,0}\). Fix \(\delta >0\), set \(\mu (\delta ) = \psi (-\delta _{i}) - u(-\delta _{i})\) and \(\alpha (\delta ) = \min \{\delta ,\frac{\mu (\delta )}{2 \delta }\}\), and let \(x_{\delta }\) be a maximum of \(u - \psi - \alpha (\delta )x\) in \(\overline{I^{\delta }_{i}} \subseteq \overline{I_{i}}\).
Observe that \(x_{\delta } \ne 0\). Indeed, if \(x_{\delta } = 0\), then
$$\begin{aligned} \xi _{i} + \lambda _{i,0} + \alpha (\delta ) =\psi _{x_{i}}(0) + \alpha (\delta ) \in J^{+}u_{i}(0), \end{aligned}$$
contradicting the definition of \(\lambda _{i,0}\).
Additionally, \(x_{\delta } \ne -\delta _{i}\) since
$$\begin{aligned} u(-\delta _{i}) - \psi (-\delta _{i}) + \alpha (\delta )\delta&\le \frac{u(-\delta _{i}) - \psi (-\delta _{i})}{2} \\&< 0 = u(0) - \psi (0) < u(x_{\delta }) - \psi (x_{\delta }) - \alpha (\delta ) x_{\delta }. \end{aligned}$$
Thus, \(x_{\delta } \in (0,\delta )\) and the sub-solution property of u gives
$$\begin{aligned} F_{i}(x_{\delta },u(x_{\delta }),\psi _{x_{i}}(x_{\delta }) + \alpha (\delta )) \le 0. \end{aligned}$$
From the inequality \(u(0) - \psi (0) < u(x_{\delta }) - \psi (x_{\delta }) - \alpha (\delta ) x_{\delta }\), the upper semi-continuity of u implies \(u(x_{\delta }) \rightarrow u(0)\). Therefore, observing that
$$\begin{aligned} \lim _{\delta \rightarrow 0^{+}} (x_{\delta },\alpha (\delta ),u(x_{\delta })) = (0,0,u(0)), \end{aligned}$$
the result follows. \(\square \)
In the proof of Proposition 10 we will use the following fact about sub-solutions of (82). In fact, the corresponding property of the Neumann problem is actually embedded in the definition in [18].
Suppose u is a sub-solution of (82), \(\varphi \in C^{2}({\mathcal {I}})\), and \(u - \varphi \) has a local maximum at 0. If \({\sum }_{i = 1}^{K} \varphi _{x_{i}}(0) \ge B\), then \(\min _{i} F_{i}(0,u(0),\varphi _{x_{i}}(0)) \le 0\).
By the definition of sub-solution, it suffices to consider the case when \({\sum }_{i = 1}^{K} \varphi _{x_{i}}(0) = B\). Define \((\xi _{1},\dots ,\xi _{K})\) by \(\xi _{i} = \varphi _{x_{i}}(0)\) and let \((\lambda _{1,0},\dots ,\lambda _{K,0})\) be defined as in Lemma 9. If \(\lambda _{j,0} = 0\) for some j, then Lemma 9 implies \(F_{j}(0,u(0),\xi _{j}) \le 0\). On the other hand, if \(\min _{i}\lambda _{i,0} > 0\), then, for small enough \(\delta > 0\), \(\xi _{i} + \delta \in J_{i}^{+}u(0)\) holds, no matter the choice of i. From \({\sum }_{i = 1}^{K} (\xi _{i} + \delta ) = B + \delta K\) and the sub-solution property, we find \(\min _{i}F_{i}(0,u(0),\xi _{i} + \delta ) \le 0\). We conclude by sending \(\delta \rightarrow 0^{+}\). \(\square \)
We continue with the
Proof of Proposition 10
Since one direction is immediate, here we prove only the "only if" statement.
Suppose \((\xi _{1},\dots ,\xi _{K})\) is a K-tuple satisfying \(\xi _{i} \in J^{+}_{i}u(0)\) for each i.
In what follows, we use the notation in the statement of Lemma 9. If there is a \(j \in \{1,2,\dots ,K\}\) such that \(\left( {\sum }_{i = 1}^{K} \xi _{i} -B\right) ^{-} < \lambda _{j,0}\), then let \({\tilde{\xi }}_{k} = \xi _{k}\) if \(k \ne j\) and \({\tilde{\xi }}_{j} = \xi _{j} + \left( {\sum }_{i = 1}^{K} \xi _{i} -B\right) ^{-}\). For each i, \({\tilde{\xi }}_{i} \in J_{i}^{+} u(0)\) and
$$\begin{aligned} \sum _{i = 1}^{K} {\tilde{\xi }}_{i} = \left( \sum _{i = 1}^{K} \xi _{i}\right) + \left( \sum _{j = 1}^{K} \xi _{j} - B\right) ^{-} \ge B. \end{aligned}$$
Thus, Proposition 11 implies \(\min _{i} F_{i} \left( 0,u(0), {\tilde{\xi }}_{i} \right) \le 0\). From this and the definition of \(({\tilde{\xi }}_{1},\dots ,{\tilde{\xi }}_{K})\), we conclude
$$\begin{aligned} \min _{j} \min _{{\tilde{\theta }} \in [0,1]} F_{j}\left( 0,u(0), \xi _{j} + {\tilde{\theta }} \left( \sum _{i = 1}^{K} \xi _{i} -B\right) ^{-} \right) \le 0. \end{aligned}$$
It only remains to consider the case when \(\lambda _{j,0} \le \left( {\sum }_{i = 1}^{K} \xi _{i}-B\right) ^{-}\) independently of the choice of j. In this case, we can fix \(({\tilde{\theta }}_{1},{\tilde{\theta }}_{2},\dots ,{\tilde{\theta }}_{K}) \in [0,1]^{K}\) such that \(\lambda _{j,0} = {\tilde{\theta }}_{j} \left( {\sum }_{i = 1}^{K} \xi _{i} - B\right) ^{-}\). Therefore, Lemma 9 yields that, for each j,
$$\begin{aligned} \min _{{\tilde{\theta }} \in [0,1]} F_{j} \left( 0,u(0), \xi _{j} + {\tilde{\theta }} \left( \sum _{i = 1}^{K} \xi _{i} - B\right) ^{-} \right) \le F_{j}(0,u(0),\xi _{j} + \lambda _{j,0}) \le 0. \end{aligned}$$
\(\square \)
The result for super-solutions is stated next. Since the proof is so similar, we omit the details.
A function \(v \in \text {LSC}({\mathcal {I}})\) is a viscosity super-solution of (82) if and only if for each \(\varphi \in C^{2}({\mathcal {I}})\) and any local minimum \(x_{0}\) of \(u - \varphi \), the following inequalities are satisfied:
$$\begin{aligned} \left\{ \begin{array}{l l} F_{i}(x_{0},v(x_{0}),\varphi _{x_{i}}(x_{0})) \ge 0 &{} \text {if} \, \, x_{0} \in I_{i}\\ \max _{i, {\tilde{\theta }} \in [0,1]} F_{i}\left( 0,v(0),\varphi _{x_{i}}(0) - {\tilde{\theta }} \left( {\sum }_{j = 1}^{K} \varphi _{x_{i}}(0) -B \right) ^{+}\right) \ge 0 &{} \text {if} \, \, x_{0} = 0 \end{array} \right. \end{aligned}$$
There is an analogous reformulation of time-dependent equations like (2). We do not prove it here since we have no immediate use for it and it does not simplify the uniqueness proof presented in Sect. 3.
Appendix B: Dimensionality reduction lemma
In this section, we show how to obtain time-independent equations from those in which time-derivatives do not appear. The following result implies Proposition 3:
Lemma 10
Assume that, for each i, \(F_{i} : [0,T] \times {\mathcal {I}} \times {\mathbb {R}}^{2} \rightarrow {\mathbb {R}}\) is a continuous function, and fix \(B \in {\mathbb {R}}\) and \(\delta > 0\). Let the upper semi-continuous function \(u : \bigcup _{i = 1}^{K} \overline{I_{i}^{\delta }} \times [0,T] \rightarrow {\mathbb {R}}\) be a sub-solution of
$$\begin{aligned} \left\{ \begin{array}{l l} F_{i}(t,x,u,u_{x_{i}}) = 0 &{} \text {in} \, \, I_{i}^{\delta } \times (0,T) \\ {\sum }_{i = 1}^{K} u_{x_{i}} = B &{} \text {on} \, \, \{0\} \times (0,T) \end{array} \right. \end{aligned}$$
For each \(t_{0} \in (0,T]\), the function \(u(\cdot ,t_{0}) : \bigcup _{i = 1}^{K} \overline{I_{i}^{\delta }} \rightarrow {\mathbb {R}}\) is a sub-solution of
$$\begin{aligned} \left\{ \begin{array}{l l} F_{i}(t_{0},x,u(\cdot ,t_{0}),u_{x_{i}}(\cdot ,t_{0})) = 0 &{} \text {in} \, \, I_{i}^{\delta } \\ {\sum }_{i = 1}^{K} u_{x_{i}}(\cdot ,t_{0}) = B &{} \text {on} \, \, \{0\} \end{array} \right. \end{aligned}$$
We remark that a version of Lemma 10 for super-solutions follows from it by replacing u by \(-u\).
Fix a \(t_{0} \in (0,T]\). Given \(\varphi \in C^{2}({\mathcal {I}})\), suppose \(u(\cdot ,t_{0})- \varphi \) has a strict global maximum in \(\bigcup _{i = 1}^{K} \overline{I_{i}^{\delta }}\) at \(x_{0} \in \bigcup _{i = 1}^{K} I_{i}^{\delta }\). We consider only the case when \(x_{0} = 0\), the other case being slightly easier.
For each \(\epsilon > 0\), let \(\Phi _{\epsilon } : \bigcup _{i = 1}^{K} \overline{I_{i}^{\delta }} \times [0,T] \rightarrow {\mathbb {R}}\) be the function given by
$$\begin{aligned} \Phi _{\epsilon }(x,t) = u(x,t) - \varphi (x) - \frac{(t- t_{0})^{2}}{2 \epsilon }. \end{aligned}$$
Write \(\Phi _{\epsilon }(x,t) = u(x,t) - \Psi _{\epsilon }(x,t)\) and note that \(\Psi _{\epsilon } \in C^{2,1}({\mathcal {I}} \times [0,T])\). Let \((x_{\epsilon }, t_{\epsilon })\) denote a maximum point of \(\Phi _{\epsilon }\) in its domain. Since 0 is a strict global maximum of \(u(\cdot ,t_{0}) - \varphi \), it follows that \(t_{\epsilon } \rightarrow t_{0}\), \(x_{\epsilon } \rightarrow 0\), and \(u(x_{\epsilon },t_{\epsilon }) \rightarrow u(0,t_{0})\) as \(\epsilon \rightarrow 0^{+}\). Fix \(\epsilon _{1} > 0\) such that \(t_{\epsilon } > 0\) if \(\epsilon \in (0,\epsilon _{1})\). If there is a sequence \(\epsilon _{n} \rightarrow 0\) such that \(x_{\epsilon _{n}} \in I_{j}\) for some j and each n, then we immediately obtain
$$\begin{aligned} F_{j}(t_{\epsilon _{n}},x_{\epsilon _{n}},u(x_{\epsilon _{j}},t_{\epsilon _{j}}),\varphi _{x_{j}}(x_{\epsilon _{n}})) = F_{j}(t_{\epsilon _{n}}, x_{\epsilon _{n}}, u(x_{\epsilon _{j}},t_{\epsilon _{j}}),\Psi _{x_{j}}(x_{\epsilon _{n}},t_{\epsilon _{n}})) \le 0. \end{aligned}$$
Sending \(n \rightarrow \infty \), we recover \(F_{j}(t_{0},0,u(x_{0},t_{0}),\varphi _{x_{j}}(0)) \le 0\).
It remains to consider the case when there is an \(\epsilon _{2} > 0\) such that \(x_{\epsilon } = 0\) for all \(\epsilon \in (0,\epsilon _{2})\). Fix such an \(\epsilon \). For each \(j \in \{1,2,\dots ,K\}\), the map
$$\begin{aligned} (x,t) \mapsto u(x,t) - \varphi (x) - \frac{(t - t_{0})^{2}}{2 \epsilon } \end{aligned}$$
defined in \(\overline{I_{j}^{\delta }} \times [0,T]\) has a local maximum at \((0,t_{\epsilon })\). Thus,
$$\begin{aligned} \min \left\{ \min _{i} F_{i}(t_{\epsilon },0,u(0,t_{\epsilon }),\varphi _{x_{i}}(0)), \sum _{i = 1}^{K} \varphi _{x_{i}}(0) - B \right\} \le 0. \end{aligned}$$
We conclude by sending \(\epsilon \rightarrow 0^{+}\) and appealing to continuity of the functions \(F_{1},\dots ,F_{K}\). \(\square \)
Appendix C: Existence of solutions of the Cauchy problems
In this section, we prove the existence of solutions of (2) and (4). The main results proved herein follow:
Theorem 11
If \(u_{0} \in \text {UC} \left( {\mathcal {I}} \right) \), then there is a \(u \in \text {UC}({\mathcal {I}} \times [0,T])\) solving (2). If, in addition, \(u_{0} \in \text {Lip}({\mathcal {I}})\), then \(u \in \text {Lip}({\mathcal {I}} \times [0,T])\), and \(\text {Lip}(u)\) depends on \(u_{0}\) only through \(\text {Lip}(u_{0})\).
Fix \(\epsilon > 0\). If \(u_{0} \in \text {UC} \left( {\mathcal {I}} \right) \), then there is a \(u^{\epsilon } \in \text {UC}({\mathcal {I}} \times [0,T])\) solving (4). Moreover, if \([u_{0}]_{1} + [u_{0}]_{2} < \infty \), then there is a \(C > 0\) depending only on \([u_{0}]_{1} + \epsilon [u_{0}]_{2}\) such that \(\text {Lip}(u^{\epsilon }) \le C\).
We prove Theorem 11 by sending \(\epsilon \rightarrow 0^{+}\) in Theorem 12. Therefore, the main thrust of this section is the proof of Theorem 12 and associated estimates.
The proof of Theorem 12 is divided into three steps. First, we use the estimate proved by von Below in [22] and Schaefer's fixed point theorem to obtain solutions of (4) when \(u_{0}\) is a smooth function satisfying some compatibility conditions. Next, we prove Lipschitz estimates when the initial data is sufficiently regular. Finally, we approximate arbitrary initial data by smooth data and use the comparison principle to pass to the limit.
Recall that in Remark 8 above we observed that an alternative proof of Theorem 11 can be obtained using the finite-difference scheme (6) and the method of half-relaxed limits.
C.1: Existence for regular data
Here we obtain solutions of (4) using a priori Hölder estimates for linear parabolic equations on networks and Schaefer's fixed point theorem.
To begin with, for a given \(R > 0\), we let \(\{{\tilde{H}}_{1}^{(R)},\dots ,{\tilde{H}}_{K}^{(R)}\}\) take the form
$$\begin{aligned} {\tilde{H}}_{i}^{(R)}(t,x,p) = \psi ^{(R)}(p) H_{i}(t,x,p) + (1 - \psi ^{(R)}(p)) R, \end{aligned}$$
where \(\psi ^{(R)} : {\mathbb {R}} \rightarrow [0,1]\) is a smooth cut-off function satisfying \(\psi ^{(R)}(p) = 1\) if \(|p| \le \frac{R}{2}\) and \(\psi ^{(R)}(p) = 0\) if \(|p| \ge R\). Notice that \(\{{\tilde{H}}_{1}^{(R)},\dots ,{\tilde{H}}_{K}^{(R)}\}\) are bounded functions on their respective domains, and the assumptions (7) and (9) continue to hold.
The result is stated next:
Suppose \(a > 0\) and \(u_{0} \in C^{3}({\mathcal {I}})\) satisfies, for each \(i \in \{1,2,\dots ,K\}\),
$$\begin{aligned} \epsilon u_{0,x_{i}x_{i}}(0) - H_{i}(0,0,u_{0,x_{i}}(0))&= \epsilon u_{0,x_{1}x_{1}}(0) - H_{1}(0,0,u_{0,x_{1}}(0)) \end{aligned}$$
$$\begin{aligned} \sum _{i = 1}^{K} u_{0,x_{i}}(0)&= 0 \end{aligned}$$
$$\begin{aligned} _{1} + [u_{0}]_{2} + [u_{0}]_{3}&< \infty \end{aligned}$$
Assume, in addition, that \(R \ge 2 [u_{0}]_{1}\). Then there is a viscosity solution \(u^{(a)}: \bigcup _{i = 1}^{K} \overline{I^{a}_{i}} \times [0,T] \rightarrow {\mathbb {R}}\) of the following equation:
$$\begin{aligned} \left\{ \begin{array}{l l} u^{(a)}_{t} - \epsilon u^{(a)}_{x_{i}x_{i}} + {\tilde{H}}^{(R)}_{i}(t,x,u^{(a)}_{x_{i}}) = 0 &{} \text {in} \, \, I_{i}^{a} \times (0,T) \\ {\sum }_{i = 1}^{K} u^{(a)}_{x_{i}} = 0 &{} \text {on} \, \, \{0\} \times (0,T) \\ u^{(a)} = u_{0} &{} \text {on} \, \, {\bigcup }_{i = 1}^{K} \overline{I_{i}^{a}} \times \{0\} \\ u^{(a)} = \beta _{i} &{} \text {on} \, \, \{-a_{i}\} \times (0,T) \end{array} \right. \end{aligned}$$
where the functions \(\beta _{1},\dots ,\beta _{K} : [0,T] \rightarrow {\mathbb {R}}\) are given by
$$\begin{aligned} \beta _{i}(t) = u_{0}(-a_{i}) + \left( \epsilon u_{0,x_{i}x_{i}}(-a_{i}) - {\tilde{H}}^{(R)}_{i}(0,-a_{i},u_{0,x_{i}}(-a_{i}))\right) t. \end{aligned}$$
For each \(i \in \{1,2,\dots ,K\}\), the functions \(u^{(a)}\), \(u^{(a)}_{t}\), \(u^{(a)}_{x_{i}}\), and \(u^{(a)}_{x_{i}x_{i}}\) are Hölder continuous in \(\overline{I_{i}^{a}} \times [0,T]\).
A similar result has been obtained in [2] starting with weak solutions in \(L^{p}\) spaces.
Our proof of Proposition 13 follows the same general outline presented in [17, Chapter 5]. As in the fixed point arguments contained there, the next remark will play a significant role here. For a proof, see, for example, [17, Lemma 3.1].
Suppose \(I \subseteq {\mathbb {R}}\) is an open interval and \(u : {\overline{I}} \times [0,T] \rightarrow {\mathbb {R}}\) is twice continuously differentiable in space and once continuously differentiable in time. Then \(u_{x}\) is \(\frac{1}{2}\)-Hölder continuous in time with a Hölder constant that only depends on I and the suprema of \(|u_{t}|\) and \(|u_{xx}|\).
It will be convenient in what follows to use the semi-norms \([\cdot ]_{\alpha }\) and \([\cdot ]_{1 + \alpha }\) on functions with domain \(\bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,T]\), abusing the notation somewhat. By this, we mean the semi-norms as defined in Sect. 1.5, but with \(\overline{I_{i}}\) replaced everywhere in the definitions with \(\overline{I_{i}^{a}}\).
First, for each \(\alpha \in (0,1)\), define a norm on functions \(v : \bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,T] \rightarrow {\mathbb {R}}\) by
$$\begin{aligned} \Vert v\Vert _{\alpha } = [v]_{0} + [v]_{\alpha } + \max _{i} \, [v_{x_{i}}]_{i,0} + [v]_{1 + \alpha }. \end{aligned}$$
Let \(V_{\alpha }\) be the Banach space of functions v with \(\Vert v\Vert _{\alpha } < \infty \). We will find the solution as the fixed point of a certain operator on \(V_{\alpha }\).
Fix \(\alpha \in (0,1)\). We claim we can define a compact, continuous operator \(T : V_{\alpha } \rightarrow V_{\alpha }\) so that \(u = T(v)\) solves the equation
$$\begin{aligned} \left\{ \begin{array}{l l} u_{t} - \epsilon u_{x_{i}x_{i}} + {\tilde{H}}^{(R)}_{i}(x,t,v_{x_{i}}) = 0 &{} \text {in} \, \, I_{i}^{a} \times (0,T) \\ {\sum }_{i = 1}^{K} u_{x_{i}} = 0 &{} \text {on} \, \, \{0\} \times (0,T) \\ u = u_{0} &{} \text {on} \, \, {\bigcup }_{i = 1}^{K} \overline{I_{i}^{a}} \times \{0\} \\ u = \beta _{i} &{} \text {on} \, \, \{-a\} \times (0,T) \end{array} \right. \end{aligned}$$
Indeed, since \([v]_{1 + \alpha } < \infty \) and \({\tilde{H}}^{(R)}_{i}(0,0,u_{0,x_{i}}(0)) = H_{i}(0,0,u_{0,x_{i}}(0))\) for all i by the choice of R, the compatibility conditions (85), (86), and (88) imply the result of [22] is applicable. In particular, a bounded solution u of (89) exists and the functions \(u_{t},u_{x_{1}x_{1}},\dots ,u_{x_{K}x_{K}}\) are bounded and continuous. Thus, Remark 9 implies \([u]_{2} < \infty \), and \(u \in V_{\alpha }\) follows. Arguing as in Theorem 9, we see that u is uniquely determined. Therefore, T is well-defined.
We claim that T is compact and continuous. Suppose \((v_{n})_{n \in {\mathbb {N}}} \subseteq V_{\alpha }\) and \(\Vert v_{n}\Vert _{\alpha } \le C\) independently of n. Let \(u^{(n)} = T(v_{n})\). The main result of [22] implies \(u^{(n)},u^{(n)}_{t},u^{(n)}_{x_{1}x_{1}},\dots ,u^{(n)}_{x_{K}x_{K}}\) are bounded continuous functions with bounds depending on \((v_{n})_{n \in {\mathbb {N}}}\) only through the constant C. Thus, Remark 9 implies \([u^{(n)}]_{1} + [u^{(n)}]_{2}\) is uniformly bounded. Since \(\alpha < 1\) and \(\{u^{(n)}(\cdot ,0)\}_{n \in {\mathbb {N}}} = \{u_{0}\}\), it follows that \((u^{(n)})_{n \in {\mathbb {N}}}\) is relatively compact in \(V_{\alpha }\). Therefore, by definition, T is compact. Since solutions of (89) in \(V_{\alpha }\) are unique, if, in addition, \(v_{n} \rightarrow v\) in \(V_{\alpha }\), then it is straightforward to check that T(v) is the only possible subsequential limit point of \((u^{(n)})_{n \in {\mathbb {N}}}\). In particular, \(u^{(n)} \rightarrow T(v)\) in \(V_{\alpha }\), which proves that T is continuous.
Finally, we check the hypotheses of Schaefer's fixed point theorem (cf. [12, Section 9.2.2]). Recall that we need to find a \(C >0\) such that if \(u \in V_{\alpha }\) satisfies \(u= \sigma T(u)\) for some \(\sigma \in [0,1]\), then \(\Vert u\Vert _{\alpha } \le C\). Indeed, arguing as in Proposition 14 below, we see that \(u_{t}\) is bounded independently of \(\sigma \). Since \(\{{\tilde{H}}_{1}^{(R)},\dots ,{\tilde{H}}_{K}^{(R)}\}\) are bounded functions, the equation implies \(u_{x_{i}x_{i}}\) is also bounded independently of \(\sigma \) and i. From this, we obtain a bound on \([u]_{2}\) by Remark 9. Finally, the regularity of \(u_{0}\) and the uniform bound on \([u]_{2}\) gives a bound on \(\max _{i} \, [u_{x_{i}}]_{i,0} + [u]_{1 + \alpha }\), and this together with the uniform bound on \(u_{t}\) provides one for \([u]_{0} + [u]_{\alpha }\). It follows that \(\Vert u\Vert _{\alpha }\) is bounded independently of \(\sigma \).
By Schaefer's theorem, we conclude there is a \(u^{(a)} \in V_{\alpha }\) such that \(T(u^{(a)}) = u^{(a)}\). The regularity of \(u^{(a)}\) and its derivatives follows directly from the result of [22]. \(\square \)
C.2: A priori bounds
In the previous subsection, we showed that smooth initial data have smooth solutions, provided certain compatibility conditions are satisfied. Now we prove some a priori estimates satisfied by these solutions.
We start with a bound on the time derivative, which follows from (9) and the maximum principle.
Let \(a> 0\). If \(u_{0}\) and \(R > 0\) satisfy the hypotheses of Proposition 13, and if \(u^{(a)}\) is the solution obtained therein, then
$$\begin{aligned} u^{(a)}_{t}(x,0) = \epsilon u_{0,x_{i}x_{i}}(x) - {\tilde{H}}^{(R)}_{i}(x,0,u_{0,x_{i}}(x)) \quad \text {if} \, \, x \in \overline{I_{i}^{a}}, \, \, i \in \{1,2,\dots ,K\}, \end{aligned}$$
and, for each \((x,t) \in {\mathcal {I}} \times [0,T]\),
$$\begin{aligned} |u^{(a)}_{t}(x,t)| \le [u^{(a)}_{t}(\cdot ,0)]_{0} + Dt. \end{aligned}$$
The claim concerning \(u^{(a)}_{t}(\cdot ,0)\) follows from the regularity established in Proposition 13.
Given \(\zeta \in (0,T)\), define \(v^{\zeta } : \bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,T - \zeta ] \rightarrow {\mathbb {R}}\) by \(v^{\zeta }(x,t) = \frac{u^{(a)}(x,t + \zeta ) - u^{(a)}(x,t)}{\zeta }\). An immediate computation shows \(v^{\zeta }\) is a classical solution of a linear parabolic equation of the following form:
$$\begin{aligned} \left\{ \begin{array}{l l} v^{\zeta }_{t} - \epsilon v^{\zeta }_{x_{i}x_{i}} + b_{i}^{\zeta }(x,t) v^{\zeta }_{x_{i}} - g_{i}^{\zeta }(x,t) = 0 &{} \text {in} \, \, I^{a}_{i} \times (0,T - \zeta ) \\ {\sum }_{i = 1}^{K} v^{\zeta }_{x_{i}} = 0 &{} \text {on} \, \, \{0\} \times (0,T - \zeta ) \\ v^{\zeta } = u^{(a)}_{t}(\cdot ,0) &{} \text {on} \, \, \{-a_{i}\} \times (0,T - \zeta ) \end{array} \right. \end{aligned}$$
Notice that \(\{b_{1}^{\zeta },\dots ,b_{K}^{\zeta }\}\) and \(\{g_{1}^{\zeta },\dots ,g_{K}^{\zeta }\}\) are bounded functions by (7) and (9). Specifically, the functions \(\{g_{1}^{\zeta },\dots ,g_{K}^{\zeta }\}\) are bounded above and below by D and \(-D\), respectively, independently of the choice of \(\zeta \).
We claim that if \( (x,t) \in \bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,T-\zeta ]\), then
$$\begin{aligned} v^{\zeta }(x,t) - Dt \le \sup \left\{ v^{\zeta }(x,0) \, \mid \, x \in \bigcup _{i = 1}^{K} \overline{I_{i}^{a}}\right\} . \end{aligned}$$
To see this, fix \(K > 0\) strictly greater than the suprema of the functions \(\{|b_{1}^{\zeta }|,\dots ,|b_{K}^{\zeta }|\}\) and notice that if \(\delta > 0\), then the function \((x,t) \mapsto v^{\zeta }(x,t) - \delta x - (D + K\delta )t\) cannot attain its maximum in \(\bigcup _{i = 1}^{K} (\overline{I^{a}_{i}} \setminus \{-a_{i}\})\times (0,T - \zeta ]\). Recalling that \(v^{\zeta }\) is constant on \(\bigcup _{i = 1}^{K} \{-a_{i}\} \times [0,T - \zeta ]\) and sending \(\delta \rightarrow 0^{+}\), we recover (91).
Notice that for each \(\zeta ' \in (0,T)\), the Hölder continuity of \(u^{(a)}_{t}\) implies \(v^{\zeta } \rightarrow u^{(a)}_{t}\) uniformly in \(\bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,T - \zeta ']\) as \(\zeta \rightarrow 0^{+}\). Thus, after sending \(\zeta \rightarrow 0^{+}\) in (91), we find \(u^{(a)}_{t}(x,t) \le [u^{(a)}_{t}(\cdot ,0)]_{0} + Dt\). To see that \(u^{(a)}_{t}(x,t) \ge -( [u^{(a)}_{t}(\cdot ,0)]_{0} + Dt)\), we repeat the previous argument, replacing \(v^{\zeta }\) by \(-v^{\zeta }\).
Next, we leverage the bound on the time derivative to obtain a matching bound on the first order space derivatives.
If \(u^{(a)}\) is the solution obtained in Proposition 13 and we define \(C_{1} = [u_{t}(\cdot ,0)]_{0} + DT\), then there is an \(L > 0\) independent of a, depending on \(u_{0}\) only through \([u_{0}]_{1} + \epsilon [u_{0}]_{2}\), and such that if \(a > 2\left( 2C_{1} + 1\right) T\) and \(R \ge 2KL\), then
$$\begin{aligned} |u^{(a)}(x,t) - u^{(a)}(y,t)| \le KLd(x,y) \quad \text {if} \, \, d(x,0), d(y,0) \le \frac{a}{2}, \, \, t \in [0,T]. \end{aligned}$$
First, let \(L_{0} = [u_{0}]_{1} + \epsilon [u_{0}]_{2}\). By (8), there is an \(L_{1} \ge 1\) such that
$$\begin{aligned} -(C_{1} + 1) + H_{i}(t,x,p) \ge 1 \quad \text {if} \, \, |p| \ge L_{1}, \, \, i \in \{1,2,\dots ,K\}. \end{aligned}$$
Let \(L_{2} = L_{0} + L_{1}\). Notice that since \(C_{1}\) depends on \(u_{0}\) only through \([u_{0}]_{1} + \epsilon [u_{0}]_{2}\), it follows that \(L_{2}\) depends on \(u_{0}\) only through that quantity. Assume in what follows that \(R \ge 2K(3L_{2} + 1)\).
Fix \(i \in \{1,2,\dots ,K\}\) and \((x,t) \in \overline{I_{i}^{a}} \times (0,T)\) such that \(d(x,0) \le \frac{a}{2}\). Define the test function \(\varphi : {\mathcal {I}} \rightarrow {\mathbb {R}}\) exactly as in (17), but with \(u^{(a)}(x,t)\) in place of u(x) and \(3L_{2} + 1\) in place of L. Finally, define \(w : {\mathcal {I}} \times [0,t] \rightarrow {\mathbb {R}}\) by
$$\begin{aligned} w(y,s) = \varphi (y) + (C_{1} + 1)(t - s). \end{aligned}$$
We claim that the function \((y,s) \mapsto u^{(a)}(y,s) - w(y,s)\) defined in \(\bigcup _{i =1}^{K} \overline{I_{i}^{a}} \times [0,t]\) is maximized at (x, t). First, note that \(u^{(a)}(x,t) - w(x,t) = 0\). Moreover, if \(s < t\), then the inequality \([u_{t}^{(a)}]_{0} \le C_{1}\) implies
$$\begin{aligned} u^{(a)}(x,s) - w(x,s) = u^{(a)}(x,s) - u^{(a)}(x,t) - (C_{1} + 1)(t - s) \le - (t -s) < 0. \end{aligned}$$
Therefore, the maximum does not occur at a point of the form (x, s), where \(s \in [0,t)\).
If (y, s) is the maximum of \(u^{(a)} - w\) in \(\bigcup _{i = 1}^{K} \overline{I^{a}_{i}} \times [0,t]\), \(y \in \bigcup _{i = 1}^{K} I_{i}^{a} \setminus \{x\}\), and \(s \in (0,t]\), then, in view of the choice of R, the equation yields
$$\begin{aligned} -(C_{1} + 1) + H_{i}(t,x, u (3L_{2} + 1)) \le 0 \quad \text {for some} \, \, u \in \{-1,1, K,-K\}, \end{aligned}$$
contradicting the choice of \(L_{1}\). We get a contradiction similarly in the case when \(y = 0\) and \(s \in (0,t]\).
If (y, s) is the maximum, \(y \ne x\), and \(s = 0\), then the inequalities \([u_{0}]_{1} \le L_{2}\) and \([u^{(a)}_{t}]_{0} \le C_{1}\) yield the following
$$\begin{aligned} 0 \le u^{(a)}(y,0) - w(y,0) \le u_{0}(y) - u_{0}(x) - (3L_{2}+1) d(x,y) < 0, \end{aligned}$$
which is a contradiction.
Finally, if (y, s) is the maximum and \(y = -a_{j}\) for some j, then the assumption \(d(x,0) \le \frac{a}{2}\), the inequalities \([u_{0}]_{1} \le L_{2}\) and \([u^{(a)}_{t}]_{0} \le C_{1}\), and the assumption \(a > 2\left( 2C_{1} + 1\right) T\) all come together to imply the following:
$$\begin{aligned} u^{(a)}(-a_{j},s)&\ge w(-a_{j},s) \\&= \varphi (-a_{j}) + (C_{1} + 1)(t - s) \\&\ge u^{(a)}(x,t) + (3L_{2} + 1)\left( \frac{a}{2}\right) + (C_{1} + 1)(t - s) \\&\ge \left( u_{0}(x) - u_{0}(-a_{j})\right) - C_{1}(t + s) + u^{(a)}(-a_{j},s) + (3L_{2} + 1) \left( \frac{a}{2}\right) \\&\quad \quad + (C_{1} + 1)(t - s) \\&\ge -L_{2} \left( \frac{3a}{2}\right) - (2C_{1} + 1) T + (3L_{2} + 1) \left( \frac{a}{2}\right) + u^{(a)}(-a_{j},s) \\&> u^{(a)}(-a_{j},s), \end{aligned}$$
which is another contradiction. Therefore, the function \((y,s) \mapsto u^{(a)}(y,s) - w(y,s)\) is maximized in \(\bigcup _{i = 1}^{K} \overline{I_{i}^{a}} \times [0,t]\) at the point (x, t).
Thus, restricting to points \((y,s) = (y,t)\), we find
$$\begin{aligned} u^{(a)}(y,t) - u^{(a)}(x,t) \le K (3L_{2} + 1) |x - y| \quad \text {if} \, \, y \in \overline{I_{j}}. \end{aligned}$$
After setting \(L = 3L_{2} + 1\), we conclude that (92) holds. \(\square \)
C.3: Viscosity solutions
Now we prove Theorems 11 and 12.
To prove these, we need to ensure that we can approximate the initial datum with a regular function that satisfies the compatibility conditions (85) and (86). That is the purpose of the next two results.
Suppose \(p : (-\infty ,0] \rightarrow {\mathbb {R}}\) is a thrice continuously differentiable function for which there is a constant \(C_{p} > 0\) such that, for each \(x \in (-\infty ,0]\),
$$\begin{aligned} |p'(x)| + |p''(x)| \le C_{p} \end{aligned}$$
and \(\sup \left\{ |p'''(x)| \, \mid \, x \in (-\infty ,0]\right\} < \infty \). Let \(b \in {\mathbb {R}}\). There is a universal constant \(C'_{p} > 0\) depending only on \(C_{p}\) and b such that if \(\zeta > 0\), then there is a thrice continuously differentiable function \(p_{\zeta } : (-\infty ,0] \rightarrow {\mathbb {R}}\) such that \(p_{\zeta }(0) = p(0)\), \(p_{\zeta }'(0) = p'(0)\), \(p_{\zeta }''(0) = b\), \(\sup \left\{ |p_{\zeta }'''(x)| \, \mid \, x \in (-\infty ,0]\right\} < \infty \), and, for each \(x \in (-\infty ,0]\),
$$\begin{aligned} |p_{\zeta }'(x)| + |p_{\zeta }''(x)|&\le C_{p}'\\ |p_{\zeta }(x) - p(x)|&\le C_{p}' \zeta ^{2} \end{aligned}$$
Given \(\zeta > 0\), choose a smooth function \(\varphi _{\zeta } : (-\infty ,0] \rightarrow {\mathbb {R}}\) such that
$$\begin{aligned} \varphi _{\zeta }(x) = 0 \quad \text {if} \, \, x \in (-\infty ,-2\zeta ]&, \quad \varphi _{\zeta }(x) = 1 \quad \text {if} \, \, x \in [-\zeta ,0], \\ \max \left\{ \zeta |\varphi _{\zeta }'(x)|, \zeta ^{2} |\varphi _{\zeta }''(x)| \right\}&\le C_{0} \quad \text {if} \, \, x \in (-\infty ,0], \end{aligned}$$
where \(C_{0} \ge 1\) is a universal constant independent of \(\zeta \), p, \(C_{p}\), and b.
Define \(q(x) = p(0) + p'(0)x + \frac{b x^{2}}{2}\) and then let \(p_{\zeta } : (-\infty ,0] \rightarrow {\mathbb {R}}\) be given by
$$\begin{aligned} p_{\zeta }(x) = (1 - \varphi _{\zeta }(x))p(x) + \varphi _{\zeta }(x)q(x). \end{aligned}$$
The choice of \(\varphi _{\zeta }\) implies \(p_{\zeta }(0) = p(0)\), \(p_{\zeta }'(0) = p'(0)\), and \(p_{\zeta }''(0) = b\). Moreover, \(\sup \left\{ |p'''_{\zeta }(x)| \, \mid \, x \in (-\infty ,0]\right\} < \infty \) holds.
Differentiating \(p_{\zeta }\), we find
$$\begin{aligned} p_{\zeta }'(x)&= (1 - \varphi _{\zeta }(x))p'(x) + \varphi _{\zeta }(x) q'(x) + \varphi _{\zeta }'(x)(q(x) - p(x)), \\ p_{\zeta }''(x)&= (1 - \varphi _{\zeta }(x))p''(x) + \varphi _{\zeta }(x) q''(x) + 2\varphi _{\zeta }'(x)(q'(x) - p'(x)) \\&\qquad + \varphi _{\zeta }''(x)(q(x) - p(x)). \end{aligned}$$
Thus, the regularity of p and the definition of \(\varphi _{\zeta }\) imply the desired bounds by Taylor expansion at 0. \(\square \)
Now we use Lemma 11 to show how to approximate a \(C^{3}({\mathcal {I}})\) function by one that satisfies the compatibility conditions.
Suppose \(u_{0} \in C^{3}({\mathcal {I}})\) satisfies \({\sum }_{i = 1}^{K} u_{0,x_{i}}(0) = 0\) and \([u_{0}]_{1} + [u_{0}]_{2} + [u_{0}]_{3} < \infty \). Then there is a universal constant \(C' > 0\) depending only on \([u_{0}]_{1} + [u_{0}]_{2}\) such that for all \(\zeta > 0\), there is a \(u_{0}^{\zeta } \in C^{3}({\mathcal {I}})\) satisfying the following conditions:
\([u_{0}^{\zeta }]_{1} + [u_{0}^{\zeta }]_{2} + [u_{0}^{\zeta }]_{3} < \infty \)
For each \(i \in \{1,2,\dots ,K\}\),
$$\begin{aligned} u^{\zeta }_{0,x_{i}}(0)&= u_{0,x_{i}}(0) \\ -\epsilon u^{\zeta }_{0,x_{i}x_{i}}(0) + H_{i}(0,0,u^{\zeta }_{0,x_{i}}(0))&= - \epsilon u^{\zeta }_{0,x_{1}x_{1}}(0) + H_{1}(0,0,u^{\zeta }_{0,x_{1}}(0)) \\ \end{aligned}$$
For each \(i \in \{1,2,\dots ,K\}\) and each \(x \in \overline{I_{i}}\),
$$\begin{aligned} |u^{\zeta }_{0,x_{i}}(x)| + |u^{\zeta }_{0,x_{i}x_{i}}(x)|&\le C' \end{aligned}$$
$$\begin{aligned} |u^{\zeta }_{0}(x) - u_{0}(x)|&\le C'\zeta ^{2} \end{aligned}$$
Define \(\{b_{1},\dots ,b_{K}\}\) by \(b_{1} = 1\) and
$$\begin{aligned} b_{i} = \epsilon ^{-1}\left( \epsilon - H_{1}(0,0,u_{0,x_{1}}(0)) + H_{i}(0,0,u_{0,x_{i}}(0))\right) . \end{aligned}$$
Notice that this immediately implies \(\{b_{1},\dots ,b_{K}\}\) satisfy, for each i,
$$\begin{aligned} -\epsilon b_{i} + H{i}(0,0,u_{0,x_{i}}(0)) = -\epsilon b_{1} + H_{1}(0,0,u_{0,x_{1}}(0)). \end{aligned}$$
Now apply Lemma 11 to obtain functions \(\{\psi ^{\zeta ,(1)},\dots ,\psi ^{\zeta ,(K)}\}\) and a constant \(C' > 0\) so that, for each \(i \in \{1,2,\dots ,K\}\), \(\psi ^{\zeta ,(i)}\) has domain \((-\infty ,0)\) and the following relations hold:
$$\begin{aligned} \sup \left\{ |\psi ^{\zeta ,(i)}_{x}(x)| + |\psi ^{\zeta ,(i)}_{xx}(x)| \, \mid \, x \le 0 \right\}&\le C' \end{aligned}$$
$$\begin{aligned} \psi ^{\zeta ,(i)}(0)&= u_{0}(0) \end{aligned}$$
$$\begin{aligned} \psi ^{\zeta ,(i)}_{x}(0)&= u_{0,x_{i}}(0) \end{aligned}$$
$$\begin{aligned} \psi ^{\zeta ,(i)}_{xx}(0)&= b_{i} \end{aligned}$$
$$\begin{aligned} \sup \left\{ |\psi ^{\zeta ,(i)}(x) - u_{0}(x)| \, \mid \, x \in \overline{I_{i}}\right\}&\le C' \zeta ^{2} \end{aligned}$$
By construction, \(\{\psi ^{\zeta ,(1)},\dots ,\psi ^{\zeta ,(K)}\}\) come together to form a function \(u^{\zeta }_{0} \in C^{3}({\mathcal {I}})\) with the desired properties. \(\square \)
In the proof that follows, we will not use the \(\epsilon \) superscript to denote solutions of (4). Since we are only dealing with (4) and not (2) in the proof, we hope this will not cause too much confusion.
Proof of Theorem 12
First, assume \(u_{0} \in C^{3}({\mathcal {I}})\) and \([u_{0}]_{1} + [u_{0}]_{2} + [u_{0}]_{3} < \infty \). For \(\zeta > 0\) sufficiently small, let \(u^{\zeta }_{0}\) be the function obtained from Proposition 16, and fix \(R \ge 2C'\), where \(C'\) is the constant defined in the proposition. For each \(a > 0\), let \(u^{(a),\zeta }\) be the solution of (89) with initial datum \(u_{0}^{\zeta }\) obtained in Proposition 13.
By Propositions 14 and 15 and the uniform bound (93), there are constants \(B, L, a_{0} > 0\), all independent of \(\zeta \), such that if \(a \ge a_{0}\) and \(R \ge 2KL\), then \([u_{t}^{(a),\zeta }]_{0} \le B\) and (92) holds with \(u^{(a)} = u^{(a),\zeta }\). Henceforth, assume \(R \ge 2KL\).
The estimates obtained in the previous paragraph imply we can fix a sequence \((a_{n})_{n \in {\mathbb {N}}} \subseteq [a_{0},\infty )\) and a function \(u^{\zeta } : {\mathcal {I}} \times [0,T] \rightarrow {\mathbb {R}}\) such that \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(u^{\zeta } = \lim _{n \rightarrow \infty } u^{(a_{n}),\zeta }\) locally uniformly in \({\mathcal {I}} \times [0,T]\). The local uniform convergence and the stability of viscosity solutions together imply \(u^{\zeta }\) is a solution of (4) with Hamiltonians \(\{{\tilde{H}}^{(R)}_{1},\dots ,{\tilde{H}}^{(R)}_{K}\}\) and initial datum \(u_{0}^{\zeta }\).
Since \(\lim _{n \rightarrow \infty } a_{n} = \infty \), (92) shows that \(u^{\zeta }\) satisfies \(\text {Lip}(u^{\zeta }(\cdot ,t)) \le KL\) for all \(t \in [0,T]\). Thus, as \({\tilde{H}}^{(R)}_{i}(t,x,p) = H_{i}(t,x,p)\) for all \(|p| \le KL\), it follows that \(u^{\zeta }\) is actually a solution of (4) with the Hamiltonians \(\{H_{1},\dots ,H_{K}\}\). By Theorem 9, we deduce that the limit is unique and, in fact, \(u^{\zeta } = \lim _{a \rightarrow \infty } u^{(a),\zeta }\) locally uniformly in \({\mathcal {I}} \times [0,T]\).
Finally, we send \(\zeta \rightarrow 0^{+}\). Since \(u^{\zeta }_{0} \rightarrow u_{0}\) uniformly in \({\mathcal {I}}\) as \(\zeta \rightarrow 0^{+}\), Remark 2 implies \((u^{\zeta })_{\zeta > 0}\) is uniformly Cauchy in \({\mathcal {I}} \times [0,T]\). In particular, \(u = \lim _{\zeta \rightarrow 0^{+}} u^{\zeta }\) exists uniformly in \({\mathcal {I}} \times [0,T]\) and the stability of viscosity solutions implies u solves (4) with initial datum \(u_{0}\).
Since L and B were independent of \(\zeta \), the uniform convergence \(u^{\zeta } \rightarrow u\) implies \(\text {Lip}(u) \le B + L\).
To remove the \(C^{3}({\mathcal {I}})\) assumption, we argue by approximation. That is, if \(u_{0} \in C^{1}({\mathcal {I}})\) and \([u_{0}]_{1} + [u_{0}]_{2} < \infty \), we obtain the solution u of (4) and show that it is in \(\text {Lip}({\mathcal {I}} \times [0,T])\) by approximating \(u_{0}\) with functions \((u_{0,n}) \subseteq C^{3}({\mathcal {I}})\) such that \(u_{0,n} \rightarrow u_{0}\) uniformly in \({\mathcal {I}}\) and \(\sup \left\{ [u_{0,n}]_{1} + [u_{0,n}]_{2} \, \mid \, n \in {\mathbb {N}} \right\} < \infty \). Since the proof that it is possible to do this is very similar to some of the arguments presented in Appendix C.5, we omit it.
Finally, if \(u_{0} \in UC({\mathcal {I}})\), then, arguing as in Remark 10 below, we can fix a sequence \((u_{0}^{(n)})_{n \in {\mathbb {N}}} \subseteq C^{1}({\mathcal {I}})\) satisfying \([u_{0}^{(n)}]_{1} + [u_{0}^{(n)}]_{2} < \infty \) for each n and such that \(u_{0}^{(n)} \rightarrow u_{0}\) uniformly in \({\mathcal {I}}\) as \(n \rightarrow \infty \). By the previous step, we can let \(u^{(n)}\) be the solution of (4) with initial datum \(u_{0}^{(n)}\), and Remark 2 shows that \((u^{(n)})_{n \in {\mathbb {N}}}\) is uniformly Cauchy in \({\mathcal {I}} \times [0,T]\). Therefore, as before, the limit \(u = \lim _{n \rightarrow \infty } u^{(n)}\) is a continuous viscosity solution of (4). In fact, \(u \in UC({\mathcal {I}} \times [0,T])\), being the uniform limit of such functions. \(\square \)
C.4: Existence of solutions of (2)
Finally, we establish the existence of solutions of (2). Here, as in the error analysis, we invoke Proposition 17.
First, assume \(u_{0} \in \text {Lip}({\mathcal {I}})\). By Proposition 17 below, there is a family \((v_{0}^{\epsilon })_{\epsilon > 0} \subseteq C^{1} \left( {\mathcal {I}} \right) \) such that \(\lim _{\epsilon \rightarrow 0^{+}} [v_{0}^{\epsilon } - u_{0}]_{0} =0\) and \(\sup \{[v_{0}^{\epsilon }]_{1} + \epsilon [v_{0}^{\epsilon }]_{2} \, \mid \, \epsilon > 0\} \le C'\), where \(C'\) only depends on \(\text {Lip}(u_{0})\).
For each \(\epsilon > 0\), let \(v^{\epsilon }\) be the solution of (4) with initial datum \(v_{0}^{\epsilon }\). Since \([v_{0}^{\epsilon }]_{1} + \epsilon [v_{0}^{\epsilon }]_{2}\) is bounded, Theorem 12 implies there is an \(L' > 0\) depending on \(C'\), but not \(\epsilon \), such that \(\text {Lip}(v^{\epsilon }) \le L'\).
In view of the uniform Lipschitz estimate, we can fix \((\epsilon _{n})_{n \in {\mathbb {N}}}\) and a function \(u : {\mathcal {I}} \times [0,T] \rightarrow {\mathbb {R}}\) such that \(\lim _{n \rightarrow \infty } \epsilon _{n} = 0\) and \(u = \lim _{n \rightarrow \infty } v^{\epsilon _{n}}\). By the stability of viscosity solutions, u solves (2) with initial datum \(u_{0}\). In fact, Theorem 8 implies u is independent of the choice of subsequence, and, thus, \(u = \lim _{\epsilon \rightarrow 0^{+}} v^{\epsilon }\). Moreover, \(\text {Lip}(u) \le L'\).
In general, if \(u_{0} \in UC({\mathcal {I}})\), then there is a sequence \((u_{0}^{(n)})_{n \in {\mathbb {N}}} \subseteq \text {Lip}({\mathcal {I}})\) such that \(u_{0}^{(n)} \rightarrow u_{0}\) uniformly in \({\mathcal {I}}\) as \(n \rightarrow \infty \). (See Remark 10.) Let \(u^{(n)}\) denote the solution of (2) with initial datum \(u_{0}^{(n)}\). By Remark 2, \((u^{(n)})_{n \in {\mathbb {N}}}\) is uniformly Cauchy in \({\mathcal {I}} \times [0,T]\). Invoking stability of viscosity solutions, we conclude that the limit \(u = \lim _{n \rightarrow \infty } u^{(n)}\) is a solution of (2) with initial datum \(u_{0}\). Moreover, as a uniform limit of such functions, \(u \in UC({\mathcal {I}} \times [0,T])\). \(\square \)
C.5: A useful approximation result
In the error analysis of Sect. 6, we used the following result:
Let \(u_{0} \in \text {Lip}\left( {\mathcal {I}} \right) \). For each \(\epsilon >0\), there is a \(v_{0}^{\epsilon } \in C^{1} \left( {\mathcal {I}}\right) \) and a universal constant \(C > 0\) such that:
$$\begin{aligned}{}[v_{0}^{\epsilon } - u_{0}]_{0}&\le \text {Lip}(u_{0})\epsilon \\ [v_{0}^{\epsilon }]_{1}&\le C \text {Lip}(u_{0}) \\ [v_{0}^{\epsilon }]_{2}&\le C \epsilon ^{-1} \text {Lip}(u_{0}) \end{aligned}$$
Moreover, \(v_{0}^{\epsilon }\) can be chosen in such a way that both \(v_{0}^{\epsilon }(0) = u_{0}(0)\) and \({\sum }_{i = 1}^{K} v^{\epsilon }_{0,x_{i}}(0) = 0\).
The same method used to prove Proposition 17 below can be used to establish more general approximation results for functions on \({\mathcal {I}}\) with varying degrees of regularity. We will not expound on those here. However, since we use an approximation result for functions in \(\text {UC}({\mathcal {I}})\) in the proof of Theorems 11 and 12, we include its statement as a remark:
Remark 10
Arguing as in the proof that follows, we can show that if \(u_{0} \in \text {UC}({\mathcal {I}})\), then there is a sequence of functions \((u^{(n)}_{0})_{n \in {\mathbb {N}}} \subseteq C^{2}({\mathcal {I}})\) satisfying \([u^{(n)}_{0}]_{1} + [u^{(n)}_{0}]_{2} < \infty \), \({\sum }_{i = 1}^{K} u_{0,x_{i}}^{(n)}(0) = 0\), and such that
$$\begin{aligned}{}[u_{0}^{(n)} - u_{0}]_{0} \le \omega (2 n^{-1}), \end{aligned}$$
where \(\omega \) is the modulus of continuity of \(u_{0}\) in \({\mathcal {I}}\).
First, given \(\epsilon > 0\), let \(\varphi _{\epsilon }\) be as in the proof of Lemma 11. Additionally, let \(\rho : {\mathbb {R}} \rightarrow [0,\infty )\) be a smooth symmetric function supported in \((-1,1)\) and satisfying \(\int _{-\infty }^{\infty } \rho (x) \, dx = 1\).
Define \({\tilde{\psi }}_{i}^{\epsilon } : \overline{I_{i}} \rightarrow {\mathbb {R}}\) by \({\tilde{\psi }}_{i}^{\epsilon }(x) = \epsilon ^{-1} \int _{-\infty }^{\infty } u_{0,i}(y) \rho (\epsilon ^{-1}(x - y)) \, dy\), where \(u_{0,i}\) is given by \(u_{0,i}(x) = u_{0}(x_{i})\) if \(x < 0\) and \(u_{0,i}(x) = u_{0}(0)\), otherwise. Recall the following well-known properties of \({\tilde{\psi }}_{i}^{\epsilon }\):
$$\begin{aligned} \sup \left\{ |{\tilde{\psi }}_{i}^{\epsilon }(x) - u_{0}(x)| \, \mid \, x \in I_{i} \right\}&\le \text {Lip}(u_{0})\epsilon \\ \sup \left\{ |{\tilde{\psi }}_{i,x_{i}}^{\epsilon }(x)| \, \mid \, x \in I_{i} \right\}&\le \text {Lip}(u_{0}) \\ \sup \left\{ |{\tilde{\psi }}_{i,x_{i}x_{i}}^{\epsilon }(x)| \, \mid \, x \in I_{i} \right\}&\le C \text {Lip}(u_{0}) \epsilon ^{-1} \end{aligned}$$
We proceed by combining \(\{{\tilde{\psi }}^{\epsilon }_{1},\dots ,{\tilde{\psi }}^{\epsilon }_{K}\}\) into a function on \({\mathcal {I}}\).
Define \(v_{0}^{\epsilon } : {\mathcal {I}} \rightarrow {\mathbb {R}}\) by
$$\begin{aligned} v_{0}^{\epsilon }(x) = (1 - \varphi _{\epsilon }(x)) {\tilde{\psi }}^{\epsilon }_{i}(x) + \varphi _{\epsilon }(x) u_{0}(0) \quad \text {if} \, \, x \in \overline{I_{i}}, \, \, i \in \{1,2,\dots ,K\}. \end{aligned}$$
Observe that \(\min \{|\varphi _{\epsilon }'(x)|, |\varphi _{\epsilon }''(x)|\} > 0\) only if \(x \in [-2\epsilon , \epsilon ]\). Moreover, for such x, the following inequality holds:
$$\begin{aligned} |{\tilde{\psi }}^{\epsilon }_{i}(x) - u_{0}(0)| \le |{\tilde{\psi }}^{\epsilon }_{i}(x) - u_{0}(x)| + |u_{0}(x) - u_{0}(0)| \le 3 \text {Lip}(u_{0}) \epsilon . \end{aligned}$$
Therefore, we can argue as in Lemma 11 to see that \(v_{0}^{\epsilon }\) satisfies the required estimates.
Finally, \(v_{0}^{\epsilon }(x) = u_{0}(0)\) if \(x \in \bigcup _{i = 1}^{K} \overline{I_{i}^{\epsilon }} \) so \({\sum }_{i = 1}^{K} v^{\epsilon }_{0,x_{i}}(0) = 0\). \(\square \)
Appendix D: Time-dependent finite-difference schemes
In this section, we show that the finite-difference scheme approximating (2) is monotone provided a CFL-type condition is satisfied. We also establish the required regularity properties of the solution.
We begin by introducing the necessary terminology. A function \(V : {\mathcal {J}} \times S \rightarrow {\mathbb {R}}\) is said to be a sub-solution of the scheme (74) if it satisfies the system of inequalities obtained by replacing all equal signs with \(\le \). Analogously, a function W on the same domain is called a super-solution of the scheme (74) if it satisfies the system of inequalities obtained by replacing all equal signs with \(\ge \). As in the stationary case, the scheme is monotone when sub- and super-solutions obey a discrete maximum principle. This is made precise in the following definition.
Definition 2
The finite-difference scheme (74) is called monotone if the following two criteria hold:
(i) If \(V,\chi : {\mathcal {J}} \times \{0,1,\dots ,N\} \rightarrow {\mathbb {R}}\), V is a sub-solution of (74), and \(V - \chi \) has a global maximum at (m, s) with \(s > 0\) and \(m \in J_{i}\), then
$$\begin{aligned} \frac{\chi (m,s) - \chi (m,s - 1)}{\Delta t} + F_{i}(D^{+}\chi (m,s - 1),D^{-}\chi (m,s - 1)) \le f_{i}((s-1)\Delta t, -m \Delta x) \end{aligned}$$
if \(m \ne 0\), and
$$\begin{aligned} \sum _{i = 1}^{K} (\chi (0,s) - \chi (1_{i},s)) \le 0, \quad \text {otherwise}. \end{aligned}$$
(ii) If \(W,\chi : {\mathcal {J}} \times \{0,1,\dots ,N\} \rightarrow {\mathbb {R}}\), W is a super-solution of (74), and \(W - \chi \) has a global minimum at (m, s) with \(s > 0\) and \(m \in J_{i}\), then
$$\begin{aligned} \frac{\chi (m,s) - \chi (m,s - 1)}{\Delta t} + F_{i}(D^{+}\chi (m,s - 1),D^{-}\chi (m,s - 1)) \ge f_{i}((s-1)\Delta t, -m \Delta x) \end{aligned}$$
$$\begin{aligned} \sum _{i = 1}^{K} (\chi (0,s) - \chi (1_{i},s)) \ge 0, \quad \text {otherwise}. \end{aligned}$$
As in the time-independent setting, when we use the term "monotone" in reference to (74), we always mean it in the sense of the previous definition.
The error analysis of (74) uses a discrete version of Lipschitz continuity. Specifically, given a function \(U : {\mathcal {J}} \times S \rightarrow {\mathbb {R}}\), we say that U is Lipschitz if
$$\begin{aligned} \text {Lip}(U) := \sup \left\{ |U(m,s) - U(k,r)| \, \mid \, \frac{d(-m\Delta x, -k \Delta x)}{\Delta x} + |s - r| \le 1 \right\} < \infty . \end{aligned}$$
The following result gives sufficient conditions under which the scheme (74) is monotone and the solution is Lipschitz. Recall that \(L_{G}\) is a uniform bound on the Lipschitz constants of the numerical Hamiltonians \(G_{1},\dots ,G_{K}\), and \(L_{c}\) is the cut-off in assumption (48).
There is an \({\tilde{L}}_{c} > 0\) depending only on \(\text {Lip}(u_{0})\), D, \(L_{G}\), \(L_{2}\), and T such that if (78) holds and \(L_{c} \ge {\tilde{L}}_{c}\), then the finite-difference scheme (74) is monotone and the solution U of (74) satisfies \(\text {Lip}(U) \le {\tilde{L}}_{c} \Delta x\).
From (78), we see that \(\epsilon \ge 2L_{G} \Delta x\) and \(\frac{\Delta x}{\Delta t} - \frac{ \epsilon }{\Delta x} - 2L_{G} \ge 0\). From this, it follows that the expression
$$\begin{aligned} \frac{\chi (k,s) - \chi (k,s-1)}{\Delta t} + F_{i}(D^{+}\chi (k,s-1),D^{-}\chi (k,s-1)) \end{aligned}$$
is non-increasing in the variables \(\chi (k,s-1)\), \(\chi (k+1,s-1)\), and \(\chi (k-1,s-1)\). We leave it to the reader to verify that this implies (74) is monotone according to Definition 2.
To see that U is Lipschitz, we argue as in the continuum case. To start with, define \(V : {\mathcal {J}} \times (S \setminus \{N\}) \rightarrow {\mathbb {R}}\) by \(V(k,s) = \frac{U(k,s + 1) - U(k,s)}{\Delta t}\). Observe that if \(s \in S \setminus \{N,N - 1\}\) and \(k \in {\mathcal {J}} \setminus \{0\}\), then
$$\begin{aligned} D_{t}V(k,s) + B_{i}^{+}(k,s) D^{+}V(k,s) + B_{i}^{-}(k,s) D^{-}V(k,s) - D_{t} \Gamma (k,s) = 0, \end{aligned}$$
where the coefficients of the equation are defined as follows:
$$\begin{aligned} B_{i}^{+}(k,s)&= \frac{F_{i}(D^{+}U(k,s + 1),D^{-}U(k,s+1)) - F_{i}(D^{+}U(k,s),D^{-}U(k,s+1))}{D^{+}U(k,s + 1) - D^{+}U(k,s)}, \\ B_{i}^{-}(k,s)&= \frac{F_{i}(D^{+}U(k,s),D^{-}U(k,s+1)) - F_{i}(D^{+}U(k,s),D^{-}U(k,s))}{D^{-}U(k,s+1) - D^{-}U(k,s)}, \\ \Gamma (k,s)&= f_{i}(s \Delta t, -k\Delta x). \end{aligned}$$
The discussion in the previous paragraph implies \(B_{i}^{+} \le 0\) and \(B_{i}^{-} \ge 0\) pointwise in \({\mathcal {J}} \times (S \setminus \{N\})\).
In addition to (102), V satisfies \({\sum }_{i = 1}^{K} D^{+}V(1_{i},s) = 0\) if \(s \in S \setminus \{1,N\}\). Notice that if we define a scheme using (102) and this discrete Kirchoff condition, then the signs of \(B_{i}^{+}\) and \(B_{i}^{-}\) imply it is monotone in \({\mathcal {J}} \times (S \setminus \{N\})\) in the sense of Definition 2.
By (9), \(|D_{t}\Gamma | \le D\) pointwise in \({\mathcal {J}} \times (S \setminus \{N\})\). Therefore, using monotonicity and arguing as in Proposition 14, we find that if \((k,s) \in {\mathcal {J}} \times (S \setminus \{N\})\), then
$$\begin{aligned} |V(k,s)| \le \sup \left\{ |V(k,0)| \, \mid \, k \in {\mathcal {J}} \right\} + DT. \end{aligned}$$
In particular, since V(k, 0) is determined by \(u_{0}\), there is a constant \(C_{0}\) depending only on \(\text {Lip}(u_{0})\) such that \(|V| \le C_{0} + DT\) pointwise. Notice that, by (78), \(C_{0}\) can be chosen independent of \(\Delta x\) and \(\epsilon \), though it does depend on \(L_{2}\).
Now we show that the finite differences \(D^{+}U\) and \(D^{-}U\) are uniformly bounded. Indeed, if we fix \(s \in S \setminus \{N\}\), then the function \(m \mapsto U(m,s)\) defined in \({\mathcal {J}}\) satisfies the stationary finite difference equation
$$\begin{aligned} V(m,s) + F_{i}(D^{+}U(m,s),D^{-}U(m,s)) = f_{i}(s\Delta t,-m\Delta x) \quad \text {in} \, \, J_{i}. \end{aligned}$$
Since V is uniformly bounded and the assumption \(\epsilon \ge 2 L_{G} \Delta x\) implies the difference equation (103) is monotone, we can argue exactly as in Theorem 10 to see that there is an \({\tilde{L}}_{c} > 0\) depending only on \(C_{0}\) and D, but not on s, such that if \(L_{c} \ge {\tilde{L}}_{c}\), then \(\text {Lip}(U(\cdot ,s)) \le {\tilde{L}}_{c}\Delta x\).
The bound we obtained through the equation only applies if \(s < N\). To get a bound at \(s = N\), observe that the assumption \(\frac{\Delta x}{\Delta t} \ge 2 L_{G}\) implies
$$\begin{aligned} |U(k + 1,N) - U(k,N)|&\le |U(k + 1,N) - U(k+1,N-1)| \\&\quad + |U(k+1,N-1)- U(k,N-1)| \\&\quad + |U(k,N-1) - U(k,N)| \\&\le 2(C_{0} + DT)\Delta t + {\tilde{L}}_{c}\Delta x \\&\le \left( \frac{C_{0} + DT}{L_{G}} + {\tilde{L}}_{c} \right) \Delta x \end{aligned}$$
Thus, making \({\tilde{L}}_{c}\) larger if necessary, we can assume that \(\text {Lip}(U(\cdot ,s)) \le {\tilde{L}}_{c}\) independently of \(s \in S\). Making \({\tilde{L}}_{c}\) larger again, we can assume that \({\tilde{L}}_{c} \ge \frac{C_{0} + DT}{L_{G}}\) and, thus, \(|V| \le {\tilde{L}}_{c}L_{G}\) pointwise. From this and the assumption that \(\frac{\Delta t}{\Delta x} \le L_{G}^{-1}\), we conclude that \(\text {Lip}(U) \le {\tilde{L}}_{c}\Delta x\) on \({\mathcal {J}} \times S\). \(\square \)
Appendix E: Proof of Theorem 7
In this section, we take on the hardest step in the comparison results presented above. In order to apply [20, Lemma 3.1], we need to understand, roughly speaking, the extent to which the equation "sees" the differentiability (or lack thereof) of a sub-solution or super-solution at the junction.
In what follows, given \(u : (-\infty ,0] \rightarrow {\mathbb {R}}\) and \(x \in (-\infty ,0]\), we define \(J^{+}u(x)\) to be the set of all \(p \in {\mathbb {R}}\) such that
$$\begin{aligned} u(y) \le u(x) + p(y - x) + o(|y -x|) \quad \text {as} \, \, y \rightarrow x. \end{aligned}$$
\(J^{-}u(x)\) is defined by \(J^{-}u(x) = -J^{+}(-u)(x)\).
Notice that this is analogous to the definitions in Appendix A. In particular, given \(x \in \overline{I_{i}}\) and \(u : {\mathcal {I}} \rightarrow {\mathbb {R}}\), if \(u_{i} : (-\infty ,0] \rightarrow {\mathbb {R}}\) is defined by restricting u to \({\overline{I}}_{i}\), then \(J^{+}_{i}u(x) = J^{+}u_{i}(x)\) and \(J^{-}_{i}u(x) = J^{-}u_{i}(x)\).
If \(u : (-\infty ,0] \rightarrow {\mathbb {R}}\) is upper semi-continuous continuous and \(u_{x}(0)\) exists, then there are sequences \((x_{n}^{+})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0)\), \((p_{n}^{+})_{n \in {\mathbb {N}}} \subseteq {\mathbb {R}}\) such that
\(p_{n}^{+} \in J^{+}u(x_{n}^{+})\) for each \(n \in {\mathbb {N}}\)
\(\lim _{n \rightarrow \infty } p_{n}^{+} = u_{x}(0)\)
\(\lim _{n \rightarrow \infty } x_{n}^{+} = 0\)
\(\lim _{n \rightarrow \infty } u(x_{n}^{+}) = u(0)\)
Similarly, if \(v : (-\infty ,0] \rightarrow {\mathbb {R}}\) is lower semi-continuous and \(v_{x}(0)\) exists, then there are sequences \((x_{n}^{-})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0)\) and \((q_{n}^{-})_{n \in {\mathbb {N}}} \subseteq {\mathbb {R}}\) such that
\(q_{n}^{-} \in J^{-}v(x_{n}^{-})\) for each \(n \in {\mathbb {N}}\)
\(\lim _{n \rightarrow \infty } q_{n}^{-} = v_{x}(0)\)
\(\lim _{n \rightarrow \infty } x_{n}^{-} = 0\)
\(\lim _{n \rightarrow \infty } v(x_{n}^{-}) = v(0)\)
Regarding \((x_{n}^{+},p_{n}^{+})\), this follows from the proof of Lemma 9 and the fact that, in this case, \(J^{+}u(0) = (-\infty ,u_{x}(0)]\). To obtain the sequences \((x_{n}^{-},p_{n}^{-})\), use the fact that \(-v\) is upper semi-continuous and \(J^{+}(-v)(x) = -J^{-}v(x)\).
When the solution is not differentiable at the junction, Lemma 12 is replaced by the following one:
Suppose \(u : (-\infty ,0] \rightarrow {\mathbb {R}}\) is continuous and \(u_{x}(0)\) does not exist. Let \(p^{+} = \limsup _{x \rightarrow 0^{-}} \frac{u(x) - u(0)}{x}\) and \(p^{-} = \liminf _{x \rightarrow 0^{-}} \frac{u(x) - u(0)}{x}\). If \(p \in (p^{-},p^{+})\), then there is a sequence \((x_{n}^{+})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0)\) such that
\(p \in J^{+}u(x_{n}^{+})\) for all \(n \in {\mathbb {N}}\)
Similarly, suppose \(v : (-\infty ,0] \rightarrow {\mathbb {R}}\) is continuous and \(v_{x}(0)\) does not exist. Let \(q^{+} = \limsup _{x \rightarrow 0^{-}} \frac{v(x) - v(0)}{x}\) and \(q^{-} = \liminf _{x \rightarrow 0^{-}} \frac{v(x) - v(0)}{x}\). If \(q \in (q^{-},q^{+})\), then there is a sequence \((x_{n}^{-})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0)\) such that
\(q \in J^{-}v(x_{n}^{-})\) for all \(n \in {\mathbb {N}}\)
We only provide the arguments in the upper semi-continuous case since the lower semi-continuous case follows by a transformation as in the previous lemma.
First, observe that since \(p^{+} > p^{-}\), u crosses the line \(x \mapsto px\) infinitely often as \(x \rightarrow 0^{-}\). Therefore, there is a sequence \((y_{n})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0)\) such that
\(y_{n} < y_{n + 1}\) for all \(n \in {\mathbb {N}}\),
\(\lim _{n \rightarrow \infty } y_{n} = 0\),
\(\frac{u(y_{n}) - u(0)}{y_{n}} \le p\) for all \(n \in {\mathbb {N}}\), and
For all \(n \in {\mathbb {N}}\), \(y \mapsto u(y) - u(0) - py\) has a positive maximum in \([y_{n},y_{n + 1}]\).
For each \(n \in {\mathbb {N}}\), let \(x_{n}^{+}\) be a point in \([y_{n},y_{n + 1}]\) where \(y \mapsto u(y) - u(0) - py\) is maximized. Notice that (iii) and (iv) imply \(x_{n}^{+} \in (y_{n},y_{n + 1})\). Therefore, \(p \in J^{+}u(x_{n}^{+})\). Moreover, \(\lim _{n \rightarrow \infty } x_{n}^{+} = 0\), and, thus, by assumption, \(\lim _{n \rightarrow \infty } u(x_{n}^{+}) = u(0)\). \(\square \)
Finally, we have the ingredients necessary to establish Theorem 7. For the sake of clarity, we begin by boiling Lemmas 12 and 13 down into the form we will use in the proof.
Fix \(i \in \{1,2,\dots ,K\}\). Suppose that \(u : (-\infty ,0] \rightarrow {\mathbb {R}}\) is a continuous sub-solution (resp. super-solution) of \(u + H_{i}(x,u_{x}) = 0\) in \((-\infty ,0)\). Let \(p^{+} = \limsup _{x \rightarrow 0^{-}} \frac{u(x) - u(0)}{x}\) and \(p^{-} = \liminf _{x \rightarrow 0^{-}} \frac{u(x) - u(0)}{x}\). If \(p \in (p^{-},p^{+})\), then \(u(0) + H_{i}(0,p) \le 0\) (resp. \(\ge 0\)).
If \(|p^{+}| < \infty \) (resp. \(|p^{-}| < \infty \)), then the conclusion holds with \(p = p^{+}\) (resp. \(p = p^{-}\)) as well.
We only provide the arguments when u is a sub-solution since the super-solution case follows in the same way.
Fix \(p \in (p^{-},p^{+})\). Notice that Lemmas 12 and 13 together imply that there is a sequence \((x_{n},p_{n})_{n \in {\mathbb {N}}} \subseteq (-\infty ,0) \times {\mathbb {R}}\) such that \(p_{n} \in J^{+}u(x_{n})\) independently of \(n \in {\mathbb {N}}\) and \(\lim _{n \rightarrow \infty } (x_{n},p_{n},u(x_{n})) = (0,p,u(0))\). Since \(x_{n} < 0\), we can invoke the sub-solution property to find
$$\begin{aligned} u(x_{n}) + H_{i}(x_{n},p_{n}) \le 0, \end{aligned}$$
which, upon sending \(n \rightarrow \infty \), becomes \(u(0) + H_{i}(0,p) \le 0\).
If \(|p^{+}| < \infty \), then \(u(0) + H_{i}(0,p^{+}) \le 0\) follows from the continuity of \(p \mapsto H_{i}(0,p)\). The same can be said if \(|p^{-}| < \infty \). \(\square \)
The proof of Theorem 7 is now an application of Proposition 19 and Remark 1:
Proof of Theorem 7
We will only give the details for sub-solutions. In addition to \((p_{1}^{+},\dots ,p_{K}^{+})\), let us also define \((p_{1}^{-},\dots ,p_{K}^{-})\) by
$$\begin{aligned} p_{i}^{-} = \liminf _{I_{i} \ni x \rightarrow 0} \frac{u(x) - u(0)}{x}. \end{aligned}$$
Proposition 19 implies (i) directly. Additionally, it shows that if \({\tilde{p}}_{i} \ge p_{i}^{-}\) for some \(i \in \{1,2,\dots ,K\}\), then (12) in (ii) holds.
It only remains to establish (ii) in the case when \({\tilde{p}}_{i} < p_{i}^{-}\) for all \(i \in \{1,2,\dots ,K\}\). Remark 1 implies that, in this case, if \(\varphi : {\mathcal {I}} \rightarrow {\mathbb {R}}\) is given by
$$\begin{aligned} \varphi (x) = u(0) + {\tilde{p}}_{i}x \quad \text {if} \, \, x \in \overline{I_{i}}, \end{aligned}$$
then \(u - \varphi \) has a local maximum at 0. Therefore, since u is a sub-solution, we find
$$\begin{aligned} \min \left\{ \sum _{i = 1}^{K} {\tilde{p}}_{i}, u(0) + \min _{i} H_{i}(0,{\tilde{p}}_{i}) \right\} \le 0. \end{aligned}$$
Thus, (12) holds, as claimed. \(\square \)
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© Springer Nature Switzerland AG 2020
1.Department of MathematicsUniversity of ChicagoChicagoUSA
Morfe, P.S. Nonlinear Differ. Equ. Appl. (2020) 27: 10. https://doi.org/10.1007/s00030-020-0615-1
Accepted 02 January 2020
DOI https://doi.org/10.1007/s00030-020-0615-1
Publisher Name Springer International Publishing
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\begin{document}
\title{On Kernelized Multi-armed Bandits} \author{\name{Sayak Ray Chowdhury} \email{[email protected]}\\ \addr Electrical Communication Engineering,\\ Indian Institute of Science,\\ Bangalore 560012, India\\ \\ \name{Aditya Gopalan} \email{[email protected]}\\ \addr Electrical Communication Engineering,\\ Indian Institute of Science,\\ Bangalore 560012, India\\ }
\maketitle
\begin{abstract}
We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization -- Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized concentration inequality for vector-valued martingales of arbitrary, possibly infinite, dimension. Finally, experimental evaluation and comparisons to existing algorithms on synthetic and real-world environments are carried out that highlight the favorable gains of the proposed strategies in many cases. \end{abstract} \section{Introduction} \label{sec:Introduction}
Optimization over large domains under uncertainty is an important subproblem arising in a variety of sequential decision making problems, such as dynamic pricing in economics \citep{BesZee09:dynpricinglearning}, reinforcement learning with continuous state/action spaces \citep{kaelbling1996reinforcement, SmaKae00:practicalRLcontinuous}, and power control in wireless communication \citep{ChiHanLanTan08:powercontrol}. A typical feature of such problems is a large, or potentially infinite, domain of decision points or covariates (prices, actions, transmit powers), together with only partial and noisy observability of the associated outcomes (demand, state/reward, communication rate); reward/loss information is revealed only for decisions that are chosen. This often makes it hard to balance exploration and exploitation, as available knowledge must be transferred efficiently from a finite set of observations so far to estimates of the values of infinitely many decisions. A classic case in point is that of the canonical stochastic MAB with finitely many arms, where the effort to optimize scales with the total number of arms or decisions; the effect of this is catastrophic for large or infinite arm sets.
With suitable structure in the values or rewards of arms, however, the challenge of sequential optimization can be efficiently addressed. Parametric bandits, especially linearly parameterized bandits \citep{RusTsi10:linbandits}, represent a well-studied class of structured decision making settings. Here, every arm corresponds to a known, finite dimensional vector (its feature vector), and its expected reward is assumed to be an unknown linear function of its feature vector. This allows for a large, or even infinite, set of arms all lying in space of finite dimension, say $d$, and a rich line of work gives algorithms that attain sublinear regret with a polynomial dependence on the dimension, e.g., Confidence Ball \citep{dani2008stochastic}, OFUL \citep{abbasi2011improved} (a strengthening of Confidence Ball) and Thompson sampling for linear bandits \citep{agrawal2013thompson}\footnote{Roughly, for rewards bounded in $[-1,1]$, these algorithms achieve optimal regret $\Otilde{d\sqrt{T}}$, where $\Otilde{\cdot}$ hides $\text{polylog}(T)$ factors.} The insight here is that even though the number of arms can be large, the number of unknown parameters (or degrees of freedom) in the problem is really only $d$, which makes it possible to learn about the values of many other arms by playing a single arm.
A different approach to modelling bandit problems with a continuum of arms is via the framework of Gaussian processes (GPs) \citep{rasmussen2006gaussian}. GPs are a flexible class of nonparametric models for expressing uncertainty over functions on rather general domain sets, which generalize multivariate Gaussian random vectors. GPs allow tractable regression for estimating an unknown function given a set of (noisy) measurements of its values at chosen domain points. The fact that GPs, being distributions on functions, can also help quantify function uncertainty makes it attractive for basing decision making strategies on them. This has been exploited to great advantage to build nonparametric bandit algorithms, such as GP-UCB \citep{srinivas2009gaussian}, GP-EI and GP-PI \citep{hoffman2011portfolio}. In fact, GP models for bandit optimization, in terms of their kernel maps, can be viewed as the parametric linear bandit paradigm pushed to the extreme, where each feature vector associated to an arm can have infinite dimension \footnote{The completion of the linear span of all feature vectors (images of the kernel map) is precisely the reproducing kernel Hilbert space (RKHS) that characterizes the GP.}.
Against this backdrop, our work revisits the problem of bandit optimization with stochastic rewards. Specifically, we consider stochastic multiarmed bandit (MAB) problems with a continuous arm set, and whose (unknown) expected reward function is assumed to lie in a reproducing kernel Hilbert space (RKHS), with bounded RKHS norm -- this effectively enforces smoothness on the function\footnote{Kernels, and their associated RKHSs, }. We make the following contributions- \begin{itemize} \item We design a new algorithm -- Improved Gaussian Process-Upper Confidence Bound (IGP-UCB) -- for stochastic bandit optimization. The algorithm can be viewed as a variant of GP-UCB \citep{srinivas2009gaussian}, but uses a significantly reduced confidence interval width resulting in an order-wise improvement in regret compared to GP-UCB. IGP-UCB also shows a markedly improved numerical performance over GP-UCB.
\item We develop a nonparametric version of Thompson sampling, called Gaussian Process Thompson sampling (GP-TS), and show that enjoys a regret bound of $\Otilde{\gamma_T \sqrt{dT}}$. Here, $T$ is the total time horizon and $\gamma_T$ is a quantity depending on the RKHS containing the reward function. This is, to our knowledge, the first known regret bound for Thompson sampling in the agnostic setup with nonparametric structure.
\item We prove a new self-normalized concentration inequality for infinite-dimensional vector-valued martingales, which is not only key to the design and analysis of the IGP-UCB and GP-TS algorithms, but also potentially of independent interest. The inequality generalizes a corresponding self-normalized bound for martingales in finite dimension proven by \citet{abbasi2011improved}.
\item Empirical comparisons of the algorithms developed above, with other GP-based algorithms, are presented, over both synthetic and real-world setups, demonstrating performance improvements of the proposed algorithms, as well as their performance under misspecification.
\end{itemize}
\section{Problem Statement} \label{sec:Problem-Statement} We consider the problem of sequentially maximizing a fixed but unknown reward function $f:D\ra \Real$ over a (potentially infinite) set of decisions $D \subset \Real^d$, also called actions or arms. An algorithm for this problem chooses, at each round $t$, an action $x_t \in D$, and subsequently observes a reward $y_t=f(x_t) + \epsilon_t$, which is a noisy version of the function value at $x_t$. The action $x_t$ is chosen causally depending upon the arms played and rewards obtained upto round $t-1$, denoted by the history $\cH_{t-1} = \lbrace (x_s,y_s): s = 1,\ldots,t-1\rbrace$. We assume that the noise sequence $\lbrace\epsilon_t\rbrace_{t=1}^{\infty}$ is conditionally $R$-sub-Gaussian for a fixed constant $R \ge 0$, i.e., \begin{equation} \forall t \ge 0,\;\;\forall \lambda \in \Real, \;\; \expect{e^{\lambda \epsilon_t} \; \big\vert \; \cF_{t-1}} \le \exp\left(\frac{\lambda^2R^2}{2}\right), \label{eqn:noise} \end{equation} where $\cF_{t-1}$ is the $\sigma$-algebra generated by the random variables $\lbrace x_s, \epsilon_s\rbrace_{s = 1}^{t-1}$ and $x_t$.
This is a mild assumption on the noise (it holds, for instance, for distributions bounded in $[-R, R]$) and is standard in the bandit literature \citep{abbasi2011improved,agrawal2013thompson}.
\textbf{Regret.} The goal of an algorithm is to maximize its cumulative reward or alternatively minimize its cumulative {\em regret} -- the loss incurred due to not knowing $f$'s maximum point beforehand. Let $x^\star \in \mathop{\mathrm{argmax}}_{x\in D}f(x)$ be a maximum point of $f$ (assuming the maximum is attained). The instantaneous regret incurred at time $t$ is $r_t = f(x^\star)-f(x_t)$, and the cumulative regret in a time horizon $T$ (not necessarily known a priori) is defined to be $R_T = \sum_{t=1}^T r_t$. A sub-linear growth of $R_T$ in $T$ signifies that $R_T/T \ra 0$ as $T\ra \infty$, or vanishing per-round regret.
\textbf{Regularity Assumptions.} Attaining sub-linear regret is impossible in general for arbitrary reward functions $f$ and domains $D$, and thus some regularity assumptions are in order. In what follows, we assume that $D$ is compact. The smoothness assumption we make on the reward function $f$ is motivated by Gaussian processes\footnote{Other work has also studied continuum-armed bandits with weaker smoothness assumptions such as Lipschitz continuity -- see Related work for details and comparison.} and their associated reproducing kernel Hilbert spaces (RKHSs, see \citet{scholkopf2002learning}). Specifically, we assume that $f$ has small norm in the RKHS of functions $D \to \mathbb{R}$, with positive semi-definite kernel function $k: D \times D \to \mathbb{R}$. This RKHS, denoted by $H_k(D)$, is completely specified by its kernel function $k(\cdot,\cdot)$ and vice-versa, with an inner product $\inner{\cdot}{\cdot}_k$ obeying the reproducing property: $f(x)=\inner{f}{k(x,\cdot)}_k$ for all $f \in H_k(D)$. In other words, the kernel plays the role of delta functions to represent the evaluation map at each point $x \in D$ via the RKHS inner product. The RKHS norm $\norm{f}_k = \sqrt{\inner{f}{f}}_k$ is a measure of the smoothness\footnote{One way to see this is that for every element $g$ in the RKHS, $|g(x) - g(y)| = |\inner{g}{k(x,\cdot) - k(y,\cdot)}| \le \norm{g}_k \norm{k(x,\cdot) - k(y,\cdot)}_k$ by Cauchy-Schwarz.} of $f$, with respect to the kernel function $k$, and satisfies: $f \in H_k(D)$ if and only if $\norm{f}_k < \infty$.
We assume a known bound on the RKHS norm of the unknown target function\footnote{This is analogous to the bound on the weight $\theta$ typically assumed in regret analyses of linear parametric bandits.}: $\norm{f}_k \le B$. Moreover, we assume bounded variance by restricting $k(x,x) \le 1$, for all $x \in D$. Two common kernels that satisfy bounded variance property are \textit{Squared Exponential} and \textit{Mat$\acute{e}$rn}, defined as \begin{eqnarray*} k_{SE}(x,x') &=& \exp\Big(-s^2/2l^2\Big), \\ k_{Mat\acute{e}rn}(x,x') &=& \frac{2^{1-\nu}}{\Gamma(\nu)}\Big(\frac{s\sqrt{2\nu}}{l}\Big)^\nu B_\nu\Big(\frac{s\sqrt{2\nu}}{l}\Big), \end{eqnarray*} where $l > 0$ and $\nu > 0$ are hyperparameters, $s = \norm{x-x'}_2$ encodes the similarity between two points $x,x'\in D$, and $B_\nu(\cdot)$ is the modified Bessel function. Generally the bounded variance property holds for any stationary kernel, i.e. kernels for which $k(x,x')=k(x-x')$ for all $x,x'\in \Real^d$. These assumptions are required to make the regret bounds scale-free and are standard in the literature \citep{agrawal2013thompson}. Instead if $k(x,x) \le c$ or $\norm{f}_k \le cB$, then our regret bounds would increase by a factor of $c$.
\section{Algorithms} \label{sec:Algorithms} {\bf Design philosophy.} Both the algorithms we propose use Gaussian likelihood models for observations, and Gaussian process (GP) priors for uncertainty over reward functions. A Gaussian process over $D$, denoted by $GP_D(\mu(\cdot),k(\cdot,\cdot))$, is a collection of random variables $(f(x))_{x \in D}$, one for each $x \in D$, such that every finite sub-collection of random variables $(f(x_i))_{i = 1}^m$ is jointly Gaussian with mean $\expect{f(x_i)} = \mu(x_i)$ and covariance $\expect{(f(x_i)-\mu(x_i))(f(x_j)-\mu(x_j))} = k(x_i, x_j)$, $1 \le i, j \le m$, $m \in \mathbb{N}$. The algorithms use $GP_D(0,v^2k(\cdot,\cdot))$, $v > 0$, as an initial prior distribution for the unknown reward function $f$ over $D$, where $k(\cdot,\cdot)$ is the kernel function associated with the RKHS $H_k(D)$ in which $f$ is assumed to have `small' norm at most $B$. The algorithms also assume that the noise variables $\epsilon_t = y_t-f(x_t)$ are drawn independently, across $t$, from $\cN(0,\lambda v^2)$, with $\lambda \ge 0$. Thus, the prior distribution for each $f(x)$, is assumed to be $\cN(0,v^2k(x,x))$, $x\in D$. Moreover, given a set of sampling points $A_t = (x_1, \ldots, x_t)$ within $D$, it follows under the assumption that the corresponding vector of observed rewards $y_{1:t}= [y_1,\ldots,y_t]^T$ has the multivariate Gaussian distribution $\cN(0,v^2(K_t+\lambda I))$, where $K_t = [k(x,x')]_{x,x'\in A_t}$ is the kernel matrix at time $t$. Then, by the properties of GPs, we have that $y_{1:t}$ and $f(x)$ are jointly Gaussian given $A_t$: \begin{equation*} \begin{bmatrix} f(x) \\ y_{1:t} \end{bmatrix} \sim \cN\left(0,\begin{bmatrix} v^2k(x,x) & v^2k_t(x)^T\\ v^2k_t(x) & v^2(K_t+\lambda I) \end{bmatrix}\right), \end{equation*} where $k_t(x) = [k(x_1,x),\ldots, k(x_t,x)]^T$. Therefore conditioned on the history $\cH_t$, the posterior distribution over $f$ is $GP_D(\mu_t(\cdot),v^2k_t(\cdot,\cdot))$, where \begin{eqnarray} \mu_t(x) &=& k_t(x)^T(K_t + \lambda I)^{-1}y_{1:t}, \label{eqn:mean update}\\ k_t(x,x') &=& k(x,x') - k_t(x)^T(K_t + \lambda I)^{-1} k_t(x'),\label{eqn:cov update}\\ \sigma_t^2(x) &=& k_t(x,x).\label{eqn:sd update} \end{eqnarray} Thus for every $x \in D$, the posterior distribution of $f(x)$, given $\cH_t$, is $\cN(\mu_t(x),v^2\sigma_t^2(x))$.
\textit{\textbf{Remark.}} Note that the GP prior and Gaussian likelihood model described above is only an aid to algorithm design, and has nothing to do with the actual reward distribution or noise model as in the problem statement (Section \ref{sec:Problem-Statement}). The reward function $f$ is a fixed, unknown, member of the RKHS $H_k(D)$, and the true sequence of noise variables $\epsilon_t$ is allowed to be a conditionally $R$-sub-Gaussian martingale difference sequence (Equation \ref{eqn:noise}). In general, thus, this represents a misspecified prior and noise model, also termed the {\em agnostic} setting by \citet{srinivas2009gaussian}.
The proposed algorithms, to follow, assume the knowledge of only the sub-Gaussianity parameter $R$, kernel function $k$ and upper bound $B$ on the RKHS norm of $f$. Note that $v, \lambda$ are free parameters (possibly time-dependent) that can be set specific to the algorithm.
\subsection{Improved GP-UCB (IGP-UCB) Algorithm} \label{subesc:UCB}
We introduce the IGP-UCB algorithm (Algorithm \ref{algo:ucb}), that uses a combination of the current posterior mean $\mu_{t-1}(x)$ and standard deviation $v \sigma_{t-1}(x)$ to (a) construct an upper confidence bound (UCB) envelope for the actual function $f$ over $D$, and (b) choose an action to maximize it. Specifically it chooses, at each round $t$, the action \begin{equation} x_t=\mathop{\mathrm{argmax}}_{x\in D}\mu_{t-1}(x)+\beta_t\sigma_{t-1}(x), \end{equation} \label{eqn:UCB-rule} with the scale parameter $v$ set to be $1$. Such a rule trades off exploration (picking points with high uncertainty $\sigma_{t-1}(x)$) with exploitation (picking points with high reward $\mu_{t-1}(x)$), with $\beta_t=B+ R\sqrt{2(\gamma_{t-1}+1+\ln(1/\delta))}$ being the parameter governing the tradeoff, which we later show is related to the width of the confidence interval for $f$ at round $t$. $\delta \in (0,1)$ is a free {\em confidence} parameter used by the algorithm, and $\gamma_t$ is the \textit{maximum information gain} at time $t$, defined as: \begin{equation*} \gamma_t := \max_{A \subset D : \abs{A}=t} I(y_A;f_A). \end{equation*} Here, $I(y_A;f_A)$ denotes the \textit{mutual information} between $f_A = [f(x)]_{x\in A}$ and $y_A = f_A + \epsilon_A$, where $\epsilon_A \sim \cN(0,\lambda v^2 I)$ and quantifies the reduction in uncertainty about $f$ after observing $y_A$ at points $A \subset D$. $\gamma_t$ is a problem dependent quantity and can be found given the knowledge of domain $D$ and kernel function $k$. For a compact subset $D$ of $\Real^d$, $\gamma_T$ is $O((\ln T)^{d+1})$ and $O(T^{d(d+1)/(2\nu+d(d+1))}\ln T)$, respectively, for the Squared Exponential and Mat$\acute{e}$rn kernels \citep{srinivas2009gaussian}, depending only polylogarithmically on the time $T$.
\begin{algorithm} \renewcommand\thealgorithm{1} \caption{Improved-GP-UCB (IGP-UCB)}\label{algo:ucb} \begin{algorithmic} \STATE \textbf{Input:} Prior $GP(0,k)$, parameters $B$, $R$, $\lambda$, $\delta$. \FOR{t = 1, 2, 3 \ldots T} \STATE Set $\beta_t=B+ R\sqrt{2(\gamma_{t-1}+ 1+\ln(1/\delta))}$. \STATE Choose $x_t = \mathop{\mathrm{argmax}}\limits_{x\in D} \mu_{t-1}(x) + \beta_t\sigma_{t-1}(x)$. \STATE Observe reward $y_t = f(x_t)+\epsilon_t$. \STATE Perform update to get $\mu_t$ and $\sigma_t$ using \ref{eqn:mean update}, \ref{eqn:cov update} and \ref{eqn:sd update}. \ENDFOR \end{algorithmic} \addtocounter{algorithm}{-1} \end{algorithm} \textit{\textbf{Discussion.}} \cite{srinivas2009gaussian} have proposed the GP-UCB algorithm, and \citet{valko2013finite} the KernelUCB algorithm, for sequentially optimizing reward functions lying in the RKHS $H_k(D)$. Both algorithms play an arm at time $t$ using the rule: $x_t=\mathop{\mathrm{argmax}}_{x\in D}\mu_{t-1}(x)+\tilde{\beta}_t\sigma_{t-1}(x)$.
GP-UCB uses the exploration parameter $\tilde{\beta}_t = \sqrt{2B^2+300\gamma_{t-1}\ln^3(t/\delta)}$, with $\lambda$ set to $\sigma^2$, where $\sigma$ is additionally assumed to be a known, uniform (i.e., almost-sure) upper bound on all noise variables $\epsilon_t$ \citep[Theorem $3$]{srinivas2009gaussian}. Compared to GP-UCB, IGP-UCB (Algorithm \ref{algo:ucb}) reduces the width of the confidence interval by a factor roughly $O(\ln^{3/2}t)$ at every round $t$, and, as we will see, this small but critical adjustment leads to much better theoretical and empirical performance compared to GP-UCB. In KernelUCB, $\tilde{\beta}_t$ is set as $\eta/\lambda^{1/2}$, where $\eta$ is the exploration parameter and $\lambda$ is the regularization constant. Thus IGP-UCB can be viewed as a special case of KernelUCB where $\eta=\beta_t$.
\subsection{Gaussian Process Thompson Sampling (GP-TS)}
\label{subsection:TS}
Our second algorithm, GP-TS (Algorithm \ref{algo:ts}), inspired by the success of Thompson sampling for standard and parametric bandits \citep{agrawal2012analysis,kaufmann2012thompson,gopalan2014thompson,agrawal2013thompson}, uses the time-varying scale parameter $v_t = B + R\sqrt{2(\gamma_{t-1}+ 1+ \ln(2/\delta))}$ and operates as follows. At each round $t$, GP-TS samples a random function $f_t(\cdot)$ from the GP with mean function $\mu_{t-1}(\cdot)$ and covariance function $v_t^2k_{t-1}(\cdot,\cdot)$. Next, it chooses a decision set $D_t \subset D$, and plays the arm $x_t \in D_t$ that maximizes $f_t$\footnote{If $D_t = D$ for all $t$, then this is simply {\em exact} Thompson sampling. For technical reasons, however, our regret bound is valid when $D_t$ is chosen as a suitable discretization of $D$, so we include $D_t$ as an algorithmic parameter.}. We call it GP-Thompson-Sampling as it falls under the general framework of Thompson Sampling, i.e., (a) assume a prior on the underlying parameters of the reward distribution, (b) play the arm according to the prior probability that it is optimal, and (c) observe the outcome and update the prior. However, note that the prior is nonparametric in this case.
\begin{algorithm} \renewcommand\thealgorithm{2} \caption{GP-Thompson-Sampling (GP-TS)}\label{algo:ts} \begin{algorithmic} \STATE \textbf{Input:} Prior $GP(0,k)$, parameters $B$, $R$, $\lambda$, $\delta$. \FOR{t = 1, 2, 3 \ldots,} \STATE Set $v_t = B + R\sqrt{2(\gamma_{t-1}+ 1+ \ln(2/\delta))}$. \STATE Sample $f_t(\cdot)$ from $GP_D(\mu_{t-1}(\cdot),v_t^2k_{t-1}(\cdot,\cdot))$. \STATE Choose the current decision set $D_t \subset D$. \STATE Choose $x_t = \mathop{\mathrm{argmax}}\limits_{x\in D_t} f_t(x)$. \STATE Observe reward $y_t = f(x_t)+\epsilon_t$. \STATE Perform update to get $\mu_t$ and $k_t$ using \ref{eqn:mean update} and \ref{eqn:cov update}. \ENDFOR \end{algorithmic} \addtocounter{algorithm}{-2} \end{algorithm}
\section{Main Results} \label{sec:main-results} We begin by presenting two key concentration inequalities which are essential in bounding the regret of the proposed algorithms. \begin{mytheorem} \label{thm:self-normalized-bound} Let $\lbrace x_t \rbrace_{t=1}^{\infty}$ be an $\Real^d$-valued discrete time stochastic process predictable with respect to the filtration $\lbrace \cF_t \rbrace_{t=0}^{\infty}$, i.e., $x_t$ is $\cF_{t-1}$-measurable $\forall t \ge 1$. Let $\lbrace\epsilon_t\rbrace_{t=1}^{\infty}$ be a real-valued stochastic process such that for some $R \ge 0$ and for all $t \ge 1$, $\epsilon_t$ is (a) $\cF_t$-measurable, and (b) $R$-sub-Gaussian conditionally on $\cF_{t-1}$. Let $k: \Real^d \times \Real^d \to \Real$ be a symmetric, positive-semidefinite kernel, and let $0 < \delta \le 1$.
For a given $\eta > 0$, with probability at least $1 -\delta$, the following holds simultaneously over all $t \ge 0$: \begin{equation} \label{eqn:thmpart1} \norm{\epsilon_{1:t}}_{((K_t+ \eta I)^{-1} + I)^{-1}}^2 \le 2R^2\ln \frac{\sqrt{\det((1+\eta)I+K_t)}}{\delta}. \end{equation} (Here, $K_t$ denotes the $t \times t$ matrix $K_t(i,j) = k(x_i, x_j)$, $1 \le i, j \le t$ and for any $x \in \Real^t$ and $A \in \Real^{t\times t}$, $\norm{x}_A := \sqrt{x^TAx}$).
Moreover, if $K_t$ is positive definite $\forall t \ge 1$ with probability 1, then the conclusion above holds with $\eta = 0$.
\end{mytheorem}
Theorem \ref{thm:self-normalized-bound} represents a self-normalized concentration inequality: the `size' of the increasing-length sequence $\{\epsilon_t\}_t$ of martingale differences is normalized by the growing quantity $((K_t+ \eta I)^{-1} + I)^{-1}$ that explicitly depends on the sequence. The following lemma helps provide an alternative, abstract, view of the self-normalized process of Theorem \ref{thm:self-normalized-bound}, based on the feature space representation induced by a kernel. \begin{mylemma} \label{lem:selfnorm} Let $k: \Real^d \times \Real^d \to \Real$ be a symmetric, positive-semidefinite kernel, with associated feature map $\phi: \Real^d \to H_k$ and the reproducing kernel Hilbert space\footnote{Such a pair $(\phi, H_k)$ always exists, see e.g., \citet{rasmussen2006gaussian}.} (RKHS) $H_k$. Letting $S_t = \sum_{s=1}^{t}\epsilon_s\phi(x_s)$ and the (possibly infinite dimensional) matrix\footnote{More formally, $V_t: H_k \to H_k$ is the linear operator defined by $V_t(z) = z + \sum_{s=1}^t \phi(x_s) \inner{\phi(x_s)}{z}$ $\forall z \in H_k$.} $V_t = I+\sum_{s=1}^{t}\phi(x_s)\phi(x_s)^T$, we have, whenever $K_t$ is positive definite, that \begin{equation*} \norm{\epsilon_{1:t}}_{\left( K_t^{-1} + I \right)^{-1}} = \norm{S_t}_{V_t^{-1}}, \end{equation*} where $ \norm{S_t}_{V_t^{-1}} := \norm{V_t^{-1/2} S_t}_{H_k}$ denotes the norm of $V_t^{-1/2} S_t$ in the RKHS $H_k$. \end{mylemma}
Observe that $S_t$ is $\cF_{t}$-measurable and also $\expect{S_t\; \big\vert \; \cF_{t-1}} = S_{t-1}$. The process $\lbrace S_t \rbrace_{t\ge0}$ is thus a martingale with values\footnote{We ignore issues of measurability here.} in the RKHS $H$, which can possibly be infinite-dimensional, and moreover, whose deviation is measured by the norm weighted by $V_t^{-1}$, which is itself derived from $S_t$. Theorem \ref{thm:self-normalized-bound} represents the kernelized generalization of the finite-dimensional result of \citet{abbasi2011improved}, and we recover their result under the special case of a linear kernel: $\phi(x)=x$ for all $x \in \Real^d$.
We remark that when $\phi$ is a mapping to a finite-dimensional Hilbert space, the argument of \citet[Theorem 1]{abbasi2011improved} can be lifted to establish Theorem \ref{thm:self-normalized-bound}, but it breaks down in the generalized, infinite-dimensional RKHS setting, as the self-normalized bound in their paper has an explicit, growing dependence on the feature dimension. Specifically, the method of mixtures \citep{de2009self} or Laplace method, as dubbed by \citet{maillard2016self} (Lemma 5.2), fails to hold in infinite dimension. The primary reason for this is that the mixture distribution for finite dimensional spaces can be chosen independently of time, but in a nonparametric setup like ours, where the dimensionality of the self-normalizing factor $\left( K_t^{-1} + I \right)^{-1}$ itself grows with time, the use of (random) stopping times, precludes using time-dependent mixtures. We get around this difficulty by applying a novel `double mixture' construction, in which a pair of mixtures on (a) the space of real-valued functions on $\Real^d$, i.e., the support of a Gaussian process, and (b) on real sequences is simultaneously used to obtain a more general result, of potentially independent interest (see Section \ref{sec:Key-Techniques} and the appendix for details).
Our next result shows that how the posterior mean is concentrated around the unknown reward function $f$. \begin{mytheorem} \label{thm:true-function-bound} Under the same hypotheses as those of Theorem \ref{thm:self-normalized-bound}, let $D \subset \Real^d$, and $f:D \to \Real$ be a member of the RKHS of real-valued functions on $D$ with kernel $k$, with RKHS norm bounded by $B$. Then, with probability at least $1-\delta$, the following holds for all $x \in D$ and $t \ge 1$: \begin{eqnarray*} \abs{\mu_{t-1}(x)-f(x)}\le \Big(B + R\sqrt{2(\gamma_{t-1}+1+ \ln(1/\delta))}\Big)\sigma_{t-1}(x), \end{eqnarray*} where $\gamma_{t-1}$ is the maximum information gain after $t-1$ rounds and $\mu_{t-1}(x)$, $\sigma^2_{t-1}(x)$ are mean and variance of posterior distribution defined as in Equation \ref{eqn:mean update}, \ref{eqn:cov update}, \ref{eqn:sd update}, with $\lambda$ set to $1+\eta$ and $\eta=2/T$. \end{mytheorem}
Theorem $3.5$ of \citet{maillard2016self} states a similar result on the estimation of the unknown reward function from the RKHS. We improve upon it in the sense that the confidence bound in Theorem \ref{thm:true-function-bound} is {\em simultaneous} over all $x \in D$, while the bound has been shown only for a single, fixed $x$ in the Kernel Least-squares setting.
We are able to achieve this result by virtue of Theorem \ref{thm:self-normalized-bound}.
\subsection{Regret Bound of IGP-UCB} \label{subsubsec:regret-UCB} \begin{mytheorem} \label{thm:regret-bound-UCB} Let $\delta\in(0,1)$, $\norm{f}_k \le B$ and $\epsilon_t$ is conditionally $R$-sub-Gaussian. Running IGP-UCB for a function $f$ lying in the RKHS $H_k(D)$, we obtain a regret bound of $O\Big(\sqrt{T}(B\sqrt{\gamma_T}+\gamma_T)\Big)$ with high probability. More precisely, with probability at least $1-\delta$, $R_T = O\Big(B\sqrt{T\gamma_T}+\sqrt{T\gamma_T(\gamma_T+\ln(1/\delta))}\Big)$. \end{mytheorem}
\textbf{Improvement over GP-UCB.} \citet{srinivas2009gaussian}, in the course of analyzing the GP-UCB algorithm, show that when the reward function lies in the RKHS $H_k(D)$, GP-UCB obtains regret $O\Big(\sqrt{T}(B\sqrt{\gamma_T}+\gamma_T\ln^{3/2}(T))\Big)$ with high probability (see Theorem $3$ therein for the exact bound). Furthermore, they assume that the noise $\epsilon_t$ is {\em uniformly bounded} by $\sigma$, while our sub-Gaussianity assumption (see Equation \ref{eqn:noise}) is slightly more general, and we are able to obtain a $O(\ln^{3/2}T)$ multiplicative factor improvement in the final regret bound thanks to the new self-normalized inequality (Theorem \ref{thm:self-normalized-bound}). Additionally, in our numerical experiments, we observe a significantly improved performance of IGP-UCB over GP-UCB, both on synthetically generated function, and on real-world sensor measurement data (see Section \ref{sec:Experiments}).
\textbf{Comparison with KernelUCB.} \citet{valko2013finite} show that the cumulative regret of KernelUCB is $\tilde{O}(\sqrt{\tilde{d}T})$, where $\tilde{d}$, defined as the \textit{effective dimension}, measures, in a sense, the number of principal directions over which the projection of the data in the RKHS is spread. They show that $\tilde{d}$ is at least as good as $\gamma_T$, precisely $\gamma_T \ge \Omega(\tilde{d}\ln \ln T)$ and thus the regret bound of KernelUCB is roughly $\tilde{O}(\sqrt{T\gamma_T})$, which is $\sqrt{\gamma_T}$ factor better than IGP-UCB. However, KernelUCB requires the number of actions to be \textit{finite}, so the regret bound is not applicable for infinite or continuum action spaces.
\subsection{Regret Bound of GP-TS} \label{subsubsec:regret-TS} For technical reasons, we will analyze the following version of GP-TS. At each round $t$, the decision set used by GP-TS is restricted to be a {unique} discretization $D_t$ of $D$ with the property that $\abs{f(x)-f([x]_t)} \le 1/t^2$ for all $x \in D$, where $[x]_t$ is the closest point to $x$ in $D_t$. This can always be achieved by choosing a compact and convex domain $D \subset [0,r]^d$ and discretization $D_t$ with size $\abs{D_t}=(BLrdt^2)^d$ such that $\norm{x-[x]_t}_1 \le rd/BLrdt^2 = 1/BLt^2$ for all $x \in D$, where $L= \sup\limits_{x\in D}\sup\limits_{j\in [d]}\Big(\frac{\partial^2 k(p,q)}{\partial p_j \partial q_j}\at{p=q=x}\Big)^{1/2}$. This implies, for every $x \in D$, \begin{equation} \abs{f(x)-f([x]_t)} \le\norm{f}_k L \norm{x-[x]_t}_1 \le 1/t^2, \label{eqn:lipschitz} \end{equation} as any $f\in H_k(D)$ is Lipschitz continuous with constant $\norm{f}_k L$ \citep[Lemma $1$]{de2012exponential}. \begin{mytheorem}[Regret bound for GP-TS] \label{thm:regret-bound-TS} Let $\delta\in(0,1)$, $D \subset [0,r]^d$ be compact and convex, $\norm{f}_k \le B$ and $\{\epsilon_t\}_t$ a conditionally $R$-sub-Gaussian sequence. Running GP-TS for a function $f$ lying in the RKHS $H_k(D)$ and with decision sets $D_t$ chosen as above, with probability at least $1-\delta$, the regret of GP-TS satisfies $R_T=O\Big(\sqrt{(\gamma_T+\ln(2/\delta))d\ln (BdT)} \Big(\sqrt{T\gamma_T}+B\sqrt{T\ln(2/\delta)}\Big)\Big)$. \end{mytheorem}
\textbf{Comparison with IGP-UCB.} Observe that regret scaling of GP-TS is $\tilde{O}(\gamma_T\sqrt{dT})$ which is a multiplicative $\sqrt{d}$ factor away from the bound $\tilde{O}(\gamma_T\sqrt{T})$ obtained for IGP-UCB and similar behavior is reflected in our simulations on synthetic data. The additional multiplicative factor of $\sqrt{d\ln(BdT)}$ in the regret bound of GP-TS is essentially a consequence of discretization. How to remove this extra logarithmic dependency, and make the analysis discretization-independent, remains an open question.
\textit{\textbf{Remark.}} The regret bound for GP-TS is inferior compared to IGP-UCB in terms of the dependency on dimension $d$, but to the best of our knowledge, Theorem \ref{thm:regret-bound-TS} is the first (frequentist) regret guarantee of Thompson Sampling in the {agnostic, non-parametric setting of infinite action spaces}.
\textbf{Linear Models and a Matching Lower Bound.} If the mean rewards are perfectly linear, i.e. if there exists a $\theta \in \Real^d$ such that $f(x)=\theta^Tx$ for all $x\in D$, then we are in the parametric setup, and one way of casting this in the kernelized framework is by using the {\em linear kernel} $k(x,x')=x^Tx'$.
For this kernel, $\gamma_T =O(d\ln T)$, and the regret scaling of IGP-UCB is $\tilde{O}(d\sqrt{T})$ and that of GP-TS is $\tilde{O}(d^{3/2}\sqrt{T})$, which recovers the regret bounds of their linear, parametric analogues OFUL \citep{abbasi2011improved} and Linear Thompson sampling \citep{agrawal2013thompson}, respectively. Moreover, in this case $\tilde{d} = d$, thus the regret of IGP-UCB is $\sqrt{d}$ factor away from that of KernelUCB. But the regret bound of KernelUCB also depends on the number of arms $N$, and if $N$ is exponential in $d$, then it also suffers $\tilde{O}(d\sqrt{T})$ regret. We remark that a similar $O(\ln^{3/2}T)$ factor improvement, as obtained by IGP-UCB over GP-UCB, was achieved in the linear parametric setting by \cite{abbasi2011improved} in the OFUL algorithm, over its predecessor ConfidenceBall \citep{dani2008stochastic}. Finally we see that the for linear bandit problem with infinitely many actions, IGP-UCB attains the information theoretic lower bound of $\Omega(d\sqrt{T})$ (see \cite{dani2008stochastic}), but GP-TS is a factor of $\sqrt{d}$ away from it.
\section{Overview of Techniques} \label{sec:Key-Techniques} We briefly outline here the key arguments for all the theorems in Section \ref{sec:main-results}. Formal proofs and auxiliary lemmas required are given in the appendix.
\textbf{Proof Sketch for Theorem \ref{thm:self-normalized-bound}.} It is convenient to assume that $K_t$, the induced kernel matrix at time $t$, is invertible, since this is where the crux of the argument lies. First we show that for any function $g:D \ra \Real$ and for all $t \ge 0$, thanks to the sub-Gaussian property (\ref{eqn:noise}), the process $\left\{M_t^g := \exp(\epsilon_{1:t}^Tg_{1:t}-\frac{1}{2}\norm{g_{1:t}}^2)\right\}_t$ is a non-negative super-martingale with respect to the filtration $\cF_t$, where $g_{1:t} := [g(x_1),\ldots,g(x_t)]^T$ and in fact satisfies $\expect{M_t^g}\le 1$. The chief difficulty is to handle the behavior of $M_t$ at a (random) stopping time, since the sizes of quantities such as $\epsilon_{1:t}$ at the stopping time will be random.
We next construct a mixture martingale $M_t$ by mixing $M_t^g$ over $g$ drawn from an independent $GP_D(0, k)$ Gaussian process, which is a measure over a large space of functions, i.e., the space $\Real^D$. Then, by a change of measure argument, we show that this induces a mixture distribution which is essentially $\cN(0,K_t)$ over {\em any} desired finite dimension $t$, thus obtaining $M_t = \frac{1}{\sqrt{\det(I+K_t)}}\exp\Big(\frac{1}{2}\norm{\epsilon_{1:t}}_{(I+K_t^{-1})^{-1}}^2\Big)$. Next from the fact that $\expect{M_\tau}\le 1$ and from Markov's inequality, for any $\delta \in (0,1)$, we obtain \begin{equation*} \prob{\norm{\epsilon_{1:\tau}}_{(K_\tau^{-1} + I)^{-1}}^2 > 2\ln\Big(\sqrt{\det(I+K_\tau)}/\delta\Big)} \le \delta. \end{equation*} Finally, we lift this bound simultaneously for all $t$ through a standard stopping time construction as in \citet{abbasi2011improved}.
\textbf{Proof Sketch for Theorem \ref{thm:true-function-bound}.} Here we sketch the special case of $\eta=0$, i.e. $\lambda =1$. Observe that $\abs{\mu_t(x)-f(x)}$ is upper bounded by sum of two terms, $P := \abs{k_t(x)^T(K_t+ I)^{-1}\eps_{1:t}}$ and $Q := \abs{k_t(x)^T(K_t+I)^{-1}f_{1:t} - f(x)}$. Now we observe that $\sigma_t^2(x)=\phi(x)^T(\Phi_t^T\Phi_t + I)^{-1}\phi(x)$ and use this observation to show that $P=\abs{\phi(x)^T(\Phi_t^T\Phi_t + I)^{-1}\Phi_t^T\eps_{1:t}}$ and $Q = \abs{\phi(x)^T(\Phi_t^T\Phi_t+ I)^{-1}f}$, which are in turn upper bounded by the terms $\sigma_t(x)\norm{S_t}_{V_t^{-1}}$ and $\norm{f}_k\sigma_t(x)$ respectively. Then the result follows using Theorem \ref{thm:self-normalized-bound}, along with the assumption that $\norm{f}_k \le B$ and the fact that $\frac{1}{2}\ln(\det(I+K_t)) \le \gamma_t$ almost surely (see Lemma \ref{lem:info-theoretic-results}) when $K_t$ is invertible.
\textbf{Proof Sketch for Theorem \ref{thm:regret-bound-UCB}.} First from Theorem \ref{thm:true-function-bound} and the choice of $x_t$ in Algorithm \ref{algo:ucb}, we show that the instantaneous regret $r_t$ at round $t$ is upper bounded by $2\beta_t\sigma_{t-1}(x_t)$ with probability at least $1-\delta$. Then the result follows by essentially upper bounding the term $\sum_{t=1}^{T}\sigma_{t-1}(x_t)$ by $O(\sqrt{T\gamma_T})$ (Lemma \ref{lem:bound-sum-sd} in the appendix).
\textbf{Proof Sketch for Theorem \ref{thm:regret-bound-TS}.}
We follow a similar approach given in \citet{agrawal2013thompson} to prove the regret bound of GP-TS. First observe that from our choice of discretization sets $D_t$, the instantaneous regret at round $t$ is given by $r_t = f(x^\star)-f([x^\star]_t)+f([x^\star]_t)-f(x_t) \le \frac{1}{t^2} + \Delta_t(x_t)$, where $\Delta_t(x):= f([x^\star]_t)-f(x)$ and $[x^\star]_t$ is the closest point to $x^\star$ in $D_t$. Now at each round $t$, after an action is chosen, our algorithm improves the confidence about true reward function $f$, via an update of $\mu_t(\cdot)$ and $k_t(\cdot,\cdot)$. However, if we play a suboptimal arm, the regret suffered can be much higher than the improvement of our knowledge. To overcome this difficulty, at any round $t$, we divide the arms (in the present discretization $D_t$) into two groups: \textit{saturated arms}, $S_t$, defined as those with $\Delta_t(x) > c_t\sigma_{t-1}(x)$ and \textit{unsaturated} otherwise, where $c_t$ is an appropriate constant
(see Definition \ref{def:constants}, \ref{def:saturated-arms}). The idea is to show that the probability of playing a saturated arm is small and then bound the regret of playing an unsaturated arm in terms of standard deviation. This is useful because the inequality $\sum_{t=1}^{T}\sigma_{t-1}(x_t)\le O(\sqrt{T\gamma_T})$ (Lemma \ref{lem:bound-sum-sd}) allows us to bound the total regret due to unsaturated arms.
First we lower bound the probability of playing an unsaturated arm at round $t$. We define a filtration $\cF^{'}_{t-1}$ as the history $\cH_{t-1}$ up to round $t-1$ and prove that for ``most" (in a high probability sense) $\cF^{'}_{t-1}$, $\prob{x_t \in D_t\setminus S_t\; \big\vert \; \cF^{'}_{t-1}} \ge p-1/t^2$, where $p=1/4e\sqrt{\pi}$ ( Lemma \ref{lem:prob-playing-saturated-arms}). This observation, along with concentration bounds for $f_t(x)$ and $f(x)$ (Lemma \ref{lem:event-concentration}) and ``smoothness" of $f$ (Equation \ref{eqn:lipschitz}), allow us to show that the expected regret at round $t$ is upper bounded in terms of $\sigma_{t-1}(x_t)$, i.e. in terms of regret due to playing an unsaturated arm. More precisely, we show that for ``most" $\cF^{'}_{t-1}$, $\expect{r_t\; \big\vert \; \cF^{'}_{t-1}} \le \frac{11c_t}{p}\expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} + \frac{2B+1}{t^2}$ (Lemma \ref{lem:bound-on-instantaneous-regret}), and use it to prove that $X_t \simeq r_t-\frac{11c_t}{p}\sigma_{t-1}(x_t) - \frac{2B+1}{t^2};t\ge 1$ is a super-martingale difference sequence adapted to filtration $\lbrace\cF^{'}_t\rbrace_{t\ge 1}$ (Lemma \ref{lem:supermartingale}). Now, using the Azuma-Hoeffding inequality (Lemma \ref{lem:total-regret}), along with the bound on $\sum_{t=1}^{T}\sigma_{t-1}(x_t)$, we obtain the desired high-probability regret bound.
\section{Experiments} \label{sec:Experiments} In this section we provide numerical results on both synthetically generated test functions and functions from real-world data. We compare GP-UCB, IGP-UCB and GP-TS with GP-EI and GP-PI\footnote{GP-EI and PI perform similarly and thus are not separately distinguishable in the plots.}.
{\bf Synthetic Test Functions.} \label{subsec:synthetic}
We use the following procedure to generate test functions from the RKHS. First we sample $100$ points uniformly from the interval $[0,1]$ and use that as our decision set. Then we compute a kernel matrix $K$ on those points and draw reward vector $y \sim \cN(0,K)$. Finally, the mean of the resulting posterior distribution is used as the test function $f$. We set noise parameter $R^2$ to be $1\%$ of function range and use $\lambda = R^2$. We used Squared Exponential kernel with lengthscale parameter $l = 0.2$ and Mat$\acute{e}$rn kernel with parameters $\nu=2.5,l=0.2$. Parameters $\beta_t,\tilde{\beta}_t,v_t$ of IGP-UCB, GP-UCB and GP-TS are chosen as given in Section \ref{sec:Algorithms}, with $\delta=0.1, B^2= f^TKf$ and $\gamma_t$ set according to theoretical upper bounds for corresponding kernels. We run each algorithm for $T=30000$ iterations, over $25$ independent trials (samples from the RKHS) and plot the average cumulative regret along with standard deviations (Figure \ref{fig:synthetic_plot_rkhs}). We see a significant improvement in the performance of IGP-UCB over GP-UCB. In fact IGP-UCB performs the best in the pool of competitors, while GP-TS also fares reasonably well compared to GP-UCB and GP-EI/GP-PI.
We next sample $25$ random functions from the $GP(0,K)$ and perform the same experiment (Figure \ref{fig:synthetic_plot_gp}) for both kernels with exactly same set of parameters. The relative performance of all methods is similar to that in the previous experiment, which is the arguably harder ``agnostic'' setting of a fixed, unknown target function. \begin{figure}
\caption{Cumulative regret for functions lying in the RKHS corresponding to (a) Squared Exponential kernel and (b) Mat$\acute{e}$rn kernel. }
\label{fig:synthetic_plot_rkhs}
\end{figure} \begin{figure}
\caption{Cumulative regret for functions lying in the GP corresponding to (a) Squared Exponential kernel and (b) Mat$\acute{e}$rn kernel.
}
\label{fig:synthetic_plot_gp}
\end{figure}
{\bf Standard Test Functions.} \label{subsec:standard} We consider $2$ well-known synthetic benchmark functions for Bayesian Optimization: \textit{Rosenbrock} and \textit{Hartman}3 (see \citet{azimi2012hybrid} for exact analytical expressions). We sample $100\;d$ points uniformly from the domain of each benchmark function, $d$ being the dimension of respective domain, as the decision set. We consider the Squared Exponential kernel with $l=0.2$ and set all parameters exactly as in previous experiment. The cumulative regret for $25$ independent trials on \textit{Rosenbrock} and \textit{Hartman3} benchmarks is shown in Figure \ref{fig:standard_plot}. We see GP-EI/PI perform better than the rest, while IGP-UCB and GP-TS show competitive performance. Here no algorithm is aware of the underlying kernel function, hence we conjecture that the UCB- and TS- based algorithms are somewhat less robust on the choice of kernel than EI/PI.
\begin{figure}
\caption{Cumulative regret for (a) \textit{Rosenbrock} and (b) \textit{Hartman3} benchmark function.
}
\label{fig:standard_plot}
\end{figure}
\textbf{Temperature Sensor Data.} \label{subsec:temp-sensor} We use temperature data\footnote{\url{http://db.csail.mit.edu/labdata/labdata.html}} collected from 54 sensors deployed in the Intel Berkeley Research lab between February 28th and April 5th, 2004 with samples collected at 30 second intervals. We tested all algorithms in the context of learning the maximum reading of the sensors collected between 8 am to 9 am. We take measurements of first 5 consecutive days (starting Feb. 28th 2004) to learn algorithm parameters. Following \citet{srinivas2009gaussian}, we calculate the empirical covariance matrix of the sensor measurements and use it as the kernel matrix in the algorithms. Here $R^2$ is set to be $5\%$ of the average empirical variance of sensor readings and other algorithm parameters is set similarly as in the previous experiment with $\gamma_t=1$ (found via cross-validation). The functions for testing consist of one set of measurements from all sensors in the two following days and the cumulative regret is plotted over all such test functions. From Figure \ref{fig:sensor_plot}, we see that IGP-UCB and GP-UCB performs the same, while GP-TS outperforms all its competitors. \begin{figure}
\caption{Cumulative regret plots for (a) temperature data and (b) light sensor data.
}
\label{fig:sensor_plot}
\end{figure}
\textbf{Light Sensor Data.} \label{subsec:light-sensor} We take light sensor data collected in the CMU Intelligent Workplace in Nov 2005, which is available online as Matlab structure\footnote{\url{http://www.cs.cmu.edu/~guestrin/Class/10708-F08/projects/lightsensor.zip}} and contains locations of $41$ sensors, $601$ train samples and $192$ test samples. We compute the kernel matrix, estimate the noise and set other algorithm parameters exactly as in the previous experiment. Here also GP-TS is found to perform better than the others, with IGP-UCB performing better than GP-EI/PI (Figure \ref{fig:sensor_plot}).
{\bf Related work.} An alternative line of work pertaining to $\mathcal{X}$-armed bandits \cite{kleinberg2008multi,bubeck2011x,carpentier2015simple,azar2014online} studies continuum-armed bandits with smoothness structure.
For instance, \cite{bubeck2011x} show that with a Lipschitzness assumption on the reward function, algorithms based on discretizing the domain yield nontrivial regret guarantees, of order $\Omega(T^{\frac{d+1}{d+2}})$ in $\Real^d$.
Other Bayesian approaches to function optimization are GP-EI \cite{movckus1975bayesian}, GP-PI \cite{kushner1964new}, GP-EST \cite{wang2016optimization} and GP-UCB, including the contextual \cite{krause2011contextual}, high-dimensional \cite{djolonga2013high,wang2013bayesian}, time-varying \cite{bogunovic2016time} safety-aware \cite{gotovos2015safe}, budget-constraint \cite{hoffman2013exploiting} and noise-free \cite{de2012exponential} settings. Other relevant work focuses on best arm identification problem in the Bayesian setup considering pure exploration \cite{grunewalder2010regret}.
For Thompson sampling (TS), \citet{russo2014learning} analyze the Bayesian regret of TS, which includes the case where the target function is sampled from a GP prior. Our work obtains the first frequentist regret of TS for unknown, fixed functions from an RKHS. \vskip -5mm
\section{Conclusion} \label{sec:Conclusion}
For bandit optimization, we have improved upon the existing GP-UCB algorithm, and introduced a new GP-TS algorithm. The proposed algorithms perform well in practice both on synthetic and real-world data. An interesting case is when the kernel function is also not known to the algorithms a priori and needs to be learnt adaptively. Moreover, one can consider classes of time varying functions from the RKHS, and general reinforcement learning with GP techniques. There are also important questions on computational aspects of optimizing functions drawn from GPs.
\section*{Appendix} \section*{A. Proof of Theorem \ref{thm:self-normalized-bound}} For a function $ g:D \ra \Real$ and a sequence of reals $n \equiv (n_t)_{t=1}^\infty$, define for any $t \ge 0 $ \begin{equation*} M_t^{g,n} = \exp\Big(\epsilon_{1:t}^Tg_{1:t,n}-\frac{R^2}{2}\norm{g_{1:t,n}}^2\Big), \end{equation*} where the vector $g_{1:t,n} := [g(x_1) + n_1,\ldots,g(x_t) + n_t]^T$.
We first establish the following technical result, which resembles \citet[Lemma 8]{abbasi2011improved}. \begin{mylemma} For fixed $g$ and $n$, $\lbrace M_t^{g,n} \rbrace_{t=0}^{\infty}$ is a super-martingale with respect to the filtration $\lbrace \cF_t\rbrace_{t=0}^{\infty}$. \end{mylemma}
\begin{proof} First, define \begin{equation*} \Delta_t^{g,n} := \exp\Big(\epsilon_t(g(x_t) + n_t)-\frac{R^2}{2}(g(x_t) + n_t)^2\Big). \end{equation*} Since $x_t$ is $\cF_{t-1}$-measurable and $\epsilon_t$ is $\cF_t$-measurable, $M_t^{g,n}$ as well as $\Delta_t^{g,n}$ are $\cF_t$ measurable. Also, by the conditional $R$-sub-Gaussianity of $\epsilon_t$, we have \begin{equation*} \forall \lambda \in \Real, \;\; \expect{e^{\lambda \epsilon_t} \; \big\vert \; \cF_{t-1}} \le \exp\left(\frac{\lambda^2R^2}{2}\right), \end{equation*} which in turn implies $\expect{\Delta_t^{g,n}\; \big\vert \; \cF_{t-1}} \le 1$. We also have \begin{eqnarray*} &&\expect{M_t^{g,n}\; \big\vert \; \cF_{t-1}}\\ &=& \expect{M_{t-1}^{g,n} \Delta_t^{g,n}\; \big\vert \; \cF_{t-1}} = M_{t-1}^{g,n} \expect{\Delta_t^{g,n} \; \big\vert \; \cF_{t-1}} \le M_{t-1}^{g,n}, \end{eqnarray*} showing that $\lbrace M_t^{g,n} \rbrace_{t=0}^{\infty}$ is a non-negative super-martingale and proving the lemma.
\end{proof} Also observe that $\expect{M_t^{g,n}} \le 1$ for all $t$, as \begin{equation*} \expect{M_t^{g,n}} \le \expect{M_{t-1}^{g,n}} \le \cdots \le \expect{M_0^{g,n}} = \expect{1} = 1. \end{equation*} Now, let $\tau$ be a stopping time with respect to the filtration $\lbrace \cF_t\rbrace_{t=0}^{\infty}$. By the convergence theorem for nonnegative super-martingales \citep{dur05:probtebook}, $M_{\infty}^{g,n} = \lim\limits_{t\ra \infty}M_t^{g,n}$ exists almost surely, and thus $M_\tau^{g,n}$ is well-defined. Now let $Q_t^{g,n} = M_{\min\lbrace \tau,t \rbrace}^{g,n}$, $t \ge 0$, be a stopped version of $\lbrace M_t^{g,n}\rbrace_t$. By Fatou's lemma \citep{dur05:probtebook}, \begin{eqnarray} \expect{M_\tau^{g,n}} &=& \expect{\lim_{t\ra \infty}Q_t^{g,n}} = \expect{\liminf_{t\ra \infty}Q_t^{g,n}}\nonumber\\ &\le & \liminf_{t \ra \infty} \expect{Q_t^{g,n}}\nonumber\\ &=& \liminf_{t \ra \infty} \expect{M_{\min\lbrace \tau,t \rbrace}^{g,n}} \le 1, \label{eqn:Fatou} \end{eqnarray} since the stopped super-martingale $\left(M_{\min\lbrace \tau,t \rbrace}^{g,n}\right)_t$ is also a super-martingale \citep{dur05:probtebook}.
Now, let $\cF_{\infty}$ be the $\sigma$-algebra generated by $\lbrace\cF_t\rbrace_{t=0}^{\infty}$, and let $N \equiv (N_t)_{t =1}^\infty$ be a sequence of independent and identically distributed Gaussian random variables with mean $0$ and variance $\eta$, independent of $\cF_\infty$. Let $h: D \to \Real$ be a random function distributed according to the Gaussian process measure $GP_D(0,k)$, and independent of both $\cF_{\infty}$ and $(N_t)_{t =1}^\infty$.
For each $t \ge 0$, define $M_t = \expect{M_t^{h,N}\; \big\vert \; \cF_{\infty}}$. In words, $(M_t)_t$ is a mixture of super-martingales of the form $M_t^{g,n}$, and it is not hard to see that $(M_t)_t$ is also a (non-negative) super-martingale w.r.t. the filtration $\{\cF_t\}_t$, hence $M_{\infty}=\lim\limits_{t\ra \infty}M_t$ is well-defined almost surely. We can write \begin{equation*} \expect{M_t} = \expect{M_t^{h,N}} = \expect{\expect{M_t^{h,N}\; \big\vert \; h, N}} \le \expect{1} = 1 \;\; \forall t. \end{equation*} An argument similar to (\ref{eqn:Fatou}) also shows that $\expect{M_\tau} \le 1$ for any stopping time $\tau$. Now, without loss of generality, we assume $R=1$ (this can always be achieved through appropriate scaling), and compute \begin{eqnarray*} M_t &=& \expect{\exp\Big(\epsilon_{1:t}^Th_{1:t,N}-\frac{1}{2}\norm{h_{1:t,N}}^2\Big)\; \big\vert \; \cF_{\infty}}\\ &=& \int_{\Real^D} \int_{\Real^t} \exp\Big(\epsilon_{1:t}^T ([h(x_1) \ldots h(x_t)]^T + z) -\frac{1}{2}\norm{[h(x_1) \ldots h(x_t)]^T + z}^2\Big) d\mu_1(h) d\mu_2(z)\\ &=& \int_{\Real^t}\exp\Big(\epsilon_{1:t}^T \lambda-\frac{1}{2}\norm{\lambda}^2\Big)f(\lambda) d\lambda, \end{eqnarray*} where $\mu_1$ is the Gaussian process measure $GP_D(0,k)$ over the function space $\Real^D \equiv \{g: D \to \Real \}$, $\mu_2$ is the multivariate Gaussian distribution on $\Real^t$ with mean $0$ and covariance $\eta I$ where $I$ is the identify, $du$ is standard Lebesgue measure on $\Real^t$, and $f$ is the density of the random vector $[h(x_1) \ldots h(x_t)]^T + z$, which is distributed as the multivariate Gaussian $\cN(0,K_t + \eta I)$ given the sampled points $x_1,\ldots, x_t$ up to round $t$, where $K_t$ is the induced kernel matrix at time $t$ given by $K_t(i,j) = k(x_i, x_j)$, $1 \le i, j \le t$. (Note: $K_t$ is not positive definite and invertible when there are repetitions among $(x_1, \ldots, x_t)$, but $K_t + \eta I$ is).
Thus, we have \begin{eqnarray*} & M_t&=\frac{1}{\sqrt{(2\pi)^t \det(K_t + \eta I)}}\int_{\Real^t}\exp\Bigg(\epsilon_{1:t}^T \lambda-\frac{\norm{\lambda}^2}{2}-\frac{\norm{\lambda}^2_{(K_t + \eta I)^{-1}}}{2}\Bigg)d\lambda\\ &=& \frac{\exp\Big(\frac{\norm{\epsilon_{1:t}}^2}{2}\Big)}{\sqrt{(2\pi)^t \det(K_t + \eta I)}}\int_{\Real^t}\exp\Bigg(-\frac{\norm{\lambda -\epsilon_{1:t}}^2}{2}-\frac{\norm{\lambda}^2_{(K_t+ \eta I)^{-1}}}{2}\Bigg)d\lambda. \end{eqnarray*} Now for positive-definite matrices $P$ and $Q$ \begin{equation*} \norm{x-a}_P^2 + \norm{x}_Q^2 = \norm{x-(P+Q)^{-1}Pa}_{P+Q}^2 + \norm{a}_P^2 -\norm{Pa}_{(P+Q)^{-1}}^2. \end{equation*} Therefore, \begin{eqnarray*} &&\norm{\lambda -\epsilon_{1:t}}_I^2 + \norm{\lambda}^2_{(K_t+ \eta I)^{-1}}\\ &=&\norm{\lambda - (I+(K_t+ \eta I)^{-1})^{-1}I\epsilon_{1:t}}_{I+(K_t+ \eta I)^{-1}}^2 + \norm{\epsilon_{1:t}}_I^2 - \norm{I\epsilon_{1:t}}_{(I+(K_t+ \eta I)^{-1})^{-1}}^2, \end{eqnarray*} which yields \begin{eqnarray*} M_t &=& \frac{1}{\sqrt{(2\pi)^t \det(K_t+ \eta I)}}\exp\Big(\frac{1}{2}\norm{\epsilon_{1:t}}_{(I+(K_t+ \eta I)^{-1})^{-1}}^2\Big)\\ &&\times\int_{\Real^t}\exp\Big(-\frac{1}{2}\norm{\lambda-(I+(K_t+ \eta I)^{-1})^{-1}\epsilon_{1:t}}^2_{I+(K_t+ \eta I)^{-1}}\Big)d\lambda\\ &=&\frac{1}{\sqrt{\det(K_t+ \eta I)\det((K_t+ \eta I)^{-1}+I)}} \exp\Big(\frac{1}{2}\norm{\epsilon_{1:t}}_{(I+(K_t+ \eta I)^{-1})^{-1}}^2\Big)\\ &=& \frac{1}{\sqrt{\det(I+K_t+ \eta I)}}\exp\Big(\frac{1}{2}\norm{\epsilon_{1:t}}_{(I+(K_t+ \eta I)^{-1})^{-1}}^2\Big), \end{eqnarray*} since for any positive definite matrix $A \in \Real^t$, \begin{eqnarray*} &&\int_{\Real^t}\exp\Big(-\frac{1}{2}(x-a)^TA(x-a)\Big)dx = \int_{\Real^t}\exp\Big(-\frac{1}{2}\norm{x-a}_A^2\Big)dx =\sqrt{(2\pi)^t/\det(A)}. \end{eqnarray*}
Now as $\expect{M_\tau} \le 1$, using Markov's inequality gives,
for any $\delta \in (0,1)$, \begin{eqnarray} &&\prob{\norm{\epsilon_{1:\tau}}_{((K_\tau + \eta I)^{-1} + I)^{-1}}^2 > 2\ln\Big(\sqrt{\det((1+\eta)I+K_\tau)}/\delta\Big)}\nonumber \\ &=& \prob{M_\tau > 1/\delta} < \delta \expect{M_\tau} \le \delta. \label{eqn:stopping-time-bound} \end{eqnarray} To complete the proof, we now employ a stopping time construction as in \citet{abbasi2011improved}. For each $t \ge 0$, define the `bad' event \begin{equation*} B_t(\delta) = \Big\lbrace \omega \in \Omega : \norm{\epsilon_{1:t}}_{((K_t + \eta I)^{-1} + I)^{-1}}^2 > 2\ln\Big(\sqrt{\det((1+\eta)I+K_t)}/\delta\Big) \Big\rbrace, \end{equation*} so that \begin{eqnarray*} &&\prob{\bigcup\limits_{t\ge0}B_t(\delta)}\\ &=& \prob{\exists t \ge 0:\norm{\epsilon_{1:t}}_{((K_t+ \eta I)^{-1} + I)^{-1}}^2 \le 2\ln\Big(\sqrt{\det((1+\eta)I+K_t)}/\delta\Big)}, \end{eqnarray*} which is the probability required to be bounded by $\delta$ in the statement of the theorem.
Let $\tau'$ be the first time when the bad event $B_t(\delta)$ happens, i.e., $\tau'(\omega) := \min\lbrace t \ge 0:\omega \in B_t(\delta)\rbrace$, with $\min\lbrace\emptyset\rbrace := \infty$ by convention. Clearly, $\tau'$ is a stopping time, and \begin{equation*} \bigcup\limits_{t\ge0}B_t(\delta) = \lbrace \omega \in \Omega : \tau'(\omega)< \infty\rbrace. \end{equation*} Therefore, we can write \begin{eqnarray*} &&\prob{\bigcup\limits_{t\ge0}B_t(\delta)}\\ &=& \prob{\tau' < \infty}\\ &=& \prob{\norm{\epsilon_{1:\tau'}}_{((K_{\tau'}+ \eta I)^{-1} + I)^{-1}}^2 > 2\ln\Big(\sqrt{\det((1+\eta)I+K_{\tau'})}/\delta\Big),\tau' < \infty}\\ &\le & \prob{\norm{\epsilon_{1:\tau'}}_{((K_{\tau'}+ \eta I)^{-1} + I)^{-1}}^2 > 2\ln\Big(\sqrt{\det((1+\eta)I+K_{\tau'})}/\delta\Big)} \le \delta, \end{eqnarray*} by the inequality (\ref{eqn:stopping-time-bound}).
When $K_t$ is positive definite (and hence invertible) for each $t \ge 1$, one can use a similar construction as in Part 1, with $\eta = 0$ (i.e., $N$ is the all-zeros sequence with probability 1), to recover the corresponding conclusion (\ref{eqn:thmpart1}) with $\eta = 0$. \qed
\section*{Proof of Lemma \ref{lem:selfnorm}} Define, for each time $t$, the $t \times \infty$ matrix $\Phi_t := [\phi(x_1) \cdots \phi(x_t)]^T$, and observe that $V_t = I + \Phi_t^T \Phi_t$ and $K_t = \Phi_t \Phi_t^T$. With this, we can compute \begin{align*}
\norm{S_t}^2_{V_t^{-1}} &= S_t^T V_t^{-1} S_t = \sum_{s=1}^{t}\epsilon_s\phi(x_s)^T \left(I + \Phi_t^T \Phi_t \right)^{-1} \sum_{s=1}^{t}\epsilon_s\phi(x_s) \\
&= \epsilon_{1:t}^T \Phi_t \left(I + \Phi_t^T \Phi_t \right)^{-1} \Phi_t^T \epsilon_{1:t} \\
&= \epsilon_{1:t}^T \Phi_t \Phi_t^T \left( \Phi_t \Phi_t^T + I \right)^{-1} \epsilon_{1:t} \\
&= \epsilon_{1:t}^T K_t \left( K_t + I \right)^{-1} \epsilon_{1:t} \\
&= \epsilon_{1:t}^T \left( K_t^{-1} + I \right)^{-1} \epsilon_{1:t} = \norm{\epsilon_{1:t}}^2_{\left( K_t^{-1} + I \right)^{-1}}, \end{align*} completing the proof.
\section*{B. Information Theoretic Results} \begin{mylemma} For every $t \ge 0$, the maximum information gain $\gamma_t$, for the points chosen by Algorithm \ref{algo:ucb} and \ref{algo:ts} satisfy, almost surely, the following : \begin{eqnarray*} \gamma_t &\ge& \frac{1}{2}\ln(\det(I+\lambda^{-1}K_t)),\\ \gamma_t &\ge& \frac{1}{2}\sum_{s=1}^{t}\ln(1 +\lambda^{-1}\sigma_{s-1}^2(x_s)). \end{eqnarray*} \label{lem:info-theoretic-results} \end{mylemma}
\begin{proof} At round $t$ after observing the reward vector $y_{1:t}$ at points $A_t = \lbrace x_1,\ldots,x_t\rbrace \subset D$, the information gain - by the algorithm - about the unknown reward function $f$ is given by the mutual information between $f_{1:t}$ and $y_{1:t}$ sampled at points $A_t$: \begin{equation*} I(y_{1:t};f_{1:t}) =H(y_{1:t})-H(y_{1:t}\; \big\vert \; f_{1:t}),
\end{equation*} where $y_{1:t}= f_{1:t}+\epsilon_{1:t} = [y_1,\ldots,y_t]^T$, $f_{1:t} = [f(x_1),\ldots,f(x_t)]^T$ and $\epsilon_{1:t}=[\epsilon_1,\ldots,\epsilon_t]^T$. Clearly, given $f_{1:t}$ the randomness - as perceived by the algorithm - in $y_{1:t}$ are only in the noise vector $\epsilon_{1:t}$ and thus \begin{equation*} H(y_{1:t}\; \big\vert \; f_{1:t}) = \frac{1}{2}\ln(\det(2\pi e \lambda v^2I))\\ = \frac{t}{2}\log(2\pi e \lambda v^2), \end{equation*} as $\epsilon_{1:t}$ is assumed to follow the distribution $\cN(0,\lambda v^2I)$ and $H(\cN(\mu,\Sigma)) = \frac{1}{2}\ln (\det(2\pi e \Sigma))$. Now $y_{1:t}$ sampled at points $A_t$ is believed to be distributed as $\cN(0,v^2(K_t+\lambda I))$, which gives $H(y_{1:t}) =\frac{1}{2}\ln(\det(2\pi e v^2(\lambda I+K_t)))= \frac{t}{2}\log(2\pi e \lambda v^2) + \frac{1}{2}\ln(\det(I+\lambda^{-1}K_t))$, and therefore \begin{equation} I(y_{1:t};f_{1:t}) = \frac{1}{2}\ln(\det(I+\lambda^{-1}K_t)). \label{eqn:information-gain-batch} \end{equation}
Again, conditioned on reward vector $y_{1:s-1}$ observed at points $A_{s-1}$, the reward $y_s$ at round $s$ observed at $x_s$ is believed to follow the distribution $\cN(\mu_{s-1}(x_s),v^2(\lambda +\sigma_{s-1}^2(x_s)))$, which gives $H(y_s\; \big\vert \; y_{1:s-1})=\frac{1}{2}\ln(2\pi ev^2(\lambda +\sigma_{s-1}^2(x_s))) = \frac{1}{2}\ln(2\pi e \lambda v^2)+\frac{1}{2}\ln(1 +\lambda^{-1}\sigma_{s-1}^2(x_s))$. Now by chain rule $H(y_{1:t}) = \sum_{s=1}^{t}H(y_s\; \big\vert \; y_{1:s-1})= \frac{t}{2}\ln(2\pi e \lambda v^2) + \frac{1}{2}\sum_{s=1}^{t}\ln(1 +\lambda^{-1}\sigma_{s-1}^2(x_s))$, and therefore
\begin{equation} I(y_{1:t};f_{1:t}) = \frac{1}{2}\sum_{s=1}^{t}\ln(1 +\lambda^{-1}\sigma_{s-1}^2(x_s)). \label{eqn:information-gain-online} \end{equation} Now $I(y_{1:t};f_{1:t})$ is a function of $A_t \subset D$, the random points chosen by the algorithm and thus \begin{equation*} I(y_{1:t};f_{1:t}) \le \max\limits_{A \subset D : \abs{A}=t} I(y_A;f_A) = \gamma_t, \;\; \text{a.s.}, \end{equation*} Now the proof follows from Equation \ref{eqn:information-gain-batch} and \ref{eqn:information-gain-online}. \end{proof} \begin{mylemma} \label{lem:bound-sum-sd} Let $x_1, \ldots x_t$ be the points selected by the algorithms. The sum of predictive standard deviation at those points can be expressed in terms of the maximum information gain. More precisely, \begin{equation*} \sum_{t=1}^{T}\sigma_{t-1}(x_t)\le \sqrt{4(T+2)\gamma_T}. \end{equation*} \end{mylemma} \begin{proof} First note that, by Cauchy-Schwartz inequality, $\sum_{t=1}^{T}\sigma_{t-1}(x_t)\le \sqrt{T\sum_{t=1}^{t}\sigma_{t-1}^2(x_t)}$. Now since $0 \le \sigma_{t-1}^2(x)\le 1$ for all $x \in D$ and by our choice of $\lambda = 1+\eta, \eta \ge 0$, we have $\lambda^{-1}\sigma_{t-1}^2(x_t) \le 2\ln(1+\lambda^{-1}\sigma_{t-1}^2(x_t))$, where in the last inequality we used the fact that for any $0 \le \alpha \le 1$, $\ln(1+\alpha) \ge \alpha/2$. Thus we get $\sigma_{t-1}^2(x_t) \le 2\lambda\ln(1+\lambda^{-1}\sigma_{t-1}^2(x_t))$. This implies \begin{equation*} \sum_{t=1}^{T}\sigma_{t-1}(x_t)\le \sqrt{2T\sum_{t=1}^{T}\lambda \ln(1+\lambda^{-1}\sigma_{t-1}^2(x_t))}\le \sqrt{4T\lambda \sum_{t=1}^{T}\frac{1}{2}\ln(1+\sigma_{t-1}^2(x_t))} \le \sqrt{4T(1+\eta)\gamma_T}, \end{equation*} where the last inequality follows from Lemma \ref{lem:info-theoretic-results}. Now the result follows by choosing $\eta = 2/T$. \end{proof} \section*{C. Proof of Theorem \ref{thm:true-function-bound}}
First define $\phi(x)$ as $k(x,\cdot)$, where $\phi : \Real^d\ra H$ maps any point $x$ in the primal space $\Real^d$ to the RKHS $H$ associated with kernel function $k$. For any two functions $g,h \in H$, define the inner product $\inner{g}{h}_k$ as $g^Th$ and the RKHS norm $\norm{g}_k$ as $\sqrt{g^Tg}$. Now as the unknown reward function $f$ lies in the RKHS $H_k(D)$, these definitions along with reproducing property of the RKHS imply $f(x)=\inner{f}{k(x,\cdot)}_k=\inner{f}{\phi(x)}_k=f^T\phi(x)$ and $k(x,x')=\inner{k(x,\cdot)}{k(x',\cdot)}_k= \inner{\phi(x)}{\phi(x')}_k=\phi(x)^T\phi(x')$ for all $x,x'\in D$. Now defining $\Phi_t :=\big[\phi(x_1)^T,\hdots,\phi(x_t)^T\big]^T$, we get the kernel matrix $K_t = \Phi_t\Phi_t^T$, $k_t(x)= \Phi_t\phi(x)$ for all $x\in D$ and $f_{1:t}=\Phi_tf$. Since the matrices $(\Phi_t^T\Phi_t + I)$ and $(\Phi_t\Phi_t^T + \lambda I)$ are strictly positive definite and $(\Phi_t^T\Phi_t + \lambda I)\Phi_t^T = \Phi_t^T(\Phi_t\Phi_t^T + \lambda I)$, we have \begin{equation} \Phi_t^T(\Phi_t\Phi_t^T + \lambda I)^{-1} = (\Phi_t^T\Phi_t + \lambda I)^{-1}\Phi_t^T. \label{eqn:dim-change} \end{equation} Also from the definitions above $(\Phi_t^T\Phi_t+\lambda I)\phi(x)=\Phi_t^Tk_t(x) + \lambda \phi(x)$, and thus from \ref{eqn:dim-change} we deduce that \begin{equation*} \phi(x)=\Phi_t^T(\Phi_t\Phi_t^T+\lambda I)^{-1}k_t(x)+\lambda (\Phi_t^T\Phi_t + \lambda I)^{-1}\phi(x), \end{equation*} which gives \begin{equation*} \phi(x)^T\phi(x)= k_t(x)^T(\Phi_t\Phi_t^T+\lambda I)^{-1}k_t(x)+\lambda \phi(x)^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\phi(x). \end{equation*} This implies \begin{equation} \lambda \phi(x)^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\phi(x) = k(x,x) -k_t(x)^T(K_t+\lambda I)^{-1}k_t(x) = \sigma_t^2(x) \label{eqn:variance} \end{equation} Now observe that \begin{eqnarray*} \abs{f(x)-k_t(x)^T(K_t+\lambda I)^{-1}f_{1:t}} &=& \abs{\phi(x)^T f- \phi(x)^T\Phi_t^T(\Phi_t\Phi_t^T+\lambda I)^{-1}\Phi_t f}\\ &=& \abs{\phi(x)^T f-\phi(x)^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\Phi_t^T\Phi_t f}\\ &=& \abs{\lambda \phi(x)^T(\Phi_t^T\Phi_t+\lambda I)^{-1}f}\\ &\le& \norm{\lambda(\Phi_t^T\Phi_t+\lambda I)^{-1}\phi(x)}_k\norm{f}_k\\ &=& \norm{f}_k \sqrt{\lambda \phi(x)^T(\Phi_t^T\Phi_t+\lambda I)^{-1}\lambda I(\Phi_t^T\Phi_t+ \lambda I)^{-1}\phi(x)}\\ &\le & B \sqrt{\lambda \phi(x)^T(\Phi_t^T\Phi_t+\lambda I)^{-1}(\Phi_t^T\Phi_t+\lambda I)(\Phi_t^T\Phi_t+\lambda I)^{-1}\phi(x)}\\ &=&B\; \sigma_t(x), \end{eqnarray*} where the second equality uses \ref{eqn:dim-change}, the first inequality is by Cauchy-Schwartz and the final equality is from \ref{eqn:variance}. Again see that \begin{eqnarray*} \abs{k_t(x)^T(K_t+\lambda I)^{-1}\eps_{1:t}} &=& \abs{\phi(x)^T\Phi_t^T(\Phi_t\Phi_t^T + \lambda I)^{-1}\eps_{1:t}}\\ &=& \abs{\phi(x)^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\Phi_t^T\eps_{1:t}} \\ &\le &\norm{(\Phi_t^T\Phi_t + \lambda I)^{-1/2}\phi(x)}_k \norm{(\Phi_t^T\Phi_t + \lambda I)^{-1/2}\Phi_t^T\eps_{1:t}}_k\\ &= &\sqrt{\phi(x)^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\phi(x)}\sqrt{(\Phi_t^T\eps_{1:t})^T(\Phi_t^T\Phi_t + \lambda I)^{-1}\Phi_t^T\eps_{1:t}}\\ &=&\lambda^{-1/2}\sigma_t(x)\sqrt{\eps_{1:t}^T\Phi_t\Phi_t^T(\Phi_t\Phi_t^T +\lambda I)^{-1}\epsilon_{1:t}}\\ &=&\lambda^{-1/2}\sigma_t(x)\sqrt{\eps_{1:t}^T K_t(K_t+\lambda I)^{-1}\eps_{1:t}} \end{eqnarray*} where the second equality is from \ref{eqn:dim-change}, the first inequality is by Cauchy-Schwartz and the fourth inequality uses both \ref{eqn:dim-change} and \ref{eqn:variance}.
Now recall that, at round $t$, the posterior mean function $\mu_t(x) = k_t(x)^T(K_t + \lambda I)^{-1}y_{1:t} = k_t(x)^T(K_t + \lambda I)^{-1}(f_{1:t}+\epsilon_{1:t})$, where $f_{1:t}=\big[f(x_1),\ldots,f(x_t)\big]^T$ and $\epsilon_{1:t} = \big[\eps_1,\ldots,\eps_t\big]^T$. Thus we have \begin{eqnarray*} \abs{\mu_t(x)-f(x)} &\le & \abs{k_t(x)^T(K_t+\lambda I)^{-1}\eps_{1:t}} + \abs{f(x)-k_t(x)^T(K_t+\lambda I)^{-1}f_{1:t}}\\&\le& \sigma_t(x)\Big(B + (1+\eta)^{-1/2}\sqrt{\eps_{1:t}^T K_t(K_t+(1+\eta) I)^{-1}\eps_{1:t}}\Big), \end{eqnarray*} where we have used $\lambda = 1+\eta$, where $\eta \ge 0$ as stated in Theorem \ref{thm:self-normalized-bound}. Now observe that when $K$ is invertible, $K(K+I)^{-1}=((K+I)K^{-1})^{-1}=(I+K^{-1})^{-1}$. Using $K=K_t+\eta I$, we get \begin{equation*} (K_t+\eta I)(K_t+(1+\eta)I)^{-1}=((K_t+\eta I)^{-1}+I)^{-1}. \end{equation*} Now see that \begin{equation*} \epsilon_{1:t}^TK_t(K_t+(1+\eta)I)^{-1}\epsilon_{1:t} \le \epsilon_{1:t}^T(K_t+\eta I)(K_t+(1+\eta)I)^{-1}\epsilon_{1:t}=\epsilon_{1:t}^T((K_t+\eta I)^{-1}+I)^{-1}\epsilon_{1:t} \end{equation*} Now using Theorem \ref{thm:self-normalized-bound}, for any $\delta \in (0,1)$, with probability at least $1-\delta$, $\forall t \ge 0, \forall x \in D$, we obtain \begin{equation*} \abs{\mu_t(x)-f(x)}\le \sigma_t(x)\Big(B+\norm{\epsilon_{1:t}}_{(K_t+\eta I)^{-1}+I)^{-1}}\Big)\\
\le \sigma_t(x)\Big(B + R\sqrt{2\ln\frac{\sqrt{\det((1+\eta)I+K_t)}}{\delta}}\Big).\\ \end{equation*} Now observe that $\det((1+\eta)I+K_t)=\det(I+(1+\eta)^{-1}K_t)\det((1+\eta)I)$. Thus we have \begin{equation*} \ln(\det((1+\eta)I+K_t))=\ln(\det(I+(1+\eta)^{-1}K_t))+t\ln(1+\eta) \le 2\gamma_t + \eta t, \end{equation*} from lemma \ref{lem:info-theoretic-results}. Now choosing $\eta = 2/T$ we have $\abs{\mu_t(x)-f(x)}\le \sigma_t(x)\Big(B + R\sqrt{2\big(\gamma_t+ 1+ \ln(1/\delta)\big)}\Big)$ and hence the result follows. \qed \section*{D. Analysis of IGP-UCB (Theorem \ref{thm:regret-bound-UCB})} Observe that at each round $t \ge 1$, by the choice of $x_t$ in Algorithm \ref{algo:ucb}, we have $\mu_{t-1}(x_t)+\beta_t\sigma_{t-1}(x_t) \ge \mu_{t-1}(x^\star)+\beta_t\sigma_{t-1}(x^\star)$ and from Lemma \ref{thm:true-function-bound}, we have $f(x^\star) \le \mu_{t-1}(x^\star)+\beta_t\sigma_{t-1}(x^\star)$ and $\mu_{t-1}(x_t)-f(x_t) \le \beta_t\sigma_{t-1}(x_t)$. Therefore for all $t \ge 1$ with probability at least $1-\delta$, \begin{eqnarray*} r_t &=& f(x^\star) - f(x_t)\\ &\le & \beta_t\sigma_{t-1}(x_t)+\mu_{t-1}(x_t)-f(x_t)\\ &\le & 2\beta_t\sigma_{t-1}(x_t), \end{eqnarray*} and hence $\sum\limits_{t=1}^{T}r_t \le 2\beta_T\sum\limits_{t=1}^{T}\sigma_{t-1}(x_t)$. Now from Lemma \ref{lem:bound-sum-sd}, $\sum\limits_{t=1}^{T}\sigma_{t-1}(x_t) = O(\sqrt{T\gamma_T}) $ and by definition $\beta_T \le B + R\sqrt{2(\gamma_T+1+\ln(1/\delta))}$. Hence with probability at least $1-\delta$, \begin{equation*} R_T = \sum\limits_{t=1}^{T}r_t = O\Big(B\sqrt{T\gamma_T}+\sqrt{T\gamma_T(\gamma_T+\ln(1/\delta))}\Big), \end{equation*} and thus with high probability, \begin{equation*} R_T = O\Big(\sqrt{T}(B\sqrt{\gamma_T}+\gamma_T)\Big). \end{equation*} \section*{E. Analysis of GP-TS (Theorem \ref{thm:regret-bound-TS})} \begin{mylemma} For any $\delta \in (0,1)$ and any finite subset $D'$ of $D$, \begin{equation*} \mathbb{P}\Big[\forall x \in D',\abs{f_t(x)-\mu_{t-1}(x)}\le v_t\sqrt{2\ln(\abs{D'}t^2)}\;\sigma_{t-1}(x) \; \big\vert \; \cH_{t-1}\Big]\ge 1- 1/t^2, \end{equation*} for all possible realizations of history $\cH_{t-1}$. \label{lem:gaussian-concentration} \end{mylemma} \begin{proof} Fix $x \in D$ and $t \ge 1$. Given history $\cH_{t-1}$, $f_t(x)\sim \cN( \mu_{t-1}(x),v_t^2\sigma_{t-1}^2(x))$. Thus using Lemma $B4$ of \cite{hoffman2013exploiting}, for any $\delta \in (0,1)$, with probability at least $1-\delta$ \begin{equation*} \abs{f_t(x)-\mu_{t-1}(x)} \le \sqrt{2\ln(1/\delta)}\;v_t\sigma_{t-1}(x), \end{equation*} and now applying union bound, \begin{equation*} \abs{f_t(x)-\mu_{t-1}(x)} \le v_t\sqrt{2\ln(\abs{D'}/\delta)}\;\sigma_{t-1}(x) \;\; \forall x \in D' \end{equation*} holds with probability at least $1-\delta$, given any possible realizations of history $\cH_{t-1}$. Now setting $\delta = 1/t^2$, the result follows. \end{proof} \begin{mydefinition} Define For all $t \ge 1$, $\tilde{c}_t= \sqrt{4\ln t + 2d\ln(BLrdt^2)}$ and $c_t=v_t(1+\tilde{c}_t)$, where $v_t = B + R\sqrt{2(\gamma_{t-1}+ 1+\ln(2/\delta))}$. Clearly, $c_t$ increases with $t$. \label{def:constants} \end{mydefinition} \begin{mydefinition} Define $E^f(t)$ as the event that for all $x\in D$, \begin{equation*} \abs{\mu_{t-1}(x)-f(x)} \le v_t\sigma_{t-1}(x), \end{equation*} and $E^{f_t}(t)$ as the event that for all $x\in D_t$, \begin{equation*} \abs{f_t(x)-\mu_{t-1}(x)} \le v_t \tilde{c}_t\sigma_{t-1}(x). \end{equation*} \label{def:two-events} \end{mydefinition} \begin{mydefinition} Define the set of saturated points $S_t$ in discretization $D_t$ at round $t$ as \begin{equation*} S_t := \lbrace x\in D_t: \Delta_t(x) > c_t\sigma_{t-1}(x)\rbrace, \end{equation*} where $\Delta_t(x) := f([x^\star]_t)-f(x)$, the difference between function values at the closest point to $x^\star$ in $D_t$ and at $x$. Clearly $\Delta_t([x^\star]_t) = 0$ for all $t$, and hence $[x^\star]_t\in D_t$ is unsaturated at every $t$. \label{def:saturated-arms} \end{mydefinition} \begin{mydefinition} Define filtration $\cF^{'}_{t-1}$ as the history until time $t$, i.e., $\cF^{'}_{t-1}=\cH_{t-1}$. By definition, $\cF^{'}_1\subseteq \cF{'}_2 \subseteq \cdots$. Observe that given $\cF^{'}_{t-1}$, the set $S_t$ and the event $E^f(t)$ are completely deterministic. \label{def:filtration} \end{mydefinition} \begin{mylemma} Given any $\delta \in (0,1)$, $\prob{\forall t \ge 1, E^f(t)}\ge 1-\delta/2 $ and for all possible filtrations $\cF^{'}_{t-1}$, $\prob{E^{f_t}(t)\; \big\vert \; \cF^{'}_{t-1}} \ge 1-1/t^2$. \label{lem:event-concentration} \end{mylemma} \begin{proof} The probability bound for the event $E^f(t)$ follows from Theorem \ref{thm:true-function-bound} by replacing $\delta$ with $\frac{\delta}{2}$ and for the event $E^{f_t}(t)$ follows from Lemma \ref{lem:gaussian-concentration} by setting $D'=D_t$ and $\cH_{t-1}=\cF^{'}_{t-1}$. \end{proof} \begin{mylemma}[Gaussian Anti-concentration] For a Gaussian random variable $X$ with mean $\mu$ and standard deviation $\sigma$, for any $\beta > 0$, \begin{equation*} \prob{\frac{X-\mu}{\sigma} > \beta} \ge \frac{e^{-\beta^2}}{4\sqrt{\pi}\beta}. \end{equation*} \label{anti-concentration} \end{mylemma} \begin{mylemma} For any filtration $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, \begin{equation*} \prob{f_t(x)>f(x)\; \big\vert \; \cF^{'}_{t-1}} \ge p, \end{equation*} for any $x \in D$, where $p = \frac{1}{4e\sqrt{\pi}}$. \label{lem:compare-sample-with-original} \end{mylemma} \begin{proof} Fix any $x\in D$. Given filtration $\cF^{'}_{t-1}$, $f_t(x)$ is a Gaussian random variable with mean $\mu_{t-1}(x)$ and standard deviation $v_t\sigma_{t-1}(x)$ and since event $E^f(t)$ is true, $\abs{\mu_{t-1}(x)-f(x)} \le c_{1,t}\sigma_{t-1}(x)$. Now using the anti-concentration inequality in Lemma \ref{anti-concentration}, we have \begin{eqnarray*} \prob{f_t(x)>f(x)\; \big\vert \; \cF^{'}_{t-1}} &=& \prob{\frac{f_t(x)-\mu_{t-1}(x)}{v_t \sigma_{t-1}(x)}>\frac{f(x)-\mu_{t-1}(x)}{v_t \sigma_{t-1}(x)}\; \big\vert \; \cF^{'}_{t-1}}\\ &\ge&\prob{\frac{f_t(x)-\mu_{t-1}(x)}{v_t \sigma_{t-1}(x)}>\frac{\abs{f(x)-\mu_{t-1}(x)}}{v_t \sigma_{t-1}(x)}\; \big\vert \; \cF^{'}_{t-1}}\\ &\ge&\frac{1}{4\sqrt{\pi}\beta_t}e^{-\theta_t^2}, \end{eqnarray*} where, from Definition \ref{def:two-events}, $\theta_t = \frac{\abs{f(x)-\mu_{t-1}(x)}}{v_t \sigma_{t-1}(x)} \le 1$. Therefore $\prob{f_t(x)>f(x)\; \big\vert \; \cF^{'}_{t-1}} \ge \frac{1}{4e\sqrt{\pi}}$, and hence the result follows. \end{proof} \begin{mylemma} For any filtration $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, \begin{equation*} \prob{x_t\in D_t\setminus S_t\; \big\vert \; \cF^{'}_{t-1}} \ge p-1/t^2. \end{equation*} \label{lem:prob-playing-saturated-arms} \end{mylemma} \begin{proof} At round $t$ our algorithm chooses the point $x_t\in D_t$, at which the highest value of $f_t$, within current decision set $D_t$, is attained. Now if $f_t([x^\star]_t)$ is greater than $f_t(x)$ for all saturated points at round $t$, i.e.,$f_t([x^\star]_t) > f_t(x), \forall x \in S_t$, then one of the unsaturated points (which includes $[x^\star]_t$) in $D_t$ must be played and hence $x_t \in D_t\setminus S_t$. This implies \begin{equation} \prob{x_t\in D_t\setminus S_t\; \big\vert \; \cF^{'}_{t-1}} \ge \prob{f_t([x^\star]_t) > f_t(x), \forall x \in S_t\; \big\vert \; \cF^{'}_{t-1}}. \label{eqn:prob-saturated-arms} \end{equation} Now form Definition \ref{def:saturated-arms}, $\Delta_t(x) > c_t\sigma_{t-1}(x)$, for all $x\in S_t$. Also if both the events $E^f(t)$ and $E^{f_t}(t)$ are true, then from Definition \ref{def:constants} and \ref{def:two-events}, $f_t(x) \le f(x) + c_t\sigma_{t-1}(x)$, for all $x \in D_t$. Thus for all $x\in S_t$, $f_t(x) < f(x) + \Delta_t(x)$. Therefore, for any filtration $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, either $E^{f_t}(t)$ is false, or else for all $x \in S_t$, $f_t(x) < f([x^\star]_t)$. Hence, for any $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, \begin{equation*} \prob{f_t([x^\star]_t) > f_t(x), \forall x \in S_t\; \big\vert \; \cF^{'}_{t-1}}\\ \ge \prob{f_t([x^\star]_t) > f([x^\star]_t)\; \big\vert \; \cF_{t-1}} - \prob{\ol{E^{f_t}(t)}\; \big\vert \; \cF^{'}_{t-1}} \\ \ge p - 1/t^2, \end{equation*} where we have used Lemma \ref{lem:event-concentration} and Lemma \ref{lem:compare-sample-with-original}. Now the proof follows from Equation \ref{eqn:prob-saturated-arms}. \end{proof} \begin{mylemma} For any filtration $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, \begin{equation*} \expect{r_t\; \big\vert \; \cF^{'}_{t-1}} \le \frac{11c_t}{p}\expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} + \frac{2B+1}{t^2}, \end{equation*} where $r_t$ is the instantaneous regret at round $t$. \label{lem:bound-on-instantaneous-regret} \end{mylemma} \begin{proof} Let $\bar{x}_t$ be the unsaturated point in $D_t$ with smallest $\sigma_{t-1}(x)$, i.e., \begin{equation} \bar{x}_t = \mathop{\mathrm{argmin}}\limits_{x \in D_t\setminus S_t}\sigma_{t-1}(x). \label{eqn:smallest-sd} \end{equation} Since $\sigma_{t-1}(\cdot)$ and $S_t$ are deterministic given $\cF^{'}_{t-1}$, so is $\bar{x}_t$. Now for any $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, \begin{eqnarray} \expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} &\ge& \expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1},x_t \in D_t\setminus S_t}\prob{x_t\in D_t\setminus S_t\; \big\vert \; \cF^{'}_{t-1}}\nonumber\\ &\ge& \sigma_{t-1}(\bar{x}_t)(p-1/t^2), \label{eqn:expectation-smallest-sd} \end{eqnarray} where we have used Equation \ref{eqn:smallest-sd} and Lemma \ref{lem:prob-playing-saturated-arms}. Now, if both the events $E^f(t)$ and $E^{f_t}(t)$ are true, then from Definition \ref{def:constants} and \ref{def:two-events}, $f(x) - c_t\sigma_{t-1}(x) \le f_t(x) \le f(x) + c_t\sigma_{t-1}(x)$, for all $x \in D_t$. Using this observation along with Definition \ref{def:saturated-arms} and the facts that $f_t(x_t) \ge f_t(x)$ for all $x \in D_t$ and $\bar{x}_t \in D_t\setminus S_t$, we have \begin{eqnarray*} \Delta_t(x_t) &=& f([x^\star]_t)-f(\bar{x}_t)+f(\bar{x}_t)-f(x_t)\\ &\le& \Delta_t(\bar{x}_t)+ f_t(\bar{x}_t)+c_t\sigma_{t-1}(\bar{x}_t)-f_t(x_t)+c_t\sigma_{t-1}(x_t)\\ &\le& c_t\sigma_{t-1}(\bar{x}_t) + c_t\sigma_{t-1}(\bar{x}_t)+c_t\sigma_{t-1}(x_t)\\ &\le& c_t\big(2\sigma_{t-1}(\bar{x}_t)+c_t\sigma_{t-1}(x_t)\big). \end{eqnarray*} Therefore, for any $\cF^{'}_{t-1}$ such that $E^f(t)$ is true, either $\Delta_t(x_t) \le c_t\big(2\sigma_{t-1}(\bar{x}_t)+c_t\sigma_{t-1}(x_t)\big)$, or $E^{f_t}(t)$ is false. Now from our assumption of bounded variance, for all $x \in D$, $\abs{f(x)} \le \norm{f}_k k(x,x) \le B$, and hence $\Delta_t(x) \le 2\sup\limits_{x\in D} \abs{f(x)} \le 2B$. Thus, using Equation \ref{eqn:expectation-smallest-sd}, we get \begin{eqnarray} \expect{\Delta_t(x_t)\; \big\vert \; \cF^{'}_{t-1}} &\le& \expect{c_t\big(2\sigma_{t-1}(\bar{x}_t)+c_t\sigma_{t-1}(x_t)\big)\; \big\vert \; \cF^{'}_{t-1}}+ 2B\prob{\ol{E^{f_t}(t)}\; \big\vert \; \cF^{'}_{t-1}}\nonumber\\ &\le& \frac{2c_t}{p-1/t^2} \expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} + c_t \expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} + \frac{2B}{t^2}\nonumber\\ &\le& \frac{11c_t}{p}\expect{\sigma_{t-1}(x_t)\; \big\vert \; \cF^{'}_{t-1}} +\frac{2B}{t^2}, \label{eqn:expectation-arm-played} \end{eqnarray} where in the last inequality we used that $1/(p-1/t^2) \le 5/p$, which holds trivially for $t \le 4$ and also holds for $t\ge 5$, as $t^2 > 5e\sqrt{\pi}$. Now using Equation \ref{eqn:lipschitz}, we have the instantaneous regret at round $t$, \begin{equation*} r_t = f(x^\star)-f([x^\star]_t)+f([x^\star]_t)-f(x_t) \le \frac{1}{t^2} + \Delta_t(x_t), \end{equation*} and then taking conditional expectation on both sides, the result follows from Equation \ref{eqn:expectation-arm-played}. \end{proof} \begin{mydefinition} Let us define $Y_0 = 0$, and for all $t=1,\ldots,T$: \begin{eqnarray*} \bar{r}_t &=& r_t \cdot \indic{E^f(t)},\\ X_t &=& \bar{r}_t - \frac{11c_t}{p}\sigma_{t-1}(x_t)-\frac{2B+1}{t^2},\\ Y_t &=& \sum_{s=1}^{t}X_s. \end{eqnarray*} \label{def:regret} \end{mydefinition} \begin{mydefinition} A sequence of random variables $(Z_t;t\ge 0)$ is called a super-martingale corresponding to a filtration $\cF_t$, if for all $t$, $Z_t$ is $\cF_t$-measurable, and for $t\ge 1$, \begin{equation*} \expect{Z_t\; \big\vert \; \cF_{t-1}} \le Z_{t-1}. \end{equation*} \label{def:supermartingale} \end{mydefinition} \begin{mylemma}[Azuma-Hoeffding Inequality] If a super-martingale $(Z_t;t \ge 0)$, corresponding to filtration $\cF_t$, satisfies $\abs{Z_t - Z_{t-1}} \le \alpha_t$ for some constant $\alpha_t$, for all $t = 1,\ldots,T$, then for any $\delta \ge 0$, \begin{equation*} \prob{Z_T - Z_0 \le \sqrt{2\ln(1/\delta)\sum_{t=1}^{T}\alpha_t^2}\;} \ge 1-\delta. \end{equation*} \label{lem:azuma-hoeffding} \end{mylemma} \begin{mylemma} $(Y_t;t = 0,...,T)$ is a super-martingale process with respect to filtration $\cF^{'}_t$. \label{lem:supermartingale} \end{mylemma} \begin{proof} From Definition \ref{def:supermartingale}, we need to prove that for all $t \in \lbrace 1,\ldots,T\rbrace$ and any possible $\cF^{'}_{t-1}$, $\expect{Y_t - Y_{t-1}\; \big\vert \; \cF^{'}_{t-1}} \le 0$, i.e. \begin{equation}
\expect{\bar{r}_t|\cF^{'}_{t-1}} \le \frac{11c_t}{p}\expect{\sigma_{t-1}(x_t)|\cF^{'}_{t-1}} + \frac{2B+1}{t^2}. \label{eqn:supermartingale} \end{equation} Now if $\cF^{'}_{t-1}$ such that $E^f(t)$ is false, then $\bar{r}_t = r_t \cdot \indic{E^f(t)} = 0$, and Equation \ref{eqn:supermartingale} holds trivially. Moreover, for $\cF^{'}_{t-1}$ such that both $E^f(t)$ is true, Equation \ref{eqn:supermartingale} follows from Lemma \ref{lem:bound-on-instantaneous-regret}. \end{proof} \begin{mylemma} Given any $\delta \in (0,1)$, with probability at least $1-\delta$, \begin{equation*} R_T=\sum_{t=1}^{T}r(t)=\frac{11c_T}{p}\sum_{t=1}^{T}\sigma_{t-1}(x_t) +\frac{(2B+1)\pi^2}{6}+\frac{(4B+11)c_T}{p}\sqrt{2T\ln(2/\delta)}, \end{equation*} where $T$ is the total number of rounds played. \label{lem:total-regret} \end{mylemma} \begin{proof} First note that from Definition \ref{def:regret} for all $t=1,\ldots,T$, \begin{equation*} \abs{Y_t - Y_{t-1}} = \abs{X_t} \le\abs{\bar{r}_t} + \frac{11c_t}{p}\sigma_{t-1}(x_t) + \frac{2B+1}{t^2}. \end{equation*} Now as $\bar{r}_t \le r_t \le 2\sup\limits_{x\in D}\abs{f(x)} \le 2B$ and $\sigma^2_{t-1}(x_t) \le \sigma^2_0(x_t) \le 1$, we have \begin{equation*} \abs{Y_t - Y_{t-1}} \le 2B + \frac{11c_t}{p} + \frac{2B+1}{t^2} \le \frac{(4B+11)c_t}{p}, \end{equation*} which follows from the fact that $2B \le 2Bc_t/p$ and also $(2B+1)/t^2 \le 2Bc_t/p$. Thus, we can apply Azuma-Hoeffding inequality (Lemma \ref{lem:azuma-hoeffding}) to obtain that with probability at least $1-\delta/2$, \begin{eqnarray*} \sum_{t=1}^{T}\bar{r}_t &\le & \sum_{t=1}^{T}\frac{11c_t}{p}\sigma_{t-1}(x_t) + \sum_{t=1}^{T}\frac{2B+1}{t^2}+ \sqrt{2\ln(2/\delta)\sum_{t=1}^{T}\frac{(4B+11)^2c_t^2}{p^2}}\\ &\le & \frac{11c_T}{p}\sum_{t=1}^{T}\sigma_{t-1}(x_t) + \frac{(2B+1)\pi^2}{6} + \frac{(4B+11)c_T}{p}\sqrt{2T\ln(2/\delta)}, \end{eqnarray*} as by definition $c_t \le c_T$ for all $t \in \lbrace 1,\ldots,T\rbrace$. Now, as the event $E^f(t)$ holds holds for all $t$ with probability at least $1-\delta/2$ (see Lemma \ref{lem:event-concentration}), then from Definition \ref{def:regret}, $\bar{r}_t=r_t$ for all $t$ with probability at least $1-\delta/2$. Now by applying union bound, the result follows. \end{proof} \subsection*{Proof of Theorem \ref{thm:regret-bound-TS}} From Lemma \ref{lem:bound-sum-sd} we have, $\sum\limits_{t=1}^{T}\sigma_{t-1}(x_t) = O(\sqrt{T\gamma_T})$. Also from Definition \ref{def:constants}, \begin{eqnarray*} C_T &\le& B + R\sqrt{2(\gamma_T+ 1+\ln(2/\delta))} + \Big(B + R\sqrt{2(\gamma_T+ 1+ \ln(2/\delta))}\Big)\sqrt{4\ln T + 2d\ln(BLrdT^2)}\\
&=& O\Big(\sqrt{(\gamma_T+\ln(2/\delta))(\ln T+d\ln(BdT))}+B\sqrt{d\ln(BdT)}\Big)\\ &=& O\Big(\sqrt{(\gamma_T+\ln(2/\delta))d\ln(BdT)}\Big). \end{eqnarray*} Hence, from Lemma \ref{lem:total-regret}, with probability at least $1-\delta$, \begin{eqnarray*} R_T=O\Bigg(\sqrt{(\gamma_T+\ln(2/\delta))d\ln (BdT)}\cdot \Big(\sqrt{T\gamma_T}+B\sqrt{T\ln(2/\delta)}\Big)\Bigg) \end{eqnarray*} and thus with high probability, \begin{eqnarray*} R_T&=& O\Big(\sqrt{T\gamma_T^2 d\ln (BdT)} + B\sqrt{T\gamma_Td\ln (BdT)}\Big)\\ &=& O\Bigg(\sqrt{Td\ln (BdT)}\Big(B\sqrt{\gamma_T}+\gamma_T\Big) \Bigg). \hspace*{75pt} \end{eqnarray*} \section*{F. Recursive Updates of Posterior Mean and Covariance}
We now describe a procedure to update the posterior mean and covariance function in a recursive fashion through the properties of Schur complement (\citet{zhang2006schur}) rather than evaluating Equation \ref{eqn:mean update} and \ref{eqn:cov update} at each round. Specifically for all $t \ge 1$ we show the following: \begin{eqnarray} \mu_t(x) &=& \mu_{t-1}(x) + \dfrac{k_{t-1}(x,x_t)}{\lambda+\sigma^2_{t-1}(x_t)}(y_t - \mu_{t-1}(x_t)),\label{eqn:mean-online}\\ k_t(x,x') &=& k_{t-1}(x,x') - \dfrac{k_{t-1}(x,x_t)k_{t-1}(x_t,x')}{\lambda+ \sigma^2_{t-1}(x_t)},\label{eqn:cov-online}\\ \sigma^2_t(x) &=& \sigma^2_{t-1}(x) - \dfrac{k^2_{t-1}(x,x_t)}{\lambda+ \sigma^2_{t-1}(x_t)}.\label{eqn:sd-online} \end{eqnarray} These update rules make our algorithms easy to implement and we are not aware of any literature which explicitly states or uses these relations.
First we write the matrix $K_t + \lambda I$ as $\begin{bmatrix} A & B\\ C & D \end{bmatrix}$, where $A = K_{t-1} + \lambda I$, $B = k_{t-1}(x_t)$, $C = B^T$ and $D = \lambda+k(x_t,x_t)$. Now using Schur's complement we get \begin{eqnarray*} \begin{bmatrix} A & B\\ C & D \end{bmatrix}^{-1} &=& \begin{bmatrix} A^{-1}+A^{-1}B\beta CA^{-1} & -A^{-1}B\beta\\ -\beta CA^{-1} & \beta \end{bmatrix}\\ &=& \begin{bmatrix} A^{-1}+\beta A^{-1}BB^TA^{-1} & -\beta A^{-1}B\\ -\beta B^TA^{-1} & \beta \end{bmatrix}\\ &=& \begin{bmatrix} A^{-1}+\beta \gamma & -\beta \alpha\\ -\beta \alpha^T & \beta \end{bmatrix}, \end{eqnarray*} where $\beta = (D - CA^{-1}B)^{-1} = 1/(D - B^TA^{-1}B)$, $\gamma = A^{-1}BB^TA^{-1}$ and $\alpha = A^{-1}B$. Therefore we have \begin{eqnarray*} \mu_t(x) &=& k_t(x)^T(K_t + \lambda I)^{-1}y_{1:t}\\ &=& \begin{bmatrix} k_{t-1}(x)^T & k(x_t,x) \end{bmatrix}\begin{bmatrix} A^{-1}+\beta \gamma & -\beta \alpha\\ -\beta \alpha^T & \beta \end{bmatrix}\begin{bmatrix} y_{1:{t-1}}\\ y_t \end{bmatrix}\\ &=& k_{t-1}(x)^T(A^{-1}+\beta \gamma)y_{1:{t-1}} - \beta k(x_t,x)\alpha^Ty_{1:{t-1}} - \beta y_t \alpha^Tk_{t-1}(x) +\beta y_t k(x_t,x)\\ &=& k_{t-1}(x)^T A^{-1}y_{1:{t-1}} + \beta \Big(k_{t-1}(x)^T\gamma y_{1:{t-1}}-k(x_t,x)\alpha^Ty_{1:{t-1}} - y_t \alpha^Tk_{t-1}(x) + y_t k(x_t,x)\Big), \end{eqnarray*} where \begin{eqnarray*} k_{t-1}(x)^T A^{-1}y_{1:{t-1}} &=& k_{t-1}(x)^T (K_{t-1}+\lambda I)^{-1}y_{1:{t-1}} = \mu_{t-1}(x),\\ k_{t-1}(x)^T\gamma y_{1:{t-1}} &=& k_{t-1}(x)^T A^{-1}k_{t-1}(x_t)k_{t-1}(x_t)^TA^{-1}y_{1:{t-1}}= \left(k_{t-1}(x_t)^T A^{-1}k_{t-1}(x)\right)\mu_{t-1}(x_t),\\ \alpha^Ty_{1:{t-1}} &=& k_{t-1}(x_t)^TA^{-1}y_{1:{t-1}} = \mu_{t-1}(x_t),\\ \alpha^Tk_{t-1}(x) &=& k_{t-1}(x_t)^T A^{-1}k_{t-1}(x). \end{eqnarray*} Thus we have \begin{eqnarray*} \mu_t(x) &=& \mu_{t-1}(x) + \beta \Big( k_{t-1}(x_t)^T A^{-1}k_{t-1}(x)\left(\mu_{t-1}(x_t)-y_t \right)+ k(x_t,x)\left(y_t - \mu_{t-1}(x_t)\right)\Big)\\ &=& \mu_{t-1}(x) + \beta\left(y_t - \mu_{t-1}(x_t)\right)\left(k(x_t,x) - k_{t-1}(x_t)^T A^{-1}k_{t-1}(x)\right)\\ &=& \mu_{t-1}(x) + \beta k_{t-1}(x_t,x)(y_t - \mu_{t-1}(x_t)). \end{eqnarray*} Now as \begin{equation*} D - B^TA^{-1}B = \lambda+k(x_t,x_t) - k_{t-1}(x_t)^T(K_{t-1}+\lambda I)^{-1}k_{t-1}(x_t) = \lambda +\sigma_{t-1}^2(x_t), \end{equation*} putting $\beta = 1/(\lambda+\sigma^2_{t-1}(x_t))$, we obtain Equation \ref{eqn:mean-online}. Again observe that \begin{eqnarray*} && k_t(x,x')\\&=& k(x,x') - k_t(x)^T(K_t + \lambda I)^{-1}k_t(x')\\ &=& k(x,x') - k_{t-1}(x)^TA^{-1}k_{t-1}(x')\\ &&+ \beta \Big(k_{t-1}(x)^T\gamma k_{t-1}(x')- k(x_t,x)\alpha^Tk_{t-1}(x')-k(x_t,x')\alpha^Tk_{t-1}(x)+k(x_t,x)k(x_t,x')\Big). \end{eqnarray*} Now we have \begin{equation*} k(x,x') - k_{t-1}(x)^TA^{-1}k_{t-1}(x')= k(x,x') - k_{t-1}(x)^T(K_{t-1} + \lambda I)^{-1}k_{t-1}(x') = k_{t-1}(x,x'), \end{equation*} also \begin{eqnarray*} && k_{t-1}(x)^T\gamma k_{t-1}(x') - k(x_t,x)\alpha^Tk_{t-1}(x') \\&=& k_{t-1}(x)^TA^{-1}k_{t-1}(x_t)k_{t-1}(x_t)^TA^{-1}k _{t-1}(x') - k(x_t,x)k_{t-1}(x_t)^TA^{-1}k_{t-1}(x')\\ &=& (k_{t-1}(x)^TA^{-1}k_{t-1}(x_t)-k(x_t,x))k_{t-1}(x_t)^TA^{-1}k_{t-1}(x')\\ &=& -k_{t-1}(x_t,x)k_{t-1}(x_t)^TA^{-1}k_{t-1}(x'), \end{eqnarray*} and \begin{eqnarray*} k(x_t,x)k(x_t,x') - k(x_t,x')\alpha^Tk_{t-1}(x) &=& k(x_t,x')(k(x_t,x)-k_{t-1}(x_t)^TA^{-1}k_{t-1}(x))\\ &=& k(x_t,x')k_{t-1}(x_t,x). \end{eqnarray*} Putting all these together we get \begin{eqnarray*} k_t(x,x') &=& k_{t-1}(x,x') - \beta \Big(k(x_t,x')k_{t-1}(x_t,x)- k_{t-1}(x_t,x)k_{t-1}(x_t)^TA^{-1}k_{t-1}(x')\Big)\\ &=&k_{t-1}(x,x') - \beta \Big(k_{t-1}(x_t,x)\Big(k(x_t,x')-k_{t-1}(x_t)^TA^{-1}k_{t-1}(x')\Big)\Big)\\ &=&k_{t-1}(x,x') - \beta k_{t-1}(x_t,x)k_{t-1}(x_t,x'). \end{eqnarray*} Now Equation \ref{eqn:cov-online} and \ref{eqn:sd-online} follows by using $\beta = 1/ (\lambda+\sigma^2_{t-1}(x))$ and $\sigma_t^2(x) = k_t(x,x)$. \qed
\end{document}
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arXiv
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Subhash Khot
Subhash Khot FRS (born 10 June 1978 in Ichalkaranji) is an Indian-American mathematician and theoretical computer scientist who is the Julius Silver Professor of Computer Science in the Courant Institute of Mathematical Sciences at New York University. Khot has contributed to the field of computational complexity, and is best known for his unique games conjecture.[1]
Subhash Khot
FRS
Born (1978-06-10) 10 June 1978
Ichalkaranji, Maharashtra, India
Alma materPrinceton University, IIT Bombay
Known forUnique games conjecture
AwardsWaterman Award (2010)
Rolf Nevanlinna Prize (2014)
MacArthur Fellow (2016)
Fellow of the Royal Society (2017)
Scientific career
FieldsComputer Science
InstitutionsGeorgia Tech
Courant Institute of Mathematical Sciences
University of Chicago
Doctoral advisorSanjeev Arora
Khot received the 2014 Rolf Nevanlinna Prize by the International Mathematical Union.Khot stood First in the highly difficult IIT Entrance Exam. He received the MacArthur Fellowship in 2016 [2] and was elected a Fellow of the Royal Society in 2017.[3]
Education
Khot obtained his bachelor's degree in computer science from the Indian Institute of Technology Bombay in 1999. He received his doctorate degree in computer science from Princeton University in 2003 under the supervision of Sanjeev Arora. His doctoral dissertation was titled "New Techniques for Probabilistically Checkable Proofs and Inapproximability Results."[4]
Honours and awards
Khot is a two time silver medallist representing India at the International Mathematical Olympiad (1994 and 1995).[5][6]
He has been awarded the Microsoft Research New Faculty Fellowship Award (2005),[7] the Alan T. Waterman Award (2010), the Rolf Nevanlinna Prize for his work on the Unique Games Conjecture (2014), and the MacArthur Fellowship (2016).[8]
He was elected a Fellow of the Royal Society in 2017.[9]
References
1. Khot, Subhash (2002), "On the power of unique 2-prover 1-round games", Proceedings of the 17th Annual IEEE Conference on Computational Complexity, p. 25, CiteSeerX 10.1.1.133.5651, doi:10.1109/CCC.2002.1004334, ISBN 978-0-7695-1468-0, S2CID 32966635.
2. "Subhash Khot - MacArthur Foundation".
3. "Subhash Khot". Royal Society. Archived from the original on 23 May 2017. Retrieved 27 May 2017.
4. "ACM Doctoral Dissertation Award 2003". Archived from the original on 3 November 2014. Retrieved 13 September 2014.
5. Subhash Khot's results at International Mathematical Olympiad
6. Shirali, S.A. (2006), "The Sierpinski problem", Resonance, 11 (2): 78–87, doi:10.1007/BF02837277, S2CID 121269449
7. Microsoft Faculty Fellowship Recipients 2005
8. "MacArthur Fellows Program". Archived from the original on 2 April 2012.
9. "Subhash Khot". Royal Society. Archived from the original on 23 May 2017. Retrieved 27 May 2017.
Nevanlinna Prize winners
• Tarjan (1982)
• Valiant (1986)
• Razborov (1990)
• Wigderson (1994)
• Shor (1998)
• Sudan (2002)
• Kleinberg (2006)
• Spielman (2010)
• Khot (2014)
• Daskalakis (2018)
Fellows of the Royal Society elected in 2017
Fellows
• Yves-Alain Barde
• Tony Bell
• Christopher Bishop
• Neil Burgess
• Keith Beven
• Wendy Bickmore
• Krishna Chatterjee
• James Durrant
• Warren East
• Tim Elliott
• Anne Ferguson-Smith
• Jonathan Gregory
• Mark Gross
• Roy Harrison
• Gabriele Hegerl
• Eddie Holmes
• Richard Houlston
• Yvonne Jones
• Subhash Khot
• Julia King
• Stafford Lightman
• Yadvinder Malhi
• Andrew McKenzie
• Gerard Milburn
• Anne Neville
• Alison Noble
• Andrew Orr-Ewing
• David Owen
• Lawrence Paulson
• Josephine Pemberton
• Sandu Popescu
• Sarah Price
• Anne Ridley
• David Rubinsztein
• Gavin Salam
• Nigel Shadbolt
• Angus Silver
• Gordon Slade
• Pete Smith
• Nicola Spaldin
• Jonathan Stoye
• John Sutherland
• J. Roy Taylor
• Jenny Thomas
• Patrick Vallance
• Susanne von Caemmerer
• Hugh Watkins
• Roger Williams
• Ken Wolfe
• Andy Woods
Honorary
• David Neuberger
Foreign
• Max Cooper
• Whitfield Diffie
• Robert Grubbs
• Hideo Hosono
• Marcia McNutt
• Ginés Morata
• Robert Ritchie
• Thomas Südhof
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• Susan Wessler
Authority control
International
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National
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Academics
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Wikipedia
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Theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by Rosenhain (1851)
History and genus 1
The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
Higher genus
For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as
22g
in number if the base field is algebraically closed.
This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of
2D = 0.
In other words, with K the canonical class and Θ any given solution of
2Θ = K,
any other solution will be of form
Θ + D.
This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g. Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety. The answer is the same, provided the characteristic of the field is not 2.
A theta characteristic Θ will be called even or odd depending on the dimension of its space of global sections $H^{0}(C,\Theta )$. It turns out that on C there are $2^{g-1}(2^{g}+1)$ even and $2^{g-1}(2^{g}-1)$ odd theta characteristics.
Classical theory
Classically the theta characteristics were divided into these two kinds, odd and even, according to the value of the Arf invariant of a certain quadratic form Q with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents, as it corresponds to the 28 bitangents of a quartic. The geometric construction of Q as an intersection form is with modern tools possible algebraically. In fact the Weil pairing applies, in its abelian variety form. Triples (θ1, θ2, θ3) of theta characteristics are called syzygetic and asyzygetic depending on whether Arf(θ1)+Arf(θ2)+Arf(θ3)+Arf(θ1+θ2+θ3) is 0 or 1.
Spin structures
Atiyah (1971) showed that, for a compact complex manifold, choices of theta characteristics correspond bijectively to spin structures.
References
• Atiyah, Michael Francis (1971), "Riemann surfaces and spin structures", Annales Scientifiques de l'École Normale Supérieure, Série 4, 4: 47–62, ISSN 0012-9593, MR 0286136
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Neural effects of controllability as a key dimension of stress exposure
Emily M. Cohodes, Paola Odriozola, Jeffrey D. Mandell, Camila Caballero, Sarah McCauley, Sadie J. Zacharek, H. R. Hodges, Jason T. Haberman, Mackenzye Smith, Janeen Thomas, Olivia C. Meisner, Cameron T. Ellis, Catherine A. Hartley, Dylan G. Gee
Journal: Development and Psychopathology , First View
Published online by Cambridge University Press: 17 January 2022, pp. 1-10
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Cross-species evidence suggests that the ability to exert control over a stressor is a key dimension of stress exposure that may sensitize frontostriatal-amygdala circuitry to promote more adaptive responses to subsequent stressors. The present study examined neural correlates of stressor controllability in young adults. Participants (N = 56; M age = 23.74, range = 18–30 years) completed either the controllable or uncontrollable stress condition of the first of two novel stressor controllability tasks during functional magnetic resonance imaging (fMRI) acquisition. Participants in the uncontrollable stress condition were yoked to age- and sex-matched participants in the controllable stress condition. All participants were subsequently exposed to uncontrollable stress in the second task, which is the focus of fMRI analyses reported here. A whole-brain searchlight classification analysis revealed that patterns of activity in the right dorsal anterior insula (dAI) during subsequent exposure to uncontrollable stress could be used to classify participants' initial exposure to either controllable or uncontrollable stress with a peak of 73% accuracy. Previous experience of exerting control over a stressor may change the computations performed within the right dAI during subsequent stress exposure, shedding further light on the neural underpinnings of stressor controllability.
The influence of fluid–structure interaction on cloud cavitation about a rigid and a flexible hydrofoil. Part 3
Yin Lu Young, Jasmine C. Chang, Samuel M. Smith, James A. Venning, Bryce W. Pearce, Paul A. Brandner
Journal: Journal of Fluid Mechanics / Volume 934 / 10 March 2022
Published online by Cambridge University Press: 11 January 2022, A2
Print publication: 10 March 2022
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Experimental studies of the influence of fluid–structure interaction on cloud cavitation about a stiff stainless steel (SS) and a flexible composite (CF) hydrofoil have been presented in Parts I (Smith et al., J. Fluid Mech., vol. 896, 2020a, p. A1) and II (Smith et al., J. Fluid Mech., vol. 897, 2020b, p. A28). This work further analyses the data and complements the measurements with reduced-order model predictions to explain the complex response. A two degrees-of-freedom steady-state model is used to explain why the tip bending and twisting deformations are much higher for the CF hydrofoil, while the hydrodynamic load coefficients are very similar. A one degree-of-freedom dynamic model, which considers the spanwise bending deflection only, is used to capture the dynamic response of both hydrofoils. Peaks in the frequency response spectrum are observed at the re-entrant jet-driven and shock-wave-driven cavity shedding frequencies, system bending frequency and heterodyne frequencies caused by the mixing of the two cavity shedding frequencies. The predictions capture the increase of the mean system bending frequency and wider bandwidth of frequency modulation with decreasing cavitation number. The results show that, in general, the amplitude of the deformation fluctuation is higher, but the amplitude of the load fluctuation is lower for the CF hydrofoil compared with the SS hydrofoil. Significant dynamic load amplification is observed at subharmonic lock-in when the shock-wave-driven cavity shedding frequency matches with the nearest subharmonic of the system bending frequency of the CF hydrofoil. Both measurements and predictions show an absence of dynamic load amplification at primary lock-in because of the low intensity of cavity load fluctuations with high cavitation number.
Transmission of severe acute respiratory coronavirus virus 2 (SARS-CoV-2), delta variant, between two fully vaccinated healthcare personnel
L. Leigh Smith, Aaron M. Milstone, Morgan Jibowu, Chun Huai Luo, C. Paul Morris, Heba H. Mostafa, Lisa L. Maragakis
Journal: Infection Control & Hospital Epidemiology , First View
Published online by Cambridge University Press: 08 November 2021, pp. 1-3
P.101 Response to high dose nocturnal diazepam in children with ESES
H Kiani, C Go, KC Jones, MB Connolly, M Smith, R RamachandranNair
Journal: Canadian Journal of Neurological Sciences / Volume 48 / Issue s3 / November 2021
Published online by Cambridge University Press: 05 January 2022, p. S48
Print publication: November 2021
Background: To assess the response to high dose daily nocturnal diazepam (HDD) in children with encephalopathy associated with electrical status epilepticus in sleep (ESES). Methods: A prospective cohort of patients (4-12 years), newly diagnosed with ESES, initiated on the first course HDD, was followed for ≤ 1-year. Sleep EEG scores (SES) pre and post HDD were evaluated. An EEG grading system based on both sleep spike wave index (sSWI) (Grade: 1-4) and distribution of epileptiform discharges (Grade: 0-4) was used and summed to yield an aggregate SES (ASES) (Grade: 1-8). Results: Eighteen eligible children (M:F 12:6; median age, 7.6 years) were initiated on first course HDD (median, 0.5 mg/kg/d). sSWI decreased from 85.7% (mean, SD 13.9) to 32.6% (mean, SD 37.1) at subsequent EEG (95% CI = -70.60- -35.62; p < 0.001). ASES decreased from 6.5 (SD 1.3) to 3.1 (SD 1.9) (95% CI = -4.17- -2.60; p < 0.001). EEG relapse after a period of improvement occurred in 10 children. Minimal response to HDD occurred in 2 children. Five patients manifested mild side effects; behavior (2), hyperactivity (2), and lethargy (1). Conclusions: HDD safely and significantly reduces both SWI and aggregate sleep EEG score in children with ESES.
P.117 Pediatric acute ischemic stroke protocols
M Gladkikh, H McMillan, A Andrade, C Boelman, I Bhatal, J Mailo, A Mineyko, M Moharir, S Perreault, J Smith, D Pohl
Background: Approximately 1,000 children present with AIS annually in North America. Most suffer from long-term disability. Childhood AIS is diagnosed after a median of 23 hours post-symptom onset, limiting thrombolytic treatment options that may improve outcomes. Pediatric stroke protocols decrease time to diagnosis. AIS treatment is not uniform across Canada, nor are pediatric stroke protocols standardized. Methods: We contacted neurologists at all 16 Canadian pediatric hospitals regarding their AIS management. Results: Response rate was 100%. Seven centers have an AIS protocol and two have a protocol under development. Seven centers do not have a protocol – two redirect patients to adult neurology, and five use a case-by-case approach for management. Analysis of the seven AIS protocols reveals differences: 1) IV-tPA dosage: age-dependent 0.75-0.9 mg/kg (n=1) versus age-independent 0.9 mg/kg (n=6), with maximum doses 75 mg (n=1) or 90 mg (n=6); 2) IV-tPA lower age cut-off: 2 years (n=4) versus 3, 4 or 10 years (n=1); 3) IV-tPA exclusion criteria: PedNIHSS score <4 (n=3), <5 (n=1), or <6 (n=3); 4) Pre-treatment neuroimaging: CT (n=3) versus MRI (n=4); 5) Intra-arterial tPA use (n=3). Conclusions: The seven Canadian pediatric AIS protocols show prominent differences. We plan a teleconference discussing a Canadian pediatric AIS consensus approach.
C.5 Musashi-1 is a master regulator of aberrant translation in MYC-amplified Group 3 medulloblastoma
MM Kameda-Smith, H Zhu, E Luo, C Venugopal, K Brown, BA Yee, S Xing, F Tan, D Bakhshinyan, AA Adile, M Subapanditha, D Picard, J Moffat, A Fleming, K Hope, J Provias, M Remke, Y Lu, J Reimand, R Wechsler-Reya, G Yeo, SK Singh
Background: Medulloblastoma (MB) is the most common solid malignant pediatric brain neoplasm. Group 3 (G3) MB, particularly MYC amplified G3 MB, is the most aggressive subgroup with the highest frequency of children presenting with metastatic disease, and is associated with a poor prognosis. To further our understanding of the role of MSI1 in MYC amplified G3 MB, we performed an unbiased integrative analysis of eCLIP binding sites, with changes observed at the transcriptome, the translatome, and the proteome after shMSI1 inhibition. Methods: Primary human pediatric MBs, SU_MB002 and HD-MB03 were kind gifts from Dr. Yoon-Jae Cho (Harvard, MS) and Dr. Till Milde (Heidelberg) and cultured for in vitro and in vivo experiments. eCLIP, RNA-seq, Polysome-seq, and TMT-MS were completed as previously described. Results: MSI1 is overexpressed in G3 MB. shRNA Msi1 interference resulted in a reduction in tumour burden conferring a survival advantage to mice injected with shMSI1 G3MB cells. Robust ranked multiomic analysis (RRA) identified an unconventional gene set directly perturbed by MSI1 in G3 MB. Conclusions: Our robust unbiased integrative analysis revealed a distinct role for MSI1 in the maintenance of the stem cell state in G3 MB through post-transcriptional modification of multiple pathways including identification of unconventional targets such as HIPK1.
Livestock-associated methicillin-resistant Staphylococcus aureus in slaughtered pigs in England
R. P. Smith, M. Sharma, D. Gilson, M. Anjum, C. J. Teale
Journal: Epidemiology & Infection / Volume 149 / 2021
Published online by Cambridge University Press: 13 October 2021, e236
This study was performed to investigate the occurrence of livestock-associated methicillin-resistant Staphylococcus aureus (LA-MRSA) in batches of pigs at slaughter and at different stages along the slaughter line. Nasal and ear skin swabs were collected from 105 batches of 10 pigs at six abattoirs. Cultures (pooled or individual) were performed for MRSA using selective media; presumptive MRSA were confirmed by mecA and nuc gene detection and a selection was spa-typed. MRSA was detected in 46 batches. All spa-types detected were those associated with LA-MRSA clonal complex 398. The proportion of positive batches varied among abattoirs (0–100%). Two abattoirs were subsequently further investigated, with samples taken at post-stunning, chiller and either at lairage or post-singe. Results suggested cross-contamination occurred between the lairage and point of post-stunning, but the slaughter processes appeared effective at reducing contamination before carcases entered the chiller. One abattoir provided only negative samples in the initial study and in the subsequent study along the slaughter line (26 batches in total), suggesting differences possibly in the MRSA status of pigs on arrival from supply farms or in its abattoir practices affecting the MRSA status of pigs at the sampling points. This study highlights that in the investigated abattoirs, MRSA was detected in 43.8% of batches of pigs at slaughter using sensitive selective culture methods.
The influence of background turbulence on Ahmed-body wake bistability
D. Burton, S. Wang, D. Tudball Smith, H. N. Scott, T. N. Crouch, M. C. Thompson
Journal: Journal of Fluid Mechanics / Volume 926 / 10 November 2021
Published online by Cambridge University Press: 06 September 2021, R1
Print publication: 10 November 2021
The discovery of wake bistability has generated an upsurge in experimental investigations into the wakes of simplified vehicle geometries. Particular focus has centred on the probabilistic switching between two asymmetrical bistable wake states of a square-back Ahmed body; however, the majority of this research has been undertaken in wind tunnels with turbulence intensities of less than $1\,\%$, considerably lower than typical atmospheric levels. To better simulate bistability under on-road conditions, in which turbulence intensities can easily reach levels of $10\,\%$ or more, this experimental study investigates the effects of free-stream turbulence on the bistability characteristics of the square-back Ahmed body. Through passive generation and quantification of the free-stream turbulent conditions, a monotonic correlation was found between the switching rate and free-stream turbulence intensity.
The GLEAM 200-MHz local radio luminosity function for AGN and star-forming galaxies
T. M. O. Franzen, N. Seymour, E. M. Sadler, T. Mauch, S. V. White, C. A. Jackson, R. Chhetri, B. Quici, M. E. Bell, J. R. Callingham, K. S. Dwarakanath, B. For, B. M. Gaensler, P. J. Hancock, L. Hindson, N. Hurley-Walker, M. Johnston-Hollitt, A. D. Kapińska, E. Lenc, B. McKinley, J. Morgan, A. R. Offringa, P. Procopio, L. Staveley-Smith, R. B. Wayth, C. Wu, Q. Zheng
Journal: Publications of the Astronomical Society of Australia / Volume 38 / 2021
Published online by Cambridge University Press: 06 September 2021, e041
The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76–227 MHz of the entire southern sky (Declination $<\!{+}30^{\circ}$ ) with an angular resolution of ${\approx}2$ arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1 590 local (median $z \approx 0.064$ ) radio sources with $S_{200\,\mathrm{MHz}} > 55$ mJy across an area of ${\approx}16\,700\,\mathrm{deg}^{2}$ . From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73% are fuelled by an active galactic nucleus (AGN) and 27% by star formation. We present the local radio luminosity function for AGN and star-forming (SF) galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and ${\sim}1$ GHz. For the AGN, the median spectral index between 200 MHz and ${\sim}1$ GHz, $\alpha_{\mathrm{high}}$ , is $-0.600 \pm 0.010$ (where $S \propto \nu^{\alpha}$ ) and the median spectral index within the GLEAM band, $\alpha_{\mathrm{low}}$ , is $-0.704 \pm 0.011$ . For the SF galaxies, the median value of $\alpha_{\mathrm{high}}$ is $-0.650 \pm 0.010$ and the median value of $\alpha_{\mathrm{low}}$ is $-0.596 \pm 0.015$ . Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4% of local radio AGN have ultra-steep radio spectra at low frequencies ( $\alpha_{\mathrm{low}} < -1.2$ ). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.
Impact of weekly asymptomatic testing for severe acute respiratory coronavirus virus 2 (SARS-CoV-2) in inpatients at an academic hospital
Leigh Smith, Sara Pau, Susan Fallon, Sara E. Cosgrove, Melanie S. Curless, Valeria Fabre, Sara M. Karaba, Lisa L. Maragakis, Aaron M. Milstone, Anna C. Sick-Samuels, Polly Trexler, Clare Rock, for the CDC Prevention Epicenter Program
Published online by Cambridge University Press: 27 August 2021, pp. 1-3
We analyzed the impact of a 7-day recurring asymptomatic SARS-CoV-2 testing protocol for all patients hospitalized at a large academic center. Overall, 40 new cases were identified, and 1 of 3 occurred after 14 days of hospitalization. Recurring testing can identify unrecognized infections, especially during periods of elevated community transmission.
Characterisation of age and polarity at onset in bipolar disorder
Janos L. Kalman, Loes M. Olde Loohuis, Annabel Vreeker, Andrew McQuillin, Eli A. Stahl, Douglas Ruderfer, Maria Grigoroiu-Serbanescu, Georgia Panagiotaropoulou, Stephan Ripke, Tim B. Bigdeli, Frederike Stein, Tina Meller, Susanne Meinert, Helena Pelin, Fabian Streit, Sergi Papiol, Mark J. Adams, Rolf Adolfsson, Kristina Adorjan, Ingrid Agartz, Sofie R. Aminoff, Heike Anderson-Schmidt, Ole A. Andreassen, Raffaella Ardau, Jean-Michel Aubry, Ceylan Balaban, Nicholas Bass, Bernhard T. Baune, Frank Bellivier, Antoni Benabarre, Susanne Bengesser, Wade H Berrettini, Marco P. Boks, Evelyn J. Bromet, Katharina Brosch, Monika Budde, William Byerley, Pablo Cervantes, Catina Chillotti, Sven Cichon, Scott R. Clark, Ashley L. Comes, Aiden Corvin, William Coryell, Nick Craddock, David W. Craig, Paul E. Croarkin, Cristiana Cruceanu, Piotr M. Czerski, Nina Dalkner, Udo Dannlowski, Franziska Degenhardt, Maria Del Zompo, J. Raymond DePaulo, Srdjan Djurovic, Howard J. Edenberg, Mariam Al Eissa, Torbjørn Elvsåshagen, Bruno Etain, Ayman H. Fanous, Frederike Fellendorf, Alessia Fiorentino, Andreas J. Forstner, Mark A. Frye, Janice M. Fullerton, Katrin Gade, Julie Garnham, Elliot Gershon, Michael Gill, Fernando S. Goes, Katherine Gordon-Smith, Paul Grof, Jose Guzman-Parra, Tim Hahn, Roland Hasler, Maria Heilbronner, Urs Heilbronner, Stephane Jamain, Esther Jimenez, Ian Jones, Lisa Jones, Lina Jonsson, Rene S. Kahn, John R. Kelsoe, James L. Kennedy, Tilo Kircher, George Kirov, Sarah Kittel-Schneider, Farah Klöhn-Saghatolislam, James A. Knowles, Thorsten M. Kranz, Trine Vik Lagerberg, Mikael Landen, William B. Lawson, Marion Leboyer, Qingqin S. Li, Mario Maj, Dolores Malaspina, Mirko Manchia, Fermin Mayoral, Susan L. McElroy, Melvin G. McInnis, Andrew M. McIntosh, Helena Medeiros, Ingrid Melle, Vihra Milanova, Philip B. Mitchell, Palmiero Monteleone, Alessio Maria Monteleone, Markus M. Nöthen, Tomas Novak, John I. Nurnberger, Niamh O'Brien, Kevin S. O'Connell, Claire O'Donovan, Michael C. O'Donovan, Nils Opel, Abigail Ortiz, Michael J. Owen, Erik Pålsson, Carlos Pato, Michele T. Pato, Joanna Pawlak, Julia-Katharina Pfarr, Claudia Pisanu, James B. Potash, Mark H Rapaport, Daniela Reich-Erkelenz, Andreas Reif, Eva Reininghaus, Jonathan Repple, Hélène Richard-Lepouriel, Marcella Rietschel, Kai Ringwald, Gloria Roberts, Guy Rouleau, Sabrina Schaupp, William A Scheftner, Simon Schmitt, Peter R. Schofield, K. Oliver Schubert, Eva C. Schulte, Barbara Schweizer, Fanny Senner, Giovanni Severino, Sally Sharp, Claire Slaney, Olav B. Smeland, Janet L. Sobell, Alessio Squassina, Pavla Stopkova, John Strauss, Alfonso Tortorella, Gustavo Turecki, Joanna Twarowska-Hauser, Marin Veldic, Eduard Vieta, John B. Vincent, Wei Xu, Clement C. Zai, Peter P. Zandi, Psychiatric Genomics Consortium (PGC) Bipolar Disorder Working Group, International Consortium on Lithium Genetics (ConLiGen), Colombia-US Cross Disorder Collaboration in Psychiatric Genetics, Arianna Di Florio, Jordan W. Smoller, Joanna M. Biernacka, Francis J. McMahon, Martin Alda, Bertram Müller-Myhsok, Nikolaos Koutsouleris, Peter Falkai, Nelson B. Freimer, Till F.M. Andlauer, Thomas G. Schulze, Roel A. Ophoff
Journal: The British Journal of Psychiatry / Volume 219 / Issue 6 / December 2021
Published online by Cambridge University Press: 25 August 2021, pp. 659-669
Print publication: December 2021
Studying phenotypic and genetic characteristics of age at onset (AAO) and polarity at onset (PAO) in bipolar disorder can provide new insights into disease pathology and facilitate the development of screening tools.
To examine the genetic architecture of AAO and PAO and their association with bipolar disorder disease characteristics.
Genome-wide association studies (GWASs) and polygenic score (PGS) analyses of AAO (n = 12 977) and PAO (n = 6773) were conducted in patients with bipolar disorder from 34 cohorts and a replication sample (n = 2237). The association of onset with disease characteristics was investigated in two of these cohorts.
Earlier AAO was associated with a higher probability of psychotic symptoms, suicidality, lower educational attainment, not living together and fewer episodes. Depressive onset correlated with suicidality and manic onset correlated with delusions and manic episodes. Systematic differences in AAO between cohorts and continents of origin were observed. This was also reflected in single-nucleotide variant-based heritability estimates, with higher heritabilities for stricter onset definitions. Increased PGS for autism spectrum disorder (β = −0.34 years, s.e. = 0.08), major depression (β = −0.34 years, s.e. = 0.08), schizophrenia (β = −0.39 years, s.e. = 0.08), and educational attainment (β = −0.31 years, s.e. = 0.08) were associated with an earlier AAO. The AAO GWAS identified one significant locus, but this finding did not replicate. Neither GWAS nor PGS analyses yielded significant associations with PAO.
AAO and PAO are associated with indicators of bipolar disorder severity. Individuals with an earlier onset show an increased polygenic liability for a broad spectrum of psychiatric traits. Systematic differences in AAO across cohorts, continents and phenotype definitions introduce significant heterogeneity, affecting analyses.
Implications of new technologies for future food supply systems
S. Asseng, C. A. Palm, J. L. Anderson, L. Fresco, P. A. Sanchez, F. Asche, T. M. Garlock, J. Fanzo, M. D. Smith, G. Knapp, A. Jarvis, A. Adesogan, I. Capua, G. Hoogenboom, D. D. Despommier, L. Conti, K. A. Garrett
Journal: The Journal of Agricultural Science / Volume 159 / Issue 5-6 / July 2021
Published online by Cambridge University Press: 06 December 2021, pp. 315-319
Print publication: July 2021
The combination of advances in knowledge, technology, changes in consumer preference and low cost of manufacturing is accelerating the next technology revolution in crop, livestock and fish production systems. This will have major implications for how, where and by whom food will be produced in the future. This next technology revolution could benefit the producer through substantial improvements in resource use and profitability, but also the environment through reduced externalities. The consumer will ultimately benefit through more nutritious, safe and affordable food diversity, which in turn will also contribute to the acceleration of the next technology. It will create new opportunities in achieving progress towards many of the Sustainable Development Goals, but it will require early recognition of trends and impact, public research and policy guidance to avoid negative trade-offs. Unfortunately, the quantitative predictability of future impacts will remain low and uncertain, while new chocks with unexpected consequences will continue to interrupt current and future outcomes. However, there is a continuing need for improving the predictability of shocks to future food systems especially for ex-ante assessment for policy and planning.
Early Science from POSSUM: Shocks, turbulence, and a massive new reservoir of ionised gas in the Fornax cluster
C. S. Anderson, G. H. Heald, J. A. Eilek, E. Lenc, B. M. Gaensler, Lawrence Rudnick, C. L. Van Eck, S. P. O'Sullivan, J. M. Stil, A. Chippendale, C. J. Riseley, E. Carretti, J. West, J. Farnes, L. Harvey-Smith, N. M. McClure-Griffiths, Douglas C. J. Bock, J. D. Bunton, B. Koribalski, C. D. Tremblay, M. A. Voronkov, K. Warhurst
Published online by Cambridge University Press: 23 April 2021, e020
We present the first Faraday rotation measure (RM) grid study of an individual low-mass cluster—the Fornax cluster—which is presently undergoing a series of mergers. Exploiting commissioning data for the POlarisation Sky Survey of the Universe's Magnetism (POSSUM) covering a ${\sim}34$ square degree sky area using the Australian Square Kilometre Array Pathfinder (ASKAP), we achieve an RM grid density of ${\sim}25$ RMs per square degree from a 280-MHz band centred at 887 MHz, which is similar to expectations for forthcoming GHz-frequency ${\sim}3\pi$-steradian sky surveys. These data allow us to probe the extended magnetoionic structure of the cluster and its surroundings in unprecedented detail. We find that the scatter in the Faraday RM of confirmed background sources is increased by $16.8\pm2.4$ rad m−2 within 1 $^\circ$ (360 kpc) projected distance to the cluster centre, which is 2–4 times larger than the spatial extent of the presently detectable X-ray-emitting intracluster medium (ICM). The mass of the Faraday-active plasma is larger than that of the X-ray-emitting ICM and exists in a density regime that broadly matches expectations for moderately dense components of the Warm-Hot Intergalactic Medium. We argue that forthcoming RM grids from both targeted and survey observations may be a singular probe of cosmic plasma in this regime. The morphology of the global Faraday depth enhancement is not uniform and isotropic but rather exhibits the classic morphology of an astrophysical bow shock on the southwest side of the main Fornax cluster, and an extended, swept-back wake on the northeastern side. Our favoured explanation for these phenomena is an ongoing merger between the main cluster and a subcluster to the southwest. The shock's Mach angle and stand-off distance lead to a self-consistent transonic merger speed with Mach 1.06. The region hosting the Faraday depth enhancement also appears to show a decrement in both total and polarised radio emission compared to the broader field. We evaluate cosmic variance and free-free absorption by a pervasive cold dense gas surrounding NGC 1399 as possible causes but find both explanations unsatisfactory, warranting further observations. Generally, our study illustrates the scientific returns that can be expected from all-sky grids of discrete sources generated by forthcoming all-sky radio surveys.
Australian square kilometre array pathfinder: I. system description
A. W. Hotan, J. D. Bunton, A. P. Chippendale, M. Whiting, J. Tuthill, V. A. Moss, D. McConnell, S. W. Amy, M. T. Huynh, J. R. Allison, C. S. Anderson, K. W. Bannister, E. Bastholm, R. Beresford, D. C.-J. Bock, R. Bolton, J. M. Chapman, K. Chow, J. D. Collier, F. R. Cooray, T. J. Cornwell, P. J. Diamond, P. G. Edwards, I. J. Feain, T. M. O. Franzen, D. George, N. Gupta, G. A. Hampson, L. Harvey-Smith, D. B. Hayman, I. Heywood, C. Jacka, C. A. Jackson, S. Jackson, K. Jeganathan, S. Johnston, M. Kesteven, D. Kleiner, B. S. Koribalski, K. Lee-Waddell, E. Lenc, E. S. Lensson, S. Mackay, E. K. Mahony, N. M. McClure-Griffiths, R. McConigley, P. Mirtschin, A. K. Ng, R. P. Norris, S. E. Pearce, C. Phillips, M. A. Pilawa, W. Raja, J. E. Reynolds, P. Roberts, D. N. Roxby, E. M. Sadler, M. Shields, A. E. T. Schinckel, P. Serra, R. D. Shaw, T. Sweetnam, E. R. Troup, A. Tzioumis, M. A. Voronkov, T. Westmeier
Published online by Cambridge University Press: 05 March 2021, e009
In this paper, we describe the system design and capabilities of the Australian Square Kilometre Array Pathfinder (ASKAP) radio telescope at the conclusion of its construction project and commencement of science operations. ASKAP is one of the first radio telescopes to deploy phased array feed (PAF) technology on a large scale, giving it an instantaneous field of view that covers $31\,\textrm{deg}^{2}$ at $800\,\textrm{MHz}$. As a two-dimensional array of 36 $\times$12 m antennas, with baselines ranging from 22 m to 6 km, ASKAP also has excellent snapshot imaging capability and 10 arcsec resolution. This, combined with 288 MHz of instantaneous bandwidth and a unique third axis of rotation on each antenna, gives ASKAP the capability to create high dynamic range images of large sky areas very quickly. It is an excellent telescope for surveys between 700 and $1800\,\textrm{MHz}$ and is expected to facilitate great advances in our understanding of galaxy formation, cosmology, and radio transients while opening new parameter space for discovery of the unknown.
The association between C-reactive protein, mood disorder, and cognitive function in UK Biobank
David C. Milton, Joey Ward, Emilia Ward, Donald M. Lyall, Rona J. Strawbridge, Daniel J. Smith, Breda Cullen
Journal: European Psychiatry / Volume 64 / Issue 1 / 2021
Published online by Cambridge University Press: 01 February 2021, e14
Systemic inflammation has been linked with mood disorder and cognitive impairment. The extent of this relationship remains uncertain, with the effects of serum inflammatory biomarkers compared to genetic predisposition toward inflammation yet to be clearly established.
We investigated the magnitude of associations between C-reactive protein (CRP) measures, lifetime history of bipolar disorder or major depression, and cognitive function (reaction time and visuospatial memory) in 84,268 UK Biobank participants. CRP was measured in serum and a polygenic risk score for CRP was calculated, based on a published genome-wide association study. Multiple regression models adjusted for sociodemographic and clinical confounders.
Increased serum CRP was significantly associated with mood disorder history (Kruskal–Wallis H = 196.06, p < 0.001, η2 = 0.002) but increased polygenic risk for CRP was not (F = 0.668, p = 0.648, η2 < 0.001). Compared to the lowest quintile, the highest serum CRP quintile was significantly associated with both negative and positive differences in cognitive performance (fully adjusted models: reaction time B = −0.030, 95% CI = −0.052, −0.008; visuospatial memory B = 0.066, 95% CI = 0.042, 0.089). More severe mood disorder categories were significantly associated with worse cognitive performance and this was not moderated by serum or genetic CRP level.
In this large cohort study, we found that measured inflammation was associated with mood disorder history, but genetic predisposition to inflammation was not. The association between mood disorder and worse cognitive performance was very small and did not vary by CRP level. The inconsistent relationship between CRP measures and cognitive performance warrants further study.
The relationship between spatial and in-store food environments and adolescent food purchasing and dietary behaviours: a systematic review
S. Shaw, M. Barrett, C. Shand, S. Crozier, C. Cooper, D. Smith, M. Barker, C. Vogel
Journal: Proceedings of the Nutrition Society / Volume 80 / Issue OCE5 / 2021
Print publication: 2021
A history of high-power laser research and development in the United Kingdom
60th Celebration of First Laser
Colin N. Danson, Malcolm White, John R. M. Barr, Thomas Bett, Peter Blyth, David Bowley, Ceri Brenner, Robert J. Collins, Neal Croxford, A. E. Bucker Dangor, Laurence Devereux, Peter E. Dyer, Anthony Dymoke-Bradshaw, Christopher B. Edwards, Paul Ewart, Allister I. Ferguson, John M. Girkin, Denis R. Hall, David C. Hanna, Wayne Harris, David I. Hillier, Christopher J. Hooker, Simon M. Hooker, Nicholas Hopps, Janet Hull, David Hunt, Dino A. Jaroszynski, Mark Kempenaars, Helmut Kessler, Sir Peter L. Knight, Steve Knight, Adrian Knowles, Ciaran L. S. Lewis, Ken S. Lipton, Abby Littlechild, John Littlechild, Peter Maggs, Graeme P. A. Malcolm, OBE, Stuart P. D. Mangles, William Martin, Paul McKenna, Richard O. Moore, Clive Morrison, Zulfikar Najmudin, David Neely, Geoff H. C. New, Michael J. Norman, Ted Paine, Anthony W. Parker, Rory R. Penman, Geoff J. Pert, Chris Pietraszewski, Andrew Randewich, Nadeem H. Rizvi, Nigel Seddon, MBE, Zheng-Ming Sheng, David Slater, Roland A. Smith, Christopher Spindloe, Roy Taylor, Gary Thomas, John W. G. Tisch, Justin S. Wark, Colin Webb, S. Mark Wiggins, Dave Willford, Trevor Winstone
Journal: High Power Laser Science and Engineering / Volume 9 / 2021
Published online by Cambridge University Press: 27 April 2021, e18
The first demonstration of laser action in ruby was made in 1960 by T. H. Maiman of Hughes Research Laboratories, USA. Many laboratories worldwide began the search for lasers using different materials, operating at different wavelengths. In the UK, academia, industry and the central laboratories took up the challenge from the earliest days to develop these systems for a broad range of applications. This historical review looks at the contribution the UK has made to the advancement of the technology, the development of systems and components and their exploitation over the last 60 years.
Recreating the OSIRIS-REx slingshot manoeuvre from a network of ground-based sensors
Trent Jansen-Sturgeon, Benjamin A. D. Hartig, Gregory J. Madsen, Philip A. Bland, Eleanor K. Sansom, Hadrien A. R. Devillepoix, Robert M. Howie, Martin Cupák, Martin C. Towner, Morgan A. Cox, Nicole D. Nevill, Zacchary N. P. Hoskins, Geoffrey P. Bonning, Josh Calcino, Jake T. Clark, Bryce M. Henson, Andrew Langendam, Samuel J. Matthews, Terence P. McClafferty, Jennifer T. Mitchell, Craig J. O'Neill, Luke T. Smith, Alastair W. Tait
Published online by Cambridge University Press: 27 November 2020, e049
Optical tracking systems typically trade off between astrometric precision and field of view. In this work, we showcase a networked approach to optical tracking using very wide field-of-view imagers that have relatively low astrometric precision on the scheduled OSIRIS-REx slingshot manoeuvre around Earth on 22 Sep 2017. As part of a trajectory designed to get OSIRIS-REx to NEO 101955 Bennu, this flyby event was viewed from 13 remote sensors spread across Australia and New Zealand to promote triangulatable observations. Each observatory in this portable network was constructed to be as lightweight and portable as possible, with hardware based off the successful design of the Desert Fireball Network. Over a 4-h collection window, we gathered 15 439 images of the night sky in the predicted direction of the OSIRIS-REx spacecraft. Using a specially developed streak detection and orbit determination data pipeline, we detected 2 090 line-of-sight observations. Our fitted orbit was determined to be within about 10 km of orbital telemetry along the observed 109 262 km length of OSIRIS-REx trajectory, and thus demonstrating the impressive capability of a networked approach to Space Surveillance and Tracking.
Neutron Star Extreme Matter Observatory: A kilohertz-band gravitational-wave detector in the global network
Gravitational Wave Astronomy
K. Ackley, V. B. Adya, P. Agrawal, P. Altin, G. Ashton, M. Bailes, E. Baltinas, A. Barbuio, D. Beniwal, C. Blair, D. Blair, G. N. Bolingbroke, V. Bossilkov, S. Shachar Boublil, D. D. Brown, B. J. Burridge, J. Calderon Bustillo, J. Cameron, H. Tuong Cao, J. B. Carlin, S. Chang, P. Charlton, C. Chatterjee, D. Chattopadhyay, X. Chen, J. Chi, J. Chow, Q. Chu, A. Ciobanu, T. Clarke, P. Clearwater, J. Cooke, D. Coward, H. Crisp, R. J. Dattatri, A. T. Deller, D. A. Dobie, L. Dunn, P. J. Easter, J. Eichholz, R. Evans, C. Flynn, G. Foran, P. Forsyth, Y. Gai, S. Galaudage, D. K. Galloway, B. Gendre, B. Goncharov, S. Goode, D. Gozzard, B. Grace, A. W. Graham, A. Heger, F. Hernandez Vivanco, R. Hirai, N. A. Holland, Z. J. Holmes, E. Howard, E. Howell, G. Howitt, M. T. Hübner, J. Hurley, C. Ingram, V. Jaberian Hamedan, K. Jenner, L. Ju, D. P. Kapasi, T. Kaur, N. Kijbunchoo, M. Kovalam, R. Kumar Choudhary, P. D. Lasky, M. Y. M. Lau, J. Leung, J. Liu, K. Loh, A. Mailvagan, I. Mandel, J. J. McCann, D. E. McClelland, K. McKenzie, D. McManus, T. McRae, A. Melatos, P. Meyers, H. Middleton, M. T. Miles, M. Millhouse, Y. Lun Mong, B. Mueller, J. Munch, J. Musiov, S. Muusse, R. S. Nathan, Y. Naveh, C. Neijssel, B. Neil, S. W. S. Ng, V. Oloworaran, D. J. Ottaway, M. Page, J. Pan, M. Pathak, E. Payne, J. Powell, J. Pritchard, E. Puckridge, A. Raidani, V. Rallabhandi, D. Reardon, J. A. Riley, L. Roberts, I. M. Romero-Shaw, T. J. Roocke, G. Rowell, N. Sahu, N. Sarin, L. Sarre, H. Sattari, M. Schiworski, S. M. Scott, R. Sengar, D. Shaddock, R. Shannon, J. SHI, P. Sibley, B. J. J. Slagmolen, T. Slaven-Blair, R. J. E. Smith, J. Spollard, L. Steed, L. Strang, H. Sun, A. Sunderland, S. Suvorova, C. Talbot, E. Thrane, D. Töyrä, P. Trahanas, A. Vajpeyi, J. V. van Heijningen, A. F. Vargas, P. J. Veitch, A. Vigna-Gomez, A. Wade, K. Walker, Z. Wang, R. L. Ward, K. Ward, S. Webb, L. Wen, K. Wette, R. Wilcox, J. Winterflood, C. Wolf, B. Wu, M. Jet Yap, Z. You, H. Yu, J. Zhang, J. Zhang, C. Zhao, X. Zhu
Gravitational waves from coalescing neutron stars encode information about nuclear matter at extreme densities, inaccessible by laboratory experiments. The late inspiral is influenced by the presence of tides, which depend on the neutron star equation of state. Neutron star mergers are expected to often produce rapidly rotating remnant neutron stars that emit gravitational waves. These will provide clues to the extremely hot post-merger environment. This signature of nuclear matter in gravitational waves contains most information in the 2–4 kHz frequency band, which is outside of the most sensitive band of current detectors. We present the design concept and science case for a Neutron Star Extreme Matter Observatory (NEMO): a gravitational-wave interferometer optimised to study nuclear physics with merging neutron stars. The concept uses high-circulating laser power, quantum squeezing, and a detector topology specifically designed to achieve the high-frequency sensitivity necessary to probe nuclear matter using gravitational waves. Above 1 kHz, the proposed strain sensitivity is comparable to full third-generation detectors at a fraction of the cost. Such sensitivity changes expected event rates for detection of post-merger remnants from approximately one per few decades with two A+ detectors to a few per year and potentially allow for the first gravitational-wave observations of supernovae, isolated neutron stars, and other exotica.
Exploring trajectories in dietary adequacy of the B vitamins folate, riboflavin, vitamins B6 and B12, with advancing older age: a systematic review
N. Gillies, D. Cameron-Smith, S. Pundir, C. R. Wall, A. M. Milan
Journal: British Journal of Nutrition / Volume 126 / Issue 3 / 14 August 2021
Published online by Cambridge University Press: 29 October 2020, pp. 449-459
Print publication: 14 August 2021
Maintaining nutritional adequacy contributes to successful ageing. B vitamins involved in one-carbon metabolism regulation (folate, riboflavin, vitamins B6 and B12) are critical nutrients contributing to homocysteine and epigenetic regulation. Although cross-sectional B vitamin intake in ageing populations is characterised, longitudinal changes are infrequently reported. This systematic review explores age-related changes in dietary adequacy of folate, riboflavin, vitamins B6 and B12 in community-dwelling older adults (≥65 years at follow-up). Following Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines, databases (MEDLINE, Embase, BIOSIS, CINAHL) were systematically screened, yielding 1579 records; eight studies were included (n 3119 participants, 2–25 years of follow-up). Quality assessment (modified Newcastle–Ottawa quality scale) rated all of moderate–high quality. The estimated average requirement cut-point method estimated the baseline and follow-up population prevalence of dietary inadequacy. Riboflavin (seven studies, n 1953) inadequacy progressively increased with age; the prevalence of inadequacy increased from baseline by up to 22·6 and 9·3 % in males and females, respectively. Dietary folate adequacy (three studies, n 2321) improved in two studies (by up to 22·4 %), but the third showed increasing (8·1 %) inadequacy. Evidence was similarly limited (two studies, respectively) and inconsistent for vitamins B6 (n 559; −9·9 to 47·9 %) and B12 (n 1410; −4·6 to 7·2 %). This review emphasises the scarcity of evidence regarding micronutrient intake changes with age, highlighting the demand for improved reporting of longitudinal changes in nutrient intake that can better direct micronutrient recommendations for older adults. This review was registered with PROSPERO (CRD42018104364).
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\begin{document}
\begin{flushleft}
{\Large\bf Spectral Theorem for definitizable normal linear operators on Krein spaces}
\\[5mm]
\textsc{Michael Kaltenb\"ack}
\\[6mm]
{\small
\textbf{Abstract:In the present note a spectral theorem for normal
definitizable linear operators on Krein spaces is derived by developing
a functional calculus $\phi \mapsto \phi(N)$ which is the
proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert space situation.}
} \end{flushleft}
\begin{flushleft}
{\small
{\bf Mathematics Subject Classification (2010):} 47A60, 47B50, 47B15.
} \end{flushleft} \begin{flushleft}
{\small
{\bf Keywords:} Krein space, definitizable operators, normal operators, spectral theorem
} \end{flushleft}
\section{Introduction}
A bounded linear operator $N$ on a Krein space $(\mc K,[.,.])$ is called normal, if $N$ commutes with its Krein space adjoint $N^*$, i.e.\ $NN^*=N^*N$. This is equivalent to the fact that its real part $A:=\frac{N+N^*}{2}$ and its imaginary part $B:=\frac{N-N^*}{2i}$ commute. We call $N$ definitizable whenever the selfadjoint operators $A$ and $B$ are both definitizable in classical sense, i.e.\ there exist so-called definitizing polynomials $p(z)$ and $q(z)$ such that $[p(A)x,x] \geq 0$ and $[q(B)x,x] \geq 0$ for all $x\in \mc K$; see \cite{langer1982}.
In the Hilbert space setting the spectral theorem for bounded linear, normal operators is a well-known functional analysis result. In fact, it is almost as as folklore as the older spectral theorem for bounded linear, selfadjoint operators.
In the Krein space world there exists no similar result for general selfadjoint operators. But assuming in addition definitizability a spectral theorem could be shown by Heinz Langer; cf.\ \cite{langer1982}. This theorem became an important starting point for various spectral results. The main difference to selfadjoint operators on Hilbert spaces is the appearance of (finitely many) critical points, where the spectral projections no longer behave like a measure.
Only a rather small number of publications dealt with the situation of a normal (definitizable) operators in a Krein space. The Pontryagin space case was studied up to a certain extent for example in \cite{xiaochao1985} and \cite{langerszafraniec2006}. Special normal operators on Krein spaces were considered for example in \cite{azizovstrauss2003} and \cite{philippstrausstrunk2013}. But until now no adequate version of a spectral theorem on normal definitizable operators in Krein spaces has been found.
In the present paper we present a spectral theorem for bounded linear, normal, definitizable operators formulated in terms of a functional calculus generalizing the functional calculus $\phi\mapsto \int \phi \, dE$ in the Hilbert space case. In order to achieve this goal, we use the methods developed in \cite{KaPr2014} for definitizable selfadjoint operators and extend them for two commuting definitizable selfadjoint operators.
Let us anticipate a little more explicitly what happens in this note. Denoting by $p(z)$ and $q(z)$ the definitizing real polynomials for $A$ and $B$, respectively, we build a Hilbert space $\mc V$ which is continuously and densely embedded in the given Krein space $\mc K$ such that $TT^* = p(A)+q(B)$, where $T: \mc V \to \mc K$ denotes that adjoint of the embedding mapping. Then we use the $*$-homomorphism $\Theta: (TT^*)' \ (\subseteq B(\mc K)) \to (T^*T)' \ (\subseteq B(\mc V))$, $C \mapsto (T\times T)^{-1}(C)$, studied in \cite{KaPr2014}, in order to drag our normal operator $N\in (TT^*)'\subseteq B(\mc K)$ into $(T^*T)' \ (\subseteq B(\mc V)$. The resulting normal operator $\Theta(N)$ acts in a Hilbert space, and therefore has a spectral measure $E(\Delta)$, where $\Delta$ are Borel subsets of $\bb C$.
The proper family $\mc F_N$ of functions suitable for the aimed functional calculus are bounded and measurable functions on \[ \big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i \ (\subseteq \bb C \dot\cup \bb C^2) \,. \] Here $Z^{\bb R}_p = p^{-1}\{0\}\cap \bb R$ and $Z^{\bb R}_q = q^{-1}\{0\}\cap \bb R$ denote the real zeros of $p(z)$ and $q(z)$, respectively, and $Z^i = (p^{-1}\{0\}\times q^{-1}\{0\}) \setminus (\bb R\times\bb R)$. Moreover, the functions $\phi\in \mc F_N$ assume values in $\bb C$ on $\sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$, values in $\bb C^{\mf d_p(\RE z)\cdot \mf d_q(\IM z) + 2}$ at $z\in Z^{\bb R}_p + i Z^{\bb R}_q$ and values in $\bb C^{\mf d_p(\xi)\cdot \mf d_q(\eta)}$ at $z=(\xi,\eta)\in Z^i$. Here $\mf d_p(w)$ ($\mf d_q(w)$) denotes $p$'s ($q$'s) degree of zero at $w$. Finally, $\phi\in \mc F_N$ satisfies a growth regularity condition at all points from $Z^{\bb R}_p + i Z^{\bb R}_q$ which are not isolated in $\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)$.
Any polynomial $s(z,w)\in \bb C[z,w]$ can be seen as a function $s_N\in \mc F_N$. The nice thing about these, somewhat tediously defined functions $\phi\in \mc F_N$ is that \begin{equation}\label{decompact}
\phi(z) = s_N(z) +
(p_N + q_N) (z)\cdot g(z), \ z \in \sigma(\Theta(N)) \,, \end{equation} where $s \in \bb C[z,w]$ is a suitable polynomial in two variables and $g: \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \to \bb C$ is bounded and measurable and $g: \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q) \to \bb C^2$.
We then define $\phi(N):=s(A,B) + T \int_{\sigma(\Theta(N))}^{R_1,R_2} g \, dE T^*$, show that this operator does not depend on the actual decomposition \eqref{decompact} and that $\phi \mapsto \phi(N)$ is indeed a $*$-homomorphism. Here $\int_{\sigma(\Theta(N))}^{R_1,R_2} g \, dE$ is the integral of $g$ with respect to the spectral measure $E$ taking into account the fact that $g$ has values in $\bb C^2$ on $\sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$.
If $\phi$ is stems from a characteristic function corresponding to a Borel subset $\Delta$ of $\bb C$ such that no point of $Z^{\bb R}_p + i Z^{\bb R}_q$ belongs to the boundary of $\Delta$, then $\phi(N)$ is a selfadjoint projection on $\mc K$. In fact, it can be seen as the corresponding special projection for $N$.
\section{Multiple embeddings}
For the present section we fix a Krein space $(\mc K,[.,.])$ and Hilbert spaces $(\mc V,[.,.])$, $(\mc V_1,[.,.])$ and $(\mc V_2,[.,.])$. Moreover, let $T_1: \mc V_1 \to \mc K$, $T_2: \mc V_2 \to \mc K$ and $T: \mc V \to \mc K$ be bounded linear, injective mappings such that \[
TT^* = T_1T_1^* + T_2T_2^* \] holds true. Since for $x\in \mc K$ we have \begin{multline*}
[T^*x,T^*x]_{\mc V} = [TT^*x,x] = \\ [T_1T_1^*x,x] + [T_2T_2^*x,x] =
[T_1^*x,T_1^*x]_{\mc V_1} + [T_2^*x,T_2^*x]_{\mc V_2} \,, \end{multline*} one easily concludes that $T^* x \mapsto T_j^*x$ constitutes a well-defined, contractive linear mapping from $\ran T^*$ onto $\ran T_j^*$ for $j=1,2$. By $(\ran T^*)^\bot = \ker T=\{0\}$ and $(\ran T_j^*)^\bot = \ker T_j=\{0\}$ these ranges are dense in the Hilbert spaces $\mc V$ and $\mc V_j$. Hence, there is a unique bounded linear continuation of $T^* x \mapsto T_j^*x$ to $\mc V$, which has dense range in $\mc V_j$.
Denoting by $R_j$ for $j=1,2$ the adjoint mapping of this continuation we clearly have $T_j = T R_j$ and $\ker R_j = (\ran R_j^*)^\bot = \{0\}$. From $TT^* = T_1T_1^* + T_2T_2^*$ we conclude \[
T( \ I \ )T^* = TT^* = T R_1 R_1^*T^* + T R_2 R_2^*T^* = T( \ R_1 R_1^* + R_2 R_2^* \ )T^* \,. \] $\ker T=\{0\}$ and the density of $\ran T^*$ yields $R_1 R_1^* + R_2 R_2^* = I$.
If $T_1T_1^*$ and $T_2T_2^*$ commute, then by $TT^* = T_1T_1^* + T_2T_2^*$ also $T_jT_j^*$ and $TT^*$ commute. Moreover, in this case \[
T ( \ T^* T R_j R_j^* \ )T^* = TT^* T_jT_j^* = T_jT_j^* TT^* = T ( \ R_j R_j^* T^* T \ )T^* \,. \] Employing again $T$'s injectivity and the density of $\ran T^*$ we see that $R_j R_j^*$ and $T^* T$ commute for $j=1,2$. From this we get \[
T_j^* T_j R_j^*R_j = R_j^* (T^* T R_j R_j^*) R_j = R_j^* (R_j R_j^* T^* T) R_j = R_j^*R_j T_j^* T_j \,. \] Thus, we showed
\begin{lemma}\thlab{existtreans}
With the above notations and assumptions there exist injective contractions $R_1: \mc V_1 \to \mc V$
and $R_2: \mc V_2 \to \mc V$ such that $T_1 = T R_1$, $T_2 = T R_2$ and
$R_1 R_1^* + R_2 R_2^* = I$.
If $T_1T_1^*$ and $T_2T_2^*$ commute, then the operators $R_j R_j^*$ and $T^* T$ on $\mc V$ commute
as well as the operators $R_j^*R_j$ and $T_j^* T_j$ on $\mc V_j$ for $j=1,2$. \end{lemma}
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\begin{tikzpicture}[grow=left, sloped]
\node[bag] (a) {$\mc V$}
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}; \draw[->,thin] (a) -- (2,0) node[right,scale=1.0] (b) {$\mc K$}; \path[thin] (a) edge node[above] {$T$} (2,0); \path (c1) edge [bend left=15,below,thin,->] node[scale=1.0,above] {$T_1$} (b); \path (c2) edge [bend right=15,below,thin,->] node[scale=1.0,below] {$T_2$} (b); \end{tikzpicture} \end{center}
By $\Theta_j: (T_jT_j^*)' \ (\subseteq B(\mc K)) \to (T_j^*T_j)' \ (\subseteq B(\mc V_j)), \ j=1,2$, and by $\Theta: (TT^*)' \ (\subseteq B(\mc K)) \to (T^*T)' \ (\subseteq B(\mc V))$ we shall denote the $*$-algebra homomorphisms mapping the identity operator to the identity operator as in \thref{thetadefeig} from \cite{KaPr2014} corresponding to the mappings $T_j, \ j=1,2$, and $T$: \[
\Theta_j(C_j) = (T_j\times T_j)^{-1}(C_j) = T_j^{-1}C_jT_j, \ C_j\in (T_jT_j^*)' \,, \] \begin{equation}\label{thetaVdef}
\Theta(C) = (T\times T)^{-1}(C) = T^{-1}CT, \ C\in (TT^*)' \,. \end{equation} We can apply \thref{thetadefeig} in \cite{KaPr2014} also to the bounded linear, injective $R_j: \mc V_j \to \mc V, \ j=1,2$, and denote the corresponding $*$-algebra homomorphisms by $\Gamma_j : (R_jR_j^*)' \ (\subseteq B(\mc V)) \to (R_j^*R_j)' \ (\subseteq B(\mc V_j))$: \[
\Gamma_j(D) = (R_j\times R_j)^{-1}(D) = R_j^{-1}DR_j, \ D \in (R_jR_j^*)' \,. \]
\begin{proposition}\thlab{comreg}
With the above notations and assumptions we have $(T_1T_1^*)' \cap (T_2T_2^*)' \subseteq (TT^*)'$
and $\Theta((T_1T_1^*)' \cap (T_2T_2^*)') \subseteq (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$,
where in fact ($j=1,2$) \begin{equation}\label{zuef}
\Theta(C) R_j R_j^* = R_j \Theta_j(C) R_j^*=R_j R_j^* \Theta(C), \ \ C\in (T_1T_1^*)' \cap (T_2T_2^*)' \,. \end{equation}
Moreover, \begin{equation}\label{zuefvor}
\Theta_j(C) = \Gamma_j \circ \Theta(C), \ \ C\in (T_1T_1^*)' \cap (T_2T_2^*)' \,. \end{equation} \end{proposition} \begin{proof}
$(T_1T_1^*)' \cap (T_2T_2^*)' \subseteq (TT^*)'$ is clear from $TT^* = T_1T_1^* + T_2T_2^*$.
According to \thref{thetadefeig} in \cite{KaPr2014} we have
$\Theta_j(C) T_j^* = T_j^* C$ and $T^* C = \Theta(C) T^*$ for $C\in (T_1T_1^*)' \cap (T_2T_2^*)'$.
Therefore, \begin{multline*}
T (\ R_j \Theta_j(C) R_j^* \ ) T^* = T_j \Theta_j(C) T_j^* = T_j T_j^* C = \\
T R_j R_j^* T^* C = T ( \ R_j R_j^* \Theta(C) \ ) T^* \,. \end{multline*} $\ker T=\{0\}$ and the density of $\ran T^*$ yield $R_j \Theta_j(C) R_j^*=R_j R_j^* \Theta(C)$ for $j=1,2$. Applying this equation to $C^*$ and taking adjoints yields $R_j \Theta_j(C) R_j^*=\Theta(C) R_j R_j^*$. In particular, $\Theta(C) \in (R_jR_j^*)'$. Therefore, we can apply $\Gamma_j$ to $\Theta(C)$ and get \[
\Gamma_j \circ \Theta(C) = R_j^{-1} T^{-1}C T R_j = T_j^{-1} C T_j = \Theta_j(C) \,. \] \end{proof}
For the following assertion note that by \eqref{zuefvor} and by the fact that $\Gamma_j$ is a $*$-algebra homomorphism mapping the identity operator to the identity operator, we have ($j=1,2$) \begin{equation}\label{4ucd75}
\sigma(\Theta(C)) \subseteq \sigma(\Theta_j(C)), \ C\in (T_1T_1^*)' \cap (T_2T_2^*)' \,. \end{equation}
\begin{corollary}\thlab{normtransf}
With the above notations and assumptions let $N\in (T_1T_1^*)' \cap (T_2T_2^*)'$ be normal.
Then $\Theta(N),\Theta_1(N), \Theta_2(N)$ are all normal operators in the Hilbert spaces
$\mc V$, $\mc V_1$, $\mc V_2$, respectively. If $E$ ($E_1$,$E_2$) denotes the spectral
measure for $\Theta(N)$ ($\Theta_1(N)$, $\Theta_2(N)$), then $E(\Delta) \in (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$ and
\[
\Gamma_j(E(\Delta)) = E_j(\Delta), \ \ j=1,2 \,,
\]
for all Borel subsets $\Delta$ of $\bb C$, where $E_j(\Delta) \in (R_j^*R_j)' \cap (T_j^*T_j)'$.
Moreover, $\int h \, dE \in (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$
and
\[
\Gamma_j\left(\int h \, dE \right) = \int h \, dE_j
\]
for any bounded and measurable $h: \sigma(\Theta(N)) \to \bb C$, where $\int h \, dE_j$
belongs to $(R_j^*R_j)' \cap (T_j^*T_j)'$. \end{corollary} \begin{proof}
The normality of $\Theta(N),\Theta_1(N)$ and $\Theta_2(N)$ is clear, since $\Theta$, $\Theta_1, \Theta_2$ are
$*$-homomorphisms. From \thref{comreg} we know that $\Theta(N) \in (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$.
According to the well known properties of $\Theta(N)$'s spectral measure we obtain
$E(\Delta) \in (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$ and, in turn,
$\int h \, dE \in (R_1R_1^*)' \cap (R_2R_2^*)' \cap (T^*T)'$. In particular, $\Gamma_j$ can be applied
to $E(\Delta)$ and $\int h \, dE$.
Similarly, $\Theta_j(N)\in (T_j^*T_j)'$ implies $E_j(\Delta), \int h \, dE_j \in (T_j^*T_j)'$
for a bounded and measurable $h$.
Recall from \thref{thetadefeig} in \cite{KaPr2014} that $\Gamma_j(D) R_j^*x = R_j^* D$
for $D \in (T^*T)'$. For $x\in \mc V$ and $y\in \mc V_j$ we therefore get
\[
[\Gamma_j(E(\Delta)) R_j^*x, y] = [R_j^* E(\Delta) x,y] = [E(\Delta) x, R_j y]
\]
and, in turn,
\begin{multline*}
\int_{\bb C} s(z,\bar z) \, d[\Gamma_j(E) R_j^*x, y] =
\int_{\bb C} s(z,\bar z) \, d[E x, R_j y] = [s(\Theta(N),\Theta(N)^*)x, R_j y] = \\
[R_j ^* s(\Theta(N),\Theta(N)^*)x, y] = [\Gamma_j(s(\Theta(N),\Theta(N)^*)) R_j^* x,y]
\end{multline*}
for any trigonometric polynomial $s(z,\bar z) \in \bb C[z,\bar z]$.
By \eqref{zuefvor} and the fact that $\Gamma_j$ is a $*$-homomorphism
we have $\Gamma_j(s(\Theta(N),\Theta(N)^*)) = s(\Theta_j(N),\Theta_j(N)^*)$.
Consequently,
\[
\int_{\bb C} s(z,\bar z) \, d[\Gamma_j(E) R_j^*x, y] = \int_{\bb C} s(z,\bar z) \, d[E_j R_j^*x, y] \,.
\]
Since $E(\bb C\setminus K)=0$ and $E_j(\bb C\setminus K)=0$ for a certain compact $K\subseteq \bb C$ and since
$\bb C[z,\bar z]$ is densely contained in $C(K)$, we obtain from the uniqueness assertion of
the Riesz Representation Theorem
\[
[\Gamma_j(E(\Delta)) R_j^*x, y] = [E_j(\Delta) R_j^*x, y], \ x\in \mc V, \, y\in \mc V_j \,,
\]
for all Borel subsets $\Delta$ of $\bb C$.
Due to the density of $\ran R_j^*$ in $\mc V_j$ we even have
$[\Gamma_j(E(\Delta)) z, y] = [E_j(\Delta) z, y], \ y,z \in \mc V_j$ and, in turn,
$\Gamma_j(E(\Delta))=E_j(\Delta)$. Since $\Gamma_j$ maps into $(R_j^*R_j)'$ we have
$E_j(\Delta)\in (R_j^*R_j)'$ and, in turn, $\int h \, dE_j\in (R_j^*R_j)'$ for any bounded and measurable $h$.
If $h: \sigma(\Theta(N)) \to \bb C$ is bounded and measurable, then, clearly, also
its restriction to $\sigma(\Theta_j(N)) = \sigma(\Gamma_j\circ\Theta(N))$ is bounded and measurable;
see \eqref{4ucd75}.
Due to $E_j(\Delta) R_j^*= \Gamma_j(E(\Delta)) R_j^* = R_j^*E(\Delta)$ for
$x\in\mc V$ and $y\in\mc V_j$ we have
\begin{multline*}
[\Gamma_j\left(\int h \, dE \right) R_j^*x,y] =
[R_j^* \left(\int h \, dE \right) x,y] =
[\left(\int h \, dE \right) x, R_j y] = \\
\int h \, d[E x,R_j y] = \int h \, d[E_j R_j^* x, y] =
[\left(\int h \, dE_j\right) R_j^*x,y] \,.
\end{multline*}
Again the density of $\ran R_j^*$ yields $\Gamma_j\left(\int h \, dE \right) = \int h \, dE_j$. \end{proof}
Recall from \thref{Xidefeig} in \cite{KaPr2014} the mappings ($j=1,2$) \begin{equation}\label{xijdef}
\Xi_j : (T_j^*T_j)' \ (\subseteq B(\mc V_j)) \to (T_jT_j^*)' \ (\subseteq B(\mc K)), \
\Xi_j(D_j) = T_j D_j T_j^{*} \,, \end{equation} and $\Xi : (T^*T)' \ (\subseteq B(\mc V)) \to (TT^*)' \ (\subseteq B(\mc K)), \ \Xi(D) = T D T^{*}$. By ($j=1,2$) \[
\Lambda_j: (R_j^*R_j)' \ (\subseteq B(\mc V_j)) \to (R_jR_j^*)' \ (\subseteq B(\mc V)),
\ \Lambda_j(D_j) = R_j D_j R_j^{*} \,, \] we shall denote the corresponding mappings outgoing from the mappings $R_j: \mc V_j \to \mc V$. By \thref{existtreans} we have \[
\Xi_j(D_j) = T_j R_j D_j R_j^* T_j^* = \Xi\circ\Lambda_j(D_j) \ \text{ for }
\ D_j\in (R_j^*R_j)' \cap (T_j^*T_j)' \,. \] According to \thref{Xidefeig} in \cite{KaPr2014}, $\Lambda_j\circ \Gamma_j(D) = D R_jR_j^*$. Hence, using the notation from \thref{normtransf} \begin{equation}\label{zuef2}
\Xi_j(\int h \, dE_j) = \Xi\circ\Lambda_j\circ \Gamma_j\left(\int h \, dE \right) = \Xi(R_jR_j^{*} \int h \, dE) \,. \end{equation} Finally, $T^{-1} T_jT_j^* T = T^{-1} T R_j R_j^* T^* T = R_j R_j^* T^* T$. In case that $T_1T_1^*$ and $T_2T_2^*$ commute we have $T_1T_1^*, T_2T_2^* \in (TT^*)'$ and the later equality can be expressed as ($j=1,2$) \begin{equation}\label{zuef3}
\Theta(T_jT_j^*) = R_j R_j^* T^* T \,. \end{equation}
\section{Normal definitizable operators}
\begin{definition}\thlab{definitnordef} We will call a bounded linear and normal operator $N$ on a Krein Space \emph{definitizable} if its real part $A:=\frac{N+N^*}{2}$ and its imaginary part $B:=\frac{N-N^*}{2i}$ are both definitizable, i.e.\ there exist real polynomials $p,q\in\bb R[z]$ such that $p$ is definitizing for $A$ ($[p(A)x,x] \geq 0, \, x\in \mc K$) and such that $q$ is definitizing for $B$ ($[q(B)x,x] \geq 0, \, x\in \mc K$); see \cite{langer1982}. \end{definition}
By \thref{anderedefinitz} in \cite{KaPr2014} the definitizability of $A$ and $B$ is equivalent to the concept of definitizability in \cite{KaPr2014}.
Also note that in Pontryagin spaces any bounded linear and normal operator is definitizable in the above sense; see \thref{pontrunit} in \cite{KaPr2014}.
\begin{proposition}\thlab{defNspaces}
Let $A$ and $B$ be commuting, bounded linear, selfadjoint and definitizable operators on a Krein space
$(\mc K,[.,.])$ with definitizing polynomials $p\in\bb R[z]$ for $A$ and $q\in\bb R[z]$ for $B$.
Then there exist Hilbert spaces $(\mc V_1,[.,.])$, $(\mc V_2,[.,.])$, $(\mc V,[.,.])$ and
bounded linear and injective operators $T_1: \mc V_1 \to \mc K$, $T_2: \mc V_2 \to \mc K$, $T: \mc V \to \mc K$
such that
\[
T_1 T_1^* = p(A), \ T_2 T_2^* = q(B), \ T T^* = p(A)+q(B) = T_1 T_1^* + T_2 T_2^*
\]
with commuting $T_1 T_1^*$ and $T_2 T_2^*$.
Moreover, if $\Theta: (TT^*)' \ (\subseteq B(\mc K)) \to (T^*T)' \ (\subseteq B(\mc V))$ is as in
\eqref{thetaVdef} and $R_j: \mc V_j \to \mc V$ ($j=1,2$) are as in \thref{existtreans}, then \begin{equation}\label{heaybab} \begin{aligned}
p(\Theta(A)) = R_1 R_1^* \big(p(\Theta(A)) + q(\Theta(B)) \big), \\
q(\Theta(B)) = R_2 R_2^* \big(p(\Theta(A)) + q(\Theta(B)) \big)
\,, \end{aligned} \end{equation}
where $R_1 R_1^*$ and $R_2 R_2^*$ commute with $p(\Theta(A)) + q(\Theta(B))$. \end{proposition} \begin{proof} Let $(\mc V_1,[.,.])$ be the Hilbert space completion of $\mc K/\ker p(A)$ with respect to $[p(A).,.]$ and let $T_1: \mc V_1 \to \mc K$ be the adjoint of the embedding of $\mc K$ into $\mc V_1$. Since $T_1^*$ has dense range, $T_1$ is injective. Analogously let $(\mc V_2,[.,.])$ be the Hilbert space completion of $\mc K/\ker q(B)$ with respect to $[q(B).,.]$ and denote by $T_2:\mc V_2 \to \mc K$ the injective adjoint of the embedding of $\mc K$ into $\mc V_2$. Finally, let $(\mc V,[.,.])$ be the Hilbert space completion of $\mc K/(\ker p(A)+q(B))$ with respect to $[(p(A)+q(B)).,.]$ and let $T: \mc V \to \mc K$ be the injective adjoint of the embedding of $\mc K$ into $\mc V$.
From $[T T^* x,y] = [T^* x,T^* y]_{\mc V} = [x,y]_{\mc V} = [(p(A)+q(B))x,y]$, $[T_1 T_1^* x,y] = [T_1^* x,T_1^* y]_{\mc V_1} = [x,y]_{\mc V_1} = [p(A)x,y]$ and $[T_2 T_2^* x,y] = [q(B)x,y]$ for all $x,y\in\mc K$ we conclude that
\[
T_1 T_1^* = p(A), \ T_2 T_2^* = q(B), \ T T^* = p(A)+q(B) \,,
\] where $p(A)=T_1T_1^*$ and $q(B)=T_2T_2^*$ commute, because $A$ and $B$ do.
From \eqref{zuef3} and \thref{thetadefeig} in \cite{KaPr2014} we get \begin{multline*}
p(\Theta(A)) = \Theta(p(A)) = \Theta(T_1T_1^*) = R_1 R_1^* T^* T = R_1 R_1^* \Theta(TT^*) = \\
R_1 R_1^* \Theta( p(A)+q(B) ) = R_1 R_1^* \big(p(\Theta(A)) + q(\Theta(B)) \big) \,. \end{multline*} Similarly, $q(\Theta(B)) = R_2 R_2^* (p(\Theta(A)) + q(\Theta(B)) )$. Finally, $R_1 R_1^*$ and $R_2 R_2^*$ commute with $T^* T=p(\Theta(A)) + q(\Theta(B))$ by \thref{existtreans}. \end{proof}
The fact that a normal operator is definitizable implies certain spectral properties of $\Theta(N)$.
\begin{lemma}\thlab{speknorm} With the notion of \thref{defNspaces} applied to the real part $A:=\frac{N+N^*}{2}$ and the imaginary part $B:=\frac{N-N^*}{2i}$ of a bounded linear, normal and definitizable operator $N$ we have \[
\{z\in \bb C : |p(\RE z)| > \| R_1 R_1^* \| \cdot |p(\RE z) + q(\IM z)| \} \subseteq \rho(\Theta(N)) \,, \] and \[
\{z\in \bb C : |q(\IM z)| > \| R_2 R_2^* \| \cdot |p(\RE z) + q(\IM z)| \} \subseteq \rho(\Theta(N)) \,. \] In particular, the zeros of $p(\RE z) + q(\IM z)$ are contained in $\rho(\Theta(N)) \cup \{z\in \bb C: p(\RE z) = 0 = q(\IM z)\}$. \end{lemma} \begin{proof} We are going to show the first inclusion. The second one is shown in the same manner. For this let $n\in \bb N$ and set \[
\Delta_n:= \{z\in \bb C : |p(\RE z)|^2 > \frac{1}{n} + \| R_1 R_1^* \|^2 \cdot |p(\RE z) + q(\IM z)|^2 \} \,. \] For $x\in E(\Delta_n)(\mc V)$ we then have \begin{multline*}
\| p(\Theta(A)) x \|^2 = \int_{\Delta_n} |p(\RE \zeta)|^2 \, d[E(\zeta)x,x] \geq \\ \int_{\Delta_n} \frac{1}{n} \, d[E(\zeta)x,x] +
\| R_1 R_1^* \|^2 \int_{\Delta_n} |p(\RE \zeta) + q(\IM \zeta)|^2 \, d[E(\zeta)x,x] \\ \geq
\frac{1}{n} \|x\|^2 + \| R_1 R_1^* \big(p(\Theta(A)) + q(\Theta(B))x \|^2 \,. \end{multline*} By \eqref{heaybab} this inequality can only hold for $x=0$. Since $\Delta_n$ is open, by the Spectral Theorem for normal operators on Hilbert spaces we have $\Delta_n \subseteq \rho(N)$. The asserted inclusion now follows from \[
\{z\in \bb C : |p(\RE z)| > \| R_1 R_1^* \| \cdot |p(\RE z) + q(\IM z)| \} = \bigcup_{n\in\bb N} \Delta_n \,. \] \end{proof}
\begin{corollary}\thlab{korvda}
With the notation and assumptions from \thref{speknorm} we have \begin{multline*}
R_1R_1^* \, E\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\} = \\
\int_{\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\}} \frac{p(\RE z)}{p(\RE z) + q(\IM z)} \, dE(z) \end{multline*}
and \begin{multline*}
R_2R_2^* \, E\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\} = \\
\int_{\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\}} \frac{q(\RE z)}{p(\RE z) + q(\IM z)} \, dE(z) \end{multline*} \end{corollary} \begin{proof}
First note that the integrals on the right hand sides exist as bounded operators, because
by \thref{speknorm} we have $|p(\RE z)| \leq \| R_1 R_1^* \| \cdot |p(\RE z) + q(\IM z)|$ and
$|q(\IM z)| \leq \| R_2 R_2^* \| \cdot |p(\RE z) + q(\IM z)|$ on $\sigma(\Theta(N))$.
Clearly, both sides vanish on the range of $E\{z\in \bb C: p(\RE z) = 0 = q(\IM z)\}$.
Its orthogonal complement \begin{multline*}
\mc H:=\ran E\{z\in \bb C: p(\RE z) = 0 = q(\IM z)\}^\bot = \\
\ran E\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\} \,, \end{multline*}
is invariant under
$\int \big(p(\RE z) + q(\IM z)\big) \, dE(z) = \big(p(\Theta(A)) + q(\Theta(B))\big)$.
By \thref{speknorm} the restriction of this operator to $\mc H$
is injective, and hence, has dense range in $\mc H$.
If $x$ belongs to this dense range, i.e.\
$x=\big(p(\Theta(A)) + q(\Theta(B))\big) y$ with $y\in \mc H$, then
\[
\int_{\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\}}
\frac{p(\RE z)}{p(\RE z) + q(\IM z)} \, dE(z) x =
\] \begin{multline*}
\int_{\{z\in \bb C: p(\RE z) \not= 0 \text{ or } q(\IM z)\not=0\}} p(\RE z) \, dE(z) y =
p(\Theta(A))y = \\ R_1 R_1^* \big(p(\Theta(A)) + q(\Theta(B))\big) y = R_1 R_1^* x \,. \end{multline*}
By a density argument the first asserted equality of the present corollary holds true on $\mc H$
and in turn on $\mc V$.
The second equality is shown in the same manner. \end{proof}
\section{The proper function class}
In order to introduce a functional calculus we have to introduce an algebra structure on $\mc A_{m,n}:=(\mathbb C^{m} \otimes \mathbb C^{n}) \times \mathbb C^2 \simeq \mathbb C^{m\cdot n + 2}$ and on $\mc B_{m,n} := \mathbb C^{m} \otimes \mathbb C^{n} \simeq \mathbb C^{m\cdot n}$ for $m,n\in \bb N$. For notational convenience we also set $\mc A_{0,0} := \mathbb C$.
\begin{definition}\thlab{muldefb1}
Firstly, let $\mc A_{0,0} = \bb C$ be provided with the usual addition, scalar multiplication,
multiplication and conjugation.
Secondly, in case that $m,n\in \bb N$ we provide $\mc A_{m,n}$ with the componentwise
addition and scalar multiplication. Moreover, for
$a=(a_{k,l})_{(k,l)\in I_{m,n}}, b=(b_{k,l})_{(k,l)\in I_{m,n}}$ with
$I_{m,n}:=(\{0,\dots,m-1\}\times \{0,\dots,n-1\})\cup\{(m,0),(0,n)\}$ we set \[
a\cdot b :=
\Big( \sum_{c=0}^k\sum_{d=0}^l a_{c,d} b_{k-c,l-d} \Big)_{(k,l)\in I_{m,n}} \ \text{ and } \
\overline{a}:=\big(\bar a_{k,l}\big)_{(k,l)\in I_{m,n}} \,. \]
On $\mc B_{m,n}$ we define addition, scalar multiplication, multiplication and conjugation in the same way
only neglecting the the entries with indices $(m,0)$ and $(0,n)$.
Finally, for $m,n\in \bb N$ we introduce the projection $\pi: \mc A_{m,n} \to \mc B_{m,n}$, $(a_{k,l})_{(k,l)\in I_{m,n}} \mapsto (a_{k,l})_{\substack{0\leq k \leq m-1 \\ 0\leq l\leq n-1}}$. On $\mc B_{m,n}$ we assume $\pi$ to be the identity. \end{definition}
\begin{remark}\thlab{z30f3} It is easy to check that $\mc A_{m,n}$ and $\mc B_{m,n}$ are commutative, unital $*$-algebras. Setting $e_{0,0}=1$ and $e_{k,l}=0, \ (k,l)\neq (0,0)$, it is easy to verify that $\big(e_{k,l}\big)_{(k,l)\in I_{m,n}}$ is the multiplicative unite in $\mc A_{m,n}$ and $\big(e_{k,l}\big)_{\substack{0\leq k \leq m-1 \\ 0\leq l\leq n-1}}$ is the multiplicative unite in $\mc B_{m,n}$. We shall denote these unites by $e$.
Moreover, it is straight forward to check that an element $(a_{k,l})$ of $\mc A_{m,n}$ (of $\mc B_{m,n}$) has a multiplicative inverse in $\mc A_{m,n}$ (in $\mc B_{m,n}$) if and only if $a_{0,0}\neq 0$. \end{remark}
For the rest of the paper assume that $N$ bounded linear, normal and definitizable operator in a Krein space $\mc K$ with real part $A$ and imaginary part $B$. Moreover, we fix definitizing polynomials $p\in\bb R[z]$ for $A$ and $q\in\bb R[z]$ for $B$.
\begin{definition}\thlab{muldefb2}
We define functions $\mf d_p, \mf d_q: \bb C \to \bb N\cup\{0\}$
such that $\mf d_p(z)$ is $p$'s degrees of the zero at $z$ and
$\mf d_q(z)$ is $q$'s degrees of the zero at $z$.
Moreover, we shall denote the set of their real zeros by $Z^{\bb R}_p$ and $Z^{\bb R}_q$, i.e.\
\[
Z^{\bb R}_p:=p^{-1}\{0\}\cap \bb R, Z^{\bb R}_q:=q^{-1}\{0\}\cap \bb R \,,
\]
and we set $Z^i:=(p^{-1}\{0\}\times q^{-1}\{0\})\setminus (\bb R\times \bb R)$.
\noindent
Now we are going to introduce class of functions:
\begin{enumerate}[$(i)$]
\item
By $\mc M_{N}$ we denote the set of functions $\phi$ defined on
\[
\big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i
\]
with $\phi(z) \in \mf C(z)$, where $\mf C(z):=\mc B_{\mf d_p(\xi),\mf d_q(\eta)}$ for $z=(\xi,\eta) \in Z^i$
and where
$\mf C(z):=\mc A_{\mf d_p(\RE z),\mf d_q(\IM z)}$
for $z \in \sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)$.
\item
We provide $\mc M_{N}$ pointwise with scalar multiplication,
addition and multiplication, where the operations on $\mc A_{\mf d_p(\RE z),\mf d_q(\IM z)}$
or $\mc B_{\mf d_p(\xi),\mf d_q(\eta)}$ are as in \thref{muldefb1}.
We also define a conjugate linear involution $.^\#$ on $\mc M_{N}$ by \begin{multline*}
\phi^\#(z) = \overline{\phi(z)}, \ z \in \sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q), \\
\phi^\#(\xi,\eta) = \overline{\phi(\bar \xi,\bar \eta)}, \ (\xi,\eta) \in Z^i \,. \end{multline*}
\item
By $\mc R$ we denote the set of all elements $\phi\in\mc M_{N}$ such that
$\pi(\phi(z)) = 0$ for all $z \in (Z^{\bb R}_p + i Z^{\bb R}_q) \dot \cup Z^i$. \end{enumerate} \end{definition}
With the operations introduced in \thref{muldefb2} $\mc M_{N}$ is a commutative $*$-algebra as can be verified in a straight forward manner. Moreover, $\mc R$ is an ideal of $\mc M_{N}$.
\begin{definition}\thlab{feinbetef}
Let $f: \dom f \to \bb C$ be a function with $\dom f \subseteq \bb C^2$ such that
$\tau\big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big) \subseteq \dom f$, where
$\tau : \bb C \to \bb C^2, \ (x+iy)\mapsto (x,y)$, such that
$f\circ \tau$ is sufficiently smooth -- more exactly, at least
$\max_{x,y\in \bb R} \mf d_p(x) + \mf d_q(y) - 1$ times continuously differentiable --
on an open neighbourhood of $Z^{\bb R}_p + i Z^{\bb R}_q$, and
such that $f$ is holomorphic on an open neighbourhood of $Z^i$.
Then $f$ can be considered as an element $f_{N}$ of $\mc M_{N}$ by setting
$f_{N}(z) := f\circ \tau(z)$ for $z\in \sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$, by \[
f_{N}(z) :=
\big(\frac{1}{k!l!} \, \frac{\partial^{k+l}}{\partial x^k\partial y^l}
f\circ \tau (z)\big)_{(k,l)\in I_{\mf d_p(\RE z),\mf d_q(\IM z)}} \]
for $z\in Z^{\bb R}_p + i Z^{\bb R}_q$, and by
\[
f_{N}(\xi,\eta) := \big(\frac{1}{k!l!} \, \frac{\partial^{k+l}}{\partial z^k\partial w^l}
f(\xi,\eta)\big)_{\substack{0\leq k \leq \mf d_p(\xi)-1 \\ 0\leq l\leq \mf d_q(\eta) - 1}} \,,
\]
for $(\xi,\eta)\in Z^i$. \end{definition}
\begin{remark}\thlab{bweuh30} By the Leibniz rule $f\mapsto f_{N}$ is compatible with multiplication. Obviously, it is also compatible with addition and scalar multiplication. If we define for a function $f$ as in \thref{feinbetef} the function $f^\#$ by $f^\#(z,w) = \overline{f(\bar z,\bar w)}, \ (z,w) \in \dom f$, then we also have $(f^\#)_{p,q} = (f_{N})^\#$. Note that in general $(\bar f)_{p,q} \not= (f_{N})^\#$.
Finally, note that $\mathds{1}_N(z)$ is the multiplicative unite in $\mf C(z)$ for all $z\in \big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i$. \end{remark}
\begin{example}\thlab{fuinm0pre}
For the constant one function $\mathds{1}$ on $\bb C^2$ we have $\mathds{1}_N(z) = e$
for all $z\in \big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i$, where
$e$ is the multiplicative unite in $\mf C(z)$; see \thref{z30f3}. \end{example}
\begin{example}\thlab{fuinm0}
$p(z)$ considered as an element of $\bb C[z,w]$ is clearly holomorphic on $\bb C^2$.
Hence, we can consider $p_N$ as defined in \thref{feinbetef}.
It satisfies $p_{N}(z)_{k,l}=0$,
$(k,l)\in I_{\mf d_p(\RE z),\mf d_q(\IM z)}\setminus \{(\mf d_p(\RE z),0)\}$, and
\[
p_{N}(z)_{\mf d_p(\RE z),0} = \frac{1}{\mf d_p(\RE z)!} p^{(\mf d_p(\RE z))}(\RE z)
\]
for all $z \in Z^{\bb R}_p + i Z^{\bb R}_q$.
Since $\RE z$ is a zero of $p$ of degree
exactly $\mf d_p(\RE z)$ the entries with index $(\mf d_p(\RE z),0)$ do not vanish.
Moreover, $p_{N}(\xi,\eta)=0$ for all $(\xi,\eta)\in Z^i$. In particular, $p_{N} \in \mc R$.
Similarly, if $q(w)$ is considered as an element of $\bb C[z,w]$, then $q_{N}(z)_{k,l}=0$,
$(k,l)\in I_{\mf d_p(\RE z),\mf d_q(\IM z)}\setminus \{(0,\mf d_q(\IM z))\}$, and
\[
q_{N}(z)_{0,\mf d_q(\IM z)} = \frac{1}{\mf d_q(\IM z)!} q^{(\mf d_q(\IM z))}(\IM z) \neq 0
\]
for all $z \in Z^{\bb R}_p + i Z^{\bb R}_q$.
Also here $q_{N}(\xi,\eta)=0$ for all $(\xi,\eta)\in Z^i$ and, in turn, $q_{N} \in \mc R$. \end{example}
We need an easy algebraic lemma based in the Euclidean algorithm.
\begin{lemma}\thlab{einbett2pre}
For $a(z), b(z)\in \bb C[z]$ we denote by $a^{-1}\{0\}$ and $b^{-1}\{0\}$ the set of all zeros of
$a$ and $b$ in $\bb C$, and by $\mf d_a(z)$ ($\mf d_b(z)$) $a$'s ($b$'s)
degree of zero at $z\in \bb C$. Denote by $m$ ($n$) the degree of the polynomial $a$ ($b$).
Then any $s\in \bb C[z,w]$ can be written as
\[
s(z,w) = a(z)u(z,w) + b(w)v(z,w) + r(z,w)
\]
with $u(z,w), v(z,w), r(z,w)\in \bb C[z,w]$
such that $r$'s $z$-degree is less than $m$ and its $w$-degree is less than $n$.
Here $u(z,w), v(z,w), r(z,w)$ can be found in $\bb R[z,w]$ if $a(z), b(z)\in \bb R[z], \ s\in \bb R[z,w]$.
If we define $\varpi: \bb C[z,w] \to \bb C^{m\cdot n}$ by
\[
\varpi(s) = \left( \Big(\frac{1}{k!l!} \, \frac{\partial^{k+l}}{\partial z^k\partial w^l}
s(z,w)\Big)_{\substack{0\leq k \leq \mf d_a(z)-1 \\ 0\leq l\leq \mf d_b(w)-1}}\right)_{
z\in a^{-1}\{0\}, w\in b^{-1}\{0\}} \,,
\]
then $s\in \ker \varpi$ if and only if $s(z,w)=a(z)u(z,w) + b(w) v(z,w)$
for some $u(z,w), v(z,w) \in \bb C[z,w]$. Moreover, $\varpi$ restricted to the space of all
polynomials from $\bb C[z,w]$ with $z$-degree less than $m$ and $w$-degree less than $n$
is bijective. \end{lemma} \begin{proof}
Applying the Euclidean algorithm to $s(z,w)\in \bb C[z,w]$ and $a(z)$ we get
$s(z,w) = a(z)u(z,w) + t(z,w)$, where $u(z,w), t(z,w)\in \bb C[z,w]$ such that
$t$'s $z$-degree is less than $m$. Applying the Euclidean algorithm to $t(z,w)$ and $b(w)$ we get
\[
s(z,w) = a(z)u(z,w) + b(w)v(z,w) + r(z,w)
\]
with $v(z,w), r(z,w)\in \bb C[z,w]$
such that $r$'s $z$-degree is less than $m$ and its $w$-degree is less than $n$.
The resulting polynomials $u(z,w), t(z,w), v(z,w), r(z,w)$ belong to $\bb R[z,w]$ if
$a(z), b(z)\in \bb R[z], \ s(z,w)\in \bb R[z,w]$.
In any case it is easy to check that then $\varpi(s)=\varpi(r)$.
Hence, $r(z,w)=0$ yields $s(z,w)\in \ker \varpi$. On the other hand, if $0=\varpi(s)=\varpi(r)$, then
for each fixed $\zeta \in a^{-1}\{0\}$ and $k\in\{0,\dots, \mf d_a(\zeta)-1\}$ the function
$w\mapsto \frac{\partial^{k}}{\partial z^k} r(\zeta,w)$ has zeros at all
$w\in b^{-1}\{0\}$ with multiplicity at least $\mf d_b(w)$.
Since $w\mapsto \frac{\partial^{k}}{\partial z^k} r(\zeta,w)$ is of
$w$-degree less than $n$, it must be identically equal to zero.
This implies that for any $\eta \in \bb C$ the polynomial $z \mapsto r(z,\eta)$ has zeros at all
$\zeta\in a^{-1}\{0\}$ with multiplicity at least $\mf d_a(\zeta)$. Since the degree of this polynomial in $z$
is less than $m$, we obtain $r(z,\eta)=0$ for any $z\in \bb C$. Thus, $r\equiv 0$.
Our description of $\ker \varpi$ shows in particular that $\varpi$ restricted to the space of all
polynomials from $\bb C[z,w]$ with $z$-degree less than $m$ and $w$-degree less than $n$
is one-to-one. Comparing dimensions shows that this restriction of $\varpi$ is also onto. \end{proof}
\begin{corollary}\thlab{existbe}
With the notation from \thref{muldefb2}
for any $\phi \in \mc M_{N}$ we find an
$s\in \bb C[z,w]$ such that $\phi - s_N \in \mc R$. \end{corollary} \begin{proof}
By \thref{einbett2pre} there exists an $s\in \bb C[z,w]$ such that
$\varpi(s)_{\RE z,\IM z} = \pi(\phi(z))$ for all
$z \in Z^{\bb R}_p + i Z^{\bb R}_q$, and
such that $\varpi(s)_{\xi,\eta} = \phi(\xi,\eta)$ for all $(\xi,\eta) \in Z^i$.
According to $\mc R$'s definition we obtain $\phi - s_N \in \mc R$. \end{proof}
\begin{remark}\thlab{durchdiv}
Recall from \thref{speknorm} that $p(\RE z) + q(\IM z)=0$ with $z\in \sigma(\Theta(N))$
implies $p(\RE z) = 0 = q(\IM z)$, i.e.\ $z\in Z^{\bb R}_p + i Z^{\bb R}_q$.
If $\phi\in \mc R$, then we find a function $g$ on $\sigma(\Theta(N))$ with
$g(z)\in \bb C$ for $z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$ and
$g(z)\in \bb C^2$ for $z\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$, such that
$\phi(z) = (p_N + q_N)(z)\cdot g(z), \ z \in \sigma(\Theta(N))$; see \thref{fuinm0}.
Here $(p_N + q_N)(z) \cdot g(z)$
is the usual multiplication on $\bb C$ for $z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$,
whereas
\[
\big((p_N + q_N)(z) \cdot g(z)\big)_{k,l} = 0, \ k = 0,\dots,\mf d_p(\RE z)-1; l = 0,\dots,\mf d_q(\IM z)-1 \,,
\]
and
\[
\big((p_N + q_N)(z)\cdot g(z)\big)_{\mf d_p(\RE z),0}
= (p_N + q_N)(z)_{\mf d_p(\RE z),0} \, \cdot g_1(z) \,,
\]\[
\big((p_N + q_N)(z)\cdot g(z)\big)_{0,\mf d_q(\IM z)}
= (p_N + q_N)(z)(z)_{0,\mf d_q(\IM z)} \, \cdot g_2(z) \,.
\]
for $z\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$.
In fact, we simply set $g(z):=\frac{\phi(z)}{p(\RE z)+q(\IM z)}$ for
$z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$ and
\[
g_1(z) := \frac{\mf d_p(\RE z)! \, \phi(z)_{(\mf d_p(\RE z),0)}}{p^{(\mf d_p(\RE z))}(\RE z)}, \
g_1(z) := \frac{\mf d_q(\IM z)! \, \phi(z)_{(0,\mf d_q(\IM z))}}{q^{(\mf d_q(\IM z))}(\IM z)}
\]
for $z\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$. \end{remark}
We are going to introduce a subclass of $\mc M_{N}$, which will be the proper class, in order to build up our functional calculus.
\begin{definition}\thlab{FdefklM}
With the notation from \thref{muldefb2} we denote by $\mc F_{N}$ the set of all elements
$\phi \in \mc M_{N}$ such that
$z\mapsto \phi(z)$ is Borel measurable and bounded on
$\sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$, and such that
for each $w \in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)$
\begin{equation}\label{fn8qw3b}
\frac{\phi(z) -
\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l}{\max(|\RE(z-w)|^{\mf d_p(\RE w)},|\IM (z-w)|^{\mf d_q(\IM w)})}
\end{equation}
is bounded for $z\in \sigma(\Theta(N))\cap U(w)\setminus \{w\}$,
where $U(w)$ is a sufficiently small neighbourhood of $w$. \end{definition}
Note that \eqref{fn8qw3b} is immaterial if $w$ is an isolated point of $\sigma(\Theta(N))$.
\begin{example}\thlab{fedela}
For $\zeta \in (Z^{\bb R}_p + i Z^{\bb R}_q)\dot\cup Z^i$ and $a\in \mf C(\zeta)$ consider the functions
$a\delta_\zeta \in \mc M_{N}$ which assumes the value $a$ at $\zeta$ and the value zero on the rest of
$\big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i$
If $\zeta$ belongs to $Z^i$ or if $\zeta$ is an isolated point of
$\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)$, then $a\delta_\zeta$
belongs to $\mc F_{N}$. \end{example}
\begin{remark}\thlab{taylormehrdimrem}
Let $h$ be defined on an open subset $D$ of $\bb R^2$ with values in $\bb C$.
Moreover, assume that for given $m,n\in\bb N$ the function $h$ is
$m+n$ times continuously differentiable. Finally, fix $w\in D$.
The well-known Taylors Approximation Theorem from multidimensional calculus then yields
\[
h(z) = \sum_{j=0}^{m+n-1} \sum_{\stackrel{k,l\in \bb N_0}{k+l=j}}
\frac{1}{k!l!}
\frac{\partial^j h}{\partial x^{k} \partial y^{l}}(w)\RE(z-w)^k \IM(z-w)^{l}
+ O(|z-w|^{m+n})
\]
for $z\to w$. Since $|z-w|^{m+n} \leq 2^{m+n}\max(|\RE(z-w)|^{m+n}, |\IM(z-w)|^{m+n})
= O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n}))$ and since
$\RE(z-w)^k \IM(z-w)^{l} = O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n}))$ for
$k \geq m$ or $l\geq n$, we also have \begin{multline*}
h(z) = \sum_{k=0}^{m-1} \sum_{l=0}^{n-1} \frac{1}{k!l!}
\frac{\partial^{k+l} h}{\partial x^{k} \partial y^{l}}(w)\RE(z-w)^k \IM(z-w)^{l} +
\\ O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n})) \,. \end{multline*} \end{remark}
\begin{lemma}\thlab{gehzuF}
Let $f: \dom f \ (\subseteq \bb C^2)\to \bb C$ be a function with the properties
mentioned in \thref{feinbetef}. Then $f_N$ belongs to $\mc F_{N}$. \end{lemma} \begin{proof}
For fixed $w \in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ and
$z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$ by \thref{taylormehrdimrem}
the expression
\[
f_N(z) - \sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
f_N(w)_{k,l} \RE(z-w)^k \IM(z-w)^l =
\]
\[
f\circ\tau(z) -
\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\frac{1}{k!l!} \frac{\partial^{k+l} f\circ \tau}{\partial x^k\partial y^l}(w) \RE(z-w)^k \IM(z-w)^l
\]
is a $O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)}))$ for $z\to w$.
Therefore, $f_N\in \mc F_{N}$. \end{proof}
In order to be able to prove spectral results for our functional calculus, we need that with $\phi$ also $z\mapsto \phi(z)^{-1}$ belongs to $\mc F_N$ if $\phi$ is bounded away from zero.
\begin{lemma}\thlab{einduF}
If $\phi \in \mc F_{N}$ is such that $\phi(z)$ is invertible in $\mf C(z)$ (see \thref{z30f3}) for all
$z\in \big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i$
and such that
$0$ does not belong to the closure of
$\phi\big(\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \big)$, then
$\phi^{-1}: z\mapsto \phi(z)^{-1}$ also belongs to $\mc F_{N}$. \end{lemma} \begin{proof}
By the first assumption $\phi^{-1}$ is a well-defined object belonging to $\mc M_{N}$.
Clearly, with $\phi$ also $z\mapsto \phi(z)^{-1}=\frac{1}{\phi(z)}$ is
measurable on $\sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$.
By the second assumption $z\mapsto \phi(z)^{-1}=\frac{1}{\phi(z)}$ is bounded on this set.
It remains to verify the boundedness of \eqref{fn8qw3b} on a certain neighbourhood of $w$
for each $w \in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ for $\phi^{-1}$. To do so,
we calculate for $z\in \sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$
\begin{equation}\label{dfbz45pre}
\phi^{-1}(z) -
\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi^{-1}(w)_{k,l} \RE(z-w)^k \IM(z-w)^l =
\end{equation}
\begin{equation}\label{dfbz45}
\frac{1}{\phi(z)} -
\frac{1}{\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l} +
\end{equation} \begin{multline}\label{fght33}
\frac{1}{\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l} - \\
\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi^{-1}(w)_{k,l} \RE(z-w)^k \IM(z-w)^l \end{multline}
The expression in \eqref{dfbz45} can be written as
\begin{multline*}
\frac{1}{\phi(z)} \cdot \frac{1}{\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l} \cdot \\
\left( \phi(z) - \sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l \right)
\end{multline*}
Here $\frac{1}{\phi(z)}$ is bounded by assumption.
The assumed invertibility of $\phi(w)$ means $\phi(w)_{0,0}\neq 0$. Hence,
\[
\frac{1}{\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l} = O(1)
\]
for $z\to w$.
From $\phi\in \mc F_N$ we then conclude that \eqref{dfbz45} is a
$O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)}))$ for $z\to w$.
Because of $\phi(w)\cdot \phi^{-1}=e$ (see \thref{z30f3}), \eqref{fght33} can be rewritten as
\begin{multline*}
- \frac{1}{\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l}\cdot \\
\Big(
\sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\RE(z-w)^k \IM(z-w)^l \cdot \sum_{c=0}^k \sum_{d=0}^l \phi(w)_{c,d} \cdot \phi^{-1}(w)_{k-c,l-d}
\\ + O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)})) - 1\Big) =
\end{multline*}
\begin{multline*}
O(1) \cdot O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)})) = \\
O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)}))
\end{multline*}
for $z\to w$. Altogether \eqref{dfbz45pre} is a
$O(\max(|\RE(z-w)|^{\mf d_p(\RE w)}, |\IM(z-w)|^{\mf d_q(\IM w)}))$. Therefore,
$\phi^{-1}\in \mc F_N$. \end{proof}
\section{Functional Calculus}
In this section we employ the same assumptions and notation as in the previous one.
\begin{lemma}\thlab{aufspaltb}
For any $\phi\in \mc F_{N}$ there exists a polynomial $s\in \mathbb C[z,w]$
and a function $g$ on $\sigma(\Theta(N))$ with values in
$\bb C$ on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$
and values in $\bb C^2$ on $\sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ such that
$\phi - s_N\in \mc R$, such that
$g$ is bounded and measurable on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$, and
such that
\begin{equation}\label{dbew66290}
\phi(z) = s_N(z) +
(p_N + q_N) (z)\cdot g(z), \ z \in \sigma(\Theta(N)) \,,
\end{equation}
where the multiplication here has to be understood in the sense of \thref{durchdiv}. \end{lemma} \begin{proof}
According to \thref{existbe} there exists an $s\in \mathbb C[z,w]$ such that
$\phi - s_N \in \mc R$, and by \thref{durchdiv} we then find a function $g$
such that \eqref{dbew66290} holds true.
The measurability of
\[
g(z)=\frac{\phi(z) - s(\RE z,\IM z)}{p(\RE z)+q(\IM z)} \ \text{ on } \
\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)
\]
follows from the assumption $\phi\in\mc F_{N}$; see \thref{FdefklM}.
In order to show $g$'s boundedness, first recall from \thref{speknorm} that
\[
\max(|p(\RE z)|,|q(\IM z)|) \leq \max(\|R_1R_1^*\|,\|R_2R_2^*\|) \, |p(\RE z) + q(\IM z)|
\]
for $z\in \sigma(\Theta(N))$.
Hence,
\begin{multline*}
\frac{\max(|p(\RE z)|,|q(\IM z)|)}{|p(\RE z) + q(\IM z)|} \leq \max(\|R_1R_1^*\|,\|R_2R_2^*\|), \\
z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \,.
\end{multline*}
As $\phi\in \mc F_{N}$ we find for each $w\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$
an open neighbourhood $U(w)$ of $w$ such that \eqref{fn8qw3b} is bounded for $z \in U(w)\setminus \{w\}$.
Clearly, we can make the neighbourhoods $U(w)$ smaller so that they are pairwise disjoint.
Since for $w\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ the real number $\RE w$ ($\IM w$)
is a zero of $p(\RE z)$ ($q(\IM z)$) with multiplicity $\mf d_p(\RE w)$ ($\mf d_q(\IM w)$), we have
\[
c |\RE(z-w)|^{\mf d_p(\RE w)} \leq |p(\RE z)|, \ \
d |\IM(z-w)|^{\mf d_q(\IM w)} \leq |q(\IM z)|
\]
for $z \in U(w)$ with constants $c, d > 0$. Hence,
\[
\frac{\max(|\RE(z-w)|^{\mf d_p(\RE w)},|\IM(z-w)|^{\mf d_q(\IM w)})}{\max(|p(\RE z)|,|q(\IM z)|)} \leq C_w
\]
on $\sigma(\Theta(N))\cap U(w)\setminus \{w\}$ for some $C_w > 0$.
By what was said in \thref{taylormehrdimrem} and we also have
\begin{multline*}
s(\RE z,\IM z) = \sum_{k=0}^{\mf d_p(\RE w)-1}\sum_{l=0}^{\mf d_q(\IM w)-1}
\phi(w)_{k,l} \RE(z-w)^k \IM(z-w)^l + \\
O\big(\max(|\RE(z-w)|^{\mf d_p(\RE w)},|\IM(z-w)|^{\mf d_q(\IM w)})\big)
\,,
\end{multline*}
because $\phi - s_N \in \mc R$ implies $\phi(w)_{k,l} =
\frac{1}{k!l!} \frac{\partial^{k+l} s}{\partial x^k\partial y^l}(\RE w,\IM w)$.
Using the boundedness of \eqref{fn8qw3b} we altogether obtain the boundedness of
\begin{equation}\label{nurx549t6}
g(z) = \frac{\phi(z) - s(\RE z,\IM z)}{p(\RE z)+q(\IM z)} =
\end{equation} \begin{multline*}
\frac{\max(|p(\RE z)|,|q(\IM z)|)}{p(\RE z) + q(\IM z)} \cdot \\
\frac{\max(|\RE(z-w)|^{\mf d_p(\RE w)},|\IM(z-w)|^{\mf d_q(\IM w)})}{\max(|p(\RE z)|,|q(\IM z)|)} \cdot \\
\frac{\phi(z) - s(\RE z,\IM z)}{\max(|\RE(z-w)|^{\mf d_p(\RE w)},|\IM(z-w)|^{\mf d_q(\IM w)})} \end{multline*}
for $z\in \sigma(\Theta(N))\cap U(w)\setminus \{w\}$.
Since by \thref{speknorm} the function
$\frac{1}{p(\RE z)+q(\IM z)}$ is continuous, and hence
bounded on $\sigma(\Theta(N)) \setminus \bigcup_{w\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)} U(w)$,
we see that
\eqref{nurx549t6} is even bounded for $z\in \sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$. \end{proof}
\begin{definition}\thlab{funcaldef}
For any $\phi\in \mc F_{N}$ we define \[
\phi(N):= s(A,B) + \Xi\left( \int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE \right) \,, \]
where $s\in \bb C[z,w]$ and $g$ is a function on $\sigma(\Theta(N))$ with the properties mentioned in
\thref{aufspaltb}, and where \begin{multline*}
\int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE :=
\int_{\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)} g \, dE
\ + \\ \hspace*{5mm}
\sum_{w\in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)}
\big(g(w)_1 R_1R_1^* E\{w\} + g(w)_2 R_2R_2^* E\{w\}\big) \,. \end{multline*} \end{definition}
First we shall show that $\phi(N)$ is well defined.
\begin{theorem}\thlab{welldef}
Let $\phi \in \mc F_{N}$, $s, \tilde s\in \mathbb C[z,w]$ and
functions $g,\tilde g$ on $\sigma(\Theta(N))$ be given, such that
the assertion of \thref{aufspaltb} holds true for $s, g$ as well as for
$\tilde s, \tilde g$. Then \[
s(A,B) + \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE \right) =
\tilde s(A,B) + \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} \tilde g \, dE \right) \,. \] \end{theorem} \begin{proof}
By assumption we have $\phi - s_N, \phi - \tilde s_N \in \mc R$. Subtracting
these functions yields $\tilde s_N - s_N \in \mc R$. Using the notation of \thref{einbett2pre} this gives
$\varpi(\tilde s - s)_{\xi,\eta} = 0$ for $(\xi,\eta) \in p^{-1}\{0\}\times q^{-1}\{0\}$.
According to \thref{einbett2pre} we then get
\begin{equation}\label{opzerlsssch}
\tilde s(z,w) - s(z,w) = p(z) u(z,w) + q(w) v(z,w)
\end{equation}
for some $u(z,w), v(z,w) \in \bb C[z,w]$.
By \thref{Xidefeig} in \cite{KaPr2014} we have
\[
\Xi_1\big(u(\Theta_1(A),\Theta_1(B))\big) = \Xi_1\big(\Theta_1(u(A,B))\big) = p(A) u(A,B) \,,
\]
\[
\Xi_2\big(v(\Theta_2(A),\Theta_2(B))\big) = \Xi_2\big(\Theta_2(v(A,B))\big) = q(B) v(A,B) \,,
\]
where $\Xi_j, \ j=1,2$, are as defined in \eqref{xijdef}.
Since $u(\Theta_1(A),\Theta_1(B)) = \int u(\RE z,\IM z) \, dE_1(z)$, we get from \eqref{zuef2} \[
\Xi_1\big(u(\Theta_1(A),\Theta_1(B))\big) =
\Xi\big(R_1R_1^* \int u(\RE z,\IM z) \, dE(z)\big) \,. \]
Similarly, $\Xi_2\big(v(\Theta_2(A),\Theta_2(B))\big) = \Xi\big(R_2R_2^* \int v(\RE z,\IM z) \, dE(z)\big)$.
Therefore, employing \thref{korvda} we get \begin{multline}\label{tismis}
\tilde s(A,B) - s(A,B) = p(A) u(A,B) + q(B) v(A,B) =
\\ \Xi\big(R_1R_1^* \int u(\RE z,\IM z) \, dE(z) + R_2R_2^* \int v(\RE z,\IM z) \, dE(z)\big) = \end{multline} \begin{multline*}
\Xi\left( \int_{\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)} \hspace*{-5mm}
\frac{p(\RE z)u(\RE z,\IM z) + q(\IM z)v(\RE z,\IM z)}{p(\RE z)+q(\IM z)} \, dE(z) + \right. \\
\left. \sum_{w\in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)} \hspace*{-5mm}
\big( u(\RE w,\IM w) R_1R_1^* E\{w\} + v(\RE w,\IM w) R_2R_2^* E\{w\}\big)
\right)\,. \end{multline*}
On the other hand, since \eqref{dbew66290} holds true for $s,g$ and $\tilde s,\tilde g$, we have
\begin{equation}\label{kkwae}
(\tilde s_N -s_N)(z) = (p_N + q_N) (z)\cdot (g(z)-\tilde g(z)), \ z \in z \in \sigma(\Theta(N)) \,.
\end{equation}
For $z\in \sigma(\Theta(N)) \setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$ by \eqref{opzerlsssch} this means \begin{multline*}
p(\RE z) u(\RE z,\IM z) + q(\IM z) v(\RE z,\IM z)
= \\ \tilde s(\RE z,\IM z) - s(\RE z,\IM z) = (p(\RE z) + q(\IM z)) \cdot (g(z)-\tilde g(z)) \end{multline*}
and, in turn,
\[
g(z)-\tilde g(z) = \frac{p(\RE z) u(\RE z,\IM z) + q(\IM z) v(\RE z,\IM z)}{p(\RE z) + q(\IM z)} \,.
\]
Considering for $z\in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ the
entries of \eqref{kkwae} with indices $(\mf d_p(\RE z),0)$ and $(0,\mf d_q(\IM z))$
together with \eqref{opzerlsssch} we get \begin{multline*}
\frac{1}{\mf d_p(\RE z)!} \, p^{(\mf d_p(\RE z))}(\RE z) \, u(\RE z,\IM z) = \\
\frac{1}{\mf d_p(\RE z)!} \, \frac{\partial^{\mf d_p(\RE z)}}{\partial x^{\mf d_p(\RE z)}}
(\tilde s(\RE z,\IM z) - s(\RE z,\IM z)) = \\ \frac{1}{\mf d_p(\RE z)!} \, p^{(\mf d_p(\RE z))}(\RE z) \,
(g(z)_1-\tilde g(z)_1) \end{multline*}
and \begin{multline*}
\frac{1}{\mf d_q(\IM z)!} \, q^{(\mf d_q(\IM z))}(\IM z) \, v(\RE z,\IM z) = \\
\frac{1}{\mf d_q(\IM z)!} \, \frac{\partial^{\mf d_q(\IM z)}}{\partial y^{\mf d_q(\IM z)}}
(\tilde s(\RE z,\IM z) - s(\RE z,\IM z)) = \\ \frac{1}{\mf d_q(\IM z)!} \, q^{(\mf d_q(\IM z))}(\IM z)
\, (g(z)_2-\tilde g(z)_2) \end{multline*}
where we employed the product rule and the fact that
$p^{(k)}(\RE z) = 0 = q^{(l)}(\IM z)$ for $0\leq k < \mf d_p(\RE z), \, 0\leq l < \mf d_q(\IM z)$.
Since $p^{(\mf d_p(\RE z))}(\RE z)$ and $q^{(\mf d_q(\IM z))}(\IM z)$ do not vanish for
$z\in \sigma(\Theta(N)) \cap (Z^{\bb R}_p + i Z^{\bb R}_q)$, we get
$u(\RE z,\IM z) = g(z)_1-\tilde g(z)_1$ and $v(\RE z,\IM z) = g(z)_2-\tilde g(z)_2$.
Therefore, we can write \eqref{tismis} as
\[
\tilde s(A,B) - s(A,B) = \Xi\left( \int^{R_1,R_2}_{\sigma(\Theta(N))} \big(g-\tilde g\big) \, dE \right)\,,
\]
showing the asserted equality. \end{proof}
\begin{theorem}\thlab{mimalvertr}
The mapping $\phi\mapsto \phi(N)$ constitutes a $*$-homomorphism from
$\mc F_{N}$ into $\{N,N^*\}'' \ (\subseteq B(\mc K))$ with $s_N(N) = s(A,B)$ for all $s\in \bb C[z,w]$. \end{theorem} \begin{proof}
$s_N(N) = s(A,B)$ for all $s\in \bb C[z,w]$ follows from \thref{welldef} because
we have $s_N = s_N + (p_N + q_N)(z)\cdot 0, \, z \in \sigma(\Theta(N))$.
Assume that for $\phi,\psi \in\mc F_{N}$ we have $s,r\in \bb C[z,w]$ and functions $g,h$ on $\sigma(\Theta(N))$
such that $\phi - s_N, \psi-r_N\in \mc R$,
such that $g$ and $h$ are bounded and measurable on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$,
and such that \eqref{dbew66290} as well as
\[
\psi(z) = r_N(z) +
(p_N + q_N)(z)\cdot h(z), \ z \in \sigma(\Theta(N)) \,,
\]
hold true; see \thref{aufspaltb}. Then for $\lambda,\mu\in\bb C$ we get from \thref{bweuh30}
\[
(\lambda \phi + \mu \psi)(z) = (\lambda s + \mu r)_N(z) +
(p_N + q_N)(z)\cdot (\lambda g(z) + \mu h(z)), \ z \in \sigma(\Theta(N)) \,,
\]
where
$\lambda \phi + \mu \psi - (\lambda s + \mu r)_N =
\lambda(\phi-s_N) + \mu(\psi-r_N) \in \mc R$, and where
$\lambda g + \mu h$ is bounded and measurable on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$.
Since the definition of $\phi(N)$ in \thref{funcaldef} depends linearly on $s$ and $g$,
we conclude from \thref{welldef} that
\[
(\lambda \phi + \mu \psi)(N)=\lambda \phi(N) + \mu \psi(N) \,.
\]
Similarly, we get $\phi^\#(z) = (s^\#)_N(z) +
(p_N + q_N)(z)\cdot \bar g(z), \ z \in \sigma(\Theta(N))$; see \thref{bweuh30}.
Thereby $\phi^\# - (s^\#)_N = (\phi - s_N)^\#\in \mc R$ holds true
due to the fact that $\mf d_p(\xi)=\mf d_p(\bar \xi)$
and $\mf d_q(\eta)=\mf d_q(\bar \eta)$ for all $(\xi,\eta)\in Z^i$.
Since $\bar g$ is bounded and measurable on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$,
and since
\[
\phi(N)^*= s^\#(A,B) + \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} \bar g \, dE \right) \,,
\]
we again obtain from \thref{welldef} that $\phi^\#(N)=\phi(N)^*$.
Concerning the compatibility with $\cdot$, first note that by \thref{bweuh30}
\[
\phi(z) \cdot \psi(z) = (s\cdot r)_N(z) +
(p_N + q_N)(z)\cdot \omega(z), \ z \in \sigma(\Theta(N)) \,.
\]
Here we have $\omega(z) = s(z) h(z) + r(z) g(z) + g(z)h(z)( p(\RE z) + q(\IM z))$ for
$z\in \sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$
and $\omega(z)_j = s(z) g(z)_j + r(z) h(z)_j, \ j=1,2$ for
$z\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$ because
$a,b \in \ker \pi$ implies $a\cdot b=0$ and, in turn,
$(p_N + q_N)(z) \cdot (p_N + q_N)(z) = 0$ for
$z\in \sigma(\Theta(N))\cap (Z^{\bb R}_p + i Z^{\bb R}_q)$.
On the other hand, by \thref{Xidefeig} in \cite{KaPr2014}
we have $\Xi(D)C = \Xi(D\Theta(C))$, $C\Xi(D) = \Xi(\Theta(C)D)$, and
$\Xi(D_1)\Xi(D_2) = \Xi(D_1D_2 T^*T)$, where $T^*T=p(A) + q(B)$.
Hence,
\[
\phi(N) \ \psi(N) =
\]\[
s(A,B) \, r(A,B) + \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE \right) r(A,B) +
s(A,B) \, \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} h \, dE \right) +
\]\[
\Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE \right) \,
\Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} h \, dE \right)
=
\] \begin{multline*}
(s \cdot r)(A,B) +
\Xi\left(
\int^{R_1,R_2}_{\sigma(\Theta(N))} (g\cdot r + h \cdot s) \, dE + \right.
\\
\left. \int_{\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)}
\big(p(\RE(.)) + q(\IM(.))\big) \cdot h\cdot g \, dE\right) = \end{multline*}
\[
(s \cdot r)(A,B) + \Xi\left(\int^{R_1,R_2}_{\sigma(\Theta(N))} \omega \, dE\right) \,.
\]
Here $\omega$ is bounded and measurable on $\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q)$
and, using the fact that $\mc R$ is an ideal,
\[
\phi \cdot \psi - (s\cdot r)_N = (\phi-s_N)\cdot \psi + (\psi-r_N)\cdot s_N \in \mc R \,.
\]
Hence, we again obtain from
\thref{welldef} that $\phi(N)\cdot \psi(N)=\big(\phi\cdot\psi\big)(N)$.
Finally, we shall show that $\phi(N)\in \{N,N^*\}''$.
Clearly, $s(A,B) \in \{A,B\}''=\{N,N^*\}''$. If $C\in \{A,B\}' \subseteq \big(p(A)+q(B)\big)' = (TT^*)'$,
then $\Theta(C) \in \{\Theta(A),\Theta(B)\}'$ because $\Theta$ is a homomorphism. By the
spectral theorem for normal operators $\Theta(C)$ commutes with
\[
D:=\int^{R_1,R_2}_{\sigma(\Theta(N))} g \, dE \,.
\]
According to \thref{Xidefeig} in \cite{KaPr2014} we then get
\[
\Xi(D) C = \Xi(D\Theta(C)) = \Xi(\Theta(C) D) = C \Xi(D) \,.
\]
Hence, $\Xi(D) \in \{A,B\}''=\{N,N^*\}''$, and altogether $\phi(N) \in \{A,B\}''=\{N,N^*\}''$. \end{proof}
\begin{remark}\thlab{rieszproj} For $\zeta \in Z^i$ or for an isolated $\zeta \in \sigma(\Theta(N)) \cup (Z^{\bb R}_p + iZ^{\bb R}_q)$ we saw in \thref{fedela} that $a\delta_\zeta \in \mc F_{N}$. If $a$ is the unite $e \in \mf C(\zeta)$ (see \thref{z30f3}), then $(e\delta_\zeta)\cdot (e\delta_\zeta) = (e\delta_\zeta)$ together with \thref{mimalvertr} shows that $(e\delta_\zeta)(N)$ is a projection. It is a kind of Riesz projection corresponding to $\zeta$.
We set $\xi:=\RE \zeta, \ \eta:=\IM \zeta$ if $\zeta \in \sigma(\Theta(N)) \cup (Z^{\bb R}_p + iZ^{\bb R}_q)$ and $(\xi,\eta):=\zeta$ if $\zeta \in Z^i$. For $\lambda \in \bb C\setminus \{\xi+i\eta\}$ and for $s(z,w):=z+iw-\lambda$ we then have $s_N \cdot (e\delta_\zeta) = \big(s_N(\zeta)\big)\delta_\zeta$, where the entry $s(\xi,\eta)$ of $s_N(\zeta)$ with index $(0,0)$ does not vanish. By \thref{z30f3} it therefore has a multiplicative inverse $b\in \mf C(\zeta)$. We then obtain \[
s_N \cdot (e\delta_\zeta) \cdot (b\delta_\zeta) = e\delta_\zeta \,. \] From $s_N(N) = N - \lambda$ we then get that $N\vert_{\ran (e\delta_\zeta)(N)} - \lambda$ has $(b\delta_\zeta)(N)\vert_{\ran (e\delta_\zeta)(N)}$ as its inverse operator. Thus, $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) \subseteq \{\xi+i\eta\}$. \end{remark}
\begin{lemma}\thlab{deth56}
If for $\phi\in \mc F_{N}$ we have $\phi(z)= 0$
for all
\[
z \in \big(\sigma(\Theta(N)) \cup ((Z^{\bb R}_p +i Z^{\bb R}_q) \cap \sigma(N))\big)
\dot\cup \{(\alpha,\beta)\in Z^i: \alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\} \,,
\]
then $\phi(N)=0$. \end{lemma} \begin{proof}
Since any $\zeta \in (Z^{\bb R}_p +i Z^{\bb R}_q) \setminus \sigma(N)$ is isolated
in $\sigma(\Theta(N)) \cup (Z^{\bb R}_p + iZ^{\bb R}_q)$, we saw in \thref{rieszproj} that
for
\[
\zeta \in \underbrace{\big((Z^{\bb R}_p +i Z^{\bb R}_q) \setminus \sigma(N)\big)}_{=:Z_1} \dot\cup
\underbrace{\{(\alpha,\beta)\in Z^i: \alpha+i\beta \in \rho(N)\}}_{=:Z_2}
\]
the expression
$(e\delta_\zeta)(N)$ is a bounded projection commuting with $N$. Hence,
$(e\delta_\zeta)(N)$ also commutes $(N - (\xi+i\eta))^{-1}$, where
$\xi:=\RE \zeta, \ \eta:=\IM \zeta$ if $\zeta \in Z_1$ and
$(\xi,\eta):=\zeta$ if $\zeta \in Z_2$.
Consequently, $N\vert_{\ran (e\delta_\zeta)(N)}- (\xi+i\eta)$ is invertible on $\ran (e\delta_\zeta)(N)$,
i.e.\ $\xi+i\eta \not\in \sigma(N\vert_{\ran (e\delta_\zeta)(N)})$. By
\thref{rieszproj} we have $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) \subseteq \{\xi+i\eta\}$.
Hence, $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) = \emptyset$, which is impossible for
$\ran (e\delta_\zeta)(N) \neq \{0\}$. Thus, $(e\delta_\zeta)(N) = 0$.
For $(\xi,\eta)\in Z_3:=\{(\alpha,\beta)\in Z^i: \bar\alpha+i\bar\beta \in \rho(N)\}$
we get $(\bar \xi,\bar \eta)\in Z_2$. Hence,
\[
0=(e\delta_{(\bar\xi,\bar\eta)})(N)^*=(e\delta_{(\xi,\eta)})(N) \,.
\]
By our assumption $\phi$ is supported on $Z_1\cup Z_2 \cup Z_3$. Hence,
\[
\phi(N) = (\hspace{-2mm}\sum_{\zeta \in Z_1\cup Z_2 \cup Z_3} \hspace{-2mm}
\phi(\zeta)\delta_\zeta \hspace{+2mm})(N) =
\sum_{\zeta \in Z_1\cup Z_2 \cup Z_3} \phi(\zeta) (e\delta_\zeta)(N) = 0 \,.
\]
\end{proof}
\begin{remark}\thlab{fhwr34} As a consequence of \thref{deth56} for $\phi \in \mc F_{N}$ the operator $\phi(N)$ only depends on $\phi$'s values on
\begin{multline*}
\sigma_N:= \big(\sigma(\Theta(N)) \cup ((Z^{\bb R}_p +i Z^{\bb R}_q) \cap \sigma(N))\big) \dot\cup \\ \{(\alpha,\beta)\in Z^i:
\alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\}
\end{multline*} Thus, we can re-define the function class $\mc F_{N}$ for our functional calculus so that the elements $\phi$ of $\mc F_{N}$ are functions on this set with $\phi(z)\in \mf C(z)$ such that $z\mapsto \phi(z)$ is measurable and bounded on $\sigma(\Theta(N) \setminus (Z^{\bb R}_p +i Z^{\bb R}_q)$ and such that \eqref{fn8qw3b} is bounded locally at $w$ for all $w\in \sigma(\Theta(N) \cap (Z^{\bb R}_p +i Z^{\bb R}_q)$. \end{remark}
\begin{lemma}\thlab{wannboundinv}
If $\phi \in \mc F_{N}$
is such that $\phi(z)$ is invertible in $\mf C(z)$ (see \thref{z30f3}) for all
$z\in \sigma_N$
and such that
$0$ does not belong to the closure of
$\phi\big(\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \big)$, then
$\phi(N)$ is a boundedly invertible operator on $\mc K$. \end{lemma} \begin{proof}
We think of $\phi$ as a function on $\big(\sigma(\Theta(N)) \cup (Z^{\bb R}_p + i Z^{\bb R}_q)\big)\dot\cup Z^i$
by setting $\phi(z)=e$ (see \thref{z30f3}) for all $z$ not belonging to $\sigma_N$.
Then all assumptions of \thref{einduF} are satisfied. Hence $\phi^{-1} \in \mc F_{N}$, and we conclude from
\thref{mimalvertr} and \thref{fuinm0pre} that
\[
\phi^{-1}(N) \phi(N) = \phi(N) \phi^{-1}(N) = (\phi\cdot \phi^{-1})(N) = \mathds{1}_N(N) = I \,.
\] \end{proof}
\begin{corollary}\thlab{sigmaN}
If $N$ is a definitizable normal operator on the Krein space $\mc K$, then $\sigma(N)$
equals to
\begin{multline}\label{specform}
\sigma(\Theta(N)) \cup
((Z^{\bb R}_p +i Z^{\bb R}_q) \cap \sigma(N))
\cup \\ \{\alpha + i\beta : (\alpha,\beta)\in Z^i,
\alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\}
\end{multline} \end{corollary} \begin{proof}
Since $\Theta$ is a homomorphism, we have $\sigma(\Theta(N))\subseteq \sigma(N)$.
Hence, \eqref{specform} is contained in $\sigma(N)$.
For the converse, consider the polynomial $s(z,w) = z+iw - \lambda$ for a $\lambda$
not belonging to \eqref{specform}. We conclude that for any $z\in \sigma_N$ the first entry
$(s_N(z))_{0,0}$ of $s_N(z)\in \mf C(z)$ does not vanish, i.e.\ is invertible in $\mf C(z)$.
$(s_N(\sigma(\Theta(N))))_{0,0} = \sigma(\Theta(N)) - \lambda$ being compact,
$0$ does not belong to the closure of
$s_N\big(\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \big)$.
Applying \thref{wannboundinv} we see that $s_N(N)=(N-\lambda)$ is invertible. \end{proof}
\begin{corollary} For $\phi\in \mc F_{N}$ we have \[
\sigma(\phi(N)) \subseteq \overline{\phi(\sigma_N)_{0,0}} \,. \] \end{corollary} \begin{proof} For $\lambda\notin \overline{\phi(\sigma_N)_{0,0}}$ and any $z\in \sigma_N$ we have $(\phi(z)- \lambda \mathds{1}_N(z))_{0,0} = \phi(z)_{0,0} - \lambda \neq 0$. Hence $\phi(z)- \lambda \mathds{1}_N(z)$ is invertible in $\mf C(z)$.
Moreover, $0$ does not belong to the closure of $\phi\big(\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \big) -\lambda = (\phi- \lambda \mathds{1}_N)\big(\sigma(\Theta(N))\setminus (Z^{\bb R}_p + i Z^{\bb R}_q) \big)_{0,0}$. Therefore, we can apply \thref{wannboundinv} to $\phi- \lambda \mathds{1}_N$, and get $\lambda \in \rho(\phi(N))$. \end{proof}
\begin{remark}\thlab{prf55o1} For any characteristic function $\mathds{1}_\Delta$ of a Borel subset $\Delta\subseteq \bb C$ such that $(Z^{\bb R}_p +i Z^{\bb R}_q) \cap \sigma(N) \cap \partial_{\bb C} \Delta = \emptyset$ the function $(\mathds{1}_{\tau(\Delta)})_{N}$ belongs to $\mc F_N$; see \thref{feinbetef} and \thref{gehzuF}. Since this function is idempotent and satisfies $(\mathds{1}_{\tau(\Delta)})_{N}^\# = (\mathds{1}_{\tau(\Delta)})_{N}$, $(\mathds{1}_{\tau(\Delta)})_{N}(N)$ is a bounded and self-adjoint projection on the Krein space $\mc K$. These projections constitute the family of spectral projections for $N$. \end{remark}
\end{document}
|
arXiv
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NCERT Solutions for Class 11 Chemistry chapter 4 Chemical Bonding and Molecular Structure PDF - eSaral
NCERT Solutions for Class 11 Chemistry chapter 4 Chemical Bonding and Molecular Structure PDF
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Question 1: Explain the formation of a chemical bond.
Solution. A chemical bond is defined as an attractive force that holds the constituents (atoms, ions etc.) together in a chemical species.
Various theories have been suggested for the formation of chemical bonds such as the electronic theory, valence shell electron pair repulsion theory, valence bond theory, and molecular orbital theory.
each other and complete their respective octets or duplets to attain the stable configuration of the nearest noble gases. This combination can occur either by sharing of electrons or by transferring one or more electrons from one atom to another. The chemical bond formed as a result of sharing of electrons between atoms is called a covalent bond. An ionic bond is formed as a result of the transference of electrons from one atom to another
Question 2: Write Lewis dot symbols for atoms of the following elements: $\mathrm{Mg}, \mathrm{Na}, \mathrm{B}, \mathrm{O}, \mathrm{N}, \mathrm{Br}$.
Solution. Mg: There are two valence electrons in $\mathrm{Mg}$ atom. Hence, the Lewis dot symbol for $\mathrm{Mg}$ is: $\ddot{\mathrm{M}}_{\mathrm{g}}$
Na: There is only one valence electron in an atom of sodium. Hence, the Lewis dot structure is: Na-
B: There are 3 valence electrons in Boron atom. Hence, the Lewis dot structure is: $\cdot$ B.
Q: There are six valence electrons in an atom of oxygen. Hence, the Lewis dot structure is: :0:
$\underline{N}$ : There are five valence electrons in an atom of nitrogen. Hence, the Lewis dot structure is: $: \ddot{\mathrm{N}}$.
$\mathrm{Br}$ : There are seven valence electrons in bromine. Hence, the Lewis dot structure is: $: \ddot{\mathrm{Br}}$ :
Question 3: Write Lewis symbols for the following atoms and ions:
$\mathrm{S}$ and $\mathrm{S}^{2-} ; \mathrm{Al}$ and $\mathrm{Al}^{3+} ; \mathrm{H}$ and $\mathrm{H}^{-}$
Solution. \text { (i) } \mathrm{S} \text { and } \mathrm{S}^{2-}
The number of valence electrons in sulphur is 6 .
The dinegative charge infers that there will be two electrons more in addition to the six valence electrons. Hence, the Lewis dot symbol of $\mathrm{S}^{2-}$ is $[\ddot{\therefore}::]$
(ii) $\mathrm{Al}$ and $\mathrm{Al}^{3+}$
The number of valence electrons in aluminium is $3 .$
The Lewis dot symbol of aluminium (AI) is $^{\circ} \cdot$
The tripositive charge on a species infers that it has donated its three electrons. Hence, the Lewis dot symbol is $[\mathrm{Al}]^{3+}$
(iii) $\mathrm{H}$ and $\mathrm{H}^{-}$
The number of valence electrons in hydrogen is 1.
The Lewis dot symbol of hydrogen $(\mathrm{H})$ is $\mathrm{H} \cdot$.
The uninegative charge infers that there will be one electron more in addition to the one valence electron. Hence, the Lewis dot symbol is $[\ddot{H}]^{-}$
Question 4: Draw the Lewis structures for the following molecules and ions:
$\mathrm{H}_{2} \mathrm{~S}, \mathrm{SiCl}_{4}, \mathrm{BeF}_{2}, \mathrm{CO}_{3}^{2-}, \mathrm{HCOOH}$
Solution. $\mathrm{H}_{2} \mathrm{~S}$
$\mathrm{SiCl}_{4}$
BeF $_{2}$
$\mathrm{CO}_{3}^{2-}$
HCOOH
Question 5: Define octet rule. Write its significance and limitations.
Solution. The octet rule or the electronic theory of chemical bonding was developed by Kossel and Lewis. According to this rule, atoms can combine either by transfer of valence electrons from one atom to another or by sharing their valence electrons in order to attain the nearest noble gas configuration by having an octet in their valence shell.
The octet rule successfully explained the formation of chemical bonds depending upon the nature of the element.
Limitations of the octet theory:
The following are the limitations of the octet rule:
(a) The rule failed to predict the shape and relative stability of molecules.
(b) It is based upon the inert nature of noble gases. However, some noble gases like xenon and krypton form compounds such as $\mathrm{XeF}_{2}, \mathrm{KrF}_{2}$ etc.
(c) The octet rule cannot be applied to the elements in and beyond the third period of the periodic table. The elements present in these periods have more than eight valence electrons around the central atom. For example: $\mathrm{PF}_{5}, \mathrm{SF}_{6,}$ etc.
(d) The octet rule is not satisfied for all atoms in a molecule having an odd number of electrons. For example, $\mathrm{NO}$ and $\mathrm{NO}_{2}$ do not satisfy the octet rule.
Question 6: Write the favourable factors for the formation of ionic bond.
Solution. neutral atoms can lose or gain electrons. Bond formation also depends upon the lattice energy of the compound formed.
Hence, favourable factors for ionic bond formation are as follows:
(i) Low ionization enthalpy of metal atom.
(ii) High electron gain enthalpy $\left(\Delta_{e g} H\right)$ of a non-metal atom.
(iii) High lattice energy of the compound formed.
Question 7: Discuss the shape of the following molecules using the VSEPR model:
$\mathrm{BeCl}_{2}, \mathrm{BCl}_{3}, \mathrm{SiCl}_{4}, \mathrm{AsF}_{5}, \mathrm{H}_{2} \mathrm{~S}, \mathrm{PH}_{3}$
Solution. $\mathrm{BeCl}_{2}$
$\mathrm{Cl}: \mathrm{Be}: \mathrm{Cl}$
The central atom has no lone pair and there are two bond pairs. i.e., $\mathrm{BeCl}_{2}$ is of the type $\mathrm{AB}_{2}$. Hence, it has a linear shape.
$\mathrm{BCl}_{3}:$
The central atom has no lone pair and there are three bond pairs. Hence, it is of the type $\mathrm{AB}_{3}$. Hence, it is trigonal planar.
The central atom has no lone pair and there are four bond pairs. Hence, the shape of $\mathrm{SiCl}_{4}$ is tetrahedral being the $\mathrm{AB}_{4}$ type molecule.
$\mathrm{AsF}_{5}$ :
The central atom has no lone pair and there are five bond pairs. Hence, $A s F_{5}$ is of the type $A B_{5}$. Therefore, the shape is_trigonal bipyramidal.
$\mathrm{H}_{2} \mathrm{~S}:$
The central atom has one lone pair and there are two bond pairs. Hence, $\mathrm{H}_{2} \mathrm{~S}$ is of the type $\mathrm{AB}_{2} \mathrm{E}$. The shape is Bent.
$\mathrm{PH}_{3}:$
The central atom has one lone pair and there are three bond pairs. Hence, $\mathrm{PH}_{3}$ is of the $\mathrm{AB}_{3} \mathrm{E}$ type. Therefore, the shape is trigonal pyramidal.
Question 8: Although geometries of $\mathrm{NH}_{3}$ and $\mathrm{H}_{2} \mathrm{O}$ molecules are distorted tetrahedral, bond angle in water is less than that of ammonia. Discuss.
Solution. The molecular geometry of $\mathrm{NH}_{3}$ and $\mathrm{H}_{2} \mathrm{O}$ can be shown as:
The central atom ( $\mathrm{N}$ ) in $\mathrm{NH}_{3}$ has one lone pair and there are three bond pairs. In $\mathrm{H}_{2} \mathrm{O}$, there are two lone pairs and two bond pairs.
The two lone pairs present in the oxygen atom of $\mathrm{H}_{2} \mathrm{O}$ molecule repels the two bond pairs. This repulsion is stronger than the repulsion between the lone pair and the three bond pairs on the nitrogen atom.
Since the repulsions on the bond pairs in $\mathrm{H}_{2} \mathrm{O}$ molecule are greater than that in $\mathrm{NH}_{3}$, the bond angle in water is less than that of ammonia.
Question 9: How do you express the bond strength in terms of bond order?
Solution. Bond strength represents the extent of bonding between two atoms forming a molecule. The larger the bond energy, the stronger is the bond and the greater is the bond order.
Question 10: Define the bond length.
Solution. Bond length is defined as the equilibrium distance between the nuclei of two bonded atoms in a molecule. Bond lengths are expressed in terms of Angstrom $\left(10^{-10} \mathrm{~m}\right)$ or picometer
Bond lengths are expressed in terms of Angstrom $\left(10^{-10} \mathrm{~m}\right)$ or picometer
$\left(10^{-12} \mathrm{~m}\right)$ and are measured by spectroscopic X-ray diffractions and electron-diffraction techniques.
In an ionic compound, the bond length is the sum of the ionic radii of the constituting atoms $\left(d=r_{+}+r_{-}\right) .$ In a covalent compound, it is the sum of their covalent radii $\left(d=r_{\mathrm{A}}+r_{\mathrm{B}}\right)$
Question 11: Explain the important aspects of resonance with reference to the $\mathrm{CO}_{3}^{2-}$ ion.
Solution. According to experimental findings, all carbon to oxygen bonds in $\mathrm{CO}_{3}^{2-}$ are equivalent. Hence, it is inadequate to represent $\mathrm{CO}_{3}^{2-}$ ion by a single Lewis structure having two single bonds and one double bond.
Therefore, carbonate ion is described as a resonance hybrid of the following structures:
Question 12: $\mathrm{H}_{3} \mathrm{PO}_{3}$ can be represented by structures 1 and 2 shown below. Can these two structures be taken as the canonical forms of the resonance hybrid representing $\mathrm{H}_{3} \mathrm{PO}_{3} ?$ If not, give reasons for the same.
Solution. The given structures cannot be taken as the canonical forms of the resonance hybrid of $\mathrm{H}_{3} \mathrm{PO}_{3}$ because the positions of the atoms have changed.
Question 13: Write the resonance structures for $\mathrm{SO}_{3}, \mathrm{NO}_{2}$ and $\mathrm{NO}_{3}^{-}$.
Solution. The resonance structures are:
(a) $\mathrm{SO}_{3}$.
(b) $\mathrm{NO}_{2}$ :
(c) $\mathrm{NO}_{3}^{-}$ :
Question 14: Use Lewis symbols to show electron transfer between the following atoms to form cations and anions: (a) $\mathrm{K}$ and $\mathrm{S}$ (b) $\mathrm{Ca}$ and $\mathrm{O}$ (c) $\mathrm{Al}$ and $\mathrm{N}$.
Solution. (a) $\mathrm{K}$ and $\mathrm{S}$.
The electronic configurations of $\mathrm{K}$ and $\mathrm{S}$ are as follows:
$\mathrm{K}: 2,8,8,1$
$S: 2,8,6$
Sulphur (S) requires 2 more electrons to complete its octet. Potassium (K) requires one electron more than the nearest noble gas i.e., Argon. Hence, the electron transfer can be shown as:
(b) $\underline{\text { Ca and }} \mathrm{O}$ :
The electronic configurations of $\mathrm{Ca}$ and $\mathrm{O}$ are as follows:
Ca: $2,8,8,2$
$\mathrm{O}: 2,6$
Oxygen requires two electrons more to complete its octet, whereas calcium has two electrons more than the nearest noble gas i.e., Argon. Hence, the electron transfer takes place as:
(c) $\mathrm{Al}$ and $\mathrm{N}$ :
The electronic configurations of $\mathrm{Al}$ and $\mathrm{N}$ are as follows:
Al: $2,8,3$
$\mathrm{N}: 2,5$
Nitrogen is three electrons short of the nearest noble gas (Neon), whereas aluminium has three electrons more than Neon. Hence, the electron transference can be shown as:
Question 15: Although both $\mathrm{CO}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$ are triatomic molecules, the shape of $\mathrm{H}_{2} \mathrm{O}$ molecule is bent while that of $\mathrm{CO}_{2}$ is linear. Explain this on the basis of dipole moment.
Solution. According to experimental results, the dipole moment of carbon dioxide is zero. This is possible only if the molecule is linear so that the dipole moments of $\mathrm{C}-\mathrm{O}$ bonds are equal and opposite to nullify each other.
Resultant $\mu=0 \mathrm{D}$
$\mathrm{H}_{2} \mathrm{O}$, on the other hand, has a dipole moment value of $1.84 \mathrm{D}$ (though it is a triatomic molecule as $\mathrm{CO}_{2}$ ). The value of the dipole moment suggests that the structure of $\mathrm{H}_{2} \mathrm{O}$ molecule is bent where the dipole moment of O-H bonds are unequal.
Question 16: Write the significance/applications of dipole moment.
Solution. In heteronuclear molecules, polarization arises due to a difference in the electronegativities of the constituents of atoms. As a result, one end of the molecule acquires a The product of the magnitude of the charge and the distance between the centres of positive-negative charges is called the dipole moment $(\mu)$ of the molecule. It is a vector quantity and is represented by an arrow with its tail at the positive centre and head pointing towards a negative centre.
Dipole moment $(\mu)=$ charge $(\mathrm{Q}) \times$ distance of separation $(r)$
The SI unit of a dipole moment is 'esu'.
1 esu $=3.335 \times 10^{-30} \mathrm{Cm}$
Dipole moment is the measure of the polarity of a bond. It is used to differentiate between polar and non-polar bonds since all non-polar molecules (e.g. $\mathrm{H}_{2}, \mathrm{O}_{2}$ ) have zero dipole moments. It is also helpful in calculating the percentage ionic character of a molecule.
Question 17: Define electronegativity. How does it differ from electron gain enthalpy?
Solution. Electronegativity is the ability of an atom in a chemical compound to attract a bond pair of electrons towards itself.
Electronegativity of any given element is not constant. It varies according to the element to which it is bound. It is not a measurable quantity. It is only a relative number.
On the other hand, electron gain enthalpy is the enthalpy change that takes place when an electron is added to a neutral gaseous atom to form an anion. It can be negative or positive depending upon whether the electron is added or removed. An element has a constant value of the electron gain enthalpy that can be measured experimentally.
Question 18: Explain with the help of suitable example polar covalent bond.
Solution. When two dissimilar atoms having different electronegativities combine to form a covalent bond, the bond pair of electrons is not shared equally. The bond pair shifts towards the nucleus of the atom having greater electronegativity. As a result, electron distribution gets distorted and the electron cloud is displaced towards the electronegative atom.
As a result, the electronegative atom becomes slightly negatively charged while the other atom becomes slightly positively charged. Thus, opposite poles are developed in the molecule and this type of a bond is called a polar covalent bond.
$\mathrm{HCl}$, for example, contains a polar covalent bond. Chlorine atom is more electronegative than hydrogen atom. Hence, the bond pair lies towards chlorine and therefore, it acquires a partial negative charge.
Question 19: Arrange the bonds in order of increasing ionic character in the molecules: $\mathrm{LiF}, \mathrm{K}_{2} \mathrm{O}, \mathrm{N}_{2}, \mathrm{SO}_{2}$ and $\mathrm{CIF}_{3}$
Solution. The ionic character in a molecule is dependent upon the electronegativity difference between the constituting atoms. The greater the difference, the greater will be the ionic character of the molecule.
On this basis, the order of increasing ionic character in the given molecules is
$\mathrm{N}_{2}<\mathrm{SO}_{2}<\mathrm{CIF}_{3}<\mathrm{K}_{2} \mathrm{O}<\mathrm{LiF}$
Question 20: The skeletal structure of $\mathrm{CH}_{3} \mathrm{COOH}$ as shown below is correct, but some of the bonds are shown incorrectly. Write the correct Lewis structure for acetic acid.
Solution. The correct Lewis structure for acetic acid is as follows:
Question 21: Apart from tetrahedral geometry, another possible geometry for $\mathrm{CH}_{4}$ is square planar with the four $\mathrm{H}$ atoms at the corners of the square and the $\mathrm{C}$ atom at its centre. Explain why $\mathrm{CH}_{4}$ is not square planar?
Solution. Electronic configuration of carbon atom:
${ }_{6} \mathrm{C}: 1 \mathrm{~s}^{2} 2 s^{2} 2 p^{2}$
In the excited state, the orbital picture of carbon can be represented as:
Hence, carbon atom undergoes $s p^{3}$ hybridization in $\mathrm{CH}_{4}$ molecule and takes a tetrahedral shape.
For a square planar shape, the hybridization of the central atom has to be $d s p^{2}$. However, an atom of carbon does not have $d$ -orbitalsto undergo $d s p^{2}$ hybridization. Hence, the structure of $\mathrm{CH}_{4}$ cannot be square planar.
Moreover, with a bond angle of $90^{\circ}$ in square planar, the stability of $\mathrm{CH}_{4}$ will be very less because of the repulsion existing between the bond pairs. Hence, VSEPR theory also supports a tetrahedral structure for $\mathrm{CH}_{4}$.
Question 22: Explain why $\mathrm{BeH}_{2}$ molecule has a zero dipole moment although the $\mathrm{Be}-\mathrm{H}$ bonds are polar.
Solution. The Lewis structure for $\mathrm{BeH}_{2}$ is as follows:
$\mathrm{H}: \mathrm{Be}: \mathrm{H}$
There is no lone pair at the central atom (Be) and there are two bond pairs. Hence, $\mathrm{BeH}_{2}$ is of the type $\mathrm{AB}_{2}$. It has a linear structure.
Dipole moments of each H-Be bond are equal and are in opposite directions. Therefore, they nullify each other. Hence, $\mathrm{BeH}_{2}$ molecule has zero dipole moment.
Question 23: Which out of $\mathrm{NH}_{3}$ and $\mathrm{NF}_{3}$ has higher dipole moment and why?
Solution. In both molecules i.e., $\mathrm{NH}_{3}$ and $\mathrm{NF}_{3}$, the central atom (N) has a lone pair electron and there are three bond pairs. Hence, both molecules have a pyramidal shape. Since fluorine is more electronegative than hydrogen, it is expected that the net dipole moment of $\mathrm{NF}_{3}$ is greater than $\mathrm{NH}_{3}$. However, the net dipole moment of $\mathrm{NH}_{3}(1.46 \mathrm{D})$ is greater than that of $\mathrm{NF}_{3}(0.24 \mathrm{D})$.
This can be explained on the basis of the directions of the dipole moments of each individual bond in $\mathrm{NF}_{3}$ and $\mathrm{NH}_{3}$. These directions can be shown as
Thus, the resultant moment of the $\mathrm{N}-\mathrm{H}$ bonds add up to the bond moment of the lone pair (the two being in the same direction), whereas that of the three $\mathrm{N}-\mathrm{F}$ bonds partly cancels the moment of the lone pair
Hence, the net dipole moment of $\mathrm{NF}_{3}$ is less than that of $\mathrm{NH}_{3}$.
Question 24: What is meant by hybridisation of atomic orbitals? Describe the shapes of $s p, s p^{2}, s p^{3}$ hybrid orbitals.
Solution. Hybridization is defined as an intermixing of a set of atomic orbitals of slightly different energies, thereby forming a new set of orbitals having equivalent energies and shapes.
For example, one 2 s-orbital hybridizes with two $2 p$ -orbitals of carbon to form three new $s p^{2}$ hybrid orbitals.
These hybrid orbitals have minimum repulsion between their electron pairs and thus, are more stable. Hybridization helps indicate the geometry of the molecule
Shape of sp hybrid orbitals: sp hybrid orbitals have a linear shape. They are formed by the intermixing of $s$ and $p$ orbitals as
Shape of $s p^{2}$ hybrid orbitals:
$s p^{2}$ hybrid orbitals are formed as a result of the intermixing of one s-orbital and two $2 p$ -orbitals. The hybrid orbitals are oriented in a trigonal planar arrangement as:
Four $s p^{3}$ hybrid orbitals are formed by intermixing one s-orbital with three $p$ -orbitals. The four $s p^{3}$ hybrid orbitals are arranged in the form of a tetrahedron as:
Question 25: Describe the change in hybridisation (if any) of the Al atom in the following reaction.
$\mathrm{AlCl}_{3}+\mathrm{Cl}^{-} \longrightarrow \mathrm{AlCl}_{4}^{-}$
Solution. The valence orbital picture of aluminium in the ground state can be represented as:
The orbital picture of aluminium in the excited state can be represented as:
Hence, it undergoes $s p^{2}$ hybridization to give a trigonal planar arrangement (in $\mathrm{AlCl}_{3}$ ).
To form $\mathrm{AlCl}_{4}^{-}$, the empty $3 p_{\mathrm{z}}$ orbital also gets involved and the hybridization changes from $s p^{2}$ to $s p^{3} .$ As a result, the shape gets changed to tetrahedral.
Question 26: Is there any change in the hybridisation of $B$ and $N$ atoms as a result of the following reaction?
$\mathrm{BF}_{3}+\mathrm{NH}_{3} \rightarrow \mathrm{F}_{3} \mathrm{~B} \cdot \mathrm{NH}_{3}$
Solution. Boron atom in $\mathrm{BF}_{3}$ is $s p^{2}$ hybridized. The orbital picture of boron in the excited state can be shown as:
Nitrogen atom in $\mathrm{NH}_{3}$ is $s p^{3}$ hybridized. The orbital picture of nitrogen can be represented as:
After the reaction has occurred, an adduct $\mathrm{F}_{3} \mathrm{~B} \cdot \mathrm{NH}_{3}$ is formed as hybridization of ' $\mathrm{B}$ ' changes to $s p^{3}$. However, the hybridization of ' $\mathrm{N}$ ' remains intact.
Question 27: Draw diagrams showing the formation of a double bond and a triple bond between carbon atoms in $\mathrm{C}_{2} \mathrm{H}_{4}$ and $\mathrm{C}_{2} \mathrm{H}_{2}$ molecules
Solution. $\mathrm{C}_{2} \mathrm{H}_{4}:$
The electronic configuration of $\mathrm{C}$ -atom in the excited state is:
${ }_{6} \mathrm{C}=1 s^{2} 2 s^{\prime} 2 p_{x}^{\prime} 2 p_{y}^{\prime} 2 p_{z}^{\prime}$
In the formation of an ethane molecule $\left(\mathrm{C}_{2} \mathrm{H}_{4}\right)$, one $s p^{2}$ hybrid orbital of carbon overlaps a $s p^{2}$ hybridized orbital of another carbon atom, thereby forming a $\mathrm{C}-\mathrm{C}$ sigma bond.
The remaining two $s p^{2}$ orbitals of each carbon atom form a $s p^{2}-s$ sigma bond with two hydrogen atoms. The unhybridized orbital of one carbon atom undergoes sidewise overlap with the orbital of a similar kind present on another carbon atom to form a weak $\Pi$ -bond.
$\mathrm{C}_{2} \mathrm{H}_{2}:$
In the formation of $\mathrm{C}_{2} \mathrm{H}_{2}$ molecule, each C-atom is sp hybridized with two $2 p$ -orbitals in an unhybridized state.
One sp orbital of each carbon atom overlaps with the other along the internuclear axis forming a C-C sigma bond. The second sp orbital of each C-atom overlaps a half-filled 1 s-orbital to form a $\sigma$ bond
The two unhybridized $2 p$ -orbitals of the first carbon undergo sidewise overlap with the $2 p$ orbital of another carbon atom, thereby forming two pi ( $\pi)$ bonds between carbon atoms. Hence, the triple bond between two carbon atoms is made up of one sigma and two m-bonds.
Question 28: What is the total number of sigma and pi bonds in the following molecules?
(a) $\mathrm{C}_{2} \mathrm{H}_{2}$ (b) $\mathrm{C}_{2} \mathrm{H}_{4}$
Solution. A single bond is a result of the axial overlap of bonding orbitals. Hence, it contributes a sigma bond. A multiple bond (double or triple bond) is always formed as a result of the sidewise overlap of orbitals. A pi-bond is always present in it. A triple bond is a combination of two pi-bonds and one sigma bond.
Structure of $\mathrm{C}_{2} \mathrm{H}_{2}$ can be represented as:
Hence, there are three sigma and two pi-bonds in $\mathrm{C}_{2} \mathrm{H}_{2}$.
The structure of $\mathrm{C}_{2} \mathrm{H}_{4}$ can be represented as:
Hence, there are five sigma bonds and one pi-bond in $\mathrm{C}_{2} \mathrm{H}_{4}$.
Question 29: Considering $x$ -axis as the internuclear axis which out of the following will not form a sigma bond and why? (a) $1 s$ and $1 s$ (b) $1 s$ and $2 p_{x}$ (c) $2 p_{y}$ and $2 p_{y}$ (d) $1 s$ and $2 s$.
Solution. $2 p_{y}$ and $2 p_{y}$ orbitals will not a form a sigma bond. Taking $x$ -axis as the internuclear axis, $2 p_{y}$ and $2 p_{y}$ orbitals will undergo lateral overlapping, thereby forming a pi (\Pi) bond.
Question 30: Which hybrid orbitals are used by carbon atoms in the following molecules?
Solution. (a) $\mathrm{CH}_{3}-\mathrm{CH}_{3}$
(b) $\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}_{2}$
(c) $\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{OH}$
(d) $\mathrm{CH}_{3}-\mathrm{CHO}$
(e) $\mathrm{CH}_{3} \mathrm{COOH}$
Solution.(a)
Both $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are $s p^{3}$ hybridized.
$\mathrm{C}_{1}$ is $s p^{3}$ hybridized, while $\mathrm{C}_{2}$ and $\mathrm{C}_{3}$ are $s p^{2}$ hybridized.
$\mathrm{C}_{1}$ is $s p^{3}$ hybridized and $\mathrm{C}_{2}$ is $s p^{2}$ hybridized.
Question 31: What do you understand by bond pairs and lone pairs of electrons? Illustrate by giving one example of each type.
Solution. When two atoms combine by sharing their one or more valence electrons, a covalent bond is formed between them.
The shared pairs of electrons present between the bonded atoms are called bond pairs. All valence electrons may not participate in bonding. The electron pairs that do not participate in bonding are called lone pairs of electrons.
For example, in $\mathrm{C}_{2} \mathrm{H}_{6}$ (ethane), there are seven bond pairs but no lone pair present.
In $\mathrm{H}_{2} \mathrm{O}$, there are two bond pairs and two lone pairs on the central atom (oxygen).
Question 32: Distinguish between a sigma and a pi bond.
Solution. The following are the differences between sigma and pi-bonds:
Question 33: Explain the formation of $\mathrm{H}_{2}$ molecule on the basis of valence bond theory.
Solution. Let us assume that two hydrogen atoms (A and B) with nuclei $\left(\mathrm{N}_{\mathrm{A}}\right.$ and $\left.\mathrm{N}_{\mathrm{B}}\right)$ and electrons $\left(\mathrm{e}_{\mathrm{A}}\right.$ and $\mathrm{e}_{\mathrm{B}}$ ) are taken to undergo a reaction to form a hydrogen molecule.
When $A$ and $B$ are at a large distance, there is no interaction between them. As they begin to approach each other, the attractive and repulsive forces start operating.
Attractive force arises between:
(a) Nucleus of one atom and its own electron i.e., $N_{A}-e_{A}$ and $N_{B}-e_{B}$.
(b) Nucleus of one atom and electron of another atom i.e., $N_{A}-e_{B}$ and $N_{B}-e_{A}$.
Repulsive force arises between:
(a) Electrons of two atoms i.e., $e_{A}-e_{B}$.
(b) Nuclei of two atoms i.e., $\mathrm{N}_{\mathrm{A}}-\mathrm{N}_{\mathrm{B}}$.
The force of attraction brings the two atoms together, whereas the force of repulsion tends to push them apart.
The magnitude of the attractive forces is more than that of the repulsive forces. Hence, the two atoms approach each other. As a result, the potential energy decreases. Finally, a state is reached when the attractive forces balance the repulsive forces and the system acquires minimum energy. This leads to the formation of a dihydrogen molecule.
Question 34: Write the important conditions required for the linear combination of atomic orbitals to form molecular orbitals.
Solution. The given conditions should be satisfied by atomic orbitals to form molecular orbitals:
(a) The combining atomic orbitals must have the same or nearly the same energy. This means that in a homonuclear molecule, the 1 s-atomic orbital of an atom can combine with the $1 \mathrm{~s}$ -atomic orbital of another atom, and not with the $2 \mathrm{~s}$ -orbital.
(b) The combining atomic orbitals must have proper orientations to ensure that the overlap is maximum.
(c) The extent of overlapping should be large.
Question 35: Use molecular orbital theory to explain why the $\mathrm{Be}_{2}$ molecule does not exist.
Solution. The electronic configuration of Beryllium is $1 s^{2} 2 s^{2}$.
The molecular orbital electronic configuration for $\mathrm{Be}_{2}$ molecule can be written as:
$\begin{array}{cccc}\sigma^{2} & \sigma^{*} & \sigma^{2} & \sigma^{* 2} \\ 1 s & 1 s & 2 s & 2 s\end{array}$
Hence, the bond order for $\mathrm{Be}_{2}$ is $\frac{1}{2}\left(N_{b}-N_{a}\right)$.
$N_{b}=$ Number of electrons in bonding orbitals
$N_{a}=$ Number of electrons in anti-bonding orbitals
$\therefore$ Bond order of $\mathrm{Be}_{2}=\frac{1}{2}(4-4)=0$
A negative or zero bond order means that the molecule is unstable. Hence, $\mathrm{Be}_{2}$ molecule does not exist.
Question 36: Compare the relative stability of the following species and indicate their magnetic properties;
$\mathrm{O}_{2}, \mathrm{O}_{2}^{+}, \mathrm{O}_{2}^{-}$ (superoxide), $\mathrm{O}_{2}^{2-}$ (peroxide)
Solution. There are 16 electrons in a molecule of dioxygen, 8 from each oxygen atom. The electronic configuration of oxygen molecule can be written as:
Since the 1 s orbital of each oxygen atom is not involved in boding, the number of bonding electrons $=8=N_{\mathrm{b}}$ and the number of anti-bonding orbitals $=4=N_{\mathrm{a}}$
Bond order $=\frac{1}{2}\left(N_{\mathrm{b}}-N_{\mathrm{a}}\right)$
$=\frac{1}{2}(8-4)$
$=2$
Similarly, the electronic configuration of $\mathrm{O}_{2}^{+}$ can be written as:
$N_{\mathrm{b}}=8$
$N_{\mathrm{a}}=3$
Bond order of $\mathrm{O}_{2}^{+}=\frac{1}{2}(8-3)$
$=2.5$
Electronic configuration of $\mathrm{O}_{2}^{-}$ ion will be:
Bond order of $\mathrm{O}_{2}^{-}=\frac{1}{2}(8-5)$
Electronic configuration of $\mathrm{O}_{2}^{2-}$ ion will be:
Bond order of $\mathrm{O}_{2}^{2-}=\frac{1}{2}(8-6)$
Bond dissociation energy is directly proportional to bond order. Thus, the higher the bond order, the greater will be the stability. On this basis, the order of stability is $\mathrm{O}_{2}^{+}>\mathrm{O}_{2}>\mathrm{O}_{2}^{-}>\mathrm{O}_{2}^{2}$
Question 37: Write the significance of a plus and a minus sign shown in representing the orbitals.
Solution. Molecular orbitals are represented by wave functions. A plus sign in an orbital indicates a positive wave function while a minus sign in an orbital represents a negative wave function.
Question 38: Describe the hybridisation in case of $\mathrm{PCl}_{5}$. Why are the axial bonds longer as compared to equatorial bonds?
Solution. The ground state and excited state outer electronic configurations of phosphorus $(Z=15)$ are:
Phosphorus atom is $s p^{3} d$ hybridized in the excited state. These orbitals are filled by the electron pairs donated by five $\mathrm{Cl}$ atoms as:
$\mathrm{PCl}_{5}$
The five $s p^{3} d$ hybrid orbitals are directed towards the five corners of the trigonal bipyramidals. Hence, the geometry of $\mathrm{PCl}_{5}$ can be represented as:
There are five $P-C l$ sigma bonds in $\mathrm{PCl}_{5}$. Three P-Cl bonds lie in one plane and make an angle of $120^{\circ}$ with each other. These bonds are called equatorial bonds.
The remaining two P-Cl bonds lie above and below the equatorial plane and make an angle of $90^{\circ}$ with the plane. These bonds are called axial bonds
As the axial bond pairs suffer more repulsion from the equatorial bond pairs, axial bonds are slightly longer than equatorial bonds.
Question 39: Define hydrogen bond. Is it weaker or stronger than the van der Waals forces?
Solution. A hydrogen bond is defined as an attractive force acting between the hydrogen attached to an electronegative atom of one molecule and an electronegative atom of a different molecule (may be of the same kind).
Due to a difference between electronegativities, the bond pair between hydrogen and the electronegative atom gets drifted far away from the hydrogen atom. As a result hydrogen atom becomes electropositive with respect to the other atom and acquires a positive charge.
The magnitude of H-bonding is maximum in the solid state and minimum in the gaseous state.
(i) Intermolecular H-bond e.g., $\mathrm{HF}, \mathrm{H}_{2} \mathrm{O}$ etc.
(ii) Intramolecular H-bond e.g., o-nitrophenol
Hydrogen bonds are stronger than Van der Walls forces since hydrogen bonds are regarded as an extreme form of dipole-dipole interaction.
Question 40: What is meant by the term bond order? Calculate the bond order of: $\mathrm{N}_{2}, \mathrm{O}_{2}, \mathrm{O}_{2}^{+}$ and $\mathrm{O}_{2}^{-}$.
Solution. Bond order is defined as one half of the difference between the number of electrons present in the bonding and anti-bonding orbitals of a molecule.
If $N_{\mathrm{a}}$ is equal to the number of electrons in an anti-bonding orbital, then $N_{\mathrm{b}}$ is equal to the number of electrons in a bonding orbital.
If $N_{\mathrm{b}}>N_{\mathrm{a}}$, then the molecule is said be stable. However, if $N_{\mathrm{b}} \leq N_{\mathrm{a}}$, then the molecule is considered to be unstable.
Bond order of $\mathrm{N}_{2}$ can be calculated from its electronic configuration as:
Number of bonding electrons, $N_{\mathrm{b}}=10$
Number of anti-bonding electrons, $N_{\mathrm{a}}=4$
Bond order of nitrogen molecule $=\frac{1}{2}(10-4)$
There are 16 electrons in a dioxygen molecule, 8 from each oxygen atom. The electronic configuration of oxygen molecule can be written as:
Since the $1 s$ orbital of each oxygen atom is not involved in boding, the number of bonding electrons $=8=N_{\mathrm{b}}$ and the number of anti-bonding electrons $=4=N_{\mathrm{a}}$
Hence, the bond order of oxygen molecule is $2 .$
Thus, the bond order of $\mathrm{O}_{2}^{+}$ is $2.5$.
The electronic configuration of $\mathrm{O}_{2}^{-}$ ion will be:
Thus, the bond order of $\mathrm{O}_{2}^{-}$ ion is $1.5$.
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René Thom
René Frédéric Thom (French: [ʁəne tɔm]; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
René Thom
Thom in 1970
Born(1923-09-02)2 September 1923
Montbéliard, France
Died25 October 2002(2002-10-25) (aged 79)
Bures-sur-Yvette, France
Alma materÉcole Normale Supérieure, University of Paris
Known forCatastrophe theory
Cobordism
Gradient conjecture
Quasi-fibration
Splitting lemma
Thom conjecture
Thom isomorphism
Thom space
Thom transversality theorem
Thom's first isotopy lemma
Thom–Porteous formula
Thom–Sebastiani Theorem
Dold–Thom theorem
ChildrenFrançoise Thom
AwardsFields Medal (1958)
Brouwer Medal (1970)
John von Neumann Lecture Prize (1976)
Scientific career
FieldsMathematics
InstitutionsUniversity of Strasbourg
Université Joseph Fourier
Institut des Hautes Études Scientifiques
ThesisEspaces fibrés en sphères et carrés de Steenrod (1951)
Doctoral advisorHenri Cartan
Doctoral studentsDavid Trotman
InfluencedVladimir Arnold[1]
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Christopher Zeeman).[2][3][4][5][6]
Life and career
René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946.[7]
He received his PhD in 1951 from the University of Paris. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan.[8]
After a fellowship at Princeton University Graduate College (1951-1952), he became Maître de conférences at the Universities of Grenoble (1953–1954) and Strasbourg (1954–1963), where he was appointed Professor in 1957. In 1964 he moved to the Institut des Hautes Études Scientifiques, in Bures-sur-Yvette, where he worked until 1990.[9]
In 1958 Thom received the Fields Medal at the International Congress of Mathematicians in Edinburgh for the foundations of cobordism theory, which were already present in his thesis.[10] He was invited speaker at the International Congress of Mathematicians two more times: in 1970 in Nice[11] and 1983 in Warsaw (which he did not attend).[12]
He was awarded the Brouwer Medal in 1970,[13] the Grand Prix Scientifique de la Ville de Paris in 1974, and the John von Neumann Lecture Prize in 1976.[14] He become the first president, together with Louis Néel, of the newly established Fondation Louis-de-Broglie In 1973 [15] and was elected Member of the Académie des Sciences of Paris in 1976.[16]
Salvador Dalí paid homage to René Thom with the paintings The Swallow's Tail and Topological Abduction of Europe.[17]
Research
While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972,[18] his academic achievements concern mostly his mathematical work on topology.[19][20]
In the early 1950s it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 1950s he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom–Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set.
Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969–70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.[21]
During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science. In 1992, he was one of eighteen academics who sent a letter to Cambridge University protesting against plans to award Jacques Derrida an honorary doctorate.[22]
Beyond Thom's contributions to algebraic topology, he studied differentiable mappings, through the study of generic properties. In his final years, he turned his attention to an effort to apply his ideas about structural topography to the questions of thought, language, and meaning in the form of a "semiophysics".
Bibliography
• Thom, René (1952), "Espaces fibrés en sphères et carrés de Steenrod" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 3, 69: 109–182, doi:10.24033/asens.998, MR 0054960
• Thom, René (1954), "Quelques propriétés globales des variétés différentiables", Commentarii Mathematici Helvetici, 28: 17–86, doi:10.1007/BF02566923, MR 0061823, S2CID 120243638
• "Ensembles et morphismes stratifiés", Bulletin of the American Mathematical Society 75 (1969), 240–284.
• "Semio Physics: A Sketch", Addison Wesley, (1990), ISBN 0-201-50060-4
• Structural Stability and Morphogenesis, W. A. Benjamin, (1972), ISBN 0-201-40685-3.
See also
• "Quelques propriétés globales des variétés differentiables"
• Reeb graph
References
1. "Archived copy" (PDF). Archived from the original (PDF) on 14 July 2015. Retrieved 22 February 2015.{{cite web}}: CS1 maint: archived copy as title (link)
2. O'Connor, John J.; Robertson, Edmund F., "René Thom", MacTutor History of Mathematics Archive, University of St Andrews
3. Wright, Pearce (2002-11-14). "Obituary: René Thom". The Guardian. Retrieved 2022-04-10. also available at "René Thom - Guardian obituary". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 2022-04-10.
4. "René Frédéric Thom". encyclopedia.com. Retrieved 2022-04-10.
5. Alberganti, Michel (2002-10-31). "René Thom". Le Monde (in French). Retrieved 2022-04-10.
6. "Thom René Frédéric". serge.mehl.free.fr. Retrieved 2022-04-10.
7. Dougnac, Sophie (30 July 2015). "René Thom: le fils d'épiciers devient prix Nobel" [René Thom: the grocers' son becomes Nobel prize] (in French). L'Est Républicain. Retrieved 2022-04-10.
8. "René Thom - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2022-04-10.
9. "René Thom, permanent professor from 1963 to 1990 - IHES". www.ihes.fr. Retrieved 2022-04-10.
10. Todd, John Arthur, ed. (1960). Proceedings of the International Congress of Mathematician 1958 (PDF). Cambridge: Cambridge University Press. pp. 248–255.
11. Proceedings of the International Congress of Mathematician 1970 (PDF) (in French). Paris: Gauthier-Villars. 1971. pp. 257–265.
12. Ciesielski, Zbigniew; Olech, Czeslaw, eds. (1984). Proceedings of the International Congress of Mathematician 1983 (PDF). Warsaw: Polish Scientific Publishers PWN. pp. XVI.
13. "The Brouwer Lecture and the Brouwer Medal". 2017-05-10. Archived from the original on 10 May 2017. Retrieved 2022-04-10.
14. "SIAM: The John von Neumann Lecture". Society for Industrial and Applied Mathematics. Retrieved 2022-04-10.
15. "Fondation Louis de Broglie". fondationlouisdebroglie.org. Retrieved 2022-04-10.
16. Connes, Alain. "René Thom - Les Membres de l'Académie des sciences". Académie des Sciences. Retrieved 2022-04-10.
17. Andrew, Masterson (2018-01-16). "René Thom: Dalí's favourite mathematician". Cosmos. Retrieved 2022-04-10.
18. E.C. Zeeman, Catastrophe Theory, Scientific American, April 1976; pp. 65–70, 75–83
19. Hopf, Heinz (1960). The Work of R. Thom (PDF) (in German). Cambridge: Cambridge University Press. pp. X–XIV.
20. "René Thom - Scholars". Institute for Advanced Study. 2019-12-09. Retrieved 2022-04-10.
21. Gibson, Christopher G.; Wirthmüller, Klaus; Du Plessis, Andrew; Looijenga, E. (1976). Topological stability of smooth mappings. Berlin: Springer-Verlag. ISBN 3-540-07997-1. OCLC 2705384.
22. "Derrida Letter, The Cambridge Affair, 1992".
• Petitot, Jean, ed. (1996). Logos et Théorie des Catastrophes: à partir de l'oeuvre de René Thom. Colloque de Cerisy-la-Salle 1982. Geneva: Patiño. ISBN 978-2-88213-010-5.
• Aubin, David (2004). "Forms of Explanations in the Catastrophe Theory of René Thom: Topology, Morphogenesis, and Structuralism" (PDF). In Wise, M. N. (ed.). Growing Explanations: Historical Perspective on the Sciences of Complexity. Durham: Duke University Press. pp. 95–130.
• Reilly, Brian J. (2006). "René Thom". In Kritzman, Lawrence D. (ed.). The Columbia History of Twentieth-Century French Thought. New York: Columbia University Press. pp. 663–666. ISBN 978-0-231-10791-4.
• Weil, Martin (November 17, 2002). "French Mathematician René Thom Dies". Washington Post. p. C10.
• Papadopoulos, Athanase (2018). "René Thom: Portrait Mathématique et philosophique". CNRS Editions, Paris. ISBN 978-2-271-11827-1.
External links
Wikimedia Commons has media related to René Thom.
• O'Connor, John J.; Robertson, Edmund F., "René Thom", MacTutor History of Mathematics Archive, University of St Andrews
• Washington Post Online edition (free registration)
• Meeting René THOM
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Farmers' self-reported value of cooperative membership: evidence from heterogeneous business and organization structures
Eeva Alho1Email author
Agricultural and Food Economics20153:23
© Alh. 2015
Received: 20 January 2015
The value of membership of an agricultural producer cooperative to a farmer is universally understood to include market access, improved bargaining power, and reduced transaction costs. As a result of consolidation in agriculture, many farmers in developed countries have found themselves in complex cooperative structures in which market orientation may elevate capital-related membership benefits over the traditional patronage and farming-related benefits. This study utilized the heterogeneity in producer organization structures to examine the significance to farmers of membership in modern agricultural cooperatives. Survey data including 682 Finnish agricultural producers in the livestock sector enabled the subjective value of cooperative membership and the relationship with transaction cost benefits to be analyzed. The effect of vertical integration in cooperatives on the self-reported value of membership benefits was assessed with a sample consisting of members in three types of cooperative organizations: dairy marketing, dairy supply, and meat cooperatives. The findings confirm that a stable market channel is still the most important benefit that producers perceive as deriving from cooperative membership. Multivariate ordered probit analysis indicated that the market channel is equally appreciated by large and small producers, but the reduced uncertainty brought by a cooperative buyer is particularly valuable to farmers who are investing in farm expansion. The survey findings indicate that the more competition for the raw material from producers there is in an area, the greater is the pressure cooperatives may be under to develop their service offering in order to attract members.
Agricultural cooperatives
Ordered probit
Innovations in the forms of producer organizations have emerged over recent decades (van Bekkum and Bijman 2006; Hendrikse and Bijman 2002). The relaxation of restrictions on the residual claims of agricultural cooperatives has been a response to the competitive pressures (Chaddad and Cook 2004). Meat production, in particular, has undergone major structural changes in Europe. The pig meat market is highly integrated, both in production and in the processing industry (Pyykkönen et al. 2012). The consolidation process has led to producer cooperatives adopting hybrid business structures, meaning the transformation of cooperatives towards investor-owned firms (IOF), when the cooperatives establish limited liability companies (Bijman et al. 2012). This process is typically motivated by the need to attract risk capital. Cooperatives are owned and controlled by the patrons. Member ownership and control imply that the choice of the organizational form reflects the decisions taken by the members based on their perception of which structure brings them the most value.
Heterogeneity in agricultural producer organizations leads to the question of which factors constitute benefits for the members in modern farmers' cooperatives. Our research strategy was to distinguish preferences at the level of cooperative types without going into farmer-level differences and individual-specific factors that produce differences in preferences. The focus on cooperative types in this paper addresses the topical question of how cooperatives have to change in terms of their member satisfaction and benefit delivery strategies when their cooperative structures change. We utilized a farmer questionnaire to examine the value of agricultural cooperative membership to Finnish milk and meat producers. The survey data, including 682 Finnish agricultural producers in the livestock sector, enabled an analysis of the relationship between the subjective value of cooperative membership and farmer-specific variables. We were also interested in whether the self-reported significance of cooperative benefits to farmers is related to vertically integrated structures. Differences in responses were analyzed with respect to three organization types: marketing, supply, and hybrid cooperatives.
Varying the degree of vertical integration means that a member of either of the first two cooperative types is positioned differently from a member of the other cooperative type. Moreover, moving further from dairy marketing and supply cooperative types to a hybrid reflects the structural shift from traditional cooperative forms towards IOF-like structures. With traditional forms, we refer to the definition of ownership rights in Chaddad and Iliopoulos (2013), in which a traditional cooperative is characterized by residual return rights assigned to member-patrons, ownership is related to patronage, voting rights are non-proportional, and the residual claim is not transferable.
The following literature review discusses the relevant background theories and empirical evidence and builds the research hypotheses. The paper proceeds by presenting the survey data and the methods. The results section presents the survey findings on the perceptions of membership benefits among farmers and the ordered probit analysis of the effect of background variables, after which the final section concludes and discusses the managerial implications.
Background literature and hypotheses
The existence of cooperatives is traditionally explained as substituting for market failures, serving as a market channel, and in the agricultural sector in particular, bringing countervailing power to farmers (Sykuta and Cook 2001; Hendrikse and Bijman 2002). By organizing market access through a cooperative, farmers are able to benefit from lower costs than they would face by bargaining independently with buyers (Staatz 1987). Transaction cost factors are also present in modern agriculture, which may explain the importance of producer cooperatives in the European food supply chain (Valentinov 2007).
Transaction costs are affected by uncertainty, the frequency of transactions, and asset specificity (Williamson 1989; Ménard 2004). Due to its perishability, a dairy farmer is dependent on the frequent and timely collection of milk from the farm, whereas meat is more flexible as a production type (Masten 2000). The obligation of dairy cooperatives to collect all of the milk produced is a valuable membership benefit that economizes on transaction costs and protects specific assets. Consequently, we hypothesized that in our sample, the dairy producers would place greater significance on the cooperative as a stable market channel than the meat producers (hypothesis 1). Large producers may be more dependent on the cooperative as a market channel due to larger asset specificity and potentially large contracting costs if they have to negotiate with several buyers. On the other hand, large producers may have more options due to their better bargaining position and reduced contracting costs for the buyer. In this light, large producers can be hypothesized to be less dependent on a particular cooperative buyer, as Chechin et al. (2013) point out. For small producers, the cooperative offers a safe transaction relationship, and they gain from an improved bargaining position. As the investments in asset specificity and the bargaining power explanations lead to contradicting predictions on the relationship between farm size and the perceived value of a cooperative buyer, we only built a hypothesis on the bargaining power being relatively more valued by small than large producers (hypothesis 2).
Empirical evidence indicates that transaction costs influence farmers' decisions to join and deliver their production to a cooperative. Pascucci et al. (2012) observed that the dependency of farmers on cooperatives increases in relation to the size of their total assets. Their finding suggests that commitment to delivering to a cooperative may strengthen among those farmers who invest a considerable amount in their own production. According to Hernández-Espallardo et al. (2013), satisfying farmer members is crucial to the survival of agricultural marketing cooperatives. They observed the perceived transaction costs to be a more important factor in creating member satisfaction than the producer price.
According to Hansmann (1988), an organizational form emerges that minimizes the transaction costs, and ownership costs explain why ownership rights are assigned to particular patrons. In this framework, the emergence of non-traditional organization models of farmer-owner cooperatives is a result of minimizing of the costs of ownership. Evidence from the field indicates that producer organizations typically aim at finding a model that retains the cooperative form and ideology but enables access to non-member equity capital (van Bekkum and Bijman 2006). Gaining access to growth capital from investors has for many been the decisive factor in departing from the traditional cooperative structure (Chaddad and Iliopoulos 2013). The polar opposite to the traditional cooperative in the typology of Chaddad and Cook (2004) is the transformation into an IOF. The majority of farmer respondents in the survey of Alsemgeest and Smit (2012) perceived profit maximization as the goal currently strived for by agricultural businesses, whereas the provision of competitive services to farmers and improving farm profitability, which were voiced as the main goals, were not in the business focus.
While milk producer cooperatives in Finland represent the traditional cooperative organizational form, i.e., their organization conforms to the cooperative principles of equal treatment of members, equal voting rights, and unallocated capital (Nilsson et al. 2009), large meat cooperatives have adopted vertically integrated hybrid structures in which ownership and control rights are separated from patronage (Pyykkönen et al. 2012). The role of the meat cooperatives is exclusively to exercise ownership and control rights in the stock exchange listed processing and marketing company. We expected the members of meat cooperatives to value the price and capital benefits more highly than the members of dairy cooperatives (hypothesis 3).
Surveys were conducted among selected Finnish milk and meat producer cooperatives. The sample was designed to encompass the heterogeneity in producer cooperative structures within Finnish agriculture. Finnish meat producer cooperatives have transformed into holding companies in which the farmers are the owners and members, but they deliver their production to a separate subsidiary corporation. The businesses of processing and marketing further downstream are incorporated in a stock-listed company. As a consequence, our sample of meat producers of the two large Finnish meat cooperatives represented so-called hybrid cooperative structures, which adopt organizational structures similar to those of investor-owned firms. Alternatively, the term IOF-like cooperative could be used. Farmers' organizations have a majority ownership in the firm, but other non-member owners can also invest in the firm's stock in a stock exchange. Altogether, these two meat cooperatives had 3259 members at the end of 2013. The Finnish meat producer population consists of 3500 beef farms, 1540 pig meat farms and 568 poultry farms, i.e., a total of 5608 meat producer farms (Niemi and Ahlstedt, 2013). Therefore, our sample covered a large proportion of the population, and it was also geographically representative, as the two cooperatives operate in the main meat production areas of Finland: cattle in northern and eastern Finland and pigs in southern and western Finland.
Conversely, Finnish dairy cooperatives operate in a traditional agricultural cooperative form, and two variants exist. The market is divided into dairy cooperatives that are part of the Valio Group, in which the role of farmers is to deliver milk and to indirectly exercise ownership in the processing company through their cooperative membership (supply cooperatives). On the other side are dairy cooperatives that are independent of the Valio Group, which take care of the whole chain from milk processing to the marketing of products under their own brand name (marketing cooperatives). Our sample included two large supply cooperatives (i.e., owners of Valio group) and three smaller independent dairy cooperatives, which are referred to as marketing cooperatives. These definitions used in Finland are slightly different from the cooperative types in many other countries. For example in the USA and Canada, supply agricultural cooperatives are those that supply their farmer-members with farm inputs and related services. Here the term supply refers to the supply of farmers' produce to the cooperative. Marketing cooperatives are generally defined as organizations that collect, process, package, and market the farmers' produce.
The total number of dairy producers in Finland was 8373 at the beginning of 2015 (The Central Union of Agricultural Producers and Forest Owners, MTK). The dairy cooperatives included in the sample had a total of 2408 members. Composing the sample of two large dairy supply cooperatives and three marketing cooperatives provided a representative sample in terms of both geography and cooperative type. Comparison of the summary statistics for our sample with the national farmer statistics (Natural Resources Institute of Finland) corroborates the representativeness of the sample in terms of farm and farmer characteristics.
The questionnaires were delivered by mail in February 2014. The response rates were 16.8 % in the milk producer survey, yielding 406 farmers in the final sample, and 14.3 % in the meat producer survey, which yielded 276 meat farmers. Table 1 presents the farmer characteristics of milk and meat producers separately for members of the dairy supply and marketing cooperatives, and for the meat sectors. Variables that measure farm size (herd size and production volume), distance from the farm to the processing unit, and the number of alternative buyers are indicators of transaction costs. The questionnaire enquired about the intentions to enlarge production or exit farming within the next five years. Dummy variables (1 = yes, 0 = no) were constructed from the responses. Similarly, a dummy indicated whether a farmer had expanded within the previous five-year period.
Descriptive statistics on dairy and meat farmers
Dairy 1)
Meat 2)
(st. dev)
Farmer age
Field area, hectares
99***
Cooperative capital, euro 3)
30,480***
7,265***
Alternative buyers
1.9**
3.0***
Distance to processing unit
Herd size 4)
Production volume, liters/kg/year 4)
Female, % in sample
Have expanded, %
Intention to expand, %
Intention to exit, %
Region North-East, N of farmers
1) Mann–Whitney test for difference between supply and marketing in scale variables
2) Kruskall-Wallis test for difference between pig, cattle, and poultry in scale variables
3) Capital suffers from a large number of missing values due to non-responses. Only 122 farmers provided capital information
4) Herd size and production volume were not tested in meat, as such a comparison is not meaningful due to different farm structures. Pigs are only reported for farms specializing in piglets, and in cattle for farms that breed calves
***(p < 0.01), **(p < 0.05), *(p < 0.1)
The Shapiro-Wilk normality test for small samples indicated that only age was normally distributed. Therefore, the analyses proceeded with non-parametric tests. Differences in the variables between the subsamples were tested under the null hypothesis that the distribution of a variable was the same across categories. As the dairy sample was divided into two categories, the Mann-Whitney test was used, while the Kruskall-Wallis test for three categories was applied to the meat sample.
In the meat sample, all the other background variables differed between the production sectors except for age. Differences between the herd size and production kilograms were not tested, because by definition they differ depending on the livestock. Herd size was not available for poultry. According to the variance tests of two independent samples, the dairy farms in supply and marketing cooperatives had similar characteristics in terms of size, field area, farmer age, and capital, but they differed in terms of alternative buyers and distance. The producers of marketing cooperatives were located closer to the processing unit and they had slightly more alternatives.
The dairy farms had an average herd size of 32 dairy cows, corresponding exactly to the national average (Tike agricultural statistics, 1.5.2014). A herd size of 20–29 cows was the most typical in the sample, which is same as in the Finnish population, as a quarter of dairy farms in Finland are of this size. Dairy farms with more than 40 cows were categorized as large for the present analysis, and 99 dairy farms fell into this category. The pig and cattle subsamples consisted of heterogeneous production. Pig farms can be of three types: pork meat production, raising piglets, or a combination of the two. The size categorization of pig farms took into account the field of specialization: A farm with over 2000 piglets or yearly meat production exceeding the sample average was classified as large. The data on cattle farms possibly included some members of meat cooperatives whose primary production sector was milk, although potential overlaps with the milk sample were screened out.
Stated preference questionnaires are suitable research methods for examining perceptions of the benefits that farmers receive from membership of agricultural producer cooperatives and delivering their production to the cooperative. Attitudinal surveys often use Likert scale scoring, in which the extent of agreement is expressed by choosing from the following: 1 = strongly disagree, 2 = disagree, 3 = neither disagree nor agree, 4 = agree, or 5 = strongly agree. Dairy and meat farmers used this scale to respond to thirteen statements on how important the mentioned factors were in their membership. The question set was designed to cover a wide range of potential benefits, from the traditional advantages of joining a cooperative, such as access to the market and bargaining power, to benefits originating from structural changes in agriculture to capital-oriented efficient business. The full list of questions is presented in the results section. The questionnaire responses need to be interpreted cautiously, as the responses only reflect the subjective valuation of the respondent's current situation. As such, the self-reported significance of the benefits can be interpreted as satisfaction, or value, scores.
In order to analyze value differences across cooperative types, the Likert scale responses were transformed to three levels, where 3 = farmer perceives the benefit as significant, 2 = the farmer is indifferent, i.e., perceives neither significant nor insignificant benefits, and 1 = insignificant benefit perceived by the farmer. The scale data were analyzed using the Mann-Whitney and Kruskall-Wallis tests of independence between the distributions of self-reported values across cooperative types (categories) in 2×3 and 3×3 cross-tabulations. Non-parametric tests are suitable when one of the samples is drawn from a skewed or peaked distribution (de Winter and Dodou 2010). These statistical tests indicate, whether two independent samples have significant differences. Under the null hypothesis the samples are identical. If the null hypothesis is rejected, the analyzed cooperative types are concluded to differ in terms of the given characteristic.
The relationships between farmer-specific variables that approximated the level of a farmer's transaction costs were analyzed with an ordered probit regression model. Ordered probit is appropriate for statistical analysis of ordinal survey responses, such as the Likert scale, in rating assignments (Greene, 2000). In this study, the estimation method was applied to analyze the effect of farmer characteristics on the likelihood of reporting a certain valuation score for a cooperative benefit. The estimations were conducted using Limdep Nlogit software.
The idea in the ordered probit model is that in addition to y i , which is an individual's i (where i =1,…,n) response to a survey question, and which takes an integer value 1, 2, 3,…, J, there is a latent index y i * , which measures the subjective scale and the propensity to agree with the statement. Once it exceeds a certain threshold, the respondent reports a value of 'significant' and then further 'very significant', along an ordinal scale. The latent index y i * is assumed to depend linearly on the vector of observed characteristics x i that explain an individual's attitude and unobserved factors ε i
$$ {y_i}^{*}={x}_i\beta +{\varepsilon}_i $$
What is observed is
$$ {y}_i = 0\kern0.37em \mathrm{if}\;{y_i}^{*}\le\ 0 $$
$$ {y}_i = 1\ \mathrm{if}\ 0\ \le {y_i}^{*}\le {\mu}_1 $$
$$ {y}_i = 2\ \mathrm{if}\;{\mu}_1\le {y_i}^{*}\le {\mu}_2 $$
where μs are unknown parameters to be estimated with β. They are referred to as the threshold parameters, which are in theory different for all respondents. The estimated threshold parameters are averages over the respondents. This presentation follows the general notation, and is applied here from Greene (2000) and Daykin and Moffatt (2002).
We estimated the ordered probit model for a set of dependent variables, which were the self-reported values (scale 1–5 recoded to 0–4 for analysis purposes) for the benefit statements and explain the preferred choice with farmer-specific characteristics
$$ \begin{array}{c}{y}_i\ast = Dair{y}_i{\beta}_1+ Pi{g}_i{\beta}_2+ Field\kern0.34em are{a}_i{\beta}_3+ Expande{d}_i{\beta}_4+ Will\kern0.34em expan{d}_i{\beta}_5+ Femal{e}_i{\beta}_6\\ {}+Ag{e}_i{\beta}_7+ Distanc{e}_i{\beta}_8+ Dairy- distanc{e}_i{\beta}_9+ Marketin{g}_i{\beta}_{10}+ Dairy- larg{e}_i{\beta}_{11}\\ {}+ Pi g- larg{e}_i{\beta}_{12}+Wes{t}_i{\beta}_{13}+ North-Eas{t}_i{\beta}_{14}+{\varepsilon}_i\end{array} $$
where Dairy, Pig, Expanded, Will expand, Female, Marketing, Dairy-large, Pig-large, Region West, and Region North-East are indicator variables taking a value of 1, or zero otherwise, and Field area is measured in hectares, a farmer's Age in years, and Distance represents a farm's distance to the processing unit in tens of kilometers. Dairy-distance is an interaction term that is intended to capture the distance effect specific to the dairy producers. These variables were selected as proxies for transaction costs, because they are common indicators for all producers, irrespective of their production specialization. Moreover, they are easy for farmers to report in a questionnaire. Location factors were studied using the distance variable and geographical indicators. Farm size was captured with both the field area and the indicator variable for large producers. Moreover, the intention to expand and recent farm expansion indicated the change in farm size, which was expected to affect the member's relationship with the cooperative.
Value of membership benefits
Farmers considered a stable channel for selling their production as the main benefit provided by cooperative membership. Both milk and meat producers indicated the highest valuations for the statements related to market access (Tables 2 and 3). The cooperative as a stable market channel was significant benefit for 89 % of dairy cooperative members and 83 % of meat cooperative members. Moreover, cooperative membership appears to bring valuable business continuity to producers. Since the dairy cooperative is obliged to collect all the milk produced by its members, the transaction costs from searching for a buyer are reduced. The timing of market access is crucial in milk production due to the perishability of the product and also due to the frequency of milk collection. The benefit of a cooperative buyer with an obligation to collect all of the production was clearly valued by milk producers in this survey.
Benefits of cooperative membership for milk producers
% of members
Statement on the benefits provided by the cooperative
1. The cooperative offers a stable market channel.
2. The cooperative is obliged to collect all of my agricultural production.
3. As a member of the cooperative, I have better possibilities to expand my agricultural production.
4. The cooperative operates in the nearby region.
5. The cooperative offers good services to the members.
6. The cooperative pays a competitive producer price.
7. I gain other, non-pecuniary benefits from the cooperative.
8. Producers benefit from a good bargaining position through the cooperative.
9. Cooperative capital is an attractive investment instrument.
10. Membership provides me taxation gains.
11. Membership provides me control in the governance of the cooperative.
12. As a member, I have an opportunity to influence business decisions that promote my own business.
13. As a member, I have an opportunity to carry out work that supports the community.
Question: What does cooperative membership mean to you? Indicate on scale of 1 to 5 for each of the statements how important the factors are to your membership. (5 = very important … 1 = not at all important)
Benefits of cooperative membership for meat producers
2. Membership secures the marketing of all of my production.a
aThe wording of statement 2 in the meat producer version of the survey slightly differs from the milk producer version. The statement is, however, intended to convey the same meaning as close as possible. All other statements are the same
On the other hand, the role of the meat cooperatives as buyers is somewhat different from dairy cooperatives, since the business is separated from the cooperative in the form of a subsidiary procurement company. However, like the dairy cooperative members, the meat producers in the study highly valued the security of having a destination for their production, which was indicated by 78 % of respondents stating this as a significant benefit.
A competitive producer price appeared to be a very significant benefit provided by cooperative membership for milk producers (Table 2). After requesting them to score the individual statements, the respondents were asked in a follow-up question to indicate which of the benefits they considered the most important. Among milk producers, a competitive producer price was the second most often stated benefit after a stable market channel. There was, however, a marked difference between the members of milk supply cooperatives and dairy marketing cooperatives in the perceptions of the price. While a competitive producer price was valued highest by 36 % of supply cooperative members, the corresponding proportion was 25 % in the sample of marketing cooperatives. Difference in the valuations of the producer price as a cooperative benefit can probably be explained by the actual price levels paid by the supply and marketing cooperatives included in the sample. Members of the cooperatives supplying milk to the Valio Group had been able to benefit from producer prices that were also relatively high in international comparison. On the other hand, smaller marketing cooperatives, which take care of the whole dairy processing chain and do not engage in profitable business on international markets in the same scale, had on average paid a lower price for the milk of their producers. The sector-wide price pressures that have strained the profitability of meat production over the past few years may explain the relatively lower value of the producer price as a cooperative benefit among the meat producers compared to the dairy farmers (Table 3).
The dairy farmers agreed quite unanimously with the statement of benefiting from good services offered by the cooperative. Access to production-related services was appreciated, bringing significant value to membership, as only 5 % of milk producers indicated such access as insignificant. However, the cooperative's services were clearly of secondary importance relative to market access and competitive remuneration, as only three respondents out of the total of 384 dairy farmers highlighted these services as the most important benefit. Variable transaction costs on the farm can potentially be lowered if the cooperative is able to arrange essential services in a cost-efficient way. Meat producers did not perceive significant value being derived from services provided by their cooperative. The result may reflect the fact that, in the case of meat, the services are not offered by the cooperative itself but by an associated subsidiary company. Thus, access to fodder advisory and other meat production-related services is indirectly an outcome of cooperative membership.
The perceived value of cooperative membership was not limited to tangible factors such as market access, the producer price, and business facilitating services, but farmers also reported other non-pecuniary benefits as important. While what are perceived as non-pecuniary benefits is subjective, correlations between statement scorings suggest that these benefits may be related to a better bargaining position through the cooperative. In the responses of dairy farmers, the value of non-pecuniary benefits was correlated with control benefits, participating in decision-making that promotes their own farm business, and the opportunity to carry out influential work in support of the community. Meat producers displayed similar interlinkages between the control and influence benefits. However, the meat producers on average placed less value on these factors in cooperative membership than the milk producers.
Effect of cooperative type
The differences in the valuations of membership benefits across cooperative types are reported in Table 4. Fisher's exact test was employed to test the independence of membership benefit valuations of the cooperative type. This test indicates whether the members of dairy supply, dairy marketing, and meat cooperatives differently value the stated benefits. The z-test values reported for 'significant' show how the groups differ. Superscripts a, b, and c in Table 5 denote the results of the z-test, which compares the column proportions. If the result of the statistical test is insignificant, the members in one of the cooperative types are interpreted to value the stated benefit as much as the members in the other cooperative types. If the performed test gives a significant result, a benefit is interpreted to be more highly valued by the members in a cooperative type whose score is higher.
Differences in the significance of membership benefits according to the cooperative type
Statement on the benefits
Supply dairy
Marketing dairy
The cooperative offers a stable market channel***
% of producers
83a,b
The cooperative is obliged to collect all of my production 1)***
Better possibilities to expand my agricultural production***
The cooperative operates in the nearby region***
The cooperative offers good services to the members***
The cooperative pays a competitive producer price***
I gain other, non-pecuniary benefits from the cooperative***
Benefit from a good bargaining position through the cooperative***
Cooperative capital is an attractive investment instrument
Membership provides me taxation gains
Control in the governance of the cooperative
Influence in decisions that promote my own business
Opportunity to carry out work that supports the community
Membership secures the marketing of all of my production. Superscripts a, b, and c denote the results of the z-test, which compares the column proportions. Different letters indicate differences and the level of significance is tested with Fisher exact test. Fisher's exact test ***(p < 0.01), **(p < 0.05), *(p < 0.1)
Differences in the significance of membership benefits according to producer size
Milk small
Milk large
Pig small
Pig large
The cooperative offers a stable market channel
Chi-squared
9.86***
The cooperative is obliged to collect all of my production 1)
Better possibilities to expand my agricultural production
10.58***
The cooperative operates in the nearby region
The cooperative offers good services to the members
The cooperative pays a competitive producer price
8.53**
I gain other, non-pecuniary benefits from the cooperative
Benefit from a good bargaining position through the cooperative
Membership secures the marketing of all of my production. Superscripts a and b denote the results of the z-test, which compares the column proportions. Different letters indicate differences and the level of significance is tested with Chi-squared test. Chi-squared test between large and small separately for milk and pig samples. Responses 'indifferent' and 'insignificant' are omitted for clarity ***(p < 0.01), **(p < 0.05), *(p < 0.1)
Milk suppliers and meat producers exhibited different valuations for the cooperative offering a stable market channel and being the destination for all of their production. These factors were relatively more important to the members of supply cooperatives associated with the Valio Group. On the other hand, the members of smaller independent dairy marketing cooperatives responded to statements 1 and 2 in a similar way to the meat farmers. All groups differed in terms of the subjective value of non-pecuniary benefits and the cooperative operating in the local region. Altogether, 85 % of the farmers in dairy marketing cooperatives perceived the proximity of the cooperative as significant, while none of them stated it as insignificant. This finding may reflect affective regionalism and a strong sense of community. Dairy producers may feel loyal to their local cooperative, which could possibly explain their decision to join the marketing cooperative and, subsequently, their high valuation of proximity. On the other hand, less than half of the meat producers reported the proximity of the cooperative as beneficial to them. This may be a result of the higher degree of concentration in the meat industry in comparison to the dairy industry. The provision of production-related services appears to have been more important to dairy farmers than to meat farmers. Services were equally appreciated by milk producers in supply and marketing cooperatives.
Size effects
Similar tests on the impact of farm size are reported in Table 5. Benefits in the form of market access were relatively more significant to large dairy producers than to those classified as small. However, the proportion of farmers reporting 'significant' in statements 1 and 2 was 85 % and 87 % respectively, which underlines the importance of the cooperative as a market channel for all milk producers, irrespective of their size. Due to high asset specificity, large farms may be more dependent on the stability of the milk deliveries and certainty over the buyer for all of their production, a phenomenon for which Pascucci et al. (2012) provided evidence among Italian agricultural producers. Large producers are likely to have more market options than smaller producers because they are relatively more attractive to the buyers, since the contracting costs of the buyer are reduced. In addition due to the economies of scale in doing business with fewer large farms than with many small ones, they can economize on searching and contracting costs when the sole business relationship is with their dairy cooperative, which collects all of their production.
The traditional explanation for farmers forming cooperatives is to gain bargaining power and improve their position in relation to the buyers of their farm production (Valentinov, 2007). We analyzed whether there were systematic differences in the stated value of control and power between small and large producers. In addition, due to the divergence of business practices in dairy and meat sectors, the significance of power factors needed to be analyzed with respect to the specialization of farm production.
The meat producer sample included farmers specialized in the pig sector, cattle, or poultry. Size effects could only be analyzed meaningfully with the sample of pig farmers. The dataset on cattle producers was subject to confounding effects, because it was also likely to include some farmers with dairy cows. Specialization in cattle meat production, or combined dairy and meat production, could not be exclusively identified in the data, and size classification based on the reported yearly meat production (kilograms) was therefore unlikely to capture the true size effect in meat production that we were interested in. The dataset on poultry farmers was small, but included quite a balanced sample of small and large producers.
According to the analysis, the valuation of control was indeed related to the size of the pig farm. Consistently with cooperative theory, farmers categorized as small scale attached higher value to gaining control through cooperative membership, the opportunity to take part in decision-making that facilitates their own business, and influence in the community. In addition, 52 % of small pig farmers perceived that cooperative membership empowers them with influence in decision-making that promotes their farm business, while only 29 % of large pig farmers shared this view. Due to the insufficient sample size, z-test statistics were not significant in the pig data, as the number of observations in each category became small. However, the percentages are indicative of direction. Poultry farmers rated the power and control factors on average as highly as the small pig farmers, but no size effect could be found in the small subsample. The opposite effect of size on the importance of control was found among dairy cooperative members. Those classified as large milk producers provided a higher rating on average for the benefits of bargaining power and control than the small milk producers.
The perception of the cooperative as a power and control mechanism is hypothesized to be related to the market and the organizational structures. This means that farmers may have better bargaining power if they have a number of alternative buyers. The governance structures may also affect the ability of producers to negotiate with the buyer. The majority of dairy producers in the marketing cooperatives reported that the number of alternative buyers operating in their region was two (43 % of respondents), while quite a large group of marketing cooperative members even had three alternatives (23 %). Only 11 % of respondents in this group reported one buyer. In contrast, 28 % of the dairy producers in supply cooperatives reported that they had only one buyer, while 26 % had two, and 16 % had three.
The presence of more alternatives on average is related to location factors and may explain why the members of marketing cooperatives reported the proximity of the cooperative as being so significant. While they reported having more alternative buyers in the region, the farmers supplying milk to the independent marketing cooperatives more frequently indicated that they had never switched cooperative compared to the farmers supplying milk to the large supply cooperatives of the Valio Group (88 versus 75 %). The disposition towards switching suggests that those dairy farmers who had the most delivery alternatives did not behave opportunistically and shop around with buyers, but rather remained committed to their local cooperative buyer.
Further analysis of the delivery alternatives provided indications of how farmers benefited from a fragmented market structure with several buyers. The total farmer sample was classified into three groups depending on the reported number of alternative buyers (one, two, or three or more), and the response distributions of membership benefits were tested with Fisher's exact test. Three observations related to the transaction cost theory were made. First, farmers who had only one buyer in the region were statistically significantly more likely to value the benefits from access to the market for their whole production through the cooperative than those farmers who had several alternatives. This finding reflects reduced uncertainty. Second, farmers who had more alternative buyers reported on average a higher value for production-related services. A competitive regional market for raw material supply (i.e., more buyers) may provide impetus for buyers to develop their service offering in order to attract and retain members. In such a situation, farmers gain from reduced transaction costs if they obtain the services as a membership benefit more easily and/or more cheaply than they would elsewhere. Third, a similar pattern was observed in the importance of non-pecuniary benefits. As discussed earlier, the statement in the questionnaire did not define in what form the benefit was received, but the result confirms that value from the presence of alternative buyers materializes to farmers as other than price or capital-related benefits.
Regression results
The multivariate ordered probit (probability) model allowed an examination of the contemporaneous effect of several farmer-specific background variables on the valuation of the membership benefits by farmers. The dependent variables were obtained from the response to the benefit statements 1–13, as described above. We concentrated on six potential membership benefits that were noted in the previous section to mark differences among farmers and that are theoretically related to transaction costs and the benefits of organizing into farmer cooperatives. The dependent variables were as follows, with the benefit statements they mapped to in parentheses: 1) market channel (statement 1), 2) proximity (statement 3), 3) services (statement 5), 4) producer price (statement 6), 5) bargaining power (statement 8), and 6) control (statement 11). The responses to the seven remaining benefit statements were also estimated in the ordered probit, but are not reported here. Table 6 presents the estimated regression coefficients of the ordered probit models. The interpretation of the coefficients is as probabilities that a farmer characteristic is relevant in explaining the self-reported importance of a benefit.
Ordered probit regressions for self-reported value of membership benefits
Market channel
Producer price
Bargaining power
Dairy dummy
1.242***
Pig dummy
Field area
−0.105***
−0.036
0.112**
Expanded dummy
Will expand dummy
Female dummy
Distance, 10 km
−0.022*
(0.013
Dairy-distance interaction
−0.03
Marketing dummy
Dairy-large dummy
Pig-large dummy
−0.914**
Region North-East
Threshold parameters
Log likelihood
−316.84
Chi 2
Pseudo R2
Dependent variables 1–6 are the Likert scale responses to selected value statements recoded to 0–4. The table reports the estimated probit coefficients and their standard errors in parentheses. The original statements are presented in Tables 2 and 3. ***(p < 0.01), **(p < 0.05), *(p < 0.1)
In model 1, we observed that the market channel opened through cooperative membership was very important to milk producers, but stable market access was also valued by pig farmers. Positive and statistically significant regression coefficients for dummy variables for recently expanded farms and those intending to expand signified the role of the cooperative buyer as a stable transaction partner when the members invested in increasing their production volume. The cooperative market channel was even more important to the farmers who planned to expand in comparison to those who had already invested in enlarging their production. The benefit may arise from reduced transaction uncertainty.
The coefficients for the size indicators were not statistically significant. This result is in line with the earlier notion that cooperative membership provides market access that is unanimously very highly rated by both large and small producers. This holds for both milk and pig sectors, although the coefficient for the pig dummy was negative. The statistically significant coefficient for field, however, suggests that those farms with more hectares were less likely to value the market channel as a benefit than smaller farms when measured in field area. A potential explanation is that the farms that receive more of their income from crop farming are not as dependent on their livestock business, and the value of membership is not therefore as tightly linked to production-related benefits. This explanation receives support from the estimations with control, cooperative capital as an attractive investment, and taxation gains as dependent variables, as the field area obtained a positive and statistically significant coefficient. The farmers who owned a large field area derived capital and control-related value from cooperative membership.
Other explanatory variables that obtained statistically significant coefficients in predicting the significance of the market channel were the number of buyers and farmer age. Age appeared to explain the response in all models and was always positive, which suggests that older farmers may be more satisfied with the cooperative relationship. The probability of agreeing with the statement that the cooperative provides benefits as a stable market channel increased with the number of alternative buyers operating in the region. However, running the same regression with statement 2 as a dependent variable (not reported) yielded an insignificant effect of buyer alternatives on the subjective value of the cooperative buying/marketing all of the production. This result is unsurprising, as when the cooperative is the destination for all of the farm production, the presence of more alternatives does not bring added value to cooperative membership.
The estimated marginal effects (Appendix) reveal how a discrete change in the farmer characteristics variable from 0 to 1 affects the prediction of a benefit being reported (very significant), holding all other characteristics as constant. Milk producers were 46 percentage points more likely to rate the market channel benefit as very significant in comparison to meat producers. The coefficient for 'Will expand' indicates that those farmers who intended to enlarge their production were 18 percentage points more likely to report the stable market channel through the cooperative as a very significant benefit compared to the farmers not planning to increase their farm size in the near future.
In model 2, a distance effect among members of the supply and marketing dairy cooperatives was observed. The estimated coefficient for the indicator variable for marketing cooperatives exhibited a positive and statistically significant coefficient. The marginal effects show that farmers of independent marketing cooperatives were 23 percentage points more likely to indicate the proximity of the cooperative as a very significant benefit. Recall that independent cooperative members are on average located closer to the processing unit than the members of dairy supply cooperatives. The ordered probit regression coefficient for distance confirms that farmers located further from the processing unit were less likely to value proximity.
The region indicator suggests that the proximity of the cooperative was valued by farmers in western Finland. This is an area characterized by higher social capital in comparison to the rest of the country (KAKS 2004). People in western Finland have in various studies been found more socially active on average and more satisfied in their relationships with other people. Latent attitudinal factors may be reflected in the self-reported values for proximity. Western farmers were 11 percentage points more likely to report this as very significant. Female farmers were also found to value proximity more highly.
Model 3, with the benefits from production-related services as the dependent variable, confirmed the size effect observed in univariate comparisons in the multivariate framework. While the estimated coefficient for the milk producer dummy was positive, the negative sign with the dummy variable for large milk producers implies that service benefits were more significant to smaller producers. The same applies to the small pig farmers. Being a large pig farmer reduced the probability of reporting service benefits as very significant by 15 percentage points. Although large producers appeared to be less satisfied with the cooperative services, those farmers who intended to expand production viewed services as valuable membership benefits. Female farmers were 10 percentage points more likely to report services as very significant compared to male farmers. Producers in the western part of Finland were indifferent or slightly negative towards the value of services as benefits. This result may be a confounding effect of the presence of large pig farms in the area.
Producer price divided the farmers most as a membership benefit. It is a variable that clearly reflects the satisfaction of farmers with the current situation, i.e., the level of the currently paid producer price. Differences between dairy and meat farmers therefore not only reflected organizational heterogeneity, but were inevitably affected by global market prices, pricing conditions of cooperatives, and the profitability pressures on farms. The estimated model 4 reveals that milk producers were more satisfied with price benefit from cooperative membership than meat producers, but the result only applied to the members of the large supply cooperatives associated with the Valio Group. The dummy variable for independent marketing cooperatives captured a negative coefficient of −0.541, which was statistically significant. Marginal effects revealed that members of small dairy cooperatives were 21 percentage points less likely to agree strongly with receiving a price benefit from their cooperative. This finding is in line with the actual price difference between the dairy cooperative types, since Valio has on average paid a better producer price to its farmers. The independent marketing cooperatives, on the other hand, have paid somewhat lower producer prices in recent years. Size was not found to be a relevant factor in explaining the attitudes of dairy producers towards price benefits. In a dairy cooperative, the price is the same for all members.
Large pig farmers exhibited strong disagreement over the producer price as a membership benefit. Overall, pig farmers were dissatisfied with the pricing of the cooperative. The marginal effects indicate that large pig farmers were 50 percentage points less likely to consider the producer price as a very significant benefit. Female farmers appeared to find the producer price paid by the cooperative acceptable. The presence of more alternative buyers increased the probability that a farmer would be satisfied with the price. This result suggests that in areas where many buyers operate, an attractive price level may hold, because the buyers compete for producers. It may also reflect the bargaining power of farmers.
The effect of farmer-specific factors on the perception of bargaining power through the cooperative is indicated in model 5. The number of alternative buyers was positively related to the probability of a farmer valuing bargaining power. The marginal effect is moderate but nonetheless indicates a positive 4 percentage points higher probability of strongly agreeing with bargaining position gains when the number of potential buyers increases by one. The farmers who had enlarged their farm production or intended to do so valued the bargaining power benefit from cooperative membership more highly. A large farm size as such did not contribute to the perception of bargaining power benefits among dairy and pig farmers. The indicator variable for farm location shows that cooperative membership was associated with the significance of bargaining power benefits in northern or eastern Finland. Potential explanations may be related to differences in structural factors in agriculture between separate regions of Finland, e.g., the production sector, differences in the organizational structures of the buyers, or the size of farms.
Finally, model 6, with control as the dependent variable, demonstrated the effect of field area and the significance of control opportunities to expanding producers. We were not able to confirm a difference in control benefits between dairy and meat sectors. On the contrary, the dummy for pig farmers received a statistically significant positive coefficient, which indicates that they were likely to value control in the governance of the cooperative more highly. However, pig farmers classified as large were 23 percentage points more likely to report control as an insignificant factor than small and medium sized pig farmers.
Analysis of a questionnaire completed by 682 Finnish milk and meat producers revealed heterogeneity in the perceived valuation of benefits that farmers receive as members of agricultural producer cooperatives. Membership endows cooperative patron-owners with various benefits, of which some are monetary, such as a competitive producer price, attractive capital investments, or taxation gains. Some benefits are tangible and specific business practices, such as access to services, a contract with the cooperative to deliver all production, or the physical proximity of the business. Cooperative membership also brings abstract and less easily perceivable benefits such as bargaining power and influence in decision making. Producers can verify the tangible member benefits, such as services or capital interest, but they may not be able to measure the value of the bargaining power they gain through cooperative membership. Irrespective of the form, member benefits are assumed to bring utility to farmers through decreased transaction costs.
The foremost finding is that producers value many membership benefits as significant. The finding validates the importance of the cooperative organizational form in modern agriculture. However, marked differences in self-reported values were observed with respect to farmer-specific factors, and also to the type of the cooperative. The comparison of the responses between the three cooperative types provided evidence of differing positions among farmers in the vertical integration of agribusinesses. The degree of vertical integration causes a cooperative's role for a producer to be very different depending whether the membership rights are defined in the organizational context of a traditional marketing cooperative, supply cooperative, or an IOF-like hybrid cooperative.
The dairy marketing cooperatives of this study represented the most traditional type of agricultural cooperative, in which the patron-owners are closely involved in the whole process from the supply of milk to processing and marketing of the end products. Members of dairy supply cooperatives included in the sample have a role of supplying and exercising ownership rights in the cooperative and indirectly in the central group company under whose brand name the products are collectively marketed. Members of the hybrid type of meat cooperatives are most distanced from the end markets. The role of the meat cooperatives is to exercise ownership and control rights in the stock exchange listed processing and marketing company. It is evident that heterogeneity in organizational structures is related to the valuation of benefits by farmers and how significant cooperative membership is to them. However, the causation could not be confirmed and is left for future studies. The perceived value of the cooperative to its members may affect their decisions as the owners to choose another organizational structure that better corresponds to their valuations.
The benefit of market access was found to be the most significant factor for both milk and meat producers. The market channel through the cooperative and the obligation to take all the milk produced saves transaction costs for dairy farmers, as they do not have to search for a buyer for the rest of the milk and continuously renegotiate contracts. This relates to the frequency of transactions and perishability of the production. The observed higher significance of the cooperative market channel to large milk producers relative to smaller producers supports the hypothesis that cooperative membership provides protection for specific investments. Distance is in theory a factor that contributes to the level of transaction costs, but only tentative evidence was observed in this study. Proximity was valued by the members of small dairy marketing cooperatives, but this finding is more likely to be related to some sort of regional spirit and support for the local community than to transaction cost benefits. In dairy cooperatives, farmers are in fact indifferent to the distance factors, because the cooperative bears the transaction costs of collecting the milk from farms. In meat cooperatives, the delivery distance may be built into the pricing policy, and we would therefore expect distance effects in the meat producer sample. However, we were unable to identify a statistically significant distance effect in the multivariate analysis.
The results of this study indicate that in parallel with the evolution of organizational structures of agricultural producers, the mechanisms for membership remuneration also need to be developed. Although the primary rationale for farmers to organize into cooperatives is still the facilitation of market access, some other factors such as bargaining power and control benefits may be losing their significance in vertically integrated hybrid structures. Satisfaction with the producer price level seems to be reflected in the overall satisfaction with cooperative membership. The findings of this study have managerial implications with respect to understanding the member perspectives. The observations emphasize the importance of the various forms of benefits, which have relatively different importance to producers depending on their farm size. Carefully designed benefit policies that cater to the farmer valuations are likely to encourage investments in the farm business and breed commitment to delivering.
Drawing from a single questionnaire, the paper has obvious limitations. The results provide a useful reference for agricultural organizations in similar situations with a closely corresponding member structure to that of the sample of this study. However, the results cannot be generalized to farmer cooperatives that are in very different stage of organizational development than those studied in the Finnish context. In addition, the sectors under study, dairy and meat production, are marked by certain features, which may not characterize some other agricultural sectors, and thus the member benefits may be valued very differently by other type of farmers. Due to data limitations in a stated preference method of this study, the measuring of transaction costs is only an approximation at best. Further studies could aim at developing more accurate survey instruments and methods to measure transaction costs at farms and benefits from cooperative membership.
Marginal effects of ordered probit estimation
Very insignificant
Very significant
(1) Market channel
(2) Proximity
(3) Services
(4) Producer price
(5) Bargaining power
(6) Control
The author declares that he/she has no competing interests.
Department of Economics, University of Helsinki and Pellervo Economic Research PTT, Eerikinkatu 28 A, Helsinki, 00180, Finland
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Scaling-laws of Radio Spike Bursts and Their Constraints on New Solar Radio Telescopes
by Baolin Tan et al.
2019-06-18 Solar Radio Science Highlights
Radio observation is one of the most important methods in solar physics and space science. Sometimes, it is almost the sole approach to observing physical processes such as the acceleration, emission, and propagation of non-thermal energetic particles, etc. Long-term observation and study have revealed that a strong solar radio burst is always composed of many small bursts with different time-scales. Among them, a radio spike burst is the smallest one, with the shortest lifetime, narrowest bandwidth, and smallest source region. Solar radio spikes are considered to be related to a single magnetic energy release process, and can be regarded as an elementary burst in solar flares. It is a basic requirement for new solar radio telescopes to observe and discriminate these solar radio spike bursts, even though their temporal and spatial scales actually vary with the observing frequency. Here, we presents the scaling laws of the lifetime and bandwidth of solar radio spike bursts with respect to the observing frequency, which provide some constraints for the next generation of solar radio telescopes, and help us to select the rational telescope observing parameters. As well as this, we propose a spectrum-image combination mode as the best observation mode for new solar radio telescopes with high temporal, spectral, and spatial resolutions, which may have an important significance for revealing the physical essence of the various non-thermal processes in violent solar eruptions.
Solar radio observation is the most important approach for obtaining information about solar energetic particles, violent energy release, and mass ejections in solar eruptions. Solar radio telescopes include solar radiometers, radiospectrometers, and radioheliographs with various frequency bandwidth, cadence, spectral and spatial resolutions. Based on new scientific assumptions and technical development, new plans of solar radio telescopes are continuously proposed. In the development of new generation solar radio telescopes, it is very important to select a set of suitable observing parameters, such as frequency range, bandwidth, cadence, spectral resolution, and spatial resolution, etc. So, how does one select a reasonable group of observing parameters of a proposed solar radio telescope?
Previous statistical studies indicate that a solar eruption lasting several tens of minutes always contains several big pulses with timescales of minutes, and each big pulse is frequently composed of a group of pulses with timescales of seconds, and each pulse is still composed of many sub-pulses with timescales of sub-seconds. Actually, a violent solar eruption always contains a great number of sub-second radio bursts, which are called fast fine spectral structures (FFS). FFS includes spike bursts, dot bursts, and narrow-band type III bursts. In the microwave range, they are called small-scale microwave bursts (SMB) (Tan 2013). They have a very short lifetime, very narrow frequency band, and very high brightness temperature. They always occur in large groups and form various kinds of complex structures, such as QPPs, Zebra patterns, and other long-lasting pulses. Each SMB may represent an elementary energy release process, which can be regarded as the elementary burst (EB) in solar eruptions. Therefore, it becomes the basic requirement to identify clearly SMBs for the new generation of solar radio telescopes.
This work investigated the previous observational results of solar radio spike bursts, dot bursts, and narrowband type III bursts, including the previous publications (Gudel & Benz 1990, Rozhansky et al. 2008, etc.), and tried to obtain a modified scaling law of solar SMBs. Such a modified scaling law will be the theoretical basis to select reasonable parameters for designing the new generation of solar radio telescopes, and help us understand the nature of solar eruptions.
Figure 1 presents the statistical relationship between the averaged lifetime and frequency among solar radio spike bursts. The frequency range covers from 210 MHz to 7.0 GHz, and the lifetime ranges from 5 ms to 91 ms. It shows that the averaged lifetime of radio spike bursts is anti-correlated to the emission frequency, the correlation coefficient is -0.58. A fitted function is near a power-law function:
\[ \tau \approx 8.2 \times 10^3 f^{-0.84\pm0.15} \]
is the averaged lifetime of SMB in units of ms, $f$ is the frequency in units of MHz.
Figure 1 – The relationship between the averaged lifetime and frequency among the solar radio spike bursts. Here, the crosses represent the results published in Gudel & Benz 1990 and Rozhansky et al. 2008 and the dashed line is obtained by least squared fitting method. The diamonds represent the results observed by the Chinese Solar Broadband Radio Spectrometers at Huairou (SBRS/Huairou) since 2006 (Wang et al. 2008, Tan 2013) and the solid line is obtained by least squared fitting method over the total sample.
Figure 2 presents the relationship between the averaged bandwidth and frequency among the solar radio spike bursts. The observing frequencies of the whole sample range from 305 MHz to 7.0 GHz. The narrowest bandwidth is 1.4 MHz at central frequency of 710 MHz, while the widest bandwidth is 115 MHz at 1250 MHz. We found that the higher the observing frequency, the wider the bandwidth of SMB. The statistical correlation coefficient between the bandwidth and the central frequency is 0.47 among the 166 samples, which is obviously positive correlation. A fitting function is also obtained:
\[ f_{bw}\approx 0.011 \times f^{0.99\pm0.018}\sim 1.1\% f \]
Here, $f_{bw}$ is the averaged bandwidth of SMB in unit of MHz.
Figure 2 – The relationship between the averaged bandwidth and frequency among the solar radio spike bursts. Here, the crosses represent the results published in Gudel & Benz 1990 and Rozhansky et al. 2008. The diamonds represent the results observed by the Chinese Solar Broadband Radio Spectrometers at Huairou (SBRS/Huairou) since 2006 (Wang et al. 2008, Tan 2013) and the dot-dashed line is obtained by least squared fitting method over the total sample.
Because SMBs, including spike, dot, and narrow band type III bursts are the smallest eruptive units in solar eruptions, their scaling laws may provide a most important and fundamental basis for understanding the nature of solar eruptions and for the designing of the next generation solar radio telescopes.
For the studies of solar radio astronomy, we always hope that the telescope has high parameters configuration as soon as possible, such as high sensitivity, high resolutions, and broad frequency coverage. However, one high parameter is always at the cost of decrement of other parameters. For example, the high frequency resolution inevitably means the frequency bandwidth of individual channel becomes narrow, and this will decrease the sensitivity. When the time resolution increases, the integration time will become short which will cause the sensitivity decrease and make the relatively weak burst to be vague and submerged in noise. The scaling laws of SMB show that the time scale of the detailed variation in solar radio bursts decreases with the increase of frequency, and the bandwidth increases with the increase of frequency. The scaling laws may help us to determine the optimal parameters configuration of the new generation solar radio telescopes, so as to ensure the scientific output from the observed data to the greatest extent. For the imaging observation, if we select too high time and frequency resolutions, it will not only face to a great challenge in techniques, but also inevitably reduce the observational sensitivity, and sacrifice the scientific objective of the relevant telescope. Therefore, we propose the spectrum-image combination mode to observe the solar radio eruptions on the basis of the scaling laws of radio spike emission, it can realize the observation simultaneously with high temporal, spatial, and spectral resolutions, as well as a high sensitivity, and can be taken as the principal mode for the future new generation solar radio observations, it will have broad prospects for the relevant studies. However, the high parameters here are relative, they will be gradually upgraded with the new development of radio and computer techniques.
Based on a recently published paper: Tan, Bao-lin, Cheng, Jun, Tan Cheng-ming, Kou, Hong-xiang, ChA&A, 2019, 43, 59-74, doi: 10.1016/j.chinastron.2019.02.005
Guedel, M., Benz, A. O.: 1990, A&A, 231, 202
Rozhansky, I. V., Fleishman, G. D., Huang, G.-L.: 2008, ApJ, 681, 1688
Tan, B.L.: 2013, ApJ, 773, 165
Tan, B. L., Cheng, J., Tan, C. M., Kou, H. X.: 2019, ChA&A, 43, 59
Wang, S. J., Yan, Y. H., Liu, Y. Y., Fu, Q. J., Tan, B. L., Zhang, Y.: 2008, SoPh, 253, 133
fine spectral structures
radio spikes
type III bursts
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\begin{document}
\providecommand{\keywords}[1] {
\small
\textsl{Keywords:} #1 }
\providecommand{\ams}[1] {
\small
\textsl{AMS subject classification:} #1 }
\title{The inverse problem of positive autoconvolution} \author{Lorenzo Finesso\thanks{Lorenzo Finesso is with the Institute of Electronics, Information Engineering and Telecommunications, National Research Council, CNR-IEIIT, Padova; email: {\tt [email protected]}}
\and
Peter Spreij\thanks{Peter Spreij is with the Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam and with the Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Nijmegen; e-mail: {\tt [email protected]}} \thanks{This work was partially funded under the STM-2020 CNR Fund, project number 123456. The material in this paper was not presented at any conference.} }
\maketitle \begin{abstract} We pose the problem of approximating optimally a given nonnegative signal with the scalar autoconvolution of a nonnegative signal. The I-divergence is chosen as the optimality criterion being well suited to incorporate nonnegativity constraints. After proving the existence of an optimal approximation we derive an iterative descent algorithm of the alternating minimization type to find a minimizer. The algorithm is based on the lifting technique developed by Csisz\'ar and Tusn\'adi and exploits the optimality properties of the related minimization problems in the larger space. We study the asymptotic behavior of the iterative algorithm and prove, among other results, that its limit points are Kuhn-Tucker points of the original minimization problem. Numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm.
\keywords{autoconvolution, inverse problem, positive system, I-di\-ver\-gen\-ce, alternating minimization}
\ams{93B30, 94A17}
\end{abstract}
\section{Introduction}
Inverse problems in system modeling and identification have a long tradition and have been the subject of a vast technical literature in applied mathematics, engineering, and specialized applied fields. The classic book~\cite{tikhonov} surveys the early contributions to the field. The focus of this paper is on the subclass of inverse problems for which the models are of \emph{autoconvolution} type. In linear time invariant systems, inputs are transformed into outputs by convolution with a kernel representing the system's impulse response. Autoconvolution systems produce the output by convolution of the input signal with itself.
A lot of work has been dedicated to the inverse problem of autoconvolution for functions on the real line, emphasizing the functional analytic aspects and motivating its interest in a variety of applications in physics and engineering. Most of the contributions analyse special cases, where exact solutions to the inverse problem exist, and propose different theoretical approaches for their construction. The paper \cite{dose1979deconvolution}, for example, focuses on inversion of autoconvolution integrals using spline functions, whereas \cite{dose1979deconvolutionII} performs inversion by polygonal approximation. In \cite{martinez1979global} inversion is studied based on the application of the FFT algorithm and digital signal processing concepts. Special cases arise when dealing with autoconvolution of probability density functions, as in \cite{hofmann2006determination}. In \cite{douglas2014autoconvolution} the autoconvolution has been introduded for continuous time processes as an alternative to autocorrelation. Ill-posedness issues and Tykhonov regularization are omnipresent, see for instance \cite{burger2015deficit}, \cite{gorenflo1994autoconvolution}.
The main differences between the cited literature and this paper are that we consider approximation problems, rather than looking for exact solutions which exist only exceptionally, and that our (time) domain is discrete rather than the real line. Moreover, the nonnegativity constraint, that we impose on all signals, is a crucial feature of the present work. Some earlier work shares, at least in part, our point of view, e.g. the papers~\cite{choilanterman2005}, \cite{choilantermanraich2006} dealing with image processing and 2D systems, contain an algorithm of the same type as ours and an analysis of its behavior. In~\cite{schulz2005signal} an algorithm similar to ours is set up to solve a problem of signal recovery using auto and cross correlations instead of autoconvolutions.
The purpose of this paper is threefold. First we pose the problem of a time-domain approximation of a nonnegative input/output systems by finite autoconvolutions when the output observations are available. Following the choice made in other optimization problems for nonnegative system, we opt for the I-divergence, which as argued in~\cite{cs1991} (see also \cite{snyderetal1992}), is the natural choice for approximation problems under nonnegativity constraints. We provide a result on the existence of the minimizer of the approximation criterion. Then we propose an iterative algorithm to find the best approximation, and finally we study the asymptotical behavior of the algorithm.
We employ techniques that have already been used in~\cite{fs2006} to analyse a nonnegative matrix factorization problem and the approach is similar to the one in \cite{finessospreij2015ieeeit}, \cite{finessospreij2019automatica}, but differs from the latter references as they treat linear convolutional problems, whereas the autoconvolution is inherently nonlinear. The algorithm that we propose is of the alternating minimization type, and the optimality conditions (the Pythagorean relations) are satisfied at each step.
The inherent non convexity, and nonlinearity of the problem make the analysis of the asymptotic behavior challenging. The main result in this respect is contained in Proposition \ref{prop:kt} which states that all limit points of the algorithm satisfy the Kuhn Tucker optimality conditions. This should be compared with other known results on the convergence of alternating minimization algorithms. In some cases it is possible to show convergence to a (unique) limit, which is also the minimizer of the criterion. This happens, in particular, when dealing with a convex criterion. Contributions in this direction are e.g.~\cite{cover1984}, \cite{snyderetal1992}, and \cite{finessospreij2015ieeeit}, \cite{finessospreij2019automatica}, \cite{Vardietal1985}. On the other hand for non convex, nonlinear problems, to the best of our knowledge, there are no asymptotic results comparable with the present Proposition \ref{prop:kt}.
It must be remarked that the nonparametric approach to the inverse problem, that we follow in this paper, is different from the one followed in identification or realization of nonnegative and linear systems, see~\cite{benvenutifarina2004} for a survey, and for instance~\cite{andersondeistler1996}, \cite{farina1995}, \cite{gurvits2007}, \cite{nagy2005}, \cite{nagy2007}, \cite{shu2008}.
A brief summary of the paper follows. In Section~\ref{section:problem} we state the problem and show the existence of a solution and give some of its properties. In Section~\ref{section:lift} the original problem is lifted into a higher dimensional setting, thus making it amenable to alternating minimization. The optimality properties (Pythagoras rules) of the ensuing partial minimization problems are discussed here. After that we derive in Section~\ref{section:algo} the iterative minimization algorithm combining the solutions of the partial minimizations, and analyse the convergence properties. In particular we show that limit points of the algorithm are Kuhn-Tucker points of the original optimization problem. In the concluding Section~\ref{section:numerics} we present numerical experiments that show the quick convergence of the algorithm and corroborate the theoretical results on its asymptotic behaviour.
\section{Problem statement and initial results}\label{section:problem}
In the paper we consider real valued signals $x: \mathbb Z \rightarrow \mathbb R$, mapping $i \mapsto x_i$, that vanish for $i<0$, i.e., $x_i=0$ for $i<0$. The \emph{support} of $x$ is the discrete time interval $[0, n]$, where $n=\inf\{\,k:\,\, x_i=0,\,\,\, \text{for $i> k$}\,\}$, if the infimum is finite (and then a minimum), and $[0, \infty)$ otherwise. The autoconvolution of $x$ is the signal $x* x$, vanishing for $i<0$, and satisfying, \begin{equation}\label{eq:xconv} (x* x)_i =\sum_{j=-\infty}^\infty x_{i-j}x_j =
\sum_{j=0}^i x_{i-j}x_j\,, \qquad i \ge 0. \end{equation} Notice that if the support of $x$ is finite $[0, n]$, the support of $x* x$ is $[0,2n]$. In this case, when computing $(x * x)_i$ for $i>n$, the summation in Equation~\eqref{eq:xconv} has non zero addends only in the range $i-n\le j \le n$, as $x_{i-j}=0$ and $x_j=0$ for $i-j>n$ and $j>n$ respectively. If the signal $x$ is nonnegative, i.e.\ $x_i \ge0$ for all $i\in \mathbb Z$, the autoconvolution (\ref{eq:xconv}) is too. Given a finite \emph{nonnegative} data sequence $$ y=(y_0,\dots , y_n), $$ the problem is finding a \emph{nonnegative} signal $x$ whose autoconvolution $x* x$ best approximates $y$. Since the signals involved are nonnegative, the approximation criterion is chosen to be the I-divergence, see~\cite{c1975,cs1991}. The I-divergence between two nonnegative vectors $u$ and $v$ of equal length is \[ \mathcal{I}(u,v)=\sum_i u_i\log\frac{u_i}{v_i}-u_i+v_i\,, \] if $u_i=0$ whenever $v_i=0$, and $\mathcal{I}(u,v)=\infty$ if there exist an index $i$ with $u_i>0$ and $v_i=0$. It is known that $\mathcal{I}(u,v)\geq 0$, with equality iff $u=v$.
Depending on the constraints imposed on the support of $x$ the basic problem splits into two different cases. The first case involves a full length signal $x=(x_0,\dots, x_n)$ and produces the approximation problem specified below, where we write $x* x\in\mathbb{R}^{n+1}$ for the restriction to $[0, n]$ of the convolution $x* x$ defined in~\eqref{eq:xconv}. \begin{problem}\label{problem} Given $y\in\mathbb{R}^{n+1}_+$ minimize, over $x\in\mathbb{R}^{n+1}_+=[0,\infty)^{n+1}$, \begin{equation} \label{cost-full}
\mathcal{I}=\mathcal{I}(x):=\mathcal{I}(y||x* x)=\sum_{i=0}^n\Big(y_i\log\frac{y_i}{(x* x)_i}-y_i+(x* x)_i\Big)\,. \end{equation} \end{problem} In alternative, recalling that, in the finite case, the support of $x* x$ is twice the support of $x$, one can consider, when $n=2m$, the problem of approximating the given data $y=(y_0,\dots, y_{2m})$ with the autoconvolution $x* x$ of a signal of half length, $x=(x_0,\dots, x_m)$. This leads to the following approximation problem. \begin{problem}\label{problemhalf} Given $y\in\mathbb{R}^{2m+1}_+$ minimize, over $x\in\mathbb{R}^{m+1}_+=[0,\infty)^{m+1}$, \begin{equation} \label{cost-half}
\mathcal{I}=\mathcal{I}(x):=\mathcal{I}(y||x* x)=\sum_{i=0}^{2m}\Big(y_i\log\frac{y_i}{(x* x)_i}-y_i+(x* x)_i\Big)\,. \end{equation} \end{problem} Notice that if the given data are $y=(y_0,\dots,y_n)$ with $n$ odd, i.e.\ $n=2m-1$ for some integer $m\geq 1$, one can still pose Problem~\ref{problemhalf} with $x\in\mathbb{R}^{m+1}_+$, simply introducing the fictitious data point $y_{2m}=0$. Hence in Problem~\ref{problemhalf}, without loss of generality, the number of data points will always be assumed odd, that is we assume $n$ even, $n=2m$.
Note that Problem~\ref{problem}, under the constraint that the support of $x$ is $[0, m]$, where $m=\lfloor \frac{n+1}{2}\rfloor$, reduces to Problem \ref{problemhalf}. Although the latter is a constrained version of the former problem and the approaches to their solutions are similar, the analysis and the results are very different. In this paper we concentrate on Problem \ref{problemhalf} which is easier to analyse and produces an algorithm with a much simpler structure. Problem~\ref{problem} will be investigated in a future publication.
The objective function \eqref{cost-half} is non convex and nonlinear in $x$, the existence of a minimizer is therefore not immediately clear. Our first result settles in the affirmative the question of the existence. The issue of uniqueness remains open, but we have evidence of the existence of multiple local minima of $\mathcal{I}(x)$. See Section~\ref{section:numerics} for numerical examples. \begin{proposition}\label{proposition:exist} Problem~\ref{problemhalf} admits a solution. \end{proposition}
\begin{proof} Let $x=x^0$ be an arbitrary vector in $\mathbb{R}^{m+1}_+$. Performing one step of Algorithm~\ref{algorithm:half}, introduced below, yields the iterate $x^1$ satisfying $\mathcal{I}(x^1)\leq \mathcal{I}(x)$ and $(\sum_{i=0}^mx^1_i)^2=\sum_{i=0}^{2m}y_i$, by virtue of Proposition~\ref{proposition:properties}. The search for a minimizer can hence be limited to the compact subset $K_0\subset \mathbb R^{m+1}_+$ of the $x$'s satisfying $(\sum_{i=0}^mx^1_i)^2=\sum_{i=0}^{2m}y_i$. Noting that $\mathcal{I}(x)=\sum_{i:y_i>0}(y_i\log\frac{y_i}{(x* x)_i}-y_i)+\sum_i(x* x)_i$, we can restrict attention even further to those $x$'s for which $(x* x)_i\geq \varepsilon$ for all $i$ such that $y_i>0$, by choosing $\varepsilon$ sufficiently small and positive. This implies that we restrict the finding of the minimizers to an even smaller compact set $K_1$ on which $\mathcal{I}$ is continuous. This proves the existence of a minimizer. \end{proof}
\noindent A basic ingredient for the minimization of the cost~(\ref{cost-half}) is its gradient which is computed below. As a preliminary step note that \[ \frac{\partial}{\partial x_j}(x* x)_i = \left\{\begin{array}{ll}
2x_{i-j}, &\text{for}\,\,\, 0\le j\le m,\,\,\,j\le i\le j+m\\
0, & \mbox{otherwise}\,,
\end{array}
\right. \] therefore \begin{align} \nabla_j\mathcal{I}(x) & :=\frac{\partial \mathcal{I}(x)}{\partial x_j} = \frac{\partial}{\partial x_j} \Big( \sum_{i=0}^{2m} -y_i\log (x* x)_i + (x* x)_i\Big) \nonumber\\ &= 2\sum_{i=j}^{j+m} \Big(-x_{i-j} \frac{y_i}{(x* x)_i} + x_{i-j} \Big)
= 2\sum_{\ell=0}^m \Big(- x_\ell \frac{y_{\ell+j}}{(x* x)_{\ell+j}} + x_\ell \Big)\,.\label{eq:gradj} \end{align} Equations \eqref{eq:gradj} are highly nonlinear in $x$ and solving the first order optimality conditions $\nabla \mathcal{I}(x)=0$, where $\nabla$ denotes the gradient vector, to find the stationary points of \eqref{cost-half}, will not result in analytic solutions except in trivial cases. This observation calls for a numerical approach to the optimization, which we will present in Section~\ref{section:algo}.
\noindent The following result shows a useful property of the minimizers of $\mathcal{I}(x)$. \begin{proposition}\label{prop:sum} For any $x\in\mathbb R^{m+1}$ it holds that \begin{equation}\label{eq:propconv} \sum_{i=0}^{2m}(x* x)_i= \Big(\sum_{i=0}^{2m} x_i\Big)^2. \end{equation} Moreover, if $x^\star\in\mathbb R^{m+1}_+$ is a minimizer of Problem \ref{problemhalf}, \begin{equation}\label{eq:propminim} \sum_{i=0}^{2m}(x^\star* x^\star)_i= \Big(\sum_{i=0}^{2m} x^\star_i\Big)^2= \sum_{i=0} ^{2m}y_i\,. \end{equation} \end{proposition}
\begin{proof} The identity (\ref{eq:propconv}) is a general property, indeed for any $x$, \begin{align*} \sum_{i=0}^{2m} (x* x)_i & = \sum_{i=0}^{2m} \sum_{j=0}^i x_{i-j} x_j = \sum_{j=0}^{2m} \sum_{i=j}^{2m} x_{i-j} x_j\\ & = \sum_{j=0}^{2m} x_j\sum_{i=j}^{2m} x_{i-j} = \Big(\sum_{j=0}^{m} x_j\Big)^2. \end{align*} \noindent To prove identity (\ref{eq:propminim}), let $x^\star$ be a minimizer of $\mathcal{I}(x)$ and define $f(\alpha)=\mathcal{I}(\alpha x^\star)$, for $\alpha>0$. It follows that $f'(1)=0$. A direct computation of $f'(\alpha)$ gives $f'(\alpha)=-\frac{2}{\alpha}\sum_{i=0} ^{2m}y_i+2\alpha\sum_{i=0}(x^\star * x^\star)_i$, hence $f'(1)=0$ yields the wanted identity. \end{proof}
\begin{remark} If $y$ is strictly positive the I-divergence in \eqref{cost-half} vanishes if and only if $y_i=(x* x)_i$ for all $i\in [0,2m]$. That is the (special) case where an exact solution to the deautoconvolution problem exists. Notice that this is a non generic case as the $2m+1$ equations $y_i=(x* x)_i$ in the $m+1$ variables $x$ specify an (at most) $(m+1)$-dimensional submanifold in the data space $\mathbb R^{2m+1}_+$. See the example below for an illustration. \end{remark}
\begin{example} \label{expl:trivial-half-case} For $m=1$, let $y=(y_0, y_1, y_2)$ be the given data. Setting the gradient $\nabla \mathcal{I}(x)=0$ one gets the unique minimizer $x^\star=(x_0^\star,x_1^\star)$ as \begin{align*} x_0^\star=\frac{2y_0+y_1}{2\sqrt{y_0+y_1+y_2}}\raisepunct{,} \quad x_1^\star=\frac{2y_2+y_1}{2\sqrt{y_0+y_1+y_2}}\raisepunct{.} \end{align*}
One easily verifies that $x^\star$ satisfies property~(\ref{eq:propminim}). Note that this solution, in general, does not give a perfect match; e.g.\ it should hold that $(x^\star * x^\star)_0=(x_0^\star)^2= y_0$. In fact, a necessary and sufficient condition on $y$ that insures the existence of the exact solution, i.e.\ $\mathcal{I}(y||x^\star* x^\star)=0$, is $y_1^2=4y_0y_2$. \end{example}
\begin{remark} Problem~\ref{problemhalf} has an interesting probabilistic interpretation when $\sum_{i=0}^{2m}y_i=1$. The $y_i$ can then be considered as the distribution of a random variable $Y$ taking on $2m+1$ different values. The problem is then to find the optimal distribution of independent and identically distributed random variable $X_1$ and $X_2$ (assuming $m+1$ values) such that $Y=X_1+X_2$. Note that Proposition~\ref{prop:sum} guarantees that the optimal vector $x^\star$ indeed has the interpretation of a distribution. In Example~\ref{expl:trivial-half-case}, with $y_0+y_1+y_2=1$, the optimal distribution is then $(x_0,x_1)=(y_0+\half y_1,y_2+\half y_1)$. As now one has $y_1=1-y_0-y_2$, it follows that $(x_0,x_1)=\half(y_0-y_2+1,y_2-y_0+1)$ and the perfect match condition reduces to $\sqrt{y_0}+\sqrt{y_2}=1$, in which case of course $X_1$ and $X_2$ can be thought of having a Bernoulli distribution and $Y$ a binomial distribution. \end{remark}
\section{Lifting and partial minimizations}\label{section:lift}
In this section Problem~\ref{problemhalf} is recast as a double minimization problem by lifting it into a larger space. The ambient spaces for the lifted problem are the subsets $\mbox{{\boldmath $\mathcal{Y}$}}$ and $\mbox{{\boldmath $\mathcal{W}$}}$, defined below, of the set of matrices $\mathbb{R}^{(2m+1)\times (m+1)}_+$, $$ \mbox{{\boldmath $\mathcal{Y}$}} := \big\{\,\mathbf{Y} : \,\, \mathbf{Y}_{ij}=0,\quad \text{for}\,\, 0\le i < j \,\, \text{and}\,\, i> j+m, \,\, \text{and}\,\, \textstyle{\sum_j}\mathbf{Y}_{ij} = y_i\, \big\}\,, $$ with $y=(y_0,\dots, y_{2m})\in\mathbb R^{2m+1}_+$ the given data vector, and $$ \mbox{{\boldmath $\mathcal{W}$}} := \big\{\mathbf{W} : \,\,\mathbf{W}_{ij} = x_{i-j}x_j,\,\text{if}\,\, 0\le j\le m,\, j\le i \le j+m;\,\, \mathbf{W}_{ij}=0\,\,\text{otherwise} \big\}\,. $$ The structure of the matrices in $\mbox{{\boldmath $\mathcal{Y}$}}$ and $\mbox{{\boldmath $\mathcal{W}$}}$ is shown below for $m=3$,
\small $$ \mathbf{Y}=\begin{bmatrix} \mathbf{Y}_{00} & 0 & 0 & 0\\ \mathbf{Y}_{10} & \mathbf{Y}_{11}& 0 & 0\\ \mathbf{Y}_{20} & \mathbf{Y}_{21}& \mathbf{Y}_{22} & 0\\ \mathbf{Y}_{30} & \mathbf{Y}_{31}& \mathbf{Y}_{32} & \mathbf{Y}_{33}\\ 0 & \mathbf{Y}_{41}& \mathbf{Y}_{42} & \mathbf{Y}_{43}\\ 0 & 0& \mathbf{Y}_{52} & \mathbf{Y}_{53}\\ 0 & 0& 0 & \mathbf{Y}_{63} \end{bmatrix},\qquad \mathbf{W}=\begin{bmatrix} x_0x_0 & 0 & 0 & 0\\ x_1x_0 & x_0x_1& 0 & 0\\ x_2x_0 & x_1x_1& x_0x_2 & 0\\ x_3x_0 & x_2x_1& x_1x_2 & x_0x_3\\ 0 & x_3x_1& x_2x_2 & x_1x_3\\ 0 & 0& x_3x_2 & x_2x_3\\ 0 & 0& 0 & x_3x_3 \end{bmatrix}\,. $$ \normalsize
\noindent The interpretation is as follows. The matrices $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$ and $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$ have common support on the the diagonal and first $m$ subdiagonals of $\mathbb{R}^{(2m+1)\times (m+1)}_+$. The row marginal (i.e.\ the column vector of row sums) of any $\mathbf{Y}\in \mbox{{\boldmath $\mathcal{Y}$}}$ coincides with the given data vector $y$. The elements of the $\mathbf{W}$ matrices factorize, equivalently their row marginal is the autoconvolution of the column marginal rescaled by $1/\sum_i x_i$.
\noindent We introduce two partial minimization problems over the subsets $\mbox{{\boldmath $\mathcal{Y}$}}$ and $\mbox{{\boldmath $\mathcal{W}$}}$. Recall that the I-divergence between two nonnegative matrices of the same sizes $M, N\in\mathbb{R}^{p\times q}_+$ is defined as \[
\mathcal{I}(M||N) := \sum_{i,j} \Big( M_{ij} \log \frac{M_{ij}}{N_{ij}} - M_{ij} + N_{ij}\Big)\,. \]
\begin{problem}\label{problemyhalf}\, Given $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$, minimize $\mathcal{I}(\mathbf{Y}||\mathbf{W})$ over $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$. \end{problem}
\begin{problem}\label{problemwhalf}\, Given $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$, minimize $\mathcal{I}(\mathbf{Y}||\mathbf{W})$ over $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$. \end{problem} The solutions to both problems can be given in closed form.
\begin{lemma}\label{lemmayhalf} Problem~\ref{problemyhalf} has the explicit minimizer $\mathbf{Y}^\star=\mathbf{Y}^\star(\mathbf{W})$ given by \begin{align} \mathbf{Y}^\star_{ij} & = \frac{\mathbf{W}_{ij}}{\sum_j\mathbf{W}_{ij}}\,y_i
= \left\{ \begin{array}{ll}\dfrac{x_{i-j}x_j}{(x* x)_i}\,y_i & \mbox{ if }\, 0\le j\leq i \leq j+ m,\\ \\ 0 & \mbox{ otherwise}. \end{array} \right. \label{eq:ystarhalf} \end{align} Moreover the \emph{Pythagorean identity} \begin{equation}\label{eq:pyth-1}
\mathcal{I}(\mathbf{Y}||\mathbf{W})=\mathcal{I}(\mathbf{Y}||\mathbf{Y}^\star)+\mathcal{I}(\mathbf{Y}^\star||\mathbf{W})\,, \end{equation} holds for any $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$, and \begin{equation}\label{eq:fallback}
\mathcal{I}(\mathbf{Y}^\star||\mathbf{W})=\mathcal{I}(y||x* x)\,. \end{equation} \end{lemma}
\begin{proof} Proceed by direct computation. The Lagrangian function is \[ L=\sum_{ij}\big(\mathbf{Y}_{ij}\log \mathbf{Y}_{ij}-\mathbf{Y}_{ij}\log \mathbf{W}_{ij}-\mathbf{Y}_{ij}+\mathbf{W}_{ij}\big)-\sum_i\lambda_i\big(\sum_j\mathbf{Y}_{ij}-y_i\big)\,, \] therefore \[ \frac{\partial L}{\partial \mathbf{Y}_{ij}}=\log\mathbf{Y}_{ij}-\log\mathbf{W}_{ij}-\lambda_i=0\,, \] yields $\mathbf{Y}_{ij}=\mathbf{W}_{ij}e^{\lambda_i}$ and imposing the marginal constraint $\sum_j \mathbf{Y}_{ij}=y_i$ one gets the asserted minimizer~\eqref{eq:ystarhalf}. Next, introducing the notation $\mathbf{W}_{i\,\cdot}=\sum_j\mathbf{W}_{ij}$ and $\mathbf{Y}_{i\,\cdot}=\sum_j\mathbf{Y}_{ij}$, substitution into the RHS of \eqref{eq:pyth-1} gives \begin{align*}
\mathcal{I}(\mathbf{Y} &||\mathbf{Y}^\star) + \mathcal{I}(\mathbf{Y}^\star||\mathbf{W}) \\ &= \sum_{ij}\Big(\mathbf{Y}_{ij}\log \frac{\mathbf{Y}_{ij}}{\mathbf{Y}^\star_{ij}}-\mathbf{Y}_{ij}+\mathbf{Y}^\star_{ij}\Big) + \Big(\mathbf{Y}^\star_{ij}\log \frac{\mathbf{Y}^\star_{ij}}{\mathbf{W}_{ij}}-\mathbf{Y}^\star_{ij}+\mathbf{W}_{ij}\Big) \\ & = \sum_{ij}\Big(\mathbf{Y}_{ij}\log \frac{\mathbf{Y}_{ij}}{\mathbf{W}_{ij}}-\mathbf{Y}_{ij}\log \frac{y_i}{\mathbf{W}_{i\,\cdot}}-\mathbf{Y}_{ij}\Big)
+ \Big(\frac{y_i}{\mathbf{W}_{i\,\cdot}}\mathbf{W}_{ij}\log\frac{y_i}{\mathbf{W}_{i\,\cdot}} +\mathbf{W}_{ij}\Big) \\
& = \mathcal{I}(\mathbf{Y}||\mathbf{W})\,, \end{align*} thus proving~\eqref{eq:pyth-1}. As a byproduct of the Pythagorean identity one gets that $\mathbf{Y}^\star$ is indeed a minimizer for Problem \ref{problemyhalf}. Finally, using $\mathbf{W}_{ij}=x_{i-j}x_j$, and $\mathbf{W}_{i\,\cdot}=(x* x)_i$ one finds that the optimal value of Problem~\ref{problemyhalf} coincides with~\eqref{eq:fallback}. Indeed, \begin{align*}
\mathcal{I}(\mathbf{Y}^\star||\mathbf{W}) & = \sum_{ij}\Big(\mathbf{W}_{ij}\frac{y_i}{\mathbf{W}_{i\,\cdot}}\log\frac{y_i}{\mathbf{W}_{i\,\cdot}}-\mathbf{W}_{ij}\frac{y_i}{\mathbf{W}_{i\,\cdot}}+\mathbf{W}_{ij}\Big) \\ & = \sum_{i} \Big(y_i\log\frac{y_i}{\mathbf{W}_{i\,\cdot}}-y_i+\mathbf{W}_{i\,\cdot}\Big) \\ & = \sum_{i}\Big(y_i\log\frac{y_i}{(x* x)_i}-y_i+(x* x)_i\Big) \\
& = \mathcal{I}(y||x* x)\,. \end{align*} \end{proof}
\begin{remark} \label{rem:sym-y} Note that the minimizer $\mathbf{Y}^\star$ in~\eqref{eq:ystarhalf} exhibits always the following symmetry \begin{equation} \label{eq:sym-y} \mathbf{Y}^\star_{j+\ell, \ell} = \mathbf{Y}^\star_{j+\ell, j}\,,\quad \text{for all}\,\,\, \ell, j=0,\dots, m\,, \end{equation} i.e., for all $j=0,\dots, m$, the $j$-th subdiagonal of $\mathbf{Y}^\star$ and the $(\mathbf{Y}_{j,j}, \dots, \mathbf{Y}_{j+m,j})^\top$ subvector of its $j$-th column coincide. \end{remark}
\begin{lemma}\label{lemmawhalf} Problem~\ref{problemwhalf} has explicit minimizer $\mathbf{W}^\star=\mathbf{W}^\star(\mathbf{Y})$ corresponding to $x^\star_j$ as follows, \begin{equation}\label{eq:xstarhalf} x_j^\star=\frac{\widehat\mathbf{Y}_j}{2\sqrt{\sum_{i=0}^{2m}y_i}}\,\raisepunct{,} \qquad j=0,\dots, m\,, \end{equation} where \begin{equation}\label{eq:yhatl} \widehat\mathbf{Y}_j:=\sum_{i=0}^m\mathbf{Y}_{i+j,i} + \sum_{i=j}^{j+m}\mathbf{Y}_{ij}\,, \qquad j=0,\dots, m\,. \end{equation} Moreover the \emph{Pythagorean identity} \begin{equation}\label{eq:pyth-2}
\mathcal{I}(\mathbf{Y}||\mathbf{W})=\mathcal{I}(\mathbf{Y}||\mathbf{W}^\star)+\mathcal{I}(\mathbf{W}^\star||\mathbf{W})\,, \end{equation} holds for any $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$. \end{lemma}
\begin{proof} Minimizing the I-divergence \[
\mathcal{I}(\mathbf{Y}||\mathbf{W})=\sum_{j=0}^m\sum_{i=j}^{j+m} \,\Big(\,\mathbf{Y}_{ij}\log\frac{\mathbf{Y}_{ij}}{\mathbf{W}_{ij}}- \mathbf{Y}_{ij}+ \mathbf{W}_{ij}\,\Big)\,, \] with respect to $\mathbf{W}\in \mbox{{\boldmath $\mathcal{W}$}}$, is equivalent, since $\mathbf{W}_{ij}=x_{i-j}x_j$, to minimizing \begin{align}\label{eq:fwhalf} F(x) &:= \sum_{j=0}^m\sum_{i=j}^{j+m}\Big(-\mathbf{Y}_{ij}\log (x_{i-j}x_j) + x_{i-j}x_j\Big) \nonumber\\ & = \sum_{j=0}^m\sum_{i=j}^{j+m} \Big(-\mathbf{Y}_{ij}\log x_{i-j} - \mathbf{Y}_{ij}\log x_j + x_{i-j}x_j \Big)\,. \end{align} Applying to the first and third double sums in~(\ref{eq:fwhalf}) the identity \begin{equation}\label{eq:fubini-sum} \sum_{j=0}^m\sum_{i=j}^{j+m} a(i,j) = \sum_{j=0}^m\sum_{\ell=0}^{m} a(\ell+j,\ell)\,, \end{equation} and recalling the definition~(\ref{eq:yhatl}), one easily gets \begin{equation} \label{eq:Fx-nice} F(x) = -\sum_{j=0}^m\widehat\mathbf{Y}_{j}\log x_j + \Big(\sum_{j=0}^m x_j\Big)^2\,. \end{equation}
\noindent The partial derivatives of $F$ immediately follow from Equation~\eqref{eq:Fx-nice} as \[ \frac{\partial F}{\partial x_j}= -\frac{\widehat \mathbf{Y}_j}{x_j}+2\sum_{\ell=0}^{m}x_\ell\,,\qquad j=0,\dots, m\,. \] Setting $\frac{\partial F}{\partial x_j}=0$ gives \[ x_j^\star=\frac{\widehat\mathbf{Y}_j}{2\sum_{\ell=0}^{m}x_\ell^\star}\,\raisepunct{,} \qquad j=0,\dots, m\,, \] and hence by summation \begin{equation}\label{eq:normal-conv} \Big(\sum_{j=0}^m x_j^\star\Big)^2=\half\sum_{j=0}^m\widehat\mathbf{Y}_j = \sum_{i=0}^{2m} y_i\,. \end{equation}
where, to prove the last identity, it is sufficient to observe that equation~(\ref{eq:yhatl}) defines $\widehat\mathbf{Y}_j$ as the sum of the $j$-th subdiagonal and $j$-th column of the matrix $\mathbf{Y}\in \mbox{{\boldmath $\mathcal{Y}$}}$. This completes the proof of~(\ref{eq:xstarhalf}). To prove the Pythagorean identity~(\ref{eq:pyth-2}) it is convenient to prove that $\mathcal{I}(\mathbf{Y}||\mathbf{W})-\mathcal{I}(\mathbf{Y}||\mathbf{W}^\star)=\mathcal{I}(\mathbf{W}^\star||\mathbf{W})$, which is equivalent to $$ \sum_{j=0}^m\sum_{i=j}^{j+m} \mathbf{Y}_{ij}\log\frac{x_{i-j}^\star x_j^\star}{x_{i-j}x_j} = \sum_{j=0}^m\sum_{i=j}^{j+m} x_{i-j}^\star x_j^\star \log\frac{x_{i-j}^\star x_j^\star}{x_{i-j}x_j}\,\raisepunct{.} $$ The last identity is easily verified by direct substitution of~(\ref{eq:xstarhalf}) and~(\ref{eq:yhatl}) to express $x^\star_j$, and using the identity~(\ref{eq:fubini-sum}). Again, as a byproduct of the Pythagorean identity one gets that $\mathbf{W}^\star$ is indeed a minimizer for Problem \ref{problemwhalf}. \end{proof}
\begin{remark} Problem~\ref{problemwhalf} admits an interesting interpretation as a symmetric (constrained) rank one approximation of a given nonnegative matrix. We introduce the square matrices $\overline Y, \overline W \in \mathbb R^{(m+1)\times (m+1)}$, as `rectifications' of the $\mathbf{Y}$ and $\mathbf{W}$ matrices, defined as $$ \overline Y_{ij}=\mathbf{Y}_{i+j,j}\,, \qquad \overline W_{ij}=W_{i+j,j}=x_i x_j\,. $$ Problem~\ref{problemwhalf} can be rephrased as $$ \min_{x\in \mathbb R^{m+1}_+} D(\overline Y \mid\mid x x^\top)\,, $$ whose solution is attained at $$ x_i^\star = \frac{1}{2}\, \frac{\overline Y_{\cdot\,i}+\overline Y_{i\,\cdot}}{\sqrt{\sum_{ij}\overline Y_{ij}}}\,. $$ In the probabilistic case ($\sum_{ij} \overline Y_{ij}=1$), the interpretation is that the best approximation of a two-dimensional distribution ($\overline Y$) by an i.i.d.\ product distribution ($x x^{\top}$) is attained at $x^\star$ equal to the average of the row and column marginals of $\overline Y$. \end{remark}
\begin{remark} In the next section, when considering Problem~\ref{problemwhalf}, the given $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$ will always exhibit symmetry~(\ref{eq:sym-y}). When this is the case, Equation~(\ref{eq:xstarhalf}) for the optimal $x^\star$ simplifies considerably. Indeed, under symmetry~(\ref{eq:sym-y}), Equation~(\ref{eq:yhatl}) becomes \begin{equation*}\label{eq:yhatlsimple} \widehat\mathbf{Y}_j=\sum_{\ell=0}^m\mathbf{Y}_{\ell+j,\ell} + \sum_{i=j}^{j+m}\mathbf{Y}_{ij}= 2 \sum_{i=j}^{j+m}\mathbf{Y}_{ij}\,, \qquad j=0,\dots, m \,, \end{equation*} Equation~(\ref{eq:xstarhalf}) then reduces to \begin{equation}\label{eq:xstarhalfsimple} x_j^\star=\frac{1}{c}\,\sum_{i=j}^{j+m}\mathbf{Y}_{ij} =\frac{1}{c}\,\sum_{\ell=0}^{m}\mathbf{Y}_{\ell+j,j}\,, \end{equation} where \begin{equation}\label{eq:def-c} c:=\sqrt{\textstyle{\sum_{i=0}^{2m}y_i}}= \sum_{j=0}^m x_j^\star\,. \end{equation} \end{remark} The connection between the original Problem~\ref{problemhalf} and the two lifted minimization problems is explained in the next proposition.
\begin{proposition} The minimum of the original Problem~\ref{problemhalf} coincides with the double minimization Problems~\ref{problemyhalf} and~\ref{problemwhalf}, i.e.\ \[
\min_{x\in\mathbb{R}^{m+1}_+}\mathcal{I}(y||x* x)=\min_{\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}},\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}}\mathcal{I}(\mathbf{Y}||\mathbf{W}). \] \end{proposition} \begin{proof} For given $x\in\mathbb{R}^{m+1}_+$, with corresponding $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$, and $\mathbf{Y}\in\mbox{{\boldmath $\mathcal{Y}$}}$ consider the optimizers $\mathbf{Y}^\star$ and $\mathbf{W}^\star$ from Lemmas~\ref{lemmayhalf} and \ref{lemmawhalf} and recall Equation~\eqref{eq:fallback}. Then
$\mathcal{I}(\mathbf{Y}||\mathbf{W})\geq \mathcal{I}(\mathbf{Y}^\star||\mathbf{W})=\mathcal{I}(y|x* x)\geq \min_x\mathcal{I}(y||x* x)$, where the use of the minimum is justified by Proposition~\ref{proposition:exist}. Taking the joint minimum on the left hand side over $\mathbf{Y}$ and $\mathbf{W}$, justified by the just cited lemmas, leads to $\min_{\mathbf{Y},\mathbf{W}}\mathcal{I}(\mathbf{Y}||\mathbf{W})\geq \min_{x}\mathcal{I}(y||x* x)$. Conversely, for given $x\in\mathbb{R}^{m+1}_+$ with corresponding $\mathbf{W}\in\mbox{{\boldmath $\mathcal{W}$}}$, recalling again~\eqref{eq:fallback}, one obtains \[
\mathcal{I}(y||x* x)= \mathcal{I}(\mathbf{Y}^\star||\mathbf{W})= \min_\mathbf{Y}\mathcal{I}(\mathbf{Y}||\mathbf{W})\geq \min_\mathbf{W}\min_\mathbf{Y}\mathcal{I}(\mathbf{Y}||\mathbf{W})\,, \]
which, taking the minimum $x$, shows that $ \min_{x}\mathcal{I}(y||x* x)\geq \min_{\mathbf{Y},\mathbf{W}}\mathcal{I}(\mathbf{Y}||\mathbf{W})$, thus concluding the proof. \end{proof}
\section{The algorithm}\label{section:algo}
This section is the core of the paper. It contains an algorithm aiming at finding a minimizer of Problem~\ref{problemhalf}, which we know to exist in view of Proposition~\ref{proposition:exist}, and an analysis of its behavior.
\subsection{Construction of the algorithm, basic properties}
Starting at an initial $\mathbf{W}^0\in \mbox{{\boldmath $\mathcal{W}$}}$ and combining the two partial minimization problems, one produces a classic alternating minimization sequence, \begin{equation}\label{eq:altminseq} \cdots \, \mathbf{W}^t\stackrel{1}{\longrightarrow} \mathbf{Y}^t\stackrel{2}{\longrightarrow} \mathbf{W}^{t+1}\stackrel{1}{\longrightarrow} \mathbf{Y}^{t+1} \, \cdots, \end{equation} where the superscript $t\in\mathbb N$ denotes the iteration step. The arrow $\stackrel{1}{\longrightarrow}$ denotes the partial minimization Problem~\ref{problemyhalf}, the matrix at the tail of the arrow is the given input, and the matrix at the head $\mathbf{Y}^t=\mathbf{Y}^\star(\mathbf{W}^t)$, is the optimal solution. The meaning of $\stackrel{2}{\longrightarrow}$ is analogous, and represents the partial minimization Problem~\ref{problemwhalf}, and $\mathbf{W}^{t+1}=\mathbf{W}^\star(\mathbf{Y}^t)$. Note that, at each iteration, $\mathbf{W}^t$ is completely specified by the fixed data $y$ and by the vector $x^t=(x_0^t,\dots, x_m^t) \in\mathbb R_+^{m+1}$. An iterative algorithm for the minimization Problem~\ref{problemhalf}, solely in terms of $x^t$, can be extracted from the sequence~(\ref{eq:altminseq}) as it immediately follows combining Lemmas~\ref{lemmayhalf} and~\ref{lemmawhalf}. The update equation, say $x^{t+1}=I(x^t)$, is given below. \begin{algorithm}\label{algorithm:half} Starting from an arbitrary vector $x^0\in \mathbb R^{m+1}_+$ the update equation $x^{t+1}=I(x^t)$ is given componentwise by \begin{equation} \label{eq:algohalf} x^{t+1}_j= x^t_{j}\, \frac{1}{c} \sum_{\ell=0}^{m}\frac{x^t_\ell \,y_{\ell+j}}{(x^t* x^t)_{\ell+j}}\,, \qquad j=0,\dots, m\,. \end{equation} \end{algorithm} To verify Equation~(\ref{eq:algohalf}) it is enough to shunt the $\mathbf{Y}^t$ step in the chain~(\ref{eq:altminseq}) and concatenate directly $\mathbf{W}^t$ to $\mathbf{W}^{t+1}$. Starting with Equation~(\ref{eq:xstarhalfsimple}) and recalling the expression of $\mathbf{Y}^t$ given by Equation~(\ref{eq:ystarhalf}) one has \begin{equation}\label{eq:update} x^{t+1}_j = \frac{1}{c}\,\sum_{\ell=0}^{m}\mathbf{Y}^t_{\ell+j,j} = \frac{1}{c} \, \sum_{\ell=0}^{m}\frac{x^t_\ell x^t_{j}}{(x^t* x^t)_{\ell+j}} y_{\ell+j}\,, \end{equation} which coincides with~(\ref{eq:algohalf}).
\begin{remark} Application of Algorithm~\ref{algorithm:half} to Example~\ref{expl:trivial-half-case} gives the exact solution in one step, starting from any initial $x^0_j>0$, as is easily verified. This is an exceptional case. \end{remark} The portmanteau proposition below summarizes some useful properties of the algorithm.
\begin{proposition}\label{proposition:properties} The iterates $x^t,\, t\ge 0$, of Algorithm~\ref{algorithm:half} satisfy the following properties. \begin{enumerate}[itemsep=-.1em,topsep=-.2em,label=(\roman{*}),labelsep=0.9em] \item\label{item:hpositive} If $x^0>0$ componentwise, then $x^t>0$ componentwise, for all $t>0$. \item\label{item:simplex} $x^t$ belongs to the simplex $\mathcal{S}=\{x\in\mathbb{R}^{m+1}_+: \sum_{i=0}^mx_i=c\}$ for all $t>0$. \item\label{item:Wt+1}
$\mathcal{I}(y||x^t* x^t)$ decreases at each iteration, in fact one has \begin{equation}\label{eq:gain1}
\mathcal{I}(y||x^t* x^t) - \mathcal{I}(y||x^{t+1}* x^{t+1}) = \mathcal{I}(\mathbf{Y}^t||\mathbf{Y}^{t+1}) + \mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)\ge 0\,, \end{equation}
and, as a corollary, $\mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)$ vanishes asymptotically. \item If $y=x^t* x^t$ then $x^{t+1}=x^t$, i.e.\ perfect matches are fixed points of the algorithm. \item The update equation~(\ref{eq:algohalf}) can be written in the form \begin{equation} \label{eq:algo-alt} x_j^{t+1} = x_j^{t} \Big(1 - \frac{1}{2c}\nabla_j \mathcal{I}(x^t)\Big)\,. \end{equation} \item If $\nabla_j \mathcal{I}(x^t)=0$ then $x^{t+1}_j =x_j^t$, and if $\nabla \mathcal{I}(x)=0$ then $x^{t+1}=x^t$, i.e.\ stationary points of $\mathcal{I}(x)$ are fixed points of the algorithm. \item If $\mathcal{I}(x^t)$ is increasing (decreasing) in $x^t_j$, then $x^{t+1}_j<x^t_j$\, ($x^{t+1}_j>x^t_j$). \end{enumerate} \end{proposition}
\begin{proof}\mbox{}
\noindent (i)\, Obvious from \eqref{eq:algohalf}.
\noindent (ii)\, Consider the first equality in \eqref{eq:update}. Summing over $j$ gives $$ \sum_{j=0}^mx^{t+1}_j = \frac{1}{c}\sum_{j=0}^m \sum_{\ell=0}^{m} \mathbf{Y}^t_{\ell+j,j}=c\,, $$ \noindent in view of the two equalities in \eqref{eq:xstarhalfsimple} and \eqref{eq:def-c}.
\noindent (iii)\, Combining the Pythagorean identities~(\ref{eq:pyth-1}), (\ref{eq:pyth-2}) for the chain~(\ref{eq:altminseq}) one gets \begin{equation*}
\mathcal{I}(\mathbf{Y}^t||\mathbf{W}^t) = \mathcal{I}(\mathbf{Y}^t||\mathbf{Y}^{t+1}) + \mathcal{I}(\mathbf{Y}^{t+1}||\mathbf{W}^{t+1}) + \mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)\,, \end{equation*}
from which Equation~\eqref{eq:gain1} follows applying Equation~(\ref{eq:fallback}). The corollary is proved noting that the decreasing sequence $\mathcal{I}(y||x^t* x^t)$ certainly has a limit therefore the LHS of the equation vanishes asymptotically and so do the terms on the RHS which are nonnegative for all $t>0$.
\noindent (iv)\, Under the assumption, \eqref{eq:algohalf} reduces to $x^{t+1}_j= x^t_j \frac{1}{c}\sum_{\ell=0}^m x^t_\ell =x^t_j$ in view of \ref{item:simplex}.
\noindent (v)\, From \eqref{eq:gradj} one gets $\nabla_j \mathcal{I}(x) = -2 \,\sum_{\ell=0}^m \frac{x_\ell\, y_{\ell+j}}{(x* x)_{\ell+j} } +2\sum_{\ell=0}^m x_\ell$, and recalling that $x^t\in \mathcal S$ it follows $\nabla_j \mathcal{I}(x^t) = -2 \,\sum_{\ell=0}^m \frac{x^t_\ell\, y_{\ell+j}}{(x^t* x^t)_{\ell+j} } +2\,c$. Hence, the update equation~(\ref{eq:algohalf}) can be written as in \eqref{eq:algo-alt}.
\noindent (vi), (vii) follow immediately from (v). \end{proof}
\subsection{Convergence analysis}
The aim of this section is to investigate the behaviour of Algorithm~\ref{algorithm:half} for large values of the iteration index $t$. We start with a technical lemma. \begin{lemma}\label{lemma:tt1} For the iterates $x^t$ and their corresponding $\mathbf{W}^t$ it holds that \begin{enumerate}[itemsep=-.1em,topsep=-.2em,label=(\roman{*}),labelsep=0.9em] \item\label{item:IWt}
$\mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)=2 c\, \mathcal{I}(x^{t+1}||x^t)$, \item
$\sum_i|x^{t+1}_i-x^t_i|\leq\big(\mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)\big)^{1/2}$, \item
$\lim_{t\rightarrow \infty} \mathcal{I}(x^{t+1}||x^t) = 0$, and hence $\sum_i|x^{t+1}_i-x^t_i|\to 0$. \end{enumerate} \end{lemma} \begin{proof} To prove (i) a direct computation gives \begin{align}
\mathcal{I}(\mathbf{W}^{t+1}&||\mathbf{W}^t)=\sum_{j=0}^m\sum_{i=j}^{j+m} \bigg(\mathbf{W}_{ij}^{t+1} \log \frac{\mathbf{W}_{ij}^{t+1}}{\mathbf{W}_{ij}^t}-\mathbf{W}^{t+1}_{ij}+\mathbf{W}^t_{ij}\bigg) \nonumber\\ &= \sum_{j=0}^m \sum_{i=j}^{j+m} x_{i-j}^{t+1}x_j^{t+1} \log \frac{x_{i-j}^{t+1}x_j^{t+1}}{x^t_{i-j}x^t_j} = \sum_{j=0}^m \sum_{\ell=0}^m x_\ell^{t+1}x_j^{t+1} \log \frac{x_\ell^{t+1}x_j^{t+1}}{x^t_\ell x^t_j} \nonumber\\
&= 2 \bigg(\sum_{\ell=0}^m x_\ell^{t+1}\bigg) \sum_{j=0}^m x_j^{t+1} \log\frac{x_j^{t+1}}{x^t_j} = 2 c\, \mathcal{I}(x^{t+1}||x^t)\,, \label{eq:IWIx} \end{align} where the last identity follows from Equation~(\ref{eq:normal-conv}).
To prove (ii) recall Pinsker's inequality which states, for probability vectors $p, q$, that $\sum_i|p_i-q_i|\leq \left(2\mathcal{I}(p||q)\right)^{1/2}$. The iterates $x^t$ and $x^{t+1}$ are not probability vectors in general, but both belong to the simplex $\mathcal{S}$ therefore, by an easy corollary to Pinsker's inequality, $\sum_i|x^{t+1}_i-x^t_i|\leq \left(2c\,\mathcal{I}(x^{t+1}||x^t)\right)^{1/2}$, from which, by direct application of (i), one gets (ii).
Finally, (iii) descends from the fact that $\mathcal{I}(\mathbf{W}^{t+1}||\mathbf{W}^t)$ vanishes asymptotically, as proved by the corollary to Equation (\ref{eq:gain1}), and therefore, applying (i) again, so does $\mathcal{I}(x^{t+1}||x^t)$ and by Pinsker's inequality also $\sum_i|x^{t+1}_i-x^t_i|$. \end{proof} The existence of limit points of the sequence $(x^t)$ of the iterates of the algorithm is obvious as all $x^t$ belong to the simplex $\mathcal{S}$, see Proposition~\ref{proposition:properties}, which is a compact set. Note that the sequence $(x^t)$ depends on the initial point $x^0$. Changing $x^0$ the sequence $(x^t)$ changes and so do, in general, its limit points. To avoid a cluttered notation the dependence of the limit points on $x^0$ will not be evidenced. We continue with establishing some properties of the limit points of $x^t$.
\begin{lemma} \label{lem:conv-2} If $x^\infty$ is a limit point of the sequence $(x^t)$ then it is a fixed point of the algorithm, i.e. $$ x^\infty = I(x^\infty)\,. $$ \end{lemma} \begin{proof}
Let $x^\infty$ be a limit point of the $x^t$. The map $x^{t+1}=I(x^t)$, given componentwise in~(\ref{eq:algohalf}), is continuous. Likewise the I-divergence $\mathcal{I}(u||v)$ is jointly continuous in $(u, v)$ for all $v>0$. It follows that $\mathcal{I}(I(x^\infty)||x^\infty)$ is a limit point of the $\mathcal{I}(x^{t+1}||x^t)$ which, by Lemma~\ref{lemma:tt1} (iii), vanishes asymptotically implying that $\mathcal{I}(I(x^\infty)||x^\infty)=0$, which yields $x^\infty=I(x^\infty)$, i.e.\ $x^\infty$ is a fixed point of the algorithm. \end{proof}
\begin{proposition}\label{prop:limitpoint}
The I-divergence $\mathcal{I}(y||x^\infty * x^\infty)$ is constant over the set of all limit points $x^\infty$ of $(x^t)$. \end{proposition}
\begin{proof} Iteration of \eqref{eq:gain1} gives for $t\leq T$ \begin{equation}\label{eq:gain2}
\mathcal{I}(y||x^t* x^t) - \mathcal{I}(y||x^{T}* x^{T}) = \sum_{k=t}^{T-1}\Big(\mathcal{I}(\mathbf{Y}^{k+1}||\mathbf{Y}^{k}) + \mathcal{I}(\mathbf{W}^{k+1}||\mathbf{W}^k)\Big)\,. \end{equation} Suppose that the $x^T$ converge along a subsequence to $x^\infty$. Then we also have \begin{equation}\label{eq:gain3}
\mathcal{I}(y||x^t* x^t) - \mathcal{I}(y||x^{\infty}* x^{\infty}) = \sum_{k=t}^{\infty}\Big(\mathcal{I}(\mathbf{Y}^{k+1}||\mathbf{Y}^{k}) + \mathcal{I}(\mathbf{W}^{k+1}||\mathbf{W}^k)\Big)\,. \end{equation}
Suppose $x'$ is another limit point and $x^t$ converges to $x'$ along a suitable subsequence indexed by $t^\prime$. Taking the limit for $t = t'\to\infty$ in Equation (\ref{eq:gain3}), one sees that the RHS vanishes, whereas the LHS gives $\mathcal{I}(y||x'* x')-\mathcal{I}(y||x^\infty * x^\infty)$, which is thus zero. \end{proof}
\begin{remark}\label{remark:cc} Proposition~\ref{prop:limitpoint} makes it clear that all limit points of $x^t$ are equivalent, in the sense that their autoconvolutions have the same informational distance to the target $y$ in Problem \ref{problemhalf}. In particular, if one limit point is a minimizer, so are all other limit points.
\noindent One can show that the set of limit points of the sequence $(x^t)$ is compact and connected. Compactness follows from Proposition~\ref{prop:limitpoint} (the set of limit points is closed and contained in the simplex $\mathcal{S}$, hence bounded), whereas connectedness is essentially a consequence of Lemma~\ref{lemma:tt1}. A similar statement can be found in \cite{cover1984}. \end{remark}
\begin{proposition}\label{prop:kt} Limit points of the sequence $(x^t)$ are Kuhn-Tucker points of the minimization Problem~\ref{problemhalf}. \end{proposition}
\begin{proof} Recall the version of the update equation of the algorithm as in \eqref{eq:algo-alt}. By Lemma~\ref{lem:conv-2}, if $x^\infty$ is a limit point of the $x^t$ then it is a fixed point of the algorithm, and Equation~(\ref{eq:algo-alt}) reduces to \begin{equation*} \label{eq:algo-alt-stat} x_j^\infty = x_j^\infty \Big(1 - \frac{1}{2c}\nabla_j\mathcal{I}(x^\infty)\Big)\,, \end{equation*} showing that, if $x^\infty_j>0$ then $\nabla_j\mathcal{I}(x^\infty)=0$. To complete the verification that $x^\infty$ satisfies the Kuhn-Tucker conditions for $\mathcal{I}(x)$ one has to check that if $x^\infty_j=0$ then $\nabla_j\mathcal{I}(x^\infty)\geq 0$. So we proceed with investigating limit points on the boundary. For a given initial condition $x^0$, let $(x^t)$ be the sequence of iterates of the algorithm and define $O=\{x\in\mathbb{R}^{m+1}_+: \nabla_j\mathcal{I}(x)<0\}$. Put $L_0=0$ and let $U_0=\inf\{t> 0: x^t \in O^c\}$. If $U_0=\infty$, then all $x^t$ belong to $O$ and the $x_j^t$ form an increasing sequence in view of Equation~\eqref{eq:algo-alt}, so certainly all $x^t_j>x^0_j>0$ and a limit point with $x_j^\infty=0$ cannot occur. If $U_0$ is finite, we put $L^1=\inf\{t>U_0: x^t\in O\}$. If $L_1=\infty$, then $x^t\in O^c$ for all $t\geq U_0$, so the $x^t_j$ form a decreasing sequence, converging to some $x^\infty_j\ge 0$. With $\nabla_j\mathcal{I}(x^t)\geq 0$ for all $t$, then necessarily also $\nabla_j\mathcal{I}(x^\infty)\geq 0$, hence $x^\infty$ satisfies the Kuhn-Tucker conditions. In case $L^1<\infty$ continue by alternating definitions, $U_1=\inf\{t > L_1: x^t\in O^c\}$, $L_2=\inf\{t > U_1: x^t \in O\}$, etc. As soon as some $L_k$ or $U_k$ is infinite, we are in either of the situations just described and in a limit point one necessarily has $x_j^\infty>0$ or $x^\infty_j\ge 0$ and $\nabla_j\mathcal{I}(x^\infty)\geq 0$ satisfying the Kuhn-Tucker conditions.
As a last case, we investigate what happens if all $L_k$ and $U_k$ are finite and the interest is in possible boundary limit points $x^\infty$ with $x^\infty_j=0$. Observe that for $t$ between the $L_k$ and $U_k$ the $x^t_j$ are increasing and for $t$ between the $U_k$ and $L_{k+1}$ the $x^t_j$ are decreasing. More precisely, for $L_k\leq t <U_k$ it holds that $x^{t+1}_j\leq x^t_j$ and for $U_k\leq t <L_{k+1}$ it holds that $x^{t+1}_j> x^t_j$. In particular $x^{L_k}_j\leq x^{L_k-1}_j$ and $x^{L_k}_j< x^{L_k+1}_j$, hence $x^{L_k}_j$ is a local minimum of the $x^t_j$. Suppose that $x^\infty$ is a limit point, with $x^\infty_j=0$. Then we have to consider the liminf of the $x^t_j$, which coincides with the liminf of the $x^{L_k}_j$. But, by Lemma~\ref{lemma:tt1}, then also $x^{L_k-1}_j$ converges along a subsequence to the same liminf, and in these points one has $\nabla_j\mathcal{I}(x^{L_k-1})\geq 0$. Hence along any convergent subsequence of the $x^t$ with $\liminf x^t_j=0$, one necessarily has $\nabla_j\mathcal{I}(x^{\infty})\geq 0$. As a side remark, in this last case, since $\nabla_j\mathcal{I}(x^{L_k})< 0$ for all $k$ one gets in fact $\nabla_j\mathcal{I}(x^{\infty})= 0$. \end{proof}
\subsection{Convergence properties, further considerations}
All empirical examples suggest that the iterates of Algorithm~\ref{algorithm:half} converge to a limit. Although a full proof cannot be given, a number of considerations make this result plausible, also from a theoretical point of view.
On a technical note, in order to prove that the algorithm converges, one would need to show that $\mathcal{I}(x^\infty||x^t)$ is decreasing in $t$, for any limit point $x^\infty$. The proof of this property would go along the arguments of Lemma~A.1 of \cite{Vardietal1985} or Lemma~24 in \cite{finessospreij2015ieeeit}, if one could prove that, in our notation, $\mathcal{I}(\mathbf{W}^\infty||\mathbf{W}^t)\leq c\, \mathcal{I}(x^\infty||x^t)$. Unfortunately it is only possible to prove the looser inequality $\mathcal{I}(\mathbf{W}^\infty||\mathbf{W}^t)\le 2c\, \mathcal{I}(x^\infty||x^t)$. The factor 2 essentially appears as a consequence of the `quadratic nature in $x$' of the autoconvolution terms $(x* x)_i$ whereas terms of type $(u* x)_i$, appearing in the context of e.g.\ \cite{finessospreij2015ieeeit} or \cite{finessospreij2019automatica}, are linear in $x$. Consequently one cannot conclude that the $x^t$ converge to a global minimizer. For completeness we present, in Proposition~\ref{proposition:decrease}, the proof of convergence of the algorithm under the proviso, empirically satisfied in all cases, that $\mathcal{I}(x^\infty||x^t)$ decreases in $t$. In the simulations Section~\ref{section:numerics} we shall see an example where convergence of the $x^t$ occurs, but not to a global minimizer of $\mathcal{I}(x)$.
\begin{proposition}\label{proposition:decrease}
Let $x^\infty$ be a limit point of the sequence $(x^t)$ and assume that $\mathcal{I}(x^\infty||x^t)$ is decreasing in $t$. Then $x^t$ converges to $x^\infty$, which is the unique limit point of $x^t$. \end{proposition} \begin{proof}
By Proposition~\ref{proposition:properties}\ref{item:simplex} the $x^t$ belong to $\mathcal S$ and therefore, along some subsequence, $x^{t_k}\rightarrow x^\infty$, for some limit point $x^\infty\in \mathcal{S}$. By continuity $\mathcal{I}(x^\infty||x^{t_k})\rightarrow 0$. On the other hand, as the divergences $\mathcal{I}(x^\infty||x^t)$ are decreasing, it must hold that $\mathcal{I}(x^\infty||x^t)\rightarrow 0$. Using Pinsker's inequality as in the proof of Lemma~\ref{lemma:tt1}, $\sum_i|x^\infty_i-x^t_i|\leq \left(2c\,\mathcal{I}(x^\infty||x^t)\right)^{1/2}$, one concludes that $x^t \rightarrow x^\infty$, and hence that $x^\infty$ is the unique limit point. \end{proof} Next to the empirically observed behavior in Proposition~\ref{proposition:decrease}, we present another argument for convergence based on an element of Morse theory, for which we need the Hessian of the criterion $\mathcal{I}(x)$. Differentiate $\frac{\partial\mathcal{I}(x)}{\partial x_j}$ as given by \eqref{eq:gradj} w.r.t.\ $x_i$ to get \[ H_{ij}(x):=-2\frac{y_{i+j}}{(x* x)_{i+j}}+4\sum_{l=0}^{2m}\frac{y_{l+j}}{(x* x)_{l+j}^2}x_lx_{l+j-i}+2. \] Note that effectively the index $l$ in the summation runs from $\max\{i-j,0\}$ to $m+\min\{i-j,0\}$, because of our convention $x_\ell=0$ for $\ell<0$ or $\ell >m$. The expression for $H_{ij}(x)$ can be rewritten as \[ H_{ij}(x):=-2\frac{y_{i+j}}{(x* x)_{i+j}}+4\sum_{k=0}^{2m}\frac{y_{k}}{(x* x)_{k}^2}x_{k-j}x_{k-i}+2, \] with the same conventions as for the previous display. Effectively the index $k$ in the summation runs from $\max\{i,j\}$ to $m+\min\{i,j\}$.
We introduce the matrices $S^{(k)}\in\mathbb R^{(m+1)\times(m+1)}$, for $k\in\{0,\ldots,2m\}$, defined by $S^{(k)}_{ij}=\delta_{k,i+j}$ for $i,j\in\{0,\dots,m\}$, where the $\delta$'s are the Kronecker $\delta$'s. Let furthermore $x=(x_0,\dots,x_m)^\top$ and $\xi_k=S^{(k)}x$. Define $P(x)\in\mathbb R^{(m+1)\times(m+1)}$ with elements $P_{ij}(x)=4\sum_{k=0}^{2m}\frac{y_{k}}{(x* x)_{k}^2}x_{k-j}x_{k-i}$, then one can write \[ P(x)=4\sum_{k=0}^{2m}\frac{y_{k}}{(x* x)_{k}^2}\xi_k\xi_k^\top. \] Note that, if $x_0> 0$, the $\{\xi_k\}_{k=0}^m$ form a basis of $\mathbb{R}^{m+1}$, therefore if $y_k>0$ for $k\in[0,m]$, the matrix $P(x)$ is strictly positive definite. Alternatively, if $x_m>0$, the $\{\xi_k\}_{k=m}^{2m}$ also form a basis of $\mathbb{R}^{m+1}$ and again, if $y_k >0$ for $k\in[m,\,2m]$, the matrix $P(x)$ is strictly positive definite. Furthermore, let $Q(x)\in\mathbb R^{(m+1)\times(m+1)}$, with elements $Q_{ij}(x)=2-2\frac{y_{i+j}}{(x* x)_{i+j}}$. Hence the Hessian $H(x)$ satisfies \[ H(x)=P(x)+Q(x). \] Note that $Q(x)$ vanishes if $y_i=(x* x)_i$, for all $i\in[0,2m]$, i.e.\ in the exact model case, making $H(x)$ strictly positive definite. To find a useful expression of the Hessian in the general case introduce the matrix $R(x)\in\mathbb R^{(m+1)\times(m+1)}$ with elements $R_{ij}(x)=\frac{y_{i+j}}{(x* x)_{i+j}}$, and note that $Q(x)=2\left(\mathbf{1}\mathbf{1}^\top-R(x)\right)$, then \[ H(x)=P(x)+2\,\left(\mathbf{1}\mathbf{1}^\top-R(x)\right)\,, \] moreover the gradient $\nabla \mathcal{I}(x)$, written as a row vector, is \[ \nabla\mathcal{I}(x)=x^\top Q(x) = 2x^\top\left(\mathbf{1}\mathbf{1}^\top- R(x)\right)\,. \]
Except in the special case of an exact model, it is not obvious that in an interior limit point $x^\infty$ of the algorithm the Hessian $H(x^\infty)$ is strictly positive definite. Even the weaker statement that $H(x^\infty)$ is non-singular is hard to prove, in spite of the rather explicit form of $H(x^\infty)$ and the fact that the gradient $\nabla\mathcal{I}(x^\infty)$ vanishes. The relevance of non singularity stems from the Morse lemma, Corollary~2.3 in \cite{milnor1963}, which states that, the interior critical points of a function where the Hessian in is non singular are isolated.
Let us now look at a boundary (local) optimizer $x^\star$ of $\mathcal{I}(x)$. By the Kuhn-Tucker conditions if $x^\star_j=0$ then $\nabla_j\mathcal{I}(x^\star)\geq 0$, while if $x^\star_j>0$ then $\nabla_j\mathcal{I}(x^\star)= 0$. Write the boundary optimizer $x^\star$ as $x^\star=(\underline{x}^\star,0)$, possibly after a permutation of the coordinates, with all elements of $\underline{x}^\star$ strictly positive. We now look at optimization of $\mathcal{I}(x)$ under the constraint that $x=(\underline{x},0)$, so of $\underline{\mathcal{I}}(\underline{x}):=\mathcal{I}(\underline{x},0)$. The optimizing $\underline{x}^\star$ is now an interior point of the restricted domain, hence the gradient vanishes, $\nabla\underline{\mathcal{I}}(\underline{x}^\star)=0$. The Hessian $\underline{H}(\underline{x}^\star)$ of $\underline{\mathcal{I}}(\underline{x})$, is strictly positive definite, certainly non-singular and likely the same is true for $\underline{H}(\underline{x}^\infty)$ for any limit point $(\underline{x}^\infty,0)$ of $x^t$. The arguments underlying this are similar to the above, although it is hard to give a proof. Again by the Morse lemma, the critical points of $\underline{\mathcal{I}}(\underline{x})$, which are now interior points of the restricted domain, will then be isolated.
\begin{proposition} Let $x^0$ be a strictly positive starting point of the algorithm and let $L(x^0)$ be the set of interior limit points produced by the algorithm and assume that $H(x)$ is non-singular for all $x\in L(x^0)$. Then $L(x^0)$ is a singleton and thus the algorithm converges to a limit (possibly depending on the starting value $x^0$). The situation is analogous for boundary limit points. In both cases the limit is a Kuhn-Tucker point. \end{proposition}
\begin{proof} By Remark~\ref{remark:cc}, the set $L(x^0)$ is connected. By the above discussion the interior limit points are isolated and the same holds for the limit points on the boundary. The combination of these two properties yields that $L(x^0)$ has to be a singleton, and hence there is convergence of $x^t$ to the (unique) limit. Its Kuhn-Tucker property follows from Proposition~\ref{prop:kt}. \end{proof}
\begin{remark} In the literature it is not uncommon to see situations where the limit points are isolated. For instance, along different lines, in \cite{lange1984reconstruction} and \cite{lange1995globally} it is shown that in their setting the set of limit points of the iterates is finite, which is there a consequence of the maximization of a concave objective function. As the objective function in our minimization problem is not convex, their arguments cannot be taken over. \end{remark}
\begin{remark} In principle, the algorithm may produce different limit points, due to different initial values $x^0$. This has been observed in various numerical experiments. In fact, different starting values may either result in an interior limit or in a limit on the boundary, some of its coordinates are zero. The Kuhn-Tucker property was seen to be verified in these experiments. \end{remark}
\noindent To summarize the discussion of this section, it is very plausible that Algorithm~\ref{algorithm:half}, given a starting value, converges to a limit. This conjecture is motivated by two considerations, for both of them there is ample numerical evidence. The first one is a decreasing criterion, of which Proposition~\ref{proposition:decrease} takes care, and the second is non-singularity of the Hessian in limit points. Yet, a formal proof of the conjecture is lacking and we have to content ourself with the Kuhn-Tucker property of limit points as in Proposition~\ref{prop:kt}.
\section{Numerical experiments}\label{section:numerics}
In this section we review the results of numerical experiments for three different data sets to illustrate the behaviour of Algorithm~\ref{algorithm:half}. For the first two data sets, with $m=25$ and $m=10$ respectively, we investigated whether the algorithm is capable of retrieving the true parameter vector $x$, when the data $y$ are actually generated by the autoconvolution $y=x* x$. In the third data set, with $m=10$, the data $y$ are randomly generated.
To evaluate the performance of the algorithm we have generated, for each data set, one figure comprising three or four graphs. In all of the figures the top graph shows, in distinct colors, the trajectories of the iterates of the components, $x^t_i$, plotted against the iteration number $t\in[1, T]$.
In the exact model case, Figures~\ref{fig:good_one_N_50_Nit_1000}, \ref{fig:bad_one_N_50_Nit_1000_Xtrue_given_IC_bad},
\ref{fig:good_one_N_20_Nit_100_Xtrue_rand}, the diamonds at the right end of the top graph show the true $x_i$ values. The second graph shows the superimposed plots of the data generating signal $x$, and of the reconstructed signal $x^T$, at the last iteration, both plotted against their component number $i=0, 1, \dots, m$. The third graph shows the decreasing sequence $\mathcal{I}(y||x^t* x^t)$. The fourth and last graph shows the superimposed plots of the data vector $y$ and of the reconstructed convolution $x^T* x^T$, at the last iteration, both against the component number $i=0,1,\dots 2m$.
Figures~\ref{fig:random_Y_N=20_Nit_100_run-1} and \ref{fig:random_Y_N=20_Nit_100_run-2}, relative to the randomly generated data set, contain only three graphs, as the graph of the data generating signal is meaningless in this case.
We have observed experimentally that the iterative algorithm always converges very fast. The precise features underlying the experiments are further detailed below. All figures are collected at the end of the paper.
\subsection{True autoconvolution systems}
For the first data set we have taken $m=25$. The components of the true vector $x$ (the target values of the algorithm) have been randomly generated from a uniform distribution on the interval $[1, 11]$, and the data computed as true autoconvolutions $y=x* x$. The algorithm has been initialized at a randomly chosen strictly positive $x^0$, with components generated from a uniform distribution in the interval $[0.1,\, 0.2]$ and run for $T=1000$ iterations. Figures~\ref{fig:good_one_N_50_Nit_1000} and \ref{fig:bad_one_N_50_Nit_1000_Xtrue_given_IC_bad} show the results for two different runs (i.e.\ with the same true vector but different initial conditions) of the algorithm. In Figure~\ref{fig:good_one_N_50_Nit_1000} we see the desired behavior of the algorithm, the iterates converge to the true values and the divergence decreases to zero (because of the perfect match of $y=x* x$). This is the behavior that has been observed in a vast majority of numerical experiments of this kind. In Figure~ \ref{fig:bad_one_N_50_Nit_1000_Xtrue_given_IC_bad} we observe a different behavior. The iterates do not converge to the true values (see the second graph) and the divergence does not decrease to zero. On the other hand the convolution $x^T* x^T$ is always close to $y$ (see the fourth graphs of both figures). In fact, the instance of running the algorithm that produced Figure~\ref{fig:bad_one_N_50_Nit_1000_Xtrue_given_IC_bad} produced iterates that converged to a non-optimal local minimum of the objective function $\mathcal{I}(x)$. Indeed, we have verified that the gradient of $\mathcal{I}(x)$ at the final iteration vanished, whereas the Hessian turned out to be strictly positive definite. The conclusion of these two experiments is that it is wise to run the algorithm for the same data $y$, and same $x$, with different initial conditions and select the outcome with the lowest divergence. For the present example, the lowest divergence is of course zero, but the conclusion is also valid for any instance with any data vector $y$.
The data set used to generate Figure~\ref{fig:good_one_N_20_Nit_100_Xtrue_rand} is again of the exact type, $y=x* x$, with $m=10$ and consequently a lower number of iterations, $T=100$. We see quick convergence of the algorithm, stabilization has already occurred at $t=30$. The general behavior is identical to that observed in Figure~\ref{fig:good_one_N_50_Nit_1000}.
\subsection{Approximation of arbitrary data}
For the third data set there is no true input signal $x$ such that $y=x* x$, rather the components of the data vector $y$, with $m=10$, have been randomly generated from a uniform distribution on the interval $[0.1,\, 2]$. Thus, here we deal with a genuine approximation problem. Figures~\ref{fig:random_Y_N=20_Nit_100_run-1} and Figure~\ref{fig:random_Y_N=20_Nit_100_run-2} show the results of two runs of the algorithm, for $T=100$ iterations, and are relative to the same $y$ vector and different initial conditions $x^0$, both with components randomly generated from a uniform distribution in $[0.1,\, 0.2]$. The aim is to find the vector $x$ which yields the best autoconvolutional approximation to $y$. Inspecting the figures we conclude that the algorithm quickly stabilises in both runs. The final values $x^T$ of the iterates and the final divergences $\mathcal{I}(y||x^T* x^T)$ differ in the two runs, indicating that (at least) in the second case (with divergence slightly higher than in the first case) the algorithm is trapped in a non-optimal local minimum. For the same $y$ several other runs have produced results that were nearly identical to those in Figure~\ref{fig:random_Y_N=20_Nit_100_run-1}, so we infer that this figure represents the optimal approximation of $y$. The observed behavior suggests again to run the algorithm with different initial conditions, possibly in parallel, and to select the best final approximation as the one with smallest divergence $\mathcal{I}(y||x^T* x^T)$.
\begin{figure}
\caption{True model, $m=25$ and $T=1000$. Top panel: $m+1$ components $x^t_i$ against iteration $t$; the diamonds at $T=1000$ are the true values $x_i$ to which the $x^t_i$ converge. Second panel: $x^T_i$ (plusses) and true values $x_i$ (circles) against $i$. Third panel: $\mathcal{I}(y||x^t* x^t)$ against $t$. Fourth panel: $y_i$ (circles) and $(x^T* x^T)_i$ (plusses) against $i$.}
\label{fig:good_one_N_50_Nit_1000}
\end{figure}
\begin{figure}
\caption{The same data as in Figure~\ref{fig:good_one_N_50_Nit_1000}, with different initial conditions $x^0$. }
\label{fig:bad_one_N_50_Nit_1000_Xtrue_given_IC_bad}
\end{figure}
\begin{figure}
\caption{A true model with $m=10$ and $T=100$. The panels are as in Figure~\ref{fig:good_one_N_50_Nit_1000} and the same conclusions can be drawn.}
\label{fig:good_one_N_20_Nit_100_Xtrue_rand}
\end{figure}
\begin{figure}
\caption{Randomly generated $y$, with $m=10$ and $T=100$. Top panel: components $x^t_i$ against iteration index $t$. Second panel: $\mathcal{I}(y||x^t* x^t)$ against $t$. Third panel: $y_i$ (circles) and final autoconvolutions $(x^T* x^T)_i$ (plusses) against $i$. Third panel: The values of the $y_i$ (circles) and the final autoconvolutions $(x^T* x^T)_i$ (plusses).}
\label{fig:random_Y_N=20_Nit_100_run-1}
\end{figure}
\begin{figure}
\caption{The same data as in Figure~\ref{fig:random_Y_N=20_Nit_100_run-1}, with different initial conditions $x^0$.}
\label{fig:random_Y_N=20_Nit_100_run-2}
\end{figure}
\end{document}
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arXiv
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\begin{document}
\title{Gluing stability conditions} \author{John Collins and Alexander Polishchuk} \thanks{This work of the second author was partially supported by the NSF grant DMS-0601034}
\begin{abstract} We define and study a gluing procedure for Bridgeland stability conditions in the situation when a triangulated category has a semiorthogonal decomposition. As an application, we construct stability conditions on the derived categories of $\mathbb{Z}_2$-equivariant sheaves associated with ramified double coverings of $\mathbb{P}^3$. Also, we study the stability space for the derived category of $\mathbb{Z}_2$-equivariant coherent sheaves on a smooth curve $X$, associated with a degree $2$ map $X\to Y$, where $Y$ is another smooth curve. In the case when the genus of $Y$ is $\ge 1$ we give a complete description of the stability space. \end{abstract}
\maketitle
\section*{Introduction}
Stability conditions on triangulated categories were introduced by Bridgeland in \cite{Bridgeland06} as a mathematical formalization of Douglas' work on $\Pi$-stability in \cite{Doug1,Doug2}. A stability condition gives a way to single out (semi)stable objects in a triangulated category ${\mathcal D}$, generalizing Mumford's definition of stability for vector bundles. The remarkable feature of Bridgeland's theory is that the set of (nice) stability conditions on ${\mathcal D}$ has a structure of complex manifold. Hypothetically this manifold, called the {\it stability space} has some interesting geometric structures, and in the case when ${\mathcal D}$ is the derived category of coherent sheaves on a Calabi-Yau threefold this space should be relevant for mirror symmetry considerations (see \cite{Bridge-survey}). However, at present we have a quite limited stock of examples of stability conditions, so it is important to come up with new techniques for constructing them. Recall that a stability condition can be described via its {\it heart}, which is an abelian category $H\subset{\mathcal D}$, together with a central charge $Z$, which is a homomorphism $K_0({\mathcal D})\to\mathbb{C}$ sending every nonzero object of $H$ either to the (open) upper half-plane or to $\mathbb{R}_{<0}$. The idea to consider non-obvious abelian categories sitting inside derived categories is historically related to the theory of perverse sheaves, where such abelian categories are defined using a certain gluing procedure associated with a stratification of a topological space (see \cite{BBD}). Thus, it seems natural to try to extend the gluing construction to stability conditions. This is the first principal goal of the present paper. Secondly, we consider examples of the gluing construction for stability conditions in particular geometric situations.
The notion of an abelian category sitting nicely inside a triangulated category ${\mathcal D}$ is axiomatized in \cite{BBD}. Recall that such categories appear as {\it hearts} of $t$-structures on ${\mathcal D}$. The natural setup for gluing of $t$-structures is the situation when ${\mathcal D}$ has a {\it semiorthogonal decomposition} ${\mathcal D}=\langle {\mathcal D}_1, {\mathcal D}_2\rangle$. By definition, this means that ${\mathcal D}_1$ and ${\mathcal D}_2$ are triangulated subcategories in ${\mathcal D}$ such that $\operatorname{Hom}(E_2,E_1)=0$ for every $E_1\in{\mathcal D}_1$ and $E_2\in{\mathcal D}_2$, and for every object $E \in {\mathcal D}$ there exists an exact triangle \begin{equation}\label{semiorth-tr-eq} E_2 \rightarrow E \rightarrow E_1\to E_2[1] \end{equation} with $E_1 \in {\mathcal D}_{1}$, $E_2 \in {\mathcal D}_{2}$. Assume we are given hearts of $t$-structures $H_1\subset{\mathcal D}_1$ and $H_2\subset{\mathcal D}_2$. Under the additional assumption that \begin{equation}\label{Hom-H1-H2} \operatorname{Hom}^{\leq 0}(H_1,H_2)=0 \end{equation} the corresponding glued heart $H$ will be the smallest full subcategory of ${\mathcal D}$, closed under extensions and containing $H_1$ and $H_2$. If we have stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$ with the above hearts then we can define a central charge $Z$ on ${\mathcal D}$ uniquely, so that it restricts to the given central charges on ${\mathcal D}_1$ and ${\mathcal D}_2$. In order for the pair $(H,Z)$ to determine a stability condition on ${\mathcal D}$ one should check the Harder-Narasimhan property (see section \ref{reason-sec}). This does not seem to follow automatically from the similar property of the original stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$. We provide two sufficient criteria for checking this property: the first (Proposition \ref{GluingCondition}(a)) imposes an additional discreteness condition on the original stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$, while the second (Theorem \ref{reason-glue-thm}) imposes a stronger orthogonality condition than \eqref{Hom-H1-H2}. We also check that under appropriate assumptions the gluing operation is continuous (see Theorem \ref{a-thm} and Corollaries \ref{glue-cor}, \ref{exc-cor}).
For technical reasons we introduce the notion of a {\it reasonable} stability condition which is slightly stronger than that of a {\it locally finite} stability condition considered by Bridgeland.
Namely, we say that a stability condition is {\it reasonable} if the infimum of $|Z(E)|$ over all nonzero semistable objects $E$, is positive. In most of our considerations we work only with reasonable stabilities. We show in section \ref{reason-sec} that all (locally finite) stability conditions considered in the works \cite{AB}, \cite{Bridgeland06}, \cite{Bridgeland-K3} and \cite{Macri07} are reasonable, so this does not seem to be much of a restriction.
In the case of the semiorthogonal decomposition associated with a full exceptional collection $(E_i)$ our gluing procedure for stabilities reduces to the construction of Macr\`i in \cite{Macri07} (the collection $(E_i)$ should be $\operatorname{Ext}$-{\it exceptional}, i.e., such that $\operatorname{Hom}^{\leq 0}(E_i,E_j)=0$ for $i\neq j$). To get new examples of stability conditions we consider the following situation. Let $X\to Y$ be a ramified double covering of smooth projective varieties. Then $X$ is equipped with an involution and we can consider the derived category ${\mathcal D}={\mathcal D}_{\mathbb{Z}_2}(X)$ of $\mathbb{Z}_2$-equivariant coherent sheaves on $X$. It turns out that this category has a semiorthogonal decomposition with one block being the category of sheaves on $Y$ and another---sheaves on $R$, the ramification divisor in $Y$ (in the case of curves these semiorthogonal decompositions were considered in \cite{P-orbifold}). This allows to glue together some stability conditions for sheaves on $Y$ and $R$ into a stability condition on ${\mathcal D}$. Using examples of stability conditions on surfaces constructed in \cite{AB} this gives examples of stability conditions on ${\mathcal D}_{\mathbb{Z}_2}(X)$, where $X$ is a ramified double cover of $\mathbb{P}^3$.
Finally, we study in detail the case when $X$ and $Y$ are curves. It turns out that in this case a lot of stability conditions on ${\mathcal D}_{\mathbb{Z}_2}(X)$ are obtained by gluing. In Theorem \ref{cover-thm} we describe an open simply connected subset $U$ of the stability space consisting of the stability conditions that are ``not too far" from the standard one (similar to the Mumford's stability for nonequivariant sheaves). We show that $U$ is the universal covering of the corresponding open subset of central charges, where the group of deck transformations is $\mathbb{Z}$. In the case when genus of $Y$ is $\geq 1$ we describe the stability space of ${\mathcal D}_{\mathbb{Z}_2}(X)$ completely and show that it is contractible (see section \ref{class-sec}). Namely, we construct an isomorphism of the stability space with an explicit open subset of $\Sigma^n\times\mathbb{C}^2$, where $n$ is the number of ramification points of $X\to Y$, and $\Sigma$ is a certain simply connected Riemann surface of parabolic type (so $\Sigma$ is isomorphic to $\mathbb{C}$). This surface $\Sigma$ naturally appears as follows: we prove that if $p\in X$ is a ramification point then a stability condition on ${\mathcal D}_{\mathbb{Z}_2}(X)$ restricts to a stability condition on the subcategory ${\mathcal D}_p$ of objects supported at $p$ (provided $g(Y)\ge 1$). The stability space corresponding to ${\mathcal D}_p$ has form $\Sigma\times\mathbb{C}$, where the central charge of ${\mathcal O}_{2p}$ is given by exponentiating the projection to the second factor $\mathbb{C}$. In the case when $Y=\mathbb{P}^1$ the stability space seems to be more complicated due to the presence of additional exceptional objects in ${\mathcal D}_{\mathbb{Z}_2}(X)$. We show in this case that our open subset $U$ contains a dense open subset consisting of stabilities constructed from exceptional collections (see Proposition \ref{exc-P1-prop}).
\noindent {\it Notation.} For subcategories ${\mathcal A}_1,\ldots,{\mathcal A}_n$ in a triangulated category ${\mathcal D}$ we denote by $[{\mathcal A}_1,\ldots, {\mathcal A}_n]$ (resp., $\langle{\mathcal A}_1,\ldots,{\mathcal A}_n\rangle$) the extension-closed full subcategory (resp., triangulated subcategory) in ${\mathcal D}$ generated by the ${\mathcal A}_i$'s. We work with algebraic varieties over a fixed algebraically closed field $k$. For a smooth projective variety $X$ we denote by ${\mathcal D}(X)$ the bounded derived category of coherent sheaves on $X$. For a complex number $z$ we denote by $\Re z$ and $\Im z$ its real and imaginary part, and we call $\phi(z):=(\arg z)/\pi$ the phase of $z$.
\section{Reasonable stability conditions} \label{reason-sec}
Throughout this section ${\mathcal D}$ denotes a triangulated category. Let us briefly recall basic definitions and results concerning local finite stability conditions on ${\mathcal D}$, referring to Bridgeland's original paper \cite{Bridgeland06} for details.
By definition, a stability condition $\sigma$ is given by a pair $(Z,P)$, where $Z:K_0({\mathcal D})\to\mathbb{C}$ is a homomorphism from the Grothendieck group $K_0({\mathcal D})$ of ${\mathcal D}$, and $P$ is a slicing. Such a slicing is given by a collection of subcategories $P(\phi)$ of semistable objects of phase $\phi$ for each $\phi\in\mathbb{R}$, where $\operatorname{Hom}(P(\phi_1),P(\phi_2))=0$ for $\phi_1>\phi_2$, and $P(\phi)[1]=P(\phi+1)$. For an object $E\in P(\phi)$ we will use the notation $\phi(E)=\phi$. Similarly to the case of vector bundles, for each object $E$ of ${\mathcal D}$ there should exist a {\it Harder-Narasimhan filtration} (HN-filtration), i.e., a collection of exact triangles building $E$ from the semistable factors $E_1,\ldots,E_n$ (called the {\it HN-factors} of $E$), where $\phi(E_1)>\ldots>\phi(E_n)$ ($E_1\to E$ is an analog of the subbundle of maximal phase, etc.). For each interval $I\subset\mathbb{R}$ we denote by $PI\subset{\mathcal D}$ the extension-closed subcategory generated by all the subcategories $P(\phi)$ for $\phi\in I$. For example, $P(0,1]$ denotes the subcategory corresponding to the interval $(0,1]$.
If $\sigma=(Z,P)$ is a stabiity condition then $P(0,1]$ is a heart of a bounded nondegenerate $t$-structure on ${\mathcal D}$ with ${\mathcal D}^{\le 0}=P(0,+\infty)$ and ${\mathcal D}^{\ge 0}=P(-\infty,1]$. We will often refer to the abelian subcategory $P(0,1]\subset{\mathcal D}$ as the {\it heart} of $\sigma$. By Proposition 5.3 of \cite{Bridgeland06}, to give a stability condition is the same as to give an abelian subcategory $H\subset {\mathcal D}$ (which should be the heart of a bounded nondegenerate $t$-structure), together with a homomorphism $Z:K_0(H)\to\mathbb{C}$ such that for every nonzero object $E\in H$ one has either $\Im Z(E)>0$ or $Z(E)\in\mathbb{R}_{<0}$. These data should satisfy the Harder-Narasimhan property, i.e., once we define (semi)stability for objects in $H$ using the slopes associated with the function $Z$, then every object of $H$ should be equipped with an analog of the Harder-Narasimhan filtration. Checking the Harder-Narasimhan property is often an important ingredient in constructing stability conditions (see section \ref{HN-sec} for examples).
A stability condition $\sigma=(Z,P)$ is called {\it locally finite} if there exists $\eta>0$ such that for every $\phi\in\mathbb{R}$ the quasi-abelian category $P(\phi-\eta,\phi+\eta)$ is of finite length. The space of all locally finite stability conditions on ${\mathcal D}$ is denoted $\operatorname{Stab}({\mathcal D})$. It can be equipped with a natural topology defined as follows (see section 6 of \cite{Bridgeland06}). For $\sigma = (Z,P) \in \operatorname{Stab}({\mathcal D})$ we define a function
$||\cdot||_{\sigma}:\operatorname{Hom}(K_0({\mathcal D}),\mathbb{C})\to[0,+\infty]$ by
$$||U||_{\sigma}=\sup_{E \text{ semistable},E\neq 0}\frac{|U(E)|}{|Z(E)|}.$$ The basis of open neighborhoods of a locally finite stability condition $\sigma=(Z,P)$ in $\operatorname{Stab}({\mathcal D})$ consists of open subsets
$$B_{\epsilon}(\sigma)=\{\tau=(U,Q)\ :\ ||U-Z||_{\sigma}<\sin(\pi\epsilon), d(P,Q)<\epsilon\},$$ where $d(P,Q)$ is a natural generalized metric on the set of slicings given by $$d(P,Q)=\inf\{\epsilon\in\mathbb{R}_{\ge 0}\ :\ Q(\phi)\subset P[\phi-\epsilon,\phi+\epsilon] \text{ for all }\phi\in\mathbb{R}\}.$$ Theorem 7.1 of \cite{Bridgeland06} states that for a given locally finite stability condition
$\sigma=(Z,P)$ there exists an $\epsilon_0>0$ such that if $0<\epsilon<\epsilon_0$ then every central charge $Z'\in\operatorname{Hom}(K_0({\mathcal D}),\mathbb{C})$ with $||Z'-Z||_{\sigma}<\sin(\pi\epsilon)$ lifts to an element of $B_{\epsilon}(\sigma)$. Let us set
$$W_{\sigma} := \{U \in \operatorname{Hom}(K_0({\mathcal D}),\mathbb{C}) : ||U||_{\sigma} < \infty \}.$$ The linear subspaces $W_{\sigma}\subset\operatorname{Hom}(K_0({\mathcal D}),\mathbb{C})$ do not change as $\sigma$ varies over a connected component $C$ of $\operatorname{Stab}({\mathcal D})$. Furthermore, the natural projection $C\to W_{\sigma}$ is a local homeomorphism (see Theorem 1.2 of \cite{Bridgeland06})
In the case when ${\mathcal D}$ is of finite type over a field one can consider the numerical Grothendieck group ${\mathcal N}({\mathcal D})$ which is the quotient of $K_0({\mathcal D})$ by the kernel of the Euler bilinear form on $K_0({\mathcal D})$ (see \cite{Bridgeland06}, 1.3). A stability condition is called {\it numerical} if the corresponding central charge factors through ${\mathcal N}({\mathcal D})$. We denote by $\operatorname{Stab}_{{\mathcal N}}({\mathcal D})$ the space of numerical locally finite stability conditions on ${\mathcal D}$. The above theorem on the structure of $\operatorname{Stab}({\mathcal D})$ implies that in a neighborhood of $\sigma\in \operatorname{Stab}_{{\mathcal N}}({\mathcal D})$ the space $\operatorname{Stab}_{{\mathcal N}}({\mathcal D})$ is modeled on the linear space $W^{{\mathcal N}}_{\sigma}=W_{\sigma}\cap\operatorname{Hom}({\mathcal N}({\mathcal D}),\mathbb{C})$. A numerical stability condition $\sigma$ is called {\it full} if $W^{{\mathcal N}}_{\sigma}= \operatorname{Hom}({\mathcal N}({\mathcal D}),\mathbb{C})$ (see \cite{Bridgeland-K3}).
The space $\operatorname{Stab}({\mathcal D})$ (resp., $\operatorname{Stab}_{{\mathcal N}}({\mathcal D})$) is equipped with a canonical action of the group $\widetilde{\operatorname{GL}_{2}^{+}(\R)}$, which is a universal covering of the group of $2\times 2$-matrices over $\mathbb{R}$ with positive determinant. For a real number $a$ let us denote by $R_a:\operatorname{Stab}({\mathcal D})\to\operatorname{Stab}({\mathcal D})$ the operation of shifting the phase by $a$ which is part of this $\widetilde{\operatorname{GL}_{2}^{+}(\R)}$-action. More explicitly, for $\sigma=(Z,P)$ one has $R_a\sigma=(r_{-\pi a}\circ Z,P')$, where $P'(t)=P(t+a)$, $r_{-\pi a}$ is the rotation in $\mathbb{C}=\mathbb{R}^2$ through the angle $-\pi a$. We refer to the transformations $R_a$ as rotations.
\begin{definition} A stability condition $\sigma=(Z,P)$ on ${\mathcal D}$ is called {\it reasonable} if
$$\inf_{E\text{ semistable},E\neq 0} |Z(E)| >0$$ where $E$ runs over all nonzero $\sigma$-semistable objects. \end{definition}
\begin{lem}\label{reason-lem} Let $\sigma=(Z,P)$ be a stability condition on ${\mathcal D}$. \begin{enumerate} \item If $\sigma$ is reasonable then for every $0<\eta<1$ one has
$$\inf_{t\in\mathbb{R}, E\in P(t,t+\eta)\setminus 0} |Z(E)| >0;$$ \item $\sigma$ is reasonable if and only if for every $t$ and every $0<\eta<1$ the point $0$ is an isolated point of $Z(P(t,t+\eta))$; \item If $\sigma$ is reasonable then every category $P(t,t+\eta)$ for $0<\eta<1$ is of finite length, hence, $\sigma$ is locally finite; \item If the image of $Z$ in $\mathbb{C}$ is discrete then $\sigma$ is reasonable. \end{enumerate} \end{lem}
\begin{proof} (1) Let
$$c=\inf_{E\text{ semistable},E\neq 0} |Z(E)|>0.$$ Given an object $E\in\mathbb{P}(t,t+\eta)$ let $E_i$ be the HN-factors of $E$. Then all numbers $Z(E_i)$ (and $Z(E)$) lie in the cone $C(t,t+\eta)$ of complex numbers with phases between $t$ and $t+\eta$. Let $h:\mathbb{C}\to\mathbb{R}$ denote the scalar product with the unit vector of phase $t+\eta/2$. Then we have
$\cos(\pi\eta/2)|z|\le h(z)\le |z|$ for all $z\in C(t,t+\eta)$. Hence,
$$|Z(E)|\ge h(Z(E))=\sum_i h(Z(E_i))\ge\cos(\pi\eta/2)c.$$
\noindent (2) The ``only if" part follows from (1). Conversely, assuming that $0$ is an isolated point of $Z(P(0,3/4))$ and of $Z(P(1/2,5/4))$ we see that there is a universal lower bound for
$|Z(E)|$, where $E$ is semistable of the phase in $(0,1]$. This implies that $\sigma$ is reasonable.
\noindent (3) This is similar to Lemma 4.4 of \cite{Bridgeland-K3}. The point is that if $h:\mathbb{C}\to R$ denotes the scalar product with the unit vector of phase $t+\eta/2$ then $h(A)>c>0$ for a fixed constant $c$, where $A$ is a nonzero object of $P(t,t+\eta)$. Since $h$ is an additive function with respect to strict short exact sequences, the assertion follows.
\noindent (4) This is clear. \end{proof}
\begin{prop} Let $\Sigma$ be a connected component of $\operatorname{Stab}({\mathcal D})$ containing some reasonable stability condition. Then every $\sigma\in\Sigma$ is reasonable. \end{prop}
\begin{proof} Let $\sigma=(Z,P)$, $\sigma'=(Z',P')$ be points of $\Sigma$. Assume first that $\sigma'$ is reasonable, and $\sigma'\in B_{\epsilon}(\sigma)$, where $\epsilon<1/4$. Then for every $\sigma$-semistable object $E$ of phase $t$ we have
$|Z'(E)-Z(E)|<\sin(\pi\epsilon)|Z(E)|$ and $E\in P'(t-\epsilon,t+\epsilon)$. Hence, by Lemma \ref{reason-lem}(1), there exists a constant $c>0$ independent of $E$ such that $|Z'(E)|>c$. Therefore,
$$|Z(E)|>(1+\sin(\pi\epsilon))^{-1}|Z'(E)|>(1+\sin(\pi\epsilon))^{-1}c,$$ so $\sigma$ is reasonable. This shows that the set of reasonable stabilities is closed. Conversely, assume that $\sigma$ is reasonable and $\sigma'\in B_{\epsilon}(\sigma)$, where $\epsilon$ is sufficiently small. Given a $\sigma'$-semistable object $E$ of phase $t$ we have $E\in P(t-\epsilon,t+\epsilon)$. Let $(E_i)$ be the HN-factors of $E$ with respect to $\sigma$. Then $E_i\in P(t-\epsilon,t+\epsilon)\subset P'(t-2\epsilon,t+2\epsilon)$. Let us denote by $h:\mathbb{C}\to\mathbb{R}$ the scalar product with the unit vector of phase $t$. Then
$$|Z'(E)|=h(Z'(E))=\sum_i h(Z'(E_i))\ge \frac{1}{2}\sum_i |Z'(E_i)|$$ provided $\epsilon$ is small enough.
But $|Z'(E_1)|>(1-\sin(\pi\epsilon))|Z(E_1)|$ which is bounded below by a positive constant depending only on $\epsilon$. Hence, $\sigma'$ is reasonable, so the set of reasonable stabilities is open. \end{proof}
\begin{cor} If $\Sigma\subset\operatorname{Stab}({\mathcal D})$ is a connected component containing some stability condition such that the corresponding central charge has discrete image, then every $\sigma\in\Sigma$ is reasonable. \end{cor}
Note that this Corollary implies that all (locally finite) stability conditions constructed in \cite{AB}, \cite{Bridgeland06}, \cite{Bridgeland-K3} and \cite{Macri07} are reasonable.
\section{Gluing construction}
The general gluing construction for $t$-structures was invented in \cite{BBD}. We start by stating a particular case of this construction (see section 3.1 of \cite{Polishchuk06} for a related construction). Let ${\mathcal D}$ be a triangulated category equipped with a semiorthogonal decomposition ${\mathcal D}=\langle{\mathcal D}_{1}, {\mathcal D}_{2}\rangle$. Note that for $E \in {\mathcal D}$ the objects $E_1\in{\mathcal D}_1$ and $E_2\in{\mathcal D}_2$ from the exact triangle \eqref{semiorth-tr-eq} depend functorially on $E$. Namely, $E_2=\rho_2(E)$, where $\rho_2$ is the right adjoint functor to the inclusion ${\mathcal D}_2\to {\mathcal D}$, and $E_1=\lambda_1(E)$, where $\lambda_1$ is the left adjoint functor to the inclusion ${\mathcal D}_1\to {\mathcal D}$.
\begin{lem} Assume we have a semiorthogonal decomposition ${\mathcal D}=\langle{\mathcal D}_{1}, {\mathcal D}_{2}\rangle$ and t-structures $({\mathcal D}_{i}^{\leq 0}, {\mathcal D}_{i}^{\geq 0})$ with the hearts $H_{i}$ on ${\mathcal D}_{i}$ (where $i=1,2$), such that $\operatorname{Hom}_{{\mathcal D}}^{\leq 0}(H_1,H_2)=0$. Then there is a t-structure on ${\mathcal D}$ with the heart \begin{equation} \label{H-for}
H=\{ X\in {\mathcal D} \ |\ \lambda_1(X)\in H_1, \rho_2(X)\in H_2\}. \end{equation} With respect to this $t$-structure on ${\mathcal D}$ the functors $\lambda_1:{\mathcal D}\to{\mathcal D}_1$ and $\rho_2:{\mathcal D}\to {\mathcal D}_2$ are $t$-exact. \end{lem}
\begin{proof} Set ${\mathcal D}^{[a,b]}=\{ X\in{\mathcal D} \|\ \lambda_1(X)\in {\mathcal D}_1^{[a,b]}, \rho_2(X)\in {\mathcal D}_2^{[a,b]}\}.$ First, we have to check that $\operatorname{Hom}({\mathcal D}^{\le 0},{\mathcal D}^{\ge 1})=0$. Note that our orthogonality assumption for the hearts is equivalent to \begin{equation}\label{t-str-orth} \operatorname{Hom}_{{\mathcal D}}({\mathcal D}_1^{\le 0}, {\mathcal D}_2^{\ge 0})=0. \end{equation} Now given $X\in{\mathcal D}^{\le 0}$ and $Y\in{\mathcal D}^{\ge 1}$, the canonical exact triangles for $X$ and $Y$ show that it is enough to check the vanishings $$\operatorname{Hom}(\rho_2(X),\rho_2(Y))=\operatorname{Hom}(\rho_2(X),\lambda_1(Y))=\operatorname{Hom}(\lambda_1(X),\rho_2(Y))= \operatorname{Hom}(\lambda_1(X),\lambda_2(Y))=0.$$ The first and the fourth groups vanish since we start with $t$-structures on ${\mathcal D}_1$ and ${\mathcal D}_2$. The second group vanishes by semiorthogonality, and the third---by \eqref{t-str-orth}.
Next, let us check that for every $E\in{\mathcal D}$ there exists an exact triangle $$A\to E\to B\to A[1]$$ with $A\in{\mathcal D}^{\le 0}$ and $B\in{\mathcal D}^{\ge 1}$. Consider the canonical triangle \eqref{semiorth-tr-eq}. We are going to construct $A$ and $B$ in such a way that $A$ (resp., $B$) will be an extension of $\tau^1_{\le 0}E_1$ by $\tau^2_{\le 0}E_2$ (resp., of $\tau^1_{\ge 1}E_1$ by $\tau^2_{\ge 1}E_2$), where $\tau^1_*$ and $\tau^2_*$ denote the truncation functors on ${\mathcal D}_1$ and ${\mathcal D}_2$, respectively. First, applying the octahedron axiom to the exact triangles $E_2\to E\to E_1\to\ldots$ and $\tau^1_{\le 0}E_1\to E_1\to \tau^1_{\ge 1}E_1\to\ldots$ we construct an exact triangle $$\widetilde{A}\to E\to\tau^1_{\ge 1}E_1\to\ldots,$$ where $\widetilde{A}$ is an extension of $\tau^1_{\le 0}E_1$ by $E_2$. Next, consider the exact triangle $$\tau^2_{\le 0}E_2\to E_2\to\tau^2_{\ge 1}E_2\to\ldots.$$ The condition \eqref{t-str-orth} implies that $\operatorname{Hom}^1(\tau^1_{\le 0}E_1, \tau^2_{\ge 1}E_2)=0$. Hence, there exists an exact triangle $$A\to\widetilde{A}\to\tau^2_{\ge 1}E_2\to\ldots,$$ where $A$ is an extension of $\tau^1_{\le 0}E_1$ by $\tau^2_{\le 0}E_2$. Applying the octahedron axiom once more we deduce the required statement.
\end{proof}
Note that in the situation of the above Lemma we have $H_1\subset H$ and $H_2\subset H$. Furthermore, every object $E \in H$ fits into an exact sequence in $H$ \begin{equation}\label{H1-H2-seq} 0 \rightarrow \rho_{2}(E) \rightarrow E \rightarrow \lambda_{1}(E) \rightarrow 0, \end{equation} where $\rho_2(E)\in H_2$ and $\lambda_1(E)\in H_1$. Therefore, we also have \begin{equation}\label{H-for2} H=\langle H_2, H_1\rangle, \end{equation} and $(H_{2}, H_{1})$ is a torsion pair in $H$ (see \cite{HRS} for the definition and basic properties of torsion pairs).
Assume now that the hearts $H_1$ and $H_2$ are equipped with stability functions $Z_i: K_0(H_i)\to \mathbb{C}$. Then the formula \begin{equation}\label{Z-for} Z(X)=Z_1(\lambda_1(X))+Z_2(\rho_2(X)) \end{equation} defines a stability function on the glued heart $H$.
\begin{definition} Suppose we have stability conditions $\sigma_1=(Z_1,P_1)$ on ${\mathcal D}_1$ and $\sigma_2=(Z_2,P_2)$ on ${\mathcal D}_2$, such that the corresponding hearts $H_1=P_1(0,1]$ and $H_2=P_2(0,1]$ satisfy $\operatorname{Hom}_{{\mathcal D}}^{\leq 0}(H_1,H_2)=0$. Then we say that a stability condition $\sigma=(Z,P)$ on ${\mathcal D}$ is {\it glued from } $\sigma_1$ and $\sigma_2$ if $Z$ is given by \eqref{Z-for}, and the heart $H=P(0,1]$ is given by \eqref{H-for} (or equivalently, by \eqref{H-for2}). \end{definition}
Note that this glued stability condition is uniquely determined by $\sigma_1$ and $\sigma_2$. It exists if and only if the Harder-Narasimhan property for the stability function $Z$ on the glued heart $H$ is satisfied. We have the following easy properties of glued stability conditions.
\begin{prop}\label{char-prop} \begin{enumerate} \item A stability condition $\sigma=(Z,P)$ on ${\mathcal D}$ is glued from $\sigma_1=(Z_1,P_1)$ on ${\mathcal D}_1$
and $\sigma_2=(Z_2,P_2)$ on ${\mathcal D}_2$ if and only if $Z_i=Z|_{{\mathcal D}_i}$ for $i=1,2$, $\operatorname{Hom}^{\leq 0}(H_1,H_2)=0$ and $H_i\subset H$ for $i=1,2$, where $H=P(0,1]$, $H_i=P_i(0,1]$. \item Let $\sigma$ be a stability condition on ${\mathcal D}$ with the central charge $Z$ and the heart $H$. Assume that $H$ is glued from the hearts $H_1\subset {\mathcal D}_1$ and $H_2\subset {\mathcal D}_2$, where $\operatorname{Hom}^{\leq 0}(H_1,H_2)=0$, so that \eqref{H-for} holds. Then for $i=1,2$ there exists a stability condition $\sigma_i$ on ${\mathcal D}_i$
with the heart $H_i$ and the central charge $Z_i=Z|_{{\mathcal D}_i}$, so that $\sigma$ is glued from $\sigma_1$ and $\sigma_2$. \item If $\sigma=(Z,P)$ is glued from $\sigma_1=(Z_1,P_1)$ and $\sigma_2=(Z_2,P_2)$ then for every $\phi\in\mathbb{R}$ one has $P_1(\phi)\subset P(\phi)$ and $P_2(\phi)\subset P(\phi)$. \end{enumerate} \end{prop}
\begin{proof} (1) Let us observe that for every $E\in {\mathcal D}$ one has the equality
$[E]=[\rho_2(E)]+[\lambda_1(E)]$ in $K_0({\mathcal D})$, so the definition \eqref{Z-for} is equivalent to the condition $Z|_{{\mathcal D}_i}=Z_i$ for $i=1,2$. It remains to note also that the embeddings $H_1,H_2\subset H$ imply that $\langle H_1,H_2\rangle\subset H$. Since both are hearts of nondegenerate $t$-structures this is equivalent to the equality \eqref{H-for2}.
\noindent
(2) The subcategory $H_1\subset H$ (resp., $H_2\subset H$) is exactly the kernel of the exact functor $\rho_2:H\to H_2$ (resp., $\lambda_1:H\to H_1$). It follows that these subcategories are closed under passing to subobjects and quotient-objects in $H$. This easily implies that the Harder-Narasimhan property holds for $Z|_{H_i}$ on $H_i$, $i=1,2$, so we obtain the stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$. The fact that $\sigma$ is glued from these stabilities follows from definition.
\noindent (3) It is enough to check this in the case when $\phi\in(0,1]$. Then this follows immediately from the fact that $H_1$ and $H_2$ are stable under subobjects and quotient-objects in $H$. \end{proof}
In the case of semiorthogonal decompositions associated with a full exceptional collection $(E_1,\ldots,E_n)$ the above gluing procedure was considered by Macr\`i in \cite{Macri07}. Namely, we can consider the semiorthogonal decomposition ${\mathcal D}=\langle\langle E_1\rangle,\ldots,\langle E_n\rangle\rangle$, and equip $\langle E_i\rangle$ with the $t$-structure for which $E_i$ belongs to the heart. Then our orthogonality condition on the hearts reduces to the condition that the collection is $\operatorname{Ext}$-exceptional, i.e., $\operatorname{Hom}^{\leq 0}(E_i,E_j)=0$ for $i<j$, and the glued heart is $H=[E_{1}, \cdots, E_{n}]$. We say that a stability condition $\sigma=(Z,P)$ on ${\mathcal D}$ is \emph{glued from an $\operatorname{Ext}$-exceptional collection} $(E_{1}, \cdots, E_{n})$ if $P(0,1]=H$. Note that in this case the Harder-Narasimhan property is automatically satisfied for any stability function on $H$. We will generalize this in Proposition \ref{GluingCondition}.
\section{Harder-Narasimhan property and gluing of stability conditions}\label{HN-sec}
In this section we show how to check the Harder-Narasimhan property for the glued stability function under different sets of additional assumptions.
We start with the following basic criterion which is a slight generalization of Proposition 2.4 of \cite{Bridgeland06} (the proof is the same as in {\it loc. cit.}, using properties of quasi-abelian categories). Recall that $\phi(z)$ denotes the phase of $z\in\mathbb{C}$.
\begin{prop}\label{HN-prop} Suppose $\mathcal{A}$ is a quasi-abelian category with a stability function $Z: K_0(\mathcal{A}) \rightarrow \mathbb{C}$. Assume that for a pair of $Z$-semistable objects $E,F\in\mathcal{A}$ such that $\phi(E)>\phi(F)$ one always has $\operatorname{Hom}_{{\mathcal A}}(E,F)=0$, where we denote $\phi(E):=\phi(Z(E))$. Assume also that the following chain conditions are satisfied: \begin{enumerate} \item there are no infinite sequences of strict monomorphisms in $\mathcal{A}$ $$\cdots \subset E_{j+1} \subset E_{j} \subset \cdots \subset E_{2} \subset E_{1}$$ with $\phi(E_{j+1}) > \phi(E_{j})$ for all $j$, \item there are no infinite sequences of strict epimorphisms in $\mathcal{A}$ $$E_{1} \twoheadrightarrow E_{2} \twoheadrightarrow \cdots \twoheadrightarrow E_{j} \twoheadrightarrow E_{j+1} \twoheadrightarrow \cdots$$ with $\phi(E_{j}) > \phi(E_{j+1})$ for all $j$. \end{enumerate} Then $Z$ has the Harder-Narasimhan property on $\mathcal{A}$. \end{prop}
Quasi-abelian categories often arise as follows. Consider an abelian category ${\mathcal A}$ equipped with a torsion pair $({\mathcal T},{\mathcal F})$. The both ${\mathcal T}$ and ${\mathcal F}$ are quasi-abelian categories. Indeed, this follows from Lemma 1.2.34 of \cite{Schn}, using the tilted abelian category ${\mathcal A}^t$. For example, to check that ${\mathcal T}$ is quasi-abelian we use the fact that the embedding of ${\mathcal T}$ into ${\mathcal A}$ is stable under quotients, while the embedding of ${\mathcal T}$ into ${\mathcal A}^t$ is stable under subobjects.
\begin{lem}\label{Z-torsion-lem} Let ${\mathcal A}$ be an abelian category equipped with a torsion pair $({\mathcal T},{\mathcal F})$. Suppose $Z$ is a stability function on ${\mathcal A}$ such that for any nonzero $T\in{\mathcal T}$ and $F\in{\mathcal F}$ one has $\phi(T)>\phi(F)$ (where as before we set $\phi(F):=\phi(Z(F))$).
Let $Z|_{{\mathcal T}}$ and $Z|_{{\mathcal F}}$ be the stability functions on the exact categories ${\mathcal T}$ and ${\mathcal F}$ induced by $Z$. Then every $Z|_{{\mathcal T}}$-semistable object of ${\mathcal T}$
(resp., $Z|_{{\mathcal F}}$-semistable object of ${\mathcal F}$) is $Z$-semistable as an object of ${\mathcal A}$. \end{lem}
\begin{proof}
We consider only the case of a $Z|_{{\mathcal T}}$-semistable object $T\in{\mathcal T}$ (the second case is similar). Suppose $T$ is not $Z$-semistable as an object of ${\mathcal A}$. Then there exists a subobject $A\subset T$ such that $\phi(A)>\phi(T)$. Consider the canonical exact sequence \begin{equation}\label{torsion-ex-seq} 0\to T(A)\to A\to F(A)\to 0 \end{equation} with $T(A)\in{\mathcal T}$, $F(A)\in{\mathcal F}$. By the assumption either $\phi(T(A))> \phi(F(A))$ or one of the objects $T(A)$, $F(A)$ is zero. Note that $T(A)\neq 0$, since otherwise $A$ would be an object of $F$, so the inequality $\phi(A)>\phi(T)$ would be impossible. It follows that $\phi(T(A))\ge\phi(A)>\phi(T)$. Thus, we found a destabilizing subobject $T(A)\subset T$ (the quotient is automatically in ${\mathcal T}$ since ${\mathcal T}$ is always closed under quotients). \end{proof}
\begin{prop}\label{Z-torsion-prop} Keep the assumptions of Lemma \ref{Z-torsion-lem}. Assume that
both $({\mathcal T}, Z|_{{\mathcal T}})$ and $({\mathcal F},Z|_{{\mathcal F}})$ satisfy chain conditions (1) and (2) from Proposition \ref{HN-prop}. Then $Z$ has the Harder-Narasimhan property on ${\mathcal A}$. \end{prop}
\begin{proof}
Suppose we have a pair of $Z|_{{\mathcal T}}$-semistable objects $E,F\in{\mathcal T}$ such that $\phi(E)>\phi(F)$. Then by Lemma \eqref{Z-torsion-lem}, $E$ and $F$ are still semistable viewed as objects of the {\it abelian} category ${\mathcal A}$ with the stability function $Z$. Hence, $\operatorname{Hom}(E,F)=0$. Therefore, by Proposition \ref{HN-prop}, the Harder-Narasimhan property holds for $({\mathcal T}, Z|_{{\mathcal T}})$. The same argument works for $({\mathcal F}, Z|_{{\mathcal F}})$. Now given an object $E\in{\mathcal A}$ we can sew together the HN-filtrations of the objects $T(A)$ and $F(A)$ from the canonical exact sequence \eqref{torsion-ex-seq}. It remains to apply Lemma \ref{Z-torsion-lem} again to see that we get a HN-filtration of $E$ in this way. \end{proof}
The following Lemma is a more precise version of Proposition 5.0.1 of \cite{AP}.
\begin{lem}\label{Noeth-lem} (a) Let $Z$ be a stability function on an abelian category ${\mathcal A}$. Assume that $0$ is an isolated point of $\Im Z({\mathcal A})\subset\mathbb{R}_{\geq 0}$, and that the category
${\mathcal A}_0=\{A\in{\mathcal A}\ |\ \Im Z(A)=0\}$ is Noetherian. Then $Z$ satisfies the Harder-Narasimhan property on ${\mathcal A}$ if and only if ${\mathcal A}$ is Noetherian.
\noindent (b) Let $\sigma=(Z,P)$ be a stability condition on ${\mathcal D}$ with Noetherian heart $P(0,1]$. Assume that $0$ is an isolated point of $\Im Z(P(0,1))\subset\mathbb{R}_{\ge 0}$. Then the category $P(0,1)$ is of finite length. Also, $\sigma$ is reasonable if and only if $0$ is an isolated point of $Z(P(1))\subset\mathbb{R}_{\le 0}$. \end{lem}
\begin{proof} (a) Assume first that ${\mathcal A}$ is Noetherian. Then condition (2) of Proposition \ref{HN-prop} is automatic. To check condition (1) we observe that if $E \rightarrow F$ is a destabilizing inclusion in ${\mathcal A}$ then $\Im Z(E) < \Im Z(F)$. Indeed, we have either $\Im Z(F/E) > 0$ or $\Re Z(F/E) < 0$. But in the latter case the phase of $Z(E)$ would be smaller than that of $Z(F)$. Thus, if we have a chain \begin{equation}\label{incl-seq} \cdots \subset E_{j+1} \subset E_{j} \subset \cdots \subset E_{2} \subset E_{1} \end{equation} of destabilizing inclusions in ${\mathcal A}$ then the sequence $(\Im Z(E_j))$ is strictly decreasing. But this implies that $\Im Z(E_j/E_{j+1})$ tends to $0$ which is a contradiction. Conversely, assume $Z$ satisfies the Harder-Narasimhan property. To check that ${\mathcal A}$ is Noetherian we have to check that every sequences of quotients in ${\mathcal A}$ \begin{equation}\label{surj-seq} E_1\twoheadrightarrow E_2\twoheadrightarrow E_3\twoheadrightarrow\cdots \end{equation} stabilizes. Note that in this situation the sequence $(\Im Z(E_i))$ is decreasing, so it has to stabilize. Without loss of generality we can assume that the sequence $(\Im Z(E_i))$ is constant. Then the kernel $K_i$ of $E_1\to E_i$ belongs to ${\mathcal A}_0$. Since $Z$ satisfies the Harder-Narasimhan property, there exists a maximal subobject $F\subset E_1$ such that $F\in{\mathcal A}_0$. Then the kernels $K_i$ form an increasing chain of subobjects in $F$. Since ${\mathcal A}_0$ is Noetherian, this sequence stabilizes, so the original sequence $(E_i)$ also stabilizes. It remains to check that in this situation ${\mathcal A}_{>0}$ is Artinian. But a sequence of inclusions \eqref{incl-seq} with $\Im Z(E_j/E_{j+1})>0$ is impossible since $\Im Z(E_j/E_{j+1})$ would tend to zero.
\noindent (b) To see that $P(0,1)$ is of finite length we observe that any increasing chain of admissible inclusions in $P(0,1)$ stabilizes since ${\mathcal A}=P(0,1]$ is Noetherian. Also, if we have a chain \eqref{incl-seq} of admissible proper inclusions in $P(0,1)$ then the sequence $\Im Z(E_j)$ is strictly decreasing, which is impossible. Under our assumptions
$|Z(E)|$ is bounded below by some positive constant, where $E$ runs through nonzero semistable objects in $P(0,1)$. Thus, $\sigma$ is reasonable if and only if
$$\inf_{E\in P(1)\setminus 0} |Z(E)|>0.$$ \end{proof}
\begin{prop} \label{GluingCondition} Let $({\mathcal D}_1, {\mathcal D}_2)$ be a semiorthogonal decomposition of a triangulated category ${\mathcal D}$, and let $\sigma_1=(Z_1,H_1)$ and $\sigma_2=(Z_2,H_2)$ be a pair of locally finite stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$, respectively. Assume that $\operatorname{Hom}_{{\mathcal D}}^{\le 0}(H_1, H_2) = 0$, and let $H$ be the heart in ${\mathcal D}$ glued from $H_1$ and $H_2$. As before, consider the stability function $Z = Z_1 \lambda_1 + Z_2 \rho_2$ on $H$. Assume in addition that one of the following two conditions hold:
\noindent (a) $0$ is an isolated point of $\Im Z_i(H_i)\subset\mathbb{R}_{\geq 0}$ for $i=1,2$;
\noindent (b) $\operatorname{Hom}_{\mathcal D}^{\le 1}(H_1,P_2(0,1))=0$.
Then $Z$ has the Harder-Narasimhan property on $H$. Furthermore, in case (a) the category $P(0,1)$ for the glued stability condition $\sigma=(Z,P)$ is of finite length. In case (b) the stability condition $\sigma$ is locally finite. \end{prop}
\begin{proof} First, assume that (a) holds. Then it is easy to see that $0$ is an isolated point of $\Im Z(H)\subset \mathbb{R}_{\geq 0}$. Also, by Lemma \ref{Noeth-lem}(a), both categories $H_1$ and $H_2$ are Noetherian (the condition on ${\mathcal A}_0$ in this Lemma follows from the assumption that $\sigma_i$'s are locally finite). Using the exact functors $\lambda_1:H\to H_1$ and $\rho_2:H\to H_2$ we easily deduce that $H$ is Noetherian. Now the assertion follows by applying Lemma \ref{Noeth-lem}(a) again.
\noindent (b) In this case for every $t\in (0,1]$ let us define the subcategory $P(t)\subset H$ by
$$P(t):=\{E\in H\ |\ \lambda_1(E)\in P_1(t), \rho_2(E)\in P_2(t)\}.$$ Note that each object of $P(t)$ is an extension of an object in $P_2(t)$ by an object in $P_1(t)$. It is enough for every $E\in H$ to construct the HN-filtration with respect to this slicing. We start with the canonical extension $$0\to E_2\to E\to E_1\to 0$$ where $E_2=\rho_2(E)\in H_2$ and $E_1=\lambda_1(E)\in H_1$. Consider also the canonical exact sequences $$0\to A_i\to E_i\to B_i\to 0$$ with $A_i\in P_i(1)$ and $B_i\in P_i(0,1)$ for $i=1,2$. Since $\operatorname{Hom}^1(E_1,B_2)=0$ by assumption, we get a splitting $E\to B_2$ which gives rise to an exact sequence $$0\to A_2\to E\to B_2\oplus E_1\to 0$$ Let $E(1)\subset E$ be the preimage of $A_1\subset E_1\subset B_2\oplus E_1$. Then $E(1)$ is an extension of $A_1$ by $A_2$, so $E(1)\in P(1)$. Also, $E/E(1)\simeq B_1\oplus B_2$, so we get the required filtration by using the HN-filtrations on $B_1$ and $B_2$. The obtained glued stability has the property that $\lambda_1(P(a,b))\subset P_1(a,b)$ and $\rho_2(P(a,b))\subset P_2(a,b)$. This easily implies that it is locally finite. \end{proof}
\begin{remark} We do not know how to check local finiteness of the glued stability condition in Proposition \ref{GluingCondition}(a) without imposing additional assumptions. \end{remark}
If we work with reasonable stability conditions, we can prove the existence of the glued stability conditions under a slightly stronger orthogonality assumption.
\begin{thm}\label{reason-glue-thm} Let $({\mathcal D}_{1}, {\mathcal D}_{2})$ be a semiorthogonal decomposition of a triangulated category ${\mathcal D}$. Suppose $(\sigma_{1}, \sigma_{2})$ is a pair of reasonable stability conditions on ${\mathcal D}_1$ and ${\mathcal D}_2$, respectively, with the slicings $P_i$ and central charges $Z_i$ ($i=1,2$), and let $a$ be a real number in $(0,1)$. Assume the following two conditions hold: \begin{enumerate} \item $\operatorname{Hom}_{\mathcal D}^{\leq 0}(P_1(0,1],P_2(0,1])=0$; \item $\operatorname{Hom}_{{\mathcal D}}^{\leq 0}(P_{1}(a,a+1], P_{2}(a,a+1]) = 0$; \end{enumerate} Then there exists a stability $\sigma$ glued from $\sigma_1$ and $\sigma_2$. Furthermore, $\sigma$ is reasonable. \end{thm}
\noindent {\it Proof of Theorem \ref{reason-glue-thm}.} Let $H\subset{\mathcal D}$ be the heart glued from $P_1(0,1]$ and $P_2(0,1]$ and let $({\mathcal D}^{\le 0},{\mathcal D}^{\ge 0})$ denote the corresponding $t$-structure. Using the second condition we can construct a $t$-structure on ${\mathcal D}$ with the heart $$H_a=\langle P_1(a,a+1], P_2(a,a+1]\rangle.$$ One immediately checks that $H\subset \langle H_a, H_a[-1]\rangle$ and $H_a\subset\langle H[1], H\rangle={\mathcal D}^{[-1,0]}$. Now for every $E\in H$ consider the canonical triangle $$A\to E\to B\to A[1]$$ with $A\in H_a$ and $B\in H_a[-1]$. We claim that $A$ and $B$ belong to $H$. Indeed, we have $A\in H_a\subset{\mathcal D}^{\le 0}$. On the other hand, $A$ is an extension of $E$ by $B[-1]\in H_a[-2]$, so $A\in {\mathcal D}^{\ge 0}$. Hence, $A\in H$. Similarly, $B\in H_a[-1]\subset {\mathcal D}^{\ge 0}$, and also $B\in{\mathcal D}^{\leq 0}$ as an extension of $A[1]\in H_a[1]$ by $E$. Therefore, if we set
$$P(0,a]=\{E\in{\mathcal D}\ |\ \lambda_1(E)\in P_1(0,a], \rho_2(E)\in P_2(0,a]\},$$ \begin{equation}\label{Pa1-eq}
P(a,1]=\{E\in{\mathcal D}\ |\ \lambda_1(E)\in P_1(a,1], \rho_2(E)\in P_2(a,1]\}, \end{equation} then $(P(a,1], P(0,a])$ is a torsion pair in $H$. Next, let $Z$ be the glued central charge given by \eqref{Z-for}. Then we have $\phi(Z(E))\leq a$ for $E\in P(0,a]$, while $\phi(Z(E))>a$ for $E\in P(a,1]$. Also, since $\sigma_1$ and $\sigma_2$ are reasonable, by Lemma \ref{reason-lem}(3), the categories $P_1(0,a]$ and $P_2(0,a]$ (resp., $P_1(a,1]$ and $P_2(a,1]$) are of finite length. This implies that both $P(0,a]$ and $P(a,1]$ are also of finite length. Therefore, we can apply Proposition \ref{Z-torsion-prop} to the torsion pair $(P(a,1], P(0,a])$ in $H$ to derive that the Harder-Narasimhan property holds for $(Z,H)$. Hence, we have the corresponding stability condition $\sigma$ on ${\mathcal D}$. It follows from the definition of $P(0,a]$ and $P(a,1]$ that $0$ is an isolated point of $Z(P(0,a])$ and of $Z(P(a,1])$. This immediately implies that $\sigma$ is reasonable. \qed
\begin{remark} It may not be easy in general to determine for a particular pair of stabilities $\sigma_{1}, \sigma_{2}$ with $\operatorname{Hom}_{\mathcal D}^{\leq 0}(P_1(>0),P_2(\leq 1)=0$ whether there exists $a \in (0,1)$ such that \[\operatorname{Hom}_{{\mathcal D}}^{\leq 0}(P_{1}(>a), P_{2}(\leq a+1) = 0.\] However, in the following two cases this is automatic.
\noindent 1. If there exists $\phi > 0$ such that $P_{2}(0,\phi] = \{ 0\}$ then any $a \in (0,\phi]$ works, since in this case $P_2(\leq a+1)=P_2(\leq 1)$. For instance, this condition is satisfied when $P_2(0,1]$ is of finite length and has finite number of simple objects.
\noindent 2. If there exists $\phi < 1$ such that $P_{1}(\phi,1] = \{ 0 \}$ then any $a \in (\phi,1] $ works, since in this case $P_1(>-a)=P_1(>0)$. For example, this condition holds when $P_{1}(0,1]$ is of finite length with finite number of simple objects and $P_{1}(1) = \{ 0 \}$. \end{remark}
\section{Continuity of gluing}
Let us recall the following basic result.
\begin{lem}\label{simple-distance-lem} (Lemma 6.4 of \cite{Bridgeland06}) Suppose $\sigma = (Z,P)$ and $\tau = (Z,Q)$ are stability conditions on $\mathcal{D}$ with the same central charge $Z$. Suppose also that $d(P,Q) < 1$. Then $\sigma = \tau$. \end{lem}
We start with the observation that the condition $d(P,Q)<1$ in the above Lemma can be weakened and use this to give a nice criterion for determining when two stability conditions are close (part (b) of the following Proposition).
\begin{prop}\label{FormalDistance} Let $\sigma_1=(Z_1,P_1)$ and $\sigma_2=(Z_2,P_2)$ be stability conditions on ${\mathcal D}$.
\noindent (a) Assume that \begin{enumerate} \item $Z_{1} = Z_{2}$ and \item $P_{1}(0,1] \subset P_{2}(-1,2]$. \end{enumerate} Then $\sigma_{1} = \sigma_{2}$.
\noindent (b) Assume that $\sigma_1$ is locally finite. There exists $\epsilon_0>0$ such that if for some $0 < \epsilon < \epsilon_0$ one has \begin{enumerate} \item
$||Z_1-Z_2||_{\sigma_1} <\sin(\pi\epsilon)$ and \item $P_{2}(0,1] \subset P_{1}(-1+\epsilon, 2-\epsilon],$ \end{enumerate} then $\sigma_2\in B_{\epsilon}(\sigma_1)$. \end{prop}
\begin{proof} (a) First, using properties of $t$-structures we can easily deduce that $P_2(0,1]\subset P_1(-1,2]$. Now given $E \in P_{1}(0,1]$, there is an exact triangle $$F \to E \to G\to F[1]$$ with $F \in P_{2}(1,2]$ and $G \in P_{2}(-1,1]$. Observe that $F \in P_{1}(>0)$ and $G \in P_{1}(\leq 2)$. Since $F$ is an extension of $E$ by $G[-1]$, we derive that $F \in P_{1}(0,1]$. But the intersection $P_{1}(0,1] \cap P_{2}(1,2]$ is trivial (since $Z_1=Z_2$), so $F = 0$. This proves that $E\in P_2(-1,1]$.
Next, consider an exact triangle $$F \to E \to G\to F[1]$$ with $F \in P_{2}(0,1]$ and $G \in P_{2}(-1,0]$. Observe that $F \in P_{1}(>-1)$ and $G \in P_{1}(\leq 1]$. Since $G$ is an extension of $F[1]$ by $E$, we get $G \in P_{2}(-1,0] \cap P_{1}(0,1] = \{ 0 \}$. Therefore, $P_{1}(0,1]\subset P_{2}(0,1]$. Since these are both hearts of bounded $t$-structures, they have to be equal, so $\sigma_{1} = \sigma_{2}$.
\noindent (b) Let $\sigma=(Z_2,P)$ be the unique stability in $B_{\epsilon}(\sigma_1)$ lifting the central charge $Z_2$
---it exists by our assumption that $||Z_2-Z_1||_{\sigma_1}<\sin(\pi\epsilon)$ (using Theorem 7.1 of \cite{Bridgeland06}). Then $$P_2(0,1]\subset P_1(-1+\epsilon,2-\epsilon]\subset P(-1,2].$$ By part (a), this implies that $\sigma=\sigma_2$. \end{proof}
Now we can show that the gluing construction of Theorem \ref{reason-glue-thm} is continuous.
\begin{thm}\label{a-thm} Let $({\mathcal D}_{1}, {\mathcal D}_{2})$ be a semiorthogonal decomposition in a triangulated category ${\mathcal D}$. For a real number $a\in (0,1)$ let $S(a)\subset\operatorname{Stab}({\mathcal D}_1)\times\operatorname{Stab}({\mathcal D}_2)$ denote the subset of $(\sigma_1,\sigma_2)$ such that $\sigma_1$ and $\sigma_2$ are reasonable stability conditions satisfying \begin{enumerate} \item $\operatorname{Hom}_{\mathcal D}^{\leq 0}(P_1(0,1],P_2(0,1])=0$, \item $\operatorname{Hom}_{{\mathcal D}}^{\leq 0}(P_{1}(a,a+1], P_{2}(a,a+1]) = 0$. \end{enumerate} Let $\operatorname{gl}:S(a)\to\operatorname{Stab}({\mathcal D})$ be the map associating to $(\sigma_1,\sigma_2)$ the corresponding glued stability condition $\sigma$ on ${\mathcal D}$ (see Theorem \ref{reason-glue-thm}). Then the map $\operatorname{gl}$ is continuous on $S(a)$. \end{thm} \begin{proof} Let $\sigma_i=(Z_i,P_i)$, $\sigma'_i=(Z'_i,P'_i)$ be stabilities on ${\mathcal D}_i$ for $i=1,2$, such that $(\sigma_1,\sigma_2)$ and $(\sigma'_1,\sigma'_2)$ are points of $S(a)$, and let us denote by $\sigma=(Z,P)$ and $\sigma'=(Z',P')$ the corresponding glued stability conditions. Assume that $\sigma'_i\in B_{\delta}(\sigma_i)$ for $i=1,2$. Then for $\epsilon\ge\delta$ we have \begin{eqnarray*} P(0,1] = \langle P_{1}(0,1], P_{2}(0,1]\rangle & \subset & \langle P'_{1}(-\epsilon,1+\epsilon], P'_{2}(-\epsilon,1+\epsilon]\rangle \\ & \subset & P'(-\epsilon,1+\epsilon]. \end{eqnarray*} Thus, we can deduce the required continuity from
Proposition \ref{FormalDistance}(b), once we show that $||Z-Z'||_{\sigma} \leq\sin(\pi\epsilon)$ provided $\delta$ is small enough. Let $\phi \in (0,1]$ and $E \in P(\phi).$ We have to prove that
\[ |Z(E)-Z'(E)| \leq |Z(E)|\sin(\pi \epsilon). \] Assume first that $\phi \in (a,1]$. Let $h: \mathbb{C} \rightarrow \mathbb{R}$ denote the scalar product with the unit vector of phase $\frac{a+1}{2}$. Then there exists a positive constant $c$ (depending only on $a$) such that
\[ h(z) \leq |z| \leq c\cdot h(z), \] for all nonzero complex numbers $z$ with phase $\theta$, where $a\leq \theta \leq 1$.
Let $F_{1}, \cdots, F_{n}$ (resp., $G_{1}, \cdots, G_{m})$ be the HN-factors of $\lambda_{1}(E)$ (resp., $\rho_{2}(E)$) with respect to $\sigma_1$ (resp., $\sigma_2$). Then we have \begin{eqnarray*}
|Z(E)-Z'(E)| & \leq & |Z_{1}(\lambda_{1}E)-Z_{1}'(\lambda_{1}E)| + |Z_{2}(\rho_{2}E)-Z_{2}'(\rho_{2}E)| \\
& \leq & \sum_{i=1}^{n} |Z_{1}(F_{i})-Z_{2}'(F_{i})| + \sum_{j=1}^{m} |Z_{2}(G_{j})-Z_{2}'(G_{j})| \\
& \leq & \sin(\pi \delta)\left[\sum_{i=1}^{n} |Z_{1}(F_{i})|+\sum_{j=1}^{m} |Z_{2}(G_{j})|\right]. \end{eqnarray*} Recall that by \eqref{Pa1-eq}, we have $\lambda_1(E)\in P_1(a,1]$ and $\rho_2(E)\in P_2(a,1]$. Hence, all the numbers $Z_1(F_i)$ and $Z_2(G_j)$ have phases between $a$ and $1$, so we derive \begin{eqnarray*}
|Z(E)-Z'(E)| & \leq c \sin(\pi \delta)[\sum_{i=1}^{n} h(Z_{1}(F_{i}))+\sum_{j=1}^{m} h(Z_{2}(G_{j}))]\\
& = c \sin(\pi \delta)h(Z(E))\leq c \sin(\pi \delta)|Z(E)|. \end{eqnarray*}
So $\delta$ must be chosen to satisfy the relation $c \sin(\pi \delta) < \sin(\pi \epsilon).$ A similar argument covers the case of objects $F \in P(0,a]$ and imposes a second condition that $c' \sin(\pi \delta) < \sin(\pi \epsilon)$ for some positive constant $c'$, depending only on $a$. Given $\delta$ satisfying both conditions, it follows that $$||Z-Z'||_{\sigma} \leq \sin(\pi \epsilon).$$ \end{proof}
The following Corollary describes an open subset of pairs of stabilities that can be glued, obtained by imposing a stronger orthogonality assumption on $(\sigma_1,\sigma_2)$.
\begin{cor}\label{glue-cor} Let $U\subset\operatorname{Stab}({\mathcal D}_1)\times\operatorname{Stab}({\mathcal D}_2)$ denote the set of pairs of reasonable stabilities $(\sigma_1=(Z_1,P_1)$ and $\sigma_2=(Z_2,P_2))$ such that for some $\epsilon>0$ one has $$\operatorname{Hom}_{{\mathcal D}}^{\le 0}(P_1(-\epsilon,1],P_2(0,1+\epsilon))=0.$$ Then $U$ is open and the gluing map $\operatorname{gl}:U\to\operatorname{Stab}({\mathcal D})$ is continuous. \end{cor}
\begin{proof} Note that our assumption on $(\sigma_1,\sigma_2)$ is equivalent to $$\operatorname{Hom}_{\mathcal D}(P_1(-\epsilon,+\infty),P_2(-\infty,1+\epsilon))=0.$$ For each $\epsilon>0$ let us denote by $T_{\epsilon}$ the set of pairs $(\sigma_1,\sigma_2)$ satisfying this condition. Note that $U=\cup_{\epsilon>0}T_{\epsilon}$. Now to check that $U$ is open suppose we have $(\sigma_1,\sigma_2)\in T_{\epsilon}$. Given a pair $(\sigma'_1=(Z'_1,P'_1),\sigma'_2=(Z'_2,P'_2))$, such that $\sigma'_i\in B_{\delta}(\sigma_i)$, for $i=1,2$, where $0<\delta<\epsilon$, we have $P'_1(>-\epsilon+\delta)\subset P_1(>-\epsilon)$ and $P'_2(<1+\epsilon-\delta)\subset P_1(<1+\epsilon)$. Hence, $(\sigma'_1,\sigma'_2)$ belongs to $T_{\epsilon-\delta}$. It remains to apply Theorem \ref{a-thm}. \end{proof}
On the other hand, in the situation when ${\mathcal D}_1$ is generated by an exceptional object, we have the following result that will be used later.
\begin{cor}\label{exc-cor} Let $({\mathcal D}_{1}, {\mathcal D}_{2})$ be a semiorthogonal decomposition in a triangulated category ${\mathcal D}$.
\noindent (i) Assume that ${\mathcal D}_1$ is generated by an exceptional object $E_1$, and $H_2\subset{\mathcal D}_2$ is a heart of some bounded $t$-structure on ${\mathcal D}_2$, such that $\operatorname{Hom}^{\leq -1}_{{\mathcal D}}(E_1,H_2)=0$. Let $S_2\subset\operatorname{Stab}({\mathcal D}_2)$ denote the set of reasonable stability conditions $\sigma_2=(Z,P)$ with $P(0,1]=H_2$. On the other hand, let $R_1\subset\operatorname{Stab}({\mathcal D}_1)$ denote the set of stability conditions such that the phase of $E_1$ is $<0$. Then there is continuous gluing map $R_1\times S_2\to\operatorname{Stab}({\mathcal D})$.
\noindent (ii) Similarly, assume that ${\mathcal D}_2$ is generated by an exceptional object $E_2$, and $H_1\subset{\mathcal D}_1$ is a heart of some bounded $t$-structure on ${\mathcal D}_2$, such that $\operatorname{Hom}^{\leq -1}_{{\mathcal D}}(H_1,E_2)=0$. Let $S_1\subset\operatorname{Stab}({\mathcal D}_1)$ denote the set of reasonable stability conditions with the heart $H_1$, and let $R_2\subset\operatorname{Stab}({\mathcal D}_2)$ denote the set of stability conditions such that the phase of $E_2$ is $>1$. Then there is continuous gluing map $S_1\times R_2\to\operatorname{Stab}({\mathcal D})$. \end{cor}
\begin{proof} We will only consider (i) since the proof of (ii) is analogous. Let $R_1(\epsilon)\subset\operatorname{Stab}({\mathcal D}_1)$ denote the set of stability conditions such that the phase of $E_1$ is $<-\epsilon$. It is enough to check that for every $\epsilon>0$ one has $R_1(\epsilon)\times S_2\subset S(1-\epsilon)$, where $S(1-\epsilon)\subset\operatorname{Stab}({\mathcal D}_1)\times\operatorname{Stab}({\mathcal D}_2)$ is the subset considered in Theorem \ref{a-thm} for $a=1-\epsilon$. Note that $P_1(0,1]=\langle E_1[n]\rangle$, where $n$ is determined by the condition that the phase of $E_1$ is in the interval $(-n,-n+1]$. Hence, $n\ge 1$, so the condition $\operatorname{Hom}^{\le 0}(P_1(0,1],H_2)=0$ is satisfied. Similarly, $P_1(-\epsilon, 1-\epsilon]=\langle E_1[m]\rangle$, where $m\ge 1$. Hence, $\operatorname{Hom}^{\le 0}(P_1(-\epsilon,1-\epsilon], P_2(\le 1))=0$ which implies the condition (2) of Theorem \ref{a-thm} for $a=1-\epsilon$. \end{proof}
\section{Semiorthogonal decompositions associated with double coverings}
Let $\pi: X \to Y$ be a double covering of smooth projective varieties $X$ and $Y$, ramified along a smooth divisor $R$ in $Y$. Then we have an action of $\mathbb{Z}_2$ on $X$ such that the nontrivial element acts by the corresponding involution $\tau:X\to X$. Let us denote by ${\mathcal D}_{\mathbb{Z}_2}(X)$ the corresponding bounded derived category of $\mathbb{Z}_2$-equivariant coherent sheaves on $X$. We denote by $\zeta$ the nontrivial character of $\mathbb{Z}_2$. Note that $\tau$-invariant stability conditions on ${\mathcal D}(X)$ correspond to stability conditions on ${\mathcal D}_{\mathbb{Z}_2}(X)$ that are invariant under the autoequivalence $F\mapsto F\otimes\zeta$ (see \cite{MMS} or \cite{Polishchuk06}). Below we will show how to construct stability conditions on ${\mathcal D}_{\mathbb{Z}_2}(X)$ starting from a pair of
stability conditions on ${\mathcal D}(Y)$ and on ${\mathcal D}(R)$, satisfying certain assumptions.
Let us denote by $i:R\to X$ (resp., $j:R\to Y$) the closed embedding of the ramification divisor into $X$ (resp., $Y$). For every sheaf $F$ on $R$ we equip $i_*F$ with the trivial $\mathbb{Z}_2$-equivariant structure. This gives a functor $i_*:{\mathcal D}(R)\to {\mathcal D}_{\mathbb{Z}_2}(X)$. On the other hand, for a coherent sheaf $F$ on $Y$ we have a natural $\mathbb{Z}_2$-equivariant structure on $\pi^*F$, so we obtain a functor $\pi^*:{\mathcal D}(Y)\to {\mathcal D}_{\mathbb{Z}_2}(X)$.
\begin{thm}\label{semiorth-thm} The functors $i_*:{\mathcal D}(R)\to {\mathcal D}_{\mathbb{Z}_2}(X)$ and $\pi^*:{\mathcal D}(Y)\to {\mathcal D}_{\mathbb{Z}_2}(X)$ are fully faithful. We have two canonical semiorthogonal decompositions of ${\mathcal D}_{\mathbb{Z}_2}(X)$: $$ {\mathcal D}_{\mathbb{Z}_{2}}(X) = \langle \pi^{*}{\mathcal D}(Y), i_{*}{\mathcal D}(R) \rangle =
\langle \zeta \otimes i_{*}{\mathcal D}(R), \pi^{*}{\mathcal D}(Y) \rangle $$ \end{thm}
\begin{proof} The case where $X$ and $Y$ are curves was considered in Theorem 1.2 of \cite{P-orbifold}, and the proof in our case is very similar. The fact that $\pi^*$ is fully faithful follows immediately from the equality $(\pi_*{\mathcal O}_X)^{\mathbb{Z}_2}={\mathcal O}_Y$ and the projection formula. Similarly, to prove that $i_*$ is fully faithful it suffices to check $(Li^*i_*F)^{\mathbb{Z}_2}=F$. We have a canonical exact triangle $$F\otimes N^{\vee}[1]\to Li^*i_*F\to F\to\ldots$$
compatible with $\mathbb{Z}_2$-action, where $N^{\vee}={\mathcal O}_X(-R)|_R$ is the conormal bundle. It remains to observe that $\mathbb{Z}_2$ acts on $N^{\vee}$ by multiplication with $-1$.
Now let $F\in {\mathcal D}(Y)$ and $G\in {\mathcal D}(R)$ be some objects. Then we have $$\operatorname{Hom}_{\mathbb{Z}_2}(\pi^*(F), \zeta\otimes i_*(G))\simeq \operatorname{Hom}_{\mathbb{Z}_2}(Lj^*F,\zeta\otimes G)=0$$ which gives one of the required orthogonality conditions. On the other hand, by Serre duality, denoting $d=\dim X$, we get $$\operatorname{Hom}_{\mathbb{Z}_2}(i_*(G),\pi^*(F))^*\simeq \operatorname{Hom}_{\mathbb{Z}_2}(\pi^*(F),\omega_X\otimes i_*(G)[d])\simeq \operatorname{Hom}_{\mathbb{Z}_2}(Lj^*F,i^*\omega_X\otimes G[d]).$$ Note that $\mathbb{Z}_2$ acts nontrivially on $i^*\omega_X\simeq \omega_Y\otimes N^{\vee}$, so the above Hom-space vanishes.
Finally, we have to check that for every $F\in{\mathcal D}_{\mathbb{Z}_2}(X)$ such that $\operatorname{Hom}_{\mathbb{Z}_2}(i_*{\mathcal D}(R),F)=0$ or $\operatorname{Hom}_{\mathbb{Z}_2}(F,\zeta\otimes i_*{\mathcal D}(R))=0$, lies in the essential image of $\pi^*:{\mathcal D}(Y)\to{\mathcal D}_{\mathbb{Z}_2}(X)$. Note that by Serre duality, these two orthogonality conditions are equivalent. Assume that $\operatorname{Hom}_{\mathbb{Z}_2}(F,\zeta\otimes i_*{\mathcal D}(R))=0$. Then $\mathbb{Z}_2$ acts trivially on $i^*F$. Now the assertion follows from the main theorem of \cite{Ter}. \end{proof}
We can use the above Theorem as a setup for gluing stability conditions. The situation seems to be especially nice when either ${\mathcal D}(R)$ or ${\mathcal D}(Y)$ admits an exceptional collection (see Remark at the end of the previous section). The former possibiity occurs when $X$ and $Y$ are curves and will be considered below. The latter possibility happens if, say, $Y$ is a projective space. In particular, we derive the following result.
\begin{prop} Let $\pi:X\to\mathbb{P}^n$ be a smooth double covering ramified along a smooth hypersurface $j:R\hookrightarrow\mathbb{P}^n$. Assume we are given a reasonable stability $\sigma^R=(Z^R,P^R)$ on ${\mathcal D}(R)$, an $\operatorname{Ext}$-exceptional collection $(E_0,\ldots,E_n)$ on $\mathbb{P}^n$, and a set of vectors $v_0,\ldots,v_n$ in the upper half-plane such that $j^*E_i\in P^R(>1)$ for $i=0,\ldots,n$. Then there exists a reasonable stability $\sigma=(Z,P)$ on ${\mathcal D}_{\mathbb{Z}_2}(X)$ with $$P(0,1]=[ i_*P^R(0,1], \pi^*E_0,\ldots,\pi^*E_n ],$$ $$Z(E)=v_0 x_0(R\pi_*(E(R))^{\mathbb{Z}_2})+\ldots+v_n x_n(R\pi_*(E(R))^{\mathbb{Z}_2})- Z^R((i^*E\otimes N)^{\mathbb{Z}_2}),$$ where $x_0,\ldots,x_n:K_0(\mathbb{P}^n)\to\mathbb{Z}$ are the coordinates dual to the basis $([E_i])$. \end{prop}
\begin{proof} This stability is obtained by gluing with respect to the semiorthogonal decomposition \begin{equation}\label{semiorth-dec} {\mathcal D}_{\mathbb{Z}_{2}}(X) = \langle \pi^{*}{\mathcal D}(Y), i_{*}{\mathcal D}(R) \rangle. \end{equation} It exists by Theorem \ref{reason-glue-thm}, where $a<1$ should be taken bigger than all of the phases of the vectors $v_i$ (see Remark after Theorem \ref{reason-glue-thm}). To get the formula for the central charge we note that for $E\in D_{\mathbb{Z}_2}(X)$ one has $$\rho_2(E)=i^!(E)^{\mathbb{Z}_2}\simeq (i^*E\otimes N)^{\mathbb{Z}_2}[-1],$$ $$\lambda_1(E)=R\pi_*(E(R))^{\mathbb{Z}_2}.$$ \end{proof}
For example, if $X\to\mathbb{P}^3$ is a double covering ramified along a smooth surface $S\subset\mathbb{P}^3$ then we can consider stabilities on $S$ constructed in \cite{AB}. Choosing an appropriate $\operatorname{Ext}$-exceptional collection on $\mathbb{P}^3$ and using the above result we get examples of stabilities on ${\mathcal D}_{\mathbb{Z}_2}(X)$.
\section{Double coverings of curves}
In section we will consider the case when $X$ and $Y$ are curves. In this case the ramification divisor $R$ consists of points $p_1,\ldots,p_n$, and the category ${\mathcal D}(R)$ is generated by the orthogonal exceptional objects ${\mathcal O}_{p_1},\ldots,{\mathcal O}_{p_n}$. Recall that the category ${\mathcal D}(X)$ has a standard stability condition $\sigma_{st}$ with $Z_{st}=-\deg+i\operatorname{rk}$ and $P_{st}(0,1]=\operatorname{Coh}(X)$. There is an induced stability condition on ${\mathcal D}_{\mathbb{Z}_2}(X)$ with the heart $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ that we still denote by $\sigma_{st}$ (see \cite{MMS}).
\begin{lem}\label{simple-obj-lem} Let $E$ be an endosimple object of the category ${\mathcal D}_{\mathbb{Z}_2}(X)$ (i.e., $\operatorname{Hom}(E,E)=k$). Then for some $n\in\mathbb{Z}$ the object $E[n]$ is one of the following types: \begin{enumerate} \item a vector bundle; \item the sheaf ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y$; \item the sheaf $\zeta\otimes{\mathcal O}_{2p_i}$ for some $i\in\{1,\ldots,n\}$; \item the sheaf ${\mathcal O}_{p_i}$ for some $i$; \item the sheaf $\zeta\otimes{\mathcal O}_{p_i}$ for some $i$. \end{enumerate} \end{lem}
\begin{proof} The category $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ has cohomological dimension $1$, so every indecomposable object in ${\mathcal D}_{\mathbb{Z}_2}(X)$ has only one nonzero cohomology. Thus, we can assume that $E$ is a $\mathbb{Z}_2$-equivariant coherent sheaf. Furthermore, since the torsion part of such a sheaf splits as a direct summand, it is enough to consider the case when $E$ is an indecomposable torsion sheaf. Then the support of $E$ is either $\pi^{-1}(y)$, where $y\in Y\setminus R$, or $\{p_i\}$ for some $i\in\{1,\ldots,n\}$. In the former case $E\simeq\pi^*E'$, where $E'$ is an endosimple sheaf on $Y$ supported at $y$, so $E'\simeq{\mathcal O}_y$. In the latter case there exists $m$ such that $E\simeq{\mathcal O}_{mp_i}$ or $E\simeq\zeta\otimes{\mathcal O}_{mp_i}$. It remains to observe that for $m\geq 3$ the sheaf ${\mathcal O}_{mp_i}$ is not endosimple, since we can construct its nonscalar endomorphism as the composition of natural maps $${\mathcal O}_{mp_i}\to {\mathcal O}_{(m-2)p_i}\to{\mathcal O}_{mp_i}.$$ \end{proof}
We are going to construct explicitly some stability conditions on ${\mathcal D}_{\mathbb{Z}_2}(X)$. For this we will use a slight variation of the semiorthogonal decompositions considered in Theorem \ref{semiorth-thm}. Namely, for every partition of $\{1,\ldots,n\}$ into two disjoint subset $I$ and $J$ we have \begin{equation}\label{semiorth-dec2}
{\mathcal D}_{\mathbb{Z}_{2}}(X) = \langle\langle\zeta\otimes{\mathcal O}_{p_j}\ |\ j\in J\rangle, \pi^{*}{\mathcal D}(Y),
\langle {\mathcal O}_{p_i}\ |\ i\in I\rangle\rangle. \end{equation}
For a subset $I\subset\{1,\ldots,n\}$ let us denote by ${\mathcal D}(I)\subset{\mathcal D}_{\mathbb{Z}_2}(X)$ the full triangulated subcategory generated by $\pi^*{\mathcal D}(Y)$ and ${\mathcal O}_{p_i}$ with $i\in I$.
\begin{lem}\label{coh-lem} For $I\subset\{1,\ldots,n\}$ set $\operatorname{Coh}(I):=\operatorname{Coh}_{\mathbb{Z}_2}(X)\cap{\mathcal D}(I)$. Then $\operatorname{Coh}(I)$ is the heart of a $t$-structure on ${\mathcal D}(I)$. The natural exact functor $\operatorname{Coh}(I)\to\operatorname{Coh}_{\mathbb{Z}_2}(X)$ gives an equivalence of $\operatorname{Coh}(I)$ with the full subcategory of $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ consisting of all successive extensions of sheaves
in $\pi^*\operatorname{Coh}(Y)$ and equivariant sheaves supported on $\{p_i \ |\ i\in I\}$. The category $\operatorname{Coh}(I)$ is Noetherian. \end{lem}
\begin{proof} Note that an object $E\in{\mathcal D}_{\mathbb{Z}_2}(X)$ belongs to ${\mathcal D}(I)$ if and only if $\operatorname{Hom}^*({\mathcal O}_{p_i},E)=0$ for each $i\not\in I$. Since the category $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ has cohomological dimension $1$, we have $E\simeq\oplus H^iE[-i]$, where $H^iE\in\operatorname{Coh}_{\mathbb{Z}_2}(X)$. Therefore, $E\in{\mathcal D}(I)$ if and only if $H^iE\in{\mathcal D}(I)$ for every $i$. This immediately implies that the standard $t$-structure restricts to a $t$-structure on ${\mathcal D}(I)$ with $\operatorname{Coh}(I)$ as the heart. We have an exact embedding $\operatorname{Coh}(I)\to\operatorname{Coh}_{\mathbb{Z}_2}(X)$, so $\operatorname{Coh}(I)$ is Noetherian. Let ${\mathcal F}\in\operatorname{Coh}(I)$. Then the torsion part (resp., torsion-free part) of ${\mathcal F}$ is also in $\operatorname{Coh}(I)$. Assume first that ${\mathcal F}$ is an indecomposable torsion sheaf with the support at $p_i$ for $i\not\in I$. Then the condition $\operatorname{Hom}^*({\mathcal O}_{p_i},E)=0$ easily implies that $E\simeq{\mathcal O}_{2np_i}$. On the other hand, if ${\mathcal F}$ is a vector bundle then we have $\operatorname{Hom}({\mathcal F},\zeta\otimes{\mathcal O}_{p_i})=0$ for $i\not\in I$, which implies that the fiber of ${\mathcal F}$ at $p_i$ has trivial $\mathbb{Z}_2$-action for $i\not\in I$. Therefore, making appropriate elementary transformations at $p_i$ for $i\in I$
we can represent ${\mathcal F}$ as an extension of a sheaf supported at $\{p_i \ |\ i\in I\}$ by the pull-back of a vector bundle from $Y$ (cf. proof of Theorem 1.8 of \cite{P-orbifold}). \end{proof}
Given a partition of $\{1,\ldots,n\}$ into three disjoint subsets $I^0$, $I^+$ and $I^-$ we obtain from \eqref{semiorth-dec2} a semiorthogonal decomposition \begin{equation}\label{semiorth-dec3}
{\mathcal D}_{\mathbb{Z}_2}(X)=\langle \langle \zeta\otimes{\mathcal O}_{p_i}\ |\ i\in I^-\rangle, {\mathcal D}(I^0), \langle {\mathcal O}_{p_i}, i\in I^+\rangle\rangle. \end{equation}
\begin{prop}\label{cover-prop} Fix a partition $\{1,\ldots,n\}=I^0\sqcup I^+\sqcup I^-$ and a collection of positive integers $(n_i)$ for $i\not\in I^0$.
\noindent (a) Let $Z:{\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X))\to\mathbb{C}$ be a homomorphism, such that \begin{enumerate} \item $\Im Z({\mathcal O}_X)>0$, and $Z({\mathcal O}_{\pi^{-1}(y)})\in \mathbb{R}_{<0}$ for any point $y\in Y$; \item $Z({\mathcal O}_{p_i}[-n_i])\in{\mathfrak h'}$ for $i\in I^+$, and $Z(\zeta\otimes{\mathcal O}_{p_i}[n_i])\in{\mathfrak h'}$ for $i\in I^-$; \item $Z({\mathcal O}_{p_i})\in \mathbb{R}_{<0}$ and $Z(\zeta\otimes{\mathcal O}_{p_i})\in\mathbb{R}_{<0}$ for $i\in I^0$, \end{enumerate} where ${\mathfrak h'}\subset\mathbb{C}$ denotes the union of the upper half-plane with $\mathbb{R}_{<0}$. Then there exists a reasonable stability condition $\sigma$ with the central charge $Z$ and the heart \begin{equation}\label{glued-heart-eq}
H(I^+,I^-;{\bf n})=[ [\zeta\otimes{\mathcal O}_{p_i}[n_i]\ |\ i\in I^-], \operatorname{Coh}(I^0), [ {\mathcal O}_{p_i}[-n_i], i\in I^+] ], \end{equation} which is glued with respect to the semiorthogonal decomposition \eqref{semiorth-dec3}. All the objects ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y$ are $\sigma$-semistable (of phase $1$). The objects
${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus \{p_i\ |\ i\in I^0\}$, as well as ${\mathcal O}_{p_i}$ for $i\in I^0\cup I^+$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in I^0\cup I^-$, are $\sigma$-stable.
\noindent (b) Assume in addition that $n_i=1$ for all $i\not\in I^0$. Then all the objects ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in\{1,\ldots,n\}$ are $\sigma$-stable.
\end{prop}
\begin{proof} (a) Using the orthogonalities $$\operatorname{Hom}^{\leq 0}(\operatorname{Coh}(I^0),{\mathcal O}_{p_i}[-n_i])=\operatorname{Hom}^{\leq 0}(\zeta\otimes{\mathcal O}_{p_j}[n_j],\operatorname{Coh}(I^0))= \operatorname{Hom}^{\leq 0}(\zeta\otimes{\mathcal O}_{p_j}[n_j],{\mathcal O}_{p_i}[-n_i])$$ for $i\in I^+$, $j\in I^-$, we get the glued heart $H=H(I^+,I^-;{\bf n})$ given by \eqref{glued-heart-eq}. Note that the restriction of $Z$ to ${\mathcal N}(\pi^*{\mathcal D}(Y))$ is determined by $Z({\mathcal O}_X)$ and by $Z({\mathcal O}_{\pi^{-1}(y)})$ for a point $y\in Y$. Thus, $\Im Z(\pi^*F)=c\operatorname{rk}(F)$ for some positive constant $c$. Since $\operatorname{Coh}(I^0)$ is generated by extensions from $\pi^*\operatorname{Coh}(Y)$ and ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in I^0$, we deduce that $Z$ is a stability function on $H$. It is also easy to see that $0$ is an isolated point of $\Im Z(H)$. Since $H$ is glued from Noetherian hearts, it is also Noetherian, so Lemma \ref{Noeth-lem}(a) implies that the Harder-Narasimhan property is satisfied for $Z$. Thus, we have a stability condition $\sigma=(Z,P)$ with $P(0,1]=H$. By Proposition \ref{char-prop}(2), it is glued from the induced stability on ${\mathcal D}(I^0)$ and the exceptional objects $\zeta\otimes{\mathcal O}_{p_i}[n_i]$, $i\in I^-$ and ${\mathcal O}_{p_i}[-n_i]$, $i\in I^+$. The fact that $\sigma$ is reasonable follows from Lemma \ref{Noeth-lem}(b). Note that $P(1)\subset H$ consists of successive extensions of sheaves of the form ${\mathcal O}_{\pi^{-1}(y)}$,
$y\in Y$, and of ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in I^0$. The simple objects in $P(1)$ are the sheaves ${\mathcal O}_{\pi^{-1}(y)}$, $y\in Y\setminus\{p_i\ |\ i\in I^0\}$, and ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in I^0$, so all these objects are $\sigma$-stable. On the other hand, Proposition \ref{char-prop}(iii) implies that the above exceptional objects in the heart corresponding to $i\in I^+\cup I^-$, are $\sigma$-stable.
\noindent (b) Let us denote
$${\mathcal C}^+:=[{\mathcal O}_{p_i}\ |\ i\in I^+]\subset\operatorname{Coh}_{\mathbb{Z}_2}(X),$$
$${\mathcal C}^-:=[\zeta\otimes{\mathcal O}_{p_i}\ |\ i\in I^-]\subset\operatorname{Coh}_{\mathbb{Z}_2}(X).$$ From the definition of $H$ one can easily deduce that for every object $C\in H$ one has \begin{align*} & H^{-1}C\in{\mathcal C}^-; & H^1C\in{\mathcal C}^+; & H^0C\simeq H^0(F_{-1}\to F_0\to F_1), \text{where } F_0\in\operatorname{Coh}(I^0), F_{-1}\in{\mathcal C}^-, F_1\in{\mathcal C}^+. \end{align*} The last condition easily implies that $\operatorname{Hom}({\mathcal C}^+,H^0C)=\operatorname{Hom}(H^0C,{\mathcal C}^-)=0$.
Now let us fix $i\in I^+$ and consider the object $E=\zeta\otimes{\mathcal O}_{p_{i}}$. Note that $\zeta\otimes{\mathcal O}_{p_i}$ belongs to $H$, as an extension of ${\mathcal O}_{2p_i}$ by ${\mathcal O}_{p_i}[-1]$. Suppose we have a short exact sequence $$0\to A\to E\to B\to 0$$ in $H$ with nonzero $A$ and $B$. Since $H^2A=H^{-2}B=0$, we derive that $H^1B=H^{-1}A=0$ and there is an exact sequence \begin{equation}\label{long-ex-seq} 0\to H^{-1}B\to H^0A\to E\to H^0B\to H^1A\to 0 \end{equation} in $\operatorname{Coh}_{\mathbb{Z}_2}(X)$. Note that since $E$ is a simple object of $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ we have one of the following two cases: (i) $H^0B\to H^1A$ is an isomorphism; (ii) $H^{-1}B\to H^0A$ is an isomorphism. In the first case we obtain that $H^0B\in{\mathcal C}^+$ which implies that $H^0B=0$. Hence, in this case $B\in{\mathcal C}^-[1]$, so $\operatorname{Hom}(E,B)=0$ which is a contradiction. Now let us consider case (ii). We have $H^0A\in{\mathcal C}^-$, hence $H^0A=0$. It follows that $A=H^1A[-1]$, and $B=H^0B$ is an extension of $H^1A$ by $E$. Since $\operatorname{Hom}(H^1A,B)=0$, this extension cannot split on any direct summands of $H^1A$, which implies that $A\simeq{\mathcal O}_{p_i}[-1]$ and $B\simeq {\mathcal O}_{2p_i}$. Since $Z({\mathcal O}_{p_i}[-1])$ has smaller phase then $Z(E)$, this shows that $\zeta\otimes{\mathcal O}_{p_i}$ is stable. Similarly one proves that all the objects ${\mathcal O}_{p_i}$ for $i\in I^-$ are stable. \end{proof}
In the case when all $n_i$'s are equal to $1$, we denote the heart $H(I^+,I^-,{\bf n})$ considered in the above Proposition simply by $H(I^+,I^-)$.
We have the following partial characterization of stability conditions constructed above.
\begin{lem}\label{phase-lem} Let $\sigma=(Z,P)$ be a stability condition such that ${\mathcal O}_{\pi^{-1}(y)}\in P(1)$ for all $y\in Y\setminus R$.
\noindent (a) Assume that ${\mathcal O}_{2p_i}\in P(1)$ for all $i$, and for every $i$ one of the following three conditions holds: \begin{enumerate} \item both ${\mathcal O}_{p_i}$ and $\zeta\otimes {\mathcal O}_{p_i}$ are $\sigma$-semistable of phase $1$; \item ${\mathcal O}_{p_i}$ is $\sigma$-semistable of phase $>1$; \item $\zeta\otimes{\mathcal O}_{p_i}$ is $\sigma$-semistable of phase $\leq 0$. \end{enumerate} Assume in addition that for every line bundle $L$ on $Y$ one has $\pi^*L\in P(0,1]$. Then $\sigma$ coincides with one of the stability conditions constructed in Proposition \ref{cover-prop}. The latter condition is uniquely determined by $Z$ and by the phases of ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ for $i\in\{1,\ldots,n\}$.
\noindent (b) Now assume that $\sigma$ is locally finite, and for all $i\in\{1,\ldots,n\}$ one has ${\mathcal O}_{p_i}\in P[1,2)$ and $\zeta\otimes{\mathcal O}_{p_i}\in P(0,1]$. Assume in addition that either all objects ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ are stable, or $\Im Z(V)>0$ for every $\mathbb{Z}_2$-equivariant vector bundle $V$. Then $\sigma$ coincides with one of stability conditions constructed in Proposition \ref{cover-prop} with $I^-=\emptyset$,
$$I^+=\{ i\ |\ \Im Z({\mathcal O}_{p_i})<0\},$$ and all $n_i$'s equal to $1$.
\end{lem}
\begin{proof} (a) Let $I^0$, $I^+$ and $I^-$ be the subsets of $i$ such that conditions (1), (2) and (3) hold, respectively. Note that since we have nonzero maps ${\mathcal O}_{p_i}\to \zeta\otimes{\mathcal O}_{p_i}[1]$, the conditions (2) and (3) (and therefore, the subsets $I^0$, $I^+$ and $I^-$) are mutually disjoint. For each $i\in I^+$ (resp., $i\in I^-$) there is a unique $n_i>0$ such that $\phi({\mathcal O}_{p_i})-n_i\in (0,1]$ (resp., $\phi({\mathcal O}_{p_i})+n_i\in (0,1]$). Then $Z$ satisfies the conditions of Proposition \ref{cover-prop}, so it remains to check that $H=H(I^+,I^-;{\bf n})\subset P(0,1]$. Note that by definition, we have ${\mathcal O}_{\pi^{-1}(y)}\in P(0,1]$ for all $y\in Y$; ${\mathcal O}_{p_i},\zeta\otimes{\mathcal O}_{p_i}\in P(1)$ for $i\in I^0$; ${\mathcal O}_{p_i}[-n_i]\in P(0,1]$ for $i\in I^+$ and $\zeta\otimes{\mathcal O}_{p_i}[n_i]\in P(0,1]$ for $i\in I^-$. It remains to show that $\pi^*V\in P(0,1]$ for every vector bundle $V$ on $Y$. But such a vector bundle can be presented as an extension of line bundles, so this follows from our assumption.
\noindent (b) It is enough the check that $P(0,1]\subset H=H(I^+,\emptyset)$ (where $I^0$ is the complement to $I^+$). First, we observe that in this case all equivariant vector bundles are in $H$, as extensions of direct sums of sheaves of the form $\zeta\otimes{\mathcal O}_{p_i}$ by a sheaf in $\pi^*\operatorname{Coh}(Y)$. Let $E$ be a $\sigma$-stable object in $P(0,1)$. Note that $E$ is endosimple. Let us consider possibilities for $E$ listed in Lemma \ref{simple-obj-lem}. Since $Z({\mathcal O}_{\pi^{-1}(y)})=Z(\zeta\otimes{\mathcal O}_{2p_i})\in\mathbb{R}_{<0}$ and $E\in P(0,1)$, we obtain that for some $m\in\mathbb{Z}$, $E[m]$ is either a vector bundle, or isomorphic to ${\mathcal O}_{p_i}[-1]$, or to $\zeta\otimes{\mathcal O}_{p_i}$. In the last two cases our assumptions on $\sigma$ imply that $m=0$, so $E\in H$. If $E[m]$ is a vector bundle then using the condition $E\in P(0,1)$ we get \begin{equation}\label{Hom-van-eq} \operatorname{Hom}^{\leq -1}(E,{\mathcal O}_{\pi^{-1}(y)})=\operatorname{Hom}^{\leq 0}({\mathcal O}_{\pi}^{-1}(y),E)=0. \end{equation} This implies that $m=0$, so $E\in H$. Next, let $E$ be a $\sigma$-stable object in $P(1)$. We can assume that $E$ is not isomorphic to ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ since these objects are in $H$. Assume that $E[m]$ is a vector bundle. Note that this case cannot occur if $\Im Z(V)>0$ for all equivariant vector bundles, so we can assume that the objects ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ are stable. Then the vanishing \eqref{Hom-van-eq} still holds, so we deduce again that $m=0$. The case when $E[m]$ is either ${\mathcal O}_{p_i}$, or $\zeta\otimes{\mathcal O}_{p_i}$ (where $i\in I^0$) is also clear. Note that for $i\in I^0$ we have ${\mathcal O}_{p_i},\zeta\otimes{\mathcal O}_{p_i}\in P(1)$. Hence, for such $i$ the objects ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ are not $\sigma$-stable. Now assume that $E[m]\simeq {\mathcal O}_{2p_i}$, where $i\in I^+$. Since ${\mathcal O}_{2p_i}\in P(0,2)$ as an extension of ${\mathcal O}_{p_i}$ by $\zeta\otimes {\mathcal O}_{p_i}$, this implies that $m=0$, so $E\in H$. Finally, we observe that for $i\in I^+$ the object $\zeta\otimes{\mathcal O}_{2p_i}$ is not semistable since it is an extension of $\zeta\otimes{\mathcal O}_{p_i}$ by ${\mathcal O}_{p_i}$, where $\phi_{\min}(\zeta\otimes{\mathcal O}_{p_i})<1$ and $\phi_{\max}({\mathcal O}_{p_i})>1$. \end{proof}
Note that the classes $[{\mathcal O}_X]$, $[{\mathcal O}_{\pi^{-1}(y)}]$, and $[{\mathcal O}_{p_i}]$, $i\in\{1,\ldots,n\}$, form a basis in ${\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X))$. Thus, we can define a norm on the vector space $\operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ by setting
$$||Z||=\max(|Z({\mathcal O}_X)|,\max_{E}|Z(E)|),$$ where $E$ runs over all endosimple torsion sheaves in $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ (see Lemma \ref{simple-obj-lem}). It is also convenient to set for $Z\in \operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ $$v_Z:=Z({\mathcal O}_{\pi^{-1}(y)})\in\mathbb{C}.$$ Let us define an open subset $\overline{U}\subset \operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ as the set of central charges $Z$ satisfying the following assumptions: \begin{enumerate} \item for every $\mathbb{Z}_2$-equivariant line bundle $L$ on $X$ one has $\det(Z(L),v_Z)>0$; \item for every $i=1,\ldots,n$ one has $Z({\mathcal O}_{p_i})\not\in\mathbb{R}_{\leq 0}\cdot v_Z$, $Z(\zeta\otimes{\mathcal O}_{p_i})\not\in\mathbb{R}_{\leq 0}\cdot v_Z$. \end{enumerate} Note that in the first condition it is enough to consider representatives in the cosets for the subgroup $\pi^*\operatorname{Pic}(Y)\subset \operatorname{Pic}_{\mathbb{Z}_2}(X)$, so there is only finite number of inequalities to check (hence, $\overline{U}$ is open). Also, this condition implies that $\det(Z(V),v_Z)>0$ for every equivariant vector bundle $V $on $X$, since they can be obtained from line bundles by successive extensions.
\begin{lem}\label{num-lem} \begin{enumerate} \item Let $Z:{\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X))\to\mathbb{C}$ be a homomorphism such that $\Im Z({\mathcal O}_X)>0$, $Z({\mathcal O}_{\pi^{-1}(y)})\in\mathbb{R}_{<0}$, and for every $i=1,\ldots,n$ one has $Z({\mathcal O}_{p_i})\neq 0$ and $\Im Z({\mathcal O}_{p_i})\leq 0$.
Then there exists a constant $r>0$ such that for every $Z'\in \operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ and every endosimple object $E\in{\mathcal D}_{\mathbb{Z}_2}(X)$ one has
$$|Z'(E)|\leq r\cdot ||Z'||\cdot |Z(E)|.$$ \item The above conclusion also holds for $Z\in\overline{U}$. \end{enumerate} \end{lem}
\begin{proof} (1) Our conditions on $Z$ imply that $Z(E)\neq 0$ for every endosimple torsion $\mathbb{Z}_2$-equivariant coherent sheaf $E$. Therefore, we can set
$$r_1=\max_E(|Z(E)|^{-1}),$$
where $E$ runs over all endosimple torsion sheaves. If $E$ is such a sheaf then $|Z'(E)|\leq ||Z'||$, so the required inequality holds for $E$ provided $r\geq r_1$. Now assume that $E$ is a $\mathbb{Z}_2$-equivariant vector bundle on $X$. Then there exists an exact sequence of the form $$0\to \pi^*E'\to E\to\oplus_i \zeta\otimes{\mathcal O}_{p_i}^{m_i}\to 0,$$ where $0\le m_i\leq\operatorname{rk}(E)$. Then
$$|Z'(E)|\leq |Z'(\pi^*E')|+n\operatorname{rk}(E)\cdot ||Z'||.$$ Note that \begin{equation}\label{E'-K0-eq} [\pi^*E']=\operatorname{rk}(E)[{\mathcal O}_X]+\deg(E')[{\mathcal O}_{\pi^{-1}(y)}] \end{equation} in ${\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X))$. Thus, we obtain \begin{equation}\label{Z'-eq}
|Z'(E)|\leq ||Z'||\cdot [(n+1)\operatorname{rk}(E)+\deg(E')]. \end{equation} On the other hand, from the above exact sequence we get $$\Im Z(E)=\Im Z(\pi^*E')+\sum_i m_i\cdot \Im Z(\zeta\otimes{\mathcal O}_{p_i}).$$ Since $\Im Z(\zeta\otimes{\mathcal O}_{p_i})\geq 0$ and $\Im Z(\pi^*E')=\Im Z({\mathcal O}_X)\cdot\operatorname{rk}(E)$, we deduce that
$$\operatorname{rk}(E)\leq \frac{|Z(E)|}{\Im Z({\mathcal O}_X)}.$$ Also, from \eqref{E'-K0-eq} we get
$$|\deg(E')Z({\mathcal O}_{\pi^{-1}(y)})|\leq |Z(\pi^*E')|+\operatorname{rk}(E)|Z({\mathcal O}_X)|\leq
|Z(E)|+(n+1)\operatorname{rk}(E)\cdot ||Z||.$$ Using our estimate for $\operatorname{rk}(E)$ we get that
$$\deg(E')\leq |Z({\mathcal O}_{\pi^{-1}(y)})|^{-1}\cdot [1+(n+1)\frac{||Z||}{\Im Z({\mathcal O}_X)}]\cdot |Z(E)|.$$ Therefore, from \eqref{Z'-eq} we obtain
$$|Z'(E)|\leq r_2 ||Z'||\cdot |Z(E)|,$$ where
$$r_2=\frac{n+1}{\Im Z({\mathcal O}_X)}+|Z({\mathcal O}_{\pi^{-1}(y)})|^{-1}\cdot [1+(n+1)\frac{||Z||}{\Im Z({\mathcal O}_X)}].$$ It remains to set $r=\max(r_1,r_2)$.
\noindent (2) The subset $\overline{U}\subset \operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ is stable under composition with
rotations of $\mathbb{C}$ and with automorphisms of ${\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X))$ given by tensoring with an equivariant line bundle $L$. Also, the norms $||\cdot ||$ and
$Z'\mapsto ||Z'\circ (\otimes L)||$ on the finite-dimensional vector space $\operatorname{Hom}({\mathcal N}({\mathcal D}_{\mathbb{Z}_2}(X)),\mathbb{C})$ are equivalent, while composing with a rotation of $\mathbb{C}$ does not change the norms. Therefore, we can modify $Z$ using these operations before checking the required inequalities. Rotating $Z$ we can assume that $v_Z\in\mathbb{R}_{<0}$. Next, let $I\subset\{1,\ldots,n\}$ be the set of $i$ such that $\Im Z({\mathcal O}_{p_i})>0$. Taking $L={\mathcal O}(\sum_{i\in I}p_i)$ we will have $$L\otimes{\mathcal O}_{p_i}\simeq \begin{cases}\zeta\otimes{\mathcal O}_{p_i}, & i\in I \\ {\mathcal O}_{p_i}, & i\not\in I. \end{cases}$$ Therefore composing $Z$ with tensoring by $L$ we get the situation considered in (1). \end{proof}
Recall that for every point $\sigma\in\operatorname{Stab}_{{\mathcal N}}({\mathcal D})$ a neighborhood of $\sigma$ in $\operatorname{Stab}_{{\mathcal N}}({\mathcal D})$ is homeomorphic to a neighborhood of the corresponding central charge in the linear subspace $W^{{\mathcal N}}_{\sigma}\subset \operatorname{Hom}({\mathcal N}({\mathcal D}),\mathbb{C})$. A stability condition $\sigma$ is called {\it full} if $W^{{\mathcal N}}_{\sigma}= \operatorname{Hom}({\mathcal N}({\mathcal D}),\mathbb{C})$. The above Lemma implies that every stability condition with the central charge in the set $\overline{U}$ is full.
\begin{thm}\label{cover-thm} Let $U\subset\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ denote the set of locally finite stability conditions $\sigma=(Z,P)$ such that \begin{enumerate} \item ${\mathcal O}_{\pi^{-1}(y)}$ is stable of phase $\phi_{\sigma}$ for every $y\in Y\setminus R$; \item ${\mathcal O}_{p_i},\zeta\otimes{\mathcal O}_{p_i}$ are semistable with the phases in $(\phi_{\sigma}-1,\phi_{\sigma}+1)$ for all $i=1,\ldots,n$. \end{enumerate} Then every point in $U$ is obtained from one of the stability conditions described in Proposition \ref{cover-prop} with $I^-=\emptyset$ and all $n_i=1$ by the action of an element of $\mathbb{R}\times\operatorname{Pic}_{\mathbb{Z}_2}(X)$, where $\mathbb{R}$ acts on $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ by rotations (shifts of phases). The subset $U$ is open in $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$. The natural map $U\to\overline{U}$ is a universal covering of $\overline{U}$, and $\overline{U}=U/\mathbb{Z}$, where $1\in\mathbb{Z}$ acts on the stability space by shifting phases by $2$. Furthermore, $U$ is contractible. \end{thm}
\begin{proof} {\bf Step 1}. If $\sigma=(Z,P)\in U$ then $\sigma$ is obtained from one of the stability conditions described in Proposition \ref{cover-prop} with $I^-=\emptyset$ and all $n_i=1$ by the action of an element of $\mathbb{R}\times\operatorname{Pic}_{\mathbb{Z}_2}(X)$. Indeed, by rotating $\sigma$ we can assume that $\phi_{\sigma}=1$. Now using tensoring with an appropriate equivariant line bundle we can assume that $\Im Z({\mathcal O}_{p_i})\leq 0$ for all $i$. It remains to apply Lemma \ref{phase-lem}(b).
Note that this step implies that for $\sigma=(Z,P)\in U$ one has $Z\in \overline{U}$.
\noindent {\bf Step 2}. Let $U'$ be the preimage of $\overline{U}$ in $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$. Then the projection $U'\to\overline{U}$ is a covering map. This is checked exactly as in Proposition 8.3 of \cite{Bridgeland-K3} using Lemma \ref{num-lem}(b).
\noindent {\bf Step 3}. $U$ is open in $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$. Let $\sigma_0=(Z_0,P_0)\in U$. We have to prove that any stability $\sigma=(Z,P)$, sufficiently close to $\sigma_0$, is still in $U$. Using rotations it is enough to consider the case when $Z({\mathcal O}_{\pi^{-1}(y)})\in\mathbb{R}_{<0}$. By Step 1 we can assume that $\sigma_0$ is a stability arising in Proposition \ref{cover-prop} with $I^-=\emptyset$ and all $n_i$'s equal to $1$. For a $\mathbb{Z}_2$-equivariant line bundle $L$ and a stability condition $\sigma'=(Z',P')$ we denote by $\sigma'\otimes L$ the stability condition with central charge $E\mapsto Z'(E\otimes L^{-1})$ and the heart $P'(0,1]\otimes L$. It is enough to check that $\sigma=\sigma'\otimes L$, where $\sigma'$ is one of stability conditions from Proposition \ref{cover-prop} (with $I^-=\emptyset$ and $n_i=1$).
Let us set $L={\mathcal O}_X(\sum_{i\in I(+)}p_i)$, where $I(+)=\{i \ |\ \Im Z({\mathcal O}_{p_i})>0\}$. We claim that the central charge $Z'(E):=Z(E\otimes L)$ satisfies the assumptions of Proposition \ref{cover-prop} with
$I^+=\{i \ |\ \Im Z({\mathcal O}_{p_i})\neq 0\}$, $I^-=\emptyset$ and all $n_i=1$. Indeed, first, note that $Z'({\mathcal O}_{\pi^{-1}(y)})=Z({\mathcal O}_{\pi^{-1}(y)})\in\mathbb{R}_{<0}$, and $Z'({\mathcal O}_X)=Z(L)$ is in the upper-half plane, provided $\sigma$ is close enough to $\sigma_0$. Next, using the fact that $${\mathcal O}_{p_i}\otimes L\simeq\begin{cases}{\mathcal O}_{p_i}, & i\not\in I(+),\\ \zeta\otimes{\mathcal O}_{p_i}, & i\in I(+)\end{cases}$$ one checks the remaining assumptions. Therefore, by Proposition \ref{cover-prop}, there exists a stability condition $\sigma'$ with the central charge $Z'$ and the heart $H(I^+,\emptyset)$. Now we claim that $\sigma=\sigma'\otimes L$. Since the corresponding central charges are the same, by Proposition \ref{FormalDistance}(a), it remains to check that $H(I^+,\emptyset)\otimes L\subset P(-1,2]$. It is easy to see that \begin{eqnarray}\label{cover-heart-eq} H(I^+,\emptyset)\otimes L =&[ {\mathcal O}_X(\sum_{i\in I(+)}p_i)\otimes\pi^*\operatorname{Coh}(Y), \nonumber\\
&[\zeta\otimes{\mathcal O}_{p_i}\ |\ i\not\in I(+)], [{\mathcal O}_{p_i}\ |\ i\not\in I(-)], \nonumber\\
&[\zeta\otimes{\mathcal O}_{p_i}[-1]\ |\ i\in I(+)], [{\mathcal O}_{p_i}[-1]\ |\ i\in I(-)] ], \end{eqnarray}
where $I(-)=\{i \ |\ \Im Z({\mathcal O}_{p_i})<0\}$. Hence, $$H(I^+,\emptyset)\otimes L\subset T_0:=[ P_0(0,1],
[ {\mathcal O}_{p_i},{\mathcal O}_{p_i}[-1],\zeta\otimes{\mathcal O}_{p_i}[-1]\ |\ i=1,\ldots,n] ].$$ Furthermore, we have ${\mathcal O}_{p_i}\in P_0[1,2)$ and $\zeta\otimes {\mathcal O}_{p_i}\in P_0(0,1]$. Hence, we have $T_0\subset P_0(-1+\epsilon,2-\epsilon)$ for some $\epsilon>0$ depending only on $\sigma_0$. Thus, for $d(P,P_0)<\epsilon$ we obtain $$H(I^+,\emptyset)\otimes L\subset P_0(-1+\epsilon,2-\epsilon)\subset P(-1,2]$$ as required.
\noindent {\bf Step 4}. $U$ is closed in $U'$. More precisely, we claim that $U$ coincides with the set of $\sigma\in U'$ such that ${\mathcal O}_{\pi^{-1}(y)}$ is semistable of phase $\phi_{\sigma}$ for every $y\in Y\setminus R$, and for every $i\in\{1,\ldots,n\}$ the objects ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ are semistable with the phases in $[\phi_{\sigma}-1,\phi_{\sigma}+1]$. (recall that the set of stability conditions such that a given object $E$ is semistable is closed). Indeed, given such $\sigma=(Z,P)$, by rotating it and using tensoring with an equivariant line bundle we can assume that $\phi_{\sigma}=1$, and $\Im Z({\mathcal O}_{p_i})\leq 0$ for all $i$. Note that the condition $Z\in\overline{U}$ implies that the phase of ${\mathcal O}_{p_i}$ (resp., $\zeta\otimes{\mathcal O}_{p_i}$) is in $[1,2)$ (resp., in $(0,1]$) for every $i$, and $\Im Z(V)>0$ for every $\mathbb{Z}_2$-equivariant vector bundle $V$. Hence, by Lemma \ref{phase-lem}(b), $\sigma$ is obtained by the construction of Proposition \ref{cover-prop}, which implies that ${\mathcal O}_{\pi^{-1}(y)}$ is stable for every $y\in Y\setminus R$. It remains to note that for $\sigma\in U'$ the phases of $Z({\mathcal O}_{p_i})$ and of $Z(\zeta\otimes{\mathcal O}_{p_i})$ never equal $\phi_{\sigma}\pm 1$.
Combining Steps 2, 3 and 4 we obtain that $U\to\overline{U}$ is a covering map.
\noindent {\bf Step 5}. Assume $\sigma_1, \sigma_2\in U$ have the same central charge $Z$. Then $\sigma_2$ is obtained from $\sigma_1$ by a shift of phase in $2\mathbb{Z}$. Indeed, applying such a shift we can assume that $\phi_{\sigma_1}=\phi_{\sigma_2}$. Furthermore, applying a rotation and tensoring with a line bundle, we reduce to the situation $\phi_{\sigma_1}=1$ and $\Im Z({\mathcal O}_{p_i})\leq 0$ for all $i$. By Lemma \ref{phase-lem}(b), in this case the hearts of $\sigma_1$ and $\sigma_2$ are the same.
\noindent {\bf Step 6}. It remains to show that $U$ is contractible. We have a free action of $\mathbb{R}$ on $U$ by the shift of phase, so it is enough to consider the section of this action consisting of $\sigma\in U$ with $\phi_{\sigma}=1$. In other words, we have to consider the subset of $\overline{U}$ consisting of $Z$ with $v_Z=Z({\mathcal O}_{\pi^{-1}(y)})\in\mathbb{R}_{<0}$. A homomorphism $Z$ in this subset is determined by the following contractible data: \begin{enumerate} \item $v_Z\in\mathbb{R}_{<0}$; \item for every $i\in\{1,\ldots,n\}$, $Z({\mathcal O}_{p_i})\in\mathbb{C}\setminus(\mathbb{R}_{\geq 0}\cup (v_Z+R_{\leq 0}))$; \item $Z({\mathcal O}_X)$ in some half-plane of the form $\Im z>c$. \end{enumerate} \end{proof}
\begin{remark} In the next section we will study more closely the case $g(Y)\geq 1$. We will show that in this case the objects ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ are automatically stable with respect to any stability on ${\mathcal D}_{\mathbb{Z}_2}(X)$, and will describe the entire space $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$. \end{remark}
We conclude this section with one observation in the case where $Y = \mathbb{P}^{1}$.
\begin{prop}\label{exc-P1-prop} Consider a stability $\sigma=(Z,P) \in U$, where $U$ is as in Theorem \ref{cover-thm}. Assume that for every $i=1,\ldots,n$ the vectors $Z({\mathcal O}_{p_i})$ and $Z({\mathcal O}_{2p_i})$ are linearly independent over $\mathbb{R}$. Then some rotation of $\sigma$ is glued from an exceptional collection. \end{prop}
\begin{proof} By Theorem \ref{cover-thm}, it is enough to check the same statement for a stability $\sigma$ arising from the construction of Proposition \ref{cover-thm} with $I^+=\{1,\ldots,n\}$, $I^-=I^0=\emptyset$ and $n_i=1$. We claim that in this situation for any sufficiently small $a>0$ the rotated stability $R_{-a}\sigma=(Z_a,P_a)$ is glued from an exceptional collection. Indeed, if $a$ is small enough then we still have $\Im Z_a({\mathcal O}_{p_i})<0$ for all $i=1,\ldots,n$. There is a unique $N\in\mathbb{Z}$ such that $\Im Z_a(\pi^*{\mathcal O}(N))<0$ and $\Im Z_a(\pi^*{\mathcal O}(N+1))>0$. Consider the following full $\operatorname{Ext}$-exceptional collection on ${\mathcal D}_{\mathbb{Z}_2}(X)$: \begin{equation}\label{cover-exc-coll} (\pi^*{\mathcal O}(N)[1],\pi^*{\mathcal O}(N+1),{\mathcal O}_{p_1}[-1],\ldots,{\mathcal O}_{p_n}[-1]). \end{equation} There exists a glued stability condition with the heart generated by this exceptional collection and with the central charge $Z_a$. To see that $R_a\sigma$ coincides with this stability condition, by Proposition \ref{FormalDistance}(a), it is enough to check that all the objects of our exceptional collection lie in $P_a(-1,2]=P(-1-a,2-a]$. Recall that
$$P(0,1]=[ \pi^*\operatorname{Coh}(\mathbb{P}^1),[ \zeta\otimes{\mathcal O}_{p_i},{\mathcal O}_{p_i}[-1] \ |\ i=1,\ldots,n] ]$$ Thus, all the objects of the collection \eqref{cover-exc-coll}, except for $\pi^*{\mathcal O}(N)[1]$, lie in $P(0,1]\subset P(-1-a,2-a]$. Note that by our assumptions, the phases of ${\mathcal O}_{p_i}[-1]$ are in $(0,1)$. Also, it is easy to see that $\pi^*{\mathcal O}(m)\in P(0,1)$ for every $m\in\mathbb{Z}$. The exact sequence $$0\to\pi^*{\mathcal O}(m-1)\to\pi^*{\mathcal O}(m)\to{\mathcal O}_{\pi^{-1}(y)}\to 0$$ in $P(0,1]$ shows that $\phi_{\max}(\pi^*{\mathcal O}(m-1))\le\phi_{\max}(\pi^*{\mathcal O}(m))$.
Now let us consider the exact sequence $$0\to F\to\pi^*{\mathcal O}(N)\to G\to 0$$ in $P(0,1]$, where $F$ is the maximal $\sigma$-destabilizing subobject in $\pi^*{\mathcal O}(N)$. The corresponding long exact cohomology sequence in $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ takes form $$0\to H^0F\to\pi^*{\mathcal O}(N)\to H^0G\to H^1F\to 0,$$ so either $H^0F=0$ or $H^0F$ is a line bundle. In the former case we have
$F=H^1F[-1]\in [{\mathcal O}_{p_i}[-1]\ |\ i=1,\ldots,n]$. In the latter case we have $H^0F\simeq \pi^*{\mathcal O}(m)(-\sum_{j\in J}p_j)$ for some $m\in\mathbb{Z}$ and $J\subset\{1,\ldots,n\}$. Hence, in the derived category $H^0F$ can be viewed as an extension of $\pi^*{\mathcal O}(m)$ by $\oplus_{j\in J}{\mathcal O}_{p_j}[-1]$. Therefore, the phase of $F$ is bounded above by the maximum of the phases of $Z({\mathcal O}_{p_i}[-1])$, $i=1,\ldots,n$ and of $Z(\pi^*{\mathcal O}(m))$. Note that we have a nonzero map from $\pi^*{\mathcal O}(m)(-\sum_{i=1}^n 2p_i)\simeq\pi^*{\mathcal O}(m-n)$ to $\pi^*{\mathcal O}(N)$, so $m\le N+n$. By making $a$ small enough we can assume that $N\leq 0$, so in this case we deduce that $\pi^*{\mathcal O}(N)\in P(0,\phi)$, where $\phi<1$ is the maximum of the phases of $Z({\mathcal O}_{p_i}[-1])$, $i=1,\ldots,n$ and of $Z(\pi^*{\mathcal O}(n))$. If in addition $a<1-\phi$ then we get $\pi^*{\mathcal O}(N)[1]\in P(1,2-a]\subset P(-1-a,2-a]$ as required. \end{proof}
\section{Classification of stability conditions in the case $Y\not\simeq\mathbb{P}^1$} \label{class-sec}
First, let us formulate an abstract version of Lemma 7.2 of \cite{GKR}. We say that an object $E$ of an abelian (or triangulated category) is {\it rigid} if $\operatorname{Hom}^1(E,E)=0$.
\begin{prop}\label{GKR-lem} Let ${\mathcal A}$ be an abelian category of homological dimension $1$, and let $$Y\to E\stackrel{f}{\to} X\to Y[1]$$ be an exact triangle in $D^b({\mathcal A})$ with $E\in{\mathcal A}$, such that $\operatorname{Hom}^{\leq 0}(Y,X)=0$. Then $X=X_0\oplus X_1[1]$, where $X_0,X_1\in{\mathcal A}$. Let $f_0:E\to X_0$ be the map induced by $f$. Then \begin{enumerate} \item $\operatorname{coker}(f_0)$ and $X_1$ are rigid; \item $\operatorname{Hom}^*(\operatorname{coker}(f_0),X_0)=\operatorname{Hom}^*(\operatorname{coker}(f_0),X_1)=0$; \item $\operatorname{Hom}^0(\operatorname{ker}(f_0), X_0)=0$, and the map $\operatorname{ker}(f_0)\to X_1[1]$ induces an isomorphism $$\operatorname{Hom}^0(X_1,X_1)\simeq\operatorname{Hom}^1(\operatorname{ker}(f_0),X_1).$$ \end{enumerate} \end{prop}
\begin{proof} The first part of the proof of Lemma 7.2 in \cite{GKR} gives the statement that $X=X_0\oplus X_1[1]$, and $$\operatorname{Hom}^{\leq 0}(H^0(Y)\oplus\operatorname{coker}(f_0)[-1],X_0\oplus X_1[1])=0,$$ which implies (2). Since $X_0$ surjects onto $\operatorname{coker}(f_0)$, the natural map $$\operatorname{Hom}^1(\operatorname{coker}(f_0),X_0)\to\operatorname{Hom}^1(\operatorname{coker}(f_0),\operatorname{coker}(f_0))$$ is surjective, so we deduce that $\operatorname{coker}(f_0)$ is rigid. Next, we have an exact sequence $$0\to X_1\to H^0(Y)\to\operatorname{ker}(f_0)\to 0$$ in ${\mathcal A}$. Thus, the natural map $$\operatorname{Hom}^1(H^0(Y),X_1)\to\operatorname{Hom}^1(X_1,X_1)$$ is surjective, and we obtain that $X_1$ is rigid. Using the same exact sequence we get (3). \end{proof}
\begin{lem}\label{rigid-lem} Assume $g(Y)\geq 1$. Then every rigid object in $\operatorname{Coh}_{\mathbb{Z}_2}(X)$ is of the form $$\bigoplus_{i\in I}{\mathcal O}_{p_i}^{\oplus m_i}\oplus \bigoplus_{j\in J}\zeta\otimes{\mathcal O}_{p_j}^{\oplus n_j},$$ where $I\cap J=\emptyset$. \end{lem}
\begin{proof} The fact that $g(Y)\geq 1$ implies that $\omega_Y$ has a nowhere vanishing section. Hence, $\omega_X$ has a $\mathbb{Z}_2$-invariant section vanishing only along $R\subset X$. Therefore, for $F\in \operatorname{Coh}_{\mathbb{Z}_2}(X)$ such that $F$ is not supported on $R$ we have $$\operatorname{Hom}^1(F,F)^*\simeq\operatorname{Hom}(F,F\otimes\omega_X)\neq 0,$$ so $F$ cannot be rigid. Thus, any indecomposable rigid object should be supported at one of the ramification points. It is easy to check that the $\mathbb{Z}_2$-sheaf ${\mathcal O}_{mp_i}$ (resp., $\zeta\otimes{\mathcal O}_{mp_i}$) is rigid only for $m=1$. The assertion follows easily from this. \end{proof}
Let us denote by ${\mathcal D}_{p_i}\subset{\mathcal D}_{\mathbb{Z}_2}(X)$ the triangulated subcategory generated by equivariant sheaves supported on $p_i$.
\begin{lem}\label{gen1-lem} Assume $g(Y)\geq 1$, and let $\sigma=(Z,P)$ be a stability condition on ${\mathcal D}_{\mathbb{Z}_2}(X)$. Then \begin{enumerate} \item the object ${\mathcal O}_{\pi^{-1}(y)}$ is $\sigma$-stable for every $y\in Y\setminus R$; \item $\sigma$ restricts to a stability condition on ${\mathcal D}_{p_i}$; \item for any exact triangle $A\to {\mathcal O}_X\to B\to A[1]$ in ${\mathcal D}_{\mathbb{Z}_2}(X)$ with $\operatorname{Hom}^{\leq 0}(A,B)=0$ and nonzero $A$ and $B$, there exists $I\subset\{1,\ldots,n\}$ such that either $A={\mathcal O}_X(-\sum_{i\in I}m_ip_i)$ and $B=\oplus_{i\in I}{\mathcal O}_{m_ip_i}$, where all $m_i$'s are odd, or $A=\oplus_{i\in I}\zeta\otimes{\mathcal O}_{p_i}[-1]$ and $B={\mathcal O}_X(\sum_{i\in I}p_i)$; \item there exists a $\sigma$-semistable equivariant line bundle. \end{enumerate} \end{lem}
\begin{proof} (1) Consider the triangle $$Y\to{\mathcal O}_{\pi^{-1}(y)}\stackrel{f}{\to} X\to Y[1]$$ with $Y\in P(-\infty,t]$, $X\in P(t,+\infty)$, and assume that $X\neq 0$. Then by Proposition \ref{GKR-lem}, we have $X=X_0\oplus X_1[1]$, where $X_0$ and $X_1$ are equivariant coherent sheaves, and $X_1$ is rigid. By Lemma \ref{rigid-lem}, $X_1$ is supported at $R$. Hence, $\operatorname{Hom}(X_1,X_1)=\operatorname{Hom}^1(\operatorname{ker}(f_0),X_1)=0$ (the isomorphism comes from Proposition \ref{GKR-lem}(3)), which implies that $X_1=0$. On the other hand, since $X_0\neq 0$, the condition $\operatorname{Hom}(\operatorname{coker}(f_0),X_0)=0$ (see Proposition \ref{GKR-lem}(2)) implies that the map $f_0:{\mathcal O}_{\pi^{-1}(y)}\to X_0$ is nonzero, so it is an embedding. But $\operatorname{coker}(f_0)$ is also rigid (see Proposition \ref{GKR-lem}(1)), so it is supported at $R$. Therefore, the extension $$0\to {\mathcal O}_{\pi^{-1}(y)}\to X_0\to \operatorname{coker}(f_0)\to 0$$ splits. Since $\operatorname{Hom}(\operatorname{coker}(f_0),X_0)=0$, this implies that $f_0$ is an isomorphism.
\noindent (2) Consider the triangle $Y\to E\stackrel{f}{\to} X\to Y[1]$ with $Y\in P(-\infty,t]$, $X\in P(t,+\infty)$, where $E$ is a sheaf supported at $p_i$, and assume that $X\neq 0$. Applying Proposition \ref{GKR-lem} and Lemma \ref{rigid-lem} again we see that $\operatorname{coker}(f_0)$ is supported at $R$, so we can write $\operatorname{coker}(f_0)=C\oplus C'$, where $C$ is supported at $p_i$ and $C'$ is supported at $R-p_i$. Since $\operatorname{im}(f_0)$ is supported at $p_i$, the extension $$0\to\operatorname{im}(f_0)\to X_0\to\operatorname{coker}(f_0)\to 0$$ splits over $C'$. Since $\operatorname{Hom}(\operatorname{coker}(f_0),X_0)=0$, it follows that $C'=0$, so $X_0$ is supported at $p_i$. Similarly, we have $X_1=A\oplus A'$, where $A$ is supported at $p_0$ and $A'$ is supported at $R-p_i$. To prove that $A'=0$ we use the fact that the map $\operatorname{ker}(f_0)\to X_1$ factors through $A$, so $\operatorname{Hom}^0(A',A')$ maps to zero under the induced map $\operatorname{Hom}^0(X_1,X_1)\to\operatorname{Hom}^1(\operatorname{ker}(f_0),X_1)$. But the latter map is an isomorphism by Proposition \ref{GKR-lem}(3), so we deduce that $A'=0$. Hence, $X$ is supported at $p_i$, and so $Y$ is also supported at $p_i$.
\noindent (3) By Proposition \ref{GKR-lem}, we have $B=B_0\oplus B_1[1]$, where $B_0$ and $B_1$ are equivariant sheaves. The fact that $B_1$ is rigid (hence, torsion) implies that $\operatorname{Hom}^1({\mathcal O}_X,B_1)=0$. Together with Proposition \ref{GKR-lem}(3) this easily leads to $B_1=0$. Let $f:{\mathcal O}_X\to B=B_0$ be the map in our exact triangle. Assume first that $f$ is injective. Then $B$ is an extension of a rigid object $\operatorname{coker}(f)$ by ${\mathcal O}_X$, such that $\operatorname{Hom}^*(\operatorname{coker}(f),B)=0$. By Lemma \ref{rigid-lem}, we have $$\operatorname{coker}(f)\simeq\bigoplus_{i\in I}P_i\oplus\bigoplus_{j\in J}Q_j,$$ where $P_i={\mathcal O}_{p_i}^{\oplus m_i}$, $Q_j=\zeta\otimes{\mathcal O}_{p_j}^{\oplus n_j}$. Since $\operatorname{Hom}^1(P_i,{\mathcal O}_X)=0$, the extension $$0\to {\mathcal O}_X\to B\to\operatorname{coker}(f)\to 0$$ splits over $P_i$, which implies that $P_i=0$ (since $\operatorname{Hom}(\operatorname{coker}(f),B)=0$). Next, the map $\operatorname{Hom}(\operatorname{coker}(f),\operatorname{coker}(f))\to\operatorname{Hom}^1(\operatorname{coker}(f),{\mathcal O}_X)$ induced by the above extension is an isomorphism. Hence, for every $j$ the induced map $\operatorname{Hom}(P_j,P_j)\to \operatorname{Hom}^1(P_j,{\mathcal O}_X)$ is an isomorphism. The source of this map has dimension $n_j^2$, while the target has dimension $n_j$, so we get that $n_j=1$. This gives the required form of $A$ and $B$ in this case.
Next, assume that $\operatorname{ker}(f)\neq 0$. Then $\operatorname{ker}(f)$ is isomorphic to ${\mathcal O}_X(-\sum_i m_ip_i)$, and $\operatorname{im}(f)\simeq\oplus_i {\mathcal O}_{m_ip_i}$. The condition $\operatorname{Hom}(\operatorname{ker}(f),B)=0$ implies that $\operatorname{Hom}(\operatorname{ker}(f),\operatorname{im}(f))=0$. Hence, all nonzero $m_i$'s are odd. Let $I$ denote the set of $i$ for which $m_i\neq 0$. The extension $$0\to\operatorname{im}(f)\to B\to \operatorname{coker}(f)\to 0$$ still has the property that $\operatorname{Hom}^*(\operatorname{coker}(f),B)=0$.
This implies that $\operatorname{coker}(f)$ is supported at $\{p_i\ |\ i\in I\}$. Hence, $B$ is also supported at this set. The condition $\operatorname{Hom}(\operatorname{ker}(f),B)=0$ implies that all indecomposable direct sumands of $B$ are of the form ${\mathcal O}_{np_i}$, where $i\in I$ and $n$ is odd (and there is at least one such factor for every $i\in I$). Now the condition $\operatorname{Hom}^*(\operatorname{coker}(f),B)=0$ together with the rigidity of $\operatorname{coker}(f)$ (using Lemma \ref{rigid-lem}) implies that $\operatorname{coker}(f)=0$.
\noindent (4) It follows easily from (3) that one of the HN-factors of ${\mathcal O}_X$ is a line bundle. \end{proof}
\begin{lem}\label{O-lem} Assume $g(Y)\geq 1$, and let $\sigma=(Z,P)$ be a locally finite stability condition on ${\mathcal D}_{\mathbb{Z}_2}(X)$ such that ${\mathcal O}_{2p_i}$ is semistable for every $i=1,\ldots,n$, and all ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ are stable of phase $1$. Then \begin{enumerate} \item ${\mathcal O}_{2p_i}\in P(1)$ for every $i$; \item for every line bundle $M$ on $Y$ one has $\pi^*M\in P(0,1)$. \end{enumerate} \end{lem}
\begin{proof} (1) By Lemma \ref{gen1-lem}(4), we know that there exists a $\sigma$-semistable equivariant line bundle $L$. Since for $y\in Y\setminus R$ we have nonzero morphisms $L\to {\mathcal O}_{\pi^{-1}(y)}$ and ${\mathcal O}_{\pi^{-1}(y)}\to L[1]$, it follows that $L\in P(t)$ for some $t\in [0,1]$. Furthermore, we cannot have $t=0$ or $t=1$, since there can be only finite number of nonisomorphic simple objects of phase $t$ with nonzero maps to (or from) $L$. Now we have nonzero maps $L\to{\mathcal O}_{2p_i}$ and ${\mathcal O}_{2p_i}\to L[1]$ which implies that the phase of ${\mathcal O}_{2p_i}$ is in the interval $[t,t+1]$. But $Z({\mathcal O}_{2p_i})$ has phase $1$, so the phase of ${\mathcal O}_{2p_i}$ is equal to $1$.
\noindent (2) Tensoring $\sigma$ with $M^{-1}$ we immediately reduce to the case $M={\mathcal O}_Y$. First, we observe that for any equivariant line bundle $L$ one cannot have $L\in P[1,+\infty)$ or $L\in P(-\infty,0]$. Indeed, this follows from the existence of nonzero morphisms $L\to{\mathcal O}_{\pi^{-1}(y)}$ and ${\mathcal O}_{\pi^{-1}(y)}\to L[1]$ as in part (1). Let us consider the canonical exact triangle $$A\to{\mathcal O}_X\to B\to A[1]$$ with $A\in P[1,+\infty)$ and $B\in P(-\infty,1)$. By Lemma \ref{gen1-lem}(3) and the above observation, we obtain that $A=\oplus_{i\in I}\zeta\otimes{\mathcal O}_{p_i}[-1]$ for some $I\subset\{1,\ldots,n\}$. But this implies that for $i\in I$ one has $\zeta\otimes{\mathcal O}_{p_i}\in P[2,+\infty)$, which contradicts to the existence of a nonzero morphism from $\zeta\otimes{\mathcal O}_{p_i}$ to ${\mathcal O}_{2p_i}\in P(1)$. Therefore, ${\mathcal O}_X\in P(-\infty,1)$. Now consider the exact triangle $$C\to{\mathcal O}_X\to D\to C[1]$$ with $C\in P(0,1)$ and $D\in P(-\infty,0]$. Using Lemma \ref{gen1-lem}(3) we obtain that $D=\oplus_{i\in I}{\mathcal O}_{m_ip_i}$, where all $m_i$'s are odd. But we have a nonzero map ${\mathcal O}_{2p_i}\to{\mathcal O}_{p_i}\hookrightarrow{\mathcal O}_{m_ip_i}$, which is a contradiction since ${\mathcal O}_{2p_i}\in P(1)$. Hence, $D=0$ and ${\mathcal O}_X\in P(0,1)$. \end{proof}
Let us set ${\mathcal S}_i=\operatorname{Stab}({\mathcal D}_{p_i})$. This is a two-dimensional complex manifold that we are going to describe explicitly below. Note that these spaces for different points $p_i$ are canonically isomorphic, so we will sometimes skip the index $i$ below.
\begin{prop}\label{point-prop} (a) Let $U^+\subset{\mathcal S}$ (resp., $U^-\subset{\mathcal S}$) denote the subset of $\sigma$ such that ${\mathcal O}_{p_i}$ (resp., $\zeta\otimes{\mathcal O}_{p_i}$) is $\sigma$-stable. Let also $W^+\subset{\mathcal S}$ (resp., $W^-\subset{\mathcal S}$) denote the subset of stabilities with respect to which ${\mathcal O}_{2p_i}$ (resp., $\zeta\otimes{\mathcal O}_{2p_i}$) is semistable. Then $U^+$ and $U^-$ are open, $W^+$ and $W^-$ are closed, and $${\mathcal S}=U^+\cup U^-=W^+\cup W^-.$$ The subset $W^+\cap W^-$ is contained in $U^+\cap U^-$ and consists of $\sigma$ such that ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ are stable of the same phase. The subset $U^+\cap U^-\cap W^+$ is characterized in $U^+\cap W^+$ by the condition $\phi({\mathcal O}_{p_i})<\phi({\mathcal O}_{2p_i})+1$. Similarly, the subset $U^+\cap U^-\cap W^-$ is characterized in $U^+\cap W^-$ by the inequality $\phi({\mathcal O}_{p_i})>\phi(\zeta\otimes{\mathcal O}_{2p_i})-1$.
\noindent (b) There is a holomorphic submersion $f_i:{\mathcal S}_i\to\mathbb{C}$ such that $\exp(\pi f_i)$ is equal to $Z({\mathcal O}_{2p_i})$, and $\Im(f_i)$ is equal to the phase of ${\mathcal O}_{2p_i}$ on $W^+$ and to the phase of $\zeta\otimes{\mathcal O}_{2p_i}$ on $W^-$.
The action of the subgroup $\mathbb{R}\times\mathbb{R}^*_{>0}\subset\widetilde{\operatorname{GL}_{2}^{+}(\R)}$ of rotations and rescalings induces an isomorphism of complex manifolds $$\mathbb{C}\times \Sigma\widetilde{\to}{\mathcal S}_i,$$ such that $f_i$ corresponds to the projection to the first factor, where $\Sigma=f_i^{-1}(0)$ is a (noncompact) Riemann surface.
\noindent (c) There is a well defined branch of $\frac{1}{\pi}\log Z({\mathcal O}_{p_i})$ (resp., $\frac{1}{\pi}\log Z(\zeta\otimes{\mathcal O}_{p_i})$) on $U^+$ (resp., $U^-$) that defines an isomorphism $\Sigma\cap U^+\simeq\mathbb{C}\setminus\mathbb{R}_{\ge 0}$
(resp., $\Sigma\cap U^-\simeq\mathbb{C}\setminus\mathbb{R}_{\ge 0}$). Under both these isomorphisms $\Sigma\cap U^+\cap U^-$ is mapped to the subset of $\mathbb{C}\setminus\mathbb{R}_{\ge 0}$ consisting of $z$ with $|\Im z|<1$.
\noindent (d) The Riemann surface $\Sigma$ is simply connected and of parabolic type. More precisely, there exists an isomorphism $\Sigma\simeq \mathbb{C}$ under which the function $Z({\mathcal O}_{p_i})$ on $\Sigma$ corresponds to the function $$z\mapsto \frac{1}{2}+\frac{1}{\sqrt{\pi}}\int_0^z e^{-t^2}dt.$$ \end{prop}
\begin{proof} (a) Recall that by Lemma \ref{simple-obj-lem}, the only endosimple objects in ${\mathcal D}_{p_i}$ are ${\mathcal O}_{p_i}$, $\zeta\otimes{\mathcal O}_{p_i}$, ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$. Given a stability condition $\sigma=(Z,P)$, each subcategory $P(t)$ is generated by stable (hence, endosimple) objects. In particular, stable objects generate ${\mathcal D}_{p_i}$. This implies that we should have at least two stable objects, and that one of the objects ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ is always stable (since the objects ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ do not generate ${\mathcal D}_{p_i}$). Thus, we have ${\mathcal S}=U^+\cup U^-$.
Next, let us check that either ${\mathcal O}_{2p_i}$ or $\zeta\otimes{\mathcal O}_{2p_i}$ is always semistable, i.e., ${\mathcal S}=W^+\cup W^-$. If either ${\mathcal O}_{p_i}$ or $\zeta\otimes{\mathcal O}_{p_i}$ is not stable then one of the objects ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ has to be stable (since there should be at least two stable objects). Now assume that both ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ are stable. If $\phi({\mathcal O}_{p_i})=\phi(\zeta\otimes{\mathcal O}_{p_i})$ then ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ are both semistable of the same phase. If $\phi({\mathcal O}_{p_i})>\phi(\zeta\otimes{\mathcal O}_{p_i})$ then ${\mathcal O}_{2p_i}$ is stable and $\zeta\otimes{\mathcal O}_{2p_i}$ is unstable (=not semistable). Similarly, if $\phi({\mathcal O}_{p_i})<\phi(\zeta\otimes{\mathcal O}_{p_i})$ then $\zeta\otimes{\mathcal O}_{2p_i}$ is stable and ${\mathcal O}_{2p_i}$ is unstable. This proves that ${\mathcal S}=W^+\cup W^-$. Note in addition that ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ cannot be both stable since we have nonzero maps ${\mathcal O}_{2p_i}\to \zeta\otimes{\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}\to{\mathcal O}_{2p_i}$.
Let us classify stabilities such that ${\mathcal O}_{p_i}$ is stable. The following $3$ cases (not mutually exclusive) can occur: (i) ${\mathcal O}_{2p_i}$ is stable; (ii) $\zeta\otimes{\mathcal O}_{2p_i}$ is stable; (iii) $\zeta\otimes{\mathcal O}_{p_i}$ is stable.
In case (i) we have $\phi({\mathcal O}_{p_i})>\phi({\mathcal O}_{2p_i})$ (since there is a nonzero map ${\mathcal O}_{2p_i}\to {\mathcal O}_{p_i}$). The exact triangle $${\mathcal O}_{p_i}[-1]\to\zeta\otimes{\mathcal O}_{p_i}\to{\mathcal O}_{2p_i}\to{\mathcal O}_{p_i}$$ shows that if $\phi({\mathcal O}_{p_i})\ge\phi({\mathcal O}_{2p_i})+1$ then $\zeta\otimes{\mathcal O}_{p_i}$ is not stable. On the other hand, if $\phi({\mathcal O}_{p_i})<\phi({\mathcal O}_{2p_i})+1$ then one can easily check that $\zeta\otimes{\mathcal O}_{p_i}$ is stable. A stability condition in this case is uniquely determined by the phases and central charges of ${\mathcal O}_{p_i}$ and ${\mathcal O}_{2p_i}$ that can be arbitrary such that $\phi({\mathcal O}_{p_i})>\phi({\mathcal O}_{2p_i})$.
In case (ii) we have $\phi({\mathcal O}_{p_i})<\phi(\zeta\otimes{\mathcal O}_{2p_i})$ (because of the nonzero map ${\mathcal O}_{p_i}\to\zeta\otimes{\mathcal O}_{2p_i}$). The exact triangle $$\zeta\otimes{\mathcal O}_{2p_i}\to \zeta\otimes{\mathcal O}_{p_i}\to {\mathcal O}_{p_i}[1]\to\ldots$$ shows that if $\phi({\mathcal O}_{p_i})\le \phi(\zeta\otimes{\mathcal O}_{2p_i})-1$ then $\zeta\otimes{\mathcal O}_{p_i}$ is not stable. One can also check that for $\phi({\mathcal O}_{p_i})>\phi(\zeta\otimes{\mathcal O}_{2p_i})-1$, the object $\zeta\otimes{\mathcal O}_{p_i}$ is stable. A stability condition in case (ii) is uniquely determined by the phases and central charges of ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ subject to the condition $\phi({\mathcal O}_{p_i})<\phi(\zeta\otimes{\mathcal O}_{2p_i})$.
In case (iii) we have \begin{equation}\label{phases-in-1}
|\phi({\mathcal O}_{p_i})-\phi(\zeta\otimes{\mathcal O}_{p_i})|<1 \end{equation} (because of nonzero maps ${\mathcal O}_{p_i}\to\zeta\otimes{\mathcal O}_{p_i}[1]$ and $\zeta\otimes{\mathcal O}_{p_i}\to{\mathcal O}_{p_i}[1]$). One can easily check that if $\phi({\mathcal O}_{p_i})>\phi(\zeta\otimes{\mathcal O}_{p_i})$ (resp., $\phi({\mathcal O}_{p_i})<\phi(\zeta\otimes{\mathcal O}_{p_i})$) then ${\mathcal O}_{2p_i}$ is stable and $\zeta\otimes{\mathcal O}_{2p_i}$ is unstable (resp., $\zeta\otimes{\mathcal O}_{2p_i}$ is stable and ${\mathcal O}_{2p_i}$ is unstable). On the other hand, if $\phi({\mathcal O}_{p_i})=\phi(\zeta\otimes{\mathcal O}_{p_i})$ then both ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ are semistable of the same phase. A stability condition in case (iii) is uniquely determined by the phases and central charges of ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ subject to \eqref{phases-in-1}.
The above classification (complemented by a similar classification in the case where $\zeta\otimes{\mathcal O}_{p_i}$ is stable) implies the required characterizations of $W^+\cap W^-$, $U^+\cap U^-\cap W^+$ and $U^+\cap U^-\cap W^-$.
Note that the subsets $W^+$ and $W^-$ are closed by general properties of stability conditions. It remains to check that $U^+$ and $U^-$ are open. We'll do this only for $U^+$ (the other case will follow by applying the autoequivalence $\otimes\zeta$). Assume first that $\sigma=(Z,P)\in U^+\cap U^-$. Then there exists an interval $(t,t+\eta)$ with $0<\eta<1$ such that all the objects ${\mathcal O}_{p_i}$, $\zeta\otimes{\mathcal O}_{p_i}$, ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ are in $P(t,t+\eta)$. Hence, if $\sigma'=(Z',P')$ is sufficiently close to $\sigma$ then these four objects are still in $P'(t',t'+\eta')$ for some $0<\eta'<1$. It follows from the above classification that in this case $\sigma'\in U^+\cap U^-$. Next, assume that $\sigma=(Z,P)\in U^+$ is such that ${\mathcal O}_{2p_i}$ is stable and $\zeta\otimes{\mathcal O}_{p_i}$ is not stable. Then we have $\zeta\otimes{\mathcal O}_{p_i}\in P[\phi_0,+\infty)$, where $\phi_0=\phi({\mathcal O}_{2p_i})$, and also $\zeta\otimes{\mathcal O}_{2p_i}$ is unstable. Hence, if $\sigma'=(Z,P')$ is sufficiently close to $\sigma$ then $\zeta\otimes{\mathcal O}_{p_i}\in P'(>\phi_0-1/3)$, ${\mathcal O}_{2p_i}\in P'(\phi_0-1/3,\phi_0+1/3)$, and $\zeta\otimes{\mathcal O}_{2p_i}$ is $\sigma'$-unstable. Suppose that ${\mathcal O}_{p_i}$ is not $\sigma'$-stable. Then $\zeta\otimes{\mathcal O}_{p_i}$ and ${\mathcal O}_{2p_i}$ have to be stable. But the above inclusions show that the difference of phases of $\zeta\otimes{\mathcal O}_{p_i}$ and ${\mathcal O}_{2p_i}$ is $<1$. Therefore, ${\mathcal O}_{p_i}$ is also $\sigma'$-stable by the above classification. Finally, assume $\sigma\in U^+$ is such that $\zeta\otimes{\mathcal O}_{2p_i}$ is stable and $\zeta\otimes{\mathcal O}_{p_i}$ is not stable. Then setting $\phi_0=\phi(\zeta\otimes{\mathcal O}_{2p_i})$ we get $\zeta\otimes{\mathcal O}_{p_i}\in P(-\infty,\phi_0]$. The same argument as in the previous case shows that this implies that ${\mathcal O}_{p_i}$ is $\sigma'$-stable for $\sigma'$ close to $\sigma$.
\noindent (b) The fact that $f_i$ is well-defined and continuous follows from the fact that the phases of ${\mathcal O}_{2p_i}$ and $\zeta\otimes{\mathcal O}_{2p_i}$ agree on $W^+\cap W^-$. Since $\exp(\pi f_i)$ is holomorphic by the definition of a complex structure on the stability space, it follows that $f_i$ is holomorphic. Now let us consider the subgroup $\mathbb{R}\times\mathbb{R}^*_{>0}\subset\widetilde{\operatorname{GL}_{2}^{+}(\R)}$ acting on the stability space, where $(a,\lambda)\in\mathbb{R}\times\mathbb{R}^*_{>0}$ acts by the phase rotation $R_a$ combined with the rescaling of the central charge by $\lambda$. Note that this action is compatible with the holomorphic action of this group on the central charges, where we identify $\mathbb{R}\times\mathbb{R}^*_{>0}$ with $\mathbb{C}$ via $(a,\lambda)\mapsto \frac{\log(\lambda)}{\pi}+ia$. Under this identification we have $$f_i(z\cdot \sigma)=f_i(\sigma)+z.$$ This gives the required splitting $\mathbb{C}\times \Sigma\widetilde{\to}{\mathcal S}_i$.
\noindent (c) The identifications of $\Sigma\cap U^+$, $\Sigma\cap U^-$ and $\Sigma\cap U^+\cap U^-$ follow easily from the proof of (a). Note that it is convenient to consider separately three regions in $\Sigma$ depending on whether $\sigma\in W^+\setminus W^-$, $\sigma\in W^-\setminus W^+$, or $\sigma\in W^+\cap W^-$. In the latter case we have $\phi({\mathcal O}_{p_i})=\phi(\zeta\otimes{\mathcal O}_{p_i})$. In the first case if in addition $\sigma\in U^+$ (resp., $\sigma\in U^-$) then $\phi({\mathcal O}_{p_i})>0$ (resp., $\phi(\zeta\otimes{\mathcal O}_{p_i})<0$), etc.
\noindent (d) As we have seen in (c), the function $Z({\mathcal O}_{p_i})$ restricts to $\exp(\pi z)$ on $\Sigma\cap U^+\simeq \mathbb{C}\setminus\mathbb{R}_{\ge 0}$, hence, it has a logarthmic ramification above $0$ and $\infty$. On the other hand, since $Z({\mathcal O}_{p_i})=1-Z(\zeta\otimes{\mathcal O}_{p_i})$, we see that the restriction of this function to $\Sigma\cap U^-$ has a logarithmic ramification above $1$ and $\infty$. Now we can easily identify $\Sigma$ with the simply connected Riemann surface that has $4$ logarithmic ramification points, two over $\infty$, one over $0$ and one over $1$. Our result follows easily from the Nevanlinna's classification of such surfaces (see \cite[sec. 45]{Nevanlinna}). \end{proof}
\begin{cor}\label{fun-cor} The function $\delta_i:{\mathcal S}_i\to\mathbb{R}$ given by $$\delta_i(\sigma)=\begin{cases} \det(Z(\zeta\otimes{\mathcal O}_{p_i}),Z({\mathcal O}_{2p_i})), & \zeta\otimes{\mathcal O}_{2p_i} \text{ is } \sigma-\text{semistable},\\ 0, & {\mathcal O}_{2p_i} \text{ is } \sigma-\text{semistable} \end{cases} $$ is continuous. \end{cor}
Let us consider the submanifold $\Theta$ of ${\mathcal S}_1\times\ldots\times{\mathcal S}_n\times\mathbb{C}$ consisting of $(\sigma_1,\ldots,\sigma_n,z)$ such that $f_1(\sigma_1)=\ldots=f_n(\sigma_n)$. Note that by Proposition \ref{point-prop}(b), we have $$\Theta\simeq\mathbb{C}\times\Sigma^n\times\mathbb{C},$$ where the first factor corresponds to $f_i(\sigma_i)$.
\begin{thm}\label{cover-class-thm} Assume that $g(Y)\geq 1$. Then natural map $$\rho:\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))\to {\mathcal S}_1\times\ldots\times{\mathcal S}_n\times\mathbb{C}:
\sigma\mapsto (\sigma|_{{\mathcal D}_{p_1}},\ldots,\sigma|_{{\mathcal D}_{p_n}},Z({\mathcal O}_X))$$ induces an isomorphism of $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ with the open subset $\Theta^0\subset\Theta$ consisting of $(\sigma_1,\ldots,\sigma_n,z)$ such that \begin{equation}\label{det-ineq} \det(z,\exp(\pi f_1(\sigma_1)))+\sum_{i=1}^n\delta_i(\sigma_i)>0. \end{equation} The space $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ is contractible. \end{thm}
\begin{proof} Note that the map $\rho$ is well defined by Lemma \ref{gen1-lem}(2). It is continuous and is compatible with the similar restriction map on the central charges and with the $\widetilde{\operatorname{GL}_{2}^{+}(\R)}$-actions.
\noindent {\bf Step 1}. Let us check that the image of $\rho$ is contained in $\Theta^0$. The fact that it is contained in $\Theta$ follows immediately from the definitions, so it remains to check that \eqref{det-ineq} holds whenever $\sigma_1,\ldots,\sigma_n$ are the restrictions of some $\sigma\in\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ to ${\mathcal D}_{p_1},\ldots,{\mathcal D}_{p_n}$. Recall that by Proposition \ref{point-prop}, for every $i\in\{1,\ldots,n\}$ either ${\mathcal O}_{2p_i}$ or $\zeta\otimes{\mathcal O}_{2p_i}$ is $\sigma$-semistable. Thus, by Lemmas \ref{gen1-lem}(1) and \ref{O-lem}(1), rotating $\sigma$ and tensoring it with an appropriate line bundle, we can get a stability with respect to which all objects ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ are stable of phase $1$, and all objects ${\mathcal O}_{2p_i}$ are semistable of phase $1$. Note that for such a stability inequality \eqref{det-ineq} is satisfied, as $\delta_i(\sigma_i)=0$ for all $i$ and the first term in \eqref{det-ineq} is equal to $\Im Z({\mathcal O}_X)$ (recall that ${\mathcal O}_X\in P(0,1)$ by Lemma \ref{O-lem}). It remains to check that the left-hand side of \eqref{det-ineq} for $\rho(\sigma)$ does not change upon tensoring $\sigma$ with an equivariant line bundle (the $\widetilde{\operatorname{GL}_{2}^{+}(\R)}$-invariance is clear). It is enough to compare the left-hand sides of \eqref{det-ineq} for $\sigma$ and $\sigma'=\sigma\otimes {\mathcal O}(-p_i)$, assuming that all ${\mathcal O}_{\pi^{-1}(y)}$ for $y\in Y\setminus R$ have phase $1$ and ${\mathcal O}_{2p_i}$ is $\sigma$-semistable. Indeed, the central charge for $\sigma'$, is given by $Z'(E)=Z(E(p_i))$, so $z=Z({\mathcal O}_X)$ will get replaced by $$Z'({\mathcal O}_X)=Z({\mathcal O}(p_i))=Z({\mathcal O}_X)+Z(\zeta\otimes{\mathcal O}_{p_i}),$$ so the first term $\Im Z({\mathcal O}_X)$ in \eqref{det-ineq} gets replaced by its sum with $\Im Z(\zeta\otimes{\mathcal O}_{p_i})$. On the other hand, since $\zeta\otimes{\mathcal O}_{2p_i}$ is $\sigma'$-semistable, the term
$\delta_i(\sigma|_{{\mathcal D}_{p_i}})=0$ gets replaced by
$$\delta_i(\sigma'|_{{\mathcal D}_{p_i}})=\Im Z'(\zeta\otimes{\mathcal O}_{p_i})=\Im Z({\mathcal O}_{p_i})=-\Im Z(\zeta\otimes {\mathcal O}_{p_i}).$$
\noindent {\bf Step 2}. Up to a rotation and tensoring with a line bundle, every stability condition $\sigma\in\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ is obtained from the construction of Proposition \ref{cover-prop}. Indeed, applying a rotation and tensoring with a line bundle we can assume that ${\mathcal O}_{\pi^{-1}(y)}$ for all $y\in Y$ are $\sigma$-semistable of phase $1$. Now we have to check the remaining conditions of Lemma \ref{phase-lem}(a). By Lemma \ref{O-lem} we know that $\pi^*L$ is in $P(0,1)$ for every $L\in\operatorname{Pic}(Y)$. Next, by Proposition \ref{point-prop}(a), for every $i$ the restriction of $\sigma$ to ${\mathcal D}_i$ belongs either to $W^+\cap W^-$, or to $(W^+\cap U^+)\setminus W^-$, or to $(W^+\cap U^-)\setminus U^+$. In the first case both ${\mathcal O}_{p_i}$ and $\zeta\otimes{\mathcal O}_{p_i}$ are stable of phase $1$. In the second case ${\mathcal O}_{p_i}$ is stable of phase $>1$. Finally, in the third case $\zeta\otimes{\mathcal O}_{p_i}$ is stable of phase $\leq 0$ (this follows from Proposition \ref{point-prop}(a)).
\noindent {\bf Step 3}. $\rho$ gives a bijection from $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ to $\Theta^0$. First, suppose we have two stability conditions $\sigma=(Z,P)$ and $\sigma'=(Z',P')$ such that $\rho(\sigma)=\rho(\sigma')$. Then $Z=Z'$ and the induced stability condition on ${\mathcal D}_{p_i}$ for $\sigma$ and $\sigma'$ are the same. This implies that for every $i$, ${\mathcal O}_{2p_i}$ is $\sigma$-semistable if and only if it is $\sigma'$-semistable (of the same phase). Therefore, rotating and tensoring with a line bundle we can assume that ${\mathcal O}_{\pi^{-1}(y)}$ for all $y$ are semistable of phase $1$ with respect to both $\sigma$ and $\sigma'$. As we have seen in Step 2 this implies that conditions of Lemma \ref{phase-lem}(a) are satisfied for $\sigma$ and $\sigma'$, which gives $\sigma=\sigma'$. On the other hand, given a point $(\sigma_1,\ldots,\sigma_n,z)\in\Theta^0$, using the $\widetilde{\operatorname{GL}_{2}^{+}(\R)}$-action and operations on $\Theta^0$ corresponding to tensoring with a line bundle on $X$, we can assume that $\Im f_i(\sigma_i)=1$ and ${\mathcal O}_{2p_i}$ is
$\sigma_i$-semistable for every $i$. We can define the central charge $Z$ uniquely so that $Z|_{{\mathcal D}_i}=Z_i$ and $Z({\mathcal O}_X)=z$. Note that inequality \eqref{det-ineq} in this case takes form $\Im z>0$. Now using Proposition \ref{cover-prop} we can easily construct the stability condition $\sigma$ on ${\mathcal D}_{\mathbb{Z}_2}(X)$ with the central charge $Z$ and the given restrictions $\sigma_i$ on ${\mathcal D}_i$.
\noindent {\bf Step 4}. $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ is connected. This follows from the continuity of gluing and Step 2. More precisely, let us first show that all stabilities constructed in Proposition \ref{cover-prop} belong to the same connected component. To this end we consider them as being glued from $(\zeta\otimes{\mathcal O}_{p_i}, i\in I^-)$ and $D(I^+\cup I^0)$. Now using Corollary \ref{exc-cor} we can find a path from our stability to the one that has the phases of all $\zeta\otimes{\mathcal O}_{p_i}$'s for $i\in I^-$ in the interval $(-1,0)$, and the phases of all ${\mathcal O}_{p_i}$'s for $i\in I^+$ in the interval $(1,2)$ (in particular, we will have $n_i=1$ for all $i\in I^-\cup I^+$). By definition, such a stability belongs to the connected set $U$ considered in Theorem \ref{cover-thm}. Thus, the set $V$ of all stabilities constructed in Proposition \ref{cover-prop} is connected. Therefore, for every equivariant line bundle $L$ the set $V\otimes L$ is still connected. Since the standard stability is contained in all of these sets, the statement follows from Step 2.
\noindent {\bf Step 5}. It follows from Step 4 that every $\sigma\in\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ is full. Therefore, the projection from $\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))$ to the space of numerical central charges is a local homeomorphism. This implies that $\rho:\operatorname{Stab}_{{\mathcal N}}({\mathcal D}_{\mathbb{Z}_2}(X))\to\Theta^0$ is a local homeomorphism. Therefore, by Step 3, it is a homeomorphism.
\noindent {\bf Step 6}. It remains to prove $\Theta^0$ is contractible. By Proposition \ref{point-prop}, the space $\Theta$ can be identified with the product $\mathbb{C}\times \Sigma^n\times\mathbb{C}$, where the first factor corresponds to $f_i(\sigma_i)$. Let us consider the projection $\Theta^0\to \mathbb{C}\times \Sigma^n$ obtained by omitting the last component. Each fiber of this projection is a half-plane. Since the target is contractible, it follows that $\Theta^0$ is also contractible. \end{proof}
\end{document}
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arXiv
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Derivation and internal validation of a multi-biomarker-based cardiovascular disease risk prediction score for rheumatoid arthritis patients
Jeffrey R. Curtis ORCID: orcid.org/0000-0002-8907-89761,
Fenglong Xie1,
Cynthia S. Crowson2,
Eric H. Sasso3,4,
Elena Hitraya3,4,
Cheryl L. Chin3,4,
Richard D. Bamford3,4,
Rotem Ben-Shachar4,
Alexander Gutin4,
Darl D. Flake II4,
Brent Mabey4 &
Jerry S. Lanchbury4
Rheumatoid arthritis (RA) patients have increased risk for cardiovascular disease (CVD). Accurate CVD risk prediction could improve care for RA patients. Our goal is to develop and validate a biomarker-based model for predicting CVD risk in RA patients.
Medicare claims data were linked to multi-biomarker disease activity (MBDA) test results to create an RA patient cohort with age ≥ 40 years that was split 2:1 for training and internal validation. Clinical and RA-related variables, MBDA score, and its 12 biomarkers were evaluated as predictors of a composite CVD outcome: myocardial infarction (MI), stroke, or fatal CVD within 3 years. Model building used Cox proportional hazard regression with backward elimination. The final MBDA-based CVD risk score was internally validated and compared to four clinical CVD risk prediction models.
30,751 RA patients (904 CVD events) were analyzed. Covariates in the final MBDA-based CVD risk score were age, diabetes, hypertension, tobacco use, history of CVD (excluding MI/stroke), MBDA score, leptin, MMP-3 and TNF-R1. In internal validation, the MBDA-based CVD risk score was a strong predictor of 3-year risk for a CVD event, with hazard ratio (95% CI) of 2.89 (2.46–3.41). The predicted 3-year CVD risk was low for 9.4% of patients, borderline for 10.2%, intermediate for 52.2%, and high for 28.2%.
Model fit was good, with mean predicted versus observed 3-year CVD risks of 4.5% versus 4.4%. The MBDA-based CVD risk score significantly improved risk discrimination by the likelihood ratio test, compared to four clinical models. The risk score also improved prediction, reclassifying 42% of patients versus the simplest clinical model (age + sex), with a net reclassification index (NRI) (95% CI) of 0.19 (0.10–0.27); and 28% of patients versus the most comprehensive clinical model (age + sex + diabetes + hypertension + tobacco use + history of CVD + CRP), with an NRI of 0.07 (0.001–0.13). C-index was 0.715 versus 0.661 to 0.696 for the four clinical models.
A prognostic score has been developed to predict 3-year CVD risk for RA patients by using clinical data, three serum biomarkers and the MBDA score. In internal validation, it had good accuracy and outperformed clinical models with and without CRP. The MBDA-based CVD risk prediction score may improve RA patient care by offering a risk stratification tool that incorporates the effect of RA inflammation.
Cardiovascular disease (CVD) is the leading cause of mortality for patients with rheumatoid arthritis (RA), accounting for 30–40% of deaths [1]. Patients with RA have approximately 50% greater risk for cardiovascular disease (CVD) compared to the general population [2]. Traditional CVD risk factors such as diabetes, hypertension, and hyperlipidemia are important in RA patients and are not difficult to assess. However, the time constraints of a busy office practice often preclude making CVD risk stratification a routine part of RA patient care. Indeed, 79% of rheumatologists cite a lack of time as a major barrier [3]. Even so, rheumatologists are well positioned to help manage CVD risk in RA patients because 30% of CVD risk in RA patients is attributable to systemic inflammation and other RA-related factors [4, 5].
CVD risk predictors developed for the general population tend to underestimate CVD risk in RA patients [6,7,8]. European League Against Rheumatism (EULAR) guidelines recommend that CVD risk predicted by tools such as the Framingham Risk Score (FRS) or the American College of Cardiology and American Heart Association (ACC/AHA) pooled cohort risk equation [9] be multiplied by 1.5 to account for the effect of RA on CVD risk [6, 10]. A limitation of this approach is that it treats all RA patients the same, regardless of the level of disease activity.
ACC/AHA guidelines recommend preventive strategies for all patients with high predicted risk of CVD. Current recommendations support managing hyperlipidemia by "treating to risk" rather than a targeted LDL [11,12,13]. It is well established that vascular inflammation has a central role in atherosclerosis and CVD, but evidence that reducing systemic inflammation has potential to lower CVD risk is more recent. Proof of principle comes from the CANTOS trial, which showed that canakinumab, an anti-IL-1β biologic drug, reduced the CVD event rate in non-RA patients with a high risk of CVD and elevated high-sensitivity C-reactive protein (CRP) [14]. Patients with greater reduction in inflammation, measured by CRP, benefited the most [15].
Synovial and systemic inflammation in RA patients contribute to CVD risk independently of traditional risk factors [4]. In observational studies, the risk for CVD events was greatest in RA patients with high disease activity [16,17,18,19,20] and effective RA treatment appeared to reduce the risk for atherosclerosis [21] and CVD events [22, 23]. Traditional CVD risk factors, such as diabetes, may be exacerbated by RA-related mechanisms [24, 25]. Thus, it may be possible to reduce the CVD risk elevation attributable to RA by treating RA inflammatory pathways.
High sensitivity CRP has prognostic value for CVD events in non-RA populations, but its role for CVD risk prediction in RA patients is less clear because CRP may be a marker for systemic inflammation in RA rather than a surrogate for the extent of vascular involvement [26]. Moreover, CRP is not elevated in some RA patients with active disease [27]. CVD risk prediction models that combine measures of RA disease activity with traditional risk factors [19, 28, 29] are not yet the standard of care. Molecular markers of inflammation other than CRP have not been incorporated into validated CVD risk predictors for RA patients. Their inclusion would be novel and may have potential to improve CVD preventive care for RA patients by making CVD risk stratification more accurate and accessible.
The multi-biomarker disease activity (MBDA) test assesses RA disease activity by measuring 12 serum protein biomarkers to provide a validated score on a scale of 1–100 that correlates with the Disease Activity Score in 28 joints with CRP (DAS28-CRP) [30]. In 2019, the American College of Rheumatology disease activity measures working group concluded that the MBDA score was one of 11 measures of RA disease activity that met the minimum standard for regular use [31]. The MBDA score is predictive of future radiographic damage, independently of other measures [32, 33]. In a large, cross-sectional observational study, the MBDA score was found to be associated with risk for CVD, suggesting that the MBDA score and at least some of its biomarkers detect inflammation that is relevant to cardiovascular pathology [16].
Building on this evidence, we now describe the development and internal validation of an RA-specific CVD risk prediction score that uses routine clinical assessments plus RA-related biomarkers to predict CVD risk. The goal of this approach was to improve preventive CVD care in RA patients by developing a prognostic score that uses biomarkers to incorporate the contribution of RA-related inflammation to individual CVD risk. The intended end result of this endeavor is to create a validated CVD risk score that will enable rheumatologists to risk stratify their RA patients efficiently in an office setting, with components associated with RA disease activity directly represented in the CV risk estimate.
A retrospective RA cohort was created for this study by linking claims data in the Medicare database with data in the MBDA test commercial database (Vectra®, formerly Crescendo Bioscience, Inc., South San Francisco, CA, USA, currently Myriad Genetics Laboratories, Salt Lake City, UT, USA), using all fee-for-service Medicare data from 2006 to 2016 for all individuals who underwent MBDA testing. Data were linked on patient date of birth, sex, MBDA test date, MBDA testing codes (defined by Current Procedural Terminology codes 81479, 83520, 84999, 86140, and 81490, submitted by Crescendo Bioscience or Myriad Genetics Laboratories), and the National Provider Identifier of the treating rheumatologist. Data were linked deterministically, using established methods [16, 34]. The University of Alabama at Birmingham institutional review board approved the study.
Participant and MBDA test eligibility criteria
The patient cohort and MBDA test results included in this study were selected by applying a series of criteria to the patients and MBDA tests in the linked database described above (Supplemental Table 1). To be eligible for inclusion in the study, patients were required to (1) be ≥ 40 years old, (2) have at least one RA diagnosis code from a rheumatologist (ICD9 714.0; ICD10 M05.*, M06.*, excluding M06.4 and M06.1, with * representing any number of digits or characters), (3) have received an RA-specific treatment (TNF-inhibitor, abatacept, rituximab, anti-IL-6R, Janus kinase inhibitor, conventional synthetic disease-modifying anti-rheumatic drug including methotrexate, sulfasalazine, leflunomide and hydroxychloroquine) anytime up to and including the date of the first MBDA test, and (4) have at least one linked MBDA test result. The accuracy of this claim-based method of identifying RA patients exceeds 85% [35] and is likely made greater here by the linkage with data from MBDA testing, which is only for patients diagnosed with RA.
The baseline period for a patient was defined as the interval preceding the date of the first MBDA test in the linked database. It included all available preceding Medicare data and was required to span at least 1 year, with patients being required to have had at least 365 days of continuous coverage with Medicare parts A (hospital coverage), B (outpatient coverage), and D (pharmacy coverage). Patients were excluded if they had any diagnosis code in the baseline period for malignancy (except non-melanoma skin cancer), myocardial infarction (MI), or stroke. MBDA test results (i.e., the MBDA score and 12 biomarker measurements) were used from the earliest MBDA test performed after the above requirements had been met, unless (1) it was performed within 14 days following any hospital discharge or (2) the patient had used anti-IL-6R treatment in the preceding 90 days (because tocilizumab treatment may affect the MBDA score in a way that might confound CVD risk prediction) [36]; in these cases, the next MBDA test meeting the above requirements was used and the baseline period was anchored to that test. The follow-up period for ascertaining CVD outcomes (see below) began on the date of the first qualifying MBDA test. The follow-up period ended at the earliest of (1) a CVD outcome, (2) diagnosis of malignancy, (3) non-CVD death, or (4) the end of study (December 31, 2016).
CVD outcome
The CVD outcome we used for the prognostic test was a composite, defined as the occurrence of hospitalized MI, stroke, or fatal CVD. This outcome definition is consistent with the outcome used in the guidelines of the ACC/AHA [9]. MI was defined as ICD-9 diagnosis code 410.x1 or ICD-10 diagnosis code I21.* from an inpatient hospitalization lasting ≥ 1 night or where the patient died. Stroke was identified using ICD-9 diagnosis codes 430.*, 431.*, 433.x1, 434.x1, 436.* or ICD-10 diagnosis codes I60.*, I61.*, I63.* or I67.89 from hospital discharge. This approach has been described previously [37,38,39]. Fatal CVD was identified using a validated algorithm that identifies fatal MIs and fatal strokes from Medicare data at a threshold yielding a positive predictive value > 80%, with greater accuracy than is obtained using hospital discharge diagnoses [40].
Biomarkers and other predictors
MBDA score
All biomarker data in this study came from the MBDA test, which measures the serum concentrations of 12 biomarkers and uses an algorithm to produce a disease activity score on a scale of 1 to 100. The MBDA score has been validated against DAS28-CRP in patients treated with a variety of RA therapies, with AUROC values of 0.77 and 0.70 observed in seropositive and seronegative RA patients, respectively [30, 41]. The MBDA score is used to assess and monitor inflammatory disease activity in RA patients and is complementary to clinical assessment. It is a stronger predictor of risk for radiographic progression than DAS28-CRP [32, 33]. The MBDA score is not intended for the diagnosis of RA but rather is for use in assessing disease activity in patients with already-diagnosed RA. The MBDA score has been available for use in clinical practice in the US since 2010. Its cost has been covered in the US by Medicare since 2013 and is also covered by some private insurers.
The biomarkers in the MBDA test reflect the biology of RA and comprise cytokine-related proteins (IL-6, TNF-R1), acute phase reactants (CRP, serum amyloid A), an adhesion molecule (VCAM-1), a skeletal-related protein (bone glycoprotein 39 [YKL-40]), growth factors (EGF, VEGF-A), matrix metalloproteinases (MMP-1, MMP-3), and adipokines (leptin, resistin). All MBDA scores analyzed here were from tests that had been ordered by practitioners in the US as part of routine patient care. All MBDA testing was performed in a Clinical Laboratory Improvements Amendment-certified commercial laboratory in South San Francisco, CA (Crescendo Bioscience), where MBDA scores were calculated and stored with related data in a secure database.
Prior to and independently of the present study, an algorithm was developed and validated to adjust the MBDA score for the effects of age, sex, and leptin (as a surrogate for adiposity) [42]. This adjustment acts on the original MBDA score without affecting the individual contributions of the 12 biomarkers. Thus, the original MBDA score is calculated as previously, then adjusted to produce a score that, like the original score, has a scale of 1–100 and RA disease activity categories of low (< 30), moderate (30–44), and high (> 44) [30, 42]. The adjusted MBDA score has been in routine use since December 2017. Original MBDA scores were converted to adjusted MBDA scores for this study. In the remainder of this report, the term "MBDA score" means the adjusted MBDA score.
Variables considered for inclusion in model building
Variables considered for use in model building that came from the MBDA database included the MBDA score and the serum concentrations of its 12 component biomarkers. This approach was non-redundant because the algorithm for the MBDA score is a non-linear combination of its component biomarkers, which were neither selected nor weighted for CVD prediction [30, 41].
Demographic and clinical predictors were obtained from the Medicare database and were considered for inclusion in model building based upon their expected association with CVD risk, informed by subject matter expertise and the medical literature. Other considerations were face validity, data quality in the Medicare database, and feasibility of collecting a variable accurately in clinical practice. These predictors included age, sex, race, tobacco use (past or present), history of CVD other than MI or stroke, diagnoses of and medications for diabetes, hypertension and hyperlipidemia, RA medications as described above, glucocorticoids, and non-steroidal anti-inflammatory drugs. A diagnosis was counted as present if any of its diagnostic codes was found for the patient. Diagnostic codes for the candidate predictors, i.e., the subset of variables that were included in the final model-building exercise, and the prevalences of CVD-related conditions, appear in Supplemental Table 2.
Clinical measurements (e.g., blood pressure or lipid levels) were not available in either database and were not considered for inclusion in model building. Current use of CV-related medications (e.g., lipid-lowering therapies) and RA medications was initially considered and was evaluated as part of baseline data assessment. However, a decision was made to not include any medications as variables in model building for two reasons: (1) without being able to account for disease-related clinical measurements, the estimated effect of medications may be counterintuitive or inaccurate and (2) suboptimal medication adherence could result in meaningful misclassification of the CV risk associated with these treatments. Race was excluded because of uncertainties related to racial heterogeneity and the reporting of race.
A principled, pre-specified approach to model building and selection was conducted that followed Transparent Reporting of a multivariable prediction model for Individual Prognosis Or Diagnosis (TRIPOD) guidelines [43]. First, the cohort was randomly split 2:1 into separate datasets for training and testing (i.e., internal validation).
Prior to model building, the independent association of the MBDA score with the CVD risk was evaluated in the training dataset with a multivariable analysis that included all non-biomarker candidate predictors [16]. Separately, the form of the relationship between MBDA score and CVD risk, on the logarithmic scale of hazard, was examined and found to be linear up to MBDA scores of approximately 60 and non-linear thereafter—a relationship that can be described with a hyperbolic tangent function (see below), which is commonly used in other fields, e.g., in models of neural networks [44].
Training: evaluation of variables and model building
Model development was conducted in the training dataset, to achieve the goal of estimating individual risk for the composite CVD outcome as a function of the candidate predictors. Individual biomarker concentrations in ng/ml were natural log transformed. MBDA scores (integers on a scale of 1 to 100) were hyperbolic tangent-transformed, as f(x) = tanh(a ∗ x), where a is a constant parameter that was based on maximum likelihood estimation and updated in each step of model building. Age in years was treated as a continuous variable. A separate age-squared term was initially included to account for possible nonlinearity between age and the composite CVD outcome, but it added no additional value to model building and was dropped. Other candidate predictors were treated as binary variables.
Association with 3-year CVD risk was assessed for each candidate variable with a hazard ratio (HR) and determined by univariable analysis in the training dataset. A 3-year time frame was chosen based on the availability of MBDA biomarker data from testing performed as part of routine care. Model building used Cox proportional hazards regression with backward elimination in the training dataset. In the first step, a model was fit by including every candidate predictor variable; in each subsequent step, the least significant variable (i.e., with the highest p value) was removed, and the model was refit with the remaining variables. This process was repeated until all remaining variables had p < 0.05.
Clinical models developed for comparison
Four prespecified models for predicting CVD risk were built in the training dataset for comparison with the MBDA-based model: (1) age + sex, (2) age + sex + CRP, (3) a clinical model (age + sex + tobacco use + diabetes + hypertension + history of CVD [excluding MI and stroke]), and (4) the clinical model + CRP. These models were chosen for the availability of their variables in routine clinical practice and in our linked database.
Derivation of categories of 3-year risk for CVD events
The thresholds for 3-year CVD risk categories that would be equivalent to the thresholds for 10-year risk categories of other CVD risk prediction equations were derived in a cohort with 10 years of longitudinal data. To create a dataset in which CVD event rates at 3 and 10 years could be bridged, a cohort of 533,139 Medicare RA patients with data available from 2006 to 2016 was selected with the same requirements as for the main cohort of this study but without requiring MBDA testing. An age + sex model was developed in this cohort to establish 10-year rates of CVD events, and 3-year cutpoints corresponding to the 10-year ACC/AHA risk thresholds of 5% (± 0.1%), 7.5% (± 0.1%), and 20% (± 0.1%) [11] were obtained by bootstrapping. The derived cutpoints were 1.3%, 1.8%, and 5.2%, defining 3-year CVD risk categories of low (0 to < 1.3%), borderline (≥ 1.3 to < 1.8%), intermediate (≥ 1.8 to < 5.2%), and high (≥ 5.2%) risk.
Internal validation
The primary analysis for establishing internal validation was to estimate the risk of a composite CVD event at 3 years (i.e., the probability of a patient having an MI, a stroke, or CVD death in the next 3 years), by using the MBDA-based CVD risk score as the only variable in a Cox proportional hazard regression model. HR (with 95% confidence interval [CI]; p value by partial likelihood ratio test [LRT]) was determined for the MBDA-based CVD risk score [45,46,47]. A risk curve was constructed to illustrate this relationship, using methods described in Supplementary Text. These and all other validation analyses were performed in the validation dataset.
To assess accuracy of the MBDA-based CVD risk score, a secondary analysis for internal validation examined goodness of fit with plots that compared observed risk (based on Kaplan-Meier estimates with 95% CI) with predicted risk across CVD event-based deciles. P values were determined using the Greenwood-Nam-D'Agostino test [48], with higher (i.e., non-significant) p values indicating better fit. Goodness of fit was also assessed among patient subgroups, based on age, sex, diagnosis of diabetes, hypertension, tobacco use (past or present), and hyperlipidemia, as well as history of CVD, statin use, oral glucocorticoid use, initiation or change of a biologic agent during follow-up, and MBDA score category. Bonferroni correction was used to adjust for multiple testing. CVD event quintiles, rather than deciles, were used for patient subgroups with fewer than 110 CVD events to avoid data sparsity. In addition, Kaplan-Meier plots of CVD event-free status over time were constructed for patients grouped into CVD risk categories by the MBDA-based CVD risk score, using the Mantel-Haenszel test [45, 46].
Validation included comparisons of the predictive abilities of the MBDA-based CVD risk score and four clinical models described above. HR (95% CI) and p value (using the partial LRT) were calculated from Cox proportional hazards models in single-score (i.e., univariable) analyses of the MBDA-based CVD risk score and each of the four clinical models. To determine the incremental contribution of the MBDA-based model to each clinical model for predicting CVD risk (and vice versa), change in model deviance was determined using the likelihood ratio statistic in sequential (i.e., bivariable) analyses for each model pair.
The MBDA-based CVD risk model was also compared to the four clinical models with reclassification tables and the Net Reclassification Index (NRI) [49, 50]. The five models were each evaluated for discrimination based on the C-index (similar to AUROC) for predicting risk at 3 years, with times weighted by the square inverse of the censoring distribution [51].
Statistical software
SAS 9.4 was used for data preparation. R version 3.4 and R packages survival, nricens, and pec were used for evaluating model performance, calculating NRIs and C-indices, and generating plots [52].
Cohort selection
30,751 RA patients with 904 CVD events (480 MI, 362 stroke, 62 CVD death) were eligible for the total cohort (Supplemental Table 1). Total follow-up from the index date was 56,684 patient-years (PY) with median (interquartile range [IQR]) follow-up duration of 1.7 (0.8–2.7) years. The overall CVD event rate (95% CI) was 15.9 (14.9–17.0) events per 1000 PY.
At baseline, the mean age was 69 years, 23% of patients were under age 65 years, 18% were men, and 8% were Black (Table 1). The prevalence of CVD-related comorbidities, such as diabetes (40%) and hypertension (79%), was high. Statin use was found in 42%. Sixty percent of patients were receiving methotrexate, 33% a TNF inhibitor (TNFi), and 15% a non-TNFi biologic. Median (IQR) CRP value was 4.5 (1.6–12.0) mg/L (or 1.5 [0.5–2.5] μg/ml natural log transformed). Median (IQR) MBDA score was 40 [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49], which is in the moderate MBDA category (range, 30–44) (Table 1).
Table 1 Patient characteristics at baseline*
Confirming the MBDA score as an independent predictor of CVD risk
In the training dataset (N = 20,476 patients with 611 CVD events), the MBDA score, untransformed, was significantly prognostic of CVD events in a multivariable analysis with age, sex, diabetes, hypertension, tobacco use, CVD history, and hyperlipidemia, but with no individual biomarker variables (HR = 1.023; 95% CI 1.017–1.029).
Training of the MBDA-based model
In univariable analyses in the training dataset, all candidate predictors except EGF and MMP-1 were individually predictive of CVD events (Table 2). In the final MBDA-based model, derived from backward elimination, the variables of age, diabetes, history of CVD, hypertension, tobacco use, MBDA score, and three biomarkers (leptin, MMP-3, TNF-R1) were significant predictors in multivariable analyses; sex, hyperlipidemia, and nine biomarkers were not. HRs were significantly > 1.0 for all predictor variables in the final MBDA-based model except leptin, for which HR was 0.84, indicating a negative relationship between leptin concentration and CVD risk (Table 2).
Table 2 Hazard ratios (HR) of predictor variables used in CVD risk models (training dataset, N = 20,476)
The equation for the final MBDA-based CVD risk score was:
$$ 0.0314\times \boldsymbol{age}\kern0.3em +\kern0.3em 0.2691\times \boldsymbol{tobacco}\;\boldsymbol{use}+\kern0.3em 0.2732\times \boldsymbol{diabetes}+0.2694\times \boldsymbol{hypertension}+0.3378\times \boldsymbol{history}\kern0.17em \boldsymbol{of}\;\boldsymbol{CVD}-0.1711\times \mathrm{In}\left(\boldsymbol{Leptin}\right)+0.1454\times \ln \left(\boldsymbol{MMP}\mathbf{3}\right)+0.5724\times \ln \left(\boldsymbol{TNFR1}\right)+1.6076\times \tanh \left(\boldsymbol{MBDA}\kern0.17em \boldsymbol{score}/33.0807\right), $$
where the age is in years, clinical variables are scored as 1 when present and zero when absent, Leptin, MMP-3, and TNF-R1 represent serum concentrations in ng/mL, the term "ln" means natural logarithm, and "tanh" means hyperbolic tangent transformation. The output of this algorithm is the MBDA-based CVD risk score. This score is used in a separate formula to calculate the predicted 3-year risk for a CVD event as a percentage value (see Supplemental Text).
In the four multivariable clinical models that were generated for comparison—i.e., an age + sex model and an age + sex + diabetes + hypertension + history of CVD + tobacco use model, each one with and without CRP—all variables in each model were significant CVD predictors (Table 2).
Internal validation of the MBDA-based model
The MBDA-based CVD risk score was a strong predictor of 3-year risk for a CVD event in the validation dataset (N = 10,275 patients with 293 CVD events), with an HR (95% CI) of 2.89 (2.46–3.41, p = 4.67 × 10− 38). The relationship between the MBDA-based CVD risk score and predicted 3-year CVD risk is shown in Fig. 1a. The proportions of patients in the low, borderline, intermediate, and high categories of predicted 3-year CVD risk in the validation dataset were 9.4%, 10.2%, 52.2%, and 28.2%, respectively (Fig. 1b).
Characterization of the MBDA CVD risk score in the validation dataset (N = 10,275). a Relationship between MBDA-based CVD risk score and predicted 3-year risk of a CVD event, with 95% confidence interval. b Distribution of predicted 3-year risks. Dotted lines, horizontal in a and vertical in b, indicate thresholds at 1.3%, 1.8%, and 5.2% separating the categories of low, borderline, intermediate, and high risk, which contained 9.4%, 10.2%, 52.2%, and 28.2% of patients, respectively. CVD event is myocardial infarction, stroke, or CVD death. CVD cardiovascular disease, MBDA multi-biomarker disease activity
Assessment of accuracy with goodness of fit
The 3-year CVD risk predictions made by the MBDA-based model were similar to the observed CVD event rates across deciles based on observed CVD events (Fig. 2). The goodness of fit test statistic indicated good fit (p = 0.39). The confidence intervals for observed risk contained the average predicted risk for all but one decile group. Overall, the mean predicted 3-year CVD risk in the validation dataset was 4.5%, compared with the observed 3-year CVD risk of 4.4%. Subanalyses showed that the MBDA-based model performed well in subgroups of interest: males and females, with/without diagnosis of diabetes, with/without diagnosis of hypertension, with/without tobacco use, with/without history of CVD, with/without hyperlipidemia, taking/not taking statins, < 65 years old, < 75 years old, and patients who had or had not used oral glucocorticoids in the baseline period, or initiated or changed a biologic drug during the follow-up period, or had low, moderate, or high disease activity (MBDA score) (Supplemental Fig. 1).
Goodness of fit: Predicted CVD risk versus observed 3-year CVD event rates. The observed 3-year CVD event rate was determined for each event-based decile and is shown vs. the average predicted 3-year risk in each decile. Analysis used the validation dataset (N = 10,275). Observed event rates were determined as Kaplan-Meier (95% log-log CI) estimates. P = 0.39 by the Greenwood-Nam-D'Agostino test, indicating good fit. CVD event is myocardial infarction, stroke, or CV death. 3-year CVD risk categories (low, borderline, intermediate, high) were derived from the 10-year risk categories of the 2018 Guidelines of the American College of Cardiology/American Heart Association [8]. Threshold between low and borderline risk categories is 1.3% (not shown). CI confidence interval, CVD cardiovascular disease, MBDA multi-biomarker disease activity
Loss of CVD outcome-free status by category of predicted risk
A Kaplan-Meier plot depicting loss of CVD outcome-free status in the validation dataset showed statistically significant separation of the low, borderline, intermediate and high predicted CVD risk groups over time (p = 1.7 × 10−32) (Fig. 3).
Kaplan-Meier plot of CVD event-free survival. Occurrence of CVD events by Kaplan-Meier survival analysis is shown for patients in the validation dataset (N = 10,275) grouped by a 3-year CVD risk category predicted by the MBDA-based CVD risk score at baseline. P = 1.7 × 10−32 by the Mantel-Haenszel test. CVD event is myocardial infarction, stroke, or CVD death. See Fig. 2 for explanation of CVD risk categories. CVD cardiovascular disease, MBDA multi-biomarker disease activity
Model evaluation and comparison by likelihood test
When analyzed alone, each of the four clinical models made statistically significant contributions to the prediction of CVD risk in terms of the likelihood ratio, which represents how well the model fits the data (Fig. 4). However, these models made smaller contributions than the MBDA-based CVD risk score (Fig. 4). Moreover, the addition of these clinical models to the MBDA-based CVD risk score in paired analyses did not improve CVD risk prediction, as indicated by the respective increments in LRT statistic (0.4–3.0), which were small and non-significant (Table 3). In contrast, the MBDA-based CVD risk score provided additional information to improve the prediction of CVD risk when it was added to each clinical model, with the increments in LRT statistic being large (35.4–83.3) and statistically significant (all p < 3 × 10− 9) (Table 3).
Contribution to CVD risk prediction by MBDA-based CVD risk score and clinical models. Likelihood ratio test statistics are shown for univariable (i.e., single-score) analyses of a CVD risk prediction by the MBDA-based CVD risk score and four comparison models, using the validation dataset (N = 10,275) (see also Table 3). P values are by the likelihood ratio test. The clinical model includes age, sex, tobacco use, diabetes, hypertension, and history of CVD. CRP C-reactive protein, CVD cardiovascular disease, MBDA multi-biomarker disease activity
Table 3 Contribution of MBDA-based CVD risk score and other models to prediction of 3-year CVD risk
Compared to the simplest of the clinical models, the age + sex model, the MBDA-based model reclassified the CVD risk for 42% of patients overall and as many as 75% of patients, depending on the age + sex model risk category (Table 4A). Compared to the most comprehensive clinical model, the clinical + CRP model, the MBDA-based model reclassified the CVD risk for 28% of patients overall and as many as 64% of patients, depending on the clinical + CRP model risk category (Table 4B). Reclassification results for the age + sex + CRP model and the clinical model (without CRP) were generally intermediate to those of the other two models (Supplemental Tables 3A and 3B).
Table 4 Reclassification of patients by the MBDA-based CVD risk score versus: A, age + sex model and B, clinical + CRP model
NRI test statistics demonstrated that the MBDA-based model significantly improved classification versus all four clinical models, with NRI test statistics (95% CI) of 0.19 (0.10–0.27) versus the age + sex model, 0.16 (0.08–0.23) versus the age + sex + CRP model, 0.10 (0.04–0.17) versus the clinical model, and 0.07 (0.001–0.13) versus the clinical + CRP model.
The C-index (95% CI) for the prediction of CVD risk at 3 years by the MBDA-based CVD risk score in the validation dataset was 0.715 (0.683–0.747), which was numerically greater than the C-index for each clinical model. The difference was greatest versus the simplest clinical model and least versus the most comprehensive clinical model, with C-indices (95% CI) of 0.661 (0.628–0.695) for the age + sex model, 0.674 (0.642–0.707) for the age + sex + CRP model, 0.688 (0.656–0.721) for the clinical model, and 0.696 (0.664–0.729) for the clinical + CRP model.
Relationship between individual biomarkers and MBDA-based CVD risk score
Scatterplots derived from the validation dataset demonstrate the positive relationships between 3-year risk predicted by the MBDA-based CVD risk score and MBDA score (r = 0.438), MMP-3 (r = 0.437), and TNF-R1 (r = 0.632); and the negative relationship with leptin (r = − 0.179). For the MBDA score and for each biomarker, at most levels a range of CVD risks was observed, consistent with variation among the other variables of the MBDA-based CVD risk score (Fig. 5).
Relationship between predicted CVD risk and molecular variables. The predicted 3-year risk for a CVD event (myocardial infarction, stroke, or fatal CVD) is shown versus (a) the MBDA score and (b–d) serum concentrations (ng/ml, natural log transformed) of the three biomarker variables in the MBDA-based CVD risk score, using the validation dataset (N = 10,275). R values are Spearman correlation coefficients. CVD cardiovascular disease, MBDA multi-biomarker disease activity
We have used a cohort of over 30,000 RA patients to derive and internally validate an MBDA-based CVD risk score for use in patients with RA. This score reflects the contribution of systemic inflammation to CVD risk by including the MBDA score and three individual biomarkers, while also incorporating age and four clinical risk factors. The MBDA-based risk score accurately predicted CVD risk in terms of goodness of fit analyses in the internal validation cohort and in clinically relevant subgroups, including patients who did or did not have prior CVD, who were already taking statins, or had different levels of RA disease activity. The MBDA-based risk score discriminated CVD risk better than clinical models, assigning some patients to higher or lower risk categories compared with clinical assessment alone.
This test is unique because it uses biomarker-based measurements to incorporate the contribution of RA inflammation to CVD risk in a more personalized way than by multiplying by a fixed value, such as 1.5 [6]. The MBDA score is a measure of RA disease activity that is also predictive of risk for radiographic progression. It was shown here and previously to be associated with the CVD risk [16], even though it was not originally developed for that purpose. MMP-3 and TNF-R1 were included in the final MBDA-based CVD risk score because in model building, they were positively associated with CVD risk independently of the MBDA score and other variables, which is consistent with previous reports of their role in cardiovascular risk [53,54,55].
The other individual biomarker in the CVD risk score was leptin. In our cohort, patients with a CVD event had less obesity and a numerically lower median leptin concentration than patients without a CVD event (Table 1). Leptin had a negative coefficient in the multivariable CVD risk prediction model. These results are consistent with evidence that leptin correlates strongly with body mass index (BMI) and that BMI has been negatively associated with CVD risk in RA patients [56], even though it is positively associated with CVD risk in the general population [57]. Our findings may reflect a contribution of RA inflammation to both weight loss and mortality, rather than a biologically protective effect of obesity [58]. They may also be a reflection of index case bias, which can lower the effect estimate for a risk factor, such as leptin, if it is associated with both the sequela of a disease and the disease itself, as with CVD events and RA [59]. IL-6, CRP, and other MBDA biomarkers were not included in the MBDA-based CVD risk score despite being individually associated with the CVD risk because none added significant information to leptin, MMP-3, TNF-R1, and the MBDA score for predicting CVD risk.
Clinical covariates that might have been expected in the final MBDA-based model, such as sex and hyperlipidemia, were associated with the CVD risk in univariable analyses but were not included because they made small incremental contributions to the multivariable model and did not survive the model building process. Sex was less significant as a univariable predictor of CVD risk than any of the variables that were included in the model (Table 2). It may have been excluded due to co-linearity with other variables, such as tobacco use, which is less common in women with RA than men with RA [4], and leptin, the levels of which tend to be greater in women [60]. It is unlikely that the MBDA score caused sex to be excluded from the model because adjustment of the MBDA score (for age, sex, and leptin) should have reduced its co-linearity with sex. The failure of hyperlipidemia to survive backward elimination may relate to it also having been a less significant univariable predictor of CVD risk than any of the predictors that survived. In addition, the "lipid paradox" [61] may make it difficult to interpret lipid values in RA patients, as they can be lower during active RA and increase with effective treatment. A practical consideration is that many RA patients have not had lipids tested recently, and co-management with primary care physicians may be needed to improve rates of screening for hyperlipidemia [62].
The cohort we used included patients with diabetes or a history of CVD and patients who were receiving statin treatment. Excluding such patients, as some CVD risk calculators do, would have greatly narrowed the utility of the score and reduced the power to see differences in the risk due to other variables. Instead, diabetes and a history of CVD were entered into model building as predictor variables and they were included in the score. Subanalyses demonstrated good fit between predicted and observed CVD events for patients with or without diabetes or a history of CVD. Statin use is not in the MBDA-based CVD risk score because we excluded drug-related variables from model building. However, the risk score demonstrated good fit in subanalyses of patients who were and were not receiving statins. The MBDA-based CVD risk score accounts for the level of inflammation, the treatment of which has potential to reduce CVD risk in RA patients [21,22,23]. The score may have utility for RA patients who are receiving statins because the statin dose may not yet have been optimized and because the non-statin treatment options for elevated CVD risk in RA patients may include DMARDs.
Other RA-specific CVD risk prediction models have been created. The expanded risk scored for CVD in RA (ERS-RA) was derived from a large RA cohort in the USA [19] and has been externally validated [28]. It quantifies RA disease activity categorically with the clinical disease activity index (CDAI) and also includes the Health Assessment Questionnaire (HAQ). A Trans-Atlantic Cardiovascular Risk Consortium for Rheumatoid Arthritis (ATACC-RA) developed two predictors that include serum lipid levels and account for RA disease activity with the 28-joint Disease Activity Score with erythrocyte sedimentation rate (DAS28-ESR) or HAQ, respectively [29]. The MBDA-based CVD risk score requires no clinical measurements and no laboratory data except results from the MBDA test. Rheumatologist preference among these predictors may depend on convenience and on which RA disease activity measures they use most routinely [63, 64]. CVD risk prediction for RA patients could be facilitated in a practical way if a risk score were to be automatically calculated—within an electronic medical record or, in the case of the MBDA-based CVD risk score, when the MBDA score is calculated by the testing laboratory—and provided to the ordering rheumatologist.
The large size of this study was made possible by linking administrative data from the Medicare database to a database of existing MBDA test results. The approach we used to capture CVD endpoint components in the Medicare database has a positive predictive value of approximately ≥ 93% for MI and 80–85% for stroke [37,38,39]. Fatal CVD events were identified using algorithms with positive predictive values ≥ 80% [40]. This study was restricted to patients ≥ 40 years old, to be aligned with the ACC/AHA guidelines [9]. A limitation of having used the Medicare cohort is that it contained predominantly older patients with high rates of CVD risk factors, and most of the 23% of patients < 65 years old were eligible for Medicare because they were disabled. In subanalyses of the patients who were < 65 years old and of patients who had or lacked each of the four clinical risk factors in the model, the MBDA-based CVD risk score had good fit with observed CVD events. In a previous report, CVD risk was relatively similar in younger disabled vs. younger non-disabled RA patients after accounting for the lower prevalence of CVD risk factors [65], suggesting that the MBDA-based CVD risk score may be applicable to patients < 65 years old who are not disabled. However, further validation of the CVD risk score in younger RA patients is needed.
Another limitation of our linked cohort is that clinical practice measurements, such as the blood pressure or lipid levels, were not available and the reasons for ordering MBDA tests were not known. Nevertheless, the MBDA-based CVD risk score demonstrated good fit with observed CVD events in patients with hypertension, hyperlipidemia, history of CVD or statin use, and in patients grouped by level of biomarker-based disease activity or according to whether a biologic DMARD treatment had been initiated or changed during follow-up. Because we lacked clinical measurements, the MBDA-based CVD risk score could not be compared with CVD risk predictors that require them, such as the ACC/AHA Pooled Cohort Equation or the Framingham Risk Score. As an alternative, the MBDA-based CVD risk score was compared to four clinical models of increasing complexity, from an age + sex model to a model that included age, sex, four traditional clinical risk factors available in the Medicare database, and CRP. The MBDA-based CVD risk score showed better fit than all four models, based on LRT. It also demonstrated statistically significantly better NRI and a numerically greater C-index. Because likelihood has been considered the most powerful means for comparing CVD risk prediction tests [66], and C-indices can fail to reflect meaningful incremental contributions of CVD-related biomarkers [67], these results suggest that the MBDA-based CVD risk score may be at least comparable to existing CVD risk calculators and potentially more practical for routine use. Direct comparison with other RA-specific calculators and general population CVD risk calculators adjusted for RA would be of interest.
The 3-year horizon used here for the composite CVD outcome reflects a constraint from the availability and uptake of the MBDA test for routine clinical practice in the US. Of more scientific relevance, however, is that RA is a dynamic disease and disease activity for many patients will fluctuate, such that a single measurement of disease activity may become less associated with true CVD risk over time. Thus, our shorter, 3-year time horizon may be preferable for predicting CVD risk in patients with RA, in that it is less subject to misclassification of RA disease activity than with the 10-year time horizon used by many existing CVD risk calculators. Indeed, the dynamic nature of RA disease activity and other factors that may be important to assessing CVD risk in RA patients is reflected in the ACC/AHA recommendation that, for adult patients with RA, "it can be useful to recheck lipid values and other major ASCVD (atherosclerotic CVD) risk factors 2 to 4 months after the patient's inflammatory disease has been controlled [11]." Among all specialists, rheumatologists are likely in the best position to assess treatment response and systemic inflammatory burden in RA patients. The MBDA-based CVD risk score may assist rheumatologists by reminding them of the need for CVD risk management in RA patients—which some may wish to co-manage with a primary care physician or cardiologist—and of the unique role rheumatologists have in treating the inflammatory disease component of CVD risk [13].
In conclusion, we have developed and internally validated an MBDA-based CVD risk score that predicts risk for MI, stroke, or fatal CVD in the next 3 years for RA patients. It is novel because it accounts for the contribution of RA inflammatory disease activity by including the MBDA score and three biomarkers that are independently associated with CVD. It performed better than prediction models that used only clinical data. The MBDA-based CVD risk prediction score provides rheumatologists with a feasible tool for assessing CVD risk to inform the management of traditional CVD risk factors and RA inflammation. Further validation with more extended time frames and more heterogeneous cohorts of RA patients will be helpful to assure its robustness as a prediction model.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
ACC/AHA:
American College of Cardiology and American Heart Association
CDAI:
CVD:
DAS28-CRP:
Disease Activity Score in 28 joints with CRP
DAS28-ESR:
Disease Activity Score in 28 joints with erythrocyte sedimentation rate
EULAR:
European League Against Rheumatism
LRT:
Likelihood ratio test
MBDA:
Multi-biomarker disease activity
NRI:
Net Reclassification Index
PY:
Patient-years
TNFi:
TNF inhibitor
TRIPOD:
Transparent Reporting of a multivariable prediction model for Individual Prognosis or Diagnosis
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The authors thank Brooke Hullinger, JD, for her assistance in preparing the figures and tables and editing the manuscript.
This work was supported by the Myriad Genetics, Inc.
Jeffrey R. Curtis & Fenglong Xie
Mayo Clinic, Rochester, MN, USA
Cynthia S. Crowson
Crescendo Bioscience, South San Francisco, CA, USA
Eric H. Sasso, Elena Hitraya, Cheryl L. Chin & Richard D. Bamford
Myriad Genetics Laboratories, Salt Lake City, UT, USA
Eric H. Sasso, Elena Hitraya, Cheryl L. Chin, Richard D. Bamford, Rotem Ben-Shachar, Alexander Gutin, Darl D. Flake II, Brent Mabey & Jerry S. Lanchbury
Jeffrey R. Curtis
Fenglong Xie
Eric H. Sasso
Elena Hitraya
Cheryl L. Chin
Richard D. Bamford
Rotem Ben-Shachar
Alexander Gutin
Darl D. Flake II
Brent Mabey
Jerry S. Lanchbury
JC and FX had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. Study conception and design: JC, FX, CSC, ES, RB-S, AG, DF, BM, and JL. Acquisition of data: JC, FX, AG, DF, and BM. Analysis and interpretation of data: JC, FX, CSC, ES, EH, CLC, RB, RB-S, AG, DF, BM, and JL. All authors were involved drafting the article or revising it critically for important intellectual content, and all authors approved the final version to be published.
Eric Sasso, M.D., is an Affiliate Professor of Medicine (Rheumatology) at the University of Washington, Seattle, WA., USA.
Correspondence to Jeffrey R. Curtis.
The University of Alabama at Birmingham institutional review board approved the study.
JC received grants and personal fees from the Abbvie, Amgen, BMS, Corrona, Eli Lilly, Jannsen, Myriad Genetics, Inc., Pfizer, Regeneron, Roche, and UCB during the conduct of the study. FX and CSC received research funding from the Myriad Genetics, Inc., during the conduct of the study. ES, EH, CLC, RB, RB-S, AG, DF, BM, and JL are employed by the Myriad Genetics, Inc., and receive salaries and stock options as compensation.
Additional file 1: Supplemental Figure 1.
Goodness of fit in patient subgroups (validation dataset, total N=10,275). Supplemental Table 1. Cohort Derivation. Supplemental Table 2. A, Diagnostic codes for candidate variables used to build the MBDA-based CVD risk score and B, Frequencies of CVD-related conditions comprising the History of CVD variable. Supplemental Table 3. Reclassification of patients based on CVD risk predicted by the MBDA-based CVD risk score versus: A, the Age + Sex + CRP model and B, the Clinical model. Supplemental Text. Conversion of the MBDA-based CVD risk score into 3-year percentage risk of a CVD event.
Curtis, J.R., Xie, F., Crowson, C.S. et al. Derivation and internal validation of a multi-biomarker-based cardiovascular disease risk prediction score for rheumatoid arthritis patients. Arthritis Res Ther 22, 282 (2020). https://doi.org/10.1186/s13075-020-02355-0
Received: 13 August 2020
Cardiovascular risk
Multi-biomarker
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CommonCrawl
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\begin{document}
\begin{abstract} Brownian motions on a metric graph are defined, their Feller property is proved, and their generators are characterized. This yields a version of Feller's theorem for metric graphs. \end{abstract}
\maketitle \thispagestyle{empty}
\section{Introduction} \label{sect_1}
In his pioneering articles~\cite{Fe52, Fe54, Fe54a}, Feller raised the problem of characterizing and constructing all Brownian motions on a finite or on a semi-infinite interval. In the sequel this problem stimulated very important research in the field of stochastic processes, and the problem of constructing all such Brownian motions found a complete solution~\cite{ItMc63, ItMc74} via the combination of the theory of the local time of Brownian motion~\cite{Le48}, and the theory of (strong) Markov processes~\cite{Bl57, Dy61, Dy65a, Dy65b, Hu56}. \iffalse In view of the paper~\cite{ItMc63} it is probably fair to say that the epoch making book~\cite{ItMc74} by It\^o and McKean had one of its roots in the problem posed by Feller. \fi
On the other hand, there is a growing interest in metric graphs, that is, piecewise linear varieties where the vertices may be viewed as singularities. Metric graphs arise naturally as models in many domains, such as physics, chemistry, computer science and engineering to mention just a few --- we refer the interested reader to~\cite{Ku04} for a review of such models and for further references.
Therefore it is natural to extend Feller's problem to metric graphs. The present paper is the first in a series of three articles~\cite{BMMG2, BMMG3} (together with a more pedagogical one~\cite{BMMG0}, in which the well-known classical cases of finite and semi-infinite intervals are revisited) on the characterization and the construction of all Brownian motions on metric graphs. Stochastic processes, in particular Brownian motions and diffusions, on locally one-dimensional structures, notably on graphs and networks, have already been studied in a number of articles, out of which we want to mention~\cite{BaCh84, DeJa93, EiKa96, FrSh00, FrWe93, Fr94, Gr99, Kr95} in this context.
Heuristically, a metric graph $(\mathcal{G},d)$ can be thought of as the union of a collection of finite or semi-infinite closed intervals which are glued together at some of their endpoints which form the vertices of the graph, while the intervals are its edges. The metric $d$ is then defined in the canonical way as the length of a shortest path between two points along the edges, and the length along the edges is measured as for usual intervals. For a more formal definition of metric graphs see section~\ref{ssect_2_1}. We will only consider \emph{finite} graphs, that is, those for which the sets of vertices and edges are finite. For the definition of a Brownian motion on the metric graph $(\mathcal{G},d)$ we take a standpoint similar to the one of Knight~\cite{Kn81} for the semi-line or a finite interval: It is a strong Markov process with c\`adl\`ag paths which are continuous up to the lifetime, and which up to the first passage time at a vertex is a standard Brownian motion on the edge where it started. For the formal definition, cf.\ section~\ref{sect_3}.
The crucial problem is then to characterize the behaviour of the stochastic process when it reaches one of the vertices of the graph $\mathcal{G}$, or in other words, the characterization of the boundary conditions at the vertices of the Laplace operator which generates the stochastic process. We want to mention in passing that in an $L^2$-setting all boundary conditions for Laplace operators on $\mathcal{G}$ which make them self-adjoint operators have been characterized in~\cite{KoSc99, KoSc00}. In this series of papers we shall work with the Banach space of continuous functions on $\mathcal{G}$ which vanish at infinity. The main result of the present paper is Feller's theorem for metric graphs (cf.\ theorem~\ref{thm_5_3}). Roughly speaking it states that all possible boundary conditions are \emph{local} boundary conditions of \emph{Wentzell} type, i.e., linear combinations of the value of the function with its first and second (directional) derivatives at each vertex, subject to certain conditions on the coefficients. We want to emphasize here, that the fact that we only obtain local boundary conditions is due to the assumption that the paths of the Brownian motion have no jumps during their lifetime. On a more technical level this assumption entails that we deal with Feller processes --- as is proved in section~\ref{sect_4} --- which is of considerable advantage when we prove the strong Markov property of the processes which we will construct in the follow-up papers~\cite{BMMG2, BMMG3}. On the other hand, in a forthcoming work we shall embed the situation into the larger framework of Ray processes, and there we shall deal also with non-local boundary conditions, allowing the processes to have jumps from the vertices into $\mathcal{G}$ in addition to the jumps to the cemetery point.
The paper is organized in the following way. In section~\ref{sect_2} we recall the pertinent notions of (finite) metric graphs, and of strong Markov processes on metric graphs, at the same time setting up our notation. In section~\ref{sect_3} Brownian motions on metric graphs are defined, and some consequences of this definition are discussed. The proof of the statement that Brownian motions on metric graphs are Feller processes is given in section~\ref{sect_4}. Finally, we state and prove Feller's theorem for metric graphs in section~\ref{sect_5}. In appendix~\ref{app_A} we give a short account of Feller semigroups in a form which we find especially convenient for the purposes of the present paper, but which we could not find in this form elsewhere.
The contents of the other two papers in this series are as follows. In the article~\cite{BMMG2} all Brownian motions on single vertex graphs (roughly speaking, $n$ semi-lines $[0,+\infty)$ glued together at the origin) are constructed, and in the article~\cite{BMMG3} these Brownian motions are pieced together pathwise to yield all possible Brownian motions on a general metric graph.
\noindent \textbf{Acknowledgement.} The authors thank Mrs.~and Mr.~Hulbert for their warm hospitality at the \textsc{Egertsm\"uhle}, Kiedrich, where part of this work was done. J.P.\ gratefully acknowledges fruitful discussions with O.~Falkenburg, A.~Lang and F.~Werner. R.S.~thanks the organizers of the \emph{Chinese--German Meeting on Stochastic Analysis and Related Fields}, Beijing, May 2010, where some of the material of this article was presented.
\section{Preliminary Definitions and Notations} \label{sect_2} \subsection{Metric Graphs} \label{ssect_2_1} Throughout this paper we consider a fixed finite \emph{metric graph} $(\mathcal{G},d)$. That is, $\mathcal{G}$ is a quadruple $(V,\mathcal{I},\mathcal{E},\partial)$, where $V$ is a finite set of \emph{vertices}, $\mathcal{I}$ is a finite set of \emph{internal edges}, $\mathcal{E}$ is a finite set of \emph{external edges}, and $\partial$ is a map from the set $\mathcal{L}=\mathcal{I}\cup\mathcal{E}$ of \emph{edges} into $(V\times V)\cup V$, which maps an internal edge $i\in\mathcal{I}$ to an ordered pair $(\partial^-(i),\partial^+(i))\in V\times V$ of vertices, called the \emph{initial} and \emph{final vertex of $i$}, while $e\in\mathcal{E}$ is mapped to $\partial(e)\in V$, called the \emph{initial vertex of $e$}. Every edge $l\in\mathcal{L}$ is assumed to be isometrically isomorphic to an interval $I_l$, namely every $e\in\mathcal{E}$ is in one-to-one correspondence with the half line $[0,+\infty)$, while for every $i\in\mathcal{I}$ there exists $\rho_i>0$ so that $i$ is isomorphic to $[0,\rho_i]$. Under these isomorphisms, for $i\in\mathcal{I}$ we have that $\partial_-(i)$ corresponds to $0$ while $\partial^+(i)$ corresponds to $\rho_i$, and for $e\in\mathcal{E}$, the vertex $\partial(e)$ corresponds to $0$. $\rho_i$ is called the \emph{length} of the internal edge $i\in\mathcal{I}$. Moreover, we suppose that the edges are sets with an ordering as induced by the isomorphisms mentioned above. $\mathcal{L}(v)=\{l\in\mathcal{L},\,v\in\partial(l)\}$ is the set of edges incident with $v$. For notational simplicity we will also use $\partial(l)$, $l\in\mathcal{L}$, to denote the set consisting of $\partial^-(l)$ and $\partial^+(l)$ if $l\in \mathcal{I}$, and of $\partial(l)$ if $l\in\mathcal{E}$.
In~\cite{KoSc06} the standard notion of a \emph{walk} on a graph (e.g., \cite{Ju05}) has been generalized to graphs of the above type, and therefore we obtain in a natural way a metric $d$ on $\mathcal{G}$ as the minimal length of all walks leading from one point to another, where the length is measured along the edges as induced by the isometry with the corresponding intervals.
In the sequel it will be convenient --- and without danger of confusion --- to identify the abstract graph $\mathcal{G}$ with its isomorphic \emph{geometric graph} (e.g., \cite{Ju05}). In other words, we also consider $\mathcal{G}$ as a union of the intervals corresponding to the edges, subject to the equivalence relation defined by the combinatorial structure of the graph which identifies those endpoints of the intervals which correspond to vertices to which the respective edges are incident. Similarly, we shall often identify the edges with the intervals they are isomorphic to.
Clearly, $(\mathcal{G},d)$ is a complete, separable metric space, and hence it is a Polish space. The Borel $\sigma$--algebra of $\mathcal{G}$ is denoted by $\mathcal{B}(\mathcal{G})$. We write $B_r(\xi)$ for the open ball with radius $r>0$ and center $\xi\in\mathcal{G}$.
For $l\in\mathcal{L}$, $l\ensuremath^\circ$ denotes the open interior of $l$, i.e., $l\ensuremath^\circ$ is the subset of $l$ being isomorphic to $(0,\rho_i)$ if $l=i\in\mathcal{I}$, and to $(0,+\infty)$ if $l\in\mathcal{E}$. We set $\ensuremath\cG\op = \mathcal{G}\setdif V$ to be the interior of $\mathcal{G}$, and hence $\mathcal{G}\ensuremath^\circ$ is the pairwise disjoint union of the open edges $l\ensuremath^\circ$, $l\in\mathcal{L}$. In particular, every $\xi\in \ensuremath\cG\op$ is in one-to-one correspondence with its \emph{local coordinate} $(l,x)$, $l\in\mathcal{L}$, $x\in I_l\ensuremath^\circ$, and we may and will write $\xi=(l,x)$.
Assume that $f$ is a real valued function on $\mathcal{G}$. Then $f$ is in one-to-one correspondence with the family of functions $(f_l,\,l\in\mathcal{L})$ where $f_l$ is the restriction of $f$ to the edge $l\in\mathcal{L}$. (Of course, if $v\in V$ is a vertex with which the edges $l$, $l'\in\mathcal{L}$ are incident, then we have to have $f_l(v)=f_{l'}(v)$.) Sometimes it will also be convenient to write $f_l(x)$ instead of $f(\xi)$, for $\xi\in\ensuremath\cG\op$ having local coordinates $(l,x)$.
The space of real valued, bounded measurable functions will be denoted by $B(\mathcal{G})$, while $C_0(\mathcal{G})$ denotes the space of continuous real valued functions on $\mathcal{G}$ which vanish at infinity. Both spaces are equipped with the sup-norm, denoted by
\mbox{$\|\cdot\|$}. \mbox{$(B(\mathcal{G}),\|\cdot\|)$} and \mbox{$(C_0(\mathcal{G}),\|\cdot\|)$} are Banach spaces, the latter being separable.
\subsection{Markov Processes on Metric Graphs} \label{ssect_2_2} Let $(\mathcal{G},d)$ be a metric graph as in the previous subsection. Furthermore let $(\Omega,\mathcal{A})$ be a measurable space, equipped with a family $(P_\xi,\, \xi\in\mathcal{G})$ of probability measures. The expectation with respect to $P_\xi$, $\xi\in\mathcal{G}$, will be denoted by $E_\xi(\,\cdot\,)$. We will say that a statement holds \emph{almost surely} (a.s.), if for all $\xi\in\mathcal{G}$ the statement holds almost surely with respect to $P_\xi$.
Let $\Delta$ be a point not in $\mathcal{G}$ which we will view as a cemetery point. By $\ensuremath \cG^\gD$ we denote the union $\mathcal{G}\cup\{\Delta\}$, where $\Delta$ is adjoined to $\mathcal{G}$ as an isolated point. We define the $\sigma$--algebra $\mathcal{B}(\ensuremath \cG^\gD)$ on $\ensuremath \cG^\gD$ as the obvious minimal extension of $\mathcal{B}(\mathcal{G})$. All real valued functions $f$ on $\mathcal{G}$ are understood as being extended to $\ensuremath \cG^\gD$ with $f(\Delta)=0$.
We consider a $\ensuremath \cG^\gD$--valued normal homogeneous Markov process $X = (X_t,\,t\ge 0)$ on $(\Omega,\mathcal{A})$ relative to a filtration $\mathcal{F}=(\mathcal{F}_t,\,t\ge 0)$ in $\mathcal{A}$ with c\`adl\`ag paths. Throughout, we will suppose that the filtration $\mathcal{F}$ is right continuous and complete for the family $(P_\xi,\, \xi\in\mathcal{G})$, that is, for all $t\ge 0$, $\mathcal{F}_t=\cap_{\epsilon>0}\mathcal{F}_{t+\epsilon}$, and $\mathcal{F}_0$ contains all subsets of $\Omega$ which are negligible for all $P_\xi$, $\xi\in \mathcal{G}$. Also $\Delta$ is a cemetery state for $X$, i.e., almost surely $X_s = \Delta$, $s\ge 0$, entails $X_t = \Delta$ for all $t\ge s$. The lifetime $\zeta$ of $X$ is defined by $\zeta = \inf\{t\ge0,\,X_t = \Delta\}$.
As in~\cite{ReYo91} we assume that the transition probabilities of $X$ are given in terms of a transition function $P = (P_t,\,t\ge 0)$, i.e., \begin{equation*}
P_\xi(X_t\in C) = P_t(\xi,C),\qquad \xi\in\mathcal{G},\,t\ge 0,\,C\in\mathcal{B}(\ensuremath \cG^\gD). \end{equation*} In particular, for all $t\ge 0$, $C\in\mathcal{B}(\ensuremath \cG^\gD)$, the mapping $\xi\mapsto P_\xi(X_t\in C)$ is measurable. In terms of the transition function $P$ the Markov property of $X$ can be written as follows: \begin{equation*}
P_\xi(X_t\in C\ensuremath\,\big|\, \mathcal{F}_s)
= P_{t-s}(\xi,C), \qquad 0\le s\le t,\,C\in\mathcal{B}(\ensuremath \cG^\gD),\,\xi\in\mathcal{G}. \end{equation*}
It will be convenient and there is no loss of generality to assume the existence of a shift operator $\theta: \mathbb{R}_+\times \Omega\rightarrow \Omega$, $(s,\omega)\mapsto \theta_s(\omega)$, so that a.s.\ for all $t$, $s\ge 0$, \begin{equation} \label{eq_2_1}
X_t \comp \theta_s = X_{t+s}. \end{equation}
Thus if $X$ is a strong Markov process with respect to $\mathcal{F}$, its strong Markov property can be expressed in the following way. Let $S$ be an $\mathcal{F}$--stopping time, and as is usual denote $\sigma$--algebra of the past of $S$ by $\mathcal{F}_S$. If $\xi\in\mathcal{G}$, and $Z$ is a positive or bounded random variable, then \begin{equation} \label{eq_2_2}
E_\xi\bigl(Z\comp \theta_S\ensuremath\,\big|\, \mathcal{F}_S\bigr) = E_{X_S}(Z), \end{equation} holds $P_\xi$--a.s.\ on the set $\{X_S\ne \Delta\} = \{S<\zeta\}$.
For a subset $A$ of $\mathcal{G}$ we shall denote its hitting time by $X$ by $H_A$, \begin{equation*}
H_A = \inf\{t>0,\,X_t\in A\} \end{equation*} and if $A=\{\xi\}$, $\xi\in\mathcal{G}$, we simply write $H_\xi$.
We shall occasionally take the liberty to write $X(t)$ for $X_t$, $t\in\mathbb{R}_+$, or $H(A)$ for $H_A$, $A\subset \mathcal{G}$, whenever it is typographically more convenient.
The semigroup $U=(U_t,\,t\ge 0)$ associated with $X$ and acting linearly on $B(\mathcal{G})$ is defined by \begin{equation} \label{eq_2_3}
U_t f(\xi) = E_\xi\bigl(f(X_t)\bigr) = \int_\mathcal{G} f(\eta)\,P_t(\xi,d\eta), \end{equation}
for $f\in B(\mathcal{G})$, $\xi\in\mathcal{G}$. The bound $\|U_t f\|\le \|f\|$ holds for all $f\in B(\mathcal{G})$, $t\ge 0$. The resolvent $R=(R_\lambda,\,\lambda>0)$ associated with $X$, and acting on $B(\mathcal{G})$, is defined by \begin{equation} \label{eq_2_4}
R_\lambda f(\xi) = \int_0^\infty e^{-\lambda t} U_t f(\xi)\,dt,\qquad \lambda>0, \end{equation} and satisfies \begin{equation} \label{eq_2_5}
\bigl\|R_\lambda f\bigr\| \le \frac{1}{\lambda}\,\|f\|. \end{equation} We shall denote the restrictions of the semigroup $U$ and the resolvent $R$ to $\ensuremath C_0(\cG)$ by the same symbols.
Assume that $X$ is a strong Markov process with respect to the filtration $\mathcal{F}$. A direct consequence of the strong Markov property is the \emph{first passage time formula} for the resolvent (e.g., \cite{Ra56} or \cite{ItMc74}): Let $\xi\in\mathcal{G}$, $f\in B(\mathcal{G})$, $\lambda>0$, and assume that $S$ is a $\mathcal{F}$--stopping time which is $P_\xi$--a.s.\ finite. Then \begin{equation} \label{eq_2_6}
R_\lambda f(\xi) = E_\xi\Bigl(\int_0^S e^{-\lambda t} f(X_t)\,dt\Bigr)
+ E_\xi\bigl(e^{-\lambda S}\,R_\lambda f(X_S)\bigr) \end{equation} holds true.
We shall often have occasion to use a standard Brownian family on the real line $\mathbb{R}$ as a family of reference processes: Let $(\Omega',\mathcal{A}')$ denote another measurable space with a family $(Q_x,\,x\in\mathbb{R})$ of probability measures, and for every $x\in\mathbb{R}$, $(B_t,\,t\in\mathbb{R}_+)$ a standard Brownian motion on $\mathbb{R}$ starting $Q_x$--a.s.\ in $x$. Expectations with respect to $Q_x$ will be denoted by $E^Q_x(\,\cdot\,)$. The Brownian family is equipped with a filtration denoted by $\mathcal{J}=(\mathcal{J}_t,\,t\ge 0)$, and throughout we assume --- as we may --- that $\mathcal{J}$ is right continuous and complete for the family $(Q_x,\,x\in\mathbb{R})$. (For example, we can always consider the natural filtration generated by $(B_t,\,t\in\mathbb{R}_+)$, and then choose its universal augmentation, e.g., \cite[Chapter~III.2]{ReYo91} or \cite[Chapter~2.7]{KaSh91}.) $H_A^B$ denotes the hitting time of the set $A\subset\mathbb{R}$ by $B$, and as above we simply write $H^B_x$ for $H^B_{\{x\}}$, and occasionally $B(t)$ for $B_t$, $H^B(A)$ for $H^B_A$, $A\subset\mathbb{R}$.
Consider an edge $l\in\mathcal{L}$, and the interval $I_l\subset \mathbb{R}_+$ that $l$ is isomorphic to. We set $\partial(I_l)=\{0,\rho_l\}$ if $l\in\mathcal{I}$, and $\partial(I_l)=\{0\}$ if $l\in\mathcal{E}$. For $t\in\mathbb{R}_+$ set \begin{equation*}
B_t^l = B\bigl(t\land H^B_{\partial(I_l)}\bigr) \end{equation*} where $s\land t = \min\{s,t\}$, $s$, $t\in\mathbb{R}_+$, i.e., $B^l$ is a Brownian motion on $\mathbb{R}$ with absorption in the endpoint(s) of $I_l$. Under the family $(Q_x,\,x\in I_l)$ we call this process the \emph{absorbed Brownian motion on $I_l$}.
\section{Definition of Brownian Motions on a Metric Graph} \label{sect_3} In analogy with~\cite[Chapter~6]{Kn81} we define a Brownian motion on a metric graph $\mathcal{G}$ as follows.
\begin{definition} \label{def_3_1} A \emph{Brownian motion on $\mathcal{G}$} is a normal strong Markov process $X=(X_t,\,t\ge 0)$ with state space $\ensuremath \cG^\gD$ and lifetime $\zeta$. The sample paths of $X$ are right continuous with left limits in $\mathcal{G}$, and they are continuous on $[0,\zeta)$. Furthermore, for every $\xi\in\ensuremath\cG\op$ with local coordinates $(l,x)$, $l\in\mathcal{L}$, $x\in I_l\ensuremath^\circ$, the process $X\ensuremath ^{\text{\itshape abs}} = \bigl(X(t\land H_V),\,t\ge 0\bigr)$, with start in $\xi$ and absorption in the set of vertices $V$, is equivalent to an absorbed Brownian motion on $I_l$ with start in $x$. \end{definition}
\begin{remark} \label{rem_3_2} According to our convention of subsection~\ref{ssect_2_2}, we consider a Brownian motion $X$ on $\mathcal{G}$ as a strong Markov process with respect to a filtration $\mathcal{F}=(\mathcal{F}_t,\,t\ge 0)$ which is right continuous and complete. \end{remark}
\begin{remark} \label{rem_3_2a} Suppose that $\xi\in l$, $l\in\mathcal{L}$, then $H_V = H_{\partial(l)}$, because paths starting at $\xi$ hit the set $\partial(l)$ before any other vertex due to the continuity assumption. \end{remark}
Consider a Brownian motion $X$ on $\mathcal{G}$. Let us state the last condition in definition~{\ref{def_3_1}} more explicitly. Fix $l\in\mathcal{L}$, and $\xi\in l\ensuremath^\circ$ with local coordinates $(l,x)$. Then for all $n\in\mathbb{N}$, $t_1$, \dots, $t_n\in\mathbb{R}_+$, with $t_1\le t_2\le\dotsb\le t_n$, and all $A_1$, \dots, $A_n$ in the Borel $\sigma$--algebra $\mathcal{B}(l)$ of~$l$, \begin{equation} \label{eq_3 1} \begin{split}
P_\xi\bigl(X_{t_1}\in A_1,&\dotsc,X_{t_n}\in A_n, t_n\le H_V\bigr)\\
&= Q_x\bigl(B_{t_1}\in A_1,
\dotsc,B_{t_n}\in A_n, t_n\le H^B_{\partial(I_l)}\bigr). \end{split} \end{equation} For simplicity we have identified the set $A_i \subset l$ with its isomorphic image in $I_l$. Observe that in particular under $P_\xi$, the stopping time $H_V = H_{\partial(l)}$ has the same law as $H^B_{\partial(I_l)}$ under $Q_x$, and especially we get that for all $\xi\in\mathcal{G}$, $P_\xi(H_V<+\infty)=1$.
It follows from definition~\ref{def_3_1} that any discontinuity of the paths of $X$ can only occur at the vertices of $\mathcal{G}$, and it consists in a jump to the cemetery state $\Delta$. Hence if the process starts in $\xi\in\ensuremath\cG\op$, it cannot reach the cemetery state $\Delta$ before hitting $V$. On the other hand, if the process starts in $v\in V$, $P_v$--a.s.\ it cannot jump right away to $\Delta$, because this would contradict the right continuity of the paths and the requirement $P_\xi(X_0=\xi)=1$ for all $\xi\in \mathcal{G}$. In particular, we have for all $\xi\in\mathcal{G}$, $P_\xi(\zeta\ge H_V)=1$.
For the following discussion we assume that the process $X$ starts at a vertex $v\in V$, and consider the exit time from $v$, i.e., the stopping time $S_v = H(\ensuremath\cG\op)$. It is well known (e.g., \cite{Kn81, ReYo91, DyJu69}) that because of the strong Markov property of $X$, $S_v$ is under $P_v$ exponentially distributed with a rate $\beta_v\in[0,+\infty]$. Thus there are three possibilities:
\noindent \emph{Case $\beta_v=0$}: In this case the process stays at $v$ forever, i.e., $v$ is a \emph{trap}, and the process is given by $(X(t\land H_v),\,t\in\mathbb{R}_+)$.
\noindent \emph{Case $0<\beta_v<+\infty$}: In this case the process stays at $v$ $P_v$--a.s.\ for a strictly positive, finite moment of time, i.e., $v$ is \emph{exponentially holding}. It is well known (cf., e.g., \cite[p.~154]{Kn81}, \cite[p.~104, Prop.~3.13]{ReYo91}) that then the process has to leave $v$ by a jump, and by our assumption of path continuity on $[0,\eta)$, the process has to jump to the cemetery $\Delta$.
\noindent \emph{Case $\beta_v=+\infty$}: In this case the $X$ leaves the vertex $v$ immediately, and it begins a Brownian excursion into one of the edges incident with the vertex $v$.
\section{Feller Property} \label{sect_4} In this section we prove that the semigroup $U$ associated with every Brownian motion on $\mathcal{G}$ has the Feller property (see, e.g., \cite{Kn81, ReYo91}, or definition~\ref{def_A_1} in appendix~\ref{app_A}).
We begin with the following simple lemma.
\begin{lemma} \label{lem_4_1} Assume that $\xi\in\ensuremath\cG\op$. Then $P_\xi$--a.s.\ $H_\eta$ converges to zero, whenever $\eta$ increases or decreases to $\xi$ along the edge to which $\xi$ belongs. \end{lemma}
\begin{proof} First we notice that because up to time $\zeta$ the paths of $X$ are continuous, $\eta\mapsto H_\eta$ is $P_\xi$--a.s.\ monotone decreasing as $\eta$ increases or decreases to $\xi$. Therefore it is enough to show that $H_\eta$ converges to zero in $P_\xi$--probability.
Let $\xi\in l\ensuremath^\circ$, $l\in\mathcal{L}$, with local coordinates $(l,x)$, $x\in I_l\ensuremath^\circ$. Fix $\epsilon>0$ small enough so that $(l,x\pm \epsilon)\in l\ensuremath^\circ$. Without loss of generality we may assume that $d(\xi,\eta)<\epsilon$. We consider first the case where $\eta\ensuremath\downarrow \xi$, i.e., for the local coordinates $(l,y)$, $y\in I_l$, of $\eta$ we have $y\ensuremath\downarrow x$. Let $\delta>0$, and write \begin{equation} \label{eq_4_1}
P_\xi(H_\eta>\delta)
= P_\xi\bigl(H_\eta >\delta, H_{(l,x-\epsilon)}\ge H_\eta\bigr)
+ P_\xi\bigl(H_\eta >\delta, H_{(l,x-\epsilon)}< H_\eta\bigr). \end{equation} We estimate the second probability on the right hand side from above by \begin{equation*} P_\xi\bigl(H_{(l,x-\epsilon)}< H_\eta\bigr). \end{equation*} But this is the probability of the event that the process leaves the set on $l$ which in local coordinates is the interval $(l, [x-\epsilon,y])$ at the end point with local coordinates $(l,x-\epsilon)$. Therefore this is an event which happens before the process hits a vertex, and hence this probability is equal to the corresponding one for a standard Brownian motion (e.g., \cite[Problem~6, p.~29]{ItMc74}): \begin{equation*}
P_\xi\bigl(H_{(l,x-\epsilon)}< H_\eta\bigl) = \frac{y-x}{y-x+\epsilon}, \end{equation*} which converges to zero as $\eta\ensuremath\downarrow \xi$. Similarly, the first probability on the right hand side of equation~\eqref{eq_4_1} is equal to \begin{align*}
Q_x\bigl(H^B_y>\delta, H^B_{x-\epsilon}\ge H^B_y\bigr)
&\le Q_x\bigl(H^B_y\ge \delta\bigr)\\[1ex]
&= \int_\delta^\infty \frac{y-x}{\sqrt{2\pi t^3}}\,e^{-(y-x)^2/2t}\,dt\\[1ex]
&= \mathop{\operator@font erf}\nolimits\Bigl(\frac{y-x}{\sqrt{2\delta}}\Bigr), \end{align*} where we used the well-known density of $H^B_y$ under $Q_x$, e.g., \cite[p.~292]{Sc15}, \cite[section~1.7]{ItMc74}, or \cite[Proposition~2.6.19]{KaSh91}. Clearly, the last expression converges to zero as $y\ensuremath\downarrow x$, i.e., as $\eta\ensuremath\downarrow \xi$.
The case $\eta\ensuremath\uparrow \xi$ is treated in an analogous way. \end{proof}
\begin{lemma} \label{lem_4_2} Let $\lambda>0$, $v\in V$, and suppose that $l\in \mathcal{L}(v)$. Then \begin{equation} \label{eq_4_2}
\lim_{\eta\to v,\, \eta\in l} E_\eta\bigl(e^{-\lambda H_v}\bigr) = 1. \end{equation} \end{lemma}
\begin{proof} Fix $\epsilon>0$ in such a way that we have for every $v'\in V$, $v'\ne v$, $d(v,v')>\epsilon$. We may assume that $d(v,\eta)<\epsilon$. Set \begin{equation} \label{eq_4_3}
H_{v,\epsilon} = H\bigl(B_\epsilon(v)^c\bigr), \end{equation} where the superscript $c$ denotes the complement of a set. Write \begin{equation} \label{eq_4_4}
1-E_\eta\bigl(e^{-\lambda H_v}\bigr)
= E_\eta\bigl(1-e^{-\lambda H_v}; H_v \le H_{v,\epsilon}\bigr)
+ E_\eta\bigl(1-e^{-\lambda H_v}; H_v > H_{v,\epsilon}\bigr), \end{equation} with the notation \begin{equation}
E_\eta(Z;C) = E_\eta(Z\,1_C) \end{equation} for positive or $P_\eta$--integrable random variables $Z$, and $C\in\mathcal{A}$. Consider the case where the vertex $v$ corresponds to the point with local coordinates $(l,0)$, the case where $v$ corresponds to $(l,\rho_l)$ can be dealt with by an analogous argument. Let $\eta$ have local coordinates $(l,y)$, $0\le y<\epsilon$. The second term on the right hand side of equation~\eqref{eq_4_4} is less or equal to \begin{align*}
P_\eta(H_v>H_{v,\epsilon})
&= Q_y(H_0^B > H_\epsilon^B)\\
&= \frac{y}{\epsilon}, \end{align*} which converges to zero with $y\ensuremath\downarrow 0$, i.e., with $\eta\to v$. On the other hand \begin{align*}
E_\eta\bigl(1-e^{-\lambda H_v}; H_v \le H_{v,\epsilon}\bigr)
&= E^Q_y\bigl(1-e^{-\lambda H_0^B}; H_0^B \le H_\epsilon^B\bigr)\\
&\le E^Q_y\bigl(1-e^{-\lambda H_0^B}\bigr)\\
&= 1-e^{-\ensuremath\sqrt{2\gl} y}, \end{align*} where we used the well-known formula for the Laplace transform of the density of $H_0^B$ under $Q_y$ (e.g., \cite[p.~26, eq.~5]{ItMc74}). Obviously this converges to zero as $y\ensuremath\downarrow 0$, i.e., as $\eta\to v$. \end{proof}
\begin{theorem} \label{thm_4_3} Every Brownian motion on $\mathcal{G}$ is a Feller process. \end{theorem}
\begin{proof} The proof is based on the first passage time formula~\eqref{eq_2_6}. By theorem~\ref{thm_A_3} in appendix~\ref{app_A} it suffices to show that for all $\lambda>0$, $R_\lambda$ maps $\ensuremath C_0(\cG)$ into itself, and that for all $f\in \ensuremath C_0(\cG)$, $\xi\in\mathcal{G}$, $U_t f(\xi)$ converges to $f(\xi)$ as $t\downarrow 0$.
First we show that for every $\lambda>0$, $R_\lambda$ maps $\ensuremath C_0(\cG)$ into itself. Assume that $f\in \ensuremath C_0(\cG)$. Consider the case $\xi\in\ensuremath\cG\op$. Then it follows from lemma~\ref{lem_4_1} as in~\cite[Section~3.6]{ItMc74} that $R_\lambda f$ is continuous at $\xi$. Consider now the case $\xi=v\in V$, let $l$ belong to the set $\mathcal{L}(v)$ of edges incident with $v$, and let $\eta\in l$. Note that $P_\eta$--a.s.\ $H_v$ is finite (cf.\ section~\ref{sect_3}). Therefore we can employ equation~\eqref{eq_2_6} with $S=H_v$: \begin{equation*}
R_\lambda f(\eta) = E_\eta\Bigl(\int_0^{H_v} e^{-\lambda t} f(X_t)\,dt\Bigr)
+ E_\eta\bigl(e^{-\lambda H_v}\bigr)\,R_\lambda f(v). \end{equation*} Using~\eqref{eq_2_5} we estimate in the following way \begin{align*}
\bigl|R_\lambda f(\eta) - R_\lambda f(v)\bigr|
&\le \Bigl|E_\eta\Bigl(\int_0^{H_v} e^{-\lambda t} f(X_t)\,dt\Bigr)\Bigr|\\
&\hspace{4em} +\Bigl(1-E_\eta\bigl(e^{-\lambda H_v}\bigr)\Bigr)\,\bigl|R_\lambda f(v)\bigr|\\
&\le \frac{2}{\lambda}\,\|f\|\,\Bigl(1-E_\eta\bigl(e^{-\lambda H_v}\bigr)\Bigr). \end{align*} By lemma~\ref{lem_4_2} this term converges to zero as $\eta$ converges to $v$ along $l$. Therefore $R_\lambda f$ is also continuous at $v$.
Next we prove that for all $\lambda>0$, $f\in \ensuremath C_0(\cG)$, $R_\lambda f$ vanishes at infinity. If $\mathcal{G}$ has no external edges there is nothing to prove, and so we assume that $e\in\mathcal{E}$ is an external edge of $\mathcal{G}$ which is incident with the vertex $v\in V$,
$\partial(e)=\{v\}$. Let $\lambda$, $\epsilon>0$ be given. We choose $r_1\ge 0$ large enough so that for all $\xi\in e$ with $d(v,\xi)>r_1$ we have $|f(\xi)|< \epsilon\lambda/2$. Choose $r_2>r_1$, and consider $\xi\in e$ with $d(v,\xi)\ge r_2$. Denote by $\xi_1$ the point on $e$ which has distance $r_1$ to $v$. Then we have that $P_\xi$--a.s., $H_{\xi_1}\le H_v$, and consequently $P_\xi(H_{\xi_1}<+\infty)=1$. Hence we can use the first passage time formula~\eqref{eq_2_6} in the form \begin{equation*}
R_\lambda f(\xi) = E_\xi\Bigl(\int_0^{H_{\xi_1}} e^{-\lambda t} f(X_t)\,dt\Bigr)
+E_\xi\bigl(e^{-\lambda H_{\xi_1}}\bigr)\,R_\lambda f(\xi_1). \end{equation*} For $t\in [0,H_{\xi_1}]$ we have $d(v,X_t)\ge r_1$, and therefore the absolute value of the first term on the right hand side is bounded from above by $\epsilon/2$. For the second term we can compute the expectation as for the corresponding expression of the standard Brownian motion $B$ on $\mathbb{R}$, and we obtain (again with~\eqref{eq_2_5}) \begin{align*}
\Bigl|E_\xi\bigl(e^{-\lambda H_{\xi_1}}\bigr)\,R_\lambda f(\xi_1)\Bigr|
&= e^{-\ensuremath\sqrt{2\gl}\,d(\xi,\xi_1)}\,\bigl|R_\lambda f(\xi_1)\bigr|\\
&\le e^{-\ensuremath\sqrt{2\gl}\,(r_2-r_1)}\,\frac{\|f\|}{\lambda}. \end{align*} Now choose $r_2$ large enough so as to make the last term less than $\epsilon/2$, and we are done. Thus we have shown that for every $\lambda>0$, $R_\lambda$ maps $\ensuremath C_0(\cG)$ into itself.
Finally, consider for $f\in \ensuremath C_0(\cG)$, $\xi\in \mathcal{G}$, $t>0$, \begin{equation*}
U_t f(\xi) = E_\xi\bigl(f(X_t)\bigr). \end{equation*} By definition, $X$ has right continuous sample paths, and $P_\xi(X_0=\xi)=1$. Since $f$ is continuous and bounded, an application of the dominated convergence theorem shows that $U_t f(\xi)$ converges to $f(\xi)$ as $t$ decreases to $0$. \end{proof}
\section{Generators and Feller's Theorem} \label{sect_5} Let $V_\mathcal{L}$ denote the subset of $V\times\mathcal{L}$ given by \begin{equation*}
V_\mathcal{L} = \bigl\{(v,l),\,v\in V \text{ and }l\in\mathcal{L}(v)\bigr\}. \end{equation*} We shall also write $v_l$ for $(v,l)\in V_\mathcal{L}$. We remark in passing that \begin{equation*}
\bigl|V_\mathcal{L}\bigr| = |\mathcal{E}| + 2\,|\mathcal{I}|. \end{equation*}
Consider a real valued function $f$ on $\mathcal{G}$, let $v\in V$ and let $l\in\mathcal{L}(v)$ be an edge incident with $v$. We define the \emph{directional derivative of $f$ at $v$ in direction $l\in\mathcal{L}(v)$} as follows: \begin{equation} \label{eq5i}
f'(v_l) = \begin{cases}
\displaystyle
\phantom{-}\lim_{\xi\to v,\,\xi\in l\ensuremath^\circ} f'(\xi),
& \text{if $v$ is an initial vertex of $l$},\\[2ex]
\displaystyle
-\lim_{\xi\to v,\,\xi\in l\ensuremath^\circ} f'(\xi),
& \text{if $v$ is a final vertex of $l$},
\end{cases} \end{equation} whenever the corresponding limit on the right hand side exists. Geometrically this directional derivative is just the inward normal derivative which makes it an intrinsic definition, independent of the orientation chosen on the edge.
\begin{definition} \label{def_5_1} $\ensuremath C_0^{0,2}(\cG)$ denotes the subspace of functions $f$ in $\ensuremath C_0(\cG)$ which are twice continuously differentiable on $\ensuremath\cG\op$, and such that for every $v\in V$ and all $l\in\mathcal{L}$ the limit \begin{equation} \label{eq5ia}
f''(v_l) = \lim_{\xi\to v,\, \xi\in l\ensuremath^\circ} f''(\xi) \end{equation} exists. $\ensuremath C_0^2(\cG)$ denotes the subspace of those functions $f$ in $\ensuremath C_0^{0,2}(\cG)$ so that $f''$ extends from $\ensuremath\cG\op$ to a continuous function on $\mathcal{G}$. \end{definition}
Thus $\ensuremath C_0^2(\cG)$ consists of all $f\in\ensuremath C_0^{0,2}(\cG)$ so that for every $v\in V$, the $f''(v_l)$, $l\in\mathcal{L}(v)$, are all equal. Assume that $f\in\ensuremath C_0^2(\cG)$, and let $v\in V$. The continuous extension of $f''$ to $v$ will simply be denoted by $f''(v)$. Consider an edge $l\in\mathcal{L}(v)$ incident with $v$. Then it is easy to see that $f'(v_l)$ exists (and is finite). On the other hand, in general for $l$, $l'\in \mathcal{L}(v)$, $l\ne l'$, we have $f'(v_l)\ne f'(v_{l'})$. In other words, in general $f'$ does \emph{not} have a continuous extension from $\ensuremath\cG\op$ to $\mathcal{G}$. Also, it is not hard to check that $f'$ vanishes at infinity.
The proof of the following lemma can be taken over with minor modifications from the standard literature, e.g., from~\cite[Chapter~6.1]{Kn81}.
\begin{lemma} \label{lem_5_2} For every Brownian motion $X$ on the metric graph $\mathcal{G}$, the generator $A$ of its semigroup $U$ acting on $\ensuremath C_0(\cG)$ has a domain $\mathcal{D}(A)$ contained in $\ensuremath C_0^2(\cG)$. Moreover, for every $f\in\mathcal{D}(A)$, $A f=1/2\,f''$. \end{lemma}
Consider data of the following form \begin{equation} \label{eq5ii} \begin{split}
a &= (a_v,\,v\in V)\in [0,1)^V\\
b &= (b_{v_l},\,v_l\in V_\mathcal{L}) \in [0,1]^{V_\mathcal{L}}\\
c &= (c_v,\,v\in V)\in [0,1]^V \end{split} \end{equation} subject to the condition \begin{equation} \label{eq5iii}
a_v + \sum_{l\in \mathcal{L}(v)} b_{v_l} + c_v =1,\qquad \text{for every $v\in V$}. \end{equation} We define a subspace $\mathcal{H}_{a,b,c}$ of $\ensuremath C_0^2(\cG)$ as the space of those functions $f$ in $\ensuremath C_0^2(\cG)$ which at every vertex $v\in V$ satisfy the Wentzell boundary condition \begin{equation}\label{eq5iv}
a_v f(v) - \sum_{l\in\mathcal{L}(v)} b_{v_l}f'(v_l)+ \frac{1}{2}\,c_v f''(v)=0. \end{equation}
Now we can state and prove the analogue of \emph{Feller's theorem} for metric graphs.
\begin{theorem} \label{thm_5_3} Suppose that $X$ is a Brownian motion on a metric graph $\mathcal{G}$, and that $\mathcal{D}(A)$ is the domain of the generator $A$ of its semigroup. Then there are $a$, $b$, $c$ as in~\eqref{eq5ii}, \eqref{eq5iii}, so that $\mathcal{D}(A)=\mathcal{H}_{a,b,c}$. \end{theorem}
\begin{remark} \label{rem_5_4} The case $a_v=1$, $v\in V$, would correspond to (zero) Dirichlet boundary conditions at the vertex $v$. The paths of the process associated with this boundary condition have to jump instantaneously to $\Delta$ when reaching the vertex, and by our requirement that the paths are right continuous this means that the process will never be at the vertex. But this is in contradiction to our assumption (cf.\ definition~\ref{def_3_1}) that the process with absorption at the vertex is equivalent to a Brownian motion with absorption in the endpoint (endpoints, resp.) of the corresponding interval. Therefore this stochastic process is \emph{not} a Brownian motion on $\mathcal{G}$ in the sense of definition~\ref{def_3_1}, and this case has to be excluded from our discussion. Also note that in this case the semigroup does not act strongly continuously on $\ensuremath C_0(\cG)$, and therefore is in particular not Feller. \end{remark}
The proof of theorem~\ref{thm_5_3} has two rather distinct parts, and therefore we split it by proving the following two lemmas:
\begin{lemma} \label{lem_5_5} Suppose that $X$ is a Brownian motion on a metric graph $\mathcal{G}$, and that $\mathcal{D}(A)$ is the domain of the generator $A$ of its semigroup. Then there are $a$, $b$, $c$ as in~\eqref{eq5ii}, \eqref{eq5iii}, so that $\mathcal{D}(A)\subset\mathcal{H}_{a,b,c}$. \end{lemma}
\begin{lemma} \label{lem_5_6} Suppose that $A$ is the generator of a Brownian motion $X$ on $\mathcal{G}$ with domain $\mathcal{D}(A)\subset \mathcal{H}_{a,b,c}$ for some $a$, $b$, $c$ as in~\eqref{eq5ii}, \eqref{eq5iii}. Then $\mathcal{D}(A)=\mathcal{H}_{a,b,c}$ \end{lemma}
\begin{proof}[Proof of lemma~\ref{lem_5_5}] Our proof follows the one in~\cite[Chapter~6.1]{Kn81} quite closely --- actually, it is sufficient to consider a special case of the proof given there.
We show that for every vertex $v\in V$ there are constants $a_v\in [0,1)$, $b_{v_l}\in [0,1]$, $l\in\mathcal{L}(v)$, $c_v\in [0,1]$ satisfying~\eqref{eq5iii}, and such that all $f$ in the domain $\mathcal{D}(A)$ of the generator satisfy the boundary condition~\eqref{eq5iv}. To this end, we let $f\in\mathcal{D}(A)$, fix a vertex $v\in V$, and compute $A f(v)$. Let us consider the three cases for $\beta$ mentioned in section~\ref{sect_3}.
If $\beta=0$, $v$ is a trap, and $U_t f(v) = f(v)$ for all $t\ge 0$. Consequently, $A f(v)=0$, and therefore $1/2\, f''(v)=0$. Thus $f$ satisfies the boundary condition~\eqref{eq5iv} at $v$ with $a_v=0$, $c_v=1$, and $b_{v_l}=0$ for all $l\in\mathcal{L}(v)$.
Next we consider the case where $\beta\in (0,+\infty)$, i.e., $v$ is exponentially holding. We know from the discussion in section~\ref{sect_3} that then after expiration of the holding time the process jumps directly to the cemetery state. Therefore we get for $t>0$, $U_t f(v) = \exp(-\beta t) f(v)$, and thus $A f(v) + \beta f(v) = 0$, and the boundary condition~\eqref{eq5iv} holds for the choice \begin{equation} \label{eq_5_2}
a_v = \frac{\beta}{1+\beta},\quad c_v=\frac{1}{1+\beta},\quad
b_{v_l} = 0,\,l\in\mathcal{L}(v). \end{equation}
Finally we consider the case that $\beta=+\infty$, i.e., the process leaves $v$ immediately, and in particular, $v$ is not a trap. Therefore we may compute $A f(v)$ in Dynkin's form, e.g., \cite[p.~140, ff.]{Dy65a}, \cite[p.~99]{ItMc74}. As in~\eqref{eq_4_3} we let $H_{v,\epsilon}$ denote the hitting time of the complement of $B_\epsilon(v)$. Then \begin{equation} \label{eq_5_3}
A f(v) = \ensuremath\lim_{\gep\da 0} \frac{E_v\Bigl(f\bigl(X(H_{v,\epsilon})\bigr)\Bigr)
-f(v)}{E_v(H_{v,\epsilon})}. \end{equation} Recall the notation $f_l(\epsilon)$ for $f(\xi)$ with $\xi\in\mathcal{G}$ having local coordinates $(l,\epsilon)$, $l\in\mathcal{L}$, $\epsilon\in I_l$. Then we get \begin{align*}
E_v\Bigl(f\bigl(X(H_{v,\epsilon})\bigr)\Bigr)
&= \sum_{l\in\mathcal{L}(v)} f_l(\epsilon)\,P_v\bigl(X(H_{v,\epsilon})\in l\bigr)
+ f(\Delta)\,P_v\bigl(X(H_{v,\epsilon})=\Delta\bigr)\nonumber\\
&= \sum_{l\in\mathcal{L}(v)} f_l(\epsilon)\,P_v\bigl(X(H_{v,\epsilon})\in l\bigr), \end{align*} where the last equality follows from $f(\Delta)=0$. Let us denote \begin{align*}
r_l(\epsilon) &= \frac{P_v\bigl(X(H_{v,\epsilon})\in l\bigr)}{E_v(H_{v,\epsilon})},
\qquad l\in\mathcal{L}(v),\\[1ex]
r_\Delta(\epsilon) &= \frac{P_v\bigl(X(H_{v,\epsilon})=\Delta\bigr)}{E_v(H_{v,\epsilon})},\\[1ex]
K(\epsilon) &= 1 + r_\Delta(\epsilon) + \epsilon \sum_{l\in\mathcal{L}(v)} r_l(\epsilon). \end{align*} The continuity of the paths of $X$ up to the lifetime $\zeta$ yields \begin{equation*}
\sum_{l\in\mathcal{L}(v)} P_v\bigl(X(H_{v,\epsilon})\in l\bigr)
+ P_v\bigl(X(H_{v,\epsilon})=\Delta\bigr)=1, \end{equation*} and therefore equation~\eqref{eq_5_3} can be rewritten as \begin{equation*}
\ensuremath\lim_{\gep\da 0}\Bigl(A f(v) + r_\Delta(\epsilon) f(v)
- \sum_{l\in\mathcal{L}(v)} r_l(\epsilon) \bigl(f_l(\epsilon)-f(v)\bigr)\Bigr)=0. \end{equation*} Since for all $\epsilon>0$, $K(\epsilon)^{-1}\le 1$, it follows that \begin{equation*}
\ensuremath\lim_{\gep\da 0}\Bigl(\frac{1}{K(\epsilon)}\,A f(v) + \frac{r_\Delta(\epsilon)}{K(\epsilon)}\,f(v)
- \sum_{l\in\mathcal{L}(v)} \frac{\epsilon\, r_l(\epsilon)}{K(\epsilon)}\,
\frac{f_l(\epsilon)-f(v)}{\epsilon}\Bigr)=0, \end{equation*} which by lemma~\ref{lem_5_2} we may rewrite as \begin{equation*}
\ensuremath\lim_{\gep\da 0}\Bigl(a_v(\epsilon) f(v) + \frac{1}{2}\,c_v(\epsilon) f''(v)
- \sum_{l\in\mathcal{L}(v)} b_{v_l}(\epsilon)\,\frac{f_l(\epsilon)-f(v)}{\epsilon}\Bigr)=0, \end{equation*} where we have introduced the non-negative quantities \begin{align*}
a_v(\epsilon) &= \frac{r_\Delta(\epsilon)}{K(\epsilon)},\\
c_v(\epsilon) &= \frac{1}{K(\epsilon)},\\
b_{v_l}(\epsilon) &= \frac{\epsilon\, r_l(\epsilon)}{K(\epsilon)},\qquad l\in\mathcal{L}(v). \end{align*} Observe that for every $\epsilon>0$, \begin{equation*}
a_v(\epsilon) + c_v(\epsilon) + \sum_{l\in\mathcal{L}(v)} b_{v_l}(\epsilon) =1. \end{equation*} Therefore every sequence $(\epsilon_n,\,n\in\mathbb{N})$ with $\epsilon_n>0$ and $\epsilon_n\ensuremath\downarrow 0$ has a subsequence so that $a_v(\epsilon)$, $c_v(\epsilon)$ and $b_{v_l}(\epsilon)$, $l\in\mathcal{L}(v)$, converge along this subsequence to numbers $a_v$, $c_v$, and $b_{v_l}$ respectively in $[0,1]$, and the relation~\eqref{eq5iii} holds true. From the remark after definition~\ref{def_5_1} it follows that \begin{equation*}
\frac{f_l(\epsilon)-f(v)}{\epsilon} \end{equation*} converges with $\epsilon\ensuremath\downarrow 0$ to $f'(v_l)$, and therefore we obtain that for every vertex $v\in V$, $f\in\mathcal{D}(A)$ satisfies the boundary condition~\eqref{eq5iv} with data $a$, $b$, $c$ as in~\eqref{eq5ii}, \eqref{eq5iii}. \end{proof}
Before we can prove lemma~\ref{lem_5_6} we have to introduce some additional formalism.
For given data $a$, $b$, $c$ as in~\eqref{eq5ii}, \eqref{eq5iii}, it will be convenient to consider $\mathcal{H}_{a,b,c}$ equivalently as being the subspace of $\ensuremath C_0^{0,2}(\cG)$ so that for its elements $f$ at every $v\in V$ the boundary conditions~\eqref{eq5iv} as well as the boundary condition \begin{equation} \label{eq5iva}
f''(v_l) = f''(v_k),\qquad \text{for all $l,\,k\in\mathcal{L}(v)$} \end{equation} hold true. Relation~\eqref{eq5iva} is just another way to express that $f''$ is continuous on $\mathcal{G}$.
We consider the sets $V$, $\mathcal{E}$, and $\mathcal{I}$ as being ordered in some arbitrary way. With the convention that in $\mathcal{L}$ the elements of $\mathcal{E}$ come first this induces also an order relation on $\mathcal{L}$.
Suppose that $f\in\ensuremath C_0^{0,2}(\cG)$. With the given ordering of $\mathcal{E}$ and $\mathcal{I}$ we define the following column vectors of length $|\mathcal{E}|+2|\mathcal{I}|$: \begin{align*}
f(V) &= \Bigl(\bigl(f_e(0),\,e\in\mathcal{E}\bigr),\bigl(f_i(0),\,i\in\mathcal{I}\bigr),
\bigl(f_i(\rho_i)\,i\in\mathcal{I}\bigr)\Bigr)^t,\\
f'(V) &= \Bigl(\bigl(f'_e(0),\,e\in\mathcal{E}\bigr),\bigl(f'_i(0),\,i\in\mathcal{I}\bigr),
\bigl(-f'_i(\rho_i)\,i\in\mathcal{I}\bigr)\Bigr)^t,\\
f''(V) &= \Bigl(\bigl(f''_e(0),\,e\in\mathcal{E}\bigr),\bigl(f''_i(0),\,i\in\mathcal{I}\bigr),
\bigl(f''_i(\rho_i)\,i\in\mathcal{I}\bigr)\Bigr)^t, \end{align*} where the superscript ``$t$'' indicates transposition.
We want to write the boundary conditions~\eqref{eq5iv}, \eqref{eq5iva} in a compact way, and to this end we introduce the following order relation on $V_\mathcal{L}$: For $v_l$, $v'_{l'}\in V_\mathcal{L}$ we set $v_l \preceq v'_{l'}$ if and only if $v \prec v'$ or $v = v'$ and $l\preceq l'$ (where for $V$ and $\mathcal{L}$ we use the order relations introduced above). For $f$ as above set \begin{align*}
\tilde f(V) &= \bigl(f(v_l),\,v_l\in V_\mathcal{L}\bigr)^t,\\
\tilde f'(V) &= \bigl(f'(v_l),\,v_l\in V_\mathcal{L}\bigr)^t,\\
\tilde f''(V) &= \bigl(f''(v_l),\,v_l\in V_\mathcal{L}\bigr)^t. \end{align*} Then there exists a permutation matrix $P$ so that \begin{equation*}
\tilde f(V) = P f(V),\qquad \tilde f'(V) = P f'V),\qquad \tilde f''(V) = P f''(V). \end{equation*} In particular, $P$ is an orthogonal matrix which has in every row and in every column exactly one entry equal to one while all other entries are zero.
For every $v\in V$ we define the following $|\mathcal{L}(v)|\times|\mathcal{L}(v)|$ matrices: \begin{align*}
\tilde A(v) &= \begin{pmatrix}
a_v & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 0
\end{pmatrix},\\[2ex]
\tilde B(v) &= \begin{pmatrix}
-b_{v_{l_1}} & -b_{v_{l_2}} & -b_{v_{l_3}} & \cdots & -b_{v_{l_{|\mathcal{L}(v)|}}}\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 0
\end{pmatrix},\\[2ex]
\tilde C(v) &= \begin{pmatrix}
1/2\,c_v & 0 & 0 & 0 & \cdots & 0 \\
1 & -1 & 0 & 0 & \cdots & 0 \\
0 & 1 & -1 & 0 & \cdots & 0 \\
0 & 0 & 1 & -1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & -1
\end{pmatrix}, \end{align*}
where we have labeled the elements in $\mathcal{L}(v)$ in such a way that in the above defined ordering we have $l_1\prec l_2 \prec \dotsb \prec l_{|\mathcal{L}(v)|}$. Observe that $\tilde C(v)$ is invertible if and only if $c_v\ne 0$. Define block matrices $\tilde A$, $\tilde B$, and $\tilde C$ by \begin{equation*}
\tilde A = \bigoplus_{v\in V} A(v),\quad \tilde B = \bigoplus_{v\in V} B(v),
\quad \tilde C = \bigoplus_{v\in V} C(v). \end{equation*} Then we can write the boundary conditions~\eqref{eq5iv}, \eqref{eq5iva} simultaneously for all vertices as \begin{equation} \label{bch}
\tilde A \tilde f(V) + \tilde B \tilde f'(V) + \tilde C \tilde f''(V) = 0. \end{equation} Consequently the boundary conditions can equivalently be written in the form \begin{equation} \label{bc}
Af(V) + Bf'(V) + Cf''(V) = 0, \end{equation} with \begin{equation}
A = P^{-1}\tilde A P,\quad B = P^{-1}\tilde B P,\quad C = P^{-1}\tilde C P. \end{equation}
We bring in the following two matrix-valued functions on the complex plane \begin{equation} \label{Zpm}
\hat Z_\pm (\kappa) = A \pm \kappa B + \kappa^2 C,\qquad \kappa\in\mathbb{C}. \end{equation}
\begin{lemma} \label{lem_5_7}
There exists $R>0$ so that for all $\kappa\in\mathbb{C}$ with $|\kappa|\ge R$ the matrices $\hat Z_\pm(\kappa)$ are invertible, and there are constants $C$, $p>0$ so that \begin{equation} \label{invnorm}
\|\hat Z_\pm(\kappa)^{-1}\|\le C\,|\kappa|^p ,\qquad |\kappa|\ge R. \end{equation} \end{lemma}
\begin{remark} \label{rem_5_8} The bound~\eqref{invnorm} is actually rather crude, but sufficient for our purposes. \end{remark}
\begin{proof}[Proof of lemma~\ref{lem_5_7}] Since we have \begin{equation} \label{ZP}
\hat Z_{\pm}(\kappa) = P^{-1}\bigl(\tilde A \pm\kappa \tilde B + \kappa^2\tilde C\bigr) P \end{equation} for an orthogonal matrix $P$, for the proof of the first statement it suffices to show that there exists $R>0$ such that \begin{equation*}
\tilde A \pm\kappa \tilde B + \kappa^2\tilde C \end{equation*} are invertible for complex $\kappa$ outside of the open ball of radius $R$. For this in turn it suffices to show that for every vertex $v\in V$ the matrices \begin{equation*} \begin{split}
\tilde A(v) &\pm\kappa \tilde B(v) + \kappa^2\tilde C(v)\\
&= \begin{pmatrix}
a_v \pm \kappa b_{v_{l_1}}+ \kappa^2/2\,c_v & \pm\kappa b_{v_{l_2}} & \pm\kappa b_{v_{l_3}}
& \pm\kappa b_{v_{l_4}} & \cdots & \pm\kappa b_{v_{l_{|\mathcal{L}(v)|}}} \\
\kappa^2 & -\kappa^2 & 0 & 0 & \cdots & 0 \\
0 & \kappa^2 & -\kappa^2 & 0 & \cdots & 0 \\
0 & 0 & \kappa^2 & -\kappa^2 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & -\kappa^2
\end{pmatrix} \end{split} \end{equation*}
are invertible for all $\kappa\in\mathbb{C}$ with $|\kappa|\ge R$. An elementary calculation gives \begin{equation*}
\det\bigl(\tilde A(v) \pm\kappa \tilde B(v) + \kappa^2\tilde C(v)\bigr)
= \Bigl(a_v \pm \kappa \sum_{l\in \mathcal{L}(v)}b_{v_l} + \frac{\kappa^2}{2}\,c_v\Bigr)
\bigl(-\kappa^2\bigr)^{|\mathcal{L}(v)|-1}. \end{equation*}
The choices $\kappa=\pm 1$ together with condition~\eqref{eq5iii} show that the polynomial of second order in $\kappa$ in the first factor on the right hand side does not vanish identically. Therefore, it is non-zero in the exterior of an open ball with some radius $R_v>0$. Hence, we obtain the first statement for the choice $R = \max_{v\in V} R_v$. Moreover, from the calculation of the determinants above we also get for every $v\in V$ and all $\kappa\in\mathbb{C}$ with $|\kappa|\ge R$ an estimate of the form \begin{equation} \label{detinv}
\bigl|\det\bigl(\tilde A(v) \pm\kappa \tilde B(v) + \kappa^2\tilde C(v)\bigr)\bigr|^{-1}
\le \text{const.} \end{equation} Thus, using the co-factor formula for \begin{equation*}
\bigl(\tilde A(v) \pm\kappa \tilde B(v) + \kappa^2\tilde C(v)\bigr)^{-1} \end{equation*} we find with~\eqref{detinv} the estimate \begin{equation*}
\bigl\|\bigl(\tilde A(v) \pm\kappa \tilde B(v) + \kappa^2\tilde C(v)\bigr)^{-1}\bigr\|
\le C_v |\kappa|^{p_v},\qquad |\kappa|\ge R, \end{equation*} for some constants $C_v$, $p_v>0$. Consequently we get \begin{equation*}
\bigl\|\bigl(\tilde A \pm\kappa \tilde B + \kappa^2\tilde C\bigr)^{-1}\bigr\|
\le C |\kappa|^p,\qquad |\kappa|\ge R, \end{equation*} for some constants $C$, $p>0$, and by~\eqref{ZP} we have proved inequality~\eqref{invnorm}. \end{proof}
With these preparations we can enter the
\begin{proof}[Proof of lemma~\ref{lem_5_6}] Let the data $a$, $b$, $c$ be given as in\eqref{eq5ii}, \eqref{eq5iii}. We have to show that the inclusion $\mathcal{D}(A)\subset \mathcal{H}_{a,b,c}$ is not strict. Let $R = (R_\lambda,\,\lambda>0)$ be the resolvent of $A$. Then for every $\lambda>0$, $R_\lambda$ is a bijection from $C_0(\mathcal{G})$ onto $\mathcal{D}(A)$, that is, $R_\lambda^{-1}$ is a bijection from $\mathcal{D}(A)$ onto $C_0(\mathcal{G})$.
Assume to the contrary that the inclusion $\mathcal{D}(A)\subset \mathcal{H}_{a,b,c}$ is strict. We will derive a contradiction. For $\lambda>0$ consider the linear mapping $H_\lambda:\,f\mapsto \lambda f - 1/2 f''$ from $\mathcal{H}_{a,b,c}$ to $C_0(\mathcal{G})$. On $\mathcal{D}(A)$ this mapping coincides with $R^{-1}_\lambda$, and $R^{-1}_\lambda$ is a bijection from $\mathcal{D}(A)$ onto $C_0(\mathcal{G})$. Therefore our assumption entails that $H_\lambda$ cannot be injective. Hence for any $\lambda>0$ there exists $f(\lambda)\in\mathcal{H}_{a,b,c}$, $f(\lambda)\ne 0$, with \begin{equation} \label{homeq}
H_\lambda f(\lambda) = \lambda f(\lambda) - \frac{1}{2}\,f''(\lambda) = 0. \end{equation} We will show that $f(\lambda)\in\mathcal{H}_{a,b,c}$ satisfying \eqref{homeq} can only hold when $f(\lambda)=0$ on $\mathcal{G}$. It will be convenient to change the variable $\lambda$ to $\kappa = \sqrt{2\lambda}$, and there will be no danger of confusion that we shall simply write $f(\kappa)$ for $f(\lambda)$ from now on. Then the solution of~\eqref{homeq} is necessarily of the form given by \begin{align}
f_e(\kappa,x) &= r_e(\kappa)\,e^{-\kappa x} & e&\in\mathcal{E},\,x\in \mathbb{R}_+,\\
f_i(\kappa,x) &= r^+_i(\kappa)\,e^{\kappa x}+r^-_i(\kappa)\,e^{\kappa(\rho_i- x)}
& i&\in\mathcal{I},\,x\in [0,\rho_i], \end{align} and we want to show that for some $\kappa > 0$, the boundary conditions~\eqref{eq5iv} and~\eqref{eq5iva} entail that $r_e(\kappa) = r^+_i(\kappa) = r^-_i(\kappa)=0$ for all $e\in\mathcal{E}$, $i\in\mathcal{I}$. For $\kappa>0$, define a column vector $r(\kappa)$ of length
$|\mathcal{E}|+2|\mathcal{I}|$ by \begin{equation*}
r(\kappa) = \bigl((r_e(\kappa),\,e\in\mathcal{E}),(r^+_i(\kappa),\,i\in\mathcal{I}),(r^-_i(\kappa),\,i\in\mathcal{I})\bigr)^t, \end{equation*}
and introduce the $(|\mathcal{E}|+2|\mathcal{I}|)\times (|\mathcal{E}|+2|\mathcal{I}|)$ matrices \begin{equation*}
X_\pm(\kappa) = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & \pm e^{\kappa \rho} \\
0 & \pm e^{\kappa \rho} & 1
\end{pmatrix} \end{equation*}
--- appropriately modified in case that $\mathcal{E}$ or $\mathcal{I}$ is the empty set --- with the $|\mathcal{I}|\times|\mathcal{I}|$ diagonal matrices \begin{equation*}
e^{\kappa \rho} = \diag{e^{\kappa \rho_i},\,i\in\mathcal{I}\bigr}. \end{equation*} Then the boundary conditions~\eqref{eq5iv}, \eqref{eq5iva} for $f(\kappa)$ read \begin{equation} \label{Zr}
Z(\kappa)r(\kappa) = 0, \end{equation} with \begin{equation} \label{defZ}
Z(\kappa) = (A+\kappa^2C)X_+(\kappa) + \kappa BX_-(\kappa). \end{equation} Thus, if we can show that for some $\kappa > 0$ the matrix $Z(\kappa)$ is invertible, the proof of the theorem is finished. Note that the matrix-valued function $Z$ is entire in $\kappa$, and therefore so is its determinant. Thus, if can show that $\kappa\mapsto \det Z(\kappa)$ does not vanish identically, then it can only vanish on a discrete subset of the complex plane, and for $\kappa$ in the complement of this set $Z(\kappa)$ is invertible. Write \begin{equation*}
X_\pm(\kappa) = 1 \pm \delta X(\kappa), \end{equation*} with \begin{equation*}
\delta X(\kappa) = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & e^{\kappa \rho} \\
0 & e^{\kappa \rho} & 0
\end{pmatrix}, \end{equation*} so that we can write \begin{equation*}
Z(\kappa) = \hat Z_+(\kappa)\bigl(1+\delta Z(\kappa)\bigr), \end{equation*} with \begin{equation*}
\delta Z(\kappa) = \hat Z_+(\kappa)^{-1}\, \hat Z_-(\kappa)\,\delta X(\kappa). \end{equation*} Observe that in case that $\mathcal{I}=\emptyset$, we obtain $\delta Z(\kappa)=0$, and in this case the invertibility of $Z(\kappa)$ for all $\kappa$ with $\kappa\ge R$ follows from lemma~\ref{lem_5_7}. Hence we assume from now on that $\mathcal{I}\ne \emptyset$. Lemma~\ref{lem_5_7} provides us with the bound \begin{equation*}
\bigl\|\hat Z_+(\kappa)^{-1}\hat Z_-(\kappa)\bigr\| \le \text{const.}\,|\kappa|^{q}, \end{equation*}
for all $\kappa\in\mathbb{C}$, $|\kappa|\ge R$, and for some $q>0$. On the other hand, we get \begin{equation*}
\|\delta X(\kappa)\| \le e^{\kappa \rho_0}, \end{equation*}
for all $\kappa\le 0$ where $\rho_0 = \min_{i\in\mathcal{I}} \rho_i$. Therefore, there exists a constant $R'>0$ so that for all $\kappa\le -R'$ we have $\|\delta Z(\kappa)\|<1$, and therefore for such $\kappa$, $Z(\kappa)$ is invertible, i.e., $\det Z(\kappa)\ne 0$. Hence there also exists $\kappa > 0$ so that $Z(\kappa)$ is invertible, and the proof is finished. \end{proof}
\begin{appendix} \section{Feller Semigroups and Resolvents} \label{app_A} In this appendix we give an account of the Feller property of semigroups and resolvents. The material here seems to be quite well-known, and our presentation of it owes very much to~\cite{Ra56}, most notably the inversion formula for the Laplace transform, equation~\eqref{inv_L} in connection with lemma~\ref{lem_A_6}. On the other hand, we were not able to locate a reference where the results are collected and stated in the form in which we employ them in the present paper. Therefore we also provide proofs for some of the statements.
Assume that $(E,d)$ is a locally compact separable metric space with Borel
$\sigma$--algebra denoted by $\mathcal{B}(E)$. $B(E)$ denotes the space of bounded measurable real valued functions on $E$, $\ensuremath C_0(E)$ the subspace of continuous functions vanishing at infinity. $B(E)$ and $\ensuremath C_0(E)$ are equipped with the sup-norm $\|\,\cdot\,\|$.
The following definition is as in~\cite{ReYo91}: \begin{definition} \label{def_A_1}
A \emph{Feller semigroup} is a family $U=(U_t,\,t\ge 0)$ of positive
linear operators on $\ensuremath C_0(E)$ such that
\begin{enum_i}
\item $U_0=\text{id}$ and $\|U_t\|\le 1$ for every $t\ge 0$;
\item $U_{t+s} = U_t\comp U_s$ for every pair $s$, $t\ge 0$;
\item $\lim_{t\downarrow 0} \|U_t f - f\|=0$ for every $f\in \ensuremath C_0(E)$.
\end{enum_i} \end{definition}
Analogously we define \begin{definition} \label{def_A_2}
A \emph{Feller resolvent} is a family $R=(R_\lambda,\,\lambda>0)$ of positive
linear operators on $\ensuremath C_0(E)$ such that
\begin{enum_i}
\item $\|R_\lambda\|\le \lambda^{-1}$ for every $\lambda>0$;
\item $R_\lambda - R_\mu = (\mu-\lambda) R_\lambda\comp R_\mu$ for every
pair $\lambda$, $\mu>0$;
\item $\lim_{\lambda\to\infty}\|\lambda R_\lambda f - f\|=0$ for every $f\in\ensuremath C_0(E)$.
\end{enum_i} \end{definition}
In the sequel we shall focus our attention on semigroups $U$ and resolvents $R$ associated with an $E$--valued Markov process, and which are \emph{a priori} defined on $B(E)$. (In our notation, we shall not distinguish between $U$ and $R$ as defined on $B(E)$ and their restrictions to $\ensuremath C_0(E)$.)
Let $X = (X_t,\,t\ge 0)$ be a Markov process with state space $E$, and let $(P_x,\,x\in E)$ denote the associated family of probability measures on some measurable space $(\Omega,\mathcal{A})$ so that $P_x(X_0=x) = 1$. $E_x(\,\cdot\,)$ denotes the expectation with respect to $P_x$. We assume throughout that for every $f\in B(E)$ the mapping \begin{equation*}
(t,x) \mapsto E_x\bigl(f(X_t)\bigr) \end{equation*} is measurable from $\mathbb{R}_+\times E$ into $\mathbb{R}$. The semigroup $U$ and resolvent $R$ associated with $X$ act on $B(E)$ as follows. For $f\in B(E)$, $x\in E$, $t\ge 0$, and $\lambda>0$ set \begin{align}
U_t f(x) &= E_x\bigl(f(X_t)\bigr), \label{eq_A_1}\\
R_\lambda f(x) &= \int_0^\infty e^{-\lambda t} U_t f (x)\,dt. \label{eq_A_2} \end{align} Property~(i) of Definitions~\ref{def_A_1} and \ref{def_A_2} is obviously satisfied. The semigroup property, (ii) in Definition~\ref{def_A_1}, follows from the Markov property of $X$, and this in turn implies the resolvent equation, (ii) of Definition~\ref{def_A_2}. Moreover, it follows also from the Markov property of $X$ that the semigroup and the resolvent commute. On the other hand, in general neither the property that $U$ or $R$ map $\ensuremath C_0(E)$ into itself, nor the strong continuity property (iii) in Definitions~\ref{def_A_1}, \ref{def_A_2} hold true on $B(E)$ or on $\ensuremath C_0(E)$.
If $W$ is a subspace of $B(E)$ the resolvent equation shows that the image of $W$ under $R_\lambda$ is independent of the choice of $\lambda>0$, and in the sequel we shall denote the image by $RW$. Furthermore, for simplicity we shall write $U\ensuremath C_0(E)\subset \ensuremath C_0(E)$, if $U_t f\in\ensuremath C_0(E)$ for all $t\ge 0$, $f\in\ensuremath C_0(E)$.
\begin{theorem} \label{thm_A_3} The following statements are equivalent: \begin{enum_i}
\item $U$ is Feller.
\item $R$ is Feller.
\item $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{t\downarrow 0} U_t f(x) = f(x)$.
\item $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{\lambda\rightarrow \infty} \lambda R_\lambda f(x) = f(x)$.
\item $R\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{t\downarrow 0} U_t f(x) = f(x)$.
\item $R\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{\lambda\rightarrow \infty} \lambda R_\lambda f(x) = f(x)$. \end{enum_i} \end{theorem}
We prepare a sequence of lemmas. The first one follows directly from the dominated convergence theorem:
\begin{lemma} \label{lem_A_4} Assume that for $f\in B(E)$, $U_t f \rightarrow f$ as $t\downarrow 0$. Then $\lambda R_\lambda f \rightarrow f$ as $\lambda\to+\infty$. \end{lemma}
\begin{lemma} \label{lem_A_5} The semigroup $U$ is strongly continuous on $RB(E)$. \end{lemma}
\begin{proof} If strong continuity at $t=0$ has been shown, strong continuity at $t>0$ follows from the semigroup property of $U$, and the fact that $U$ and $R$ commute. Therefore it is enough to show strong continuity at $t=0$.
Let $f\in B(E)$, $\lambda>0$, $t>0$, and consider for $x\in E$ the following computation \begin{align*}
U_t R_\lambda f(x) &- R_\lambda f(x)\\
&= \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_{t+s})\bigr)\,ds
- \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\
&= e^{\lambda t} \int_t^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds
- \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\
&= \bigl(e^{\lambda t}-1\bigr) \int_t^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds
- \int_0^t e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\ \end{align*} where we used Fubini's theorem and the Markov property of $X$. Thus we get the following estimation \begin{align*}
\bigl\|U_t R_\lambda f - R_\lambda f\bigr\|
&\le \biggl(\bigl(e^{\lambda t} - 1\bigr)\int_t^\infty e^{-\lambda s}\,ds
+\int_0^t e^{-\lambda s}\,ds\biggl)\, \|f\|\\
&= \frac{2}{\lambda}\, \bigl(1-e^{-\lambda t}\bigr)\,\|f\|, \end{align*} which converges to zero as $t$ decreases to zero. \end{proof}
For $\lambda>0$, $t\ge 0$, $f\in B(E)$, $x\in E$ set \begin{equation} \label{inv_L}
U^\lambda_t f(x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n!}
\,n\lambda\, e^{n\lambda t}\, R_{n\lambda} f(x). \end{equation}
Observe that, because of $n\lambda\|R_{n\lambda} f\| \le \|f\|$, the last sum converges in $B(E)$.
For the proof of the next lemma we refer the reader to~\cite[p.~477~f]{Ra56}:
\begin{lemma} \label{lem_A_6} For all $t\ge 0$, $f\in RB(E)$, $U^\lambda_t f$ converges in $B(E)$ to $U_t f$ as $\lambda$ tends to infinity. \end{lemma}
\begin{lemma} \label{lem_A_7} If $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$, then $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for all $\lambda>0$. If $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for some $\lambda>0$, and $R_\lambda\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, then $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$. \end{lemma}
\begin{proof} Assume that $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$, let $f\in\ensuremath C_0(E)$, $x\in E$, and suppose that $(x_n,\,n\in\mathbb{N})$ is a sequence converging in $(E,d)$ to $x$. Then a straightforward application of the dominated convergence theorem shows that for every $\lambda>0$, $R_\lambda f(x_n)$ converges to $R_\lambda f(x)$. Hence $R_\lambda f\in \ensuremath C_0(E)$.
Now assume that that $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for some and therefore for all $\lambda>0$, and that $R_\lambda\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. Consider $f\in R\ensuremath C_0(E)$, $t>0$, and for $\lambda>0$ define $U^\lambda_t f$ as in equation~\eqref{inv_L}. Because $R_{n\lambda}f\in\ensuremath C_0(E)$ and the series in formula~\eqref{inv_L} converges uniformly in $x\in E$, we get $U^\lambda_t f\in\ensuremath C_0(E)$. By lemma~\ref{lem_A_6}, we find that $U^\lambda_t f$ converges uniformly to $U_t f$ as $\lambda\to+\infty$. Hence $U_t f\in\ensuremath C_0(E)$. Since $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, $U_t$ is a contraction and $\ensuremath C_0(E)$ is closed, we get that $U_t\ensuremath C_0(E)\subset\ensuremath C_0(E)$ for every $t\ge 0$. \end{proof}
The following lemma is proved as a part of Theorem~17.4 in~\cite{Ka97} (cf.\ also the proof of Proposition~2.4 in~\cite{ReYo91}).
\begin{lemma} \label{lem_A_8} Assume that $R\ensuremath C_0(E)\subset \ensuremath C_0(E)$, and that for all $x\in E$, $f\in\ensuremath C_0(E)$, $\lim_{\lambda\to\infty} \lambda R_\lambda f(x) = f(x)$. Then $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. \end{lemma}
If for all $f\in\ensuremath C_0(E)$, $x\in E$, $U_t f(x)$ converges to $f(x)$ as $t$ decreases to zero, then similarly as in the proof of lemma~\ref{lem_A_4} we get that $\lambda R_\lambda f(x)$ converges to $f(x)$ as $\lambda\to+\infty$. Thus we obtain the following
\begin{corollary} \label{cor_A_9} Assume that $R\ensuremath C_0(E)\subset \ensuremath C_0(E)$, and that for all $x\in E$, $f\in\ensuremath C_0(E)$, $\lim_{t\downarrow 0} U_t f(x) = f(x)$. Then $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. \end{corollary}
Now we can come to the
\begin{proof}[Proof of theorem~\ref{thm_A_3}] We show first the equivalence of statements~(i), (ii), (iv), and (vi):
\noindent ``(i)$\Rightarrow$\space(ii)'' Assume that $U$ is Feller. From lemma~\ref{lem_A_7} it follows that $R_\lambda\ensuremath C_0(E)\subset\ensuremath C_0(E)$, $\lambda>0$. Let $f\in\ensuremath C_0(E)$. Since $U$ is strongly continuous on $\ensuremath C_0(E)$, lemma~\ref{lem_A_4} implies that $\lambda R_\lambda f$ converges to $f$ as $\lambda$ tends to $+\infty$. Hence $R$ is Feller.
\noindent ``(ii)$\Rightarrow$\space(vi)'' This is trivial.
\noindent ``(vi)$\Rightarrow$\space(iv)'' By lemma~\ref{lem_A_8}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, and therefore lemma~\ref{lem_A_7} entails that $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$.
\noindent ``(iv)$\Rightarrow$\space(i)'' By lemmas~\ref{lem_A_7} and \ref{lem_A_8}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, and therefore by lemma~\ref{lem_A_5} $U$ is strongly continuous on $\ensuremath C_0(E)$. Thus $U$ is Feller.
Now we prove the equivalence of~(i), (iii), and (v):
\noindent ``(i)$\Rightarrow$\space(iii)'' This is trivial.
\noindent ``(iii)$\Rightarrow$\space(v)'' This follows directly from Lemma~\ref{lem_A_7}.
\noindent ``(v)$\Rightarrow$\space(i)'' By corollary~\ref{cor_A_9}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, hence it follows from lem\-ma~\ref{lem_A_7} that $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$. Furthermore, lemma~\ref{lem_A_5} implies the strong continuity of $U$ on $R\ensuremath C_0(E)$, and by density therefore on $\ensuremath C_0(E)$. (i) follows. \end{proof}
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\begin{document}
\title[The logarithmic Cauchy quotient means]{The logarithmic Cauchy quotient mean} \author{Martin Himmel} \address{Technical University Mountain Academy Freiberg, Faculty of Mathematics and Computer Science, Intitute of Applied Analysis, Nonnengasse 22, 09596 Freiberg, Germany} \email{[email protected]} \author{Janusz Matkowski} \curraddr{Institute of Mathematics, University of Zielona G\'{o}ra, Szafrana 4A, PL 65-516 Zielona G\'{o}ra, Poland} \email{[email protected]}
\begin{abstract} Motivated by recent results on beta-type functions, a new family of means, which are of logarithmic Cauchy quotient type, are determined and characterized. \end{abstract}
\maketitle
\section{Introduction}
\footnotetext{\textit{2010 Mathematics Subject Classification. }Primary: 26E60, 39B12. \par \textit{Keywords and phrases:} mean, premean, logarithmic Cauchy quotient mean, functional equation. \par {}}
The relationship between the Euler Gamma function and the Beta function inspired to introduce the beta-type function \cite{MatHim4}. Here we propose the $k$-variable logarithmic Cauchy quotients, the logarithmic counterpart of beta-type functions, as follows. Given a positive integer $k\geq 2$, and a function $f:I\rightarrow \left( 0,+\infty \right) $ (or $f:I\rightarrow \left( -\infty ,0\right) $) where $I\subset \left( 0,+\infty \right) $ is an interval that is closed under multiplication, we define the $k$-variable logarithmic Cauchy quotient $L_{f,k}:I^{k}\rightarrow \left( 0,+\infty \right) $ by \begin{equation} L_{f,k}\left( x_{1},...,x_{k}\right) =\frac{f\left( x_{1}\right) +\cdots+f\left( x_{k}\right) }{f\left( x_{1} \cdots x_{k}\right) }, \label{eq:Lfk} \end{equation} and we refer to $f$ as its generator (Section 2, Definition 1). Similarly to the case of beta-type functions (\cite{MatHim4}), we give conditions under which $L_{f,k}$ is a premean or a mean (see Lemma 2, Theorem 2, Theorem 3 and Theorem 4).
In Section 3, assuming that $1\in I$, we prove that two $k$-variable logarithmic Cauchy quotients coincide if and only if their generators are proportional (Theorem 1).
In Section 4, applying the theory of iterative functional equations \cite {Kuczma}, we determine the general solution of the functional equation \begin{equation} f\left( x\right) =\frac{x}{k}f\left( x^{k}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \label{eq:Lfk_Reflexivity} \end{equation} that is the reflexivity condition of $L_{f,k}$ (Lemma 2). Based on this lemma, in Section 5, we prove Theorem 2, our main result, which says that $ L_{f,k}$ is a $k$-variable mean in $\left( 1,+\infty \right) $\ iff there is $c\neq 0$ such that \begin{equation} f\left( x\right) =c\frac{\log x}{\sqrt[k-1]{x}}, \label{eq:meanGenLfk} \end{equation} for all $x \in (1, +\infty)$, or, equivalently, that $L_{f,k}=\mathcal{L}_{k},$ where $ \mathcal{L}_{k}$ is a new $k$-variable mean, called the $k$-variable \textit{ logarithmic} \textit{Cauchy quotient mean }(Definition 2), and defined by \begin{equation} \mathcal{L}_{k}\left( x_{1},\ldots,x_{k}\right) :=\sum\limits_{i=1}^{k}\frac{ \log x_{i}}{\sum\limits_{l=1}^{k}\log x_{l}}\mathcal{G}_{k-1}\left( x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k}\right), \label{eq:Lfk_mean} \end{equation} for all $x_{1},\ldots,x_{k} \in (1, + \infty)$, where $\mathcal{G}_{k-1}:\left( 0,+\infty \right) ^{k-1}\rightarrow \left( 0,+\infty \right) $ is the $\left( k-1\right) $-variable geometric mean, \begin{equation*} \mathcal{G}_{k-1}\left( x_{1},\ldots,x_{k-1}\right) =\sqrt[k-1]{x_{1}\cdots x_{k-1}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} Moreover, some properties of $\mathcal{L}_{k}$ are discussed, the results for the interval $\left( 0,1\right) $ is formulated, as well as the corresponding extension of the logarithmic Cauchy quotient mean on the interval $ \left( 0,+\infty \right) $ (denoted by $\mathfrak{L}_{k}$) is proposed.
We end our paper with two characterizations of the mean $\mathcal{L}_{k}$. In section 6, applying a variant of the Krull theorem on difference equations (\cite{Krull}) given in Kuczma \cite{{Kuczma}}, we show that $L_{f,k}= \mathcal{L}_{k}$ iff $L_{f,k}$ is reflexive in $\left( 1,+\infty \right) $ and the function $\log \circ f\circ \exp \circ \exp $ is convex. \ In Section 7, assuming that $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ is extendable to a function of the class $C^{2}$ in $\left[ 1,+\infty \right) $, we prove that $L_{f,k}$ is a premean in $\left( 1,+\infty \right) $ iff it coincides with the mean $ \mathcal{L}_{k}$.
\section{Some basic notions}
Throughout this paper $I\subset \mathbb{R}$ stands for an interval.
Let $k\in \mathbb{N}$, $k\geq 2.$ A function $M:I^{k}\rightarrow \mathbb{R}$ is called a $k$-\textit{variable mean in} $I$, if \begin{equation*} \min \left( x_{1},\ldots ,x_{k}\right) \leq M\left( x_{1},\ldots ,x_{k}\right) \leq \max \left( x_{1},\ldots ,x_{k}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ } x_{1},\ldots ,x_{k}\in I; \end{equation*} and it is called \textit{strict}, if these inequalities are strict for all nonconstant $k$-tuples $\left( x_{1},\ldots ,x_{k}\right) \in I^{k}.$
Let us note the following easy to verify properties of means.
\begin{remark} \label{PropertiesOfAMean} If $M$ is a $k$-variable mean in an interval $I,$ then \begin{enumerate}
\item[(i)] \ \ for every subinterval $J\subset I$, $M$ restricted to $J^{k}$ is a mean in $J$, and $M\left( J^{k}\right) =J,$ in particular, $ M:I^{k}\rightarrow I;$
\item[(ii)] $\ M$ is reflexive, i.e. \begin{equation*} M\left( x,...,x\right) =x,\ \ \ \ \ x\in I. \end{equation*} \end{enumerate}
\end{remark}
A function $M:I^{k}\rightarrow \mathbb{R}$ is called $k$-variable \textit{ premean} in $I,$ if it reflexive and $M\left( I^{k}\right) =I$ (see \cite {JM2006}, also \cite{Toader}, p. 29).
\begin{remark} If a reflexive function $M:I^{k}\rightarrow \mathbb{R}$ is (strictly) increasing in each variable, then it is a (strict) $k$-variable mean in $I$. \end{remark}
Let us introduce some notion playing here a significant role.
\begin{definition}\label{def:Lfk} Let $k\in \mathbb{N}$, $k\geq 2,$ be fixed, and let $I\subset \left( 0,+\infty \right) $ be an interval that is closed under multiplication. For a
function $f:I\rightarrow \left( 0,+\infty \right) $ (or $ f:I\rightarrow \left( -\infty ,0\right) $), the function $ L_{f,k}:I^{k}\rightarrow \left( 0,+\infty \right) $ defined by \eqref{eq:Lfk} \iffalse \begin{equation*} L_{f,k}\left( x_{1},\ldots ,x_{k}\right) :=\frac{\sum\limits_{j=1}^{k}f \left( x_{j}\right) }{f\left( \prod\limits_{j=1}^{k}x_{j}\right) },\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x_{1},\ldots ,x_{k}\in I, \end{equation*} \fi is called \textit{\ }$k$\textit{-variable logarithmic Cauchy quotient}, \textit{\ and }$f$ is called a\textit{\ generator of }$L_{f,k}$. \end{definition}
\begin{remark} An open interval $I \subset \mathbb{R}$ is closed under multiplication iff
$I=(p, +\infty)$ for some $p \in [1,+\infty)$; or $I=(0, p)$ for some $p \in (0,1]$, or $I=\mathbb{R}$. \end{remark}
From the definitions of the logarithmic Cauchy quotient $L_{f,k}$ and the reflexivity we obtain
\begin{remark} \label{LfkReflexivity} Under the assumptions of this definition, the logarithmic Cauchy quotient $ L_{f,k}:I^{k}\rightarrow \left( 0,+\infty \right) $ of a generator $ f:I\rightarrow \left( 0,+\infty \right) $ is reflexive (or it is a mean or a premean) if its generator $f$ satisfies the iterative functional equation \eqref{eq:Lfk_Reflexivity}. \end{remark}
\section{Equality of two logarithmic Cauchy quotients and a functional equation}
\begin{remark} Let $k\in \mathbb{N}$, $k\geq 2$, and interval $I\subset \left( 0,+\infty \right) $ satisfy conditions of Definition 1 and let $f,g:I\rightarrow \left( 0,+\infty \right) $. Then $L_{g,k}=L_{f,k}$ iff the functions\ $f$ and $g$ satisfy the functional equation \begin{equation} \label{eq:Lfk=Lgk} \frac{g\left( x_{1}\cdots x_{k}\right) }{f\left( x_{1}\cdots x_{k}\right) }=\frac{g\left( x_{1}\right) +\cdots +g\left( x_{k}\right) }{ f\left( x_{1}\right) +\cdots+f\left( x_{k}\right) }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ } x_{1},\ldots,x_{k}\in I. \end{equation} Moreover, if \thinspace $1\in I$, then $f$ and $g$ satisfy this equation if, and only if, $g=cf$ for some $c>0$. \end{remark}
\begin{proof} The first fact is an immediate consequence of Definition 1. To show the remaining one, assume that $f$ and $g$ satisfy this equation. Putting $ x_{1}=x$ and $x_{2}=x_{3}=\ldots=x_{k}=1$ gives \begin{equation*} \frac{g\left( x\right) }{f\left( x\right) }=\frac{g\left( x\right) +\left( k-1\right) g\left( 1\right) }{f\left( x\right) +\left( k-1\right) f\left( 1\right) },\ \ \ \ \ \ x\in I, \end{equation*} whence \begin{equation*} \left( k-1\right) f\left( 1\right) g\left( x\right) =\left( k-1\right) g\left( 1\right) f\left( x\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x\in I. \end{equation*} Since $f$ and $g$ are positive functions, it follows that $f\left( 1\right) \neq 0$ and $g\left( 1\right) \neq 0$. Setting $c:=\frac{g\left( 1\right) }{f\left( 1\right) }$ we hence get $g=cf$. The converse implication is obvious. \end{proof}
In the sequel we have to exclude $1$ from the interval $I$, as we are mainly interested in the case when $f\left( 1\right) =0=g\left( 1\right) $. It turns out that in this case the above functional equation is not trivial. We prove
\begin{lemma} Let\ $k\in \mathbb{N}$, $k\geq 2$, be fixed. If the functions $f,g:\left(
1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ \verb|[|or $f,g:\left(
0,1\right) \rightarrow \left( 0,+\infty \right) $\verb|]| satisfy equation \eqref{eq:Lfk=Lgk} with $
I=\left( 1,+\infty \right) $ \ \verb|[|or $I=\left( 0,1\right) $\verb|]|, and \begin{equation*} c:=\lim_{x\rightarrow 1}\frac{g\left( x\right) }{f\left( x\right) }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } \end{equation*} exists, then $g=cf$. \end{lemma}
\begin{proof} Put $h:=\frac{g}{f}$. Setting $x_{1}=x_{2}=...=x_{k}=x$ in \eqref{eq:Lfk=Lgk}, we get $ h\left( x^{k}\right) =h\left( x\right) $ for all$\ x\in(1,+\infty)$, or equivalently, \begin{equation*} h\left( x\right) =h\left( x^{\frac{1}{k}}\right) ,\ \ \ \ \ \ x\in \left( 1,+\infty \right) , \end{equation*} whence, by induction, \begin{equation*} h\left( x\right) =h\left( x^{\frac{1}{k^{n}}}\right) ,\ \ \ \ \ \ n\in \mathbb{N}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ }x\in \left( 1,+\infty \right) . \end{equation*} Letting $n\rightarrow +\infty $ we hence get $h\left( x\right) =c$ for all $ x\in \left( 1,+\infty \right) .$ \end{proof}
From this lemma we obtain
\begin{theorem} Let\ $k\in \mathbb{N}$, $k\geq 2$, be fixed. Assume that $f,g:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ (or $f,g:\left( 0,1\right) \rightarrow \left( 0,+\infty \right) $ are such that the limit $ \lim_{x\rightarrow 1}\frac{g\left( x\right) }{f\left( x\right) }$ exists. Then $L_{f,k}=L_{g,k}$ if and only if $g=cf$ for some $c>0$. \end{theorem}
\section{Reflexivity of the logarithmic Cauchy quotient}
Applying the theory of the iterative functional equations (see\ \cite{Kuczma} , p. 46, Theorem 2.1) one gets
\begin{lemma} Fix an integer $k \geq 2$ and $p \in (1, + \infty)$. Then \begin{enumerate}
\item[(i)] a function $f:\left[ p,+\infty \right) \rightarrow \left( 0,+\infty \right) $ satisfies equation \eqref{eq:Lfk_Reflexivity} for all $x \in \left[ p,+\infty \right)$
\iffalse \begin{equation*} f\left( x\right) =\frac{x}{k}f\left( x^{k}\right) ,\ \ \ \ \ x\geq a, \end{equation*} \fi if and only if \begin{equation} f\left( x\right) =k^{n}x^{^{\frac{k^{-n}-1}{k-1}}}f_{0}\left( x^{k^{-n}}\right) \label{eq:Lfk_Reflexivity_Solution} \end{equation} for all $x\in \left[ p^{k^{n}},p^{k^{n+1}}\right)$ and $n\in \mathbb{N}_{0}$, where $f_{0}:=f\mid _{_{\left[ p,p^{k}\right) }};$ moreover, $f$ is continuous if and only if so is $f_{0}$ and \begin{equation} \lim_{x\rightarrow p^{k}-}f_{0}\left( x\right) =\frac{k}{p}f_{0}\left( p\right) \label{eq:Lfk_Reflexivity_Solution_Continuity}. \end{equation}
\item[(ii)] a function $f:\left( p,+\infty \right) \rightarrow \left( 0,+\infty \right) $ satisfies equation \eqref{eq:Lfk_Reflexivity} for all $x \in (p, + \infty)$ \iffalse \begin{equation*} f\left( x\right) =\frac{x}{k}f\left( x^{k}\right) ,\ \ \ \ \ x>a, \end{equation*} \fi if and only if condition \eqref{eq:Lfk_Reflexivity_Solution} holds for all $x\in \left[ p^{k^{n}},p^{k^{n+1}}\right)$ and $n\in \mathbb{N}_{0}$, \iffalse \begin{equation*} f\left( x\right) =k^{n}x^{^{\frac{k^{-n}-1}{k-1}}}f_{0}\left( x^{k^{-n}}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \left( a^{k^{n}},a^{k^{n+1}}\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ }n\in \mathbb{N}_{0}, \end{equation*} \fi where $f_{0}:=f\mid _{_{\left[ p,p^{k}\right) }};$ moreover, $f$ is continuous if and only if so is $f_{0}$ and \eqref{eq:Lfk_Reflexivity_Solution_Continuity} holds true.
\iffalse \begin{equation*} \lim_{x\rightarrow p+}f_{0}\left( x\right) =\frac{k}{p}f_{0}\left( p^{k}\right) . \end{equation*} \fi
\item[(iii)] a function $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ satisfies equation \eqref{eq:Lfk_Reflexivity} for all $x \in (1, + \infty)$ \iffalse \begin{equation} f\left( x\right) =\frac{x}{k}f\left( x^{k}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x>1, \@ifnextchar*{\@tagstar}{\@tag}{2} \end{equation} \fi if and only if condition \eqref{eq:Lfk_Reflexivity_Solution} holds for all $x\in \left[ p^{k^{n}},p^{k^{n+1}}\right)$ and $n\in \mathbb{Z}$,
\iffalse \begin{equation*} f\left( x\right) =k^{n}x^{^{\frac{k^{-n}-1}{k-1}}}f_{0}\left( x^{k^{-n}}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \left[ a^{k^{n}},a^{k^{n+1}} \right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }n\in \mathbb{Z}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \end{equation*} \fi where $f_{0}:=f\mid _{_{\left[ p,p^{k}\right) }};$ moreover, $f$ is continuous if and only if so is $f_{0}$ and \eqref{eq:Lfk_Reflexivity_Solution_Continuity} holds true. \end{enumerate}
\iffalse \begin{equation*} \lim_{x\rightarrow p^{k}-}f_{0}\left( x\right) =\frac{k}{p}f_{0}\left( p\right) . \end{equation*} \fi \end{lemma}
\section{Means of the logarithmic Cauchy quotient type}
\begin{definition} The function $\mathcal{L}_{k}:\left( 1,+\infty \right) ^{k}\rightarrow \left( 1,+\infty \right) $, given by \begin{equation} \mathcal{L}_{k}\left( x_{1},\ldots ,x_{k}\right) :=\sum\limits_{i=1}^{k} \frac{\log x_{i}}{\sum\limits_{l=1}^{k}\log x_{l}}\left( \prod\limits_{j=1,j\neq i}^{k}x_{j}\right) ^{\frac{1}{k-1}}, \label{eq:Lfkmean} \end{equation}
that is, \begin{equation*} \mathcal{L}_{k}\left( x_{1},\ldots ,x_{k}\right) =\sum\limits_{i=1}^{k}\frac{ \log x_{i}}{\sum\limits_{l=1}^{k}\log x_{l}}\mathcal{G}_{k-1}\left( x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k}\right) \end{equation*} for all $x_{1},\ldots ,x_{k} \in (1,+\infty)$, where $\mathcal{G}_{k-1}$ is the $\left( k-1\right) $-variable symmetric geometric mean in $\left( 1,+\infty \right) $, is called $k$-variable logarithmic Cauchy quotient mean in $\left( 1,+\infty \right) $.
\end{definition}
We also use some elementary fact on the Jensen equation of two or more variables.
\begin{lemma} \label{kJensen} \textit{Let }$C$\textit{\ be a convex set of a linear space. A function }$ f:C\rightarrow \mathbb{R}$\textit{\ is a Jensen function of }$k$\textit{\ variables for some }$k\in \mathbb{N},$\textit{\ }$k\geq 2$\textit{, i.e., it satisfies the equality} \begin{equation} f\left( \frac{x_{1}+\cdots+x_{k}}{k}\right) =\frac{f\left( x_{1}\right) +\cdots+f\left( x_{k}\right) }{k}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ }x_1,\ldots,x_k\in C, \label{thm:kJensen} \end{equation} \textit{if and only if it is a Jensen function of two variables, i.e.,} \begin{equation*} f\left( \frac{x+y}{2}\right) =\frac{f\left( x\right) +f\left( y\right) }{2} \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ }x,y\in C. \end{equation*} \end{lemma}
\begin{proof}
Indeed, for arbitrary $x,y\in C,$ using \eqref{thm:kJensen}, we have \begin{equation*} f\left( \frac{x+y}{2}\right) =f\left( \frac{x+y+\sum\limits_{i=1}^{k-2}\frac{x+y}{2} }{k}\right) =\frac{f\left( x\right) +f\left( y\right) +\sum\limits_{i=1}^{k-2}f\left( \frac{x+y}{2}\right) }{k}, \end{equation*} whence $f\left( \frac{x+y}{2}\right) =\frac{f\left( x\right) +f\left( y\right) }{2}$, so $f$ is a Jensen function of two variables.
If $f$ is a Jensen function of two variables, then (see Kuczma \cite{Kuczma2}, p. 126, Lemma 1, where the Jensen convexity is considered), by induction, for every $ n\in \mathbb{N}$, we get \begin{equation*} f\left( \frac{x_{1}+\cdots+x_{2^{n}}}{2^{n}}\right) =\frac{f\left( x_{1}\right) +\cdots+f\left( x_{2^{n}}\right) }{2^{n}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ } x_{1},\ldots,x_{2^{n}}\in C\RIfM@\expandafter\text@\else\expandafter\mbox\fi{. } \end{equation*} Let \ $x_{1},\ldots,x_{k}\in C$ be arbitrarily fixed$.$ Choosing $n$ such that $ k\leq 2^{n}$ and setting here \begin{equation*} x_{k+1}=x_{k+2}=\cdots=x_{2^{n}}:=\frac{x_{1}+\cdots+x_{k}}{k}, \end{equation*} we get \begin{eqnarray*} f\left( \frac{x_{1}+\cdots+x_{k}}{k}\right) &=&f\left( \frac{ x_{1}+\cdots+x_{k}+\left( 2^{n}-k\right) \frac{x_{1}+\cdots+x_{k}}{k}}{2^{n}} \right) \\ &=&f\left( \frac{x_{1}+\cdots+x_{k}+\sum\limits_{j=k+1}^{2^{n}}\frac{x_{1}+\cdots+x_{k}}{k }}{2^{n}}\right) \\ &=&\frac{f\left( x_{1}\right) +\cdots+f\left( x_{k}\right) +\sum\limits_{j=k+1}^{2^{n}}f\left( \frac{x_{1}+\cdots+x_{k}}{k}\right) }{2^{n}} \\ &=&\frac{f\left( x_{1}\right) +\cdots+f\left( x_{k}\right) +\left( 2^{n}-k\right) f\left( \frac{x_{1}+\cdots+x_{k}}{k}\right) }{2^{n}}, \end{eqnarray*} whence \begin{equation*} kf\left( \frac{x_{1}+\cdots+x_{k}}{k}\right) =f\left( x_{1}\right) +\cdots+f\left( x_{k}\right), \end{equation*} which shows that $f$ is a Jensen function of $k$ variables.
\end{proof}
The main result of this paper reads as follows.
\begin{theorem} Fix an integer $k \geq 2$ and a function $f:\left( 1,+\infty \right) \rightarrow
\left( 0,+\infty \right) $ \verb|[| $f:\left( 0,1 \right) \rightarrow \left(
0, +\infty \right) $ \verb|]|. The following statements are pairwise equivalent:
\begin{enumerate}
\item[(i)] the logarithmic Cauchy quotient function $L_{f,k}:\left( 1,+\infty
\right) ^{k}\rightarrow \left( 0,+\infty \right) $ \verb|[| $L_{f,k}:\left( 0,1
\right) ^{k}\rightarrow \left( 0,+\infty \right) $ \verb|]| is a $k$-variable mean in $
\left( 1,+\infty \right) $ \verb|[| in $
\left( 0,1 \right) $ \verb|]|;
\item[(ii)] there is a positive \verb|[| negative \verb|]| $c$ such that equality \eqref{eq:meanGenLfk} holds for all $x\in \left( 1,+\infty
\right)$ \verb|[| for all $x\in \left( 0,1
\right)$ \verb|]|;
\iffalse \begin{equation*} f\left( x\right) =\frac{c\log x}{\sqrt[k-1]{x}}, \end{equation*} \fi
\item[(iii)] the equality \begin{equation*} L_{f,k}=\mathcal{L}_{k} \end{equation*} holds in $ \left( 1,+\infty
\right)^k$ \verb|[| in $ \left( 0,1
\right)^k$ \verb|]|. \end{enumerate}
\end{theorem}
\begin{proof} To prove the implication (i)$\Longrightarrow $(ii), assume that $L_{f,k}$ is a mean in $\left( 1,+\infty \right) $. Fix arbitrarily $p>1$ and put $ f_{0}:=f\mid _{_{\left[ p,p^{k}\right) }}.$ It follows from Remark \ref{PropertiesOfAMean} (ii) and Remark \ref{LfkReflexivity} that , the function $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ satisfies \eqref{eq:Lfk_Reflexivity}. Thus, by part (iii) of Lemma 2, for every $n\in \mathbb{Z},$
\begin{equation*} f\left( x\right) =k^{-n}x^{^{\frac{k^{n}-1}{k-1}}}f_{0}\left( x^{k^{n}}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \left[ p^{k^{-n}},p^{k^{-n+1}} \right). \end{equation*}
Hence, for all $x_{1},\ldots ,x_{k}\in \left[ p^{k^{-n}},p^{k^{-n+1}}\right) ,$ we have \begin{equation*} x_{1}\cdot \ldots \cdot x_{k}\in \left[ p^{k^{-(n-1)}},p^{k^{-(n-2)}}\right), \end{equation*} and, by Definition \ref{def:Lfk}, for all $x_{1},\ldots ,x_{k}\in \left[ p^{k^{-n}},p^{k^{-n+1}}\right) ,$ \begin{eqnarray*} L_{f,k}\left( x_{1},\ldots ,x_{k}\right) = \frac{k^{-n}x_1^{\frac{k^{n-1}}{k-1}} f_0{(x_1^{k^n})}+\cdots +k^{-n}x_k^{\frac{k^{n-1}}{k-1}}f_0{(x_k^{k^n})}}{k^{-(n-1)}(x_1 \cdot \cdots \cdot x_k)^{\frac{k^{n-2}}{k-1}} f_0{((x_1 \cdot \cdots \cdot x_k)^{k^{n-1}})}}\\ =\frac{1}{k}\frac{ \sum\limits_{j=1}^{k}x_{j}^{\frac{k^{n}-1}{k-1}}f_{0}\left( x_j^{k^{n}}\right) }{\left( \prod\limits_{j=1}^{k}x_{j}\right) ^{\frac{k^{n-2}}{k-1} }f_{0}\left( \left( \prod\limits_{j=1}^{k}x_{j}\right) ^{k^{n-1}}\right) }. \end{eqnarray*} Since $L_{f,k}$ is a $k$-variable mean in the interval $I,\ $we have, for all $x_{1},\ldots ,x_{k}\in \left[ p^{k^{-n}},p^{k^{-n+1}}\right) $, \begin{equation*} \min \left( x_{1},\ldots ,x_{k}\right) \leq \frac{1}{k}\frac{ \sum\limits_{j=1}^{k}x_{j}^{\frac{k^{n}-1}{k-1}}f_{0}\left( x_j^{k^{n}}\right) }{\left( \prod\limits_{j=1}^{k}x_{j}\right) ^{\frac{k^{n-2}}{k-1} }f_{0}\left( \left( \prod\limits_{j=1}^{k}x_{j}\right) ^{k^{n-1}}\right) } \leq \max \left( x_{1},\ldots ,x_{k}\right) . \end{equation*} Choosing $y_{1},\ldots ,y_{k}\in \left[ p,p^{k}\right) ${\LARGE \ } arbitrarily, we have, for every $n\in \mathbb{Z}$, \begin{equation*} x_{j}=y_{j}^{k^{-n}}\in \left[ p^{k^{-n}},p^{k^{-n+1}}\right) ~\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ \ for \ \ }j=1,\ldots ,k. \end{equation*} Setting these numbers into the above inequalities, and, assuming that \begin{equation*} y_{1}=\min \left( y_{1},...,y_{k}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ and \ \ \ \ \ } y_{k}=\max \left( y_{1},...,y_{k}\right) , \end{equation*} (which can be done without any loss of generality), we get \begin{equation*} y_{1}^{k^{-n}}\leq \frac{1}{k}\frac{\sum\limits_{j=1}^{k}y_{j}^{\frac{ 1-k^{-n}}{k-1}}f_{0}\left( y_{j}\right) }{\left( \prod\limits_{j=1}^{k}y_{j}\right) ^{\frac{k^{-1}-k^{-n}}{k-1}}f_{0}\left( \left( \prod\limits_{j=1}^{k}y_{j}\right) ^{k^{-1}}\right) }\leq y_{k}^{k^{-n}},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }y_{2},\ldots ,y_{k-1}\in \left[ p,p^{k}\right) . \end{equation*} Letting here $n\rightarrow +\infty ,$ we obtain \begin{equation*} 1\leq \frac{1}{k}\frac{\sum\limits_{j=1}^{k}y_{j}^{\frac{1}{k-1}}f_{0}\left( y_{j}\right) }{\left( \prod\limits_{j=1}^{k}y_{j}\right) ^{\frac{k^{-1}}{k-1} }f_{0}\left( \left( \prod\limits_{j=1}^{k}y_{j}\right) ^{k^{-1}}\right) } \leq 1,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }y_{1},\ldots ,y_{k}\in \left[ p,p^{k}\right) , \end{equation*} whence, \begin{equation*} \frac{1}{k}\sum\limits_{j=1}^{k}y_{j}^{\frac{1}{k-1}}f_{0}\left( y_{j}\right) =\left( \left( \prod\limits_{j=1}^{k}y_{j}\right) ^{k^{-1}}\right) ^{\frac{1}{k-1}}f_{0}\left( \left( \prod\limits_{j=1}^{k}y_{j}\right) ^{k^{-1}}\right) ,\ \ \ \ \ \ y_{1},\ldots ,y_{k}\in \left[ p,p^{k}\right) . \end{equation*} Defining $g:\left[ a,a^{k}\right) \rightarrow \left( 0,+\infty \right) $ by \begin{equation*} g\left( y\right) :=y^{\frac{1}{k-1}}f_{0}\left( y\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ } y\in \left[ p,p^{k}\right) , \end{equation*} we can write this equality as follows \begin{equation*} g\left( \left( \prod\limits_{j=1}^{k}y_{j}\right) ^{k^{-1}}\right) =\frac{1}{ k}\sum\limits_{j=1}^{k}g\left( y_{j}\right) ,\ \ \ \ \ \ y_{1},\ldots ,y_{k}\in \left[ p,p^{k}\right) . \end{equation*} Since, for arbitrary $s_{j}\in \left[ \log p,\log p^{k}\right) ,$ $ j=1,\ldots ,k,$ we have \begin{equation*} y_{j}=e^{s_{j}}\in \left[ p,p^{k}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ \ }j=1,\ldots ,k, \end{equation*} we hence get \begin{equation*} g\left( e^{\frac{1}{k}\left( s_{1}+\cdots +s_{k}\right) }\right) =\frac{1}{k} \left[ g\left( e^{s_{1}}\right) +\cdots +g\left( e^{s_{k}}\right) \right] ,\ \ \ \ \ \ s_{1},\ldots ,s_{k}\in \left[ \log p,\log{ p^{k}}\right) . \end{equation*} Thus, the function \begin{equation*} h:=g\circ \exp \end{equation*} satisfies the Jensen functional equation \begin{equation*} h\left( \frac{1}{k}\left( s_{1}+\cdots +s_{k}\right) \right) =\frac{1}{k} \left[ h\left( s_{1}\right) +\cdots +h\left( s_{k}\right) \right] ,\ \ \ \ \ \ s_{1},\ldots ,s_{k}\in \left[ \log p,\log {p^{k}}\right) . \end{equation*} By \cite{Kuczma2}, p. 315, Theorem 1, and Lemma \ref{kJensen} there exists an additive function ${a}:\mathbb{R}\rightarrow \mathbb{R}$ and $b\in \mathbb{R}$ such that \begin{equation*} h\left( s\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{a}\left( s\right) +b,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ }\ s\in \left[ \log a,\log a^{k}\right) . \end{equation*} From the definitions of the functions $h,$ $g$ and $f_0$, we obtain \begin{equation*} g\left( y\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{a}\left( \log y\right) +b,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ } y\in \left[ p,p^{k}\right), \end{equation*} and, using the $\mathbb{Q}$-homogeneity of the additive function $a$, \begin{equation*} f_{0}\left( y\right) =\frac{ a\left( \log{y} \right) +b }{y^{\frac{1}{k-1}}},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ }y\in \left[ p,p^{k}\right) . \end{equation*} Hence, by Lemma 2 (iii), we have, for every $n\in \mathbb{Z},$ \begin{equation*} f\left( x\right) =\frac{1}{x^{\frac{1}{k-1}}}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{a}\left( \log x\right) +\frac{b}{k^{n}}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \left[ p^{k^{n}},p^{k^{n+1}}\right) . \end{equation*} Setting this into equation \eqref{eq:Lfk_Reflexivity}, we get \begin{equation*} \frac{1}{x^\frac{1}{k-1}} \left(a(\log{x})+\frac{b}{k^n}\right)= \frac{1}{x^\frac{1}{k-1}} \left(a(\log{x})+\frac{b}{k^{n+1}}\right), \end{equation*} and thus \begin{equation*} b=0. \end{equation*} Since $f$ is assumed to be positive, the function $a$ must be continuous, i.e. there is $c>0$ such that \begin{equation*} {a}\left( x\right) =cx,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \mathbb{R}. \end{equation*} Consequently, for every $n \in \mathbb{Z}$, \begin{equation*} f\left( x\right) =\frac{c}{x^{\frac{1}{k-1}}}\log x,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in \left[ p^{k^{n}},p^{k^{n+1}}\right) . \end{equation*} This proves the implication $(i)\Longrightarrow (ii).$
Assume $(ii)$ holds. Then, by Definition 1, we get, for all $x_{1},\ldots ,x_{k}\in \left( 1,+\infty \right) ,$ \begin{eqnarray*} L_{f,k}\left( x_{1},\ldots ,x_{k}\right) &=&\frac{\frac{c}{x_1^{\frac{1}{k-1}} }\log{x_1}+ \cdots + \frac{c}{x_k^{\frac{1}{k-1}}}\log{x_k}}{\frac{c}{(x_1 \cdots x_k)^\frac{1}{k-1} }\log{(x_1 \cdots x_k)}}\\ &=& {(x_1 \cdots x_k)^\frac{1}{k-1} } \frac{ \frac{\log{x_1}}{x_1^{\frac{1}{k-1}} }+ \cdots + \frac{\log{x_k}}{x_k^{\frac{1}{k-1}}} }{{\log{x_1}+ \cdots +\log{x_k}}}\\ &=&\frac{\sum\limits_{i=1}^{k} \left( \prod\limits_{j=1,j\neq i}^{k}x_{j}^{\frac{1}{k-1}}\right) \log x_{i} }{\sum\limits_{l=1}^{k}\log x_{l}}=\sum\limits_{i=1}^{k}\frac{\log x_{i}}{ \sum\limits_{l=1}^{k}\log x_{l}}\left( \prod\limits_{j=1,j\neq i}^{k}x_{j}\right) ^{\frac{1}{k-1}} \\ &=&\sum\limits_{i=1}^{k}\frac{\log x_{i}}{\sum\limits_{l=1}^{k}\log x_{l}} \mathcal{G}_{k-1}\left( x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k}\right) \\ &=&\mathcal{L}_{k}\left( x_{1},\ldots ,x_{k}\right) , \end{eqnarray*} where $\mathcal{L}_{k}:\left( 1,+\infty \right) ^{k}\rightarrow \left( 0,+\infty \right) $ is defined by formula \eqref{eq:Lfk_mean}, and $\mathcal{G}_{k-1}$ the $\left( k-1\right) $-variable geometric mean, \begin{equation*} \mathcal{G}_{k-1}\left( x_{1},...,x_{i-1},x_{i+1},...,x_{k}\right) =\left( \prod\limits_{j=1,j\neq i}^{k}x_{j}\right) ^{\frac{1}{k-1}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }i=1,...,{k}. \end{equation*} For arbitrary $x_{1},\ldots ,x_{k}\in \left( 1,+\infty \right) $ put $x_{\min }:=\min \left( x_{1},\ldots ,x_{k}\right) $ and $x_{\max }:=\left( x_{1},\ldots ,x_{k}\right) $. Since \begin{eqnarray*} x_{\min } &=&\sum\limits_{i=1}^{k}\frac{\log x_{i}}{\sum\limits_{l=1}^{k} \log x_{l}}x_{\min }\leq \sum\limits_{i=1}^{k}\frac{\log x_{i}}{ \sum\limits_{i=l}^{k}\log x_{l}}\mathcal{G}_{k-1}\left( x_{1},...,x_{i-1},x_{i+1},...,x_{k}\right) \\ &\leq &\sum\limits_{i=1}^{k}\frac{\log x_{i}}{\sum\limits_{l=1}^{k}\log x_{l} }x_{\max }=x_{\max }, \end{eqnarray*} we have $x_{\min }\leq \mathcal{L}_{k}\left( x_{1},\ldots ,x_{k}\right) \leq x_{\max }$ (and these inequalities are strict if the $k$-tuple $\left( x_{1},\ldots ,x_{k}\right) $ is not constant)\ which shows that $\mathcal{L} _{k}$ is a $k$-variable mean in $\left( 1,+\infty \right) $. Thus $ (ii)\Longrightarrow (iii).$
The implication $(iii)\Longrightarrow (i)$ is obvious. This completes the proof. \end{proof}
\iffalse In the same way as Theorem 2, by replacing the interval $\left( 1,+\infty \right) $ by $\left( 0,1\right) $, we can prove
\begin{theorem} Let $k\in \mathbb{N},$ $k\geq 2$, and $f:\left( 0,1\right) \rightarrow \left( 0,+\infty \right) $ (or $f:\left( 0,1\right) \rightarrow \left( -\infty ,0\right) $) be fixed. The following conditions are equivalent:
(i) the logarithmic Cauchy quotient function $L_{f,k}:\left( 0,1\right) ^{k}\rightarrow \left( 0,1\right) $ is a $k$-variable mean in $\left( 0,1\right) $;
(ii) there is $c\neq 0$ such that the generator of the logarithmic Cauchy quotient $L_{f,k}$ has the form \eqref{eq:meanGenLfk};
\iffalse \begin{equation*} f\left( x\right) =\frac{c\log x}{\sqrt[k-1]{x}},\qquad x\in \left( 0,1\right) ; \end{equation*} \fi
(iii) the following equality holds: \begin{equation*} L_{f,k}=\mathcal{L}_{k} \end{equation*} where $\mathcal{L}_{k}:\left( 0,1\right) ^{k}\rightarrow \left( 0,1\right) $ is a $k$-variable mean given by \begin{equation*} \mathcal{L}_{k}\left( x_{1},\ldots ,x_{k}\right) =\sum\limits_{i=1}^{k}\frac{ \log x_{i}}{\sum\limits_{i=1}^{k}\log x_{i}}\mathcal{G}_{k-1}\left( x_{1},...,x_{i-1},x_{i+1},...,x_{k}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x_{1},\ldots ,x_{k}\in \left( 0,1\right) , \end{equation*} where $\mathcal{G}_{k-1}$ is a $\left( k-1\right) $-variable symmetric geometric mean in $\left( 0,1\right) $. \end{theorem} \fi
In the context of Theorem 2 the natural question arises if it is possible to extend the mean $\mathcal{L}_{k}$ onto $\left( 0,+\infty \right) ^{k}$. An answer gives the following
\begin{remark} The function $\mathfrak{L}_{k}:\left( 0,+\infty \right) ^{k}\rightarrow \left( 0,+\infty \right) $ defined by \begin{equation*} \mathfrak{L}_{k}\left( x_{1},\ldots ,x_{k}\right) :=\left\{ \begin{array}{ccc} \frac{\sum_{i=1}\mathcal{G}_{k-1}\left( x_{1},...,x_{i-1},x_{i+1},...,x_{k}\right) \log x_{i}}{\sum_{l=1}^{k}\log x_{l} } & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{if} & \left( x_{1},\ldots ,x_{k}\right) \in \left( 0,1\right) ^{k}\cup \left( 1,+\infty \right) ^{k} \\ 1 & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{if} & \left( x_{1},\ldots ,x_{k}\right) \notin \left( 0,1\right) ^{k}\cup \left( 1,+\infty \right) ^{k} \end{array} \right. \end{equation*} is a $k$-variable mean in $\left( 0,+\infty \right) $, and it is the only increasing extension of the means $\mathcal{L}_{k}:\left( 1,+\infty \right) ^{k}\rightarrow \left( 1,+\infty \right) $ and $\mathcal{L}_{k}:\left( 0,1\right) ^{k}\rightarrow \left( 0,1\right) .$ \end{remark}
\begin{proof}
By Theorem 2, the restriction $\mathfrak{L}_{k}|_{\left( 0,1\right) ^{k}}$ is a mean in $\left( 0,1\right) $, and $\mathfrak{L}
_{k}|_{\left( 1,+\infty \right) ^{k}}$ is a mean in $\left( 1,\infty \right) $. \ If $\left( x_{1},...,x_{k}\right) \notin \left( \left( 0,1\right) ^{k}\cup \left( 1,+\infty \right) ^{k}\right) $ then, \begin{equation*} \min \left( x_{1},...,x_{k}\right) \leq 1\leq \max \left( x_{1},...,x_{k}\right) , \end{equation*} and, clearly, the number $1$ is the only possible value for an increasing mean at such a point $\left( x_{1},...,x_{k}\right) .$
\end{proof}
To get an involutory counterpart of $\mathfrak{L}_{k},$ which could be denoted by $\mathfrak{L}_{k}^{\limfunc{inv}}$, consider the following
\begin{remark} Let $k\in \mathbb{N}$, $k\geq 2$. A function $M:\left( 1,+\infty \right)
^{k}\rightarrow \left( 1,+\infty \right) $ \verb|[| resp., $M:\left(
0,1\right) ^{k}\rightarrow \left( 0,1\right) $ \verb|]| is a $k$-variable mean in $(1,+\infty)$ \verb|[| resp. in $
\left( 0,1\right) $ \verb|]| iff the function $M^{\limfunc{inv}}:\left( 0,1\right)
^{k}\rightarrow \left( 0,1\right) $ \verb|[| resp. $M^{\limfunc{inv}}:\left(
1,+\infty \right) ^{k}\rightarrow \left( 1,+\infty \right) $ \verb|]| defined by \begin{equation*} M^{\limfunc{inv}}\left( x_{1},\ldots ,x_{k}\right) :=\frac{1}{M\left( \frac{1 }{x_{1}},\ldots ,\frac{1}{x_{k}}\right) } \end{equation*}
is a $k$-variable mean in $\left( 0,1\right) $ \verb|[| resp. in $\left(
1,+\infty \right) $ \verb|]|$.$ \end{remark}
It easy to verify
\begin{remark} The mean $\mathcal{L}_{k}^{\limfunc{inv}}:\left( 0,1\right) ^{k}\rightarrow \left( 0,1\right) ,$ the involutory conjugate mean to $\mathcal{L}_{k},$ is of the form \begin{equation*} \mathcal{L}_{k}^{\limfunc{inv}}\left( x_{1},\ldots ,x_{k}\right) =\frac{ \sum\limits_{i=1}^{k}\left( x_{i}\log x_{i}\right) \mathcal{G}_{k-1}\left( x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k}\right) }{\sum\limits_{l=1}^{k}x_{l}\log x_{l}}, \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x_{1},\ldots ,x_{k}\in \left( 0,1\right) . \end{equation*} \end{remark}
Let us note some properties of the mean $\mathcal{L}_{k}$ in
\begin{proposition} \begin{enumerate}
\item[(i)] $\mathcal{L}_{k}$ is a symmetric and strict mean, but is neither homogeneous nor translative.
\item[(ii)] $\mathcal{L}_{2}$ is the Beckenbach-Gini mean of generator $\log $, i.e. \begin{equation*} \mathcal{L}_{2}\left( x,y\right) =\frac{y\log x+x\log y}{\log x+\log y}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ , \ \ \ \ \ }x,y \in (1,+\infty); \end{equation*} and its involutory conjugate mean \begin{equation*} \mathcal{L}_{2}^{\limfunc{inv}}\left( x,y\right) =xy\frac{\log x+\log y}{ x\log x+y\log y}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x,y \in (0,1); \end{equation*}
\item[(iii)] the bivariable geometric mean $\mathcal{G}$ is invariant with respect to the mean-type mapping $\left( \mathcal{L}_{2}^{\limfunc{inv}},\mathcal{L} _{2}\right) $, i.e. $\mathcal{G\circ }\left( \mathcal{L}_{2}^{\limfunc{inv}}, \mathcal{L}_{2}\right) =\mathcal{G}$, and the sequence $\left( \left( \mathcal{L}_{2}^{\limfunc{inv}},\mathcal{L}_{2}\right) ^{n}:n\in \mathbb{N} \right) $ of iterates of $\left( \mathcal{L}_{2}^{\limfunc{inv}},\mathcal{L} _{2}\right) $ converges uniformly on compact subsets of $\left( 1,+\infty \right) ^{2}$ to $\left( \mathcal{G},\mathcal{G}\right) $ (see Theorem 1 in \cite{JM1999} ). \ \end{enumerate}
\end{proposition}
\begin{example} Indeed, for $k=2$, we have \begin{equation*} \mathcal{L}_{2}\left( 2,3\right) =\frac{3\log 2+2\log 3}{\log 2+\log 3} =\frac{\log{72}}{\log{6}},
\end{equation*} \begin{equation*} \mathcal{L}_{2}\left( 2t,3t\right) =\frac{3t\log{ 2t}+2t\log{ 3t}}{\log {2t}+\log {3t}},
\end{equation*} and \begin{equation*} \mathcal{L}_{2}\left( 2+t,3+t\right) =\frac{(3+t)\log{ (2+t)}+(2+t)\log{ (3+t)}}{\log {(2+t)}+\log {(3+t)}}.
\end{equation*} Setting $t=2$, we get $2\mathcal{L}_{2}\left( 2,3\right) =\frac{\log{144}}{\log 6} \neq \mathcal{L}_{2}\left( 4,6\right)=\frac{\log{5308416}}{\log{24}}$, and $2+\mathcal{L}_{2}\left( 2,3\right)=\frac{\log{5184}}{6} \neq \mathcal{L}_{2}\left( 4,5\right)=\frac{\log{640000}}{\log{20}}$. Thus $\mathcal{L}_{2}$ is neither homogeneous nor translative. A similar argument gives (i) of Proposition 1. \end{example}
\section{A characterization of $\mathcal{L}_{k}$ with the aid of reflexivity of $L_{f,k}$ and a special type of convexity of its generator}
Applying a generalized version of the Krull theorem on linear difference equations (\cite{Krull}) given in Kuczma \cite{Kuczma} p. 114, Theorem 5.11), we give the following characterization of the logarithmic Cauchy quotient mean $\mathcal{L}_{k}$.
\begin{theorem} Let $k\in \mathbb{N}$, $k\geq 2,$ be fixed, and assume that $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ is differentiable and such that the function $\log \circ f\circ \exp \circ \exp $ is convex. Then the following conditions are pairwise equivalent: \begin{enumerate}
\item[(i)]
\ the function $L_{f,k}$ is reflexive in $\left( 1,+\infty \right) ;$
\item[(ii)] \ there is $c>0$ such that $f$ is given by \eqref{eq:meanGenLfk} for all $x\in(1,+\infty)$;
\iffalse \begin{equation*} f\left( x\right) =\frac{c\log x}{\sqrt[k-1]{x}},\qquad x>1; \end{equation*} \fi
\item[(iii)] $\ L_{f,k}=\mathcal{L}_{k}$. \end{enumerate}
\end{theorem}
\begin{proof} Assume (i). By Definition 1 and Remark 4, the function $f$ satisfies the iterative functional equation: \begin{equation*} f\left( x\right) =\frac{x}{k}f\left( x^{k}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x \in (1,+\infty). \end{equation*} Taking $\log $ on both sides gives us \begin{equation*} \log f\left( x\right) =\log f\left( x^{k}\right) +\log x-\log k,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }x\in (1,+\infty). \end{equation*}
Putting $t=\log{x}$ here we come to the equivalent equality \begin{equation*} \log f\left( e^{t}\right) =\log f\left( e^{kt}\right) +t-\log k,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ }t\in (0,+\infty). \end{equation*} Setting $g:\left( 0,+\infty \right) \rightarrow \mathbb{R}$, defined by \begin{equation*} g=\log \circ f\circ \exp, \end{equation*} we can write this equation in the form \begin{equation*} g\left( t\right) =g\left( kt\right) +t-\log k\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }t\in (0,+\infty), \end{equation*} that is \begin{equation*} g\left( e^{\log t}\right) =g\left( e^{\log t+\log k}\right) +e^{\log t}-\log k,\ \ \ \ \ t\in (0,+\infty). \end{equation*} Setting $\Greekmath 011C =\log t$ we get \begin{equation*} g\left( e^{\Greekmath 011C }\right) =g\left( e^{\Greekmath 011C +\log k}\right) +e^{\Greekmath 011C }-\log k,\ \ \ \ \ \Greekmath 011C \in \mathbb{R}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \end{equation*} and, consequently, the function $h:\mathbb{R\rightarrow R}$, defined by \begin{equation*} h:=g\circ \exp =\log \circ f\circ \exp \circ \exp, \end{equation*} satisfies the functional equation \begin{equation*} h\left( \Greekmath 011C +\log k\right) =h\left( \Greekmath 011C \right) +\log k-e^{\Greekmath 011C },\ \ \ \ \ \Greekmath 011C \in \mathbb{R}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} Differentiating both sides with respect to $\Greekmath 011C ,$ we obtain \begin{equation*} h^{\prime }\left( \Greekmath 011C +\log k\right) =h^{\prime }\left( \Greekmath 011C \right) -e^{\Greekmath 011C },\ \ \ \ \ \Greekmath 011C \in \mathbb{R}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} Put \begin{equation*} \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ }F\left( \Greekmath 011C \right) :=-e^{\Greekmath 011C },\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ }\Greekmath 011C \in \mathbb{R} \RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} Note that $F$ is concave, and \begin{equation*} \lim_{\Greekmath 011C \rightarrow -\infty }\left[ F\left( \Greekmath 011C +\log k\right) -F\left( \Greekmath 011C \right) \right] =\lim_{\Greekmath 011C \rightarrow -\infty }\left[ -e^{\Greekmath 011C +\log k}-\left( -e^{\Greekmath 011C }\right) \right] =\lim_{\Greekmath 011C \rightarrow -\infty }\left[ e^{\Greekmath 011C }\left( -k+1\right) \right] =0. \end{equation*} Therefore, in view of the theorem of Krull (\cite{Kuczma}, p. 114, Theorem 5.11), there exists exactly one, up to an additive constant, convex solution $h^{\prime }:\mathbb{R\rightarrow R}$ of the functional equation \begin{equation*} h^{\prime }\left( \Greekmath 011C +\log k\right) =h^{\prime }\left( \Greekmath 011C \right) +F\left( \Greekmath 011C \right) ,\ \ \ \ \ \Greekmath 011C \in \mathbb{R}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} It is easy to verify that, if $f$ is given by formula \eqref{eq:meanGenLfk} in part (ii), then $h=\log \circ f\circ \exp \circ \exp $ satisfies this equation, as \begin{equation*} \log \circ f\circ \exp \circ \exp (\Greekmath 011C )=\log{c}+\Greekmath 011C -\frac{1}{k-1} e^{\Greekmath 011C }, \quad \Greekmath 011C \in \mathbb{R}.
\end{equation*} Since $ \left( \log \circ f\circ \exp \circ \exp \right) ^{\prime }$ is decreasing, the function $\log \circ f\circ \exp \circ \exp $ is concave. Indeed, we have \begin{equation} (\log \circ f\circ \exp \circ \exp (\Greekmath 011C ))^{\prime \prime} =-\frac{1}{k-1}e^{\Greekmath 011C }, \quad \Greekmath 011C \in \mathbb{R},
\end{equation} implying the concavity of the function $\log \circ f\circ \exp \circ \exp $. Thus we have shown (ii). Since logarithmic Cauchy quotients for a given generator $f$ are uniquely determined, the implication (ii)$\Rightarrow $(iii) follows. The remaining implication is due to part (ii) of Remark 1. This finishes the proof. \end{proof}
Weakening the assumption on $\mathcal{L}_{f,k}$ while adding some regularity assumption on the generator $f$, and making use of the idea applied in \cite{JM1972}, one gets the following characterization of the logarithmic Cauchy mean.
\begin{theorem} Let $k\in \mathbb{N}$, $k\geq 2$ be fixed. Assume that $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ is such that, for some $c>0,$ the function \begin{equation} \left( 0,+\infty \right) \ni x\longmapsto \frac{f\left( x\right) -c\left( x-1\right) }{\left( x-1\right) ^{2}} \label{eq:BdLfk} \end{equation} is bounded in a right vicinity of $1$.
Then the following conditions are pairwise equivalent \begin{enumerate}
\item[(i)] \ the function $L_{f,k}$ is reflexive in $\left( 1,+\infty \right) ;$
\item[(ii)] \ there is $c>0$ such that $f$satisfies \eqref{eq:meanGenLfk} for all $x\in (1,+\infty)$;
\iffalse \begin{equation*} f\left( x\right) =\frac{c\log x}{\sqrt[k-1]{x}}, \end{equation*} \fi
\item[(iii)] $\ L_{f,k}=\mathcal{L}_{k}$. \end{enumerate} \end{theorem}
\begin{proof} From \eqref{eq:BdLfk} we have \begin{equation} f\left( x\right) =c\left( x-1\right) +\Greekmath 0127 \left( x\right) \left( x-1\right) ^{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x\in (1,+\infty), \label{eq:f(x)} \end{equation} where the function $\Greekmath 0127 :\left( 1,+\infty \right) \rightarrow \mathbb{R}$ defined by \begin{equation*} \Greekmath 0127 \left( x\right) :=\frac{f\left( x\right) -c\left( x-1\right) }{ \left( x-1\right) ^{2}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x\in (1,+\infty)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \end{equation*} is bounded in an interval $\left( 1,1+r\right) $, for some $r>0.$
Assume (i). In view of Remark 4, the generator $f$ of $L_{f,k}$ satisfies the functional equation \eqref{eq:Lfk_Reflexivity}, that is equivalent to the functional equation \begin{equation} f\left( x\right) =\frac{k}{x^{\frac{1}{k}}}f\left( x^{\frac{1}{k}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x \in (1, +\infty). \label{eq:LfkReflexivity2} \end{equation} Taking into account \eqref{eq:f(x)}, we conclude that $\Greekmath 0127 $ satisfies the functional equation \begin{equation*} c\left( x-1\right) +\Greekmath 0127 \left( x\right) \left( x-1\right) ^{2}=\frac{k}{ x^{\frac{1}{k}}}\left[ c\left( x^{\frac{1}{k}}-1\right) +\left( x^{\frac{1}{k }}-1\right) ^{2}\Greekmath 0127 \left( x^{\frac{1}{k}}\right) \right] \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x \in (1, +\infty), \end{equation*}
which can be written in the form
\begin{equation} \Greekmath 0127 \left( x\right) =\frac{c\left( 1+k-x-kx^{-\frac{1}{k}}\right) }{ \left( x-1\right) ^{2}}+kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1} \right) ^{2}\Greekmath 0127 \left( x^{\frac{1}{k}}\right) , \label{eq:BoundGenLfk} \end{equation} and, moreover, $\Greekmath 0127 $ is bounded in an interval $\left( 1,1+r\right) $.
Assume that the functions $\Greekmath 0127 _{1},\Greekmath 0127 _{2}:\left( 1,+\infty \right) \rightarrow \mathbb{R}$ are bounded in $\left( 1,1+r\right) $ for some $r>0$, and satisfy equation \eqref{eq:BoundGenLfk}, that is \begin{equation*} \Greekmath 0127 _{i}\left( x\right) =\frac{c\left( 1+k-x-kx^{-\frac{1}{k}}\right) }{ \left( x-1\right) ^{2}}+kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1} \right) ^{2}\Greekmath 0127 _{i}\left( x^{\frac{1}{k}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x\in (1,+\infty); \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ }i=1,2. \end{equation*} Hence, putting \begin{equation*} \Greekmath 0120 :=\left\vert \Greekmath 0127 _{1}-\Greekmath 0127 _{2}\right\vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ and \ \ \ \ }\Greekmath 010B \left( x\right) :=x^{\frac{1}{k}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ for \ }x\in (1,+\infty),\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ } \end{equation*} we see that $\Greekmath 0120 $ is nonnegative and bounded solution of the functional equation \begin{equation} \Greekmath 0120 \left( x\right) =kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1} \right) ^{2}\Greekmath 0120 \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ }\Greekmath 010B \left( x\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x\in (1,+\infty). \label{eq:BoundGenLfk!} \end{equation} Note that
\begin{eqnarray*} \frac{x^{\frac{1}{k}}-1}{x-1} &=&\frac{x^{\frac{1}{k}}-1}{\left( x^{\frac{1}{ k}}\right) ^{k}-1}=\frac{x^{\frac{1}{k}}-1}{\left( x^{\frac{1}{k}}-1\right) \left( \left( x^{\frac{1}{k}}\right) ^{k-1}+\left( x^{\frac{1}{k}}\right) ^{k-2}+\cdots+x^{\frac{1}{k}}+1\right) } \\ &=&\frac{1}{\left( x^{\frac{1}{k}}\right) ^{k-1}+\left( x^{\frac{1}{k} }\right) ^{k-2}+\cdots+x^{\frac{1}{k}}+1}, \end{eqnarray*} so, for all $x\in (1,+\infty)$, we have \begin{equation*} kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1}\right) ^{2}=\frac{kx^{- \frac{1}{k}}}{\left( \left( x^{\frac{1}{k}}\right) ^{k-1}+\left( x^{\frac{1}{ k}}\right) ^{k-2}+\cdots+x^{\frac{1}{k}}+1\right) ^{2}} \end{equation*} Hence \begin{equation*} \lim_{x\rightarrow 1}kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1} \right) ^{2}=\lim_{x\rightarrow 1}\frac{kx^{-\frac{1}{k}}}{\left( \left( x^{ \frac{1}{k}}\right) ^{k-1}+\left( x^{\frac{1}{k}}\right) ^{k-2}+\cdots+x^{\frac{ 1}{k}}+1\right) ^{2}}=\frac{k}{k^{2}}=\frac{1}{k}, \end{equation*} and, as $k\geq 2$, \ there is $r>0$ such that \begin{equation*} kx^{-\frac{1}{k}}\left( \frac{x^{\frac{1}{k}}-1}{x-1}\right) ^{2}\leq \frac{1 }{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x\in \left( 1,1+r\right) . \end{equation*} Since $\Greekmath 010B \left( \left( 1,1+r\right) \right) \subset \left( 1,1+r\right) $, in view of \eqref{eq:BoundGenLfk!}, \begin{equation*} 0\leq \Greekmath 0120 \left( x\right) \leq \frac{1}{2}\Greekmath 0120 \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ }\Greekmath 010B \left( x\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x\in \left( 1,1+r\right) , \end{equation*} the boundedness of $\Greekmath 0120 $ implies that \begin{equation*} \Greekmath 0120 \left( x\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x\in \left( 1,1+r\right) . \end{equation*} Now, from \eqref{eq:BoundGenLfk!}, taking into account that \begin{equation*} \lim_{n\rightarrow \infty }\Greekmath 010B ^{n}\left( x\right) =1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x\in (1,+\infty) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \end{equation*} we conclude that $\Greekmath 0120 \left( x\right) =0$ for every $x \in(0,+\infty)$ which shows that $ \Greekmath 0127 _{1}=\Greekmath 0127 _{2}$. This proves that there is at most one solution of equation \eqref{eq:LfkReflexivity2} satisfying condition \eqref{eq:BdLfk}.
\iffalse which can be written in the form \begin{equation} \Greekmath 0127 \left( x\right) =\frac{c \left(1+k-x -kx^{-\frac{1}{k}} \right) }{\left( x-1\right) ^{2}}+{k}{x^{- \frac{1}{k}}}\left( \frac{x^{\frac{1}{k}}-1}{x-1}\right) ^{2}\Greekmath 0127 \left( x^{\frac{1}{k}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ }x>1, \@ifnextchar*{\@tagstar}{\@tag}{7} \end{equation} and, moreover, $\Greekmath 0127 $ is bounded in $\left( 1,1+r\right) $.
Assume that the functions $\Greekmath 0127 _{1},\Greekmath 0127 _{2}:\left( 1,+\infty \right) \rightarrow \mathbb{R}$ are bounded in an interval $\left( 1,1+r\right) $ for some $r>0$, and satisfy equation (7), that is \begin{equation*} \Greekmath 0127 _{i}\left( x\right) =\frac{c \left(1+k-x-kx^{-\frac{1}{k}}\right)}{\left( x-1\right) ^{2}}+ k{x^{-\frac{1}{k}}}\left(\frac{x^{\frac{1}{k}}-1}{x-1}\right) \Greekmath 0127 _i \left( x^{\frac{1}{k}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x\in (1, +\infty) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{; \ \ \ }i=1,2. \end{equation*} Hence, putting \begin{equation*} \Greekmath 0120 :=\left\vert \Greekmath 0127 _{1}-\Greekmath 0127 _{2}\right\vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ and \ \ \ }\Greekmath 010B \left( x\right) :=x^{\frac{1}{k}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ for }x \in (1, +\infty), \end{equation*} we see that $\Psi$ is nonnegative and bounded solution of the functional equation \begin{equation*} \Greekmath 0120 (x)=kx^{-\frac{1}{k}} \left( \frac{x^{\frac{1}{k}-1}}{x-1}\right) \Greekmath 0120 ({\Greekmath 010B (x)}), \quad x \in (1, + \infty).
\end{equation*} Note that \begin{eqnarray*} \frac{x^\frac{1}{k}-1}{x-1}= \frac{x^{\frac{1}{k}}-1}{\left(x^\frac{1}{k}\right)^k-1}\\ = \frac{x^{\frac{1}{k}}-1}{\left( x^{\frac{1}{k}}-1\right) \left(\left(x^\frac{1}{k}\right)^{k-1}+\left(x^\frac{1}{k}\right)^{k-2}+\cdots +x^\frac{1}{k}+1\right)}\\ = \frac{1}{ \left(\left(x^\frac{1}{k}\right)^{k-1}+\left(x^\frac{1}{k}\right)^{k-2}+\cdots +x^\frac{1}{k}+1\right)}, \end{eqnarray*} so for all $x \in(1, + \infty)$, we have \begin{equation*} kx^{-\frac{1}{k}} \left(\frac{x^\frac{1}{k}-1}{x-1}\right)^2 = \frac{kx^{-\frac{1}{k}} }{\left({\left( x^\frac{1}{k}\right)^{k-1}+\left( x^\frac{1}{k}\right)^{k-2}+\cdots+x^\frac{1}{k}+1}\right)^2}.
\end{equation*} Hence, .... \fi
\iffalse
\end{equation*}
It follows that \begin{equation*} \left\vert \Greekmath 0127 _{1}\left( x\right) -\Greekmath 0127 _{2}\left( x\right) \right\vert =\frac{k}{x^{\frac{1}{k}}}\left( \frac{x^{\frac{1}{k}}-1}{x-1} \right) ^{2}\left\vert \Greekmath 0127 _{1}\left( x^{\frac{1}{k}}\right) -\Greekmath 0127 _{2}\left( x^{\frac{1}{k}}\right) \right\vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x>1. \end{equation*}
Since \begin{equation*} 0<\frac{k}{x^{\frac{1}{k}}}\left( \frac{x^{\frac{1}{k}}-1}{x-1}\right) ^{2}= \frac{k}{x^{\frac{1}{k}}}\frac{1}{\left( x^{\frac{k-1}{k}}+x^{\frac{k-2}{k} }+...+x^{\frac{1}{k}}+1\right) ^{2}}<\frac{k}{1}\frac{1}{k^{2}}=\frac{1}{k} \leq \frac{1}{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x>1, \end{equation*} we hence get \begin{equation*} \left\vert \Greekmath 0127 _{1}\left( x\right) -\Greekmath 0127 _{2}\left( x\right) \right\vert \leq \frac{1}{2}\left\vert \Greekmath 0127 _{1}\left( x^{\frac{1}{k} }\right) -\Greekmath 0127 _{2}\left( x^{\frac{1}{k}}\right) \right\vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ \ \ }x>1. \end{equation*} Thus, putting
\begin{equation*} 0\leq \Greekmath 0120 \left( x\right) \leq \frac{1}{2}\Greekmath 0120 \left( \Greekmath 010B \left( x\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x>1. \end{equation*} Hence, by induction, we obtain \begin{equation} 0\leq \Greekmath 0120 \left( x\right) \leq \frac{1}{2^{n}}\Greekmath 0120 \left( \Greekmath 010B ^{n}\left( x\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x \in (1, +\infty)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ }n\in \mathbb{N}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \@ifnextchar*{\@tagstar}{\@tag}{8} \end{equation} where $\Greekmath 010B ^{n}$ is the $n$th iterate of the function $\Greekmath 010B $.
Take arbitrary $b>0$. \ Since $\lim_{n\rightarrow +\infty }\Greekmath 010B ^{n}\left( b\right) =1$ and $\Greekmath 010B $ is increasing, there is an $N\in \mathbb{N}$ such that \begin{equation*} 1<\Greekmath 010B ^{N}\left( x\right) \leq \Greekmath 010B ^{N}\left( b\right) <1+r\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ \ \ for all \ \ \ }x\in \left( 0,b\right] . \end{equation*} Hence, making use of (8), we obtain \begin{equation*} 0\leq \Greekmath 0120 \left( x\right) \leq \frac{1}{2^{N}}\Greekmath 0120 \left( \Greekmath 010B ^{N}\left( x\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ \ \ \ }x\in \left( 0,b\right] \RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} As $\Greekmath 010B ^{N}\left( \left( 0,b\right] \right) \subset \left( 1,1+r\right) , $ and $\Greekmath 0120 $ is bounded on $\left( 1,1+r\right) ,$ we conclude that $ M:=\sup \left\{ \Greekmath 0120 \left( x\right) :x\in \left( 0,b\right] \right\} $ is finite, and that \begin{equation*} 0\leq M\leq \frac{1}{2^{N}}M, \end{equation*} whence $M=0$. From the definitions of $M$ and $\Greekmath 0120 $ we conclude that $ \Greekmath 0127 _{1}=\Greekmath 0127 _{2}$, which proves that there is at most one solution of equation (6) satisfying condition (4).
Now the implication $(i)\Longrightarrow (ii)$ follows from the fact that the function \begin{equation*} \left( 1,+\infty \right) \ni x\longmapsto \frac{c\log x}{\sqrt[k-1]{x}}, \end{equation*} is a solution of the reflexivity equation (6) and satisfies condition (4).
The remaining implications are obvious. \fi
Now the implication $(i)\Longrightarrow (ii)$ follows from the fact that the function \begin{equation*} \left( 1,+\infty \right) \ni x\longmapsto \frac{c\log x}{\sqrt[k-1]{x}}, \end{equation*} is a solution of the reflexivity equation \eqref{eq:LfkReflexivity2} and satisfies condition \eqref{eq:BdLfk}.
The remaining implications are obvious. \end{proof}
Since twice continuously differentiable functions satisfy condition \eqref{eq:BdLfk}, the following result is an immediate consequence of the above result.
\begin{corollary} Let $k\in \mathbb{N}$, $k\geq 2$ be fixed. Assume that $f:\left( 1,+\infty \right) \rightarrow \left( 0,+\infty \right) $ is of the class $C^{2}$ and the function \begin{equation*} \left( 1,+\infty \right) \ni x\longmapsto f\left( x\right) \end{equation*} has an extension that is of the class $C^{2}$ in the interval $\left[ 1,+\infty \right) $.
Then the following conditions are pairwise equivalent: \begin{enumerate}
\item[(i)] \ the function $L_{f,k}$ is a premean in $\left( 1,+\infty \right) ;$
\item[(ii)] \ there is $c>0$ such that $f$ satisfies \eqref{eq:meanGenLfk} for all $x\in (1,+\infty)$
\iffalse \begin{equation*} f\left( x\right) =\frac{c\log x}{\sqrt[k-1]{x}},\qquad x>1; \end{equation*} \fi
\item[(iii)] $\ L_{f,k}=\mathcal{L}_{k}$. \end{enumerate}
\end{corollary}
\end{document}
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arXiv
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Two fair eight-sided dice have their faces numbered from 1 to 8. What is the expected value of the sum of the rolls of both dice?
To find the expected value of a double roll, we can simply add the expected values of the individual rolls, giving $4.5 + 4.5 = \boxed{9}$.
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Math Dataset
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Chandra Observations of the Spectacular A3411-12 Merger Event
Andrade-Santos, Felipe and Weeren, Reinout J. van and Gennaro, Gabriella Di and Wittman, David and Ryu, Dongsu and Lal, Dharam Vir and Placco, Vinicius M. and Fogarty, Kevin and Jee, M. James and Stroe, Andra and Sobral, David and Forman, William R. and Jones, Christine and Kraft, Ralph P. and Murray, Stephen S. and Brüggen, Marcus and Kang, Hyesung and Santucci, Rafael and Golovich, Nathan and Dawson, William (2019) Chandra Observations of the Spectacular A3411-12 Merger Event. The Astrophysical Journal, 887 (1). ISSN 0004-637X
Text (1910.07405v1)
1910.07405v1.pdf - Accepted Version
Official URL: https://doi.org/10.3847/1538-4357/ab4ce5
We present deep Chandra observations of A3411-12, a remarkable merging cluster that hosts the most compelling evidence for electron re-acceleration at cluster shocks to date. Using the $Y_X-M$ scaling relation, we find $r_{500} \sim 1.3$ Mpc, $M_{500} = (7.1 \pm 0.7) \times 10^{14} \ M_{\rm{\odot}}$, $kT=6.5\pm 0.1$ keV, and a gas mass of $M_{\rm g,500} = (9.7 \pm 0.1) \times 10^{13} M_\odot$. The gas mass fraction within $r_{500}$ is $f_{\rm g} = 0.14 \pm 0.01$. We compute the shock strength using density jumps to conclude that the Mach number of the merging subcluster is small ($M \leq 1.15_{-0.09}^{+0.14}$). We also present pseudo-density, projected temperature, pseudo-pressure, and pseudo-entropy maps. Based on the pseudo-entropy map we conclude that the cluster is undergoing a mild merger, consistent with the small Mach number. On the other hand, radio relics extend over Mpc scale in the A3411-12 system, which strongly suggests that a population of energetic electrons already existed over extended regions of the cluster.
The Astrophysical Journal
This is an author-created, un-copyedited version of an article accepted for publication/published in The Astrophysical Journal. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi: This is an author-created, un-copyedited version of an article accepted for publication/published in The Astrophysical Journal. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi: https://doi.org/10.3847/1538-4357/ab4ce5
Faculty of Science and Technology > Physics
https://eprints.lancs.ac.uk/id/eprint/138073
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CommonCrawl
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\begin{document}
\title{First-order tree-to-tree functions}
\author{Miko\l{}aj Boja\'nczyk and Amina Doumane}
\begin{abstract}
We study tree-to-tree transformations that can be defined in first-order logic or monadic second-order logic. We prove a decomposition theorem, which shows that every transformation can be obtained from prime transformations, such as tree-to-tree homomorphisms or pre-order traversal, by using combinators such as function composition. \end{abstract}
\maketitle
\section*{Erratum} In an early version of this paper, Theorem~\ref{thm:normalise} was stated without the restriction that $\lambda$-terms to be normalized need to use a unique variable as a bound variable. This old version is not correct, as pointed to us by Lê Thành Dũng (Tito) Nguy\~{\^e}n. His counter-example can be found in Example~\ref{ex:tito}.
\section{Introduction}
The purpose of this paper is to decompose tree transformations into simple building blocks. An important inspiration is the Krohn-Rhodes theorem~\cite[p.~454]{Krohn1965}, which says that every string-to-string function recognised by a Mealy machine can be decomposed into certain prime functions.
\paragraph*{Regular functions.} The transformations studied in this paper are the regular functions.
In~\cite[Theorem 13]{engelfrietMSODefinableString2001}, Engelfriet and Hoogeboom proved that deterministic two-way transducers recognise the same string-to-string functions as \mso transductions. Because of this and other properties -- such as closure under composition~\cite[Theorem 1]{chytilSerialComposition2Way1977} and decidable equivalence~\cite[Th.~1]{gurariEquivalenceProblemDeterministic1982} -- this class of functions is now called the \emph{regular string-to-string functions}. Other equivalent descriptions of the regular functions include: string transducers of Alur and {\v C}ern{\'y}~\cite{alurExpressivenessStreamingString2010}, and several models based on combinators~\cite{alur2014regular,daveGastinKrishna18, bojanczykRegularFirstOrderList2018}.
There are also regular functions for trees, which can be defined using any of the following equivalent models: \mso tree-to-tree transductions~\cite[Section 3]{bloem_comparison_2000}, single use attributed tree grammars~\cite{bloem_comparison_2000}, macro tree transducers that are single use~\cite{ENGELFRIET199934} or of linear size increase~\cite[Theorem 7.1]{engelfriet_macro_2003}, and streaming tree transducers~\cite[Theorem 4.6]{alur2017streaming}.
The goal of this paper is to prove a decomposition result for regular tree-to-tree functions. As in the Krohn-Rhodes theorem, we want to show that every such function can be obtained by combining certain prime functions.
\paragraph*{First-order transductions. } Although \mso transductions are the more popular model, we work mainly with the less expressive model of first-order transductions. Why?
As we explain in Section~\ref{sec:mso-trans}, every \mso tree-to-tree transduction can be decomposed as: (a) first, a relabelling defined in \mso, which does not change the tree structure; followed by (b) a first-order tree-to-tree transduction. In this sense, as far as transformations of the tree structure are concerned, first-order and \mso transductions have the same expressive power. Another argument for the importance of first-order tree-to-tree transductions is a connection with the $\lambda$-calculus. As we explain in Section~\ref{sec:stt-derivable}, first-order tree-to-tree transductions are expressive enough to capture evaluation of $\lambda$-terms (assuming the use of a single variable and linearity), and such evaluation turns out to be one of the core computational steps implicit in a tree-to-tree transduction.
Another advantage of first-order logic on trees, compared to \mso, is a better decomposition theory, in the sense of decomposing formulas into simpler ones~\cite{haferthomas,bojanczykDecidablePropertiesTree2004,esik-weil1}.
For our paper, the most useful decomposition is a remarkable theorem of Schlingloff, which says that first-order logic on trees is equivalent to a certain two-way variant of {\sc ctl}~\cite[Th.~4.5]{schlingloff1992expressive}. In contrast, there are no such results for \mso.
Summing up, we believe that first-order tree transformations are expressive, have a strong theory, and deserve to leave the shadow of their better known \mso cousin.
\paragraph*{Structured datatypes.} We present our main decomposition result in a formalism based on functional programming (in a combinatory variant, i.e.~without variables), with structured datatypes such as pairs or co-pairs. The motivation behind this approach -- which is inspired by~\cite{bojanczykRegularFirstOrderList2018} -- is to avoid encoding datatypes in our constructions using syntactic annotation such as endmarkers and separators. Thanks to the structured datatypes, we can use established operations such as {\tt map}, and we can assign informative types to our functions, such as $\Sigma_1 \times \Sigma_2 \to \Sigma_i$ for projection, as opposed to saying that all functions input and output trees.
The choice of datatypes for trees is harder than for the string case that was studied in~\cite{bojanczykRegularFirstOrderList2018}. The difficulty is in splitting the input into smaller pieces. A piece of a string is also a string, but this is no longer true for trees, where the pieces have dangling edges (or variables). As a result, more complicated datatypes are needed; and our design choices lead us to functions that operate on ranked sets, where each element has an associated arity.
This is a long paper. Given the limited space, we have decided to prioritise explaining design choices and intuitions, with examples and many pictures. As a result, almost all of the proofs are in the appendix.
\section{Trees and tree-to-tree functions} \label{sec:trees-transductions}
In this section, we describe the trees and tree-to-tree functions that are discussed in this paper.
A \emph{ranked set} is a set where each element has an associated \emph{arity} in $\set{0,1,2,\ldots}$. If $a$ of a ranked set has arity $n$, then elements of $\set{1,\ldots,n}$ are called \emph{ports of $a$}. We adopt the convention that ranked sets are red, e.g.~$\rSigma$ or $\rGamma$, and other objects (elements of ranked sets, or unranked sets) are black. We use ranked sets as building blocks for trees. The following picture describes the notion of trees that we use and some terminology: \mypic{1}
We use standard tree terminology, such as ancestor, descendant, child, parent. We write $\trees \rSigma$ for the (unranked) set of trees over a ranked set $\rSigma$. This paper is about \emph{tree-to-tree functions}, which are functions of the type \begin{align*} f : \trees \rSigma \to \trees \rGamma. \end{align*}
\subsection{First-order logic and transductions} To define tree-to-tree functions and tree languages, we use logic, mainly first-order logic and monadic second-order logic \mso. The idea is to view a tree as a model, and to use logic to describe properties and transformations of such models.
A \emph{vocabulary} is defined to be a set of relation names, each one with associated arity. We do not use function symbols in this paper. A vocabulary can be formalised as a ranked set, which is why we use red letters like $\ranked \sigma $ or $\ranked \tau$ for vocabularies.
\begin{definition}[Tree as a model]\label{def:tree-model}
For a tree $t$ over a ranked alphabet $\rSigma$, its \emph{associated model}
is defined as follows. The universe is the nodes of the tree, and it is equipped with the following relations:
$$\begin{array}{lcll}
x<y & & \text{$x$ is an ancestor of $y$} & \text{arity 2}\\
\mathrm{child}_i(x) & & \text{$x$ is an $i$-th child ($i\in \set{1,2,\ldots}$)} & \text{arity 1} \\
a(x) & & \text{$x$ has label $a$ ($a \in \rSigma$)} & \text{arity 1}
\end{array}$$
\end{definition}
The $i$-th child predicates are only needed for $i$ up to the maximal arity of letters in the ranked alphabet, and hence the vocabulary in the above definition is finite. We refer to this vocabulary as \emph{the vocabulary of trees over $\rSigma$}.
A sentence of first-order logic (or \mso) over this vocabulary describes a tree language, namely the set of trees whose associated models satisfy the sentence. For example, the sentence
\begin{align*}
\forall x \ a(x) \Rightarrow \exists y \ x < y \land b(x)
\end{align*}
is true in (the models associated to) trees $t$ where every node with label $a$ has a descendant with label $b$. For more background about defining properties of trees using logic, see the survey of Thomas~\cite{thomas1997languages}.
The regular tree languages are exactly those that can be defined in \mso, which was proved by Doner~\cite[Corollary 3.11]{Doner70}, and also Thatcher and Wright~\cite[p.~74]{thatcherGeneralizedFiniteAutomata1968}. The tree languages definable in first-order logic are a proper subset of those definable in \mso, and it is an open problem whether or not one can decide if a regular tree language can be defined in first-order logic~\cite[Section 3]{bojanczyk2015automata}. This is in contrast to the case of words, where the decidable characterisation of first-order logic by Sch\"utzenberger-McNaughton-Papert~\cite[Theorem 10.5]{McNaughtonPapert71} is a cornerstone of algebraic language theory.
\paragraph*{Tree-to-tree functions.}
Apart from defining tree languages, logic can also be used to define transformations on models. In the context of this paper, we are interested mainly in first-order transductions, defined below. Roughly speaking, a first-order transduction uses first-order logic to define a new tree structure on the input tree.
\begin{definition}[First-order tree-to-tree transduction] \label{def:fo-transduction} A tree-to-tree function is called a \emph{first-order transduction} if it can be obtained by composing any number of operations\footnote{There is a normal form of first-order transductions, where at two phases are used: first item 1, then item 2. We do not need the normal form, so we do not prove it, but it can be shown similarly to~\cite[Section 7.1.5]{courcelle1991}. } of the following two kinds: \begin{enumerate}
\item \emph{Copying.} Let $k \in \set{1,2,\ldots}$. Define $k$-copying to be the operation which inputs a tree and outputs a tree where every node is preceded by a chain of $k-1$ unary nodes with a fresh label $\blueball$, as in the following picture:
\mypic{94}
After $k$-copying, the number of nodes grows $k$ times.
\item \emph{Non-copying first-order transductions.} This is a tree-to-tree function which uses first-order logic to define a new tree structure over the nodes of the input tree. The syntax of such a transduction is given by:
\begin{enumerate}
\item \emph{Input and output alphabets} $\rSigma$ and $\rGamma$, which are finite ranked sets. We use the name \emph{input vocabulary} for the vocabulary of trees over the input alphabet $\rSigma$, likewise we define the \emph{output vocabulary}.
\item \label{it:universe-formula} A first-order formula over the input vocabulary, with one free variable, called the \emph{universe formula}.
\item \label{it:tree-structure} For each relation of the output vocabulary, of arity $n$, a corresponding first-order formula over the input vocabulary with $n$ free variables.
\end{enumerate}
The transduction inputs a tree over the input alphabet, and outputs a tree over the output alphabet where:
\begin{itemize}
\item the nodes are those nodes of the input tree that satisfy the universe formula in item~\ref{it:universe-formula};
\item the labels, descendant, and child relations are defined by the formulas in item~\ref{it:tree-structure}.
\end{itemize}
In order for the transduction to be well defined, the formulas in item~\ref{it:tree-structure} must be such that they produce a tree model for every input tree.
\end{enumerate} \end{definition}
If we allowed monadic second-order logic \mso in items~\ref{it:universe-formula} and~\ref {it:tree-structure} (the free variables of the formulas would still be first-order variables ranging over tree nodes), then we would get the \mso tree-to-tree transductions of Bloem and Ensgelfriet~\cite[Section 3]{bloem_comparison_2000}. We discuss these in Section~\ref{sec:mso-trans}.
We conclude this section with two examples of first-order tree-to-tree transductions.
\begin{example}\label{ex:filter-first}
Let the input and output alphabets be:
\mypic{17}
and consider the function which removes the unary nodes: \begin{center} \includegraphics[scale=.35, page=19]{pics.pdf} \end{center} This is a non-copying first-order transduction. The universe formula selects nodes which have non-unary labels. The descendant relation is inherited from the input tree. To define the child relation on the output tree, we use the descendant relation in the input tree. A node $x$ satisfies the unary $i$-th child predicate in the output tree if it satisfies the following first-order formula in the input tree: \begin{align*}
\exists y \ \child i (y) \land \underbrace{y \le x \land \forall z\ (y \le z < x \Rightarrow \blueball(z))}_{\substack{\text{$y$ is the farthest ancestor that can be} \\ \text{reached from $x$ using only unary nodes}}}. \end{align*} This example shows the usefulness of first-order logic with descendant, as opposed to child only as used in~\cite{benediktSegoufin2009}.
\end{example}
\begin{example}\label{ex:pre-order-main} Define \emph{pre-order} on nodes in a tree as follows: $x$ is before $y$ if either $x \le y$, or there exist nodes $x'$ and $y'$ such that $x' \le x$, $y' \le y$, and $x'$ is a sibling of $y'$ with a smaller child number. Consider the tree-to-tree function which transforms a tree into a list of its nodes in pre-order traversal, as explained in the following picture:
\mypic{112}
This function is a first-order tree-to-tree transduction,
because the pre-order is first-order definable. Unlike Example~\ref{ex:filter-first}, we need copying, because a node of arity $n$ in the input tree corresponds to $n+2$ nodes in the output tree. \end{example}
\section{Derivable functions}\label{sec:derivable-functions} In this section, we state the main result of this paper, which says that the first-order tree-to-tree transductions are exactly those that can be obtained by starting with certain prime functions (such as pre-order traversal from Example~\ref{ex:pre-order-main}) and applying certain combinators (such as function composition).
The guiding principle behind our approach is to describe tree-to-tree functions without using any iteration mechanisms, such as states or {\tt fold} functions. This principle validates the choice of first-order logic. If we were to use \mso, at the very least we would need to have some mechanism for groups, which are a basic building block for Krohn-Rhodes decompositions, or for evaluating Boolean formulas.
\subsection{Datatypes} \label{sec:datatype-constructors} The prime functions and combinators use datatypes such as pairs of trees, or pairs of trees of pairs, etc. Although these datatypes could be encoded in trees, we avoid this encoding and use explicit datatype constructors.
An important property of our datatypes is that they represent ranked sets, i.e.~each element of a datatype has an arity. The datatypes are obtained from the atomic datatypes by applying four datatype constructors, as described below.
\paragraph*{Atomic datatypes.} Every finite ranked set is an atomic datatype. Apart from finite ranked sets, we allow one more atomic datatype: the \emph{terminal ranked set} $\termset$ which contains exactly one element of every arity.
The set is called terminal because it admits a unique arity preserving function from every ranked set. We use $\termset$ for partial functions: a partial function with output type $\rSigma$ can be seen as a total function of output type $\ranked{\rSigma + \termset}$, which uses $\termset$ for undefined values.
\paragraph*{Terms.} The central datatype constructor is the \emph{term} constructor, which is a generalisation of trees to higher arities. A term is a tree with dangling edges, called ports. The dangling edges ares used to decompose trees (and other terms) into smaller pieces, as illustrated by the figure below.
\mypic{15} Formally speaking, terms are defined by induction as follows. As term over a ranked set $\rSigma$ is either the \emph{identity term} denoted by $\portletter$, which consists of a port and nothing else, or otherwise it is an expression of the form $a \tensorpair{t_1,\ldots,t_n}$ where $a \in \rSigma$ has arity $n$, and $t_1,\ldots,t_n$ are already defined terms. The arity of a term is the number of ports. Terms of arity zero are the same as trees. We write $\tmonad \rSigma$ for the ranked set of terms over a ranked set $\rSigma$. Because the term constructor -- like other datatype constructors -- outputs a ranked set, it makes sense to talk about terms of terms, etc.
Terms are a monad, in the category of ranked sets and arity preserving functions\footnote{An almost identical monad is used in ~\cite[Section 9.2]{bojanczykRecognisableLanguagesMonads2015}, which differs from ours in that it allows multiple uses of a single port.}. The unit of the monad, an operation of type $\ranked{\rSigma \to \tmonad \rSigma}$, is illustrated in the following picture: \mypic{98} The product of the monad, an operation of type $\ranked{\tmonad \tmonad \rSigma \to \tmonad \rSigma}$ that we call \emph{flattening}, is illustrated in the following picture: \mypic{97} This monad structure will be part of our prime functions. \paragraph*{Products and coproducts.} There are two binary datatype constructors \begin{align*} \underbrace{\ranked{\Sigma_1 \product \Sigma_2}}_{\text{product}} \qquad \underbrace{\ranked{\Sigma_1 + \Sigma_2}}_{\text{coproduct}}. \end{align*} An element of the product is a pair $\tensorpair{a_1,a_2}$ where $a_i \in \ranked{\Sigma_i}$. The arity of the pair is the sum of arities of its two coordinates $a_1$ and $a_2$.
An element of the coproduct is a pair $(i,a)$ where $i \in \set{1,2}$ and $a \in \ranked{\Sigma_i}$. The arity is inherited from $a$.
The set of terms can be defined in terms of products and coproducts, as the least solution of the equation: \begin{align*} \tmonad \rSigma = \redset{\portletter} \ranked{+\coprod_{\black {a \in} \rSigma}} \powersmall{(\tmonad \rSigma)} {\text{arity of $a$}} \end{align*} where $\ranked \coprod$ denotes possibly infinite coproduct and $\powersmall X n$ denotes the $n$-fold product of a ranked set $\ranked X$ with itself.
\paragraph*{Folding.} The final -- and maybe least natural -- datatype constructor called \emph{folding}. Folding has two main purposes: (1) reordering ports in a term; and (2) reducing arities by grouping ports into groups.
Folding is not one constructor, but a family of unary constructors $\reduce k \rSigma$, one for every $k \in \set{1,2,3,\ldots}$. An $n$-ary element of $\reduce k \rSigma$, which is called a \emph{$k$-fold}, consists of an element $a \in \rSigma$ together with an injective \emph{grouping} function \begin{align*}
f : \underbrace{\set{1,\ldots,\text{arity of $a$}}}_{\substack{\text{an element of this set is}\\\text{called a port of $a$}}} \to \underbrace{\set{1,\ldots,n} \times \set{1,\ldots,k} }_{\text{these pairs are called \emph{sub-ports}}} \end{align*} We denote such an element as $a/f$ and draw it like this: \mypic{53}
Already for $k=1$, the constructor $\reduce 1$ is non-trivial. For example, $\reduce 1 \tmonad \rSigma$ is a generalisation of terms where ports are not necessarily ordered left-to-right (because the grouping function need not be monotone), and some ports need not appear (because the grouping function need not be total); in other words this is the same as terms in the usual sense of universal algebra, with the restriction that each variable is used at most once (sometimes called linearity).
When viewed as a family of datatype constructors, folds have a monad-like structure: they are a graded monad in the sense of~\cite[p. 518]{fujiShinyaMellies2016}. The unit is the operation \mypic{98} of type $\ranked{\Sigma \to \reduce 1 \Sigma}$, while the product (or flattening) in the graded monad is the family of operations of type \begin{align*}
\ranked{\reduce {k_2} \reduce {k_1} \Sigma \to \reduce {k_1 \cdot k_2} \Sigma}, \end{align*} indexed by $k_1,k_2 \in \set{1,2,\ldots}$, that is illustrated below: \mypic{100} More formally, the flattening of a double fold $(a/{f_1})/{f_2}$ has the grouping function defined by
\begin{align*}
i \mapsto (i_2, \pi(p_1,p_2)) \qquad \text{where} \begin{cases}
(i_1,p_1) &= f_1(i)\\
(i_2,p_2) &= f_2(i_1) \end{cases} \end{align*} and $\pi$ is the natural bijection between $\set{1,\ldots,k_1} \times \set{1\ldots,k_2}$ and $\set{1,\ldots,k_1k_2}$.
\newcommand{\funcitem}[3]{\ranked{#1 } &:& \ranked{#2} \rto \ranked{#3}}
This completes the list of datatype constructors.
\begin{definition}[Datatypes]
\label{def:types}
The \emph{datatypes} are the least class of ranked sets which contains all finite ranked sets, the terminal set, and which is closed under applying the constructors
\begin{align*}
\ranked{\tmonad \rSigma \qquad \rSigma_1 \product \rSigma_2 \qquad \rSigma_1 + \rSigma_2 \qquad \reduce k \rSigma}.
\end{align*}
\end{definition}
\newcommand{\combfunc}[4]
{
\ranked{#1} : & \ranked{#2 \to #3} & \text{ for }\ranked{#4}
}
\begin{figure}
\caption{Combinators}
\label{eq:liftterm}
\label{fig:combinators}
\end{figure}
\subsection{Derivable functions} We now present the central definition of this paper.
\begin{definition}[Derivable function]\label{def:derivable-function}
An arity preserving function between two datatypes is called \emph{derivable} if it can be generated, by using the combinators in Figure~\ref{fig:combinators}, from the following prime functions:
\begin{itemize}
\item for every $\rSigma$, the unique arity preserving function $\ranked{\Sigma \to \termset}$;
\item all arity preserving functions with finite domain;
\item the prime functions in Figures~\ref{fig:monad},\ref{fig:product} and \ref{fig:not-explained};
\end{itemize}
\end{definition}
\input{functions-reduced}
The combinators in Figure~\ref{fig:combinators} are function composition, and the obvious liftings of functions along the datatype constructors. The prime functions in Figure~\ref{fig:monad} describe the monad structure of terms and folds, and were explained in Section~\ref{sec:datatype-constructors}. The prime functions in Figure~\ref{fig:product} are simple syntactic transformations, which are intended to have no computational content. Figure~\ref{fig:not-explained} contains less obvious operations, whose definitions are deferred to Section~\ref{sec:prime-and-combinators}.
\input{example-derivable}
We are now ready to state the main theorem of this paper. We say that a tree-to-tree function
\begin{align*}
f : \trees \rSigma \to \trees \rGamma
\end{align*}
is \emph{derivable} if it agrees on arguments that are trees with some derivable partial function
\begin{align*}
\ranked {f : \tmonad \Sigma \to \tmonad \Gamma + \termset}.
\end{align*} The main result of this paper is the following theorem. \begin{theorem}\label{thm:main} A tree-to-tree function is a first-order transduction if and only if it is derivable. \end{theorem}
The right-to-left implication in the above theorem is proved by a relatively straightforward induction on the derivation. The general idea is that we associate to each datatype a relational structure; for example the relational structure associated to a pair $\tensorpair {a_1,a_2}$ is the disjoint union of the relational structures associated to $a_1$ and $a_2$. In the appendix, we show that all prime functions are first-order transductions (adapted suitably to structures other than trees); and that this property is preserved under applying the combinators. There is one nontrivial step in the proof, which concerns monotone unfolding, and will be discussed below.
The left-to-right implication in the theorem, which says that every first-order transduction is derivable, is the main contribution of this paper, and is discussed in Sections~\ref{sec:stt}--\ref{sec:one-register}.
\subsection{The prime functions from Figure~\ref{fig:not-explained}} \label{sec:prime-and-combinators} In this section, we describe the prime functions from Figure~\ref{fig:not-explained}. Each of these functions will play a key role in one of the main results of the paper.
\subsubsection{Factorisations} \label{sec:factorisation-functions} We begin with the two factorisation functions \begin{align*}
\ancfact,\decfact : \ranked{\tmonad(\Sigma_1+\Sigma_2) \to \tmonad(\tmonad \Sigma_1 + \tmonad \Sigma_2)}, \end{align*}
which are used to cut terms into smaller parts. Define a \emph{factorisation} of a term to be any term of terms that flattens to it. An alternative view is that a factorisation is an equivalence relation on nodes in a term, where every equivalence class is connected via the parent-child relation.
Consider a term $t \in \ranked{\tmonad(\rSigma_1 + \rSigma_2)}$.
We say that two nodes have the \emph{same type} if both have labels in the same $\ranked{\Sigma_i}$; otherwise we say that nodes have \emph{opposing type}. Define two equivalence relations on nodes in a term as follows: (a) nodes are called \emph{$\uparrow$-equivalent} if they have the same type and the same proper ancestors of opposing type; (b) nodes are called \emph{$\downarrow$-equivalent} if they are $\uparrow$-equivalent and have the same proper descendants of opposing type.
Here is a picture of the equivalence classes, with $\ranked{\Sigma_1}$ being red and $\ranked{\Sigma_2}$ being blue:
\mypic{111}
For both equivalence relations, the equivalence classes are connected under the parent-child relation, and therefore the equivalences can be seen as factorisations. These are the factorisations produced by the functions $\ancfact$ and $\decfact$.
\subsubsection{Pre-order traversal.} The pre-order traversal function
\begin{align*}
\ranked{\preorder : \tmonad \Sigma \to \reduce 1 \tmonad (\rSigma + \redset{\grayball,\grayballbin})}
\end{align*}
is the natural extension -- from trees to terms -- of the pre-order function in Example~\ref{ex:pre-order-main}. The fold in the output type is used to reorder the ports in a way which matches the input term, as illustrated in the following picture:
\mypic{113}
\subsubsection{Unfolding of the matrix power} \label{sec:unfolding} The final prime function is called monotone unfolding. The general idea is that unfolding unpacks a representation of several trees inside a single tree. Before describing this function in more detail, we introduce some notation, inspired by the matrix power in universal algebra~\citep[p.~268]{Taylor1975}. \begin{definition}
[Matrix power] For $k \in \set{1,2,\ldots}$ define the $k$-th matrix power of a ranked set $\rSigma$, denoted by $\mati k \rSigma$, to be the ranked set $\reduce k \powersmall \rSigma k$. \end{definition} Here is a picture of elements in the third matrix power: \mypic{102}
\begin{figure}
\caption{Unfolding the matrix power}
\label{fig:unfold}
\end{figure}
An element of the $k$-th matrix power can be seen as having a group of $k$ incoming edges, and each of its ports can be seen as a group of $k$ outgoing edges. The \emph{general unfolding} operation, which has type \begin{align*}
\ranked{\tmonad \mati k{\Sigma} \to \mati k{( \tmonad \Sigma)}}, \end{align*} matches the $k$ incoming edges in a node with the $k$ outgoing edges in the parent port; it also removes the unreachable nodes. This operation is illustrated in Figure~\ref{fig:unfold}, and a formal definition is in the appendix.
\paragraph*{Chain logic.} The general unfolding operation is too powerful to be included in the derivable functions, as we explain below. It does, however, admit a characterisation in terms of a fragment of \mso called \emph{chain logic}, see~\cite[Section 2]{thomas1992} or~\cite[Section 2.5.3]{bojanczykDecidablePropertiesTree2004}, whose expressive power is strictly between first-order logic and \mso. Chain logic is defined to be the fragment of \mso where set quantification is restricted to sets where all nodes are comparable by the descendant relation.
\begin{theorem}\label{thm:chain-transductions}
The following conditions are equivalent for tree-to-tree functions: \begin{itemize}
\item is derivable, as in Definition~\ref{def:derivable-function}, except that general unfold is used instead of monotone unfold;
\item is a transduction, as in Definition~\ref{def:fo-transduction}, except that chain logic is used instead of first-order logic. \end{itemize} \end{theorem}
To see why chain logic is needed to describe general unfolding, consider the following unfolding, where two coordinates are swapped in each node of the input tree:
\mypic{108} For inputs with an odd number of swaps, the output of unfolding has a white leaf in the first coordinate, and for inputs with an even number of swaps, the output has a white leaf in the first coordinate. Checking if a path has even length can be done in chain logic, but not in first-order logic.
\paragraph*{Monotone unfolding} To avoid the problems with cyclic swaps, the unfolding function in Figure~\ref{fig:not-explained} imposes a monotonicity requirement on the matrix power, described below.
Let $a \in \mati k \rSigma$ be an element of the matrix power, let $p,q \in \set{1,\ldots,k}$, and let $i$ be a port of $a$. Define the \emph{twist function of port $i$}, denoted by $\to_i$, as follows:
$q \to_i p$ if coordinate $q$ in the $i$-th outgoing edge is connected to coordinate $p$ in root, as described in the following picture: \mypic{125} The twist function is partial. Call an element of the matrix power \emph{monotone} if for every port, its twist functions is monotone (when restricted to inputs where it is defined). In the picture above, $\to_1$ is monotone, while $\to_2$ is not. Also, the problems with an even number of swaps discussed earlier arise from a non-monotone twist function: \mypic{110} The \emph{monotone unfolding} operation in Figure~\ref{fig:unfold} defined to be the restriction of general unfolding, which is undefined if the input contains at least one label which is non-monotone, and otherwise returns the output of the general unfolding.
\paragraph*{Is unfolding derivable?} The prime functions in our main theorem are meant to be simple syntactic rewritings. It is debatable whether the unfolding operation -- even in its monotone variant -- is of this kind. For example, our proof that monotone unfolding is a first-order transduction requires an invocation of the Sch\"utzenberger-McNaughton-Papert theorem about first-order logic on words being the same as counter-free automata.
Is it possible to break down monotone unfolding into simpler primitives? In the appendix, we devote considerable resources to answering this question. We propose one new datatype
and seventeen additional prime functions, which can be called syntactic rewriting without straining the reader's patience. Then, we show that monotone unfolding can be derived using the new datatype and functions. The proof of this result is one of the main technical contributions of this paper.
\section{Register tree transducers} \label{sec:stt} We now begin the proof of the harder implication in Theorem~\ref{thm:main}, which says that every first-order tree-to-tree transduction is derivable. Our proof passes through an automaton model, which is roughly based on existing transducer models for \mso transductions from~\cite{ENGELFRIET199934,alur2017streaming}. The automaton uses registers to store parts of the output tree. The semantics of the automaton involves two phases: (a) mapping the input tree to an expression that uses register updates; (b) evaluating the expression. These phases are described in more detail below.
\paragraph*{Register valuations and updates.} We begin by explaining how the registers work. The registers store terms that are used to construct the output tree. Each register has an arity: registers of arity zero store trees, registers of arity one store unary terms, etc.
Fix two finite ranked sets: the \emph{register names} $\regnames$ and the \emph{output alphabet} $\rGamma$. A \emph{register valuation} is defined to be any arity preserving function from the register names $\regnames$ to terms $\tmonad \rGamma$.
To transform register valuations, we use \emph{register updates}. A register update is an operation which inputs several register valuations and outputs a single register valuation. For $n \in \set{0,1,\ldots}$, an \emph{$n$-ary register update} is defined to be any arity-preserving function \begin{align*}
\ranked {u : \regnames \rto \tmonad ( \rGamma + n\regnames)}, \end{align*} where $\ranked{nR}$ stands for the disjoint union of $n$ copies of $\regnames$. The $i$-th copy of $\regnames$ represents the register contents in the $i$-th argument.
Here is a picture of a register update which has arity 3 and uses two registers $r$ and $s$: \mypic{32} An $n$-ary register update $\ranked u$ induces a operation, which inputs $n$ register valuations and outputs the register valuation obtained by taking $\ranked u$ and replacing the $i$-th copy of a register name with the contents of that register in the $i$-th input register valuation. Register updates have arities, and therefore the ranked set of register updates is written in red, and can be used for labels in a tree. For such a tree \begin{align*}
t \in \trees(\ranked{\text{register updates}}), \end{align*} define its \emph{evaluation} to be the register valuation defined by induction in the natural way. Note that register updates of arity zero are the same as register valuations, which gives the induction base.
\paragraph*{First-order relabellings.} Our automaton model has no states. Instead, it uses a first-order relabelling, as defined below, to directly assign to each node of the input tree a register update that will be applied in that node. A similar model is used by Bloem and Engelfriet~\cite[Theorem 17]{bloem_comparison_2000}, except that in their case, the first phase uses \mso relabellings, and the second phase is an attribute grammar.
\begin{definition}[First-order relabelling] \label{def:forat} A \emph{first-order relabelling} is given by two finite ranked sets $\rSigma$ and $\rGamma$, called the \emph{input and output alphabets}, and a family
\begin{align*}
\set{\varphi_a(x)}_{a \in \rGamma}
\end{align*}
of first-order formulas over the vocabulary of trees over $\rSigma$. These formulas need to satisfy the following restriction:
\begin{enumerate}
\item[(*)] for every tree over the input alphabet and node in that tree, there is a unique output letter $a \in \rGamma$ such that $\varphi_a(x)$ selects the node; furthermore, the arity of $a$ is the same as the arity of (the label of) the node.
\end{enumerate} The semantics of a first-order tree relabelling is a function \begin{align*} \trees \rSigma \to \trees \rGamma, \end{align*} which changes the label of every node in the input tree to the unique letter described in (*).
\end{definition}
A first-order tree relabelling is a very special case of a first-order tree-to-tree transduction, where only the labelling of the input tree is changed, while the universe as well as the child and descendant relations are not affected.
\paragraph*{Register transducers.} Having defined registers, register updates, and first-order tree relabellings, we are now ready to define our automaton model.
\begin{definition}[First-order register transducer]\label{def:stt} The syntax of a \emph{first-order register transducer} consists of: \begin{itemize}
\item An \emph{input alphabet $\rSigma$}, which is a finite ranked set;
\item An \emph{output alphabet $\rGamma$}, which is a finite ranked set;
\item A set $\regnames$ of \emph{registers}, which is a finite ranked set;
\item A total order on the registers.
\item A designated \emph{output register} in $\regnames$, of arity zero.
\item A \emph{transition function}, which is a first-order relabelling
\begin{align*}
\trees{\rSigma} \to \trees{\rDelta},
\end{align*}for some finite set $\rDelta$ of register updates over registers $\regnames$ and output alphabet $\rGamma$. We require all register updates in $\rDelta$ to be single-use and monotone, as defined below:
\begin{enumerate}
\item \emph{Single-use\footnote{The single-use restriction is a standard feature of transducer models with linear size increase~\cite{bloem_comparison_2000, alurStreamingStringTransducers2011,alur2017streaming}. It prohibits iterated duplication of registers, which would lead to exponential size outputs.
}.} An $n$-ary register update $\ranked{u}$ is called \emph{single-use} if every $r \in \ranked{n \regnames}$
appears in at most one term from $\set{\ranked u(s)}_{s \in \regnames}$, and it appears at most once in that term.
\item \emph{Monotone\footnote{This is notion of monotonicity corresponds to the one used in Section~\ref{sec:unfolding}, see the comments on page~\pageref{page:monotone-discussed}. A similar notion appears in~\cite[p. 7]{bojanczykRegularFirstOrderList2018}.}.} This condition uses the total order on registers. An $n$-ary register update $\ranked u$ is called monotone if for every $i \in \set{1,\ldots,n}$, the binary relation $\to_i$ on register names $r,s \in \regnames$ defined by
\begin{align*}
r \to_i s \quad \text{if} \quad \text{the $i$-th copy of $r$ appears in $\ranked u(s)$},
\end{align*}
which is a partial function from $r$ to $s$ when $\ranked u$ is single-use, is monotone:
\begin{align*}
r_1 \leq r_2 \land r_1 \to_i s_1 \land r_2 \to_i s_2 \quad \Rightarrow \quad s_1 \leq s_2
\end{align*}
\end{enumerate}
\end{itemize} \end{definition}
The semantics of the transducer is a tree-to-tree function, defined as follows. The input is a tree over the input alphabet. To this tree, apply the transition function, yielding a tree of register updates. Next, evaluate the tree of register updates, yielding a register valuation. The output tree is defined to be the contents of the designated output register.
The main difference of our model with respect to prior work is that we want to capture tree transformations defined in first-order logic, as opposed to \mso used in~\cite{bloem_comparison_2000,alurStreamingStringTransducers2011,alur2017streaming}. This is why we use first-order relabellings instead of \mso relabellings. For the same reason, we require the register updates to be monotone, see the discussion in Section~\ref{sec:unfolding}.
\begin{proposition}\label{prop:unary-register-stt} For every first-order register transducer, there is a first-order register transducer defining the same function, and whose registers are all unary. \end{proposition}
The main result of this section is that first-order register transducers are expressively complete for first-order tree-to-tree transductions.
\begin{theorem}\label{thm:stt}
Every first-order tree-to-tree transduction is recognised by a first-order register transducer. \end{theorem}
The proof, which is in Appendix~\ref{sec:stt-appendix}, uses the composition method for logic, like similar proofs for~\cite[Theorem 4.6]{alur2017streaming} and~\cite[Theorem 14]{bloem_comparison_2000}. The converse inclusion in the theorem is also true. This is can be shown directly without much difficulty, following the same lines as in~\cite[Section 5]{bloem_comparison_2000}. The converse inclusion also follows from other results in this paper: (a) we show in the following sections that every function computed by the transducer is derivable; and (b) derivable functions are first-order tree-to-tree transductions by the easy implication in Theorem~\ref{thm:main}.
\paragraph*{Proof strategy for Sections~\ref{sec:fo-translation}--\ref{sec:stt-derivable}.} By Theorem~\ref{thm:stt}, to prove derivability of every first-order tree-to-tree transduction, and thus finish the proof of our main theorem, it suffices to prove derivability for first-order register transducers. In a first-order register transducer, the computation has two steps: a first-order relabelling, followed by evaluation of the register updates. The first step is handled in Section~\ref{sec:fo-translation}, and the second step is handled in Section~\ref{sec:stt-derivable}.
\newcommand{\Port}[1]{\mathsf{port}_{#1}} \newcommand{\Interface}[1]{\mathsf{Interface}_{#1}} \section{Derivable functions are first-order transductions} \label{sec:to-transductions} In this section, we give an overview on the proof of the right-to-left implication in Theorem~\ref{thm:main}, namely that if $\rSigma,\rGamma$ are finite ranked sets, then for every derivable function $\ranked {f : \tmonad \rSigma \to \tmonad \rGamma}$, its restriction to trees $f : \trees \rSigma \to \trees \rGamma$ is a first-order tree-to-tree transduction. The proof is a simple induction on the derivation of $\ranked f$. In the induction, we must deal with functions that manipulate types that are more complex than trees, e.g.~tensor pairs of terms of coproducts. To operate on such types with first-order transductions, we need to show how such types can be modeled as relational structures. For this purpose, we associate to each type $\rSigma$ \begin{itemize} \item a relational vocabulary denoted \emph{voc$ \rSigma$}; \item a map $\underline{\bullet}: a \in \rSigma \mapsto \underline a \in \text{models over voc$ \rSigma$}$;
which interprets every inhabitant of $\rSigma$ as a relational structure over the vocabulary of $\rSigma$, \end{itemize} both defined in a natural way by induction on types. Their precise definition are given in Definition~\ref{def:type-model}.
The following proposition immediately yields the right-to-left implication in Theorem~\ref{thm:main}. Its proof, which is a straightforward case analysis of all prime functions and combinators, is in the appendix.
\begin{proposition}\label{prop:to-logic} If $f : \rSigma \to \rGamma$ is derivable, then there is a first-order transduction $g$
which makes the following diagram commute
\begin{align*}
\xymatrix@C=3cm{
\rSigma \ar[d]_{a \mapsto \underline a}\ar[r]^f & \rGamma \ar[d]^{a \mapsto \underline a} \\
\text{models over Voc$\rSigma$} \ar[r]_g & \text{models over Voc$ \rGamma$}.
}
\end{align*} \end{proposition}
\section{First-order relabellings}\label{sec:fo-translation} In this section we prove derivability of the first computation step used in first-order register transducers. \begin{proposition} \label{prop:forat}
Every first-order relabelling is derivable. \end{proposition} To prove the proposition, we use a decomposition of first-order relabellings into simpler functions, in the style of the Krohn-Rhodes theorem.
We use the name \emph{unary query} for a first-order formula with one free variable over the vocabulary of trees. This assumes some implicit alphabet $\rSigma$. For a unary query, define its \emph{characteristic function}, of type \begin{align*}
\trees \rSigma \to \trees (\ranked{\rSigma + \rSigma}), \end{align*} to be the function which replaces the label of each node by its first or second copy, depending on whether the node is selected by the query. This is a special case of a first-order relabelling. The key to Proposition~\ref{prop:forat} is the following lemma, which decomposes first-order relabellings into characteristic functions of certain basic unary queries.
\begin{lemma}\label{lem:schlingloff} Every first-order relabelling can be obtained by composing the following functions:
\begin{enumerate}
\item \label{it:relabelling} \emph{Letter-to-letter homomorphisms}. For every finite $\rGamma,\rSigma$ and $\ranked {f : \rSigma \to \rGamma}$, its tree lifting $\trees \ranked f : \trees \rSigma \to \trees \rGamma$.
\item \label{it:temporal-operators} For every finite $\rSigma$ and its subsets $\rDelta, \rGamma \subseteq \rSigma$, the characteristic functions of the following unary queries over alphabet $\rSigma$:
\begin{enumerate}
\item \label{it:child} \emph{Child:} $x$ is an $i$-th child, for $i \in \set{1,2,\ldots}$
\begin{align*}
\child i (x);
\end{align*}
\item \label{it:until} \emph{Until:} $x$ has a descendant $y$ with label in $\rDelta$, such that all nodes strictly between $x$ and $y$ have label in $\rGamma$
\begin{align*}
\exists y\ y > x \land \rDelta(y) \land \forall z \ (x < z < y \Rightarrow \rGamma(z));
\end{align*}
\item \label{it:since}\emph{Since:} $x$ has an ancestor $y$ with label in $\rDelta$, such that all nodes strictly between $x$ and $y$ have label in $\rGamma$
\begin{align*}
\exists y\ y < x \land \rDelta(y) \land \forall z \ (y < z < x \Rightarrow \rGamma(z)).
\end{align*}
\end{enumerate}
\end{enumerate}
\end{lemma}
The lemma uses a theorem of Schlingloff~\cite[Theorem 2.6]{schlingloff1992expressive}, which says that all first-order definable tree properties can be defined using a temporal logic with operators similar to the ones used in items~\ref{it:temporal-operators} of the lemma. Note that the temporal logic is a two-way logic, because \emph{until} depends on the descendants of the node $x$, while \emph{since} depends on the ancestors. In fact, there is no temporal logic which characterises first-order logic, uses only descendants, and has finitely many operators~\cite[Theorem 5.5]{bojanczykWreathProductsForest2012}. The exact reduction to Schlingloff's theorem is in Appendix~\ref{sec:AppendixForat}.
It remains to show that all of the functions from Lemma~\ref{lem:schlingloff} are derivable. The letter-to-letter homomorphisms from item~\ref{it:relabelling} are a special case of homomorphisms discussed in Example~\ref{ex:filter}, and hence derivable. In Appendix~\ref{sec:AppendixForat}, we show that the functions from item~\ref{it:temporal-operators} are also derivable. In the proof, a key role is played by the factorisation functions discussed in Section~\ref{sec:factorisation-functions}.
\section{Matrix power} \label{sec:matrix-power} In this section, we define the matrix power, which can be seen as a way of representing several trees in one.
\begin{definition}
[Matrix power] For $k \in \set{1,2,\ldots}$ define the $k$-th matrix power\footnote{
The name matrix power is based the matrix power in universal algebra (for the latter, see~\cite{Taylor1975} or~\cite{szendrei1990simple}). Roughly speaking, the restrictions that we place on the original definition correspond to the single-use and monotone conditions from Definition~\ref{def:stt}.
} of a ranked set $\rSigma$ to be \begin{align*}
\mati k \rSigma \quad \eqdef \quad \ranked{\reduce k \rSigma^k}. \end{align*} \end{definition}
Here is a picture of a binary element in the matrix power, where $k=3$: \begin{center} \includegraphics[scale=.4, page=85]{pics.pdf} \end{center}
A matrix power of arity $n$ is used as an abstraction of an $n$-ary register update $\ranked{R\to \tmonad(\Gamma+R+\dots+R)}$. The number $k$ represents the number of registers (that is the size of $\ranked {R}$), and each letter from $\rSigma$ represents a register update (that is, an element of $\ranked{\tmonad(\Gamma+R+\dots+R)}$). When two letters share the same port in the matrix power, this means that the corresponding register updates call the same component of $\ranked{R+\dots+R}$ in their definition, the position in the fold indicates which precise register is called. For instance, the register update above is an abstraction of the following binary register update of registers $r, s$ and $t$:
\begin{center}
\includegraphics[scale=.3]{register-update-matrix-power.pdf}
\end{center} When we run a STT on an input tree, we decorate it with register updates following the transition function. In order to evaluate this tree, the first thing to do is to determine, for each register update of the root, its "dependency tree" that is to say the register updates they call, inductively. When register updates are abstracted by matrix powers, this operation of determining the dependency tree is what we call \emph{unfolding}. We illustrate it by the following example \begin{center} \includegraphics[scale=.38]{unfold-matrix-power} \end{center} Formally, \emph{unfolding} is a function of type \begin{align*}
\ranked{\unfold : \tmonad \mati k \rSigma \to \mati k {(\tmonad \Sigma)} }
\end{align*}
which is defined as follows by induction on the size of the input term. If the input is an empty term, then the output is this term: \begin{center} \includegraphics[scale=.3, page=83]{pics.pdf} \end{center} Otherwise, if the input is a nonempty term $a(t_1,\ldots,t_n)$ then the output is obtained by first applying unfolding to to the smaller terms $t_1,\ldots,t_n$, and then applying the following derivable function, which we call \emph{shallow unfold}. \begin{align*}
\ranked{
\xymatrix{
\shallowterm{\mati k \rSigma} {\mati k \rGamma} \ar[r] & \mati k {(\shallowterm \Sigma \Gamma)}.
}
} \end{align*} Here is a picture of unfolding for shallow terms: \begin{center} \includegraphics[scale=.4]{unfold-shallow} \end{center} More formally, unfolding for shallow terms is defined to be the composition of the following functions
\begin{align*}
\xymatrix@R-1pc@C=3.2cm{
{\begin{array}{c}
\shallowterm{\mati k \rSigma} {\mati k \rGamma} \\{=\shallowterm{\reduce k {\Sigma^k}}{\reduce k {\Gamma^k}}}
\end{array}} \ar[d]_{{\mathrm{Commutativity}}}\ar[r]^{\mathrm{Shallow\ unfold}} & {\begin{array}{c}
\ranked{\mati k {(\shallowterm \Sigma \Gamma)}} \\= \ranked{ \reduce k(\shallowterm{\Sigma}{ {\Gamma}})^k}
\end{array}} \\
\reduce k(\shallowterm{\reduce k {\Sigma^k}}{ {\Gamma^k}}) \ar[r]_{{\mathrm{Matching}}} & \ranked{ \reduce k(\shallowterm{\Sigma^k}{ {\Gamma}}) } \ar[u]_{\mathrm{Commutativity}}
}
\end{align*} Since our prime functions and combinators do not have any general mechanisms for induction, it is not clear how to derive the unfolding operation. In fact, it is not derivable in general, but it will become derivable after imposing a monotonicity requirement. The need for this condition is shown in Example~\ref{eq:twist}. Let us define it in the following.
For a port $i$ in an element of the matrix power $a/f \in \mati k \rSigma$ define its \emph{twist} to be the partial function from $\set{1,\ldots,k}$ to itself which is the composition of \begin{align*} \xymatrix{
[1,k] \ar[r]^-{j \mapsto {(j,i)}} & [1,k] \times [1,n] \ar[r]^-{f^{-1}} & [1,\arity a] \ar[r] & [1,k] } \end{align*} where the last function maps each port of the tuple $a$ to the coordinate of the tuple that created that port. Here is a picture: \begin{center} \includegraphics[scale=.3, page=87]{pics.pdf} \end{center}
We say that an element of the $k$-matrix power is \emph{monotone} if all of its ports have monotone twist. For instance, the right example above is not monotone.
We are now ready to state the principal result of this section.
\begin{proposition}\label{prop:monotone-unfold}
For every finite ranked set $\rSigma$ and every $k \in \set{1,2,\ldots}$ there is a derivable function
\begin{align*}
\ranked{ f : \tmonad \mati k \Sigma \to \mati k {(\tmonad \Sigma)}}
\end{align*}
which coincides with term unfolding for monotone inputs. \end{proposition} The proof of the above lemma is one of the main technical contributions of this paper, and it is given in the appendix. One of the ingredients of the proof is a first-order (in fact, derivable) version of the Factorisation Forest Theorem for trees, which is based in Colcombet's splits from~\cite{colcombetCombinatorialTheoremTrees2007}.
\section{Evaluation of register updates} \label{sec:stt-derivable} In this section, we deal with the second computation phase in a first-order register transducer, namely evaluating register updates. As discussed in the end of Section~\ref{sec:stt}, this completes the proof of our main theorem.
Our proof uses the language of $\lambda$-calculus. In Section~\ref{sec:one-register}, we discuss derivability of normalisation of $\lambda$-terms. In Section~\ref{sec:updates-endgame}, we reduce evaluation of register updates to unfolding the matrix power and normalisation of $\lambda$-terms.
\input{one-register}
\subsection{Evaluation of register updates} \label{sec:updates-endgame} Equipped with Theorem~\ref{thm:normalise}, we prove derivability of evaluation of register updates. Fix a first-order register transducer. We suppose from now on that: \begin{align} \label{assumption} \text{all its registers are unary} \end{align} which is possible by Proposition~\ref{prop:unary-register-stt}.
From now on, when speaking about register updates or register valuations, we mean those of the fixed transducer. Our goal is to prove the following lemma, which completes the proof of our main theorem. \begin{lemma}\label{lem:derive-register-updates}
Consider
the tree-to-tree function, which inputs a tree of register updates, evaluates it, and outputs the contents of the designated output register. This function is derivable.
\end{lemma}
\paragraph*{Output letters in $\lambda$-terms.} We will use $\lambda$-terms to represent register updates, which involve letters of the output alphabet $\rGamma$. Therefore, for the rest of Section~\ref{sec:updates-endgame}, we use an extended notion of $\lambda$-terms, which allows building $\lambda$-terms of the form \begin{align}\label{eq:non-pure} a(M_1,\ldots,M_n) \qquad \text{for every $a \in \rGamma$ of arity $n$.} \end{align} The typing rules are extended as follows: if the arguments $M_1,\ldots,M_n$ all have type $\otype$ (no other type is allowed for arguments of $a$), then~\eqref{eq:non-pure} has type $\otype$. These $\lambda$-terms can be represented as trees, as in the following picture: \mypic{118}
Theorem~\ref{thm:normalise} works without change for the extended notion of $\lambda$-terms used in this section. Note that there is no $\beta$-reduction rule for $\lambda$-terms of the form~\eqref{eq:non-pure}.
\paragraph*{$\lambda$-representations of register updates.}
To prove Lemma~\ref{lem:derive-register-updates}, we represent register updates using a matrix power of $\lambda$-terms. The idea is that the matrix power handles the parallel evaluation of registers.
Let $x$ be a variable of type $\otype$. Define $\ranked{\Gamma_\lambda}$ to be the output alphabet $\rGamma$ plus the following ranked alphabet: \begin{align}
\label{eq:alphabet-for-lambda-terms}
\overbrace{\set{x }}^{\text{arity 0}} \cup \overbrace{\set{\lambda x }}^{\text{arity 1}} \cup \overbrace{\set @}^{\text{arity 2}} \end{align}
Recall that a register update -- of arity say $n$ -- consists of a family of terms over alphabet $\ranked{\Gamma + n\regnames}$, one for each register $r \in \regnames$. We begin by explaining the $\lambda$-representation for terms in the family, which is a function
of type \begin{align}\label{eq:placeholders} \xymatrix@C=2cm{
\ranked{\tmonad(\rGamma+n R)} \ar[r]^-{\text{$\lambda$-representation}} &
\ranked{\tmonad {\Gamma_\lambda}} }. \end{align} This function is not arity preserving, which is why it is not written in red. Define a \emph{placeholder} to be an element of $\ranked{n \regnames}$; we write placeholders as $r_i$ with $r \in \regnames$ and $i \in \set{1,\ldots,n}$. The function~\eqref{eq:placeholders} is explained in the following picture: \begin{center} \includegraphics[scale=0.9]{pics103} \end{center} Note how the
arities need not be preserved: the arity of the output is the number of placeholders in the input, but the input have always at most one port by assumption~(\ref{assumption}). The correspondence of ports in the output term with placeholders in the input term is defined with respect to some arbitrary order on the set $\ranked{n \regnames}$ of placeholders, say lexicographic with respect to the order on registers and $\set{1,\ldots,n}$.
Having defined the $\lambda$-representation of terms with placeholders, we lift it to a $\lambda$-representation of register updates \begin{align}\label{eq:lambda-representation-regup} \ranked{
\xymatrix@C=2cm{
\text{register updates} \ar[r]^-{\text{$\lambda$-representation}} &
\mati k {(\tmonad\Gamma_\lambda)} } }, \end{align} where $k$ is the number of registers. This function is arity preserving.
For a register update $(t_1,\ldots,t_k)$, where $t_i$ is the term with placeholders used in the $i$-th register, its $\lambda$-representation is defined to be \begin{align*} (\text{$\lambda$-representation of $t_1$},\ldots,\text{$\lambda$-representation of $t_k$})/f , \end{align*}
where the grouping function $f$ connects a placeholder $r_i$ to the $r$-th sub-port of port $i$. Here is a picture \begin{center} \includegraphics[scale=0.9]{pic119} \end{center}
The following three properties of the $\lambda$-representation for register updates will be used later in the proof: \begin{enumerate} \item[(P1)] If we restrict the domain to a finite set of register updates, e.g.~those used in the transducer, then it is a prime function, by virtue of having finite domain. \item[(P2)] A register update is monotone (as in Definition~\ref{def:stt}) if and only if its $\lambda$-representation is monotone (as defined in Section~\ref{sec:unfolding} for the matrix power). \item[(P3)] The $\lambda$-representation uses the unique variable $x$, every binder $\lambda x$ binds a unique occurrence of $x$, and the types that appear are of the form \begin{align*} \overbrace{\otype \to \otype \to \cdots \to \otype \to \otype}^{\text{at most (maximal arity in $\rGamma$) times}} \to \otype, \end{align*} hence Theorem~\ref{thm:normalise} can be applied. \end{enumerate}
\label{page:monotone-discussed}
\paragraph*{Putting it all together.} To finish the proof of Lemma~\ref{lem:derive-register-updates}, we observe that the semantics of a register automaton are translated -- under the $\lambda$-representation -- to unfolding the matrix power and normalising a $\lambda$-term. This observation is formalised by saying that the diagram in Figure~\ref{fig:lambda-representation-diagram} commutes, and it follows directly from the definitions. Instead of giving a proof, we illustrate it on an example in Figure~\ref{fig:lambda-representation-proof}.
\begin{figure}\label{fig:lambda-representation-diagram}
\end{figure} \begin{figure}
\caption{Example for Figure~\ref{fig:lambda-representation-diagram}.}
\label{fig:lambda-representation-proof}
\end{figure}
\pagebreak We claim that all of the arrows (c), (d) and (e) on the right-down path in Figure~\ref{fig:lambda-representation-diagram} are derivable: \begin{itemize}
\item[(c)] Since we work with a fixed register transducer, there is a finite subset $\rDelta$ of register updates used, and therefore operation (a) in the figure is derivable by property (P1).
\item[(d)] Arrow (d) represents the unfolding of the matrix power. By property (P2), the outputs of arrow (c) are monotone, and so we can use the monotone unfolding operation, which is a prime function and therefore derivable.
\item[(e)] Finally, arrow (e) represents normalisation of $\lambda$-terms. This arrow is derivable by Theorem~\ref{thm:normalise}. The assumptions of this theorem are met by property (P3). \end{itemize} Since the arrows (c), (d), (e) are derivable, and the diagram commutes, it follows that the composition of the arrows (a) and (b) is derivable. In other words, there is a derivable function which maps a tree of register updates to the $\lambda$-representation of the resulting register valuation (when viewing a register valuation as a special case of a register update of arity zero). Finally, to get the contents of the output register, we get rid of the fold in the matrix power by using the last function from Figure~\ref{fig:monad}, and project onto the coordinate for the output register.
This completes the proof of Lemma~\ref{lem:derive-register-updates}, and therefore also of the main theorem.
\label{sec:strategy} In this section, we present an overview of the top-down implication in the proof of Theorem~\ref{thm:main}, which says that every first-order tree-to-tree transduction is (the restriction to trees of) a derivable function. The proof has five steps, which are spread across Sections~\ref{sec:fo-translation}--\ref{sec:stt-derivable}, and summarised below.
\newcommand{\announce}[2]{ \begin{center}
{\bf #1.} #2 \end{center} }
\paragraph*{Section~\ref{sec:fo-translation}: first-order tree relabellings.} We begin the proof of the left-to-right implication with a special case of first-order tree-to-tree transductions, which we call \emph{first-order tree relabellings.} In such a function, every node of the input tree is given a new label, depending on first-order definable properties of the node. An example is: ``if a leaf $x$ in the input tree has at least one ancestor with label $a$, then replace the label of $x$ with the letter $c$''. First-order tree relabellings do not cover all first-order transductions, because they do not change the shape of the input tree, but they are an important subcase.
The main result of Section~\ref{sec:fo-translation} is:
\announce
{Proposition~\ref{prop:forat}}
{Every first-order tree relabelling is derivable.}
The main tool used in the proof of Proposition~\ref{prop:forat} is a theorem of Schlingloff~\cite[Theorem 2.6]{schlingloff1992expressive}, which can be seen as tree version of Kamp's theorem about {\sc ltl} being expressively complete for first-order logic.
\paragraph*{Section~\ref{sec:stt}: register transducers.} To prove that first-order tree-to-tree transductions are derivable, it will be more convenient to use an automaton model. In Section~\ref{sec:stt}, we present this automaton model, which we call \emph{register transducers}\footnote{These are the only kind of register transducer used in this paper. A more complete name would be first-order tree-to-tree register transducers.}. This is a bottom-up tree automaton, which uses registers to store parts of the output tree. The model is based on streaming tree transducers from~\cite{alur2017streaming}, but it is appropriately restricted so that it matches first-order logic, as opposed to monadic second-order logic which was used in~\cite{alur2017streaming}. The main result of Section~\ref{sec:stt} is:
\announce
{Theorem~\ref{thm:stt}}
{Every first-order tree-to-tree transduction can be computed by a register transducer.}
The proof of the above theorem, like for similar results in~\cite{alur2017streaming} or~\cite[Theorem 23]{engelfrietMSODefinableString2001}, is based on the composition method in logic.
The converse implication is also true, and follows from Proposition~\ref{prop:many-register} and the bottom-up implication in Theorem~\ref{thm:main}.
\paragraph*{Section~\ref{sec:one-register}: register transducers with one register.} By Theorem~\ref{thm:stt}, in order to prove the left-to-right implication in Theorem~\ref{thm:main}, it is enough to show that register transducers can only compute derivable functions. As a first step, we consider register transducers with only one register. When there is only one register, the computed function turns out to be a composition of a first-order tree relabelling, as treated Proposition~\ref{prop:forat}, followed by the evaluation (i.e.~computation of the $\beta$-normal form) of a simply typed $\lambda$-term which is affine (i.e.~uses each variable at most once). The main result of Section~\ref{sec:one-register} is that evaluation of affine $\lambda$-terms is derivable, as stated in the following theorem:
\announce
{Theorem~\ref{thm:normalise}}
{The following function is derivable
\begin{align*}
\text{affine $\lambda$-term} \qquad \mapsto \qquad \text{its $\beta$-normal form,}
\end{align*}
if we restrict inputs so that they use a fixed number of variables, and all subterms have types from a fixed finite set of simple types. }
\paragraph*{Section~\ref{sec:stt-derivable}: deriving a register transducer.} In Section~\ref{sec:stt-derivable}, we complete the proof of Theorem~\ref{thm:main}:
\announce
{Proposition~\ref{prop:many-register}}
{Every function computed by a register transducer is derivable.}
The above result, together with Theorem~\ref{thm:stt}, completes the proof of the left-to-right implication in Theorem~\ref{thm:main}. The proof of Proposition~\ref{prop:many-register} is the observation that register automata with many registers can be reduced to register automata with one register, using the unfolding operation. For register transducers with one register only, the only computation that is done is iterated substitution, which is derivable thanks to Theorem~\ref{thm:normalise}.
\section{Functions computed by register transducers are derivable} \label{sec:stt-derivable} We now have all of the ingredients needed to derive functions computed by register transducers, as stated in the following proposition. \begin{proposition}
\label{prop:many-register} For every first-order tree transducer, the computed function is derivable. \end{proposition} The above proposition completes the proof of the left-to-right implication in Theorem~\ref{thm:main}: we have seen, by Proposition~\ref{prop:many-register}, that first-order tree-to-tree transductions can be computed by first-order tree transducers, and Proposition~\ref{prop:many-register} shows that the later are derivable, which concludes the proof.
The rest of this section is devoted to proving Proposition~\ref{prop:many-register}. There will be four main ingredients, namely derivability for: first-order tree relabelings (Proposition~\ref{prop:forat}), normalisation of affine $\lambda$-terms (Theorem~\ref{thm:normalise}), and unfolding of the matrix power (Proposition~\ref{prop:monotone-unfold}). \label{sec:proof-of-prop} To put these ingredients together, we use terminology inspired by universal algebra, as given in the following definition.
\begin{definition}
An \emph{algebra} $\alg$ consists of two sets
\begin{align*}
\overbrace{\algdom \alg}^{\text{unranked}} \qquad \overbrace{\algops \alg}^{\text{ranked}},
\end{align*} equipped with a \emph{shallow product}\footnote{
A more standard approach would be to use a family of operations indexed by the signature
\begin{align*}
\set{f^\alg : (\algdom \alg)^{\arity f} \to \algdom \alg}_{f \in \algops \alg}.
\end{align*}
Our approach is equivalent, i.e.~defining a shallow product is the same as defining a family of operations. }, which is function of type \begin{align*}
\shallowterm {(\algops \alg)}{\underbrace{(\algdom \alg)}_{\substack{\text{viewed as a ranked}\\ \text{set, with all elements}\\ \text{having arity zero}}}} \rto \redpar {\algdom \alg}. \end{align*}
For an algebra $\alg$, define its \emph{product} to be the function of type
\begin{align*}
\treepar {\algops \alg} \to \algdom \alg
\end{align*}
defined in the natural way.
\end{definition}
One example of an algebra is when the signature is some ranked set $\rSigma$, the domain is $\trees \rSigma$, and the interpretation is defined in the natural way. In this case, the product operation is the identity. A variant of this algebra is when the signature is replaced by $\tmonad \rSigma$; in this case the product operation becomes flattening.
Let us begin the proof of Proposition~\ref{prop:many-register}. Fix a register transducer with $k$ registers. For the rest of this section, when talking about register valuations and register updates, we mean the registers of the fixed register transducer. Another example of an algebra was implicit in the definition of register transducers: the domain is the register valuations and the signature is the register updates. To prove Proposition~\ref{prop:many-register}, we will show that the algebra of register updates embeds -- via the matrix power -- in an algebra, call it $\alg$, whose product operation is derivable as a special case of normalisation of affine $\lambda$-terms.
The idea behind the algebra $\alg$ is to use $\lambda$-terms to represent register contents, according to the following picture:
\begin{center}
\includegraphics[scale=.3, page=59]{pics.pdf}
\end{center}
Let $\nmax$ be the maximal arity of registers in the fixed register transducer, and let $\rGamma$ be its output alphabet. For a term $t \in \tmonad \rGamma$ of arity at most $\nmax$, its representation as a $\lambda$-term, which we denote by $t^\lambda$, uses variables from the following finite set (we use the notation $\otype^i \to \otype$ defined in Example~\ref{ex:affine-not-enough}): \begin{align*} X \quad \eqdef \quad \set{\typevar {x_a} {\otype^i \to \otype} : a \in \rGamma \text{ of arity $i$}} \cup \set{\typevar {x_i} \otype : i \in \set{1,2,\ldots,\nmax }}. \end{align*} Furthermore, $t^\lambda$ is affine and it can be typed using simple types from the following set finite set : \begin{align*}
\Tt \quad \eqdef \quad \set{\otype^i \to \otype : i \in \set{0,\ldots,\nmax}} \end{align*} This means that if $t$ has arity at most $\nmax$ -- which is true for any term that can appear in a register of our fixed register transducer -- then its $\lambda$-term representation $t^\lambda$ satisfies constraints as in Theorem~\ref{thm:normalise}. On its own, $t^\lambda$ is already in normal form, so there is no need to normalise it, but we intend to compose such $\lambda$-terms, leading to terms that are not in normal form (but which still use variables from $X$ and types from $\Tt$). This motivates the following definition of an algebra, call it $\alg$. Its domain and signature are given by \begin{align*} \overbrace{\set \bot + \trees \lamrank X}^{\algdom \alg} \qquad \overbrace{ \tmonad \lamrank X}^{\algops \alg} \end{align*} \begin{align*} \text{where }\lamrank X = {\overbrace{\set{x : x \in X}}^{\text{arity 0}} \cup \overbrace{\set{\lambda x : x \in X}}^{\text{arity 1}} \cup \overbrace{\set @}^{\text{arity 2}}} \end{align*} In other words, the domain is $\lambda$-terms plus an error element, and the signature is $\lambda$-terms with ports. The product operation is \begin{align*} t^\alg(t_1,\ldots,t_n) = \begin{cases}
\text{normal form of $t(t_1,\ldots,t_n)$} & \text{if $t(t_1,\ldots,t_n)$ is affine and can be typed using $\Tt$}\\
\bot & \text{otherwise} \end{cases} \end{align*} In particular, if one of the inputs $t_1,\ldots,t_n$ is $\bot$, then the output is also $\bot$. The algebra $\alg$ is designed so that we can apply Theorem~\ref{thm:normalise} about normalising $\lambda$-terms to see that the product operation in the algebra $\alg$ is derivable. Next, by observing that the matrix power is a form of syntactic sugar for manipulating registers in a register automaton, we see that there is an arity preserving
\begin{align*}
\ranked h : \regups \to \mati k {(\algops \alg)}
\end{align*}
which has only monotone elements in its image, and which makes the following diagram commute
\begin{align*}
\xymatrix@C=2cm{
\treepar {\regups} \ar[r]^-{\trees \ranked h}\ar[d]_{\text{product}} &
\treepar {\mati k {(\algops \alg)}}\ar[r]^{\text{Corollary~\ref{cor:matrix-power}}} &
\treepar {\algops \alg} \ar[d]^{\text{product}} \\
\regvalss \ar[rr]_{(t_1,\ldots,t_k) \mapsto t^\lambda_1}& & \algdom {\alg^{[k]}}
}
\end{align*} Putting this all together, we see that there is a derivable function which inputs a tree of register updates, and outputs the value of the first register, represented as a $\lambda$-term. The last remaining observation is that function $t^\lambda \mapsto t$, i.e.~the inverse of the $\lambda$-term representation, is derivable. This inverse can be achieved, for example, by composing a first-order tree relabelling with a tree homomorphism. This completes the proof of Proposition~\ref{prop:many-register}, and therefore also of Theorem~\ref{thm:main}. \section{Monadic second-order transductions} \label{sec:mso-trans} We finish the paper by discussing a variant of our main theorem for \mso tree-to-tree transductions. We simply add, as prime functions, all \mso relabellings, which are defined the same way as the first-order relabellings from Definition~\ref{def:forat}, except that the unary queries can use \mso logic instead of first-order logic.
\begin{theorem}\label{thm:mso-transductions}
A tree-to-tree function is an \mso transduction if and only if it can be derived using Definition~\ref{def:derivable-function} extended by adding all \mso relabellings as prime functions. \end{theorem} \begin{proof}
In~\cite[Corollary 1]{colcombetCombinatorialTheoremTrees2007}, Colcombet shows that every \mso formula on trees can be replaced by a first-order formula that runs on an \mso relabelling of the input tree. Applying that result to transductions, we see that every \mso tree-to-tree transduction can be decomposed as: (a) an \mso relabelling; followed by (b) a first-order tree-to-tree transduction. The theorem follows. \end{proof} The solution above is not particularly subtle, and contrasts our results for first-order logic and chain logic, where we took care to have a small number of primitives. This was possible thanks in part to the decomposition of first-order queries into simpler ones that was is in Section~\ref{sec:fo-translation}, and the Krohn-Rhodes theorem that is used in the proof of Theorem~\ref{thm:chain-transductions} about chain logic. In principle, a decomposition of \mso relabellings could be possible, but proving it would likely require developing a new decomposition theory for regular tree languages, in the style of the Krohn-Rhodes theorem, which we feel is beyond the scope of this paper. One would expect a Krohn-Rhodes theorem for trees to yield an effective characterisation of first-order logic -- as it does for words -- but finding such a characterisation remains a major open problem~\cite[Section 3]{bojanczyk2015automata}.
\label{sec:unfolding} The final prime function is called monotone unfolding. The general idea is that unfolding unpacks a representation of several trees inside a single tree. Before describing this function in more detail, we introduce some notation, inspired by the matrix power in universal algebra~\citep[p.~268]{Taylor1975}. \begin{definition}
[Matrix power] For $k \in \set{1,2,\ldots}$ define the $k$-th matrix power of a ranked set $\rSigma$, denoted by $\mati k \rSigma$, to be the ranked set $\reduce k \powersmall \rSigma k$. \end{definition} Here is a picture of elements in the third matrix power: \mypic{102}
\begin{figure}
\caption{Unfolding the matrix power}
\label{fig:unfold}
\end{figure}
An element of the $k$-th matrix power can be seen as having a group of $k$ incoming edges, and each of its ports can be seen as a group of $k$ outgoing edges. The \emph{general unfolding} operation, which has type \begin{align*}
\ranked{\tmonad \mati k{\Sigma} \to \mati k{( \tmonad \Sigma)}}, \end{align*} matches the $k$ incoming edges in a node with the $k$ outgoing edges in the parent port; it also removes the unreachable nodes. This operation is illustrated in Figure~\ref{fig:unfold}, and a formal definition is in the appendix.
\paragraph*{Chain logic.} The general unfolding operation is too powerful to be included in the derivable functions, as we explain below. It does, however, admit a characterisation in terms of a fragment of \mso called \emph{chain logic}, see~\cite[Section 2]{thomas1992} or~\cite[Section 2.5.3]{bojanczykDecidablePropertiesTree2004}, whose expressive power is strictly between first-order logic and \mso. Chain logic is defined to be the fragment of \mso where set quantification is restricted to sets where all nodes are comparable by the descendant relation.
\begin{theorem}\label{thm:chain-transductions}
The following conditions are equivalent for tree-to-tree functions: \begin{itemize}
\item is derivable, as in Definition~\ref{def:derivable-function}, except that general unfold is used instead of monotone unfold;
\item is a transduction, as in Definition~\ref{def:fo-transduction}, except that chain logic is used instead of first-order logic. \end{itemize} \end{theorem}
To see why chain logic is needed to describe general unfolding, consider the following unfolding, where two coordinates are swapped in each node of the input tree:
\mypic{108} For inputs with an odd number of swaps, the output of unfolding has a white leaf in the first coordinate, and for inputs with an even number of swaps, the output has a white leaf in the first coordinate. Checking if a path has even length can be done in chain logic, but not in first-order logic.
\paragraph*{Monotone unfolding} To avoid the problems with cyclic swaps, the unfolding function in Figure~\ref{fig:not-explained} imposes a monotonicity requirement on the matrix power, described below.
Let $a \in \mati k \rSigma$ be an element of the matrix power, let $p,q \in \set{1,\ldots,k}$, and let $i$ be a port of $a$. Define the \emph{twist function of port $i$}, denoted by $\to_i$, as follows:
$q \to_i p$ if coordinate $q$ in the $i$-th outgoing edge is connected to coordinate $p$ in root, as described in the following picture: \mypic{125} The twist function is partial. Call an element of the matrix power \emph{monotone} if for every port, its twist functions is monotone (when restricted to inputs where it is defined). In the picture above, $\to_1$ is monotone, while $\to_2$ is not. Also, the problems with an even number of swaps discussed earlier arise from a non-monotone twist function: \mypic{110} The \emph{monotone unfolding} operation in Figure~\ref{fig:unfold} defined to be the restriction of general unfolding, which is undefined if the input contains at least one label which is non-monotone, and otherwise returns the output of the general unfolding.
\paragraph*{Is unfolding derivable?} The prime functions in our main theorem are meant to be simple syntactic rewritings. It is debatable whether the unfolding operation -- even in its monotone variant -- is of this kind. For example, our proof that monotone unfolding is a first-order transduction requires an invocation of the Sch\"utzenberger-McNaughton-Papert theorem about first-order logic on words being the same as counter-free automata.
Is it possible to break down monotone unfolding into simpler primitives? In the appendix, we devote considerable resources to answering this question. We propose one new datatype
and seventeen additional prime functions, which can be called syntactic rewriting without straining the reader's patience. Then, we show that monotone unfolding can be derived using the new datatype and functions. The proof of this result is one of the main technical contributions of this paper.
\pagebreak \appendix \section{Unfolding the matrix power} \label{sec:appendix-unfold} In this part of the appendix, we define formally the unfolding function \begin{align*}
\ranked{\tmonad \mati k{\Sigma} \to \mati k{( \tmonad \Sigma)}} \end{align*} that was described in Section~\ref{sec:unfolding}. We present the definition in a slightly verbose manner, by decomposing unfolding into simpler operations. The presentation highlights the inductive character of unfolding, and the reasons why we are uneasy about it being a prime operation.
\subsection{Shallow terms} \label{sec:shallow-terms} We begin by defining unfolding for terms of depth two, called \emph{shallow terms}. Later, we extend the definition to all other terms by induction. We describe shallow terms as a separate datatype, since this datatype will also be used later, in Section~\ref{ap:matrix-power}, to derive the (monotone) unfolding operation.
For now, shallow terms are just an intermediate type used to define formally the unfolding function.
Let $\rSigma$ and $\rGamma$ be two ranked sets. The shallow terms datatype, which is denoted $\shallowterm \rSigma \rGamma$, consists of expressions of the form $a\tensorpair{b_1,\dots,b_n}$ where $a$ is an $n$-ary element of $\rSigma$ and $b_1,\dots, b_n$ are elements of $\rGamma$. The arity of such an expression is the sum of arities of $b_1,\ldots,b_n$. We draw shallow terms as terms of depth two, where the root is from $\rSigma$ and the children are from $\rGamma$: \mypic{54} An equivalent definition of shallow terms, in terms of products and co-products, is \begin{align}\label{eq:shallowterm-definition} \shallowterm \rSigma \rGamma \quad \eqdef \quad \ranked{\coprod_{\black{a \in} \rSigma} } \overbrace{\ranked{\Gamma \product \cdots \product \Gamma},}^{\text{arity of $a$ times}} \end{align}
\subsection{Terms as an inductive datatype} \label{sec:terms-induction-principle} Using shallow terms, we can define the set of terms as the least solution of the equation \begin{align*} \ranked{\tmonad \rSigma = \set{\portletter} + \shallowterm \Sigma {(\tmonad \rSigma)}}. \end{align*} With this inductive definition, in order to define an operation of type $\ranked{ \tmonad \rSigma \to \Gamma}$ on terms, it is enough to explain the induction base for the identity term and the induction step for shallow unfolding, as captured by two operations of types \begin{align*}
\underbrace{\ranked{\set \portletter \to \Gamma}}_{\text{induction base}}
\qquad
\underbrace{\ranked{\shallowterm \Sigma \Gamma \to \Gamma}.}_{\text{induction step}} \end{align*} We use such an induction below to define general unfolding. The crucial step is defining the induction step, which the unfolding for shallow terms defined in Section~\ref{sec:definition-of-shallow-unfolding} below.
As mentioned at the beginning of Section~\ref{sec:derivable-functions}, the guiding principle behind our approach is to avoid iteration mechanisms. The inductive definition of general unfolding could be seen as such an iteration mechanism; this is the reason for Section~\ref{ap:matrix-power}, where (monotone) unfolding is derived using simpler operations. In contrast, we believe that iteration is indeed avoided by the operations used in the induction step that are presented in Section~\ref{sec:definition-of-shallow-unfolding} below.
We do not formalise what we mean by ``avoiding iteration''. One possible direction would be to say that an operation ``avoids iteration'' if it can be computed by a family of bounded depth circuits, as in the circuit class AC$^0$. A further requirement could be that the family of circuits not only exists, but it is also easy to see.
\subsection{Unfolding for shallow terms} \label{sec:definition-of-shallow-unfolding} The induction step in general unfolding is the operation \begin{align*}
\ranked{
\xymatrix{
\shallowterm{\mati k \rSigma} {\mati k \rGamma} \ar[r] & \mati k {(\shallowterm \Sigma \Gamma)},
}
} \end{align*} which we call shallow unfolding, and which is explained in the following picture: \mypic{121} To define this operation formally, we further decompose it using three functions manipulating shallow terms. These functions, which are used here as intermediate functions in the definition of shallow unfolding, will become prime functions when we decompose the unfolding function in Appendix~\ref{ap:matrix-power}.
\subsubsection{Distribute shallow terms over fold}
Let $\rGamma$ and $\rSigma$ be two datatypes. Consider the function $\ranked{f_1}$ \begin{align*} \ranked{\shallowterm \Gamma {\reduce k \Sigma}} \ranked{\xrightarrow{\quad f_1 \quad}} \ranked{\reduce k(\shallowterm \Gamma \Sigma)} \end{align*} which distributes shallow terms over folding. This function is illustrated by the following picture \mypic{55} and defined by $$\begin{array}{rcl} a(b_1/g_1,\dots,b_n/g_n)&\mapsto& a(b_1,\dots,b_n)/g \end{array}$$ where $g$ is the function defined as follows. For every $i \in\set{1,\dots,n}$, if $j\in\set{1,\dots,\arity{b_i}}$ then
\begin{align*}
\xymatrix@C=5cm{
\begin{array}{c}
\overbrace{j+\underset{l<i}{\Sigma} \arity{b_l}}^{\text{Position of the $j$-th port of $b_i$ is shifted}}
\end{array}
\ar@{|->}[d]^{g}\\
\begin{array}{c}
\biggl(\ \ \ \underbrace{\pi_2(g_i(j))+\underset{l<i}{\Sigma} \arity{b_l/g_l}}_{\text{Position of the group is shifted}}\ \ , \underbrace{\pi_1(g_i(j))}_{\begin{array}{c} { \scriptsize\text{Position inside}}\\[-5pt]{\scriptsize \text{the group is unchaged}}
\end{array}}\biggr)
\end{array}}
\end{align*}
\subsubsection{Matching function} We now define a function \begin{align*} \ranked{\shallowterm {(\reduce k \Gamma)} {(\Sigma^k)} \xrightarrow{\quad f_2\quad} \reduce 1 (\shallowterm \Gamma \Sigma)} \end{align*} which matches the $k$-th fold with the $k$-th power\footnote{In order to reduce the number of parentheses, in the rest of the paper we assume a notational convention where the unary datatype constructors -- like folding, terms or powering -- have priority over the binary shallow term constructor. Under this convention, the operation $\ranked{f_2}$ is written as \begin{align*}
\ranked{\shallowterm {\reduce k \Gamma} {\Sigma^k} \xrightarrow{\quad f_2\quad} \reduce 1 (\shallowterm \Gamma \Sigma)}
\end{align*}}.
The function $\ranked{f_2}$ is illustrated by the following picture \begin{center} \includegraphics[scale=.35]{pictures/shallow-unfold} \end{center} and defined by \begin{eqnarray*}
\xymatrix{
(a/g)((b_{1,1},\dots,b_{1,k}),\dots, (b_{n,1},\dots,b_{n,k}))
\ar@{|->}[d]^{\ranked{f_2}} \\
a(b_{g(1)},\dots,b_{g(m)})/g'
} \end{eqnarray*} where $m$ is the arity of $a$ and the grouping function $g'$ is the natural embedding of ports
\begin{align*} \xymatrix{
\text{ports of $a(b_{g(1)},\dots,b_{g(m)}))$}
\ar[d]
\\
\displaystyle{\biggl(\coprod_{\substack{i \in \set{1,\ldots,n}\\ j \in \set{1,\ldots,k}}} \text{ports of $b_{i,j}$}, 1 \biggr)} } \end{align*}
\subsubsection{Distribute shallow terms over product} Finally, consider the function \begin{align*} \ranked{\shallowterm {\Gamma^k} {\Sigma} \xrightarrow{\quad f_3\quad} (\shallowterm \Gamma \Sigma)^k} \end{align*} which distributes shallow terms over the $k$-th power. This function is illustrated by the following picture \mypic{67} and defined by \begin{eqnarray*}
\xymatrix{
(a_1,\dots,a_k)(b_1,\dots,b_n)
\ar@{|->}[d]^{\ranked{f_2}} \\
(a_1(b_1,\dots,b_{\text{ar}_1}), a_2(b_{\text{ar}_1+1},\dots,b_{\text{ar}_2}),\dots ,a_k(b_{\text{ar}_{k-1}+1},\dots,b_{\text{ar}_k}))
} \end{eqnarray*} where $\text{ar}_i$ is the arity of $a_i$ for $i\in\set{1,\dots,k}$.
\subsubsection{Unfolding shallow terms.} The following diagram defines unfolding of shallow terms in terms of the operations $\ranked{f_1}, \ranked{f_2}, \ranked{f_3}$ defined above:
\begin{align*}
\xymatrix@C=2.5cm{
\ranked{\shallowterm{\mati k \rSigma} {\mati k \rGamma} = {\shallowterm{\reduce k {\Sigma^k}}{\reduce k {\Gamma^k}}}
\ar[d]_{\ranked{\substack{f_1}}}
\ar[r]^-{\ranked{\text{Shallow unfold}}}}
&
\ranked{ \reduce k(\shallowterm{\Sigma}{ {\Gamma}})^k = \mati k {(\shallowterm \Sigma \Gamma)}}
\\
\ranked{ \reduce k(\shallowterm{\reduce k {\Sigma^k}}{ {\Gamma^k}})}
\ar[r]_-{\ranked{\flatt\circ \reduce k f_2}}
&
\ranked{ \reduce k(\shallowterm{\Sigma^k}{ {\Gamma}}) } \ar[u]^{\ranked{\reduce k f_3}}
} \end{align*}
\subsection{Definition of unfolding} Having defined shallow unfolding, we apply the induction principle described in Section~\ref{sec:terms-induction-principle} to define unfolding for general terms \begin{align*}
\ranked{\unfold : \tmonad \mati k \rSigma \to \mati k {(\tmonad \Sigma)} }.
\end{align*} If the input to general unfolding is the identity term $\portletter$, then the output is: \mypic{83} Otherwise, if the input is a nonempty term $a(t_1,\ldots,t_n)$ then the output is obtained by first applying term unfolding to to the smaller terms $t_1,\ldots,t_n$, and then applying the shallow unfold.
\section{Examples}\label{sec:AppendixExamples}
To illustrate derivable functions, we present a series of examples, some of them will be useful later. In the rest of this section, for every $k\in\set{1,2,\dots}$ the set $\ranked{k}$ designates the ranked set containing a single element of arity $k$ that we denote by simply by $k$.
\noindent \begin{example}[Parent and children]\label{ex:sibling} Let $\rGamma$ be a finite type. We define $\ranked{\rGamma_0}$ to be the ranked set obtained from $\rGamma$ by setting the arity of every element to $0$.
Consider the function: $$\ranked{ \mathsf{Parent}: \tmonad \rGamma \to \tmonad (\rGamma\product (\rGamma_0+0))}$$ which adds to every node of a term in $\tmonad \rSigma$ the label of its parent if it has one, and $0$ if it is the root.
Let us explain how $\ranked{\mathsf{Parent}}$ can be derived. To illustrate this construction, we use the following alphabet $\rGamma$ \begin{center}
\includegraphics[scale=.4]{pictures/parent-alphabet.pdf}
\end{center} and the following term as a running example. \begin{center}
\includegraphics[scale=.4]{pictures/parent-example.pdf}
\end{center} We denote by $\ranked{\Gamma_1}$ the ranked set obtained from $\rGamma$ by setting the arity of every element to $1$. If $a$ is a element of $\rGamma$, we denote by $a_1$ the corresponding element of $\ranked{\Gamma_1}$. In our example, the alphabet $\ranked{\Gamma_1}$ is \begin{center}
\includegraphics[scale=.4]{pictures/parent-unary-alphabet.pdf}
\end{center} \begin{enumerate} \item First, we apply the homomorphism \begin{align*} \ranked{\mathsf{Hom}_g:\tmonad\Gamma\to \tmonad(\Gamma+\Gamma_1+1)} \end{align*} where $\ranked{g}$ is defined on the elements of $\rGamma$ as follows \begin{align*} \ranked{g: \Gamma} & \ranked{\to \tmonad(\Gamma+\Gamma_1+1 )}\\
a & \mapsto a\tensorpair{1\tensorpair{a_1\tensorpair{\portletter}},\dots, 1\tensorpair{a_1\tensorpair{\portletter}}} \end{align*} In our example, the action of $\ranked{g}$ on the elements of $\rGamma$ looks like this \begin{center}
\includegraphics[scale=.4]{pictures/parent-function-g.pdf}
\end{center} Hence, after the application of the homomorphism $\ranked{\mathsf{Hom}_g}$, our initial term becomes \begin{center}
\includegraphics[scale=.4]{pictures/parent-hom.pdf}
\end{center} \item We apply the factorization \begin{align*} \ranked{\ancfact: \tmonad(\Gamma+\Gamma_1+1) \to \tmonad(\tmonad(\Gamma+\Gamma_1)+\tmonad 1)} \end{align*}
to separate the symbol $1$ form the other symbols. After this operation, each node lies in the same factor as (the element of $\ranked{\Gamma_1}$ representing) its parent. In our example, the obtained term is the following \begin{center}
\includegraphics[scale=.4]{pictures/parent-block.pdf}
\end{center} \item Consider the function \begin{align*} \ranked{h: \tmonad 1 \to \tmonad((\Gamma+\Gamma_1)\product(\Gamma_0+0))} \end{align*} which is the empty term constant function. It is derivable by lifting the empty term constant function over $1$ to terms. And let $k$ be the function \begin{align*} \ranked{k: \tmonad(\Gamma+\Gamma_1) \to \tmonad((\Gamma+\Gamma_1)\product(\Gamma_0+0))} \end{align*} which is the identity function, except for the following terms in which it is defined as follows $$\begin{array}{rll} a\tensorpair{\portletter,\dots, \portletter}& \mapsto & \tensorpair{a,0}\tensorpair{\portletter,\dots,\portletter}\\ b_1\tensorpair{a\tensorpair{\portletter,\dots,\portletter}}&\mapsto& \tensorpair{a,b_0}\tensorpair{\portletter,\dots,\portletter}\\ b_1\tensorpair{\portletter} &\mapsto& \portletter \end{array}$$ \end{enumerate} We apply the function $\ranked{h}$ to the factors $\ranked{\tmonad 1}$ and the function $\ranked{k}$ to the factors $\ranked{\tmonad(\rGamma+\rGamma_1)}$. Doing so, we obtain a term in $\ranked{\tmonad\tmonad((\Gamma+\Gamma_1)\product(\Gamma_0+0))}$, which we flatten, then we erase the symbols $\ranked{\Gamma_1}$ using the function $\mathsf{Filter}$ of Example~\ref{ex:filter} to obtain the desired term.
If $\rGamma$ is a finite ranked set, we define $\ranked{\Gamma^*}$ as $$\coprod_{i \leq \text{ maximal arity in } \rGamma} \underbrace{\ranked{\rGamma\product \cdots \product \rGamma}}_{i\text{ times}}$$ Now consider the function $$\ranked{\mathsf{Children}:\tmonad\rGamma\to \tmonad (\rGamma\product (\rGamma_0+0)^*)}$$ which tags every node of a term in $\tmonad \rGamma$ by the list of its children symbols. When a child is a port, it is marked by $0$ in the list. The function $\ranked{\mathsf{Children}}$ can be derived using a similar construction as above. \end{example}
\noindent \begin{example}[Root and leaves] Let $\rSigma$ be a finite type and $\ranked{f:\rSigma \to \rGamma}$, $\ranked{g: \rSigma \to \rGamma}$ be derivable functions. The function $$\ranked{\mathsf{Root}_{f,g} : \tmonad\rSigma \to \tmonad\rGamma}$$ which applies $\ranked{f}$ to the root and $\ranked{g}$ to the rest of the tree is a derivable function. To show this, we first start by applying the function $\ranked{\mathsf{Parent}}$. Doing so, the root can be distinguished from the other nodes since it will be tagged by $0$.
The function $\ranked{h}$ defined below is derivable since its domain is finite. \begin{align*} \ranked{h:\rSigma\product(\rSigma_0+0)}&\ranked{\to \rGamma}\\
\tensorpair{a,0} &\mapsto f(a) \\
\tensorpair{a,b} &\mapsto g(a) \text{ if } b\neq 0. \end{align*} We lift $\ranked{h}$ to terms to conclude.
Similarly, the function $$\ranked{\mathsf{Leaves}_{f,g} : \tmonad\rSigma \to \tmonad\rGamma}$$
which applies $\ranked{f}$ to the leaves and $\ranked{g}$ to the rest of the tree is derivable. This is done using the same ideas as before, but invoking the function $\ranked{\mathsf{Children}}$ instead of the function $\ranked{\mathsf{Parent}}$: leaves can be distinguished from the other nodes since they are tagged either by a list of $0$ or the empty list. \end{example}
\noindent\begin{example}[Descendants and ancestors]\label{ex:descendant} If $\rSigma$ is a finite type and $\ranked{\rGamma\subseteq \rSigma}$, then the functions \begin{itemize} \item $\ranked{\mathsf{Descendant}_\rGamma: \tmonad \rSigma \to \tmonad (\rSigma+\rSigma)}$ which replaces the label of each node by its first or second copy, depending on whether it has a descendant in $\rGamma$, \item $\ranked{\mathsf{Ancestor}_\rGamma: \tmonad \rSigma \to \tmonad (\rSigma+\rSigma)}$ which replaces the label of each node by its first or second copy, depending on whether it has a descendant in $\rGamma$, \end{itemize} are derivable.
To derive $\ranked{\mathsf{Descendant}_\rGamma}$, we start by applying the factorization $$\ranked{\decfact: \tmonad\rSigma\to \tmonad(\tmonad\rGamma+\tmonad(\rSigma\setminus\rGamma))}$$ which regroups the elements of $\rSigma$ and the elements of $\ranked{\rSigma\setminus\rGamma}$ into factors depending on whether they have the same ancestors of the same type.
Obviously, all the nodes of the $\rGamma$ factors have a descendant in $\rGamma$. In the $\ranked{\rSigma\setminus\rGamma}$ factors which are not leaves in the factorized term, all the nodes have a $\rGamma$ descendant in the original term. To show this, take $f$ to be one of these factors, and suppose by contradiction that one of its nodes does not have a descendant in $\rGamma$. By definition of $\ancfact$, all the elements of $f$ do not have a descendant in $\rGamma$ as well. Since $f$ is not a leaf, it has a child $g$. The factor $g$ cannot be a $\rGamma$ factor as the nodes of $f$ would have a descendant in $\rGamma$. The factor $g$ is then necessarily a $\ranked{\rSigma\setminus \rGamma}$ factor. If a node of $g$ has a descendant in $\rGamma$, this would give a $\rGamma$ descendant to one of the node of $f$. Thus all the nodes of $g$ are in $\ranked{\rSigma\setminus \rGamma}$ and do not have a descendant in $\rGamma$, meaning that $f$ and $g$ are actually the same factor, which gives a contradiction. Finally, the $\ranked{\rSigma\setminus\rGamma}$ factors which are leaves do not have a descendant in $\rGamma$. With these observations, we can now implement $\mathsf{Descendant}_\rGamma$.
Let us consider the functions $$\begin{array}{llll} \ranked{\mathsf{Yes}_\rGamma :} & \rGamma &\ranked{\to} &\ranked{ \rSigma+\rSigma}\\%\noindent\begin{example}[Filter]\label{ex:filter} Consider the types $\rGamma, \rSigma$ where $\rGamma$ is a finite type of unary symbols. Consider the function:
\ranked{\mathsf{Yes}_{\ranked{\rSigma\setminus\rGamma}:}}& \ranked{\rSigma\setminus\rGamma}&\ranked{\to} &\ranked{ \rSigma+\rSigma}\\ \ranked{\mathsf{No}_{\ranked{\rSigma\setminus\rGamma}: }}&\ranked{\rSigma\setminus\rGamma} &\ranked{\to}& \ranked{ \rSigma+\rSigma} \end{array}$$ which replaces the label of each node by its first copy for $\ranked{\mathsf{Yes}_\Gamma}$ and $\ranked{\mathsf{Yes}_{\rSigma\setminus\rGamma}}$, and by its second copy for $\ranked{\mathsf{No}_{\rSigma\setminus\rGamma}}$. The three functions are derivable as their domains are finite. Consider the functions \begin{align*} \ranked{f:=} &\ranked{ \tmonad\mathsf{Yes}_\rGamma +\tmonad\mathsf{No}_{\rSigma\setminus \rGamma}}\ranked{: \tmonad\rGamma+\tmonad(\rSigma\setminus\rGamma) \to \tmonad (\rSigma+\rSigma)} \\ \ranked{g:=} & \ranked{ \tmonad\mathsf{Yes}_\rGamma +\tmonad\mathsf{Yes}_{\rSigma\setminus \rGamma}}\ranked{: \tmonad\rGamma+\tmonad(\rSigma\setminus\rGamma) \to \tmonad (\rSigma+\rSigma)} \end{align*} The descendant function is obtained by applying $\ranked{\mathsf{leaves}_{f,g}}$ followed by a flattening.
To derive the function $\ranked{\mathsf{Ancestor}_\Gamma}$, we apply first a the factorization \begin{align*} \ranked{\ancfact: \tmonad\rSigma\to \tmonad(\tmonad\rGamma+\tmonad(\rSigma\setminus\rGamma))} \end{align*} which regroups the elements of $\rSigma$ and the elements of $\ranked{\rSigma\setminus\rGamma}$ into factors depending on whether they have the same descendants of the same type. Using similar arguments as before, we can conclude that: \begin{itemize} \item The nodes inside $\rGamma$ factors have $\rGamma$ ancestors. \item If a $\ranked{\rSigma\setminus\rGamma}$ factor is the root of the factorized term, then its nodes do not have a $\rGamma$ ancestor. \item If a $\ranked{\rSigma\setminus\rGamma}$ factor is not the root of the factorized term, then its nodes do have a $\rGamma[Descendants and ancestors]\label{ex:descendant}$ ancestor. \end{itemize} The ancestor function is obtained by applying $\ranked{\mathsf{root}_{f,g}}$ followed by a flattening. \end{example}
\begin{example}[Error raising.]\label{ex:error-raising} We can think of the type $\ranked{\bot}$ as an error type. Indeed, the following raising error functions are derivable. \begin{lemma}\label{lem:error-raising} Let $\rSigma$ and $\rGamma$ be two datatypes. The functions $$\begin{array}{rll} \ranked{\tmonad(\Sigma+\bot)} &\ranked{\to }&\ranked{\tmonad \Sigma +\bot}\\ \ranked{(\Sigma+\bot) \product (\Gamma+\bot)} &\ranked{\to} &\ranked{\Sigma\product \Gamma+\bot }\\ \ranked{\shallowterm {(\Sigma+\bot)}{(\Gamma+\bot)}} &\ranked{\to} &\ranked{\shallowterm\Sigma\Gamma +\bot}\\ \ranked{\reduce k (\Sigma+\bot)} &\ranked{\to} &\ranked{\reduce k \Sigma +\bot} \end{array}$$ which are defined as follows \begin{align*}
t \text{ of arity } n \ \ \mapsto \ \begin{cases}
t &
\text{if $t$ does not contain any element of $\ranked{\bot}$,}\\
n & \text{otherwise.} \end{cases} \end{align*} are derivable. \end{lemma} These functions can be easily derived using Proposition~\ref{prop:forat} and distributivity prime functions. The details of the proof are left as an exercise to the reader. \end{example}
\begin{example}[Partial functions.] Thinking of $\ranked{\bot}$ as an error datatype, a function of type $\ranked{\Sigma \to \Gamma +\bot}$ can be seen as a partial function from $\rSigma$ to $\rGamma$. We write \begin{align*} \ranked{\Sigma\rightharpoonup \Gamma} \end{align*} as a notation for the function type $\ranked{\Sigma \to \Gamma +\bot}$. Using the error raising mechanisms discussed earlier, we can manipulate transparently partial function. Indeed, all datatype constructors can be lifted to partial functions, by composing the liftings (1)--(4) with the error raising functions from Lemma~\ref{lem:error-raising}. For example, if $\ranked{f:\Sigma\rightharpoonup \Gamma}$ is a partial function, then $\ranked{\tmonad f:\tmonad\Sigma\rightharpoonup \tmonad\Gamma}$ is defined as the composition \begin{align*} \ranked{\tmonad\Sigma\xrightarrow{\tmonad f} \tmonad(\Gamma+\bot) \xrightarrow{\text{Error raising}} \tmonad \Gamma +\bot}. \end{align*} \end{example}
\section{Derivable functions can be described in first-order logic} \label{sec:to-logic} The goal of this section is to show the right-to-left implication of Theorem~\ref{thm:main}, which says that derivable functions can be implemented by first-order transductions.
As discussed in the body of the paper, we proceed by induction on the derivation. During this induction, we will need to show that every prime function is a first-order transduction. Prime functions are not tree-to-tree functions, instead they transform dataypes into datatypes. This is the reason why we need \begin{itemize} \item to generalize tree-to-tree transductions into transductions that can transform models over arbitrary vocabularies (and not only the vocabulary of trees). \item show how datatypes (terms, pairs, copairs and folds) can be encoded as models over a well chosen vocabulary. More precisely, we will associate to every datatype $\rSigma$ a relational vocabulary that we call \emph{vocabulary of $\rSigma$}. Structures over this vocabulary will be called \emph{models over $\rSigma$}. Then we will define a function \begin{align*}
\xymatrix@C=2cm{
\rSigma \ar[r]^-{x \mapsto \underline x} &
\text{models over $\rSigma$}
} \end{align*} which assigns to each element $x \in \rSigma$ a corresponding model over $\rSigma$, which is denoted by $\underline x$. \end{itemize}
Right-to-left implication of Theorem~\ref{thm:main} can be then generalized to the following statement, more suited to a proof by induction:
\begin{proposition}\label{prop:main-right-to-left} Let $\rGamma$ and $\rSigma$ be two datatype. For every derivable function $\ranked{f}$, there is a first-order transduction $g$ such that the following diagram commutes
\begin{align*}
\xymatrix@C=2.8cm{
\rSigma
\ar[d]_{x \mapsto \underline x}
\ar[r]^-{\ranked{f}}
&
\rGamma \ar[d]^{x \mapsto \underline x}
\\
\text{models over $\rSigma$}
\ar[r]_-{g}
&
\text{models over $\rGamma$}
} \end{align*} \end{proposition}
The rest of this section is organized as follows. We define first-order transductions transforming arbitrary models in Section~\ref{sec:fo-transduction-def}. In Section~\ref{sec:data-as-models} we define the vocabularies for the datatypes and the model representation $x \mapsto \underline x$. Finally, we prove Proposition~\ref{prop:main-right-to-left} which gives as a corollary the right-to-left implication of Theorem~\ref{thm:main}.
\subsection{First-order transductions}\label{sec:fo-transduction-def} The following definition introduces first-order transductions, which generalizes tree-to-tree transductions given in Definition~\ref{def:fo-transduction} to arbitrary models.
\begin{definition}[First-order transduction]\label{def:fo-transduction-gen}\
A \emph{first-order transduction} is defined to be any composition of the following two kinds of transformations on structures: \begin{enumerate}
\item \emph{Copying.} Fix some relational vocabulary $\ranked \sigma$ and let $k \in \set{1,2,\ldots}$. Define $k$-copying to be the operation of type
\begin{align*}
\xymatrix{
\text{models over $\ranked \sigma$} \ar[d]\\
\txt{models over $\ranked \sigma$ extended\\ with a $k$-ary relation $\mathrm{copy}$}
}
\end{align*} which inputs a model $\mathbb A$, and outputs $k$ disjoint copies of $\mathbb A$, where the $\mathrm{copy}$ relation is interpreted as the set of tuples $(a_1,\ldots,a_k)$ such that, for some $a \in \mathbb A$, the first copy of $a$ is $a_1$, the second copy of $a$ is $a_2$, etc. The $\mathrm{copy}$ relation is not commutative, because we distinguish the copies. \item \emph{Non-copying first-order transduction.} The syntax of a \emph{non-copying first-order transduction} is given by: \begin{enumerate}
\item Input relational vocabulary $\ranked\sigma$ and output relational vocalbulary $\ranked{\gamma}$.
\item A first-order \emph{universe formula} $\varphi(x)$ over $\ranked{\sigma}$.
\item For every relation $R$ in vacubulary $\ranked{\gamma}$, a first-order formula $\varphi_R(x_1,\ldots,x_{\arity R})$ over $\ranked{\sigma}$. \end{enumerate} The semantics of a non-copying first-order transduction is a function \begin{align*}
\xymatrix{
\text{models over $\ranked \sigma$} \ar[d]\\
\text{models over $\ranked \gamma$}
} \end{align*} defined as follows. If the input model is $\mathbb A$, then the output model is defined as follows: the universe is elements of $\mathbb A$ which satisfy the universe formula, and each relation $R$ is interpreted as those tuples that satisfy $\varphi_R$.
\end{enumerate} \end{definition}
The notion of copying used in the above definition is slightly different from the notion of copying used for tree-to-tree transductions in Definition~\ref{def:fo-transduction}, which was specifically tailored to stay within the realm of trees. Nevertheless, the two definitions are easily seen to define the same class of tree-to-tree functions.
\subsection{Datatypes as models.}\label{sec:data-as-models} Let us show how to encode datatypes as relational vocabularies and data as models over these vocabularies. \begin{definition}[Associated models for terms, pairs, co-pairs, folds.] \label{def:type-model} To each type $\rSigma$ we associate a vocabulary, called the \emph{vocabulary of $\rSigma$}, and a map
\begin{align*}
a \in \rSigma \qquad \mapsto \qquad \underbrace{\underline a \in \text{models over the vocabulary of $\rSigma$}}_{\text{associated model of $a$}}.
\end{align*}
Furthermore, for each $a \in \rSigma$ we distinguish a sequence (whose length is the arity of $a$) of elements in $\underline a$, which are called the ports of $\underline a$. The definitions are by induction on the structure of $\rSigma$, as given below.
\begin{itemize}
\item \emph{Finite ranked sets.} Elements of a ranked set \begin{align*}
\rSigma = \set{a_1,\ldots,a_k}
\end{align*} are modelled using a vocabulary which has unary relations $a_1,\ldots,a_k$ and $P_1,\ldots,P_m$ where $m$ is the maximal arity of elements in $\rSigma$.
For $a \in \rSigma$ of arity $n$, the universe of $\underline a$ is $\set{0,1,\ldots,n}$, with the ports being $1,\ldots,n$.
The relation $P_i$ is interpreted as $\set i$ when $i \in \set{1,\ldots,n}$ and as the empty set otherwise. The relation $a_i$ is interpreted as $\set 0$ when $a = a_i$ and as the empty set otherwise.
\item \emph{Coproduct.} Elements of the coproduct $\ranked{\Sigma_1 + \Sigma_2}$ are modelled using the disjoint union of the vocabularies of $\ranked{\Sigma_1}$ and $\ranked{\Sigma_2}$.
If an element of the coproduct comes from $\ranked{\Sigma_1}$, then its associated model is defined as for the type $\ranked{\Sigma_1}$, with the remaining relations from the vocabulary of $\ranked{\Sigma_2}$ interpreted as empty sets. The definition is analogous for elements from $\ranked{\Sigma_2}$.
\item \emph{Product.} Pairs in $\ranked{\Sigma_1 \product \Sigma_2}$ are modelled
using the disjoint union of the vocabularies of $\ranked{\Sigma_1}$ and $\ranked{\Sigma_2}$.
For $\tensorpair{a_1,a_2}$, the associated model is the disjoint union of models $\underline{a_1} + \underline {a_2}$, with the relations of $\underline {a_1}$ using the vocabulary of ${\ranked{\Sigma_1}}$, and the relations of $\underline {a_2}$ using the vocabulary ${\ranked{\Sigma_1}}$.
If $n_1$ is the arity of $a_1$, then the first $n_1$ ports are inherited from $\underline {a_1}$ and the remaining ports are inherited from $\underline {a_2}$.
\item \emph{Folding.} For $k \in \set{1,2,\ldots}$, elements of $\reduce k \rSigma$ are modelled using the vocabulary of $\rSigma$ plus two extra binary relations $\portord$ and $R$. If $a \in \rSigma$ has arity $nk$, then the model associated to $a/f$ -- which has arity $n$ -- is obtained from $\underline{a}$ by adding a copy of the model below, where $\sqsubset$ is the natural ordering on integers
\begin{align*}
(\set{1,\ldots,n}, \portord),
\end{align*} whose elements are used as the ports, and interpreting the binary relation $R$ as
\begin{align*}
\set{(\text{$i$-th port of $\underline a$},f(i)) : i \in \set{1,\ldots,nk}}
\end{align*}
\item \emph{Terms.} Terms in $\tmonad \rSigma$ are modelled using vocabulary of $\rSigma$ extended with two fresh binary relations $\anceord$ and $\portord$.
Let $t \in \tmonad \rSigma$. Consider the disjoint union of models
\begin{align}\label{eq:non-port}
\coprod_{x \in \text{non-port nodes in $t$}} \underline{a(x)},
\end{align}
where $\underline a(x)$ is the model over vocabulary of $\rSigma$ that is defined by induction assumption. In the above disjoint union, the same vocabulary, namely the vocabulary of $\rSigma$, is used for all parts of the disjoint union. Next, consider the model
\begin{align}\label{eq:ports}
(\set{1,\ldots,n}, \portord)
\end{align}
where $\portord$ is the natural ordering on $\set{1,\ldots,n}$.
The model of $t$ is defined by taking the disjoint union of the models in~\eqref{eq:non-port} and~\eqref{eq:ports}, and defining the descendent relation $\anceord$ as the set of pairs $(u,v)$ such that:
\begin{itemize}
\item either $u$ is the $i$-th port of $\underline{a(x)}$ for some node $x$ of $a$, $v$ is a port of $\underline{a(y)}$ for some node $y$ which is a descendent of the $i$-th child of $x$.
\item or $u$ is the $i$-th port of $\underline{a(x)}$ for some node $x$ of $a$, $v=j\in\{1,\dots,n\}$ and the $j$-th port of $a$ is a descendent of the $i$-th child of $x$.
\end{itemize}
\end{itemize} \end{definition}
The above definition creates a certain ambiguity for trees, because if $t$ is a tree over a finite ranked set $\rSigma$, then $\underline t$ can be understood in two ways: as per Definition~\ref{def:tree-model} for trees, or as per Definition~\ref{def:type-model} when $t$ is viewed as a special case of a term $t \in \tmonad \rSigma$. Since we only use first-order transductions to transform relational structures, this ambiguity is not a problem, because one can easily define first-order transductions which map one definition of $\underline t$ to the other.
\subsection{Proof of Proposition~\ref{prop:main-right-to-left}} The proof proceeds by induction, following the definition of derivable functions. In the induction step, we have to deal with function composition and the lifting of function along the datatype constructors. First-order transductions are closed under composition by definition, while the liftings are immediate.
In the induction base, we need to show that all of the prime functions are first-order transductions. All the cases are easy, and consist mainly on unfolding the definitions; this is the point of calling these functions prime. There is one exception, which requires some more explanation, namely monotone unfolding. We explain below just one of the easy functions, the unit function $\ranked{\Sigma \to \tmonad \Sigma}$, and the monotone unfolding. The other prime functions are left as an exercise.
\subsubsection{A first-order transduction for the term unit} \label{sec:transduction-unit}
In the following, it will be convenient to use, as part of the vocabulary of $\rSigma$, a unary relation $\mathsf{Port}_\rSigma$ which selects the ports of the structures over the vocabulary of $\rSigma$; and a binary relation $\sqsubset_\rSigma$ which orders these ports. By induction on $\rSigma$, we can show that both relations are definable by first-order formulas over the vocabulary of $\rSigma$.
Given an element $x$ of $\rSigma$, let us show how $\unit(x)$ can be implemented using a first-order transduction. The copying constant is 2,
the first copy will contain the whole structure $\underline{x}$ and the second copy will select only the ports of $\underline{x}$ which will serve as the ports of the structure $\underline{\unit(x)}$, as illustrated by the following picture \begin{center}
\includegraphics[scale=.18]{pictures/to-logic-unit.pdf}
\end{center}
The universe formulas are then:
\begin{align*}
\varphi_1(x)=\mathsf{True} \qquad \varphi_2(x)=\mathsf{Port}_\rSigma(x)
\end{align*}
In the first copy, the vocabulary of $\rSigma$ will be interpreted as in the original structure, and as the empty set in the second copy. That is, for every unary relation $R$ and for every binary relation $S$ in the vocabulary of $\rSigma$, we set:
\begin{align*}
\varphi_R^{1}(x)=R(x) \quad&\quad \varphi_S^{1,1}(x,y)=S(x,y)\\
\varphi_R^{2}(x)=\mathsf{False} \quad&\quad \varphi_S^{2,2}(x,y)=\mathsf{False} \end{align*} Let us interpret the relations $<$ and $\sqsubset$ of the vocabulary of $\tmonad\rSigma$. The ports of $\underline{\unit(x)}$ inherit the order of the ports of $\underline{x}$, this is why we set: \begin{align*} \varphi_\sqsubset^{2,2}(x,y)=x\sqsubset_\rSigma y \end{align*} The descendant relation $<$ connects the $i^{th}$ port of $\underline{x}$ to the $i^{th}$ port of $\underline{\unit(x)}$. Since these nodes come from the same node in the original structure, we set: \begin{align*} \varphi_<^{1,2}(x,y)=x=y \end{align*}
\subsubsection{A first-order transduction for monotone unfolding} \label{sec:fo-transduction-for-unfolding} Having illustrated the syntax of first-order transductions on the example of the unit function, we describe a first-order transduction for the monotone unfolding operation \begin{align*}
\ranked{\tmonad \mati k{\Sigma} \to \mati k{( \tmonad \Sigma)} + \termset}. \end{align*} This is the only prime function whose corresponding first-order transduction is not obvious. Unlike in Section~\ref{sec:transduction-unit}, we focus more on the underlying conceptual difficulties than on the syntax of first-order transductions.
Recall that when defining the monotone unfolding operation, for each element $a \in \ranked{\tmonad \mati k{\Sigma}}$ of the matrix power, we used a family of (partial) twist functions \begin{align*} \to_i : \set{1,\ldots,k} \to \set{1,\ldots,k}, \end{align*} one for each port $i$ of $a$. For the reader's convenience, we repeat a picture from Section~\ref{sec:unfolding}, which explains the twist functions: \mypic{125}
In this example, the twist function $\to_1$ is monotone, but $\to_2$ is not. The monotone unfolding operation works in the same way as general unfolding, except that it uses the undefined value $\termset$ if the input term has at least one letter which uses at least one non-monotone twist.
The following lemma, whose simple proof is left to the reader, shows that the twist functions can be defined using first-order logic. \begin{lemma}
Let $\rSigma$ be a datatype and let $k \in \set{1,2,\ldots}$. For every partial function
\begin{align*}
\tau : \set{1,\ldots,k} \to \set{1,\ldots,k}
\end{align*}
there is a first-order formula $\varphi_\tau(x)$ such that for every $a \in \mati k \rSigma$,
\begin{align*}
\underline a \models \varphi_\tau(x)
\end{align*}
if and only if $x$ represents a port with twist function $\tau$. \end{lemma} By using the formulas from the above lemma, one can construct a first-order formula which checks if a term in $\tmonad \mati k \rSigma$ uses only monotone twists, i.e.~whether or not the output of monotone unfolding should be $\termset$.
We now proceed to the more interesting part of monotone unfolding, i.e.~actually doing the unfolding for monotone inputs. Consider an input $t \in \tmonad \mati k \rSigma$ to monotone unfolding. Define a \emph{sub-node} of $t$ to be a pair (node of $t$, number in $\set{1,\ldots,k$}), as explained in the following picture: \mypic{124}
In the output of the monotone unfolding, which is of the form \begin{align*} (t_1,\ldots,t_k)/f \in \mati k {\tmonad \rSigma}, \end{align*} the nodes of the output terms $t_1,\ldots,t_k$ will correspond to the sub-nodes in the input $t$. The sub-nodes can be produced by copying the input term $k$-times.
The most interesting part of the structure in the output is the descendant relation in the terms $t_1,\ldots,t_k$. This relation can be viewed as a descendant relation on the sub-nodes. We only describe how the descendant relation on the sub-nodes can be defined in first-order logic, and the rest of the transduction is left to the reader.
When defining the descendant relation on sub-nodes, the crucial part is composing the twist functions. Suppose that we want to check the descendant relationship between two sub-nodes \begin{align}\label{eq:descendant-relationship} (x,i) \stackrel?{\le} (y,j), \end{align} where $x,y$ are nodes on the input term and $i,j \in \set{1,\ldots,k}$. We will show that the descendant relationship~\eqref{eq:descendant-relationship} holds if and only if $x$ is an ancestor of $y$ in the input term, and the twist functions on the path connecting $x$ and $y$ maps $j$ to $i$, as explained below.
Consider a path in the input term, which connects node $x$ with $y$, as illustrated in the following picture \mypic{123} Each edge in the input term corresponds to a chosen port in some node, which in turn corresponds to some twist function, and therefore it makes sense to talk about the twist function associated to an edge in the input term. Define \begin{align*} \tau_y^x : \set{1,\ldots,k} \to \set{1,\ldots,k} \end{align*} to be the partial function, which is obtained by composing all of the twist functions corresponding to edges on the path connecting $y$ to $x$, starting with $y$ and ending with $x$. In the example from the above picture, we compose two twist functions, which correspond to edges marked in yellow.
Equipped with the above definitions, we can now characterise the descendant ordering on sub-nodes by \begin{align*}
(x,i) \le (y,j) \qquad \text{iff} \qquad x \le y \land \tau_y^x(j)=i. \end{align*} Therefore, to complete the proof, it remains to show the following lemma. This is where we use the monotonicity assumption.
\begin{lemma}\label{lem:counter-free}
For every $i,j \in \set{1,\ldots,k}$
there is a first-order formula $\psi_j^i(x,y)$ such that for every $t \in \tmonad \mati k \rSigma$
\begin{align*}
\underline t \models \psi_j^i(x,y) \qquad \text{iff} \qquad \tau_y^x(j)=i.
\end{align*} \end{lemma} \begin{proof}
Let $F$ be the set of monotone partial functions from $\set{1,\ldots,k}$ to itself. Define $L \subseteq F^*$ to be the set of those words $f_1 \cdots f_n$ such that the composition of functions $f_n\circ \cdots \circ f_1$ maps $i$ to $j$. We will show that -- thanks to the monotonicity assumption -- the language $L$ is definable in first-order logic. To get the conclusion of the lemma, we check if the sequence of twist functions on the path from $x$ to $y$ satisfies the first-order formula defining the language $L$.
The language $L$ is recognised by a finite automaton, which has states $\set{1,\ldots,k,\bot}$, and which simply applies the function in its input letter to the present state. We show below that this automaton is counter-free, in the sense of McNaughton and Papert~\cite[p.~6]{McNaughtonPapert71}, and therefore it can be defined in first-order logic.
Recall that a counter in an automaton is a sequence of at least two pairwise distinct states $q_1,\ldots,q_n$ such that
\begin{align*}
q_1 \stackrel w \to q_2 \stackrel w \to \cdots \stackrel w \to q_n \stackrel w \to q_1
\end{align*}
holds for some common input string $w$. In the automaton for the language $L$ that we have discussed above, there is no counter. Indeed, if we would have $q_1 \le q_2$, then by monotonicity of the function $w \in F$ we would have
\begin{align*}
q_1 \le q_2 \le \cdots \le q_n \le q_1
\end{align*}
and therefore all of $q_1,\ldots,q_n$ would be equal, contradicting the assumption that they are pairwise distinct. The same argument would work when $q_1 \ge q_2$. By~\cite[Theorem 10.5]{McNaughtonPapert71}, if an automaton has no counter, then its language is definable in first-order logic.
\end{proof} \section{Appendix on first-order relabelling}~\label{sec:AppendixForat}
The goal of this section is to show Proposition~\ref{prop:forat}, which says that first-order relabeling are derivable. As discussed in the body of the paper, the proof of this proposition is based on an equivalence result between first-order queries on trees and a temporal logic, as stated in Lemma~\ref{lem:schlingloff}. While this result is deaply inspired from a similar result of Schlingloff \cite{schlingloff1992expressive}, our frameworks are not exactly the same (he uses for ainstance unranked trees). In the rest of this section, we provide more details about the reduction from Schilgloff's result to our lemma (Section~\ref{sec:reduction-schilingloff}). Then we show in Section~\ref{sec:relabeling} how to use it in order to prove Proposition~\ref{prop:forat}.
\subsection{Reduction to Schilgloff's theorem}\label{sec:reduction-schilingloff}
Let us proceed to the proof of Lemma~\ref{lem:schlingloff}. Clearly the functions in the lemma are first-order tree relabeling, and first-order tree relabeling are easily seen to be closed under composition, which gives the right-to-left inclusion in the lemma. The hard part is the left-to-right inclusion, which says that every first-order tree relabeling can be decomposed into functions as in items~\ref{it:relabelling},\ref{it:child}--\ref{it:since}. The first step in the proof of the right-to-left inclusion is the observation that every first-order tree relabeling can be decomposed as \begin{align*}
g \circ f_1 \circ \cdots \circ f_n \end{align*} where $g$ is a relabeling as in item~\ref{it:relabelling} of the lemma and each $f_i$ is a characteristic function of some unary query (not necessarily of the simple form indicated in items~\ref{it:child} -- \ref{it:since} in the lemma). This is a simple observation: the functions $f_1,\ldots,f_n$ annotate the tree with the truth values of the unary queries used in the definition of the first-order relabeling, and $g$ uses these truth values to select the appropriate output label. The hard part of the lemma is showing that each $f_i$ can be further decomposed into functions as indicated in the lemma. This is where we us the result of Schlingloff~\cite[Theorem 2.6]{schlingloff1992expressive}, which says that all first-order definable tree properties can be defined using a temporal logic that has operators similar to the ones used in items~\ref{it:child} -- \ref{it:since} of the lemma.
The following table summarizes our framework (first column) and Schlingloff's one (second column). The first row describes the models under consideration, the second row the corresponding version of first-order logic, and the third row the corresponding temporal logic. \begin{center}
\begin{tabular}{|C{1.8cm}|C{2.9cm}|C{2.8cm}|} \cline{1-3} \textbf{Models} & \textbf{Trees over a finite ranked alphabet $\rGamma$}: finitely branching, ranked trees, labeled from $\rGamma$. & \textbf{Models over a set of propositions $P$}: finitely branching, unranked trees, labeled from $2^P$.\\ \hline
\multirow{3}{1.8cm}{\begin{center}\textbf{First-order logic}\end{center}}&\multicolumn{2}{C{5.6cm}|}{Usual first-order connectives ($\exists, \vee, \neg$) with the descendant predicate $x\leq y$ and the following predicates:}\\
\cline{2-3}
& $a(x)$: $x$ is labeled $a$ $(a \in \rGamma)$. & $p(x)$: label of $x$ contains $p$
($p\in P$). \\
&\hspace{-.14cm}$\child i(x)$: $x$ is an $i$-th child. &\\
& \textbf{We call it $\rGamma$-\fo.} & \textbf{We call it $P$-\fo. } \\
\hline
\multirow{4}{1.8cm}{\begin{center}
\textbf{Temporal logic}
\end{center}}& \multicolumn{2}{C{5.6cm}|}{Usual CTL connectives ($S$ (Since), $U$ (Until), $\vee$, $\neg$) together with:}\\
\cline{2-3}
& $a \in \rGamma$, & $p \in P$, \\
& $\odot_i \phi$: the $i$-th child satisfies $\phi$. & $X_i \phi$: at least $i$ children satisfy $\phi$.\\
& \textbf{ We call it 2-CTL. } & \textbf{We call it 4-CTL. } \\
\hline \end{tabular} \end{center} What is named $\rGamma$-\fo in the table is what we simply called first-order logic along the paper. The operators of 2-CTL are those of Lemma~\ref{lem:schlingloff}. Using the notation of the table, Schlingloff's theorem says that $P$-\fo formulas are equivalent to 4-CTL formulas, and Lemma~\ref{lem:schlingloff} states that $\rGamma$-\fo formulas are equivalent to 2-CTL ones. To deduce the later from the former, we will show how to translate every ranked tree $t$ over $\rGamma$ into a model $[t]$ over a well chosen set of propositions $P$, then we will apply the following scheme
$$\xymatrix@C=2.5cm{
\varphi \in \rGamma\text{-\fo}
\ar@{<->}[r]^{\forall t, \ t\models \varphi \ \leftrightarrow \ [t] \models \psi}_{\text{Lemma}~\ref{lem:from-Gamma-to-P-FO}}
&
\psi \in P \text{-\fo}
\ar@{<->}[d]^{\substack{\text{Schlingloff's}\\\text{theorem}}} \\
\theta \in \text{2-CTL}
\ar@{<->}[r]^{\forall t, \ t\models \theta \ \leftrightarrow \ [t] \models \delta}_{\text{Lemma}~\ref{lem:from-4-CTL-to-2CTL}}
&
\delta \in \text{4-CTL}
}$$ The translation $[\_]$ and Lemmas~\ref{lem:from-Gamma-to-P-FO} and \ref{lem:from-4-CTL-to-2CTL} are explained below.
\paragraph{From ranked trees to Schlingloff's models.} Let us fix a ranked alphabet $\rGamma$. Let $P$ be the following set of propositions \begin{align*}
P\eqdef \rGamma\cup\set{i\text{-th-child}\ |\ i\in[1, \text{max arity of }\rGamma]} \end{align*}
Let $t$ be a ranked tree over $\rGamma$. The translation $[t]$ of $t$ is the model defined as follows. It has the same set of nodes and the same descendant relation as $t$. The label of a node contains $a$ if its label in $t$ is $a$. It contains the proposition $i\text{-th-child}$ if it is an $i$-th child in $t$.
\paragraph{First-order logic for ranked and unranked trees.} Let $\rGamma$ and $P$ be as above. Let us show that $\rGamma$-\fo and $P$-\fo are equivalent. \begin{lemma}\label{lem:from-Gamma-to-P-FO} For every $\rGamma$-\fo formula $\phi$, there is a $P$-\fo formula $\psi$ such that $$ \forall t\in \trees\rGamma, \qquad t\models \phi \leftrightarrow [t]\models \psi$$ and conversely. \end{lemma} \begin{proof} To show this lemma its is enough to show how to translate the specific predicates of each formalism into the other. The predicate $a(x)$ of $\rGamma$-\fo can be translated by the same predicate in $P$-\fo and conversely. The predicate $\child i(x)$ can be translated by $i\text{-th-child}(x)$ and conversely. It is clear that these translations preserve the semantics. \end{proof}
\paragraph{Temporal logic for ranked and unranked trees.} Let $\rGamma$ and $P$ be as above. We show that 2-CTL and 4-CTL are equivalent. \begin{lemma}\label{lem:from-4-CTL-to-2CTL} For every $\rGamma$-\fo formula $\phi$, there is a $P$-\fo formula $\psi$ such that $$ \forall t\in \trees\rGamma, \qquad t\models \phi \leftrightarrow [t]\models \psi$$ and conversely. \end{lemma} \begin{proof} Here again, it is enough to translate the specific connectives of each formalism into the other. The connective $X_i$ can be encoded in 2-CTL as follows, where $b$ is the maximal arity of $\rGamma$ \begin{align*} \underset{\begin{array}{c} {\scriptstyle I\subseteq [1,b]}\\ {\scriptstyle\#I=i} \end{array}}{\bigvee} \underset{j\in I}{\bigwedge} \odot_j \phi \end{align*} Conversely, the connective $\odot_i$ can be encoded in 4-CTL as follows: \begin{align*} X_1(i\text{-th-child}\wedge \phi) \end{align*} \end{proof}
\subsection{First-order relabelling are derivable}\label{sec:relabeling} To show Proposition~\ref{prop:forat}, saying that first-order relabelling are derivable, we will show that each function appearing in Lemma~\ref{lem:schlingloff} and corresponding to each operator of 2-CTL is derivable. This is the role of Lemmas\ref{lem:nextmod}--\ref{lem:sincemod} presented below. \begin{lemma}\label{lem:nextmod}
For every finite $\rSigma$, $\rGamma\subseteq \rSigma$ and $i \in \set{1,2,\ldots}$, the characteristic function $\ranked{f:\tmonad \rSigma\to\tmonad(\rSigma+\rSigma)}$ of the unary query
\begin{align*} \text{``The }i\text{-th child of }x\text{ is in }\rGamma\text{''}
\end{align*}
is derivable. \end{lemma} \begin{proof} To show that $\ranked{f}$ is derivable, we start applying the children function \begin{align*} \ranked{\mathsf{Children}:\tmonad\rSigma\to \tmonad (\rSigma\product (\rSigma_0+0)^*)} \end{align*} from Example~\ref{ex:sibling} which tags every nodes by the list of its children. Consider the function $g$ $$\begin{array}{rlll} \ranked{g:} \ranked{\rSigma\product (\rSigma_0+0)^*} &\ranked{\to}& \ranked{\Sigma +\Sigma}&\\
\tensorpair{a,l} &
\mapsto& \tensorpair{a,1} &\text{if } l[i]\in\ranked{\rGamma_0}, \\
& \mapsto& \tensorpair{a,2} &\text{otherwise.} \end{array}$$ which maps an element of $\rSigma$ tagged by a list to the first copy of $\rSigma$ if the $i$-th element of the list is in $\rGamma$ and to the second copy otherwise. The function $g$ is derivable since its domain is finite. We finally get $\ranked{f}$ by lifting $\ranked{g}$ to terms. \end{proof}
\begin{lemma}\label{lem:untilmod} For every finite $\rGamma, \rDelta \subseteq \rSigma$, the characteristic function $\ranked{f:\tmonad\rSigma\to\tmonad(\rSigma+\rSigma)}$ of the unary query
\begin{align*}
\underbrace{\exists y\ y \geq x \land \rDelta(y) \land \forall z \ (x < z < y \Rightarrow \rGamma(z)).}_{\substack{\text{$x$ has a descendant $y$ with label in $\rDelta$, such that}\\ \text{all nodes between $x$ and $y$ have label in $\rGamma$}}}
\end{align*} is derivable. \end{lemma} \begin{proof} We start by applying the factorization \begin{align*} \ranked{\ancfact : \tmonad \Sigma \to \tmonad(\tmonad(\Sigma\setminus(\Gamma\cup\Delta)) +\tmonad(\Gamma\cup\Delta))} \end{align*} which decomposes our terms into factors, depending on whether their node labels are in $\ranked{\Gamma\cup\Delta}$ or not. Note that the value of a node w.r.t. the until query depends only on the node labels of its factor.
The nodes of the $\ranked{\tmonad(\rSigma\setminus(\rGamma\cup\rDelta))}$ factors do not satisfy the query, thus we will apply to them the function $\ranked{\tmonad g}$ obtained by lifting the function $$\begin{array}{rrll}
\ranked{g:}& \ranked{\rSigma\setminus(\rGamma\cup\rDelta)}& \ranked{\to} &\ranked{\rSigma+\rSigma}\\ &a&\mapsto& \tensorpair{a,2}. \end{array}$$
Nodes of the $\ranked{\tmonad(\Gamma\cup\Delta)}$ factors satisfy the query if and only if they have a descendant in $\rDelta$. Consider the function $\ranked{h}$ obtained by composing the descendant function $\ranked{\mathsf{Descendant}_\Delta}$ from Example~\ref{ex:descendant} with an injection $\ranked{\tmonad(\iota+\iota)}$ \begin{align*} \underbrace{\ranked{\tmonad(\Gamma\cup\Delta) \xrightarrow{\ranked{\mathsf{Descendant}_\Delta}} \tmonad(\Gamma\cup\Delta+\Gamma\cup\Delta) \xrightarrow{\ranked{\tmonad(\iota+\iota)}}\tmonad(\Sigma+\Sigma)}}_{\ranked{h}} \end{align*} Finally, to get the characteristic function $\ranked{f}$, we apply $\ranked{\tmonad{g}}$ to the $\ranked{\tmonad(\Sigma\setminus(\Gamma\cup\Delta))}$ factors and $\ranked{h}$ to the other factors using the co-pairing combinator, then we flat the obtained term. \end{proof}
\begin{lemma}\label{lem:sincemod} For every finite $\rGamma, \rDelta \subseteq \rSigma$, the characteristic function of the unary query
\begin{align*}
\underbrace{\exists y\ y \leq x \land \rDelta(y) \land \forall z \ (y < z < x \Rightarrow \rGamma(z)).}_{\substack{\text{$x$ has a descendant $y$ with label in $\rDelta$, such that}\\ \text{all nodes strictly between $x$ and $y$ have label in $\rGamma$}}}
\end{align*}
is derivable. \end{lemma} \begin{proof} The same proof as above, one only needs to replace the use of the function $\ranked{\mathsf{Descendant}_\Delta}$ by that of $\ranked{\mathsf{Ancestor}_\Delta}$, introduced in Example~\ref{ex:descendant}. \end{proof}
\section{Proof of Theorem~\ref{thm:stt}} \label{sec:stt-appendix} In this part of the appendix, we prove Theorem~\ref{thm:stt}, which says that every first-order tree-to-tree transduction is recognised by a register transducer.
According to Definition~\ref{def:fo-transduction}, a first-order transductions is a composition of any number of functions each of which is either copying (item 1) or a non-copying first-order transduction (item 2). In other words: $$\begin{array}{c} \text{first-order transductions} \\ \eqdef (\text{copying} \cup \text{(non-copying first-order transductions)})^* \end{array}$$ where the star denotes closure under composition. Although register transducers are closed under composition, this is not very easy to show directly, and therefore we begin by simplifying the function composition in the definition of first-order transductions. It is not hard to see that copying commutes with non-copying first-order transductions in the following sense: \begin{align*}
\text{copying} \circ \text{(non-copying first-order transductions)} \\ \subseteq \text{(non-copying first-order transductions)} \circ \text{copying}.
\end{align*} Furthermore, since the class of copying functions is closed under composition, and the same is true for non-copying first-order transductions, we get the following normal form of first-order transductions: $$\begin{array}{c}
\text{first-order transductions} \\= \text{(non-copying first-order transductions)} \circ \text{copying}.
\end{array}$$ Therefore, in order to prove Theorem~\ref{thm:stt}, it suffices to show that a register transducer can compute any function which first copies the nodes of the input tree a fixed number of times, and then applies a non-copying first-order transduction.
For the rest of this section, fix a tree-to-tree function \begin{align*} f : \trees \rSigma \to \trees \rGamma \end{align*} which is a composition of first copying (some fixed number of times), followed by a non-copying first-order transduction. We will show that $f$ is computed by some register transducer.
\newcommand{\origin}[1]{\mathrm{origin}_{#1}} \newcommand{\orcol}[2]{\mathrm{orcol}_{#1}^{#2}} In the proof, we use the origin information associated to $f$, i.e.~how nodes of the output tree can be traced back to nodes in the input tree. For an input tree $t \in \trees \rSigma$, define its origin map to be the function of type \begin{align*}
\text{nodes in $f(t)$} \to \text{nodes in $t$}
\end{align*} which maps a node $x$ of the output tree to the node of the input tree that was used to define it. (The origin in a copying function is the node that is being copied, while the node in a non-copying transduction is the node of the input structure that represents the node of the output structure.) For a node $x$ in an input tree $t$, define the origin colouring of $x$ to be the function \begin{align*}
y \in \text{nodes in $f(t)$} \ \ \mapsto \ \begin{cases}
\text{below} & \begin{array}{l}
\text{if the origin of $y$ is $x$}\\ \text{or a descendant of $x$} \end{array} \\
\text{not below} & \text{otherwise.} \end{cases} \end{align*} Define the name \emph{origin factorisation of $x$ in $t$}, which is an element of $\trees \tmonad \rGamma$, to be the factorisation of the output tree where the factors are connected parts of same type (``below '' or ``not below''). The origin factorisation is obtained by applying the ancestor factorisation $\ancfact$ to the output tree extended with its origin colouring.
The general idea behind the register transducer is that, after processing the subtree of a node $x$ in the input tree, its registers will store the ``below'' factors in the origin factorisation of $x$. We only store the ``below'' factors, and not the ``not below'' factors, because only the ``below'' factors can be computed using register updates based on the subtree of the node $x$ in the input tree. The key observation is the following lemma, which shows that the a constant number of registers will be enough.
\begin{lemma}\label{lem:composition-method}
For every input tree $t$ and node $x$ in $t$, the origin factorisation of $x$ in $t$ has at most a constant (i.e.~depending only on the fixed transduction) number of factors. \end{lemma} \begin{proof}
For an input tree $t$ and a node $x$ in it, we say that an edge in the output tree $f(t)$ is \emph{$x$-sensitive} if its the two endpoints are in different factors of the origin colouring of $x$ in $t$. The number of factors in the origin factorisation is one plus the number of sensitive edges, and therefore to prove the lemma, it is enough to show that:
\begin{itemize}
\item[(*)] for every input tree $t$ and node $x$ in $t$, there is at most a constant number of $x$-sensitive edges.
\end{itemize}
Let us write $\to$ for the image -- along the origin mapping -- of the child relation in the output tree. In other words, nodes $y,z$ in the input tree satisfy $y \to z$ if some node in the output tree with origin $z$ is a child of some node in the output tree with origin $y$.
It is not hard to see that $\to$ can be defined in first-order logic, using the formulas from the transduction. Let $r$ be the quantifier rank of the first-order formula used to define $\to$. Using Ehrenfeucht-Fraisse argument, one can show that if $x,y,z$ are nodes in the input tree such that $y$ and $z$ are on different sides of $x$ (i.e.~any path connecting $y$ and $z$ must necessarily pass through $x$),
then the truth value of any rank $r$ first-order formula $\varphi(y,z)$ depends only on the following information:
\begin{itemize}
\item the $r$-type of $(y,x)$ in the input tree, i.e.~the rank $r$ first-order formulas satisfied by $(y,x)$; and
\item the $r$-type of $(z,x)$ in the input tree, i.e.~the rank $r$ first-order formulas satisfied by $(y,x)$.
\end{itemize}
Since the relation $\to$ has constant outdegree and indegree, and it can be defined using quantifier rank $r$, follows that if $y\to z$ are on different sides of $x$ then there can only be a constant number of nodes $y'$ such that $(y,x)$ and $(y',x)$ have the same $r$-type in the input tree. Since the number of $r$-types is constant, it follows that number of pairs $y \to z$ which are on different sides of $x$ is constant; these pairs are the sensitive edges. \end{proof}
Apply the above lemma, yielding an upper bound $k \in \set{1,2,\ldots}$ on the number of factors in the origin factorisations. Note that each of the factors in the origin factorisation has arity $<k$, since the ports of the factors must lead to the other factors. It follows that, in order to store the ``below'' factors in registers, it is enough to have $k$ groups of registers, with each group having one registers for every arity in $\set{0,\ldots,k-1}$: \newcommand{\sttregval}[2]{\text{{(register valuation of $#1$ in $#2$)}}} \newcommand{\rootnode}[1]{\mathrm{root}(#1)} \begin{align*} \regnames \quad \eqdef \quad \set{r_i^j \text{ of arity $j$} : i \in \set{1,\ldots,k}, j \in \set{0,\ldots,k-1}} \end{align*}
We now define the invariant that will be satisfied by the register transducer. (We use a slightly extended model of register transducers, where some register contents can be undefined; this model is easily seen to reduce to the original one, by filling the undefined registers with some fixed nonces. \begin{itemize}
\item {\bf Invariant.} Let $t$ be an input tree, let $x$ be a node in $t$, and let $s_1,\ldots,s_n$ be the ``below'' factors of $x$, viewed as subsets of nodes in the output tree, ordered so that
\begin{align*}
\rootnode{s_1} \preceq \cdots \preceq \rootnode{s_n}
\end{align*}
where $\preceq$ is the pre-order on nodes in the output tree and $\rootnode{s_i}$ denotes the unique node in $s_i$ which is an ancestor of all other nodes in $s_i$. After processing the subtree of $x$ in the input tree, the register valuation of the register transducer is
\begin{align*}
r_i^j \mapsto \begin{cases}
\begin{array}{l}
\text{$s_i$ viewed as a term} \\[-2pt] \text{over the output alphabet}
\end{array} & \text{if $s_i$ has arity $j$}\\[10pt]
\text{undefined} & \text{otherwise}.
\end{cases}
\end{align*} \end{itemize}
The output register of the transducer is $r_1^0$. When $x$ is the root of the input tree, then there is only one ``below'' factor, namely the entire output tree (which has arity $0$) and therefore -- thanks to the invariant -- the output tree will be found in the output register.
The following lemma gives the register updates of the transducer.
\begin{lemma}\label{lem:register-updates-in-stt}
There is a finite set $\rDelta$ of register updates with the following property. For every input tree $t$ and every node $x$ in $t$, there is some $u \in \rDelta$ such that the register valuation of $x$ (as defined in the invariant) is obtained by applying $u$ to the register valuations of the children $x_1,\ldots,x_n$ of $x$, in listed in left-to-right order. Furthermore, there is a family $\set{\varphi_u(x)}_{u \in \rDelta}$ of unary queries over the input alphabet such that the update associated to a node $x$ is $u$ if and only if the node satisfies $\varphi_u(x)$ in the input tree. \end{lemma} \begin{proof}
The crucial observation is that each of the ``below'' factors in the origin factorisation for $x$ -- seen as subsets of nodes in the output tree -- is a (disjoint) set union of the the ``below'' factors in the origin factorisations for the children of $x$, plus the nodes in the output tree which have origin in $x$. Since there is at most a constant number of children and nodes with origin $x$, there is a finite number -- depending only on the transduction -- of ways in which these factors can be combined; this finite set of possible combinations is the set $\rDelta$. The ``furthermore'' part of the lemma, about computing the update using first-order queries, follows from a simple inspection of the first-order formulas used in defining the transduction. \end{proof}
The above lemma completes the definition of the register transducer. Its register updates are $\rDelta$ as in the lemma, and its transition function assigns label $u \in \rDelta$ to each node that satisfies $\varphi_u(x)$. The final part of the proof is showing that the register updates are monotone. We use the following order on the registers: \begin{align*} \underbrace{r_1^0 < r_1^1 < \cdots < r_1^{k-1} < r_2^0 < r_2^1 < \cdots < r_k^{k-2} < r_k^{k-1}}_{\text{lexicographic, with the lower index having priority}}. \end{align*}
Let $x$ be a node in an input tree $t$, and let $s_1,s_2$ be a ``below'' factor in the origin factorisation of $x$, which are register contents in register valuation of $x$. The registers storing $s_1$ and $s_2$ will be ordered -- according to the invariant -- with respect to the pre-order on the root nodes of $s_1$ and $s_2$. Let $x'$ be the parent of $x$. By the reasoning in the proof of Lemma~\ref{lem:register-updates-in-stt}, there are ``below'' factors in the origin factorisation of $x'$ which contain the factors $s_1$ and $s_2$; call these factors $s'_1$ and $s'_2$ (possibly $s'_1=s'_2$). Since $s'_1$ contains $s_1$ (as a set of nodes in the output tree), and the same is true for $s'_2$ and $s_2$, we have \begin{align*} \rootnode{s_1} \preceq \rootnode{s_2} \quad \text{implies} \quad \rootnode{s'_1} \preceq \rootnode{s'2} \end{align*} which establishes monotonicity of the register updates.
This completes the proof of Theorem~\ref{thm:stt}. \section{Normalisation of $\lambda$-terms is a first-order transduction} \label{sec:eval}
\newcommand{$\lambda$-term }{$\lambda$-term } \newcommand{$\lambda$-terms }{$\lambda$-terms }
\newcommand{\NonLinTerms}[2]{\Lambda_{#1} #2} \newcommand{\LinTerms}[2]{\mathsf{Lin}_{#1} #2}
\newcommand{\ranked{\Lambda}}{\ranked{\Lambda}}
\newcommand{\ranked{\Lambda^{\sf{lin}}}}{\ranked{\Lambda^{\sf{lin}}}}
\newcommand{\ranked{\Lambda^{\sf{thin}}}}{\ranked{\Lambda^{\sf{thin}}}}
\newcommand{\thinterm}[1]{\ranked{\mathsf{Thin}_{#1}}}
In this part of the appendix, we show Theorem~\ref{thm:normalise}, which says that under some restrictions, normalisation of $\lambda$-terms is a first-order transduction. Before proving this result in~\ref{sec:evaluation-lambda}, we will first explain in~\ref{sec:explaining-restrictions} why these restrictions are unavoidable. Then we show in \ref{sec:restrictions-are-fo} that the set of $\lambda$-terms satisfying these restrictions form a first-order tree language. This result will be useful for the proof of Theorem~\ref{thm:normalise}.
\subsection{Explaining the restrictions}\label{sec:explaining-restrictions}
Recall that Theorem~\ref{thm:normalise} says that normalisation of $\lambda$-terms is derivable under three assumptions: the input term should be linear, uses a unique variable $x$ and could be typed using a fixed finite set of types.
If the linearity condition is removed, and because of iterated duplication, the normal form of a well-typed $\lambda$-term can be exponential (or worse, see~\cite[Section 3.6]{sorensen_lectures_2006}), as shown by the following example.
\begin{example}\label{ex:exponential}
Assume that we have two variables $\typevar x \otype$ and $\typevar y {\otype \to \otype \to \otype}$ and consider the $\lambda$-terms defined by:
\begin{align*}
M_0 \eqdef \typevar x \otype \qquad M_{n+1} = (\lambda \typevar x \otype . \typevar y {\otype \to \otype \to \otype} \typevar x \otype \typevar x \otype)M_n.
\end{align*}
The $\lambda$-term $M_n$ is well-typed and of type $\otype$. It has size linear in $n$, but its normal form has size at least $2^n$.
\end{example} If there was a first-order transduction normalising these terms, it would be exponential-size increase, which is not possible since all first-order transductions are linear-size increase.
Being linear alone is not enough to normalise terms with first-order transductions. Another obstacle is terms that use types of unbounded complexity, as illustrated in the following example.
\begin{example}\label{ex:affine-not-enough} Consider the following $\lambda$-terms, which have types of unbounded size: \begin{align*} M_n = \overbrace{\lambda \typevar x \otype. \lambda \typevar x \otype. \cdots \lambda \typevar x \otype.}^{\text{$n$ times}} \typevar x \otype \end{align*} This is a well-typed affine term, whose type is \begin{align*} \otype^n \to \otype \qquad \eqdef \qquad \overbrace{\otype \to \otype \to \cdots \to \otype}^{\text{$n+1$ arrows}} \end{align*}
To $M_n$, apply $m$ arguments of type $\otype$:
\begin{align}\label{eq:complicated-term}
M_n \overbrace{\typevar y \otype \ \typevar y {\otype} \cdots \typevar y{\otype} }^{\text{$m$ times}}.
\end{align}
We claim that the above $\lambda$-term cannot be normalised using a first-order transduction, or even a monadic second-order transduction. In order to normalise, a transduction would need to be able to compare the numbers $n$ and $m$ as follows: if $m < n$ the normal form contains $\lambda$, if $m=n$ the normal form does not contain $\lambda$, and if $m > n$ then the normal form is undefined because the $\lambda$-term is not well-typed. Whether or not a $\lambda$-term (seen as a tree over a finite alphabet) contains $\lambda$ is a first-order definable property, and first-order definable properties are preserved under inverse images of first-order transductions. Therefore, if normalisation would be a first-order transduction, then there would be a first-order formula which would be true for terms of the form~\eqref{eq:complicated-term} with $m>n$ and which would be false for terms of the form~\eqref{eq:complicated-term} with $m=n$. Such a formula cannot exist, which can be shown using a pumping argument or Ehrenfeucht-Fra\"iss\'e games. \end{example}
\begin{example}\label{ex:tito} Let $\mathsf{not}$ and $\mathsf{id}$ be the following terms: \begin{align*} \mathsf{not} \eqdef \lambda \typevar b {\otype \to \otype \to \otype} . \lambda \typevar x {\otype} . \lambda \typevar y {\otype} . \typevar b {\otype \to \otype \to \otype} \typevar y {\otype} \typevar x {\otype} \\
\mathsf{id} \eqdef \lambda \typevar b {\otype \to \otype \to \otype}. \lambda \typevar x {\otype}. \lambda \typevar y {\otype}. \typevar b {\otype \to \otype \to \otype} \typevar x {\otype} \typevar y {\otype} \end{align*} For every $n\geq 1$, we let $\mathsf{not}_n$ be the following term: \begin{align*} \mathsf{not}_n = \lambda \typevar b {\otype \to \otype \to \otype}. \overbrace{\mathsf{not} \cdots \mathsf{not}}^{\text{$n$ times}} \typevar b {\otype \to \otype \to \otype} \end{align*} The normal form of $\mathsf{not}_n$ is $\mathsf{not}$ when $n$ is odd, and $\mathsf{id}$ when $n$ is even.
The $\lambda$-terms $\mathsf{not}_n$ cannot be normalised using a first-order transduction. Otherwise we would have a first-order formula which is true for those terms $\mathsf{not}_n$ where $n$ is even and false when $n$ is odd. This formula cannot exist for the same reason as the example above.
Note that $\mathsf{not}_n$ is linear and its subterms can be typed using only the types $\otype, \otype\to\otype$ and ${\otype \to \otype \to \otype}$. By restricting ourselves to terms which uses a unique bound variable, we avoid this situation. \end{example} \subsection{Restrictions of Theorem~\ref{thm:normalise} are first-order definable} \label{sec:restrictions-are-fo}
In this section, we show that the restrictions of Theorem~\ref{thm:normalise} discussed above, are first-order definable, as stated in the following theorem.
\begin{proposition}\label{prop:WellTypedFo}
Let $X$ be a finite set of simply typed variables and let $\Tt$ be a finite set of simple types. The tree language of linear $\lambda$-terms which can be typed using $\Tt$ is first-order definable. \end{proposition}
In the rest of this appendix, we denote by $\NonLinTerms \Tt X$ this tree language. To prove Proposition~\ref{prop:WellTypedFo}, we first show that for $\lambda$-terms in $\NonLinTerms \Tt X$, checking if their type is $\tau$, where $\tau$ is a type in $\Tt$, is a \fo property: \begin{lemma}\label{lem:IsTypeTauFo} For every type $\tau$ in $\Tt$, there is a first-order query $\varphi_\tau$ such that:
$$ \forall M\in \NonLinTerms \Tt X \qquad\quad M,u \models \varphi_\tau \longleftrightarrow M|_u:\tau$$
where $M|_u$ is the sub-tree of $M$ rooted in $u$. \end{lemma} Before establishing this lemma, let us see how Proposition~\ref{prop:WellTypedFo} can be derived from it. Linearity can be easily seen as a first-order property. The hard part is to show that the set of $\lambda$-terms which can be typed using $\Tt$ is first-order. Suppose for convenience that $\Tt$ is downward closed. For every type $\tau$ in $\Tt$, let $\varphi_\tau$ be the formula given by Lemma~\ref{lem:IsTypeTauFo}. In the following, we use the binary formula
$\mathsf{Succ}_{i}(u,v)$ which is valid when $v$ is the $i$-th child of $u$, and which is easily expressible in first-order logic.
Consider the unary formula $\mathsf{Lambda}(u)$, which expresses that $u$ is a binder node, that its type and the type of its child match well and both belong to $\Tt$: $$\begin{array}{ll} \mathsf{Lambda}(u) &:= \underbrace{\lambda x(u)}_{\substack{\text{$u$ has label $\lambda x$}}} \wedge\bigvee_{\substack{\sigma\rightarrow\tau \in \Tt\\x:\sigma}} \underbrace{\varphi_{\sigma\rightarrow\tau}(u)}_{\substack{\text{$u$ has type $\sigma\rightarrow\tau$}}} \\ & \wedge \underbrace{\exists v\ \ \mathrm{Succ}_1(u, v) \wedge \varphi_\tau (v)}_{\substack{\text{the child of $u$ has type $\tau$}}} \end{array}$$ Similarly, consider the unary formula $\mathsf{Application}(u)$ which checks that a node is an application node, that the type of its children match well and that both belong to $\Tt$: $$\begin{array}{l} \mathsf{Application}(u) := \underbrace{@(u)}_{\substack{\text{$u$ has label @}}} \wedge \\ \exists v, w \underset{\sigma\rightarrow\tau \in \Tt}{\bigvee} \underbrace{\mathrm{Succ}_1(u, v) \wedge \varphi_{\sigma\rightarrow\tau} (v)}_{\substack{\text{the left child of $u$ has type $\sigma\rightarrow\tau$}}} \wedge \underbrace{\mathrm{Succ}_2(u, w) \wedge \varphi_{\sigma} (w)}_{\substack{\text{the right child of $u$ has type $\sigma$}}} \end{array}$$ Finally, consider the formula $\mathsf{Variable}(u)$, which expresses that $u$ is a variable node, whose type is in $\Tt$: \begin{align*} \mathsf{Variable}(u) := \bigvee_{x:\sigma\in \Tt}\underbrace{x(u)}_{\substack{\text{$u$ has label $x$}}} \end{align*} We claim that the following (nullary) formula $\phi$ recognizes the tree language $\NonLinTerms \Tt X$ \begin{align*} \phi = \forall u.\ \mathsf{Variable}(u)\ \vee\ \mathsf{Lambda}(u)\ \vee\ \mathsf{Application}(u) \end{align*}
If a $\lambda$-term is in $\NonLinTerms \Tt X$, then it clearly satisfies $\phi$. Suppose by contradiction that there is a $\lambda$-term $M$ which is not in $\NonLinTerms \Tt X$ and yet satisfies $\phi$. Let $u$ be the deepest node of $M$ which is not in $\NonLinTerms \Tt X$ (we identify in this proof a node $u$ and the sub-term $M|_u$). In particular, the descendants of $u$ are all in $\NonLinTerms \Tt X$. The node $u$ cannot be a variable, since variable nodes are well-typed and their type is in $\Tt$ by the first disjunct of $\phi$. If $u$ was labeled by $\lambda x$, where $x$ is of type $\sigma$, then by the second disjunct of $\phi$ there is a type $\tau$ such that $\sigma\rightarrow\tau\in \Tt$ and the child $v$ of $u$ satisfies $\varphi_\tau$. Since $v$ is in $\NonLinTerms \Tt X$, its type is $\tau$ by Lemma~\ref{lem:IsTypeTauFo}. Hence $u$ is well-typed and its type is $\sigma\rightarrow\tau\in \Tt$. As a consequence $u$ is in $\NonLinTerms \Tt X$ which is a contradiction. Finally, if $u$ was labeled by $@$, then by the third disjunct of $\phi$, its two children $u_1$ and $u_2$ would satisfy respectively $\varphi_{\sigma\rightarrow\tau}$ and $\varphi_{\sigma}$ and by Lemma~\ref{lem:IsTypeTauFo} they are of type $\sigma\rightarrow\tau$ and $\sigma$ respectively. The node $u$ is then well-typed and its type is $\tau$ (which is a type of $\Tt$ thanks to downward closeness). As a consequence, $u$ is in $\NonLinTerms \Tt X$, which gives a contradiction and concludes the proof.
We can go back now to the proof of Lemma~\ref{lem:IsTypeTauFo}. \begin{proof}[Proof of Lemma~\ref{lem:IsTypeTauFo}] Let us show that the following unary query is expressible in first-order logic \begin{center} ''if $t$ is a $\lambda$-term of $\NonLinTerms s X$, then its type is $\tau$ ``: \end{center}
For that, notice that the type of a well-typed term depends only on its left-most branch. In fact, the type of a term is exactly the type of its left-most branch in the following sens.
Consider the (unranked) alphabet $ X^\lambda:= X\cup \{@, \lambda x | x\in X\}$. We can equip the words over $X^\lambda$ with the following typing rules: $$\frac{}{x: \sigma} \qquad \frac{u:\tau}{u\lambda x: \sigma\rightarrow \tau} \qquad \frac{u:\sigma\rightarrow\tau}{u@:\tau}$$ where $x$ is of type $\sigma$ and $\sigma,\tau \in \Tt$.
We say that $w$ is of type $\tau$ and write $w:\tau$ if there is a typing derivation for $w:\tau$.
We can associate to every branch of a $\lambda$-term a word over $X^\lambda$ corresponding to the sequence of its labels read bottom-up. By induction on $\lambda$-terms, we can easily show that the type of a $\lambda$-term is the type of the word corresponding to its leftmost branch.
By this last observation, we can reduce the query asking if the type of a term is $\tau$, to the same query but on $X^\lambda$ words. To show that the former is a first-order query, it is then sufficient to show that the following word language \begin{align*}
W_\tau = \{w\in X.\{@, \lambda x | x\in X\}^*\ |\ w:\tau \} \end{align*} is first-order definable, or equivalently that $W_\tau$ is recognized by a counter-free finite automaton. For that we proceed as follows: first, we show that $W_\tau$ is recognized by a pushdown automaton $P_\tau$. Then we will show that the stack height of $P_\tau$ is bounded, thus it can be turned into a deterministic finite automaton $D_\tau$. Finally, we show that the obtained automaton $D_\tau$ is actually counter-free.
Consider the pushdown automaton $P_\tau$ whose \begin{itemize} \item set of states is $\{i, p, f\}$, where $i$ is the initial state and $f$ the accepting state; \item input alphabet is the alphabet $X^\lambda$; \item stack alphabet is the set of types $\Tt$; \item and whose transition function is described as follows: \begin{itemize} \item If the automaton is in the initial state $i$ with an empty stack, and if the symbol it reads is a variable $x$ of type $\sigma_1\rightarrow\dots\rightarrow\sigma_n$, then we go to the state $p$ and push the symbols $\sigma_n,\dots,\sigma_1$ in the stack in this order. The top-level symbol of the stack is then $\sigma_1$. \item If the automaton is in the state $p$ and it reads the symbol $\lambda y$, where $y$ is of type $\sigma$, then push the symbol $\sigma$ in the stack, and stay in the state $p$. \item If the automaton is in the state $p$, if it reads the symbol $@$ and if the stack is non empty, then pop the top-level symbol and stay in the state $p$. \item If the automaton reaches the end of the word being in state $p$, and if the stack contains the symbols $\tau_1,\dots\tau_m$ in this order, $\tau_1$ being the top-level symbol, where $\tau_1\rightarrow\dots\rightarrow\tau_m$ is the type $\tau$, then pop them all and go to the final state $f$. \end{itemize} \end{itemize} A word $w$ is accepted by $P_\tau$ if there is a run that reaches the end of $w$ in the accepting state $f$ with an empty stack. We write $(r, s)\xrightarrow{w} (r', s')$ if there is a run over the word $w$ which starts in the state $r\in\set{i, p,f}$ and with a stack $s$ and ends up in the state $r'\in\set{i, p,f}$ and with a stack $s'$.
By induction on the length of the word $w$, we can easily show that: \begin{lemma}
For every word $w\in X.\{@,\lambda x | x\in X \}^*$, we have that:
$$(i, \epsilon)\xrightarrow{w} (p, \sigma_n\dots\sigma_1) \qquad\text{iff} \qquad w:\sigma_1\rightarrow\dots\rightarrow\sigma_n$$ \end{lemma}
A direct consequence of this lemma is that $P_\tau$ recognizes $W_\tau$. Another direct consequence is that the stack height of $P_\tau$ is bounded by $m$, the size of the longest type in $\Tt$. Thus $P_\tau$ can be turned into a DFA $D_\tau$, by encoding the stack information in the states. More precisely, the states of $D_\tau$ are pairs $(r,s)$ where $r\in \set{i,p,f}$ and $s$ is a stack of height at most $m$, the initial state is $(i,\epsilon)$ and there is a transition $(r,s)\xrightarrow{a}(r',s')$ where $a\in X^\lambda\cup\{\epsilon\}$ if there is a corresponding run in $P_\tau$. We show in the following that $D_\tau$ is counter-free.
Let us start with some observations. In the pushdown automaton $P_\tau$, the effect of a word $w$ on a stack $s$, starting from the state $p$ is the following: it erases the first $n$ top level elements of $s$, and replaces them by a word $u$. The number $n$ and the word $u$ do not depend on the stack $s$ but only on the word $w$. This is exactly what the following lemma claims.
\begin{lemma} For every word $w$ over ${X^\lambda}^*$, there is a natural number $n$ and a word $u\in \Tt^*$ such that if $(p,s)\xrightarrow{w}(p,s')$ then $s$ and $s'$ can be decomposed as follows:
$$s=t.v,\qquad s'=t.u\qquad \text{ and }\qquad |v|=n.$$ \end{lemma} The proof is an easy induction on the length of $w$. As a consequence we have that: \begin{itemize}
\item If $(p,s_1)\xrightarrow{w}(p,s_2)\xrightarrow{w}(p,s_3)$ and $|s_2|>|s_1|$ then $|s_3|>|s_2|$.
\item If $(p,s_1)\xrightarrow{w}(p,s_2)\xrightarrow{w}(p,s_3)$ and $|s_2|<|s_1|$ then $|s_3|<|s_2|$.
\item If $(p,s_1)\xrightarrow{w}(p,s_2)\xrightarrow{w}(p,s_3)$ and $|s_2|=|s_1|$ then $s_3 =s_2$. \end{itemize}
Let us show that $D_\tau$ is counter-free. Suppose by contradiction that there is a word $w$ and pairwise distinct stacks $s_1,\dots, s_n$ such that \begin{align*} (p,s_1)\xrightarrow{w}(p,s_2)\xrightarrow{w}\dots(p,s_n)\xrightarrow{w}(p,s_1). \end{align*} By the first two properties above, we have necessarily that \begin{align*}
|s_1|=\dots=|s_n| \end{align*}
Thus by the third property, we have that \begin{align*}
s_1=\dots=s_n \end{align*}
which concludes the proof. \end{proof}
\subsection{Normalisation of $\lambda$-terms}\label{sec:evaluation-lambda}
This section is dedicated to the proof of Theorem~\ref{thm:normalise}. Let us first introduce some terminology. In a $\lambda$-term, we call \emph{redex} a pattern of the following form \begin{center} \includegraphics[scale=.5]{pictures/redex.pdf} \end{center} that is, an application node whose left child is an abstraction node. In a linear $\lambda$-term, we call \emph{full redex} a set of nodes containing a redex, the node of its variable, together with the set of nodes between them, as illustrated below \begin{center} \includegraphics[scale=1.5]{pictures/full-redex.pdf} \end{center}
Let us go back to the proof of Theorem~\ref{thm:normalise}. Before normalising $\lambda$-terms, the first thing to do is to discriminate those $\lambda$-terms satisfying the restrictions of Theorem~\ref{thm:normalise} from the others. This amounts to pre-processing the normalisation process by the function \begin{align*} \trees \ranked{X^\lambda} \to \trees \ranked{X^\lambda} +\bot \end{align*} which is the identity for inputs satisfying the restrictions and is undefined otherwise. Let us see how this function can be derived. Thanks to Proposition~\ref{prop:WellTypedFo}, the restrictions of Theorem~\ref{thm:normalise} are first-order definable, say by a first-order query $\phi$. By virtue of Proposition~\ref{prop:forat}, the characteristic function of $\phi$ is derivable. Now following the label's root of the input, we either output the input tree if the label says that it satisfies the query $\phi$, or outputs the undefined symbol otherwise. This last function can be easily derived. From now on, we suppose that our $\lambda$-terms satify the restrictions of Theorem~\ref{thm:normalise}.
To normalise $\lambda$-terms satisfying the conditions of Thm.~\ref{thm:normalise}, the main observation is that the evaluation of a redex does not create new redexes. Hence, it suffices to reduce all the available redexes in such term to reach the normal form.
To show Thm.~\ref{thm:normalise}, we will factorise (via a derivable function) our $\lambda$-terms into factors satisfying the following properties: \begin{itemize} \item[(P1)] Every full redex falls into one of the factors.
\item[(P2)] Each factor have a a very specific shape called \emph{thin}. These factors are those $\lambda$-terms with ports whose normal form have the shape of a word (by opposition to trees, which is the general case). \end{itemize} By properties (P1) and (P2), it is enough to show that normalisation of thin $\lambda$-terms (with ports) is derivable. For this purpose, our strategy will be to prove that the word obtained by normalising a thin $\lambda$-term results from a pre-order traversal. Since pre-order traversal is a prime function, this implies that normalisation of thin $\lambda$-terms is derivable.
The last ingredient to conclude the proof is to notice that $\beta$-reducing the factors of (a factorisation of) a $\lambda$-term, then applying a flattening, is the same thing as $\beta$-reducing the original $\lambda$-term, which follows directly from the fact that $\beta$-reduction is a congruence on terms. This concludes the proof.
In the rest of this section, we develop on each of the two main steps of the proof, namely proving Properties (P1) and (P2). In Section~\ref{subsub:thin} we present thin $\lambda$-terms with ports and show how to normalise them. Then we show in Section~\ref{subsub:facto} how to factorise a $\lambda$-term into thin factors.
\subsubsection{Normalisation of thin $\lambda$-terms}\label{subsub:thin}
As discussed earlier, we will need to normalise $\lambda$-terms with ports (the factors of our factorisation). In the following, we will denote by $\ranked{\ranked{\lamrank X}}$ the ranked set \begin{align*}
\overbrace{\set{y : y \in X}}^{\text{arity 0}} \cup \overbrace{\set{\lambda x }}^{\text{arity 1}} \cup \overbrace{\set @}^{\text{arity 2}} \end{align*} With this notation, $\lambda$-terms with ports are the inhabitants of $\ranked{\tmonad \ranked{\lamrank X}}$. Normalisation of these terms generalizes that of usual $\lambda$-terms in a straightforward way: the $i$-th port is replaced by a fresh variable $x_i$, the obtained $\lambda$-term (without ports) is evaluated as usual, then the variable $x_i$ is replaced back by the port $i$, as one can see in the following example.
\begin{center} \includegraphics[scale=.4]{pictures/normalization-with-ports.pdf} \end{center}
Note that when a $\lambda$-term is linear, its normal form has the same number of ports. Note also that respecting the original order of ports in the normal form (which is important for compositionality) may twist ports, as in the example above. As a consequence, normalisation of linear $\lambda$-terms with ports is an arity preserving function of type:
\begin{align*}
\ranked{\tmonad \ranked{\lamrank X} \to \reduce 1 \tmonad \ranked{\lamrank X} +\bot}
\end{align*}
Let us present now the class of \emph{thin $\lambda$-terms with ports}.
\begin{definition} We say that the node of a $\lambda$-term is \emph{branching} if its has at two distinct children which are not ports.
A \emph{thin $\lambda$-term with ports} is a term from $\tmonad \ranked{\lamrank X}$ in which every branching node is the application node of a redex.
\end{definition} In the remaining of this section we will omit the mention ``with ports'' if clear from the context.
Since thin $\lambda$-terms branch only on redexes, the result of their normalisation is a ``word'', in the sens that every node has at most one non-port child. We will show that this word can actually be obtained by a pre-order traversal of the original $\lambda$-term. We will then use the prime $\preorder$ function to show that normalisation of thin $\lambda$-terms is derivable.
The left $\lambda$-term below is linear and thin. The rednodes are the ones which are not redexes nor the variables of these redexes. The right $\lambda$-term is its normal form: we can see that nodes appear top-down in the pre-order of the original $\lambda$-term. \begin{center}
\includegraphics[scale=.45]{pictures/thin}
\end{center}
\begin{proposition}\label{prop:EvaluateThin}
Let $X$ be a finite set of simply typed variables, $\typeset$ be a finite set of simple types.
The following tree-to-tree function is derivable:
\begin{itemize}
\item{\bf Input.} A $\lambda$-term $t$ over variables $X$.
\item {\bf Output.} The normal form of $t$, if it is thin and satisfies the conditions of Thm.~\ref{thm:normalise}, and undefined otherwise.
\end{itemize} \end{proposition}
Let $t$ be a thin $\lambda$-term and let $u$ be its normal form. As noticed before, $u$ has the shape of a word. Moreover, since $t$ is linear, the nodes of $u$ are exactly the nodes of $t$ which are not redexes, nor their variables.
\begin{proposition}\label{prop:normal-form-depth-first} Let $t$ be a linear thin $\lambda$-term and let $u$ be its normal form. The order in which the inner nodes (ie. non ports) of $u$ appear top-down is the pre-order of $t$. \end{proposition}
\begin{proof} To establish this proposition, we need the following lemma. \begin{lemma}\label{lem:internalLemma} Let $t$ be a linear thin $\lambda$-term which binds only the variable $x$ and let $r$ be one of its redexes. Consider $m$ to be the binder node of $r$ and $n$ to be the node of the variable it binds.
The node $n$ is the greatest (that is the right-most) node in the sub-term $t|_m$ w.r.t. the pre-order. \end{lemma}
\begin{proof} Suppose by contradiction that there is a node $o$ which is strictly greater than $n$ in the subterm $t|_m$. Since $t$ is thin, the least common ancestor $l$ between $n$ and $o$ is an application node of a redex. By hypothesis on the term $t$, the binder of this redex is $\lambda x$. Since $n$ is smaller than $o$, $n$ is the left descendant of $l$, in other words it is the descendant of the left child $p$ of $l$, which is a binder $\lambda x$. The node $m, n, o$ and $p$ are illustrated below: \begin{center}
\includegraphics[scale=.3]{pictures/lemma-thin.pdf} \end{center}
The variable of the node $n$ is under the scope of the binder of the node $p$, which contradicts the fact that it is bound by the binder of the node $m$, which concludes the proof. \end{proof}
Let us go back to the proof of our proposition. Consider two inner nodes $n, m$ of $t$ which are also nodes of $u$, and such that $m$ is smaller than $n$ in the pre-order of $t$ (we well call it simply pre-order in the rest of the proof). We show that $n$ is a descendant of $m$ in $u$. There is two cases to consider: \begin{itemize} \item Either $n$ is a descendant of $m$ in $t$, in this case we can conclude easily since $\beta$-reduction preserves the descendant relation. Indeed, by a small analysis of $\beta$-reduction, one can notice that a reduction step may extend the descendant relation, but can never change (or break) the order of two comparable nodes in the original $\lambda$-term. \item Otherwise, let us consider the lowest common ancestor $p$ of $m$ and $n$. We proceed by induction on the length of the path between $m$ and $p$. By definition of thin $\lambda$-terms, since $p$ is branching it is necessarily an application node, whose left child $q$ is a binder node $\lambda x$. By Lemma~\ref{lem:internalLemma}, $m$ is smaller w.r.t. the pre-order than the node $r$ of the variable bound by $q$ . We are then left with the following two situations. The first case, illustrated by the left figure below, is when $r$ is a descendant of $m$ in $t$. In this case, after one reduction step, $n$ will be a descendant of $m$. The other case is when $m$ is in the left of $r$ in $t$, as illustrated by the right figure below. In this case, after one reduction step, the lowest common ancestor between $m$ and $n$ will be a descendant of $p$, and we can conclude by induction hypothesis. \begin{center} \includegraphics[scale=.3]{pictures/cases-lemma} \end{center} \end{itemize} This concludes the proof of the first claim.
\end{proof}
Let us construct now a derivable function which computes the normal form of linear thin $\lambda$-terms binding a single variable $x$. We illustrate this construction on the term $t$ below which will be our running example in this proof. \begin{center} \includegraphics[scale=.4]{pictures/running-thin} \end{center}
\begin{proof}[Proof of Proposition~\ref{prop:EvaluateThin}] Let $t$ be a linear thin $\lambda$-term in $\tmonad \ranked{\lamrank X}$.
\begin{enumerate} \item We start by distinguishing the redexes of $t$ and their variables from the other nodes. For that, we apply the characteristic function of the following first-order query $\varphi$:
\begin{center}
``The node $u$ is a redex or a variable of a redex''
\end{center}
This query is first-order expressible. Indeed it is the disjunction of the following queries $$\begin{array}{rl} @\mathsf{Redex}(u) = & @(u) \wedge \exists v \ \mathrm{Child}_1(u,v) \wedge \lambda x(v)\\[8pt] \lambda\mathsf{Redex}(u)=& \lambda x(u) \wedge \exists v \ \mathrm{Child}_1(v,u) \wedge @(v) \\[8pt] X\mathsf{Redex}(u) = & x(u) \wedge \exists v\ \lambda\mathsf{Redex}(v) \wedge v\ \mathsf{binds}\ u \end{array}$$ where $@\mathsf{Redex}(u)$ says that $u$ is the application node of a redex, $\lambda\mathsf{Redex}(u)$ says that it is the abstraction node of a redex and $X\mathsf{Redex}(u)$ says that it is the variable of a redex. The formula $u\ \mathsf{binds}\ v$, defined below, is a binary first-order query expressing that the node $u$ is an abstraction node that binds $v$.
\begin{align*}
&\lambda x(u) \wedge x(v) \wedge(u<v)\ \\
\wedge \ & \forall u,v,w.\ u<w<v\Rightarrow \neg \lambda x(w)
\end{align*}
The formula $\varphi$ being a first-order query, its characteristic function is derivable thanks to Proposition~\ref{prop:forat}.
When we apply this function to $t$, we get a term in $\ranked{\tmonad(\ranked{\lamrank X}+\ranked{\lamrank X})}$. Below is the effect of this first step on our running example. We colored in red the nodes belonging to the first copy of $\ranked{\lamrank X}$, that is the nodes satisfying the query $\varphi$. These nodes are the ones that will disappear in the normal form of $t$. \begin{center} \includegraphics[scale=.4]{pictures/running-thin-2} \end{center} \item After that, we apply the $\preorder$ function \begin{align*}
\ranked{\preorder : \tmonad (\ranked{\lamrank X}+\ranked{\lamrank X}) \to \reduce 1\tmonad( \ranked{\lamrank X}+\ranked{\lamrank X} + 0 +2)} \end{align*} After this step, our initial term becomes \begin{center} \includegraphics[scale=.3]{pictures/running-thin-3} \end{center} In this term, the nodes of the normal form appear in the right order thanks to Prop.~\ref{prop:normal-form-depth-first}. Now, we only need to get rid of the redexes and the variable nodes that participated in the computation of the normal form (that is the ones colored in red) together with the nodes $\grayball$ and $\grayballbin$ introduced by the $\preorder$ function. \item For this purpose, we apply the function \begin{align*} \ranked{\ranked{\tmonad (X^\lambda+X^\lambda + 0+ 2)} \to \ranked{\tmonad (X^\lambda+X^\lambda + 0+2+1) }} \end{align*}
which adds the unary symbol $1$ as the parent of every node $2$. This function can be easily derivable. Then we apply the factorisation $\ancfact$
to separate the symbol $1$ from the others:
\begin{align*}
\ranked{\ancfact : \tmonad (X^\lambda+X^\lambda + 0+2+1) \to \tmonad (\tmonad(X^\lambda+X^\lambda + 0+2)+\tmonad 1))}
\end{align*}
After this step, our example term becomes like this \begin{center} \includegraphics[scale=.3]{pictures/running-thin-4} \end{center}
\item Now consider the function
\begin{align*}
\ranked{g: \tmonad(X^\lambda+X^\lambda+0+2) \to \reduce 1\tmonad(X^\lambda+X^\lambda+0+2)}
\end{align*}
which is the identity function, except for the following finite set of terms for which it is defined in figure~\ref{fig:definition-g}. \begin{figure*}
\caption{Definition of the function $g$.}
\label{fig:definition-g}
\end{figure*} The red elements are those belonging to the first copy of $\ranked{\lamrank X}$.
Now back to our term, we replace the $\ranked{\tmonad 1}$ factors by the empty term, and to the other factors we apply the function $g$. After that, we apply the function \begin{align*} \ranked{\reduce 1\reduce 1 \Sigma \to \reduce 1\Sigma} \end{align*} which untwists two consecutive applications of $\reduce 1$. Doing so, we get a term of type $$\ranked{\reduce 1\tmonad(X^\lambda+X^\lambda+0+2)}$$ which is the normal form of $t$. Our running example becomes then \begin{center} \includegraphics[scale=.4]{pictures/running-thin-5} \end{center}
\item Note that we obtained the desired term, but not with the desired type. To obtain a term in $\ranked{\reduce 1\tmonad{X^\lambda}}$, we get rid of the labels $0+2$ by transforming them respectively into variables and application nodes. The choice of which variables to choose is not important, since the only terms that will actually have $0+2$ in their results are $\lambda$-terms which are not thin or do not satisfy the conditions of Thm.~\ref{thm:normalise}. \end{enumerate} \end{proof}
\subsubsection{Factorising $\lambda$-terms into blocks of thin $\lambda$-terms}\label{subsub:facto}
\begin{proposition}\label{prop:FactoIntoThin} For every finite set of typed variables $X$, for every finite set of types $\Tt$ and for every $x\in X$, there is a factorisation $$\ranked{f:\tmonad \ranked{X^\lambda} \to \tmonad\tmonad \ranked{X^\lambda}+\bot}$$ which satisfies, for every $\lambda$-term $t$ satisfying the conditions of Thm.~\ref{thm:normalise}, that \begin{itemize} \item[(1)] every full redex of $t$ is entirely contained in one of the factors of $f(t)$; \item[(2)] the factors of $f(t)$ are thin. \end{itemize} and is undefined otherwise. \end{proposition}
\begin{proof} We define the function $\ranked{f}$ as the composition of the following three functions \begin{align*} \ranked{\tmonad \ranked{X^\lambda} \xrightarrow{\ g\ } \tmonad (\ranked{X^\lambda}+1) \xrightarrow{\ \mathsf{block}^\uparrow\ } \tmonad (\tmonad\ranked{X^\lambda}+\tmonad 1) \xrightarrow{\ \mathsf{erase}\ } \tmonad \tmonad\ranked{X^\lambda}} \end{align*} The function $\ranked{g}$ will indicate, using the unary symbol $1$, the places wheres two distinct blocks of $\ranked{f}$ will be separated. We will describe it more precisely a bit later. The function $\mathsf{block}^\uparrow$ will create these blocks and finally, we erase all the factors $\ranked{\tmonad 1}$.
The function $\mathsf{block}^\uparrow$ is a prime function and $\mathsf{erase}$ can be easily derivable. Let us show how to derive the function $\ranked{g}$, so that the $1$-nodes it introduces creates blocks satisfying the conditions (1) and (2) of Proposition~\ref{prop:FactoIntoThin} (when the input is a linear $\lambda$-term).
We define $\ranked{g}$ as the composition of the characteristic function of three first-order unary queries: $\mathsf{@redex}, \mathsf{Right}$ and $\mathsf{Left}$, followed by a homomorphims $\ranked{h}$. We define them in the following: \begin{itemize} \item The property $\mathsf{App}$ checks whether a node is the application node of a redex. It can be easily expressed by a first-order formula. \item The query $\mathsf{Right}$ (resp. $\mathsf{Left}$) checks if the node is an application node, which lies, together with his right (resp. left) child, between the binder of a redex and the node it binds. Those properties can also be easily expressed by a first-order formula. \end{itemize}
When we apply the characteristic functions of these queries to a term in $\ranked{\tmonad X^\lambda}$, each node will be decorated by three informations: whether is satisfies or not $\mathsf{App}$, whether is satisfies or not $\mathsf{Right}$ and whether is satisfies or not $\mathsf{Left}$. Note that for linear $\lambda$-terms, some combinations of these properties cannot hold in the same node. For instance, a node cannot satisfy $\mathsf{Right}$ and $\mathsf{Left}$ simultaneously, as this would contradict linearity.
Now we define the homomorphism $\ranked{h}$, which maps the $\lambda$-terms with these three informations to terms of $\ranked{\tmonad (\ranked{X^\lambda}+1)}$. We define the action of $\ranked{h}$ on each node, depending on its label and the three informations it contains: \begin{itemize} \item If the label of the node is $y$ for some variable $y\in X$, or if the label is $@$ and satisfies $\mathsf{App}$, then $\ranked{h}$ returns the same node (seen as a term), forgetting the extra three informations. \item If the node is an application node satisfying \begin{align*} \neg \mathsf{App} \wedge \neg \mathsf{Right} \wedge\neg \mathsf{Left} \end{align*} then $\ranked{h}$ adds $1$ to the two children of the node. \item If the node is an application node satisfying \begin{align*} \neg \mathsf{App} \wedge \mathsf{Right} \qquad\text{(resp. } \neg \mathsf{App} \wedge \mathsf{Left} \text{)} \end{align*} then $\ranked{h}$ adds $1$ to the left (resp. right) child of the node. \end{itemize} Let $t$ be a linear $\lambda$-term. We show that the factors induced by $g$ satisfy the two conditions of Proposition~\ref{prop:FactoIntoThin}. First of all, by analyzing the action of $\ranked{h}$ on each node, note that every application node will receive $1$ as one of its children, except when it is satisfies $\mathsf{App}$. Thus the only branching nodes in a factor are redexes, hence the factors are thin. Now suppose by contradiction that there is some full redex of $t$ which is not entirely contained in a factor. This means that in $\ranked{g}(t)$ there is a $1$ between the application node of some redex and its variable. By construction of $\ranked{h}$, $1$ is the child of an application node (call it $n$). Suppose w.l.o.g. that it is the right child of $n$. The node $n$ cannot satisfy $\mathsf{App}$ because it got $1$ as a child by $\ranked{h}$. Is satisfies $\mathsf{Right}$ by the contradiction hypothesis. Thus its satisfies $\neg \mathsf{App} \wedge \mathsf{Right}$, therefore it receives also $1$ as its left child by $h$. This means that $n$ received $1$ for its both children, and the only way to get that is to satisfy $\neg \mathsf{App} \wedge \neg \mathsf{Right} \wedge\neg \mathsf{Left}$, which gives a contradiction. \end{proof}
\section{Decomposing the unfolding function} \label{ap:matrix-power} \newcommand{\mathrm{unfold}}{\mathrm{unfold}}
As discussed in the main body of the paper, the unfolding function may be regarded as unsatisfactory. In this section, we will decompose it into a collection of small functions containing no form of iteration.
We present these new prime functions in Section~\ref{sec:functions-decomposing-unfolding}, and state the main result of this section which is that term unfolding can be derived from these new prime functions (and the other prime functions of Section~\ref{sec:derivable-functions}). To prove this result, our strategy is to show that term unfolding can be derived for a restricted class of terms that we call \emph{homogeneous}, and then to show that every term can be factorised into homogeneous terms.
The notion of homogeneous terms, and the result about decomposing arbitrary terms into homogeneous ones, are presented in Section~\ref{sec:factfor}. Next, in Section~\ref{sec:homo-unfold}, we show how term unfolding can be done for homogeneous inputs. Finally, in Section~\ref{sec:monotone-unfold-proof} we prove the main result of the section by combining the results of Sections~\ref{sec:factfor} and~\ref{sec:homo-unfold}.
\subsection{New prime functions replacing the unfolding}\label{sec:functions-decomposing-unfolding} In order to decompose the unfolding function, we enrich datatypes with the constructor of shallow terms introduced in Section~\ref{sec:shallow-terms}.
We present the prime functions which will replace the unfolding in Figures~\ref{fig:prime-for-shallow-terms}--\ref{fig:weak-unfolding}. Prime functions of Figure~\ref{fig:prime-for-shallow-terms} describe the behaviour of the shallow term datatype. Figure~\ref{fig:additional-prime-for-fold} contains some additional laws for the fold datatype and Figure~\ref{fig:additional-distrib-prime} contains some new ditributivity laws. Prime functions of Figure~\ref{fig:weak-unfolding} are weak versions of the unfolding function, containing no form of iteration.
Note that some of these functions were already presented in Appendix~\ref{sec:definition-of-shallow-unfolding} to define formally the unfolding function: distributivity of shallow terms over fold, distributivity of shallow terms over product (Figure~\ref{fig:additional-distrib-prime}), and the matching function (Figure~\ref{fig:weak-unfolding}). In appendix~\ref{sec:definition-of-shallow-unfolding}, those functions were introduced in a very formal (hence verbose) way. In this appendix, we made the opposite choice of giving only informal definitions trough some hopefully clear and unambiguous pictures. \input{functions}
The main result of this section is that the unfolding can be replaced by the more atomic functions of Figures~\ref{fig:prime-for-shallow-terms}--\ref{fig:weak-unfolding}, in presence of the prime functions presented in Section~\ref{sec:derivable-functions}, as stated in the following theorem
\begin{theorem}\label{thm:decompose-unfolding} The unfolding function can be derived using the functions of Figures~\ref{fig:prime-for-shallow-terms}--\ref{fig:weak-unfolding} and the prime functions of Section~\ref{sec:derivable-functions}. \end{theorem}
In the rest of Appendix~\ref{ap:matrix-power}, derivable means derivable from the prime functions of Figures~\ref{fig:prime-for-shallow-terms}--\ref{fig:weak-unfolding} and the prime functions of Section~\ref{sec:derivable-functions} except from unfolding.
\input{factfor} \input{homo-unfold}
\subsection{Proof of Theorem~\ref{thm:decompose-unfolding}} \label{sec:monotone-unfold-proof} In this section, we complete the proof of Theorem~\ref{thm:decompose-unfolding}. We say that a nested factorisation in $\tmonadn n \mati k \rSigma$ is \emph{monotone} if all of the labels from $\mati k \rSigma$ that appear in it are monotone. Consider the homomorphism which maps a branch to its corresponding twist, and which gives the completely undefined function in case the twist is not monotone. The homomorphism uses an aperiodic monoid, as discussed in Example~\ref{ex:partial-monoton-functions}.
Apply the Factorisation Forest Theorem with respect to this homomorphism, yielding a derivable function \begin{align*} \ranked{ f : \tmonad \mati k \rSigma \to \tmonadn n \mati k \rSigma} \end{align*} which produces only nested factorisations that are hereditarily homogeneous. (Also, because monotone functions are closed under composition, it follows that if an input to $\ranked f$ is monotone, then the same is true for the output.) Therefore, Theorem~\ref{thm:decompose-unfolding} follows by composing the function $\ranked f$ with the function $\ranked {g_n}$ from the following lemma.
\begin{lemma}\label{lem:ind-homo-twist}
For every finite ranked set $\rSigma$ and $n \in \set{1,2,\ldots}$ there is a derivable function
\begin{align*}
\ranked{ g_n : \tmonadn n \mati k \rSigma \to \mati k \tmonad \rSigma}
\end{align*}
which makes the following diagram commute for inputs that are monotone and hereditarily homogeneous:
\begin{align*}
\ranked{
\xymatrix{
\tmonadn n \mati k\rSigma \ar[d]_{\flatn n} \ar[rd]^g\\
\tmonad \mati k \rSigma \ar[r]_{\unfold}& \mati k \tmonad \rSigma
}
}
\end{align*} \end{lemma} \begin{proof}
Induction on $n$. To make the induction pass through, we also show that each function $\ranked{g_n}$ is consistent wit the twist homomorphism in the following sense: for every input $t \in \tmonadn n \mati k \rSigma$, and every port $i \in \set{1,\ldots,\arity t}$, the same value is obtained by: (a) recursive flattening $t$ and then composing all of the twists that are found on the path from the root to port $i$; (b) applying $\ranked{g_n}$ and then computing the twist corresponding to port $i$.
For the induction base $n=1$, hereditarily homogeneous inputs are units, and there are finitely many of them and the function can be derived on a case by case basis.
Consider the induction step, where the lemma has already been proved for $n$ and we want to prove it for $n+1$. The function is the composition
\begin{align*}
\ranked{
\xymatrix@C=1cm{
\tmonadn {n+1} \mati k \Sigma \ar[r]^{\tmonad g_n} & \tmonad \mati k{(\tmonad \Sigma)} \ar[r]^{\text{Lemma~\ref{lem:homo-twist}}}& \mati k {(\tmonad \tmonad \rSigma)} \ar[r]^{\mati k \flatt} & \mati k \rSigma
}
}
\end{align*}
Consider a hereditarily homogeneous input $t \in \ranked{\tmonadn{n+1} \mati k \Sigma}$.
\begin{enumerate}
\item Apply the function from the induction assumption to every label of $t$, i.e.~apply
\begin{align*}
\ranked{
\xymatrix{
\tmonadn {n+1} \mati k \Sigma \ar[r]^{\tmonad g_n} & \tmonad \mati k{(\tmonad \Sigma)}
}
}
\end{align*}
\item Let $t_1$ be the output from the previous step. Because $\ranked {g_n}$ is consistent with twists, and $t$ is hereditarily homogeneous, it follows that $t_1$ is either a shallow term, or it is homogeneous with respect to the twist homomorphism. If $t_1$ is a shallow term, then we apply the shallow unfolding operation from .. . Otherwise, we $t_1$ is homogeneous
, because $t$ is hereditarily homogeneous and $\ranked{g_n}$ is consistent with twists. Therefore, we can apply the function
from Lemma~\ref{lem:homo-twist}, with the alphabet being $\tmonad \rSigma$.
\item The result of the previous step is a term $t_2 \in \mati k {(\tmonad \tmonad \rSigma)}$. To this term, we apply $\mati k \flatt$, yielding the final result.
\end{enumerate}
A routine check shows that the function $\ranked{g_{n+1}}$ defined above satisfies the property in the statement of the lemma, and that it is furthermore consistent with the twist homomorphism. \end{proof} \section{Chain logic and general unfold} \label{sec:appendix-chain} In this section, we prove Theorem~\ref{thm:chain-transductions}, which says that adding general unfold to \mso yields exactly the chain logic tree-to-tree transductions. For the rest of this section, we use the word ``derivable'' to mean derivable in the extension of Definition~\ref{def:derivable-function} where general unfold is used instead of monotone unfold.
To prove that every derivable function is a chain logic transduction, we use the same proof as in Appendix~\ref{sec:to-logic}. The only difference is that we need to deal with general unfolding instead of monotone unfolding. For general unfolding, we use the same proof as in Section~\ref{sec:fo-transduction-for-unfolding}, with the only difference being in Lemma~\ref{lem:counter-free}. As opposed to the monotone case in Lemma~\ref{lem:counter-free}, we need to compose not necessarily monotone partial functions. In the presence of non-monotone functions, the language corresponding to $L$ from Lemma~\ref{lem:counter-free} is no longer first-order definable, but it is still a regular language, and therefore it is definable in \mso. Chain logic can evaluate arbitrary \mso properties on paths in a tree, and therefore a formula of chain logic can be used to compute the twist function between two nodes in an input tree.
There rest of this appendix is devoted to the converse implication in Theorem~\ref{thm:chain-transductions}, which says that every chain logic tree-to-tree transduction is derivable, in the presence of general unfolding.
\paragraph*{Chain logic relabellings.} Define \emph{chain logic relabellings} in the same way as the first-order relabellings from Definition~\ref{def:forat}, except that chain logic is used instead of first-order logic. As in Theorem~\ref{thm:mso-transductions} about \mso transductions, we push all of the power of chain logic into tree relabellings.
\begin{lemma}\label{lem:chain-colcombet}
Every chain logic tree-to-tree transduction can be decomposed as: (a) a chain logic relabelling; followed by (b) a first-order tree-to-tree transduction. \end{lemma} \begin{proof}[Proof sketch.]
Same proof as in~\cite[Corollary 1]{colcombetCombinatorialTheoremTrees2007}, except that \mso is replaced by chain logic. The key property is that the compositionality method, which is used in Lemmas 1 and 2 of~\cite{colcombetCombinatorialTheoremTrees2007}, also works for chain logic. \end{proof}
Thanks to the above lemma, and derivability of first-order tree-to-tree transductions from our main theorem, in order to finish the proof of Theorem~\ref{thm:chain-transductions}, it suffices to show that every chain logic relabelling is derivable. To prove derivability of chain logic relabellings, we decompose them into simpler pieces. Unlike for first-order relabellings, where the decomposition was based on Schlingloff's theorem about temporal logic, in the case of chain logic we use an approach based on top-down tree automata\footnote{The results of this section could be translated into an apparently new result, which says that chain logic has the same expressive power as an extension of Schlingloff's logic obtained by adding group modalities as defined by Baziramwabo, McKenzie and Th{\'e}rien in~\cite[Section 4]{baziramwabo1999modular}.}.
\paragraph*{Top-down tree automata.} We begin by defining top-down tree automata. These are automata which process the input tree in a deterministic top-down (i.e.~root-to-leaves) pass. Since we do not use nondeterministic top-down tree automata, we implicitly assume that the automata are deterministic.
\begin{definition}
A \emph{top-down tree automaton} is given by:
\begin{enumerate}
\item an \emph{input alphabet} $\rSigma$, which is a finite ranked set;
\item a finite unranked set of \emph{states} $Q$;
\item a designated initial state in $Q$;
\item \label{it:top-down-transition} for each input letter $a \in \rSigma$, a transition function
\begin{align*}
\delta_q : Q \to Q^{\text{arity of $a$}};
\end{align*}
\item an \emph{accepting set}, which is a subset of
\begin{align*}
Q \times \text{(input letters of arity zero)}.
\end{align*}
\end{enumerate} \end{definition} For an input tree $t \in \trees \rSigma$, the \emph{run} of the automaton is defined to be the labelling of the nodes by states, which is defined as follows by induction on the distance from the root. The state in the root is the initial state. Suppose that we have already defined the state $q$ in a node $x$ of the input tree. Apply the transition function, corresponding to the label of node $x$, to the state $q$, yielding a tuple of states $q_1,\ldots,q_n$. These are the states of the run in the children of node $x$. An input tree is accepted if for every leaf, the accepting set contains the pair (state in the leaf, label of the leaf).
\begin{definition}[Tree relabellings associated to a top-down tree automaton]
\label{def:tree-relabellings-for-a-top-down-tree-automaton}
We associate two tree-to-tree functions to a top-down tree automaton $\Aa$ with input alphabet $\rSigma$. Each of these is a special cases of a chain logic relabelling. \begin{itemize}
\item The \emph{ancestor relabelling}, is denoted by
\begin{align*}
\Aa^\uparrow: \trees \rSigma \to \trees \ranked{(\black Q \times \rSigma)},
\end{align*}
where $Q \ranked{\times \Sigma}$ is be the ranked set which consists of one copy of the alphabet $\rSigma$ for each state. The ancestor relabelling simply extends the input tree with the run of the automaton. Note that the accepting set of the automaton does not play a role in the definition of the ancestors relabelling.
\item The \emph{descendant relabelling}, denoted by
\begin{align*}
\Aa^\downarrow : \trees \rSigma \to \trees\ranked{(\rSigma + \rSigma)},
\end{align*}
is the characteristic function, in the sense of Section~\ref{sec:fo-translation}, of the query which selects nodes whose subtree is accepted by $\Aa$. In other words, for each node $x$ in the input tree, its label is replaced by the corresponding label in the first copy of $\rSigma$ if the subtree of $x$ is accepted by $\Aa$, and otherwise it is replaced by the corresponding label in the second copy of $\rSigma$. \end{itemize} \end{definition}
The reason for notation in the above definition is that, in the ancestor relabelling, the label of a node depends on its ancestors, while in the descendant relabelling, the label of a node depends on its descendants. It is worth pointing out that many different runs of the automaton are used in the descendant relabelling, because for each node the automaton is started again with the initial state in that node.
We begin with the following lemma, which states a connection between chain logic and (nestings of) top-down tree automata that was described in~\cite{bojanczykDecidablePropertiesTree2004}. \begin{lemma}\label{lem:chain-phd}
Every chain logic relabelling is a composition of functions which are either:
\begin{itemize}
\item[(a)] a letter-to-letter homomorphism; or
\item[(b)] the descendant relabelling of a top-down tree automaton.
\end{itemize} \end{lemma} \begin{proof}
Adjusting for a slightly different terminology, this lemma is the same as ~\cite[Theorem 2.5.9]{bojanczykDecidablePropertiesTree2004}. To help with the terminology, we note that the wordsum automata (WS) from~\cite{bojanczykDecidablePropertiesTree2004} are the same as top-down tree automata here, while the cascade product of wordsum automata is the same as composing descendant relabellings. \end{proof} Since letter-to-letter homomorphisms are derivable, in order to finish the proof of Theorem~\ref{thm:chain-transductions}, it remains to prove that every function of kind (b) in the above lemma is derivable. We prove this by doing a further decomposition, which reduces the descendant relabelling to the ancestor relabelling.
\begin{lemma}\label{lem:reduce-descendant-to-ancestor}
For every top-down tree automaton, its descendant relabelling is a composition of functions which are either:
\begin{itemize}
\item[(c)] a first-order relabelling; or
\item[(d)] the ancestor relabelling of a top-down tree automaton.
\end{itemize} \end{lemma} \begin{proof} In this proof, we use the forward Ramseyan splits of Colcombet~\cite{colcombetCombinatorialTheoremTrees2007}.
Fix a top-down tree automaton $\Aa$.
For the proof of this lemma, as well as for subsequent results, it will be convenient to use a different perspective on top-down tree automata, which uses automata on words. Recall the set of branches $\branches \rSigma$ that was defined in page~\pageref{page:branches}: a branch is a letter together with a distinguished port. Define the \emph{branch automaton} of $\Aa$ to be the deterministic word automaton, where the input alphabet is $\branches \rSigma$, the states are the same, the initial state is the same as in $\Aa$, and the transition function is defined by
\begin{align*}
(\overbrace q^{Q}, \overbrace{(a,i)}^{\branches \rSigma}) \qquad \mapsto \qquad \text{$i$-th state in the tuple $\delta_a(q)$.}
\end{align*}
The branch automaton does not have accepting states. Roughly speaking, the run of a top-down tree automaton corresponds to running the branch automaton on every root-to-leaf path in the tree. This correspondence is spelled out in more detail below.
Consider two nodes in a tree, called the \emph{source} and \emph{target}, such that the source is an ancestor of the target. The source can be equal to the target. The \emph{path} between these two nodes is defined to be the set of edges in the tree which connects them. We can view the path as a word over the alphabet $\branches \rSigma$, as illustrated in the following picture:
\mypic{114}
The correspondence between the top-down tree automaton $\Aa$ and its branch automaton can now be phrased as follows: for a node $x$, the state of the top-down tree automaton in node $x$ is the same as the state of the branch automaton after reading the (word corresponding to the) path from the root to node $x$.
Equipped with the above terminology, we complete the proof of the lemma. For a path in an input tree, define its \emph{state transformation} to be the function of type $Q \to Q$ which describes the state transformation of the branch automaton over the (word corresponding to the) path. By Colcombet's results on forward Ramseyan splits~\cite[Lemma 3]{colcombetCombinatorialTheoremTrees2007}, there is a top-down tree automaton $\Bb$ with input alphabet $\rSigma$ and a family of first-order formulas \begin{align*} \set{\varphi_f(x,y)}_{f : Q \to Q} \end{align*} with the following property. For every input tree $t \in \trees \rSigma$ and nodes $x \le y$ in that tree, the state transformation for the path from $x$ to $y$ is equal to $f$ if and only if \begin{align*} \Bb^{\uparrow}(t) \models \varphi_f(x,y). \end{align*} The idea is that the top-down tree automaton $\Bb$ computes the forward Ramseyan split associated to state transformations in the branch automaton of $\Aa$. It follows that there is a formula $\varphi(x)$ of first-order logic such that for every $t \in \trees \rSigma$, \begin{align*} \Bb^{\uparrow}(t) \models \varphi(x) \end{align*} holds if and only if the subtree of node $x$ is accepted by the automaton $\Aa$. The formula says that for all leaves $y \le x$, the corresponding state transformation of the branch automaton leads to an accepting state. Therefore, the descendant relabelling of $\Aa$ can be computed by first applying the ancestor relabelling of $\Bb$, and then a first-order relabelling. \end{proof}
We now show that the ancestor relabellings produced by the previous lemma can be further decomposed, so that the underlying automata are reversible. Call a top-down tree automaton \emph{reversible} if the corresponding branch automaton, as defined in the proof of Lemma~\ref{lem:reduce-descendant-to-ancestor}, is reversible, which means that for every input letter the corresponding transition function is a permutation of the states. \begin{lemma}\label{lem:reduce-to-reversible}
For every top-down automaton, its ancestor function is a composition of functions which are either:
\begin{itemize}
\item[(c)] a first-order relabelling; or
\item[(e)] the ancestor relabelling of a reversible top-down automaton.
\end{itemize} \end{lemma} \begin{proof}
A corollary of the original Krohn-Rhodes theorem.
Define a \emph{Mealy machine} to be a string-to-string transducer, which is obtained from a deterministic word automaton by adding an output function, which maps every transition to a letter of an output alphabet. The original Krohn-Rhodes theorem says that every Mealy machine is a composition of Mealy machines where the underlying automaton is either aperiodic or reversible.
Take a top-down tree automaton. We can view its associated branch automaton as a Mealy machine which decorates each position in the input word by the state after reading the input word up to and including that position. To this Mealy machine apply the Krohn-Rhodes theorem. The relabellings for the aperiodic Mealy machines can be computed by the functions of kind (c), while the relabellings for the reversible ones correspond to kind (e). \end{proof}
Putting together Lemmas~\ref{lem:chain-phd}, \ref{lem:reduce-descendant-to-ancestor} and~\ref{lem:reduce-to-reversible}, we see that every chain logic relabelling is a composition of functions which have kinds (c) or (e) as in the statement of Lemma~\ref{lem:reduce-to-reversible}. Since first-order relabellings are derivable, it remains to derive the functions of kind (e).
\begin{lemma}
For every reversible top-down tree automaton, its ancestor relabelling is derivable (in the presence of general unfolding). \end{lemma} \begin{proof}
Let the states of the automaton be $Q = \set{q_1,\ldots,q_k}$. We assume that $q_1$ is the initial state.
Consider an input letter $a$, and its associated transition function as in item~\ref{it:top-down-transition}. Here is a picture of such a transition function, where the letter $a$ is binary and the number of states is $k=3$.
\mypic{116}
In terms of the above picture, the reversibility of the automaton can be described as follows:
\mypic{117}
We can represent the above transition function as an element of the $k$-th matrix power of $Q$ copies of the states, denoted by
\begin{align*}
\hat a \in \ranked{\mati k {(\black Q \times \rSigma)}},
\end{align*}
which is illustrated in the following picture:
\mypic{115}
More formally, $\hat a$ is defined so that for every port $i$ of the letter $a$, the $i$-th twist function (see Section~\ref{sec:unfolding}) is equal to the state transformation of the branch automaton when reading the letter $(a,i) \in \branches \rSigma$. The twist functions need not be monotone, since the branch automaton need not be monotone.
The reversibility of the automaton is crucial here; for a non-reversible automaton we might need to use a sub-port of the matrix power several times.
The transformation $a \mapsto \hat a$ is defined so that unfolding the matrix power captures exactly run computation in the top-down tree automaton, as described in the following commuting diagram
\begin{align*}
\xymatrix@C=3cm{
\trees \rSigma \ar[r]^{\trees(a \mapsto \hat a)} \ar[dr]_{\Aa^\uparrow}&
\trees \ranked{(\mati k {(\black Q \times \rSigma)})} \ar[d]^{
\substack{
\text{unfold and}\\
\text{take coordinate $1$}
}}\\
& \trees{\ranked{(\black Q \times \rSigma)}}
}
\end{align*} This completes the proof of the lemma. Note how general unfolding is used, since the twists involved need not be monotone. \end{proof}
\begin{lemma}\label{lem:homo-unfold}
Let $h : \branches \ranked{\Sigma^{[k]}} \to M$ be the function .
There is a derivable function \begin{align*}
\ranked{
\xymatrix{
\tmonad (\rSigma^{[k]}) \ar[r]^{g} & \rSigma^{[k]}
}
} \end{align*} which agrees with unfolding over inputs that are $h$-homogeneous. \end{lemma} \begin{proof} Consider an input $t \in \ranked{\tmonad (\Sigma^{[k]})}$ that is $h$-homogeneous. By definition, either $t$ has depth at most two, or there is some
\begin{align*}
m : \set{1,\ldots,k} \to \set{1,\ldots,k}
\end{align*}
such that all internal subbranches of $t$ have value $m$ under $h$. In the second case, the function $m$ might be surjective or not. We treat these three cases separately, i.e.~depth two, depth at least three and $m$ surjective, and depth at most three and $m$ non-surjective.
\begin{center}
(todo fill in)
\end{center}
\end{proof}
\begin{lemma}
For every $m \in \set{1,2,\ldots}$ there is a derivable function $\ranked {g^m}$ such that the diagram \begin{align*}
\ranked{
\xymatrix{
\tmonadn m (\rSigma^{[k]}) \ar[d]_{\flatn m} \ar[dr]^{g^m}\\
\tmonad (\rSigma^{[k]}) \ar[r]_{\unfold_\rSigma} & \rSigma^{[k]}
}
} \end{align*} commutes for inputs which are hereditarily $h$-homogeneous. \end{lemma} \begin{proof} Induction on $m$. For $m=1$ we use the function from Lemma~\ref{lem:homo-unfold}. For $m >2$, we define $\ranked{g^m}$ to be the composition \begin{align*}
\ranked{
\xymatrix{
\tmonad^m (\rSigma^{[k]}) \ar[r]^{\tmonad g^{m-1}} &
\tmonad (\rSigma^{[k]}) \ar[r]^g &
\rSigma^{[k]}
}
} \end{align*} apply first $\tmonad \ranked{g^{m-1}}$, yielding to every label \end{proof}
\begin{align*}
\ranked{
\xymatrix{ \tmonad (\rSigma^{[k]}) \ar[r]^f \ar[rd]_{id}&
\tmonadn m (\rSigma^{[k]}) \ar[d]_{\flatn m} \ar[dr]^{g^m}\\ &
\tmonad (\rSigma^{[k]}) \ar[r]_{\unfold_\rSigma} & (\tmonad \rSigma)^{[k]}
}
} \end{align*}
\end{document}
|
arXiv
|
\begin{document}
\title{Martin's Maximum and the Diagonal Reflection Principle ootnote{
This research is supported by Simons Foundation Grant 318467 and JSPS Kakenhi Grant Number 18K03397.
}
\begin{abstract} We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of $[ \omega_2 ]^\omega$. \end{abstract}
\section{Introduction} \label{sec:intro}
In Foreman-Magidor-Shelah \cite{FMS}, it was shown that Martin's Maximum $\mathsf{MM}$ implies the following stationary reflection principle, which is called the Weak Reflection Principle: \begin{list}{}{\setlength{\labelwidth}{50pt}\setlength{\leftmargin}{55pt}} \item[$\mathsf{WRP} \equiv$] For any cardinal $\lambda \geq \omega_2$ and any stationary $X \subseteq [ \lambda ]^\omega$, there is $R \in [ \lambda ]^{\omega_1}$ with $R \supseteq \omega_1$ such that $X \cap [R]^\omega$ is stationary in $[R]^\omega$. \end{list} $\mathsf{WRP}$ is known to have many interesting cosequences such as Chang's Conjecture (Foreman-Magidor-Shelah \cite{FMS}), the presaturation of the non-stationary ideal over $\omega_1$ (Feng-Magidor \cite{FM}), $2^\omega \leq \omega_2$ (folklore) and the Singular Cardinal Hypothesis (Shelah \cite{Sh:RP_SCH}).
As for stationary reflection principles, simultaneous reflection is often discussed. Larson \cite{Larson} proved that $\mathsf{MM}$ also implies the following simultaneous reflection principle of $\omega_1$-many stationary sets: \begin{list}{}{\setlength{\labelwidth}{55pt}\setlength{\leftmargin}{60pt}} \item[$\mathsf{WRP}_{\omega_1} \equiv$] For any cardinal $\lambda \geq \omega_2$ and any sequence $\langle X_\xi \mid \xi < \omega_1 \rangle$ of stationary subsets of $[ \lambda ]^\omega$, there is $R \in [ \lambda ]^{\omega_1}$ with $R \supseteq \omega_1$ such that $X_\xi \cap [R]^\omega$ is stationary in $[R]^\omega$ for all $\xi < \omega_1$. \end{list}
Cox \cite{Cox:DRP} formulated the following strengthening of $\mathsf{WRP}_{\omega_1}$, which is called the Diagonal Reflection Principle: \begin{list}{}{\setlength{\labelwidth}{50pt}\setlength{\leftmargin}{55pt}} \item[$\mathsf{DRP} \equiv$] For any cardinal $\lambda \geq \omega_2$ and any sequence $\langle X_\alpha \mid \alpha < \lambda \rangle$ of stationary subsets of $[ \lambda ]^\omega$, there is $R \in [ \lambda ]^{\omega_1}$ with $R \supseteq \omega_1$ such that $X_\alpha \cap [R]^\omega$ is stationary in $[R]^\omega$ for all $\alpha \in R$. \end{list} Recently, Fuchino-Ottenbreit-Sakai \cite{FOS} proved that a variation of $\mathsf{DRP}$ is equivalent to some variation of the downward L\"{o}wenheim-Skolem theorem of the stationary logic. Cox \cite{Cox:DRP} also introduced the following weakning of $\mathsf{DRP}$, where $X \subseteq [ \lambda ]^\omega$ is said to be \emph{projectively stationary} if the set $\{ x \in X \mid x \cap \omega_1 \in S \}$ is stationary in $[ \lambda ]^\omega$ for any stationary $S \subseteq \omega_1$: \begin{list}{}{\setlength{\labelwidth}{50pt}\setlength{\leftmargin}{55pt}} \item[$\mathsf{wDRP} \equiv$] For any cardinal $\lambda \geq \omega_2$ and any sequence $\langle X_\alpha \mid \alpha < \lambda \rangle$ of projectively stationary subsets of $[ \lambda ]^\omega$, there is $R \subseteq [ \lambda ]^{\omega_1}$ with $R \supseteq \omega_1$ such that $X_\alpha \cap [R]^\omega$ is stationary in $[R]^\omega$ for all $\alpha \in R$. \end{list}
Cox \cite{Cox:DRP} proved that $\mathsf{MM}$ implies $\mathsf{wDRP}$, but it remained open whether $\mathsf{MM}$ implies $\mathsf{DRP}$. In this paper, we prove that $\mathsf{MM}$ does not imply $\mathsf{DRP}$. In fact, we prove slightly more.
To state our result, we recall $+$-versions of the forcing axiom. For a class $\Gamma$ of forcing notions and a cardinal $\mu \leq \omega_1$, $\mathsf{MA}^{+ \mu} ( \Gamma )$ is the following statement: \begin{list}{}{\setlength{\labelwidth}{65pt}\setlength{\leftmargin}{65pt}} \item[$\mathsf{MA}^{+ \mu} ( \Gamma ) \equiv$] For any $\mathbb{P} \in \Gamma$, any sequence $\langle D_\xi \mid \xi < \omega_1 \rangle$ of dense subsets of $\mathbb{P}$ and any sequence $\langle \dot{S}_\eta \mid \eta < \mu \rangle$ of $\mathbb{P}$-names of stationary subsets of $\omega_1$, there is a filter $g \subseteq \mathbb{P}$ such that \begin{renumerate} \item $g \cap D_\xi \neq \emptyset$ for any $\xi < \omega_1$, \item $\dot{S}_\eta^g = \{ \alpha < \omega_1 \mid \exists p \in g , \ p \Vdash_\mathbb{P} \textrm{``} \alpha \in \dot{S}_\eta \textrm{''} \}$ is stationary in $\omega_1$ for all $\eta < \mu$. \end{renumerate} \end{list} Let $\mathsf{MA}^{+ \mu} ( \sigma\mbox{-closed} )$ denote $\mathsf{MA}^{+ \mu} ( \Gamma )$ for the class $\Gamma$ of all $\sigma$-closed forcing notions. Also, let $\mathsf{MM}^{+ \mu}$ denote $\mathsf{MA}^{+ \mu} ( \Gamma )$ for the class $\Gamma$ of all $\omega_1$-stationary preserving forcing notions. It is well-known that $\mathsf{MA}^{+ \omega_1} ( \sigma\mbox{-closed} )$ holds if a supercompact cardinal is L\'{e}vy collapsed to $\omega_2$ and that $\mathsf{MM}^{+ \omega_1}$ holds in the standard model of $\mathsf{MM}$ constructed in Foreman-Magidor-Shelah \cite{FMS}.
Cox \cite{Cox:DRP} proved that $\mathsf{MA}^{+ \omega_1} ( \sigma\mbox{-closed} )$ implies $\mathsf{DRP}$. So $\mathsf{MM}^{+ \omega_1}$ implies $\mathsf{DRP}$. In this paper, we prove that $\mathsf{MM}^{+ \omega}$ does not imply $\mathsf{DRP}$:
\begin{mthm} Assume $\mathsf{MM}^{+ \omega}$ holds. Then there is a forcing extension in which $\mathsf{MM}^{+ \omega}$ remains to hold, but $\mathsf{DRP}$ fails at $[ \omega_2 ]^\omega$. \end{mthm}
Our proof of the Main Theorem is based on the proof of the classical result, due to Beaudoin \cite{Beaudoin} and Magidor, that the Proper Forcing Axiom does not imply the reflection of stationary subsets of the set $\{ \alpha \in \omega_2 \mid \mathrm{cof} ( \alpha ) = \omega \}$. Similar arguments are used in K\"{o}nig-Yoshinobu \cite{KY}, Yoshinobu \cite{Y1}, \cite{Y2} and Cox \cite{Cox:sep}, to separate reflection principles from strong forcing axioms.
We will prove the Main Theorem in Section \ref{sec:proof}. In Section \ref{sec:preliminaries}, we will present our notation and basic facts used in this paper.
\section{Preliminaries} \label{sec:preliminaries}
Here we present our notation and basic facts. See Jech \cite{Jech} for those which are not mentioned here.
First, we recall the notion of stationary sets in $[W]^\omega$. Let $W$ be a set with $\omega_1 \subseteq W$. $Z \subseteq [W]^\omega$ is said to be \emph{club} in $[W]^\omega$ if $Z$ is $\subseteq$-cofinal in $[W]^\omega$, and $\bigcup_{n \in \omega} x_n \in Z$ for any $\subseteq$-increasing sequence $\langle x_n \mid n < \omega \rangle$ of elements of $Z$. $X \subseteq [W]^\omega$ is said to be \emph{stationary} in $[W]^\omega$ if $X \cap Z \neq \emptyset$ for any club $Z \subseteq [W]^\omega$. For $S \subseteq \omega_1$, $S$ is stationary in $\omega_1$ in the usual sense if and only if $S$ is stationary in $[ \omega_1 ]^\omega$ in the above sense.
We will use the following standard facts without any reference. Proofs can be found also in Jech \cite{Jech}.
\begin{fact}[(1) Kueker \cite{Kueker}, (2) Menas \cite{Menas}] \label{fact:stat_basic} Suppose $W$ is a set $\supseteq \omega_1$ and $X$ is a subset of $[W]^\omega$. \begin{aenumerate} \item $X$ is stationary if and only if for any function $F : [W]^{< \omega} \to W$ there is a non-empty $x \in X$ which is closed under $F$, i.e.~$F(a) \in x$ for all $a \in [x]^{< \omega}$. \item Suppose $W' \supseteq W$. Then $X$ is stationary in $[W]^\omega$ if and only if the set $\{ x' \in [W']^\omega \mid x' \cap W \in X \}$ is stationary in $[W']^\omega$. \end{aenumerate} \end{fact}
Here we slightly simplify $\mathsf{DRP}$ at $[ \omega_2 ]^\omega$.
\begin{lemma} \label{lem:DRP_easy} Assume $\mathsf{DRP}$ at $[ \omega_2 ]^\omega$. Then, for any sequence $\langle X_\alpha \mid \alpha < \omega_2 \rangle$ of stationary subsets of $[ \omega_2 ]^\omega$, there is $\delta \in \omega_2 \setminus \omega_1$ such that $X_\alpha \cap [ \delta ]^\omega$ is stationary in $[ \delta ]^\omega$ for all $\alpha < \delta$. \end{lemma}
\begin{proof} Suppose $\langle X_\alpha \mid \alpha < \omega_2 \rangle$ is a sequence of stationary subsets of $[ \omega_2 ]^\omega$. We find $\delta$ as in the lemma.
For each $\beta < \omega_2$, take a surjection $\pi_\beta : \omega_1 \to \beta$. Let $Z$ be the set of all $x \in [ \omega_2 ]^\omega$ such that $x \cap \omega_1 \in \omega_1$ and $x$ is closed under $\pi_\beta$ for all $\beta \in x$. Then, $Z$ is club in $[ \omega_2 ]^\omega$. Moreover, it is easy to see that if $\omega_1 \subseteq R \in [ \omega_2 ]^{\omega_1}$, and $Z \cap [R]^\omega$ is $\subseteq$-cofinal in $[R]^\omega$, then $R \in \omega_2 \setminus \omega_1$.
By shrinking $X_0$ if necessary, we may assume that $X_0 \subseteq Z$. By $\mathsf{DRP}$ at $[ \omega_2 ]^\omega$, take $R \in [ \omega_2 ]^{\omega_1}$ including $\omega_1$ such that $X_\alpha \cap [R]^\omega$ is stationary for all $\alpha \in R$. Then, $R \in \omega_2 \setminus \omega_1$ since $Z \cap [R]^\omega$ is $\subseteq$-cofinal in $[R]^\omega$. So, $\delta := R$ is as desired. \end{proof}
Next, we present our notation and basic facts about forcing. Suppose $\mathbb{P}$ is a forcing notion and $M$ is a set. We say that $g \subseteq \mathbb{P} \cap M$ is $M$-\emph{generic} if $g \cap D \neq \emptyset$ for any dense $D \subseteq \mathbb{P}$ with $D \in M$.
We will use the following well-known fact about forcing axioms:
\begin{fact}[Woodin \cite{Woodin}] \label{fact:forcing_axiom} Let $\Gamma$ be a class of forcing notions and $\mu$ be a cardinal $\leq \omega_1$, and assume $\mathsf{MA}^{+ \mu} ( \Gamma )$ holds. Suppose $\mathbb{P} \in \Gamma$ and $\langle \dot{T}_\xi \mid \xi < \mu \rangle$ is a sequence of $\mathbb{P}$-names for stationary subsets of $\omega_1$. Then, for any regular cardinal $\theta$ with $\mathbb{P} \in \mathcal{H}_\theta$ and any $A \in [ \mathcal{H}_\theta ]^{\omega_1}$, there are $M \in [ \mathcal{H}_\theta ]^{\omega_1}$ and $g \subseteq \mathbb{P} \cap M$ with the following properties. \begin{renumerate} \item $A \subseteq M \prec \langle \mathcal{H}_\theta , \in \rangle$. \item $g$ is an $M$-generic filter on $\mathbb{P} \cap M$. \item $\dot{T}_\xi^g$ is stationary in $\omega_1$ for any $\xi < \mu$. \end{renumerate} \end{fact}
We will also use forcing notions for shooting club sets. For an ordinal $\lambda \geq \omega_1$ and a subset $X$ of $[ \lambda ]^\omega$, let $\mathbb{R} ( X )$ denote the poset of all $\subseteq$-increasing continuous function from some countable successor ordinal to $X$, which is ordered by reverse inclusions. The following is standard:
\begin{lemma} \label{lem:club_shoot_basic} Suppose $X$ is a stationary subset of $[ \lambda ]^\omega$ for some ordinal $\lambda \geq \omega_1$. \begin{aenumerate} \item A forcing extension by $\mathbb{R} ( X )$ adds no new countable sequences of ordinals. So it preserves $\omega_1$. \item In $V^{\mathbb{R} (X)}$, $X$ contains a club subset of $[ \lambda ]^\omega$. \item In $V$, suppose $Y \subseteq X$ and $Y$ is stationary in $[ \lambda ]^\omega$. Then $Y$ remains stationary in $V^{\mathbb{R} (X)}$. \end{aenumerate} \end{lemma}
\begin{proof} Let $\mathbb{R}$ denote $\mathbb{R} (X)$. Before starting, note that the set $\{ r \in \mathbb{R} \mid \exists \xi \in \mathop{\mathrm{dom}} \nolimits (r) , \ r( \xi ) \supseteq x \}$ is dense in $\mathbb{R}$ for any $x \in [ \lambda ]^\omega$, since $X$ is $\subseteq$-cofinal in $[ \lambda ]^\omega$.
First, we prove (1) and (3). We work in $V$. Suppose $r \in \mathbb{R}$, $\mathcal{D}$ is a countable family of dense open subsets of $\mathbb{R}$ and $\dot{F}$ is an $\mathbb{R}$-name for a function from $[ \lambda ]^{< \omega}$ to $\lambda$. It suffices to find $r^* \leq r$ and $y \in Y$ such that $r^* \in \bigcap \mathcal{D}$ and $r^*$ forces $y$ to be closed under $\dot{F}$.
Take a sufficiently large regular cardinal $\theta$. Since $Y$ is stationary, there is a countable $M \prec \langle \mathcal{H}_\theta , \in \rangle$ such that $\{ \lambda , X , r , \dot{F} \} \cup \mathcal{D} \subseteq M$ and $y := M \cap \lambda \in Y$. Then, we can construct a descending sequence $\langle r_n \mid n < \omega \rangle$ in $\mathbb{R} \cap M$ such that $r_0 = r$ and $\{ r_n \mid n < \omega \}$ is $M$-generic. Note that any lower bound of $\{ r_n \mid n < \omega \}$ forces $y$ to be closed under $\dot{F}$ by the $M$-genericity of $\{ r_n \mid n < \omega \}$.
Let $r' := \bigcup_{n < \omega} r_n$ and $\zeta : = \mathop{\mathrm{dom}} \nolimits ( r' )$. Then, using the fact mentioned at the beginning, it is easy to check that $\zeta$ is a limit ordinal and $\bigcup_{\xi < \zeta} r' ( \xi ) = y$. Let $r^*$ be an extension of $r'$ such that $\mathop{\mathrm{dom}} \nolimits ( r^* ) = \zeta + 1$ and $r^* ( \zeta ) = y$. Then $r^* \in \mathbb{R}$, and $r^*$ is a lower bound of $\{ r_n \mid n < \omega \}$. So $r^*$ and $y$ are as desired.
Next, we check (2). By (1), the definition of $\mathbb{R}$ and the fact mentioned at the beginning, if $G$ is an $\mathbb{R}$-generic filter over $V$, then $\mathrm{range} ( \bigcup G )$ is a club subset of $[ \lambda ]^\omega$ consisting of elements of $X$. So (2) holds. \end{proof}
\section{Proof of Main Theorem} \label{sec:proof}
Here we prove the Main Theorem. Throughout this section, assume that $\mathsf{MM}^{+ \omega}$ holds in the ground model $V$.
We construct a forcing notion which preserves $\mathsf{MM}^{+ \omega}$ and adds a counter-example $\langle X_\alpha \mid \alpha < \omega_2 \rangle$ of the consequence of Lemma \ref{lem:DRP_easy}. Here recall that $\mathsf{MM}$ implies $\mathsf{wDRP}$. So we must arrange our forcing notion so that each $X_\alpha$ is not projectively stationary. For some technical reason, we also make $\langle X_\alpha \mid \alpha < \omega_2 \rangle$ pairwise disjoint.
Recall the fact, due to Foreman-Magidor-Shelah \cite{FMS}, that $\mathsf{MM}$ implies $2^{\omega_1} = \omega_2$. In $V$, fix an enumeration $\langle S_\alpha \mid \alpha < \omega_2 \rangle$ of all stationary subsets of $\omega_1$. Let $\mathbb{P}$ be the following forcing notion: \begin{itemize} \item $\mathbb{P}$ consists of all functions $p$ such that \begin{renumerate} \item $p : \delta_p \times [ \delta_p ]^\omega \to 2$ for some $\delta_p < \omega_2$, \item for any $\alpha < \delta_p$, $X_{p,\alpha} := \{ x \in [ \delta_p ]^\omega \mid p( \alpha , x ) = 1 \}$ has size $\leq \omega_1$, \item $x \cap \omega_1 \in S_\alpha$ for any $\alpha < \delta_p$ and any $x \in X_{p, \alpha}$, \item $X_{p, \alpha} \cap X_{p , \beta} = \emptyset$ for any distinct $\alpha , \beta < \delta_p$, \item for any $\delta \in \delta_p + 1 \setminus \omega_1$, there is $\alpha < \delta$ with $X_{p , \alpha} \cap [ \delta ]^\omega$ non-stationary in $[ \delta ]^\omega$. \end{renumerate} \item $p \leq p'$ in $\mathbb{P}$ if $p \supseteq p'$. \end{itemize}
We observe basic properties of $\mathbb{P}$. Note that a forcing extension by $\mathbb{P}$ preserves all cardinals by (1) and (3) of the following lemma.
\begin{lemma} \label{lem:P_basic} \begin{aenumerate}
\item $| \mathbb{P} | = \omega_2$. \item $\mathbb{P}$ is $\sigma$-closed. \item A forcing extension by $\mathbb{P}$ adds no new sequences of ordinals of length $\omega_1$. \item For any $p \in \mathbb{P}$ and any $\delta < \omega_2$, there is $p' \leq p$ with $\delta \leq \delta_{p'}$. \end{aenumerate} \end{lemma}
\begin{proof} (1) This is clear from the definition of $\mathbb{P}$, especially the property (ii) of its conditions, and the fact that $2^{\omega_1} = \omega_2$ in $V$.
\noindent (4) Suppose $p \in \mathbb{P}$ and $\delta < \omega_2$. We may assume $\delta_p \leq \delta$. Let $p' : \delta \times [ \delta ]^\omega \to 2$ be an extension of $p'$ such that $p' ( \alpha , x ) = 0$ for all $\langle \alpha , x \rangle \notin \delta_p \times [ \delta_p ]^\omega$. It suffices to prove that $p' \in \mathbb{P}$. We only check that $p'$ satisfies the property (v) of conditions of $\mathbb{P}$. The other properties are easily checked.
Take an arbitrary $\gamma \in \delta + 1 \setminus \omega_1$. We find $\alpha < \delta$ with $X_{p' , \alpha} \cap [ \gamma ]^\omega$ is non-stationary. If $\gamma \leq \delta_p$, then we can find such $\alpha$ since $p \in \mathbb{P}$ and $p \subseteq p'$. Suppose $\gamma > \delta_p$. Then $Z := [ \gamma ]^\omega \setminus [ \delta_p ]^\omega$ is club in $[ \gamma ]^\omega$, and $X_{p' , \alpha} \cap Z = \emptyset$ for any $\alpha < \gamma$. So any $\alpha < \gamma$ is as desired in this case.
\noindent (2) Suppose $\langle p_n \mid n < \omega \rangle$ is a descending sequence in $\mathbb{P}$. We find a lower bound $p^*$ of $\{ p_n \mid n < \omega \}$ in $\mathbb{P}$. We may assume that $\langle p_n \mid n < \omega \rangle$ is not eventually constant.
Let $\delta_n := \delta_{p_n}$ for each $n < \omega$. Let $\delta^* := \bigcup_{n < \omega} \delta_n$, and let $p^* : \delta^* \times [ \delta^* ]^\omega \to 2$ be an extension of $\bigcup_{n \in \omega} p_n$ such that $p^* ( \alpha , x ) = 0$ for all $\alpha < \delta^*$ and all $x \in [ \delta^* ]^\omega \setminus \bigcup_{n \in \omega} [ \delta_n ]^\omega$. Note that $X_{p^* , \alpha}$ is non-stationary in $[ \delta^* ]^\omega$ for any $\alpha < \delta^*$ since $Z := [ \delta^* ]^\omega \setminus \bigcup_{n < \omega} [ \delta_n ]^\omega$ is club in $[ \delta^* ]^\omega$ and $X_{p^* , \alpha} \cap Z = \emptyset$. Then it is easy to see that $p^*$ is as desired.
\noindent (3) Suppose $p \in \mathbb{P}$ and $\langle D_\xi \mid \xi < \omega_1 \rangle$ is a sequence of dense open subsets of $\mathbb{P}$. It suffices to find $p^* \leq p$ with $p^* \in \bigcap_{\xi < \omega_1} D_\xi$.
We recursively construct a strictly descending sequence $\langle p_\xi \mid \xi < \omega_1 \rangle$ in $\mathbb{P}$ as follows. For each $\xi < \omega_1$, we let $\delta_\xi$ denote $\delta_{p_\xi}$. First, let $p_0 := p$. If $p_\xi$ has been taken, then take $p_{\xi + 1} < p_\xi$ with $p_{\xi + 1} \in D_\xi$. Suppose $\xi$ is a limit ordinal $< \omega_1$ and $\langle p_\eta \mid \eta < \xi \rangle$ has been constructed. Then define $p_\xi$ as in the proof of (2). That is, let $\delta_\xi := \bigcup_{\eta < \xi} \delta_\eta$, and let $p_\xi : \delta_\xi \times [ \delta_\xi ]^\omega \to 2$ be an extension of $\bigcup_{\eta < \xi} p_\eta$ such that $p_\xi ( \alpha , x ) = 0$ for all $\alpha < \delta_\xi$ and all $x \in [ \delta_\xi ]^\omega \setminus \bigcup_{\eta < \xi} [ \delta_\eta ]^\omega$. Then $p_\xi$ is a lower bound of $\{ p_\eta \mid \eta < \xi \}$ in $\mathbb{P}$.
We have constructed $\langle p_\xi \mid \xi < \omega_1 \rangle$. Let $\delta^* := \sup_{\xi < \omega_1} \delta_\xi$ and $p^* := \bigcup_{\xi < \omega_1} p_\xi$. Here note that $[ \delta^* ]^\omega = \bigcup_{\xi < \omega_1} [ \delta_\xi ]^\omega$. So $p^* : \delta^* \times [ \delta^* ]^\omega \to 2$. Note also that $X_{p^* , 0}$ is non-stationary in $[ \delta^* ]^\omega$ since \[ Z := \{ x \in [ \delta^* ]^\omega \mid \mbox{$x \subseteq \delta_\xi = \sup (x)$ for some limit $\xi < \omega_1$} \} \] is club in $[ \delta^* ]^\omega$ and that $X_{p^* , 0} \cap Z = \emptyset$ by the construction of $p_\xi$ for a limit $\xi < \omega_1$. Then, it is easy to check that $p^*$ is as desired. \end{proof}
Let $\dot{G}$ be the canonical $\mathbb{P}$-name for a $\mathbb{P}$-generic filter. For $\alpha < \omega_2$, let $\dot{X}_\alpha$ be the $\mathbb{P}$-name for the set \[ \{ x \in [ \omega_2 ]^\omega \mid \exists p \in \dot{G} , \ p( \alpha , x ) = 1 \} \, . \]
\begin{lemma} \label{lem:X_alpha_stationary} For each $\alpha < \omega_2$, $\dot{X}_\alpha$ is stationary in $[ \omega_2 ]^\omega$ in $V^\mathbb{P}$. \end{lemma}
\begin{proof} We work in $V$. Take an arbitrary $\alpha < \omega_2$. Suppose $p \in \mathbb{P}$ and $\dot{F}$ is a $\mathbb{P}$-name for a function from $[ \omega_2 ]^{< \omega}$ to $\omega_2$. It suffices to find $p^* \leq p$ and $x \in [ \omega_2 ]^{< \omega}$ such that $p^* \Vdash_\mathbb{P} \textrm{``}\, x \in \dot{X}_\alpha \wedge \mbox{$x$ is closed under $\dot{F}$} \,\textrm{''}$.
Take a sufficiently large regular cardinal $\theta$ and a countable $M \prec \langle \mathcal{H}_\theta , \in \rangle$ such that $\alpha , \mathbb{P} , p , \dot{F} \in M$ and $M \cap \omega_1 = \alpha$. Let $x := M \cap \omega_2$. We can take a descending sequence $\langle p_n \mid n < \omega \rangle$ in $\mathbb{P} \cap M$ such that $p_0 = p$ and $\{ p_n \mid n < \omega \}$ is $M$-generic. Note that any lower bound of $\{ p_n \mid n < \omega \}$ forces $x$ to be closed under $\dot{F}$ by the $M$-genericity. For each $n < \omega$, let $\delta_n := \delta_{p_n}$. Note that $\delta_n \in M \cap \omega_2$ for each $n < \omega$ and that $\delta^* := \sup_{n < \omega} \delta_n = \sup ( M \cap \omega_2 )$ by Lemma \ref{lem:P_basic} (4).
Let $p^* : \delta^* \times [ \delta^* ]^\omega \to 2$ be an extension of $\bigcup_{n < \omega} p_n$ such that $p^* ( \alpha , x ) = 1$ and $p^* ( \beta , y ) = 0$ for any $\beta < \delta^*$ and any $y \in [ \delta^* ]^\omega \setminus \bigcup_{n < \omega} [ \delta_n ]^\omega$ with $\langle \beta , y \rangle \neq \langle \alpha , x \rangle$. Then, it is easy to check that $p^*$ and $x$ are as desired. \end{proof}
The following is immediate from Lemma \ref{lem:DRP_easy}, \ref{lem:P_basic}, \ref{lem:X_alpha_stationary} and the property (v) of conditions of $\mathbb{P}$:
\begin{cor} \label{cor:DRP_fail} $\mathsf{DRP}$ at $\omega_2$ fails in $V^\mathbb{P}$. \end{cor}
We must show that $\mathbb{P}$ preserves $\mathsf{MM}^{+ \omega}$. The following lemma is a key:
\begin{lemma} \label{lem:triple_stat_pres} Let $\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\omega_1$-stationary preserving forcing notion and $\langle \dot{T}_n \mid n < \omega \rangle$ be a sequence of $\mathbb{P} * \dot{\mathbb{Q}}$-names for stationary subsets of $\omega_1$. Then there is a $\mathbb{P} * \dot{\mathbb{Q}}$-name $\dot{\gamma}$ of an ordinal $< \omega_2^V$ such that if we let \[ \mathbb{S} := \mathbb{P} * \dot{\mathbb{Q}} * \mathbb{R} ( [ \omega_2^V ]^\omega \setminus \dot{X}_{\dot{\gamma}} ) \, , \] then all elements of $\{ S_\alpha \mid \alpha < \omega_2^V \} \cup \{ \dot{T}_n \mid n < \omega \}$ remain stationary in $V^\mathbb{S}$. \end{lemma}
\begin{proof} Let $\lambda := \omega_2^V$. Suppose $G*H$ is a $\mathbb{P} * \dot{\mathbb{Q}}$-generic filter over $V$. In $V[G*H]$, let $X_\alpha := \dot{X}_\alpha^G$ for $\alpha < \lambda$ and $T_n := \dot{T}_n^{G*H}$ for $n < \omega$. Moreover, let $\mathbb{R}_\alpha$ denote $\mathbb{R} ( [ \lambda ]^\omega \setminus X_\alpha )$ for $\alpha < \lambda$. In $V[G*H]$, we find $\gamma < \lambda$ such that $\mathbb{R}_\gamma$ forces all elements of $\{ S_\alpha \mid \alpha < \lambda \} \cup \{ T_n \mid n < \omega \}$ stationary. Here note that all $S_\alpha$ and $T_n$ are stationary in $V[G*H]$ by the fact that $\mathbb{P} * \dot{\mathbb{Q}}$ is $\omega_1$-stationary preserving and the assumption on $\langle \dot{T}_n \mid n < \omega \rangle$.
We work in $V[ G*H ]$. For $S \subseteq \omega_1$, let $\bar{S} := \{ x \in [ \lambda ]^\omega \mid x \cap \omega_1 \in S \}$. For $X,Y \subseteq [ \lambda ]^\omega$, we write $X \subseteq^* Y$ if $X \setminus Y$ is non-stationary in $[ \lambda ]^\omega$. By Lemma \ref{lem:club_shoot_basic}, for $S \subseteq \omega_1$ and $\alpha < \lambda$, $\mathbb{R}_\alpha$ does not force $S \subseteq \omega_1$ stationary if and only if $\bar{S} \subseteq^* X_\alpha$.
Since $\langle X_\alpha \mid \alpha < \lambda \rangle$ is pairwise disjoint, for each $n < \omega$ there is at most one $\alpha < \lambda$ with $\bar{T}_n \subseteq^* X_\alpha$. Since $| \lambda | \geq \omega_1$, we can take $\beta < \lambda$ such that $\bar{T}_n \not\subseteq^* X_\beta$ for any $n < \omega$. Then $\mathbb{R}_\beta$ forces $T_n$ stationary for all $n < \omega$. Thus, if $\mathbb{R}_\beta$ also forces $S_\alpha$ stationary for all $\alpha < \lambda$, then $\gamma := \beta$ is as desired.
Assume there is $\alpha < \lambda$ such that $\mathbb{R}_\beta$ does not force $S_\alpha$ stationary. By replacing $\alpha$ with $\alpha '$ such that $S_{\alpha '} \subseteq S_\alpha$ if necessary, we may assume that $\alpha \neq \beta$. Here note that $X_\alpha \subseteq \bar{S}_\alpha$ by the property (iii) of conditions of $\mathbb{P}$. Then, $X_\alpha \subseteq \bar{S}_\alpha \subseteq^* X_\beta$ and $X_\alpha \cap X_\beta = \emptyset$. Hence $X_\alpha$ is non-stationary in $[ \lambda ]^\omega$. Thus $\mathbb{R}_\alpha$ is $\omega_1$-stationary preserving, and so $\gamma := \alpha$ is as desired. \end{proof}
Now, we can prove that $\mathbb{P}$ preserves $\mathsf{MM}^{+ \omega}$ by a similar argument as Beaudoin \cite{Beaudoin}:
\begin{lemma} \label{lem:pres_MM+omega} $\mathsf{MM}^{+ \omega}$ holds in $V^\mathbb{P}$. \end{lemma}
\begin{proof} Let $\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\omega_1$-stationary preserving foricng notion. For each $\xi < \omega_1$, let $\dot{D}_\xi$ be a $\mathbb{P}$-name for a dense subset of $\dot{\mathbb{Q}}$, and for each $n < \omega$, let $\ddot{T}_n$ be a $\mathbb{P}$-name for a $\dot{\mathbb{Q}}$-name for a stationary subset of $\omega_1$. Take an arbitrary $p_0 \in \mathbb{P}$. It suffices to find $p^* \leq p_0$ in $\mathbb{P}$ such that if $G$ is a $\mathbb{P}$-generic filter over $V$ with $p^* \in G$, then in $V[ G ]$ there is a filter $h \subseteq \mathbb{Q}$ with the following properties: \begin{renumerate} \item $h \cap D_\xi \neq \emptyset$ for any $\xi < \omega_1$. \item $\dot{T}_n^h$ is stationary in $\omega_1$ for all $n < \omega$. \end{renumerate} Here $\mathbb{Q}$, $D_\xi$ and $\dot{T}_n$ denote $\dot{\mathbb{Q}}^G$, $\dot{D}_\xi^G$ and $\ddot{T}_n^G$, respectively.
First, we find $p^*$ as above. We work in $V$. We identify each $\ddot{T}_n$ with a $\mathbb{P} * \dot{\mathbb{Q}}$-name. Let $\dot{\gamma}$ and $\mathbb{S}$ be as in Lemma \ref{lem:triple_stat_pres}. Note that $\mathbb{S}$ is $\omega_1$-stationary preserving and each $\ddot{T}_n$ is stationary in $\omega_1$ in $V^\mathbb{S}$. Let $\dot{\mathbb{R}}$ be a $\mathbb{P} * \dot{\mathbb{Q}}$-name for $\mathbb{R} ( [ \omega_2^V ]^\omega \setminus \dot{X}_{\dot{\gamma}} )$.
Take a sufficiently large regular cardinal $\theta$. By Fact \ref{fact:forcing_axiom}, there are $M \in [ \mathcal{H}_\theta ]^{\omega_1}$ and $k \subseteq \mathbb{S} \cap M$ such that \begin{renumerate} \addtocounter{enumi}{2} \item $\omega_1 \cup \{ p_0 , \dot{\mathbb{Q}} , \dot{\gamma} , \mathbb{S} \} \cup \{ \dot{D}_\xi \mid \xi < \omega_1 \} \cup \{ \ddot{T}_n \mid n < \omega \} \subseteq M \prec \langle \mathcal{H}_\theta , \in \rangle$, \item $k$ is an $M$-generic filter on $\mathbb{S} \cap M$ with $p_0 * 1_{\dot{\mathbb{Q}}} * 1_{\dot{\mathbb{R}}} \in k$, \item $\ddot{T}_n^k$ is stationary in $\omega_1$ for any $\xi < \mu$. \end{renumerate} Let $\delta^* := M \cap \omega_2 \in \omega_2$, and let \[ g := \{ p \in \mathbb{P} \cap M \mid \exists \dot{q} \, \exists \dot{r} , \ p * \dot{q} * \dot{r} \in k \} \, . \] Then, $g$ is an $M$-generic filter on $\mathbb{P} \cap M$.
Note that $\sup_{p \in g} \delta_p = \delta^*$ by Lemma \ref{lem:P_basic} (4) and the $M$-genericity of $g_0$. Let $p^* : \delta^* \times [ \delta^* ]^\omega \to 2$ be an extension of $\bigcup g$ such that $p^* ( \alpha , x ) = 0$ for all $\langle \alpha , x \rangle \notin \mathop{\mathrm{dom}} \nolimits ( \bigcup g )$. We claim that $p^*$ is as desired. For this, we use the transitive collapse of $M$. First, we make some preliminaries on it.
Let $\pi : M \to M'$ be the transitive collapse of $M$, and let $\mathbb{P}'$, $\dot{\mathbb{Q}}'$, $\dot{\mathbb{R}}'$, $\mathbb{S}'$, $k'$ and $g'$ be $\pi ( \mathbb{P} )$, $\pi ( \dot{\mathbb{Q}} )$, $\pi ( \dot{\mathbb{R}} )$, $\pi ( \mathbb{S} )$, $\pi [k]$ and $\pi [g]$, respectively. Note that $\mathbb{S} ' = \mathbb{P} ' * \dot{\mathbb{Q}} ' * \dot{\mathbb{R}} '$ in $M'$. Moreover, $k'$ is an $\mathbb{S}'$-generic filter over $M'$, and $g'$ is the $\mathbb{P}'$-generic filter over $M'$ naturally obtained from $k'$. Let $h'$ be the $( \dot{\mathbb{Q}} ' )^{g '}$-generic filter over $M'[ g' ]$ naturally obtained from $k'$, and let $i'$ be the $( \dot{\mathbb{R}} ' )^{g' * h'}$-generic filter over $M' [ g' * h' ]$ naturally obtained from $k'$.
Now, we start to prove that $p^*$ is as desired. First, we prove that $p^* \in \mathbb{P}$. We only check that $X_{p^* , \gamma}$ is non-stationary in $[ \delta^* ]^\omega$ for some $\gamma < \delta^*$. The other properties are easily checked.
First of all, note that $\pi \!\upharpoonright\! ( \mathcal{H}_{\omega_2} \cap M )$ is the identity map since $\mathcal{H}_{\omega_2} \cap M$ is transitive and that $\pi ( \omega_2 ) = \delta^*$. Let $\gamma := \pi ( \dot{\gamma} )^{g' * h'} < \pi ( \omega_2 ) = \delta^*$. Then $\mathrm{range} ( \bigcup i' )$ is a club subset of $[ \delta^* ]^\omega$ which does not intersect $\bigcup_{p' \in g'} X_{p' , \gamma} = \bigcup_{p \in g} \pi ( X_{p , \gamma} )$. Here note that $X_{p , \gamma} \in \mathcal{H}_{\omega_2} \cap M$ for all $p \in g$ by the property (ii) of conditions in $\mathbb{P}$. So $\bigcup_{p \in g} \pi ( X_{p , \gamma} ) = \bigcup_{p \in g} X_{p , \gamma} = X_{p^* , \gamma}$. Hence $X_{p^* , \gamma}$ is non-stationary in $[ \delta^* ]^\omega$
We have shown that $p^* \in \mathbb{P}$. Note that $p^*$ is a lower bound of $g$. Then $p^* \leq p$ since $p \in g$ by (iv). Suppose $G$ is a $\mathbb{P}$-generic filter over $V$ with $p^* \in G$. Working in $V[G]$, we find a filter $h \subseteq \mathbb{Q}$ satisfying (i) and (ii).
Let $M[G]$ denote the collection of $\dot{a}^G$ for all $\mathbb{P}$-names $\dot{a} \in M$, and define $\hat{\pi} : M[G] \to M'[g']$ by $\hat{\pi} ( \dot{a}^G ) := \pi ( \dot{a} )^{g'}$. It is easy to see that $\hat{\pi}$ coincides with the transitive collapse of $M[G]$ and that $\hat{\pi}$ extends $\pi$. Let $h$ be the filter on $\mathbb{Q}$ generated by $\hat{\pi}^{-1} [ h' ]$. Then $h$ satisfies (i) since $D_\xi \in M[G]$ and $h' \cap \hat{\pi} ( D_\xi ) \neq \emptyset$ for all $\xi < \omega_1$. As for (ii), it is easy to see that $\dot{T}_n^h = \ddot{T}_n^k$ for each $n < \omega$. Then, $h$ satisfies (ii) by (v). \end{proof}
\end{document}
|
arXiv
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\begin{document}
\title[Nonautonomous Hindmarsh-Rose Equations]{Global Dynamics of Nonautonomous Hindmarsh-Rose Equations}
\author[C. Phan]{Chi Phan} \address{Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA} \email{[email protected]} \thanks{}
\author[Y. You]{Yuncheng You} \address{Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA} \email{[email protected]} \thanks{}
\subjclass[2000]{Primary: 35K57, 37L30, 37L55, 37N25; Secondary: 35B40, 35K55, 92C20}
\keywords{Nonautonomous Hindmarsh-Rose equations, global dynamics, pullback attractor, smoothing Lipschitz continuity, pullback exponential attractor}
\begin{abstract} Global dynamics of nonautonomous diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process. \end{abstract}
\maketitle
\section{\textbf{Introduction}}
The Hindmarsh-Rose equations for neuronal bursting of the intracellular membrane potential observed in experiments was initially proposed in \cite{HR1, HR2}. The original model is composed of three ordinary differential equations and has been studied by numerical simulations and mathematical analysis, cf. \cite{HR1, HR2, IG, Ko, MFL, SC, SPH, Tr, Su, ZZL} and the references therein. The solutions of this model exhibit interesting bursting patterns, especially chaotic bursting and dynamics.
Very recently, the authors in \cite{PYS} and \cite{Phan} proved the existence of global attractors for the diffusive and partly diffusive Hindmarsh-Rose equations as well as the existence of a random attractor for the stochastic Hindmarsh-Rose equations with multiplicative noise.
In this work, we shall study the global dynamics for the nonautonomous diffusive Hindmarsh-Rose equations with time-dependent external inputs: \begin{align}
\frac{\partial u}{\partial t} & = d_1 \Delta u + \varphi (u) + v - w + J + p_1 (t, x)\, \label{uq} \\
\frac{\partial v}{\partial t} & = d_2 \Delta v + \psi (u) - v + p_2 (t, x), \label{vq} \\
\frac{\partial w}{\partial t} & = d_3 \Delta w + q (u - c) - rw + p_3 (t, x), \label{wq} \end{align} for $t > \tau \in \mathbb{R},\; x \in \Omega \subset \mathbb{R}^{n}$ ($n \leq 3$), where $\Omega$ is a bounded domain with locally Lipschitz continuous boundary, the stimulation inject current $J $ is assumed to be a constant, and the nonlinear terms in \eqref{uq} and \eqref{vq} are \begin{equation} \label{pp}
\varphi (u) = au^2 - bu^3, \quad \text{and} \quad \psi (u) = \alpha - \beta u^2. \end{equation} Assume that the external input terms $p_i \in L^2_{loc} (\mathbb{R}, L^2 (\Omega)), i = 1, 2, 3$, satisfy the condition of translation boundedness \cite{CV}, \begin{equation} \label{tbd}
\|p_i \|^2_{L_b^2} = \sup_{t\, \in \,\mathbb{R}}\, \int_t^{t+1} \int_\Omega |p_i (t, x)|^2 \, dx\, ds < \infty, \quad i = 1, 2, 3. \end{equation}
All the involved parameters $d_1, d_2, d_3, a, b, \gamma, \beta, q, r$, and $J$ are positive constants except $c \,(= u_R) \in \mathbb{R}$, which is a reference value of the membrane potential of a neuron cell. We impose the homogeneous Neumann boundary conditions \begin{equation} \label{nbc}
\frac{\partial u}{\partial \nu} (t, x) = 0, \; \; \frac{\partial v}{\partial \nu} (t, x)= 0, \; \; \frac{\partial w}{\partial \nu} (t, x)= 0,\quad t > \tau \in \mathbb{R}, \; x \in \partial \Omega , \end{equation} and the initial conditions to be specified are denoted by \begin{equation} \label{inc}
u(\tau, x) = u_\tau (x), \; v(\tau, x) = v_\tau (x), \; w(\tau, x) = w_\tau (x), \quad x \in \Omega. \end{equation}
In this system \eqref{uq}-\eqref{wq}, the variable $u(t,x)$ refers to the membrane electric potential of a neuronal cell, the variable $v(t, x)$ represents the transport rate of the ions of sodium and potassium through the fast ion channels and is called the spiking variable, while the variables $w(t, x)$ represents the transport rate across the neuronal cell membrane through slow channels of calcium and other ions correlated to the bursting phenomenon and is called the bursting variable.
We start with formulation of the aforementioned initial-boundary value problem of \eqref{uq}--\eqref{inc}. Define the Hilbert spaces $H = [L^2 (\Omega)]^3 = L^2 (\Omega, \mathbb{R}^3)$ and $E = [H^{1}(\Omega)]^3 = H^1 (\Omega, \mathbb{R}^3)$. The norm and inner-product of $H$ or $L^2 (\Omega)$ will be denoted by $\| \, \cdot \, \|$ and $\inpt{\,\cdot , \cdot\,}$, respectively. The norm of $E$ will be denoted by $\| \, \cdot \, \|_E$. The norm of $L^p (\Omega)$ or $L^p (\Omega, \mathbb{R}^3)$ will be dented by $\| \cdot \|_{L^p}$ if $p \neq 2$. We use $| \, \cdot \, |$ to denote a vector norm in a Euclidean space.
The nonautonomous system \eqref{uq}-\eqref{wq} with the initial-boundary conditions \eqref{nbc} and \eqref{inc} can be written in the vector form \begin{equation} \label{napb}
\begin{split}
\frac{\partial g}{\partial t} = Ag &+ f(g) + p(t,x), \quad t > \tau \in \mathbb{R}, \\[3pt]
&g(\tau) = g_\tau,
\end{split} \end{equation} where $$
g(t) = \textup{col} \, (u(t, \cdot), v(t, \cdot), w(t, \cdot)), \quad g_\tau = \textup{col}\, (u_\tau, v_\tau, w_\tau), $$ and $p(t, x) = \textup{col}\, (p_1 (t,x), p_2 (t,x), p(t,x))$, the nonpositive self-adjoint operator \begin{equation} \label{opA}
A =
\begin{pmatrix}
d_1 \Delta & 0 & 0 \\[3pt]
0 & d_2 \Delta & 0 \\[3pt]
0 & 0 & d_3 \Delta
\end{pmatrix}
: D(A) \rightarrow H, \end{equation} where $$
D(A) = \{g \in H^2(\Omega, \mathbb{R}^3): \partial g /\partial \nu = 0 \}, $$ is the generator of an analytic $C_0$-semigroup $\{e^{At}\}_{t \geq 0}$ on the Hilbert space $H$ \cite{SY}. By the fact that $H^{1}(\Omega) \hookrightarrow L^6(\Omega)$ is a continuous imbedding for space dimension $n \leq 3$ and by the H\"{o}lder inequality, there is a constant $C_0 > 0$ such that $$
\| \varphi (u) \| \leq C_0 \| u \|_{L^6}^3 \quad \textup{and} \quad \|\psi (u) \| \leq C_0 \| u \|_{L^4}^2 \quad \textup{for} \; u \in L^6 (\Omega). $$ Therefore, the nonlinear mapping \begin{equation} \label{opf}
f(u,v, w) =
\begin{pmatrix}
\varphi (u) + v - w + J \\[4pt]
\psi (u) - v, \\[4pt]
q (u - c) - rw
\end{pmatrix}
: E \longrightarrow H \end{equation} is a locally Lipschitz continuous mapping.
\subsection{\textbf{Hindmarsh-Rose Models in Neurodynamics}}
In 1982-1984, Hindmarsh and Rose developed the mathematical model \cite{HR1, HR2} to describe neuronal dynamics \begin{equation} \label{HR}
\begin{split}
\frac{du}{dt} & = au^2 - bu^3 + v - w + J, \\
\frac{dv}{dt} & = \alpha - \beta u^2 - v, \\
\frac{dw}{dt} & = q (u - u_R) - rw.
\end{split} \end{equation} This model characterizes the phenomena of synaptic bursting and especially chaotic bursting. Neuronal signals are short electrical pulses called spike or action potential. Bursting shows alternating phases of rapid firing spikes and then quiescence. It is a mechanism to modulate brain functionalities and to communicate signals with the neighbor neurons. Bursting patterns occur in a variety of bio-systems such as pituitary melanotropic gland, thalamic neurons, respiratory pacemaker neurons, and insulin-secreting pancreatic $\beta$-cells, cf. \cite{BRS, CK,CS, HR2}.
The mathematical analysis mainly using bifurcations together with numerical simulations of several models in ODEs on bursting behavior has been done by many authors, cf. \cite{BB, ET, EI, MFL, Ri, SPH, Tr, WS, Su}.
Neurons burst through synaptic coupling or diffusive coupling. Synaptic coupling has to reach certain threshold for release of quantal vesicles and synchronization \cite{DJ, Ru, Rv, SC}.
It is known that Hodgkin-Huxley equations \cite{HH} (1952) provided a four-dimensional model for the dynamics of membrane potential taking into account of the sodium, potassium and leak ions current. It is a highly nonlinear system if without simplification assumptions. The FitzHugh-Nagumo equations \cite{FH} (1961-1962) is a two-dimensional model for an excitable neuron with the membrane potential and the current variable in a lump. This 2D model admits an exquisite phase plane analysis, but it excludes any chaotic solutions and chaotic dynamics so that no chaotic bursting can be generated by the FitzHugh-Nagumo equations.
In contrast, the Hindmarsh-Rose equations contribute a three-dimensional model with cubic nonlinearity to generate a significant mechanism for rapid firing and busting in the research of neurodynamics. The chaotic coupling exhibited in the simulations and analysis of this Hindmarsh-Rose model shows more rapid synchronization and more effective regularization of neurons due to \emph{lower threshold} than the synaptic coupling \cite{IG, Tr, Rv, SK, SPH, Su}. The research on this Hindmarsh-Rose model also indicated \cite{SC} that it allows for spikes with varying interspike-interval. Therefore, this 3D model is a suitable choice for the investigation of both the regular bursting and the chaotic bursting when the parameters vary.
In general, neurons are immersed in aqueous biochemical solutions consisting of different ions electrically charged. The axon of a neuron is a long branch to propagate signals and the neuron cell membrane is the conductor along which the voltage signals travel. As pointed out in \cite{EI}, neuron is a distributed dynamical system.
From physical and mathematical point of view, it is reasonable and useful to consider the diffusive Hindmarsh-Rose model in terms of partial differential equations with the spatial variables $x$ involved, at least in $\mathbb{R}^1$. Here in the abstract extent, we shall study the diffusive Hindmarsh-Rose equations \eqref{uq}-\eqref{wq} with time-dependent external stimulations in a bounded domain of space $\mathbb{R}^3$ and we shall focus on the global dynamics of the solution processes. The chaotic bursting and dynamical properties from the nonautonomous diffusive Hindmarsh-Rose equations are expected to demonstrate a wide range of applications in neuroscience.
\subsection{\textbf{Preliminaries}}
In this work we shall consider the weak solutions of this initial value problem \eqref{napb}. \begin{definition} \label{D:wksn}
A function $g(t, x), (t, x) \in [\tau, T] \times \Omega$, is called a weak solution to the initial value problem \eqref{napb}, if the following conditions are satisfied:
\textup{(i)} $\frac{d}{dt} (g, \zeta) = (Ag, \zeta) + (f(g), \zeta)$ is satisfied for a.e. $t \in [\tau, T]$ and for any $\zeta \in E$.
\textup{(ii)} $g(t, \cdot) \in C ([\tau, T]; H) \cap L^2 ([\tau, T], E)$ and $g(\tau ) = g_0$.
\noindent Here $(\cdot , \cdot)$ stands for the dual product of the dual space $E^*$ and $E$. \end{definition}
\begin{lemma} \label{Lwn}
For any given initial data $g_0 \in H$, there exists a unique local weak solution $g(t, g_0) = (u(t), v(t), w(t)), \, t \in [\tau, T]$ for some $T > 0$, of the initial value problem \eqref{napb}, such that \begin{equation} \label{soln}
g \in C([\tau, T_{max}); H) \cap L_{loc}^2 ([\tau, T_{max}); E), \end{equation} where $I_{max} = [\tau, T_{max})$ is the maximal interval of existence. And the weak solution becomes a strong solution on $(\tau, T_{max})$, which satisfies the evolutionary equation \eqref{napb} in $H$ almost everywhere and with the regularity \begin{equation} \label{Strs}
g \in C((\tau, T_{max}); E) \cap C^1 ((\tau, T_{max}); H) \cap L_{loc}^2 ((\tau, T_{max}); H^2 (\Omega, \mathbb{R}^3)). \end{equation} \end{lemma}
\begin{proof} The proof of the local existence and uniqueness of weak solutions is made by \emph{a priori} estimates on the Galerkin approximate solutions obtained by spectral projections of the initial value problem \eqref{napb}, similar to what we shall present in Section 3, and then by the Lions-Magenes type of weak and weak$^*$ compactness and convergence argument. It is an adaptation of the treatment for local solutions of the generic reaction-diffusion system in \cite[Chapter XV, Theorem 3.1 and Proposition 3.1]{CV}. The details are omitted here. \end{proof}
The Gagliardo-Nirenberg inequalities \cite[Appendix B]{SY} of interpolation shown below will be used in several sharp estimates of this work, \begin{equation} \label{GN}
\| y \|_{W^{k, p} (\Omega)} \leq C \| y \|^\theta_{W^{m, q} (\Omega)}\, \| y \|^{1 - \theta}_{L^r (\Omega)}, \quad \text {for all} \;\; y \in W^{m, q} (\Omega), \end{equation} where $C > 0$ is a constant, provided that $p, q, r \geq 1, 0 < \theta < 1$, and $$
k - \frac{n}{p}\, \leq \, \theta \left(m - \frac{n}{q}\right) - (1 - \theta)\, \frac{n}{r}, \quad n = \text{dim}\, (\Omega). $$
In Section 2, we shall recall the basic concepts and the relevant existing results on the topics of global dynamics for nonautonomous dynamical systems. In Section 3, we prove the existence of a pullback attractor for the solution process of the nonautonomous Hindmarsh-Rose equations. In Section 4, the existence of pullback exponential attractors will be proved for this nonautonomous Hindmarsh-Rose process.
\section{\textbf{Pullback Attractor and Pullback Exponential Attractor}}
We refer to \cite{CCLF, CLR1, CLR2, CE1, CE2, EYY, Kl, LMR, LWZ, ZZX} for the concepts and some of the existing results in the theory of nonautonomous dynamical systems, especially on the topics of pullback attractors and pullback exponential attractors. Recall that these concepts are rooted in the theory of global attractors and other invariant attracting sets for the autonomous infinite-dimensional dynamical systems \cite{CV, Milani, Rb, SY, Tm, Y08, Y10, Y12} and the theory of exponential attractors or sometimes called inertial sets \cite{Eden, EMZ, Li, Milani, Yagi}.
Let $X$ be a Banach space and suppose that a nonautonomous partial differential equations with initial-boundary conditions, which usually involves a time-dependent forcing term, has global solutions in space-time. Then the solution operator $$
\{S(t, \tau): X \to X\}_{t \geq \tau \in \mathbb{R}} $$ is called a \emph{nonautonomous process} \cite{CLR2, CV}), which satisfies the three conditions:
1 ) $S(\tau, \tau) = I $ (the identity) for any $\tau \in \mathbb{R}$.
2) The cocycle property is satisfied: $$
S(t, s) S(s, \tau) = S(t, \tau) \quad \text{for any} \;\; -\infty < \tau \leq s \leq t < \infty. $$
3) The mapping $(t, \tau, g) \to S(t, \tau) g \in X$ is continuous with respect to $(t, \tau, g) \in \mathcal{T} \times X$ for any given $\tau \in \mathbb{R}$, where $\mathcal{T} = \{(t, \tau) \in \mathbb{R}^2: t \geq \tau\}$.
\begin{definition}[Nonautonomous semiflow] \label{nas}
A mapping $\Phi (t, \tau, g): \mathbb{R}^+ \times \mathbb{R} \times X \to X$ is called a \emph{nonautonomous semiflow} (or called nonautonomous dynamical system) on a Banach space $X$ over $\mathbb{R}$, if the following conditions are satisfied:
1) $\Phi (0, \tau, \cdot)$ is the identity on $X$, for any $\tau \in \mathbb{R}$.
2) $\Phi (t + s, \tau, \cdot) = \Phi (t, \tau + s, \Phi(s, \tau, \cdot))$, for any $t, s \geq 0$ and $\tau \in \mathbb{R}$.
3) $\Phi (t, \tau, g): \mathcal{T} \times X \to X$ is continuous. \end{definition}
If $\{S(t, \tau): X \to X\}_{(t, \tau) \in \mathcal{T}}$ is a continuous evolution process on $X$, then it generates a nonautonomous semiflow defined by \begin{equation} \label{PhiS}
\Phi (t, \tau, g) = S(t + \tau, \tau, g), \quad (t, \tau, g) \in \mathcal{T} \times X. \end{equation} This relation in the pullback sense is the following important identity \begin{equation} \label{PSR}
\Phi (t, \tau - t, g) = S(\tau, \tau - t)g, \quad (t, \tau, g) \in \mathbb{R}^+ \times \mathbb{R} \times X. \end{equation}
\begin{definition}[Pullback Attractor] \label{PA}
A time-parametrized set $\mathcal{A} = \{\mathcal{A}(\tau)\}_{\tau \in \mathbb{R}}$ in a Banach space $X$ is called a pullback attractor for the nonautonomous semiflow $\{\Phi (t, \tau, \cdot)\}_{(t, \tau) \in \mathcal{T}}$ generated by a continuous evolution process $\{S(t, \tau): X \to X\}_{(t, \tau) \in \mathcal{T}}$, if the following conditions are satisfied:
1) $\mathcal{A}$ is compact in the sense that for each $\tau \in \mathbb{R}$ the set $\mathcal{A}(\tau)$ is compact in $X$.
2) $\mathcal{A}$ is invariant, $$
S (t, \tau) \, \mathcal{A} (\tau) = \mathcal{A}(t), \quad t \geq 0, \; \tau \in \mathbb{R}. $$ it is equivalent to $\Phi (t, \tau, \mathcal{A}(\tau)) = \mathcal{A} (t + \tau)$ for $t \geq \tau$.
3) $\mathcal{A}$ pullback attracts every bounded set $B \subset X$ with respect to the semi-Hausdorff distance, $$
\lim_{t \to \infty} dist_X (\Phi (t, \, \tau - t, B), \, \mathcal{A} (\tau)) = \lim_{t \to \infty} dist_X (S (\tau, \, \tau - t) B, \, \mathcal{A} (\tau)) = 0. $$ \end{definition}
\begin{definition}[Pullback Exponential Attractor] \label{PEA}
A time-parametrized set $\mathscr{M} = \{\mathscr{M} (t)\}_{t \in \mathbb{R}} \subset X$, where $X$ is a Banach space, is called a pullback exponential attractor of a continuous evolution process $\{S(t, \tau)\}_{t \geq \tau \in \mathbb{R}}$ on $X$, if the following conditions are satisfied:
1) For any $t \in \mathbb{R}$, the set $\mathscr{M} (t)$ is a compact and positively invariant sel in $X$ with respect to this process, $$
S(t, \tau) \mathscr{M}(\tau) \subset \mathscr{M}(t) \quad \text{for any}\;\; \infty < \tau \leq t < \infty. $$
2) The fractal dimension $\text{dim}_{F} \mathscr{M}(t)$ for all $t \in \mathbb{R}$ is finite and $$
\sup_{t \in \mathbb{R}} \, \text{dim}_{F} \mathscr{M}(t) < \infty. $$
3) $\mathscr{M} = \{\mathscr{M} (t)\}_{t \in \mathbb{R}}$ exponentially attracts every bounded set $B \subset X$ in the sense that there exists a constant rate $\sigma > 0$, a constant $T_B > 0$ depending on $B$, and a positive function $C(\|B\|, T_B)$ where $\|B\| = \sup_{x \in B} \|x\|$, such that $$
\text{dist}_X (S(\tau, \tau - t)B, \mathscr{M}(\tau)) \leq C(\|B\|, T_B) e^{-\sigma (t - \tau)} \quad \text{for any} \; t > T_B, \; \tau \in \mathbb{R}. $$ \end{definition}
Below we present two existing results on the sufficient conditions for the existence of pullback attractor and for the existence of pullback exponential attractor, respectively.
\begin{proposition}\cite{CCLF, CLR1, CLR2, Kl} \label{PA}
A nonautonomous process $\{S(t, \tau)\}_{t \geq \tau \in \mathbb{R}}$ on a Banach space $X$ has a unique pullback attractor $\mathcal{A} = \{\mathcal{A} (\tau)\}_{\tau \in \mathbb{R}}$, if the following two conditions are satisfied\textup{:}
\textup{(i)} There is a pullback absorbing set $M$ in $X$, which means that for any given bounded set $B \subset X$, there is a finite time $T_B > 0$ such that
\begin{equation} \label{pbab}
S(\tau, \tau - t)B \subset M, \quad \textup{for all} \;\; t > T_B.
\end{equation}
\textup{(ii)} The nonautonomous process $S(t, \tau)$ is pullback asymptotically compact in the sense that for any sequences $t_k \to \infty$ and $\{x_k\} \subset B$, where $B$ is any given bounded set in $X$, the sequence $\{ S(\tau, \tau - t_k) x_k)\}_{k=1}^\infty$ has a convergent subsequence. Moreover, the pullback attractor is given by \begin{equation} \label{pbat}
\mathcal{A} (\tau) = \bigcap_{s \geq 0} \, \overline{\bigcup_{t \geq s} S (\tau, \tau - t) M}. \end{equation} \end{proposition}
\begin{proposition} \cite{CE1, CE2} \label{PeA}
Let $ X$ and $Y$ be Banach spaces and $Y$ compactly embedded in $X$. Assume that $\{S(t, \tau) \in \mathcal{L} (X) \cap \mathcal{L} (Y): t \geq \tau \in \mathbb{R}\}$ be a nonautonomous process such that the following three conditions are satisfied\textup{:}
\textup{1)} There exists a bounded pullback absorbing set $M^* \subset Y$ uniformly in time in the sense that, for any bounded set $B \subset X$, there is a finite time $T_B > 0$ such that \begin{equation} \label{ab}
\bigcup_{\tau \in \mathbb{R}} S(\tau, \tau - t)B \subset M^*, \quad \textup{for all} \;\; t > T_B. \end{equation}
\textup{2)} The smoothing Lipschitz continuity is satisfied\textup{:} There is a constant $\kappa > 0$ such that for the aforementioned bounded pullback absorbing set $M^* \subset Y$, \begin{equation} \label{smL}
\sup_{\tau \in \mathbb{R}} \|S(\tau, \tau - T_{M^*}) g_1 - S(\tau, \tau - T_{M^*}) g_2 \|_Y \leq \kappa \|g_1 - g_2\|_X, \;\; \text{for any} \;\; g_1, g_2 \in M^*. \end{equation}
\textup{3)} The H\"{o}lder/Lipschitz continuity in time is satisfied\textup{:} There exist two exponents $\gamma_1, \gamma_2 \in (0, 1]$ such that for the aforementioned set $M^* \subset Y$, \begin{align}
\sup_{\tau \in \mathbb{R}} &\, \|S(\tau, \tau - T_{M^*}) g - S(\tau, \tau - T_{M^*} - t) g \|_X \leq c_1 |t|^{\gamma_1}, \;\, t \in [0, T_{M^*}], \, g \in M^*, \label{Lpt1} \\
\sup_{\tau \in \mathbb{R}} &\, \|S(\tau, \tau - t_1) g - S(\tau, \tau - t_2) g \|_X \leq c_2 |t_1 - t_2|^{\gamma_2}, \; \,t_1, t_2 \in [T_{M^*}, 2T_{M^*}], \,g \in M^*. \label{Lpt2} \end{align} In \eqref{smL}-\eqref{Lpt2}, $T_{M^*} > 0$ is the time when all the pullback trajectories starting from $M^*$ permanently enter the absorbing set $M^*$ itself, and $c_1 = c_1 (M^*), c_2 = c_2(M^*)$ are two positive constants. Then there exists a pullback exponential attractor $\mathscr{M} = \{\mathscr{M}(\tau)\}_{\tau \in \mathbb{R}}$ in $X$ for this process. \end{proposition}
\emph{Remark} 1. The pullback absorbing set can be a time-parametrized set $M(\tau)$ in $X$ or in $Y$. Here the pullback absorbing sets specified in the above Proposition \eqref{PA} and Proposition \eqref{PeA} are time-invariant, which is what we only need.
\emph{Remark} 2. Another concept to describe the asymptotic global dynamics of a nonautonomous PDE is a skew-product dynamical systems \cite{SY}. It is to embed a nonautonomous semiflow into an augmented autonomous semiflow. The corresponding topic is uniform attractor \cite[Chapter IV]{CV}.
Although a uniform attractor is not a time-parametrized set, the major drawback is that its fractal dimension and Hausdorff dimension of a uniform attractor are in general infinite. The finite dimensionality reduction is lost. Moreover, it is usually difficult to estimate the oftentimes slow rate of attraction for a uniform attractor in terms of physical parameters in the mathematical model. Therefore, pullback attractor and pullback exponential attractor are favorable pursuit of the asymptotic behavior of nonautonomous dynamical systems generated by PDEs.
\section{\textbf{Pullback Attractor for Nonautonomous Hindmarsh-Rose Process}}
In this section, we shall first prove the global existence in time of the weak solutions to the system \eqref{napb} and then show the pullback absorbing property of the nonautonomous Hindmarsh-Rose process in the space $H$ and also in the space $E$, which leads to the existence of a pullback attractor for this nonautonomous semiflow.
\begin{lemma} \label{naab}
The weak solution of the nonautonomous system \eqref{napb} for any initial time $\tau \in \mathbb{R}$ and any initial data $g_\tau \in H$ exists globally for $t \in [\tau, \infty)$ and it generates a continuous evolution process $\{S(t,\tau) \in \mathcal{L}(H) \cap \mathcal{L}(E): t \geq \tau \in \mathbb{R}\}$, \begin{equation} \label{CP}
S(t,\tau) g_\tau = g(t, \tau, g_\tau) = \textup{col}\, (u, v, w)(t, \tau, g_\tau) \end{equation} which is called the nonautonomous Hindmaersh-Rose process. Moreover, there exists a time-invariant pullback absorbing set in the space $H$, \begin{equation} \label{naB}
M^*_H = \{g \in H: \|g \|^2 \leq K_1\} \end{equation} where $K_1$ is a positive constant independent of $\tau$ and $t$ in the sense that for any given bounded set $B \subset H$, \begin{equation} \label{pkab}
S(\tau, \tau - t)B \subset M^*_H, \quad \textup{for}\; \; t \geq T_B,
\end{equation} where the constant $T_B > 0$ depend only on $\|B\| =\sup_{g \in B} \|g\|$. \end{lemma}
\begin{proof}
Take the $H$ inner-product $\langle \eqref{napb}, (c_1 u, v, w) \rangle$ with constant $c_1 > 0$ to obtain \begin{equation} \label{Sab}
\begin{split}
&\frac{1}{2}\frac{d}{dt} \left(c_1 \|u\|^2+ \|v\|^2 + \|w\|^2\right) + \left(c_1 d_1 \|\nabla u\|^2 + d_2\|\nabla v\|^2 + d_3\|\nabla w\|^2\right) \\
= &\, \int_\Omega c_1 (au^3 - bu^4 +uv - uw + Ju + u p_1 (t, x))\,dx\\
+ &\, \int_\Omega (\alpha v - \beta u^2 v - v^2 + v p_2(t, x) + q (u - c)w - rw^2 + w p_3 (t, x))\, dx \\
\leq &\, \int_\Omega c_1 (au^3 - bu^4 +uv - uw + Ju + u p_1 (t, x))\,dx\\
+ &\, \int_\Omega \left[\left(2\alpha^2 + \frac{1}{2}\beta^2 u^4 - \frac{3}{8}v^2 \right) + \left(\frac{q^2}{r}(u^2 + c^2) - \frac{1}{2}rw^2\right)+ vp_2 + wp_3\right] dx \\
\leq &\, \int_\Omega c_1 (au^3 - bu^4 +uv - uw + Ju + u p_1 (t, x))\,dx\\
+ &\, \int_\Omega \left[\left(2\alpha^2 + \frac{1}{2}\beta^2 u^4 - \frac{3}{8}v^2 \right) + \left(\frac{q^2}{r}(u^2 + c^2) - \frac{1}{2}rw^2\right) \right] dx \\
+&\,\int_\Omega \left[ \frac{1}{8}v^2 + 2|p_2(t,x)|^2 + \frac{1}{8}rw^2 + \frac{2}{r}|p_3(t,x)|^2\right] dx.
\end{split} \end{equation} Choose the positive constant in \eqref{Sab} to be $c_1 = \frac{1}{b}(\beta^2 + 3)$ so that $$
- c_1\int_\Omega bu^4\, dx + \int_\Omega \beta^2 u^4\, dx \leq -3 \int_\Omega u^4\, dx. $$ Note that $$
\int_\Omega c_1 a u^3\,dx \leq \frac{3}{4}\int_\Omega u^4\,dx+\frac{1}{4} (c_1 a)^4 |\Omega | \leq \int_\Omega u^4 \,dx + (c_1 a)^4 |\Omega |, $$ and \begin{gather*}
\int_\Omega c_1 (uv - uw + Ju + up_1 (t,x))\,dx \leq \int_\Omega \left[2(c_1 u)^2 + \frac{1}{8}v^2 + \frac{(c_1 u)^2}{r}+\frac{1}{4} rw^2 \right. \\
\left. +\frac{1}{2}\left((c_1 u)^2 + J^2 +(c_1 u)^2 +|p_1(t,x)|^2\right)\right] dx. \end{gather*} The collection of all integral terms of $u^2$ in the above inequality and in \eqref{Sab} satisfies $$
\int_\Omega \left(2(c_1u)^2 + \frac{(c_1 u)^2}{r}+ (c_1 u)^2 + \frac{q^2}{r}u^2 \right)dx \leq \int_\Omega u^4 \,dx+ \left[c_1^2 \left(3 + \frac{1}{r} \right) + \frac{q^2}{r}\right]^2 |\Omega |. $$ Substitute these inequalities into \eqref{Sab}. Then we get \begin{equation} \label{SAB} \begin{split}
&\frac{1}{2}\frac{d}{dt} \left(c_1 \|u\|^2+ \|v\|^2 + \|w\|^2\right) + \left(c_1 d_1 \|\nabla u\|^2 + d_2\|\nabla v\|^2 + d_3\|\nabla w\|^2\right) \\[3pt]
\leq &\, \int_\Omega c_1 (au^3 - bu^4 +uv - uw + Ju + u p_1 (t, x))\,dx\\
+ &\, \int_\Omega \left[\left(2\alpha^2 + \frac{1}{2}\beta^2 u^4 - \frac{3}{8}v^2 \right) + \left(\frac{q^2}{r}(u^2 + c^2) - \frac{1}{2}rw^2\right) \right] dx \\
+&\,\int_\Omega \left[ \frac{1}{8}v^2 + 2|p_2(t,x)|^2 + \frac{1}{8}rw^2 + \frac{2}{r}|p_3(t,x)|^2\right] dx \\
\leq &\, \int_\Omega (2 - 3)u^4\,dx + \int_\Omega\left(\frac{1}{8} - \frac{3}{8} + \frac{1}{8}\right) v^2\,dx + \int_\Omega \left( \frac{1}{4} - \frac{1}{2} + \frac{1}{8}\right) rw^2\,dx \\
+ &\, \int_\Omega \left[\frac{1}{2}|p_1 (t,x)|^2+ 2|p_2 (t,x)|^2 + \frac{2}{r}|p_3 (t,x)|^2 \right] dx \\
+ &\,\left((c_1 a)^4 + J^2 + \left[c_1^2 \left(3 + \frac{1}{r}\right) + \frac{q^2}{r} \right]^2 + 2\alpha^2 + \frac{q^2c^2}{r}\right) |\Omega| \\
\leq &\, - \int_\Omega \left(u^4 (t,x) + \frac{1}{8} v^2 (t,x) + \frac{1}{8} rw^2(t,x)\right) dx + \left(2 + \frac{2}{r}\right)\| p(t)\|^2 + c_2 |\Omega |,
\end{split} \end{equation} where $$
c_2 = (c_1 a)^4 + J^2 + \left[c_1^2 \left(3 + \frac{1}{r}\right) + \frac{q^2}{r}\right]^2 + 2\alpha^2 + \frac{q^2c^2}{r}. $$ It follows that \begin{equation*}
\begin{split}
&\frac{d}{dt} (c_1 \|u(t)\|^2 + \|v(t)\|^2 +\|w(t)\|^2) + 2d (c_1 \|\nabla u\|^2 + \|\nabla v\|^2 + \|\nabla w\|^2) \\[5pt]
+ &\,\int_\Omega \left(2u^4 (t, x) + \frac{1}{4}v^2 (t,x) + \frac{1}{4}rw^2(t,x)\right) dx \leq 4 \left(1 + \frac{1}{r}\right)\| p(t)\|^2 + 2c_2 |\Omega |,
\end{split} \end{equation*} where $d = \min \{d_1, d_2, d_3\}$ and we used $$2u^4 \geq \frac{1}{4}\left(c_1 u^2 - \frac{c_1^2}{32}\right).$$ Therefore, \begin{equation} \label{nap} \begin{split}
&\frac{d}{dt} (c_1 \|u(t)\|^2 + \|v(t)\|^2 +\|w(t)\|^2) + 2d (c_1 \|\nabla u\|^2 + \|\nabla v\|^2 + \|\nabla w\|^2) \\[8pt]
+ \frac{1}{4}&\,(c_1 \|u(t)\|^2 + \|v(t)\|^2 + r\|w(t)\|^2) \leq 4 \left(1 + \frac{1}{r}\right)\| p(t)\|^2 + \left(\frac{c_1^2}{128} + 2c_2\right) |\Omega| \end{split} \end{equation} for $t \in [\tau, T_{max})$, the maximum time interval of existence. Set $$
\delta = \frac{1}{4} \min \{1, r\}. $$ Then the Gronwall inequality applied to the inequality reduced from \eqref{nap}, \begin{equation*}
\begin{split}
&\frac{d}{dt} (c_1 \|u(t)\|^2 + \|v(t)\|^2 +\|w(t)\|^2) + \delta (c_1 \|u(t)\|^2 + \|v(t)\|^2 + \|w(t)\|^2) \\[8pt]
\leq &\, 4 \left(1 + \frac{1}{r}\right)\| p(t)\|^2 + \left(\frac{c_1^2}{128} + 2c_2\right) |\Omega|,
\end{split} \end{equation*} shows that \begin{equation} \label{glab}
\begin{split}
&c_1 \|u(t)\|^2 + \|v(t)\|^2 +\|w(t)\|^2 \leq e^{- \delta t}(c_1 \|u_\tau \|^2 + \|v_\tau \|^2 +\|w_\tau \|^2) \\[8pt]
+ 4 &\, \left(1 + \frac{1}{r}\right) \int_\tau^t e^{- \delta (t-s)}\| p(s)\|^2\, ds + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega|, \quad t \in [\tau, T_{max}).
\end{split} \end{equation} By the assumption \eqref{tbd} on the translation boundedness of the external input terms and the upper bound estimate \eqref{glab}, the weak solutions will never blow up at any finite time so that $T_{max} = +\infty$ for all $\tau \in \mathbb{R}$ and any initial data $g_\tau \in H$. Thus the global existence in time of the weak solutions in the space $H$ is proved. Together with the uniqueness and the continuous dependence of $(t, \tau, g_\tau)$ which can be shown, the statement of the continuous evolution process $S(t, \tau)$ in \eqref{CP} is proved.
In order to prove the claimed existence of a pullback absorbing set, we can exploit the bounded translation property \eqref{tbd} of the time-dependent forcing terms to treat the integral in \eqref{glab} on the time interval $[\tau, t + \tau]$, or equivalently the time interval $[\tau - t, \tau]$, for $t > 0$, as follows:
\begin{equation} \label{pullab}
\begin{split}
&c_1\|u(t+\tau)\|^2 + \|v(t+\tau)\|^2 + \|w(t+\tau)\|^2 \\[5pt]
\leq &\, e^{-\delta t}(c_1\|u(\tau)\|^2 + \|v(\tau)\|^2 + \|w(\tau)\|^2) + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega |\\
+ &\, 4 \left(1 + \frac{1}{r}\right) \int_\tau^{t+\tau} e^{- \delta (t+\tau - s)}\| p(s)\|^2\, ds \\
\leq &\, e^{-\delta t}(c_1\|u(\tau)\|^2 + \|v(\tau)\|^2 + \|w(\tau)\|^2) + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega |\\
+ &\, 4 \left(1 + \frac{1}{r}\right) \sum_{k = 0}^{\infty} \int_{t+\tau-k-1}^{t+\tau -k} e^{- \delta (t+\tau - s)}\| p(s)\|^2\,ds \\
\leq &\, e^{-\delta t}(c_1\|u(\tau)\|^2 + \|v(\tau)\|^2 + \|w(\tau)\|^2) + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega |\\
+ &\, 4 \left(1 + \frac{1}{r}\right) \sum_{k = 0}^{\infty} e^{-k \delta}\left(\|p_1 \|^2_{L^2_b} + \|p_2 \|^2_{L^2_b} + \|p_3 \|^2_{L^2_b}\right) \\
= &\, e^{-\delta t}(c_1\|u_\tau\|^2 + \|v_\tau\|^2 + \|w_\tau\|^2) + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega |\\
+ &\, 4 \left(1 + \frac{1}{r}\right) \frac{1}{1 - e^{-\delta}}\left(\|p_1 \|^2_{L^2_b} + \|p_2 \|^2_{L^2_b} + \|p_3 \|^2_{L^2_b}\right).
\end{split} \end{equation} It implies that the global weak solutions of the nonautonomous diffusive Hindmarsh-Rose system \eqref{napb} admit the estimate that, for any $t \geq \tau \in \mathbb{R}$, \begin{equation} \label{gest}
\|g(t)\|^2 \leq \frac{\max \{1, c_1\}}{\min \{1,c_1\}} e^{-\delta (t - \tau)}\|g(\tau)\|^2 + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega|
+ 4 \left(1 + \frac{1}{r}\right) \frac{\|p\|^2_{L^2_b}}{1 - e^{-\delta}}. \end{equation} Hence, for any $\tau - t \leq \tau \in \mathbb{R}$ with $t > 0$, it holds that \begin{equation} \label{ttau}
\|g(\tau)\|^2 \leq \frac{\max \{1, c_1\}}{\min \{1,c_1\}} e^{-\delta t}\|g(\tau - t)\|^2 + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega|
+ 4 \left(1 + \frac{1}{r}\right) \frac{\|p\|^2_{L^2_b}}{1 - e^{-\delta}}. \end{equation} Since $$
\lim_{t \to \infty} e^{-\delta (t - \tau)}\|g(\tau)\|^2 = 0 $$ uniformly for $g(\tau) = g_\tau$ in any given bounded set $B \subset H$ in regard to \eqref{gest}, and $$
\lim_{t \to \infty} e^{-\delta t}\|g(\tau - t)\|^2 = 0 $$ uniformly for $g(\tau - t)$ in any given bounded set $B \subset H$ in regard to \eqref{ttau}, there exists a pullback absorbing set as claimed in \eqref{naB} with the constant \begin{equation} \label{M1}
K_1 = 1 + \frac{1}{\delta}\left(\frac{c_1^2}{128} + 2c_2\right) |\Omega|
+ 4 \left(1 + \frac{1}{r}\right) \frac{\|p\|^2_{L^2_b}}{1 - e^{-\delta}} \end{equation} which is independent of initial time and initial state in $H$. Therefore, the pullback absorbing property \eqref{pkab} for any given bounded set $B \subset H$ is proved: $$
S(\tau, \tau- t)B \subset M^*_H, \quad \textup{for all}\;\; t \geq T_B, $$ and \begin{equation} \label{TB}
T_B = \frac{1}{\delta}\log^+ \left(\frac{\max \{1, c_1\}}{\min \{1,c_1\}} \|B\|^2 \right) > 0 \end{equation}
depends only on $\|B\|$. The proof is completed. \end{proof}
\begin{lemma} \label{naac}
For the nonautonomous diffusive Hindmarsh-Rose system \eqref{napb}, there also exists a time-invariant pullback absorbing set in the space $E$, \begin{equation} \label{acB}
M^*_E = \{g \in E: \|g\|^2_E \leq K_2\}, \end{equation} where $K_2$ is a positive constant, such that for any given bounded set $B \subset H$, \begin{equation} \label{naE}
S(\tau, \tau - t)B \subset M^*_E \quad \textup{for all}\;\;\, t \geq T_B + 1, \end{equation} for any $\tau \in \mathbb{R}$, where the constant $T_B$ is given in \eqref{TB}. \end{lemma}
\begin{proof}
Take the $H$ inner-product $\langle \eqref{napb}, - \Delta g(t) \rangle$ to obtain \begin{equation*}
\begin{split}
&\frac{1}{2}\frac{d}{dt} \left(\|\nabla u \|^2 + \|\nabla v\|^2 + \|\nabla w\|^2\right) + d_1 \|\Delta u\|^2 + d_2 \|\Delta v\|^2 + d_3 \|\Delta w\|^2 \\[3pt]
=&\,\int_\Omega (-au^2 \Delta u - 3bu^2|\nabla u\|^2 -v\Delta u + w \Delta u - J\Delta u - p_1(t,x)\Delta u)\, dx \\
+&\, \int_\Omega (-\alpha \Delta v + \beta u^2 \Delta v - |\nabla v|^2)\, dx + \int_\Omega (qc \Delta w- qu \Delta w - r|\nabla w |^2)\, dx \\
- &\, \int_\Omega (p_2(t, x) \Delta v + p_3(t, x) \Delta w)\,dx.
\end{split} \end{equation*} By using Young's inequality appropriately to treat the integral terms on the right-hand side of the above inequality, we can get \begin{equation} \label{acg}
\begin{split}
&\frac{d}{dt} \left(\|\nabla u \|^2 + \|\nabla v\|^2 + \|\nabla w\|^2\right) + d_1 \|\Delta u\|^2 + d_2 \|\Delta v\|^2 + d_3 \|\Delta w\|^2 \\[5pt]
+ &\, 6b \|u \nabla u\|^2 + 2\|\nabla v \|^2 + 2r \|\nabla w \|^2 \\
\leq &\, \frac{4}{d_1} \|v\|^2 + \frac{4}{d_1}\| w \|^2+ \frac{4a^2}{d_1} \|u\|^4_{L^4} + \frac{8J^2}{d_1} |\Omega | + \frac{8}{d_1} \|p_1 (t)\|^2 \\
+ &\, \frac{2\beta^2}{d_2} \|u \|^4_{L^4} + \frac{4\alpha^2}{d_2} |\Omega| + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{2q^2}{d_3} \|u\|^2 + \frac{4q^2c^2}{d_3} |\Omega| +\frac{4}{d_3} \|p_3(t)\|^2 \\
= &\, \frac{4}{d_1} \|v\|^2 + \frac{4}{d_1}\| w \|^2 + \frac{2q^2}{d_3} \|u\|^2 + \left(\frac{4a^2}{d_1} + \frac{2\beta^2}{d_2}\right) \|u\|^4_{L^4} \\
+&\, \left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega| + \frac{8}{d_1} \|p_1 (t)\|^2 + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{4}{d_3} \|p_3(t)\|^2.
\end{split} \end{equation} The Sobolev imbedding $H^1(\Omega) \hookrightarrow L^4 (\Omega)$ tells us that there is a positive constant $\rho > 0$ such that \begin{equation} \label{L4H1}
\|u\|^4_{L^4} \leq \rho (\|u\|^2 + \|\nabla u\|^2)^2 \leq 2\rho (\|u\|^4 +\|\nabla u \|^4). \end{equation} According to Lemma \ref{naab}, for any given bounded set $B \subset H$, we have \begin{equation} \label{rho}
\|u(t)\|^2 + \|v(t))\|^2 + \|w(t)\|^2 \leq K_1, \quad \textup{for any}\;\;t \geq T_B, \; g_\tau \in B. \end{equation} Then \eqref{acg} yields the following inequality that for any $t \geq T_B$ and $g_\tau \in B$, \begin{equation} \label{uniGW}
\begin{split}
&\frac{d}{dt} \left(\|\nabla u \|^2 + \|\nabla v\|^2 + \|\nabla w\|^2\right) + d_1 \|\Delta u\|^2 + d_2 \|\Delta v\|^2 + d_3 \|\Delta w\|^2 \\[8pt]
+ &\, 6b \|u \nabla u\|^2 + 2\|\nabla v \|^2 + 2r \|\nabla w \|^2 \\[7pt]
\leq &\, \max \left\{\frac{4}{d_1}, \frac{4q^2 c^2}{d_3}\right\} K_1 + \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \rho K_1^2 + \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \rho \|\nabla u\|^4 \\
+&\, \left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega| + \frac{8}{d_1} \|p_1 (t)\|^2 + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{4}{d_3} \|p_3(t)\|^2 .
\end{split} \end{equation} Hence we can apply the uniform Gronwall inequality \cite[Lemma D.3]{SY} to the following inequality reduced from \eqref{uniGW} on $\nabla g(t)= \text{col} \,(\nabla u(t), \nabla v(t), \nabla w(t))$, \begin{equation} \label{uGW}
\begin{split}
&\frac{d}{dt}\|\nabla g(t)\|^2 \leq \rho \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \|\nabla g\|^2 \|\nabla g\|^2 \\
+&\, \max \left\{\frac{4}{d_1}, \frac{4q^2 c^2}{d_3}\right\} K_1 + \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \rho K_1^2 \\
+&\,\left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega| + \frac{8}{d_1} \|p_1 (t)\|^2 + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{4}{d_3} \|p_3(t)\|^2
\end{split} \end{equation} which is written in the form \begin{equation} \label{rs}
\frac{d\sigma}{dt} \leq \xi \,\sigma + h, \quad \text{for}\;\; t \geq T_B, \; g_\tau \in B, \end{equation} where \begin{gather*}
\sigma (t) = \|\nabla g(t)\|^2, \quad \xi (t) = \rho \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \|\nabla g\|^2, \quad \text{and} \\
h(t) = \max \left\{\frac{4}{d_1}, \frac{4q^2 c^2}{d_3}\right\} K_1 + \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \rho K_1^2 \\
+ \left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega| + \frac{8}{d_1} \|p_1 (t)\|^2 + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{4}{d_3} \|p_3(t)\|^2. \end{gather*} For $t \geq T_B$, by integration of the inequality \eqref{nap} we can deduce that \begin{gather*}
\int_t^{t+1} 2d(c_1 \|\nabla u(s)\|^2 +\|\nabla v(s)\|^2+ \|\nabla w(s)\|^2)\, ds\\
\leq c_1 \|u(t)\|^2 + \|v(t)\|^2 + \|w(t)\|^2 + 4\left(1 + \frac{1}{r}\right) \int_t^{t+1} \|p(s)\|^2 ds + \left(\frac{c_1^2}{128} + 2c_2\right)|\Omega | \\
\leq \max \{1, c_1\} K_1+ 4\left(1 + \frac{1}{r}\right) \| p \|^2_{L^2_b} + \left(\frac{c_1^2}{128} + 2c_2\right)|\Omega |, \end{gather*}
where $ \| p \|^2_{L^2_b} = \sum_{i=1}^3 \| p_i \|^2_{L^2_b}$. Denote by $$
N_1 = \frac{1}{2d \min \{1, c_1\}} \left[\max \{1, c_1\} K_1+ 4\left(1 + \frac{1}{r}\right) \| p \|^2_{L^2_b} + \left(\frac{c_1^2}{128} + 2c_2\right)|\Omega |\right] $$ and \begin{align*}
N_2 = \max \left\{\frac{4}{d_1}, \frac{4q^2 c^2}{d_3}\right\} K_1 + \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right) \rho K_1^2 + \left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega|. \end{align*} Then we have \begin{equation} \label{gsphih}
\begin{split}
\int_t^{t+1} \sigma (s)\,ds &\leq N_1, \\
\int_t^{t+1} \xi (s)\, ds &\leq \rho \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right)N_1, \\
\int_t^{t+1} h(s)\,ds & \leq N_2 + \max \left\{\frac{8}{d_1}, \, \frac{4}{d_2}, \,\frac{4}{d_3}\right\}\|p\|^2_{L^2_b}.
\end{split} \end{equation} Thus the uniform Gronwall inequality applied to \eqref{rs} shows that \begin{equation} \label{BE}
\|\nabla g(t)\|^2 \leq \left(N_1 + N_2 + \max \left\{\frac{8}{d_1}, \, \frac{4}{d_2}, \,\frac{4}{d_3}\right\}\|p\|^2_{L^2_b} \right) \exp \left\{ \rho \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right)N_1\right\}, \end{equation} for all $t \geq T_B + 1$ and all $g_\tau \in B$. Therefore, the claim \eqref{acB} of a pullback absorbing ball $M^*_E$ in the space $E$ is proved and the constant $K_2$ is given by $$
K_2 = K_1 + \left(N_1 + N_2 + \max \left\{\frac{8}{d_1}, \, \frac{4}{d_2}, \,\frac{4}{d_3}\right\}\|p\|^2_{L^2_b} \right) \exp \left\{ \rho \left(\frac{8a^2}{d_1} + \frac{4\beta^2}{d_2}\right)N_1\right\}. $$ Indeed, for any given bounded set $B \subset H$, we have $$
S(\tau, \tau - t)B \subset M^*_E \quad \text{for all}\;\; t \geq T_B + 1. $$ The proof is completed. \end{proof} Now we prove the first main result of this paper. \begin{theorem} \label{PAC}
Under the assumption \eqref{tbd}, for any positive parameters and $c \in \mathbb{R}$ in the Hindmarsh-Rose equations \eqref{uq}-\eqref{wq}, there exists a pullback attractor $\mathcal{A} = \{\mathcal{A} (\tau)\}_{\tau \in \mathbb{R}}$ in $H$ for the nonautonomous Hindmarsh-Rose process $\{S(t, \tau)\}_{t \geq \tau \in \mathbb{R}}$. \end{theorem}
\begin{proof}
By Lemma \ref{naab}, there exists a pullback absorbing set $M^*_H$ in $H$ for the solution process $\{S(t, \tau): t \geq \tau \in \mathbb{R}\}$ of the nonautononous Hindmarsh-Rose system \eqref{napb} so that the first condition in Proposition \ref{PA} is satisfied.
By Lemma \ref{naac} and the compact embedding $E \hookrightarrow H$, the existence of a pullback absorbing set $M^*_E$ in $E$ for this nonautonomous process shows that any sequence $\{S(\tau, \tau - t_k)g_k\}_{k=1}^\infty$, where $t_k \to \infty$ and $\{g_k\}$ in any given bounded set of $H$ has a convergent subsequence.
Thus the second condition of the pullback asymptotic compactness in Proposition \ref{PA} is also satisfied.
Then by Proposition \ref{PA}, there exists a pullback attractor $\mathcal{A} = \{\mathcal{A}(\tau)\}_{\tau \in \mathbb{R}}$, $$
\mathcal{A}(\tau) = \bigcap_{s \geq 0} \, \overline{\bigcup_{t \geq s} S (\tau, \tau - t)M^*_H} \, , $$ for this nonautonomous Hindmarsh-Rose process. \end{proof}
\section{\textbf{The Existence of Pullback Exponential Attractor}}
In this section, we shall prove the existence of a pullback exponential attractor for the nonautonomous Hindmarsh-Rose process based on Proposition \ref{PeA}. The key leverage is to prove the smoothing Lipschitz continuity of this nonautonomous process with respect to the initial data.
\begin{theorem}[Smoothing Lipschitz Continuity] \label{LpHE}
For the nonautonomous Hindmarsh-Rose process $\{S(t, \tau)\}_{t \geq \tau \in \mathbb{R}}$ in \eqref{CP} generated by the weak solutions of the nonautonomous Hinsmarsh-Rose system \eqref{napb}, there exists a constant $\kappa > 0$ such that \begin{equation} \label{lpH}
\sup_{\tau \in \mathbb{R}} \|S(\tau, \tau - T_{M^*_E})g_\tau - S(\tau,\tau- T_{M^*_E})\Tilde{g}_\tau \|_E \leq \kappa \|g_\tau - \Tilde{g}_\tau\|, \quad \text{for}\;\, g_\tau, \Tilde{g}_\tau \in M^*_E , \end{equation} where $T_{M^*_E} > 0$ is the time when all the pullback solution trajectories of \eqref{napb} starting from the set $M^*_E$ in \eqref{acB} permanently enter the set $M^*_E$ itself shown in Lemma \ref{naac}. \end{theorem}
\begin{proof}
It is equivalent to prove that \begin{equation} \label{lpHe}
\sup_{\tau \in \mathbb{R}} \|S(\tau + T_{M^*_E}, \tau)g_\tau - S(\tau + T_{M^*_E}, \tau)\Tilde{g}_\tau \|_E \leq \kappa \|g_\tau - \Tilde{g}_\tau\|, \quad g_\tau, \Tilde{g}_\tau \in M^*_E. \end{equation} Denote two solutions with any given initial data $g_\tau$ and $\Tilde{g}_\tau$ by $g(t) = (u(t), v(t), w(t))$ and $\Tilde{g} (t) = (\Tilde{u} (t), \Tilde{v} (t), \Tilde{w} (t))$, respectively. Denote the difference by $\Pi (t) = g(t) - \Tilde{g} (t) = (U(t), V(t), W(t))$. Then $\Pi (t)$ is the solution of the following intial value problem \begin{equation} \label{vpeq}
\begin{split}
\frac{d\Pi}{dt} = &\, A\Pi + f(g) -f(\Tilde{g}), \quad t \geq \tau \in \mathbb{R}, \\[3pt]
&\Pi (\tau) = g_\tau - \Tilde{g}_\tau.
\end{split} \end{equation}
Step 1. Take the inner-product $\langle \eqref{vpeq}, \Pi (t) \rangle$ through three component equations \eqref{uq}-\eqref{wq}. For the first component equation of $\Pi (t) = g(t) - \Tilde{g} (t)$, we get \begin{equation} \label{Uq}
\begin{split}
&\frac{1}{2} \frac{d}{dt} \|U(t)\|^2 + d_1\|\nabla U(t)\|^2 = \langle f_1 (g) - f_1 (\Tilde{g}), u - \Tilde{u} \rangle \\[2pt]
= &\, \int_\Omega \left(a(u - \Tilde{u})^2 (u + \Tilde{u}) - b(u - \Tilde{u})^2 (u^2 \Tilde{u} + u \Tilde{u} + \Tilde{u}^2)\right) dx \\
&\, + \int_\Omega ((v - \Tilde{v})(u - \Tilde{u}) - (w - \Tilde{w})(u - \Tilde{u})) \, dx \\
\leq &\, \int_\Omega (u - \Tilde{u})^2 \left[ a(u + \Tilde{u}) - b (u^2 + u \Tilde{u} + \Tilde{u}^2)\right] dx \\[5pt]
&\, + \| u - \Tilde{u} \| (\|v - \Tilde{v} \| + \|w - \Tilde{w} \|) \\[5pt]
\leq &\, \int_\Omega (u - \Tilde{u})^2 \left[ a(u + \Tilde{u}) - b (u^2 + u \Tilde{u} + \Tilde{u}^2)\right] dx + 2 \|g - \Tilde{g} \|^2
\end{split} \end{equation} and by Young's inequality we have \begin{gather*}
a (u + \Tilde{u}) - - b (u^2 + u \Tilde{u} + \Tilde{u}^2) = [a (u + \Tilde{u}) - b u\Tilde{u}] - b(u^2 + \Tilde{u}^2) \\[5pt]
\leq \left(\frac{b}{4} u^2 + \frac{a^2}{b}\right) + \left(\frac{b}{4} \Tilde{u}^2 + \frac{a^2}{b}\right) + \frac{b}{2} (u^2 + \Tilde{u}^2) - b(u^2 + \Tilde{u}^2) \leq - \frac{b}{4} (u^2 + \Tilde{u}^2) + \frac{2a^2}{b}. \end{gather*} It follows that \begin{equation} \label{Ue}
\begin{split}
& \frac{d}{dt} \|U(t)\|^2 \leq \frac{d}{dt} \|U(t)\|^2 + 2d_1\|\nabla U(t)\|^2 \\[5pt]
\leq &\, 2\int_\Omega (u - \Tilde{u})^2 \left( - \frac{b}{4} (u^2 + \Tilde{u}^2) + \frac{2a^2}{b}\right) dx + 4 \|g - \Tilde{g}\|^2 \\
\leq &\, \int_\Omega (u - \Tilde{u})^2 \left( - \frac{b}{2} (u^2 + \Tilde{u}^2)\right) dx + \frac{4a^2}{b} \|u - \Tilde{u} \|^2 + 4 \|g - \Tilde{g}\|^2 \\
\leq &\, - \frac{b}{2} \int_\Omega (u - \Tilde{u})^2 (u^2 + \Tilde{u}^2)\, dx + 4\left( 1+ \frac{a^2}{b}\right) \|\Pi (t) \|^2.
\end{split} \end{equation} Similarly, for the second and third components of $\Pi(t) = g(t) - \Tilde{g} (t))$, we get \begin{equation} \label{Ve}
\begin{split}
\frac{d}{dt} &\, \|V(t)\|^2 \leq \frac{d}{dt} \|V(t)\|^2 + 2d_2\|\nabla V(t)\|^2 \leq 2\langle \psi (u) - \psi (\Tilde{u}) - (v - \Tilde{v}), v - \Tilde{v} \rangle \\[3pt]
= &\, 2\int_\Omega \left( - \beta (u^2- \Tilde{u}^2) - (v - \Tilde{v}) \right) (v - \Tilde{v})\, dx \\
\leq &\, 2\int_\Omega \left(- \beta (u - \Tilde{u}) u (v - \Tilde{v}) - \beta (u - \Tilde{u})\Tilde{u} (v - \Tilde{v})\right) dx \\
\leq &\, \int_\Omega \left(\frac{bu^2}{2} (u - \Tilde{u})^2 + \frac{b\Tilde{u}^2}{2} (u - \Tilde{u})^2\right) dx + \frac{4\beta}{b} \|v - \Tilde{v} \|^2 \\
\leq &\; \frac{b}{2} \int_\Omega \, (u^2 + \Tilde{u}^2) (u - \Tilde{u})^2 \, dx + \frac{4\beta}{b}\, \|\Pi (t) \|^2
\end{split} \end{equation} and \begin{equation} \label{We}
\begin{split}
\frac{d}{dt} &\, \|W(t)\|^2 \leq \frac{d}{dt} \|W(t)\|^2 + 2d_3\|\nabla W(t)\|^2 \leq 2\langle q(u - \Tilde{u}) - r(w - \Tilde{w}), w - \Tilde{w} \rangle \\[3pt]
= &\, 2\int_\Omega \left(q (u- \Tilde{u}) - r(w - \Tilde{w}) \right) (w - \Tilde{w})\, dx \\[3pt]
\leq &\, q \|u - \Tilde{u}\|^2 + (q + 2r) \|w - \Tilde{w} \|^2 \leq 2 (q + r)\, \|\Pi(t) \|^2.
\end{split} \end{equation} Sum up the inequalities \eqref{Ue}, \eqref{Ve} and \eqref{We} with a cancellation of the first terms on the rightmost side of \eqref{Ue} and \eqref{Ve}. Then we obtain \begin{equation} \label{vpp}
\begin{split}
&\frac{d}{dt} \|\Pi \|^2 + 2(d_1 \|\nabla U \|^2 + d_2 \|\nabla V \|^2 + d_3\|\nabla W \|^2) = 2\langle f(g) - f(\Tilde{g}), \Pi \rangle \\[3pt]
\leq &\, \left(4\left[1 + \frac{1}{b} (a^2 + \beta)\right] + 2(q + r) \right)\|\Pi \|^2.
\end{split} \end{equation} It follows that, for any $g_\tau, \Tilde{g}_\tau \in M^*_E$ and indeed for any $g_\tau, \Tilde{g}_\tau \in H$, \begin{equation} \label{piq}
\frac{d}{dt} \, \|\Pi\|^2 \leq C_*\|\Pi \|^2 \end{equation} where the constant $C_* = 4\left(1 + \frac{1}{b} (a^2 + \beta)\right) +2(q + r)$. Consequently, \begin{equation} \label{PiH}
\begin{split}
&\|S(t + \tau, \tau) g_\tau - S(t + \tau,\tau)\Tilde{g}_\tau \|^2 = \|\Pi (t + \tau)\|^2 \\[6pt]
\leq &\, e^{C_*t}\|\Pi (\tau)\|^2 = e^{C_*t}\|g_\tau-\Tilde{g}_\tau\|^2, \quad t \geq 0, \;\, \tau \in \mathbb{R}.
\end{split} \end{equation}
Step 2. In oder to prove \eqref{lpHe}, we express the weak solution of \eqref{vpeq} by using the mild solution formula, \begin{equation} \label{milds}
\Pi (t + \tau) = e^{At} \, \Pi (\tau) + \int_\tau^{t+\tau} e^{A(t +\tau - s)} (f(g(s)) - f(\Tilde{g} (s))\,ds, \quad t \geq 0, \end{equation} where the $C_0$-semigroup $\{e^{At}\}_{t \geq 0}$ is generated by the operator $A$ defined in \eqref{opA}. By the regularity property of the analytic $C_0$-semigroup $\{e^{At}\}_{t \geq0}$ \cite{Rb,SY}, it holds that $e^{At}: H \to E$ for $t > 0$ and there is a constant $C_1 > 0$ such that \begin{equation} \label{eAt}
\|e^{At}\|_{\mathcal{L}(H, \,E)} \leq C_1 \, t^{-1/2}, \quad t > 0. \end{equation} Thus we have \begin{equation} \label{mdineq}
\begin{split}
\|\Pi (t + \tau)\|_E \leq &\, \|e^{At}\|_{\mathcal{L}(H, \,E)} \|\Pi (\tau)\| + \int_\tau^{t+\tau} \|e^{A(t +\tau - s)}\|_{\mathcal{L}(H, E)}\| (f(g(s)) - f(\Tilde{g} (s))\|\,ds \\[4pt]
\leq &\, \frac{C_1}{\sqrt{t}} \,\|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \, \| (f(g(s)) - f(\Tilde{g} (s))\|\,ds, \quad t > 0.
\end{split} \end{equation} Here we estimate the norm of the difference in the last integral of \eqref{mdineq}, \begin{align*}
&\|f(g) - f(\Tilde{g})\|^2 = \|\varphi (u) - \varphi(\Tilde{u}) + (v - \Tilde{v}) -(w-\Tilde{w})\|^2 \\[3pt]
+ &\, \|\psi (u) - \psi (\Tilde{u})- (v - \Tilde{v})\|^2 + \|q(u - \Tilde{u}) - r(w-\Tilde{w})\|^2 \\[3pt]
\leq &\, 3\|\varphi (u) - \varphi (\Tilde{u})\|^2 + 2\|\psi (u) - \psi (\Tilde{u})\|^2 + 2q \|u -\Tilde{u}\|^2 + 5 \|v -\Tilde{v} \|^2 + (3+2r) \|w-\Tilde{w}\|^2 \\[3pt]
= &\, (3a^2 + 2\beta^2)\|u^2 - \Tilde{u}^2\|^2 + 3b^2 \|u^3 - \Tilde{u}^3 \|^2 \\[3pt]
+ &\, 2q \|u -\Tilde{u}\|^2 + 5 \|v -\Tilde{v} \|^2 + (3+2r) \|w-\Tilde{w}\|^2 \\[3pt]
\leq &\, (6a^2 + 4\beta^2)(\|u\|^2 +\|\Tilde{u}\|^2)\|u - \Tilde{u}\|^2 + 3b^2 \|u^2 + u\Tilde{u} + \Tilde{u}^2\|^2 \|u - \Tilde{u}\|^2 \\[3pt]
+ &\, 2q \|u -\Tilde{u}\|^2 + 5 \|v -\Tilde{v} \|^2 + (3+2r) \|w-\Tilde{w}\|^2 \\[3pt]
\leq &\, (6a^2 + 4\beta^2)(\|u\|^2 +\|\Tilde{u}\|^2)\|u - \Tilde{u}\|^2 + 3b^2 \|u^2 + u\Tilde{u} + \Tilde{u}^2\|^2 \|u - \Tilde{u}\|^2 \\[3pt]
+ &\, 2q \|u -\Tilde{u}\|^2 + 5 \|v -\Tilde{v} \|^2 + (3+2r) \|w-\Tilde{w}\|^2, \end{align*}
where in the term $3b^2 \|u^2 + u\Tilde{u} + \Tilde{u}^2\|^2 \|u - \Tilde{u}\|^2$, we deduce that \begin{align*}
&\|u^2 + u\Tilde{u} + \Tilde{u}^2\|^2 = \int_\Omega (u^2 + u\Tilde{u} + u^2)^2 \, dx \\
=&\, \int_\Omega (u^4 + 3u^2\Tilde{u}^2 + \Tilde{u}^4 + 2u\Tilde{u} (u^2 + \Tilde{u}^2))\, dx \\
\leq &\, \left(u^4 + \Tilde{u}^4 + \frac{3}{2} (u^4 + \Tilde{u}^4) + u^2 \Tilde{u}^2 + (u^2 + \Tilde{u}^2)^2 \right) dx \\
\leq &\, 5\int_\Omega (u^4 + \Tilde{u}^4)\, dx = 5 \left(\|u\|^4_{L^4} + \|\Tilde{u}\|^4_{L^4}\right). \end{align*} Substitute the above inequalities into the integral term in \eqref{mdineq} and use the embedding inequality \eqref{L4H1} to obtain \begin{equation} \label{mdest} \begin{split}
&\|\Pi (t + \tau)\|_E \leq \frac{C_1}{\sqrt{t}} \, \|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}}\, \| (f(g(s)) - f(\Tilde{g} (s))\|\,ds \\
\leq &\, \frac{C_1}{\sqrt{t}} \,\|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \,(6a^2 + 4\beta^2)(\|u\|^2 +\|\Tilde{u}\|^2)\|u - \Tilde{u}\|^2\, ds \\
+ &\,\int_\tau^{t+\tau} \frac{C_1}{\sqrt{t + \tau - s}}\, 30b^2 \rho \left(\|u\|^4_{H^1} + \|\Tilde{u}\|^4_{H^1}\right)\|u - \Tilde{u}\|^2\,ds \\
+ &\, \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \,( 2q \|u -\Tilde{u}\|^2 + 5 \|v -\Tilde{v} \|^2 + (3+2r) \|w-\Tilde{w}\|^2 )\,ds, \: t \geq 0, \end{split} \end{equation} for any $\tau \in \mathbb{R}$.
Note that from \eqref{gest} and \eqref{acB}, since both $g_\tau$ and $\Tilde{g}_\tau$ are in $M^*_E$, we have \begin{equation} \label{gbd}
\|u(t+\tau)\|^2 \leq \|g(t+\tau)\|^2 \leq G_1 = \frac{\max \{1,c_1\}}{\min \{1, c_1\}} K_2 + K_1, \quad \text{for} \;\, t \geq 0,\tau \in \mathbb{R}, \end{equation} where the positive constants $K_1$ and $K_2$ are given in \eqref{M1} and \eqref{acB} respectively, and independent of $t$ and $\tau$.
Step 3. We want to improve the inequality \eqref{acg}: \begin{equation*} \begin{split}
&\frac{d}{dt} \left(\|\nabla u \|^2 + \|\nabla v\|^2 + \|\nabla w\|^2\right) + d_1 \|\Delta u\|^2 + d_2 \|\Delta v\|^2 + d_3 \|\Delta w\|^2 \\[7pt]
+ &\, 6b \|u \nabla u\|^2 + 2\|\nabla v \|^2 + 2r \|\nabla w \|^2 \\
\leq &\, \frac{4}{d_1} \|v\|^2 + \frac{4}{d_1}\| w \|^2 + \frac{2q^2}{d_3} \|u\|^2 + \left(\frac{4a^2}{d_1} + \frac{2\beta^2}{d_2}\right) \|u\|^4_{L^4} \\
+&\, \left(\frac{8J^2}{d_1} + \frac{4\alpha^2}{d_2} + \frac{4q^2c^2}{d_3}\right) |\Omega| + \frac{8}{d_1} \|p_1 (t)\|^2 + \frac{4}{d_2} \|p_2 (t)\|^2 + \frac{4}{d_3} \|p_3(t)\|^2. \end{split} \end{equation*} Specifically we need to further treat the following term on the right-hand side of the above \eqref{acg}, $$
\left(\frac{4a^2}{d_1} +\frac{2\beta^2}{d_2}\right) \|u\|^4_{L^4} $$ by using the Gagliardo-Nirenberg inequality \eqref{GN} for the interpolation spaces $$
L^1(\Omega) \hookrightarrow L^2 (\Omega) \hookrightarrow H^1 (\Omega). $$ It implies that there is a constant $C > 0$ and \begin{equation} \label{L12}
\|u^2\|^2 \leq C \|\nabla (u^2)\|^{6/5}\|u^2\|^{4/5}_{L^1}. \end{equation} because $-\frac{3}{2} =\theta (1 - \frac{3}{2}) - 3(1- \theta)$ with $\theta = 3/5$ and $1 - \theta = 2/5$. Therefore, the inequality \eqref{L12} and the Young's inequality imply that there exists a constant $0 < \varepsilon < b$ such that \begin{equation} \label{u4}
\begin{split}
&\left(\frac{4a^2}{d_1} + \frac{2\beta^2}{d_2}\right)\|u\|^4_{L^4} = \left(\frac{4a^2}{d_1} + \frac{2\beta^2}{d_2}\right) \|u^2\|^2 \\[2pt]
\leq &\, C\left(\frac{4a^2}{d_1} + \frac{2\beta^2}{d_2}\right)\|\nabla (u^2)\|^{6/5}\|u^2\|^{4/5}_{L^1}\leq \varepsilon \|\nabla u^2\|^2 + C_\varepsilon \|u^2\|_{L^1}^2 \\
= &\, 4\varepsilon \,\|u \nabla u\|^2 +C_\varepsilon \|u\|^4 \leq 4b \,\|u \nabla u\|^2 +C_\varepsilon \|u\|^4,
\end{split} \end{equation} where $C_\varepsilon > 0$ is a constant only depending on $\varepsilon$. Substitute \eqref{u4} into the above inequality \eqref{acg} to obtain \begin{equation} \label{nbg} \begin{split}
&\frac{d}{dt} \left(\|\nabla u \|^2 + \|\nabla v\|^2 + \|\nabla w\|^2\right) + d\left( \|\Delta u\|^2 + \|\Delta v\|^2 + \|\Delta w\|^2 \right) \\[9pt]
+ &\, 2b\, \|u \nabla u\|^2 + 2\|\nabla v \|^2 + 2r \|\nabla w \|^2 \leq G_2 (\|u\|^2 +\|v\|^2 + \| w \|^2) \\[7pt]
+&\, G_3 |\Omega| + \frac{8}{d} \left(\|p_1 (t)\|^2 + \|p_2 (t)\|^2 + \|p_3(t)\|^2 \right) + C_\varepsilon \|u\|^4, \quad t \geq \tau \in \mathbb{R}, \end{split} \end{equation} where $d = \min \{d_1, d_2, d_3\}$, $$
G_2 = \frac{1}{d} \max \{4, 2q^2\} \quad \text{and} \quad G_3 = \frac{1}{d} \, (8J^2 + 4\alpha^2 + 4q^2c^2). $$ Then the inequality \eqref{nbg} with \eqref{gbd} infers that \begin{equation} \label{gG} \begin{split}
\frac{d}{dt} \|\nabla (u, v, w) \|^2 &\; \leq G_2 \|(u, v, w)\|^2 + G_3 |\Omega| + \frac{8}{d} \,\|p(t)\|^2 + C_\varepsilon \|u\|^4\\[3pt]
&\; \leq G_1G_2 + G_3 |\Omega| + \frac{8}{d} \,\|p(t)\|^2 + C_\varepsilon K_1^2 \end{split} \end{equation} for any $t \geq \tau \in \mathbb{R}$. It follows that for any $0 \leq t \leq T_{M^*_E}$ we have \begin{equation} \label{gB}\
\begin{split}
&\|u(t+\tau)\|^2_{H^1 (\Omega)} = \|u(t+\tau)\|^2 + \|\nabla u(t+\tau)\|^2 \\[5pt]
\leq &\, \|u(t+\tau)\|^2 + \|\nabla u(\tau)||^2 + \int_\tau^{\tau + t} \left(G_1G_2 + G_3 |\Omega| + \frac{8}{d}\, \|p(s)\|^2 + C_\varepsilon K_1^2\right) ds \\
\leq &\, G_1 + \|\nabla u(\tau)\|^2 + t(G_1G_2 + G_3 |\Omega| + C_\varepsilon K_1^2) + \frac{8}{d} \int_\tau^{t+\tau} \|p(s)\|^2\, ds \\
\leq &\, G_1 + K_2 + T_{M^*_E}(G_1G_2 + G_3 |\Omega| + C_\varepsilon K_1^2) + \frac{8}{d} \left(T_{M^*_E} + 1\right) \|p\|^2_{L^2_b}.
\end{split} \end{equation}
Step 4. Finally we substitute \eqref{gbd} and \eqref{gB} into the inequality \eqref{mdest} for any two solutions $g(t)$ and $\Tilde{g} (t)$ of the nonautonomous system \eqref{napb} with initial states in $M^*_E$. Then for any $t > 0$ and $\tau \in \mathbb{R}$, it holds that \begin{equation} \label{PEst} \begin{split}
&\|\Pi (t + \tau)\|_E \leq \frac{C_1}{\sqrt{t}}\, \|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \,\| (f(g(s)) - f(\Tilde{g} (s))\|\,ds \\
\leq &\, \frac{C_1}{\sqrt{t}} \,\|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \, G_1(12a^2 + 8\beta^2)\|u - \Tilde{u}\|^2\, ds \\
+ &\,\int_\tau^{t+\tau} \frac{C_1}{\sqrt{t + \tau - s}}\, 30b^2 \rho \left(\|u\|^4_{H^1} + \|\Tilde{u}\|^4_{H^1}\right)\|u - \Tilde{u}\|^2\,ds \\
+ &\, \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \max \{2q, 5, 3+ 2r\}(\|u -\Tilde{u}\|^2 + \|v -\Tilde{v} \|^2 + \|w-\Tilde{w}\|^2)\,ds \\
\leq &\, \frac{C_1}{\sqrt{t}} \,\|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}}\, G_p \,\|g(s) - \Tilde{g} (s)\|^2\, ds \\[2pt]
\leq &\, \frac{C_1}{\sqrt{t}} \,\|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}}\, G_p \, e^{C_*(s-\tau)} \,\|g_\tau - \Tilde{g}_\tau\|^2\, ds \\[2pt]
\leq &\, \frac{C_1}{\sqrt{t}} \, \|g_\tau - \Tilde{g}_\tau\| + \int_{\tau}^{t + \tau} \frac{C_1}{\sqrt{t + \tau - s}} \, G_p \, e^{C_*(s-\tau)} 2\sqrt{G_1} \|g_\tau - \Tilde{g}_\tau\|\, ds \end{split} \end{equation} where we used \eqref{PiH} and \eqref{gbd} in the last two steps, and the positive constant $G_p$ is given by \begin{equation} \label{Gp}
\begin{split}
G_p = &\, G_1(12a^2 + 8\beta^2)+ \max \{2q, 5, 3+ 2r\} \\
+ &\, 60 b^2 \rho \left[ G_1 + K_2 + T_{M^*_E}(G_1G_2 + G_3 |\Omega|) + \frac{8}{d} (T_{M^*_E} + 1)\|p\|^2_{L^2_b}\right]^2
\end{split} \end{equation} which depends on the nonautonomous terms $p_i (t, x), i = 1, 2, 3$, and the permanently entering time $T_{M^*_E}$. Integrating the inequality \eqref{PEst} on the time interval $[\tau, \tau + T_{M^*_E}]$, here without of generality $T_{M^*_E} > 0$, we then obtain the result that \begin{equation} \label{PiE}
\begin{split}
&\|S(\tau + T_{M^*_E}, \tau) g_\tau - S(\tau + T_{M^*_E},\tau) \Tilde{g}_\tau \|_E = \|\Pi (\tau + T_{M^*_E})\|_E \\
\leq &\, C_1 \left(\frac{1}{\sqrt{T_{M^*_E}}} + 4\sqrt{G_1 T_{M^*_E}}\, \exp \left\{{C_* T_{M^*_E}}\right\} G_p \right) \|g_\tau - \Tilde{g}_\tau\|
\end{split} \end{equation} for any $g_\tau, \Tilde{g}_\tau \in M^*_E$ and any $\tau \in\mathbb{R}$. Therefore, \eqref{lpHe} and then \eqref{lpH} are proved with the uniform Lipschitz constant $$
\kappa = C_1 \left(\frac{1}{\sqrt{T_{M^*_E}}} + 4\sqrt{G_1 T_{M^*_E}}\, \exp \left\{C_* T_{M^*_E}\right\} G_p \right). $$ The proof of this theorem is completed. \end{proof}
After the challenging Theorem \ref{LpHE} has been proved, now we can prove the second main result of this paper.
\begin{theorem} \label{pbexA}
For the nonautonomous Hindmarsh-Rose process $\{S(t, \tau)_{t \geq \tau \in \mathbb{R}}$ generated by the nonautonomous Hindmarsh-Rose equations \eqref{uq}-\eqref{wq}, there exists a pullback exponential attractor $\mathscr{M} = \{\mathscr{M}(\tau)\}_{\tau\in \mathbb{R}}$ in the space H. \end{theorem}
\begin{proof}
We can apply Proposition \ref{PeA} to prove this theorem. Indeed Lemma \ref{naac} and Theorem \ref{LpHE} have shown that the first two conditions in that Proposition \ref{PeA} are satisfied with the pullback absorbing set $M^* = M^*_E$ in \eqref{ab} by the nonautonomous Hindmarsh-Rose process $S(t, \tau)_{t \geq \tau \in \mathbb{R}}$. Thus it suffices to show that the third condition of \eqref{Lpt1} and \eqref{Lpt2} in Proposition \ref{PeA} is satisfied.
Recall that the Hindmarsh-Rose process $S(t, \tau)$ is defined by \eqref{CP} and let $g(t, \tau, g_\tau)$ be the weak solution to the initial value problem of the nonautonomous Hindmarsh-Rose evolutionary equation \eqref{napb}. For any $t_1 < t_2$ with $|t_1 - t_2| \leq I$, where $I$ is any given positive constant, we can estimate the $H$-norm of the difference of two pullback solution trajectories $$
g^1 (t) = S(t, \tau - t_1)g_0 \quad \text{and} \quad g^2 (t) = S(t, \tau - t_2)g_0, \quad 0 \leq t_1 \leq t_2, \; g_0 \in H, $$ as follows.
Using the notation in \eqref{vpeq} but here $\Pi (t) = g^1 (t) - g^2 (t)$. Then $\Pi (t)$ is the solution of the initial value problem \begin{equation} \label{Steq}
\begin{split}
\frac{d\Pi}{dt} &\, = A\Pi + f(g^1) -f(g^2), \quad t \geq \tau - t_1 \in \mathbb{R}, \\[3pt]
&\Pi (\tau - t_1) = g_0 - S(\tau - t_1, \tau - t_2)g_0.
\end{split} \end{equation} By \eqref{piq}, we have \begin{equation} \label{Stpt}
\begin{split}
\frac{d\|\Pi\|^2}{dt} \leq C_* \|\Pi \|^2, \quad t \geq \tau,
\end{split} \end{equation} where $C_*$ is the same constant as in \eqref{piq}.
The Lipschitz and H\"{o}lder continuity associated with the regularity property of the parabolic $C_0$-semigroup of contraction $\{e^{At}\}_{t \geq 0}$, cf. \cite{SY}, gives rise to \begin{equation} \label{LH}
\|e^{A(t_0 + h)} g_0 - e^{At_0} g_0\| \leq \|e^{At_0} \| \|e^{Ah} g_0 - g_0\| \leq C_0 |h| \|g_0\|, \quad \text{for all} \;\; t_0 \geq 0, \end{equation} where $C_0 > 0$ is a constant depending only on the contraction operator semigroup $e^{At}$. Then, it follows from \eqref{Stpt} and \eqref{LH} that
\begin{equation} \label{St12}
\begin{split}
&\|S(t, \tau - t_1) g_0 - S(t, \tau - t_2)g_0\| = \|g^1 (t, \tau - t_1, g_0) - g^2 (t, \tau - t_2, g_0)\| \\[2pt]
= &\, \left\|e^{A(t - (\tau - t_1))}g_0 + \int_{\tau - t_1}^t e^{A(t-s)} [f(g^1 (s, \tau - t_1, g_0)) + p(s, x)]\, ds \right. \\
- &\, \left. e^{A(t - (\tau - t_2))}g_0 - \int_{\tau - t_2}^t e^{A(t-s)} [f(g^2 (s, \tau - t_2, g_0)) + p(s, x)]\, ds \right\| \\
\leq &\, \|\Pi (t, \tau - t_1, \Pi (\tau - t_1)\| \leq e^{\frac{1}{2}C_* |t - (\tau - t_1)|} \|\Pi (\tau - t_1, \tau - t_2, g_0) \| \\[6pt]
\leq &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} \, \|e^{A(t_2 - t_1)}g_0 - g_0)\| \\[2pt]
+ &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} \int_{\tau - t_2}^{\tau - t_1} \| e^{A(\tau - t_1-s)} [f(g^2 (s, \tau - t_2, g_0)) + p(s, x)]\|\, ds \\
\leq &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} \, C_0 \, |t_1 - t_2| \|g_0\| \\
+ &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} \int_{\tau - t_2}^{\tau - t_1} \|e^{A(t-s)} [f(g^2 (s, \tau - t_2, g_0)) + p(s, x)]\|\, ds.
\end{split} \end{equation} Denote by $T^* = T_{M^*_E} > 0$, which is the finite time when all the pullback solution trajectories started from the pullback absorbing set $M_E^*$ in Lemma \ref{naac} permanently enter into itself. Define the following set, where the closure is taken in the space $E$, \begin{equation} \label{GB}
\Gamma = \overline{\bigcup_{0 \leq t \leq T^*} S(\tau, \tau - t) M_E^*}
\end{equation} Lemma \ref{naac} demonstrated that $M_E^*$ and $T^*$ are independent of $\tau \in \mathbb{R}$ and $t \geq 0$. Denote by $D_\Gamma = \max_{g \in \Gamma} \|f(g) \|_H$, since the Nemytskii operator $f: E \to H$ is bounded on the bounded set $\Gamma$ in $E$. Here $\|e^{At}\|_{\mathcal{L}(H)} \leq 1$ and by H\"{o}lder inequality, \begin{equation} \label{fg2}
\begin{split}
&\, \int_{\tau - t_2}^{\tau - t_1} \|e^{A(t-s)}\|_{\mathcal{L}(H)} (\| f(g^2 (s, \tau - t_2, g_0)) \| + \| p(s, x)\|)\, ds \\
\leq &\, (D_\Gamma + K_2) |t_1 - t_2| + \int_{\tau - t_2}^{\tau - t_1} \| p(s, \cdot )\|\, ds \\
\leq &\, (D_\Gamma + K_2) |t_1 - t_2| + |t_1 - t_2|^{1/2} \sqrt{\int_{\tau - t_2}^{\tau - t_1} \|p(s, \cdot )\|^2\, ds} \\
\leq &\, (D_\Gamma + K_2) |t_1 - t_2| + |t_1 - t_2|^{1/2} \sqrt{(|t_1 - t_2| +1) \, \Sigma_{i=1}^3 \|p_i \|^2_{L_b^2}} \\[2pt]
\leq &\, (D_\Gamma + K_2) |t_1 - t_2| + |t_1 - t_2|^{1/2} ( |t_1 - t_2|^{1/2} + 1) \|p \|_{L_b^2} \\[6pt]
\leq &\,( |t_1 - t_2| + |t_1 - t_2|^{1/2}) (D_\Gamma + K_2 + \|p \|_{L_b^2}),
\end{split} \end{equation} for any $t_1 \geq T^*$ and $g_0 \in M_E^*$, where $K_2$ is given in \eqref{acB}.
Substituting \eqref{fg2} into \eqref{St12} we obtain \begin{equation} \label{G12}
\begin{split}
&\| S(t, \tau - t_1) g_0 - S(t, \tau - t_2) g_0\| = \|g^1 (t, \tau - t_1, g_0) - g^2 (t, \tau - t_2, g_0) \| \\[5pt]
\leq &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} C_0 \, |t_1 - t_2| \|g_0\| \\[5pt]
+ &\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} ( |t_1 - t_2| + |t_1 - t_2|^{1/2}) (D_\Gamma + K_2 + \|p \|_{L_b^2}) \\[5pt]
\leq &\, \lambda (M_E^*)\, e^{\frac{1}{2}C_* |t - (\tau - t_1)|} |t_1 - t_2|^\gamma, \quad \text{for} \;\, t \geq \tau - t_1, \; t_1 \geq T^*, \; g_0 \in M_E^*.
\end{split} \end{equation} where $$
\lambda (M_E^*) = C_0 K_2 + 2(D_\Gamma + K_2 + \|p \|_{L^2_b}) $$ and $$
\gamma = \begin{cases}
\frac{1}{2}, &\text{if $|t_1 - t_2| < 1$;} \\[5pt]
1, &\text{if $|t_1 - t_2| \geq 1$.}
\end{cases} $$
For any given $\tau \in \mathbb{R}$, in the above inequality \eqref{G12} take $$
t = \tau, \quad t_1 = T_{M^*_E}, \quad \text{and} \quad t_2 = T_{M^*_E} + t \quad \text{for} \; \;t \in [0, \,T^*_{M^*_E}]. $$ Then we obtain \begin{equation} \label{C59}
\sup_{\tau \in \mathbb{R}} \, \|S(\tau, \tau - T_{M^*_E}) g_0 - S(\tau, \tau - T_{M^*_E} - t )g_0\| \leq \lambda (M_E^*)\, \exp \left\{\frac{C_*}{2}\,T_{M^*_E}\right\} | t |^\gamma \end{equation} for $t \in [0, T_{M^*_E}], \, g_0 \in M^*_E$. It shows that the Lipschitz condition \eqref{Lpt1} with $M^* = M^*_E$ in Proposition \ref{PeA} is satisfied. Moreover, for any given $\tau \in \mathbb{R}$, take $t = \tau$ and $t_1, t_2 \in [T_{M^*_E}, 2T_{M^*_E}]$ in \eqref{G12}, we see that \begin{equation} \label{f510}
\|S(\tau, \tau - t_1) g_0 - S(\tau, \tau - t_2)g_0\| \leq \lambda (M_E^*)\, \exp \left\{\frac{C_*}{2}\,T_{M^*_E}\right\}\, |t_1 - t_2|^\gamma \end{equation} for any $g_0 \in M^*_E$. It shows that the Lipschitz condition \eqref{Lpt2} with $M^* = M^*_E$ is also satisfied by the nonautonomouss Hindmarsh-Rose process. According to Proposition \ref{PeA}, there exists a pullback exponential attractor $\mathscr{M} = \{\mathscr{M}(\tau)\}_{\tau\in \mathbb{R}}$ in the space $H$. The proof of this theorem is completed. \end{proof}
\end{document}
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Coherently dedispersed gated imaging of millisecond pulsars
Jayanta Roy,Bhaswati Bhattacharyya
Physics , 2013, DOI: 10.1088/2041-8205/765/2/L45
Abstract: Motivated by the need for rapid localisation of newly discovered faint millisecond pulsars (MSPs) we have developed a coherently dedispersed gating correlator. This gating correlator accounts for the orbital motions of MSPs in binaries while folding the visibilities with best-fit topocentric rotational model derived from periodicity search in simultaneously generated beamformer output. Unique applications of the gating correlator for sensitive interferometric studies of MSPs are illustrated using the Giant Metrewave Radio Telescope (GMRT) interferometric array. We could unambiguously localise five newly discovered Fermi MSPs in the on-off gated image plane with an accuracy of +-1". Immediate knowledge of such precise position allows the use of sensitive coherent beams of array telescopes for follow-up timing observations, which substantially reduces the use of telescope time (~ 20X for the GMRT). In addition, precise a-priori astrometric position reduces the effect of large covariances in timing fit (with discovery position, pulsar period derivative and unknown binary model), which in-turn accelerates the convergence to initial timing model. For example, while fitting with precise a-priori position (+-1"), timing model converges in about 100 days, accounting the effect of covariance between position and pulsar period derivative. Moreover, such accurate positions allows for rapid identification of pulsar counterpart at other wave-bands. We also report a new methodology of in-beam phase calibration using the on-off gated image of the target pulsar, which provides the optimal sensitivity of the coherent array removing the possible temporal and spacial decoherences.
Weak Nil Clean Rings
Dhiren Kumar Basnet,Jayanta Bhattacharyya
Abstract: We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all $n\in \mathbb{N}$, for which $\mathbb{Z}_n$ is weak nil clean but not nil clean. We show that if $R$ is a weak* nil clean and $e$ is an idempotent in $R$, then the corner ring $eRe$ is also weak* nil clean. Also we discuss $S$-weak nil clean rings and their properties, where $S$ is a set of idempotents and show that if $S=\{0, 1\}$, then a $S$-weak nil clean ring contains a unique maximal ideal. Finally we show that weak* nil clean rings are exchange rings and strongly nil clean rings provided $2\in R$ is nilpotent in the later case. We have ended the paper with introduction of weak J-clean rings.
Perturbative and non-perturbative studies with the delta function potential
Nabakumar Bera,Kamal Bhattacharyya,Jayanta K. Bhattacharjee
Physics , 2008, DOI: 10.1119/1.2830531
Abstract: We show that the delta function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential if coupled with a delta function potential remains exactly soluble. We use the strength of the delta function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.
A multi-pixel beamformer using an interferometric array and its application towards localisations of newly discovered pulsars
Jayanta Roy,Bhaswati Bhattacharyya,Yashwant Gupta
Physics , 2012, DOI: 10.1111/j.1745-3933.2012.01351.x
Abstract: We have developed a multi-pixel beamformer technique, which can be used for enhancing the capabilities for studying pulsars using an interferometric array. Using the Giant Metrewave Radio Telescope (GMRT), we illustrate the application of this efficient technique, which combines the enhanced sensitivity of a coherent array beamformer with the wide field-of-view seen by an incoherent array beamformer. Multi-pixel beamformer algorithm is implemented using the recorded base-band data. With the optimisations in multi-pixelisation described in this paper, it is now possible to form 16 directed beams in real-time. We discuss a special application of this technique, where we use continuum imaging followed by the multi-pixel beamformer to obtain the precise locations of newly discovered millisecond pulsars with the GMRT. Accurate positions measured with single observations enable highly sensitive follow-up studies using coherent array beamformer and rapid follow up at higher radio frequencies and other wavelengths. Normally, such accurate positions can only be obtained from a long-term pulsar timing program. The multi-pixel beamformer technique can also be used for highly sensitive targeted pulsar searches in extended supernova remnants. In addition this method can provide optimal performance for the large scale pulsar surveys using multi-element arrays.
The study of multi-frequency scattering of ten radio pulsars
Wojciech Lewandowski,Karolina Rozko,Jaroslaw Kijak,Bhaswati Bhattacharyya,Jayanta Roy
Physics , 2015, DOI: 10.1093/mnras/stv2159
Abstract: We present the results of the multi-frequency scatter time measurements for ten radio pulsars that were relatively less studied in this regard. The observations were performed using the Giant Meterwave Radio Telescope at the observing frequencies of 150, 235, 325, 610 and 1060~MHz. The data we collected, in conjunction with the results from other frequencies published earlier, allowed us to estimate the scatter time frequency scaling indices for eight of these sources. For PSR J1852$-$0635 it occurred that its profile undergoes a strong evolution with frequency, which makes the scatter time measurements difficult to perform, and for PSR J1835$-$1020 we were able to obtain reliable pulse broadening estimates at only two frequencies. We used the eight frequency scaling indices to estimate both: the electron density fluctuation strengths along the respective lines-of-sight, and the standardized amount of scattering at the frequency of 1 GHz. Combining the new data with the results published earlier by Lewandowski et al., we revisited the scaling index versus the dispersion measure (DM) relation, and similarly to some of the earlier studies we show that the average value of the scaling index deviates from the theoretical predictions for large DM pulsars, however it reaches the magnitude claimed by L\"ohmer et al. only for pulsars with very large DMs ($>$650 pc cm$^{-3}$). We also investigated the dependence of the scattering strength indicators on the pulsar distance, DM, and the position of the source in the Milky Way Galaxy.
Electronic structure of helium atom in a quantum dot
Jayanta K. Saha,S. Bhattacharyya,T. K. Mukherjee
Abstract: Bound and resonance states of helium atom have been investigated inside a quantum dot by using explicitly correlated Hylleraas type basis set within the framework of stabilization method. To be specific, precise energy eigenvalues of bound 1sns (1Se) [n = 1-6] states and the resonance parameters i.e. positions and widths of 1Se states due to 2sns [n = 2-5] and 2pnp [n = 2-5] configuration of confined helium below N = 2 ionization threshold of He+ have been estimated. The two-parameter (Depth and Width) finite oscillator potential is used to represent the confining potential representing the quantum dot. It has been explicitly demonstrated that electronic structure properties become a sensitive function of the dot size. It is observed from the calculations of ionization potential that the stability of an impurity ion within quantum dot may be manipulated by varying the confinement parameters. A possibility of controlling the autoionization lifetime of doubly excited states of two-electron ions by tuning the width of the quantum cavity is also discussed here.
Domestic Violence: A Social Issue in Rural Tripura [PDF]
Jayanta Choudhury, Moutoshi Deb
Advances in Applied Sociology (AASoci) , 2015, DOI: 10.4236/aasoci.2015.510024
Abstract: Domestic violence is a wide spread problem, its actual extent is difficult to measure. According to available statistics throughout the world, about 33 percent women have experienced violence at some point of their life (WHO, 1997). In India, women form about half of the population and enjoy various freedom and rights but simultaneously, like other developing countries, violence against women is overwhelming and is a matter of concern. Domestic violence leads to violence of human rights and prevents them from enjoying their fundamental rights. Though the types of violence differs from society to society, nations to nations, religion to religion, but it prevails in underdeveloped, developing and developed countries, too. Domestic violence refers to "assaultive and coercive behaviours that adults use against their intimate partners" (Holden, 2003). In India, there are a set of well equipped legislations which protect women against violence. And Protection of Women against Domestic Violence Act, 2005 is the milestone in the history of legal control and judicial response in domestic violence affairs. But the rate of violence against women increased 44 percent during 1993 to 2011. The findings of the paper are expected to enlighten individuals and the community on the causes and consequences of violence against women. The paper comes out with policy prescription for government and non-government organizations towards addressing the problem.
Banter: An Alternative Strategy in Creating a Learning Community [PDF]
Sumita Bhattacharyya
Creative Education (CE) , 2013, DOI: 10.4236/ce.2013.43030
Abstract: In this qualitative study we investigated the role of bantering in creating a learning community for science education. The curriculum was centered on a technology-integrated Project Based Approach (PBA). We examined the pattern of in-service teachers' interaction with such a learning environment and perceptions of their future instructional practices that result from collaborative reflection on the use of Banter throughout the semester. The findings suggest that exposure to bantering interaction not only helped the in-service teachers to make decisions about the scientific issues they will face in the future but also helped to construct a more inquiry based understanding of the issues in science teaching. Methodological limits and possibilities were explored through the use of data analysis software such as Inspiration and NVivo.
The GMRT High Resolution Southern Sky Survey for pulsars and transients -I. Survey description and initial discoveries
Bhaswati Bhattacharyya,Sally Cooper,Mateusz Malenta,Jayanta Roy,Jayaram N. Chengalur,Michael Keith,Sanjay Kudale,Maura McLaughlin,Scott M. Ransom,Paul S. Ray,Benjamin W. Stappers
Abstract: We are conducting a survey for pulsars and transients using the Giant Metrewave Radio Telescope (GMRT). The GMRT High Resolution Southern Sky (GHRSS) survey is an off-Galactic-plane (|b|>5) survey in the declination range -40 deg to -54 deg at 322 MHz. With the high time (up to 30.72 micro-sec) and frequency (up to 0.016275 MHz) resolution observing modes, the 5-sigma detection limit is 0.5 mJy for a 2 ms pulsar with 10% duty cycle at 322 MHz. Total GHRSS sky coverage of 2866 square-deg, will result from 1953 pointing, each covering 1.8 square-deg. The 10-sigma detection limit for a 5 milli-sec transient burst is 1.6 Jy for the GHRSS survey. In addition, the GHRSS survey can reveal transient events like rotating radio transients or fast radio bursts. With 35% of the survey completed (i.e. 1000 square-deg), we report the discovery of 10 pulsars, one of which is a millisecond pulsar (MSP), one of the highest pulsar per square degree discovery rates for any off-Galactic plane survey. We re-detected 23 known in-beam pulsars. Utilising the imaging capability of the GMRT we also localised 4 of the GHRSS pulsars (including the MSP) in the gated image plane within +/- 10 arc-second. We demonstrated rapid convergence in pulsar timing with a more precise position than is possible with single dish discoveries. We also exhibited that we can localise the brightest transient sources with simultaneously obtained lower time resolution imaging data, demonstrating a technique that may have application in the SKA.
Probability and variability analysis of rainfall characteristics of Dinhata in Koch Behar district of West Bengal
Jayanta Das
Golden Research Thoughts , 2012, DOI: 10.9780/22315063
Abstract: The historical daily rainfall data for the period of 40 years (1972-2011) of Dinhata in Koch Behar district of W.B. were analyzed to know weekly, monthly, seasonal, annual and decadal rainfall variability and probabilities at different level for suitable crop planning. The overall mean annual rainfall was 2909.88 mm, with standard deviation 658.39 mm and coefficient of variation 22.62 %. The annual rainfall of 2800.5 mm with 105 rainy days and 2487.2 mm rainfall and 91 rainy days may be expected with 51.24% and 75.64 % probability level, respectively. The seasonal rainfall was distributed as 2158.6 mm, 535.86 mm and 215.39 mm in kharif (June to September), summer (March to May) and rabi (October to February) seasons, respectively.
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CommonCrawl
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Characterizations of the $E$-Benson proper efficiency in vector optimization problems
Some properties of a class of $(F,E)$-$G$ generalized convex functions
2013, 3(4): 627-641. doi: 10.3934/naco.2013.3.627
Error bounds for symmetric cone complementarity problems
Xin-He Miao 1, and Jein-Shan Chen 2,
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
Department of Mathematics, National Taiwan Normal University, Taipei 11677
Received May 2013 Revised August 2013 Published October 2013
In this paper, we investigate the issue of error bounds for symmetric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called $R_0$-type conditions, which are new and weaker than existing ones in the literature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to linear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed $R_0$-type conditions cannot be replaced by $P$-type conditions which include monotone condition as special cases.
Keywords: Error bounds, $R_0$-type functions, symmetric cone complementarity problem., merit function.
Mathematics Subject Classification: Primary: 65K10; Secondary: 90C3.
Citation: Xin-He Miao, Jein-Shan Chen. Error bounds for symmetric cone complementarity problems. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 627-641. doi: 10.3934/naco.2013.3.627
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Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1261-1272. doi: 10.3934/jimo.2019001
Chengxiang Wang, Li Zeng. Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections. Inverse Problems & Imaging, 2016, 10 (3) : 829-853. doi: 10.3934/ipi.2016023
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Xin-He Miao Jein-Shan Chen
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CommonCrawl
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\begin{document}
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\end{document}
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arXiv
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Regular Polytopes (book)
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948,[1][2][3][4][5][6][7][8] with a second edition published by Macmillan in 1963[9][10][11][12] and a third edition by Dover Publications in 1973.[13][14][15] The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.[15]
Cover of the Dover edition, 1973
AuthorHarold Scott MacDonald Coxeter
LanguageEnglish
SubjectGeometry
Published1947, 1973, 1973
PublisherMethuen, Pitman, Macmillan, Dover
Pages321
ISBN0-486-61480-8
OCLC798003
Overview
The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations.[1][2] It has 14 chapters, along with multiple appendices,[3] providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers.[1] It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.[1][2]
The first chapter discusses regular polygons, regular polyhedra, basic concepts of graph theory, and the Euler characteristic.[3] Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their duals to generate related polyhedra,[1] including the semiregular polyhedra, and discusses zonohedra and Petrie polygons.[3] Here and throughout the book, the shapes it discusses are identified and classified by their Schläfli symbols.[1]
Chapters 3 through 5 describe the symmetries of polyhedra, first as permutation groups[3] and later, in the most innovative part of the book,[1] as the Coxeter groups, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra.[3]
The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds.[3]
Later editions
The second edition was published in paperback;[9][11] it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups,[9][12] appends a new definition of polytopes at the end of the book, and makes minor corrections throughout.[9] The photographic plates were also enlarged for this printing,[10][12] and some figures were redrawn.[12] The nomenclature of these editions was occasionally cumbersome,[2] and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the electron microscope.[13][14]
Reception
The book only assumes a high-school understanding of algebra, geometry, and trigonometry,[2][3] but it is primarily aimed at professionals in this area,[2] and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers.[3] Nevertheless, reviewer J. C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects",[4] and (despite complaining about the omission of regular skew polyhedra) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy.[7] Geologist A. J. Frueh Jr., describing the book as a textbook rather than a monograph, suggests that the parts of the book on the symmetries of space would likely be of great interest to crystallographers; however, Frueh complains of the lack of rigor in its proofs and the lack of clarity in its descriptions.[6]
Already in its first edition the book was described as "long awaited",[3] and "what is, and what will probably be for many years, the only organized treatment of the subject".[7] In a review of the second edition, Michael Goldberg (who also reviewed the first edition)[1] called it "the most extensive and authoritative summary" of its area of mathematics.[10] By the time of Tricia Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the four color theorem, proved after its last update. However, she still evaluated it as "well-written and comprehensive".[15]
See also
• List of books about polyhedra
References
1. Goldberg, M., "Review of Regular Polytopes", Mathematical Reviews, MR 0027148
2. Allendoerfer, C.B. (1949), "Review of Regular Polytopes", Bulletin of the American Mathematical Society, 55 (7): 721–722, doi:10.1090/S0002-9904-1949-09258-3
3. Cundy, H. Martyn (February 1949), "Review of Regular Polytopes", The Mathematical Gazette, 33 (303): 47–49, doi:10.2307/3608432, JSTOR 3608432
4. Miller, J. C. P. (July 1949), "Review of Regular Polytopes", Science Progress, 37 (147): 563–564, JSTOR 43413146
5. Walsh, J. L. (August 1949), "Review of Regular Polytopes", Scientific American, 181 (2): 58–59, JSTOR 24967260
6. Frueh, Jr., A. J. (November 1950), "Review of Regular Polytopes", The Journal of Geology, 58 (6): 672, JSTOR 30071213{{citation}}: CS1 maint: multiple names: authors list (link)
7. Wolfe, H. E. (February 1951), "Review of Regular Polytopes", American Mathematical Monthly, 58 (2): 119–120, doi:10.2307/2308393, JSTOR 2308393
8. Tóth, L. Fejes, "Review of Regular Polytopes", zbMATH (in German), Zbl 0031.06502
9. Robinson, G. de B., "Review of Regular Polytopes", Mathematical Reviews, MR 0151873
10. Goldberg, Michael (January 1964), "Review of Regular Polytopes", Mathematics of Computation, 18 (85): 166, doi:10.2307/2003446, JSTOR 2003446
11. Primrose, E.J.F (October 1964), "Review of Regular Polytopes", The Mathematical Gazette, 48 (365): 344–344, doi:10.1017/s0025557200072995
12. Yff, P. (February 1965), "Review of Regular Polytopes", Canadian Mathematical Bulletin, 8 (1): 124–124, doi:10.1017/s0008439500024413
13. Peak, Philip (March 1975), "Review of Regular Polytopes", The Mathematics Teacher, 68 (3): 230, JSTOR 27960095
14. Wenninger, Magnus J. (Winter 1976), "Review of Regular Polytopes", Leonardo, 9 (1): 83, doi:10.2307/1573335, JSTOR 1573335
15. Brown, Tricia Muldoon (October 2016), "Review of Regular Polytopes", MAA Reviews, Mathematical Association of America
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Wikipedia
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Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms?
Just counting the number of cousins staying in each room, there are the following possibilities: (4,0,0,0), (3,1,0,0), (2,2,0,0), (2,1,1,0), (1,1,1,1).
(4,0,0,0): There is only $1$ way to put all the cousins in the same room (since the rooms are identical).
(3,1,0,0): There are $4$ ways to choose which cousin will be in a different room than the others.
(2,2,0,0): Let us consider one of the cousins in one of the rooms. There are $3$ ways to choose which of the other cousins will also stay in that room, and then the other two are automatically in the other room.
(2,1,1,0): There are $\binom{4}{2}=6$ ways to choose which cousins stay the same room.
(1,1,1,1): There is one way for all the cousins to each stay in a different room.
The total number of possible arrangements is $1+4+3+6+1=\boxed{15}$.
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Math Dataset
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Are we failing young people not in employment, education or training (NEETs)? A systematic review and meta-analysis of re-engagement interventions
Lauren Mawn†1Email author,
Emily J. Oliver†2,
Nasima Akhter3,
Clare L. Bambra4,
Carole Torgerson5,
Chris Bridle6 and
Helen J. Stain7
Systematic Reviews20176:16
Youth comprise 40% of the world's unemployed, a status associated with adverse wellbeing and social, health, and economic costs. This systematic review and meta-analysis review synthesises the literature on the effectiveness of interventions targeting young people not in employment, education, or training (NEET).
Randomised and quasi-randomised trials with a concurrent or counterfactual control group and baseline equivalence are included. Cochrane collaboration tools are used to assess quality, and a narrative synthesis was undertaken. The primary outcome is employment; secondary outcomes were health, earnings, welfare receipt, and education.
Eighteen trials are included (9 experimental and 9 quasi-experimental), sample sizes range from 32 to 54,923. Interventions include social skills, vocational, or educational classroom-based training, counselling or one-to-one support, internships, placements, on-the-job or occupational training, financial incentives, case management, and individual support. Meta-analysis of three high-quality trials demonstrates a 4% (CI 0.0–0.7) difference between intervention and control groups on employment. Evidence for other outcomes lacks consistency; however, more intensive programmes increase employment and wages over the longer term.
There is some evidence that intensive multi-component interventions effectively decrease unemployment amongst NEETs. The quality of current evidence is limited, leaving policy makers under-served when designing and implementing new programmes, and a vulnerable population neglected.
PROSPERO CRD42014007535
Most young people succeed in education and make a positive transition to the world of work. However, global youth unemployment is estimated at 13.1%, three times that of adult rates [1] and equating to nearly 75 million individuals. This is a challenge faced by many high-income countries especially since the financial crisis of 2007/2008, with rates of a specific subgroup of young people aged 16–24 years and not in employment, education, or training (NEET) reported at 23.4% in the European Union, 15.5% in the USA, 12.2% in Australia, and 22.2% in the UK. Further, global youth unemployment has increased by 3.4 million since 2007 [2] and rates of NEET individuals and those in vulnerable employment continue to rise [1].
NEET individuals result in substantial economic costs to each country. For example, in the UK, there were an estimated 943,000 identified NEETs in 2015, despite claims of an economic recovery [3]. For each of these young people, the average lifetime direct cost to the public sector is £56,500 and the wider resource cost to the economy, including lost output, is estimated at £104,300 [4]. As a population, this has been projected to potentially cost the UK up to £77 billion in lost taxes, public service costs, and associated impacts such as crime and poor health [5].
In addition to the societal costs of NEETs, there are of course stark effects on the individuals concerned. Social inclusion, health, and wellbeing are all negatively impacted by unemployment from young adulthood and throughout life [6–9]. Unemployment increases the likelihood of medical consultations, taking medication and admission to hospital, and increases the risk of mortality [10]. The risk for psychiatric disorders, substance use, and suicidal behaviour is also increased for unemployed persons [11].
Reducing youth unemployment has been, and remains, a policy priority in many high-income countries including the UK, USA, and in Europe. Over the past few decades, there have been a number of initiatives and programmes implemented. In the UK, there have been specific programmes for youth since the introduction of the 1983 Youth Training Scheme [12]. More recent programmes include the New Deal for Young People (1998 to 2002), which provided work placements, vocational training, job search, and curriculum vitae support, plus the Educational Maintenance Allowance (1999 to 2011) which paid 16–18-year-olds an allowance to remain in full-time education. Recently, the UK Government announced apprentice schemes whereby, three million apprenticeships will be created by 2020 [13]. Additionally, much recent welfare reform has targeted NEETs, restricting their entitlements to key out-of-work benefits (such as housing benefits) and making participation in welfare schemes compulsory with non-participation leading to benefit sanctions and loss of income [14]. Across the world, NEET young people are considered to face particular barriers including: a lack of work experience, poor qualifications, heightened employer uncertainty, and—by some policymakers—considered to represent certain negative typologies (e.g. poor work, lazy, quitters; cf. [15]). As such, specific programmes are considered to be a way of providing additional support for the challenges faced by this group.
Past and present interventions targeting the NEET population are diverse. Intervention approaches include educational (academic, basic, or social skills; advice and guidance: [16, 17]); vocational (work placements, career planning, volunteering: [18]); counselling or mentoring [18, 19]; or service-based (case management, monitoring). Given that education is the most important risk factor for the development of NEET status, educational interventions target not only this established deficit but also the increase of work-related skills, knowledge, and aspirations. Thus, education serves as both an outcome and as the pathway through which engagement in work is achieved. In the present review, we focus on employment as our primary outcome, and do not exclude interventions targeting education, but recognise that our focus is primarily on this former aspect of NEET status.
Interventions working with the unemployed target a wide range of mechanisms theorised to influence engagement and wellbeing, for example developing efficacy, attitudes and perceived social norms [20], or enhancing social support and coping strategies [21]. The relative effectiveness of these, and other, different intervention approaches, however, is not known.
A lack of rigorous trial designs in evaluations of potentially effective interventions, rapid fluctuations in political and economic climates, and a diverse research base contributed to by scientific, statutory, and voluntary organisations, are potential factors leading to the paucity of knowledge of the effectiveness of interventions. However, given the longstanding and ongoing development of programmes in this area, it is important that evidence of effectiveness is examined. The aim of this systematic review was to identify, synthesise, and evaluate experimental or quasi-experimental evidence of the effects of any interventions, on employment, attainment, behavioural and health-related for youth classified as not in education, employment, or training.
The protocol for the review was published ([name deleted to maintain the integrity of the review process]) and registered with the PROSPERO database, and a PRISMA checklist is available as Additional file 1.
Trial identification and search strategy
A standardised search strategy [22] was used to search English language papers from 1990 to present. We justify narrowing as our focus given that, first, the vast majority of scientific articles are published in English and comprehension of literature would potentially be compromised by translation. Second, we suggest that target interventions are best understood in a contemporary context, hence use of the conventional inclusion threshold consistent with previous topical reviews [23].
The following databases were searched in June 2014 (replicated in May 2016): Medline, Embase, PsycINFO, ERIC, EPPI-Centre (Bibliomap), Social Science Citation Index, British Education Index, Conference Proceedings Index, Dissertation Abstracts, Popline, and grey literature collections (e.g. GLADNET). This was supplemented with internet searching (e.g. Google Scholar), forward and backward citation tracking from systematic reviews and included trials, and contact with trial authors and research groups. In addition, aid organisations with an interest in the target population were approached for internal reports (e.g. Barnardo's). Together, these approaches identified some relevant papers outside of our original search restrictions.
Eligibility criteria were constructed around population, intervention, comparison, and outcomes (PICO). The population of interest was young people aged between 16 and 24 years who were not in employment or education (or training) at the time of the intervention commencing. We included trials for which the mean sample age was between 16 and 24 years, and those that reported analyses for NEET subgroups where the total population contained NEET and non-NEET individuals. There were no restrictions placed on trial inclusion in regards to country of population. Given one of our aims was to identify the full range of interventions that have been trialled with this group, we had no restrictions by intervention type. Any intervention that was delivered to the NEET population was included, whether targeted solely at NEET individuals or targeted at a larger group of unemployed individuals but reporting effects on NEET individuals separately. In terms of study designs, only randomised or quasi-randomised (i.e. where the method of group allocation is not truly random, such as matching, or alternate allocation) controlled trials, with a concurrent control or comparison group (including usual treatment controls) were included. We were not interested in excluding at this stage on the basis of the nature of the control or comparison group. Where a quasi-randomised design was used, groups had to demonstrate baseline equivalence or a valid matching protocol. Pre/post, cross-sectional, and non-comparison group designs were excluded. The primary outcome was employment; secondary outcomes included earnings, welfare receipt, education, health, and other behaviours (e.g. drug use).
Search results were downloaded into Endnote. Following the removal of duplicate citations, a three-phase quality assurance process was conducted, using previously stated inclusion criteria. In phase 1, titles and abstracts were screened independently by two reviewers against the inclusion criteria. Agreement was high, with full consensus reached through discussion. To add rigour, 10% of trials excluded in this phase were cross-checked by a third author; no discrepancies emerged therefore we progressed to phase 2 screening. In phase 2, full text papers were again screened by two reviewers independently, with discrepancies resolved through discussion or, if necessary, by recourse to a third reviewer. Again, 10% of trials excluded in this phase were cross-checked with a third author; no discrepancies emerged and we progressed to phase 3. In phase 3, all papers were screened by a third author, and any disagreements resolved through group discussion ([initials deleted to maintain the integrity of the review process]). Search results, screening outcomes, and selection decisions are presented in a PRISMA flow chart in Fig. 1.
PRISMA flow chart
Data extraction and quality assessment
Data were extracted using a standardised form, including methodological characteristics (e.g. unit of randomisation, length of follow-up), sample characteristics (e.g. prior length of NEET status), description of the intervention and control conditions (e.g. structure, theoretical basis, type, frequency, duration, provider and setting), measures and outcomes for baseline, and all follow-up periods and process-related outcomes (e.g. recruitment approach, uptake). The data extractions were completed by two authors ([initials deleted to maintain the integrity of the review process]) independently, cross-checked, and then quantitative extractions were verified by a researcher with statistical expertise ([initials deleted to maintain the integrity of the review process]). Where required data were missing, first or corresponding authors were contacted to request this information.
The assessment of trial quality and risk of bias was conducted independently by two authors ([initials deleted to maintain the integrity of the review process]) using the Cochrane Collaboration's risk of bias assessment tool [24]. Each trial's risk was rated as high, low, or unclear for sequence generation; allocation concealment; blinding of participants, assessors, and providers; selective outcome reporting; and incomplete data.
Data synthesis and statistical analyses
Summary measures of intervention effect size with associated estimates of precision (95% CI) were calculated for outcomes where minimal adequate data was available. There was insufficient quality of data to enable sub-group analyses, by either intervention type or participant characteristics. Where estimates could be extracted for sub-groups (e.g. males, females), these are reported separately. Meta-analysis was performed using a random effect model using post-intervention mean difference and standard error between intervention and control groups. There was insufficient data to consider statistical indicators of publication or small trial bias. Data were synthesised narratively by outcome.
Deviations from protocol
Two important deviations from the protocol should be noted. First, not all of the stated analyses were conducted. Meta-analysis was only conducted on the primary outcome variable, employment. This was due to insufficient data reported within included trials for either meta-analysis or estimating publication bias. Mean difference was used as an effect measure instead of odds ratio for employment due to not having pre- and post data for intervention and control group for most trials. Second, due to the complexity and range of included analyses, an additional phase of quality assurance was conducted, with quantitative extractions reviewed by a researcher with statistical expertise ([initials deleted to maintain the integrity of the review process]).
Trial flow
Of the 1767 citations identified, 1219 non-duplicate papers were retrieved. Nine hundred ninety-five were excluded in phase 1 screening (abstract), and 139 at phase 2 screening (full text) for not meeting the eligibility criteria. The most common rationale for exclusion was that the paper did not examine or report data for a NEET population. Six trials were removed following phase 3 screening (independent quality assurance). These included trials that used secondary data, and those with problematic control and/or for which baseline equivalence could not be established. Thus, 18 papers were retained: 13 journal articles, 3 reports (retrieved from ERIC, and 2 theses (see Table 1 for a summary of all included trials, including ID numbers). Of these trials that met the criteria for inclusion, nine were experimental randomised controlled trials (1, 2, 3, 4, 6, 7, 8, 15, 16) and nine were quasi-randomised (5, 9, 10, 11, 12, 13, 14, 17, 18). For two trials, subsample data that met the inclusion criteria were used (4, 6); and one author provided additional unpublished data for analysis.
Characteristics of included trials
Participant characteristics
Outcomes measured
Effect size (d)
Alzua, Cruces, & Lopez-Erazo [30]
Entra21
Below poverty line
\( \overline{x}\ \mathrm{age} \) = 23.55 (I), 23.80 (C)
33% male
407 randomised.
407 analysed.
Receipt of welfare
Credit standing
N credit enquiries
Attanasio, Kugler, & Meghir [29]
On-the-job
6 months, 5 h per day
Wait-list
Lowest deciles of income distribution/Poor youth in urban areas
\( \overline{x}\ \mathrm{age} \) = 21.1 (I), 21.22 (C)
44.4%male
4353 randomised
3549 analysed.
Bloom, Orr, Cave, Bell, & Doolittle [33]
JTPA II-A
Some received classroom training only.
Economically disadvantaged, facing barriers to employment.
\( \overline{x}\ \mathrm{age} \) = 19
4793 randomised.
Achieved HSD or GED
Borland, Tseng, & Wilkins [31]
23 meetings, 2 years
Standard service delivery
Homeless (or history of homelessness/disadvantage) job seekers
445 recruited.
208-355 analysed.
Employment status N days income support
DEEWR programme expenditure Health and Wellbeing Community activities Housing
Borland, & Tseng [34]
Job seekers diary
Work search verification
Fortnightly, 3 months
54,923 analysed (whole sample).
Card, Ibarrarán, Regalia, Rosas-Shady, & Soares [35]
Juventud y Empleo
Basic skills Internship
44.5% male
Lowest income members of working age population
5723 realised treatment group, 1623 realised control group.
Cave, Bos, Doolittle & Toussaint [28]
Jobstart Basic skills
Occupational training
800 h, 6.6 months
400 hours, not Jobstart.
School drop-outs
Low skilled 17-21 53.5% male
Receipt of education
Receipt of qualification
Criminal activity
Chen [36]
Academic Vocational Social skills
8 months residential
Wait list (3 years)
Low income household
\( \overline{x}\ \mathrm{age} \) 18.42 (I), 18.38 (C)
15,386 analysed.
Earnings(weekly)
Creed, Machin, & Hicks [37]
Youth Conservation Corps Work experience
Unemployed >6 months
67% male (I), 52% (C)
82 analysed.
Psyc. distress
Unemployed ≥12 months
Eligible for government sponsored programmes
65 randomised.
32 analysed at F3.
Donovan, Oddy, Pardoe, & Ades [39]
Youth Opportunity Programme Work experience6–12 months
Did not access programme; unemployed.
\( \overline{x}\ \mathrm{age} \) = 15.93 at T1
Health status
Stafford [40]
Youth Opportunity Programme
16–18 54% male
Health (GHQ)
Mounsey [41]
Youth training scheme
Duration not stated
No treatment; matched.
16–17 at T1
Varied by analysis: 972 to 8885.
NEET status
Expected earnings and reservation wages
Nafilyan, & Speckesser [42]
Youth Contract
Individually tailored support
12 months est.
Matched (counterfactual); same educational attainment and probability of receiving intervention.
11,144 received intervention.
Schochet, McConnell, & Burghardt [43]
Other services; not Job Corps.
Disadvantaged—living in a household that receives welfare or is below the poverty line, and living in an environment that impairs prospects for participating in other programmes. Free of serious behavioural and medical problems. 60% males>70% members of racial or ethnic minoritygroups 16–24
15, 406 randomised.
Schochet, Burghardt, & Glazerman [44]
As trial 15
Tanner, Purdon, D'Souza, & Finch [32]
Activity Agreement pilots
One-to-one support
Individually tailored contract
Standard service delivery; matched from non-participating areas.
58% males
NEET for >20 weeks
Not receiving JSA.
1018 analysed at F1, 229 analysed at F2.
Confidence and independence
Grace & Gill [45]
422 assigned, 370 analysed.
Welfare receipt
Trial characteristics
The 18 included trials analysed between 122,488 and 131,337 participants (depending on outcome) with a median analysed sample size of 1 232, (range from 32 to 54, 923). The median of the mean ages was 19 years old (range = 15.93–23.67), of the trials that reported mean age (n = 8; 1, 2, 3, 4, 8, 9, 10, 11). Of the 12 trials that reported gender, percentage males in the sample ranged from 33 to 67%. The inclusion criteria varied across trials; please see Table 1 for full list of included trials and design, participants, location/country, intervention, and outcome characteristics.
The interventions reported in the trials included basic or social skills training (1, 6, 7, 8), vocational training (8, 15, 16), educational classroom-based training (1, 2, 3, 8, 9, 13, 15, 16), counselling or one-to-one support (15, 16, 17), internships, placements, work experience, on-the-job or occupational training (1, 2, 3, 6, 7, 9, 10, 11, 12, 13, 15, 16), financial incentives (17), work search verification (5), case management (4, 18), and individually tailored support (14, 17). The duration and intensity of interventions varied considerably. Three interventions lasted 12 months or more (4, 14, 18); nine lasted between 6 and 12 months (1, 2, 7, 8, 10, 11, 12, 15, 16); and five less than 6 months (3, 5, 6, 9, 17). The intensity of the interventions ranged from 23 sessions over a 2-year period to an 8-month full-time residential programme.
The control group interventions included no contact (1), standard service delivery (4, 5, 17, 18), use of other support services or restricted use of intervention programme services (3, 7, 15, 16), or placement on a wait list (2, 8, 9, 10). Matched data were used by four trials (11, 12, 13, 14), while one trial did not describe the control group/condition (6).
The outcomes measured were clustered in to six general domains: the primary outcome, employment; and secondary outcomes of earnings, welfare, education, health, and other. Twelve trials reported effects on employment status (1, 2, 3, 4, 6, 7, 8, 15, 16, 17), and two on NEET status specifically (13, 14). Eight trials reported effects on actual (2, 3, 6, 7, 8, 15, 16, 18) or expected (13) earnings. Seven trials reported receipt of welfare (e.g. income support; child support: 1, 4, 7, 8, 16, 18), and four trials reported either receipt of education (7, 16, 17) or educational attainment (3, 7). Health-related outcomes included general health status (5, 11, 12, 16) and psychological health (e.g. self-esteem, distress, confidence: 9, 10, 17). Other variables reported included credit standing (1), pregnancy rates (7), housing and community engagement (4, 18), health insurance provision (6), and criminal activity (7, 11, 16). Due to the diversity of outcomes, summary findings for these have not been collectively synthesised in this paper.
Risk of bias ratings for each trial (see Table 2) was examined using the Cochrane risk of bias tool [24]. Eight trials were at high risk of bias for sequence generation (4, 5, 6, 9, 10, 14, 16, 18), and the method of randomization was unclear in two trials (8, 15). For four trials, the risk of bias was not applicable due to matched counterfactual control groups (11, 12, 13, 17). Risk of bias owing to poor allocation concealment was high in five trials (4, 8, 14, 16, 18), not applicable in four trials (11, 12, 13, 17), and unclear in three trials (1, 10, 15). Lack of blinding created a high risk of bias for some outcomes in four trials (3, 6, 8, 17), was unclear in seven trials (4, 9, 10, 13, 14, 16, 18), and was not applicable to three trials (7, 11, 12).
Risk of bias assessments for included trials
Total classification
Sequence generation (selection bias)
Allocation concealment (selection bias)
Blinding (performance bias)
Outcome completeness
Selective outcome reporting
Other biases
Unclear
+ low risk of bias; ? unclear risk of bias; - high risk of bias; NA not applicable
There was a high risk of bias due to incomplete outcome data for nine trials (3, 5, 6, 7, 11, 13, 14, 16, 17), and an unclear risk of bias for further five trials (2, 4, 9, 15, 18). This could be indicative of both a high rate of attrition in trials of this type of population and/or methodological deficiency in the trials themselves. Only four trials were clearly free of selective outcome reporting (3, 4, 7, 17), eight trials did not report all outcomes (1, 8, 9, 10, 11, 13, 14, 16), and it was unclear whether six trials reported all outcomes (2, 5, 6, 12, 15, 18). The quality of data reporting was also varied. For example, six trials reported means but not standard deviations (2, 3, 4, 7, 15, 16). Due to the small number of included trials, and small samples within some trials, we were unable to assess publication bias formally. Given that any additional unpublished trials could be sufficient to change estimates of the relative benefits and harms of these interventions, we considered that there was a high risk of publication bias.
Main analysis
The findings are presented by outcome below (please see Table 3). Where possible, we have separated out findings by intervention type; however, this was challenging. All interventions featured direct contact with the population (i.e. none were indirect economic interventions). Most contained multiple elements (e.g. education, training and work placements, advice, support, and incentives); therefore, we were not able to create robust sub-groups by intervention type. The only meaningful division of interventions was comparing multi-component to single-component interventions. Even within these clusters, there was wide variation in terms of the intensity of delivery, rendering interpretation of effects based on intervention type problematic.
Outcome data summary
Mean difference (SE)
.113 (.049)
Receipt of welfare (F)
−.056 (.002)
Sum of post treatment
Employment status (F)
Employment status (M)
−.032
Earnings (F)
Columbian pesos
Earnings (M)
−182
Achieved HSD or GED (F)
Achieved HSD or GED (M)
No SE reported. 2-year follow-up.
N days income support
3-year follow-up
$AUD; 3-year follow-up
DEEWR expenditure
−.09
2-year follow-up; self reported
12-month follow-up; percentage chance in participants only (no control data)
4.0% (3.9)
.4%
Ever employed; 4-year follow-up totals:
Hours in education
−4.9%
−.3%
−.038 (.01)
22.19 (4.65)
−84.29 (38.27)
−1.93 (5.45)
−2.68 (.92)
Adjusted for T1 and gender
Cohort measured varied therefore comparison not possible
NEET status:
Estimates using nearest neighbour matching
YTS1 (M)
YTS1 (F)
YT (M)
YT (F)
Expected earnings
reservation wages
−11.01
No SE presented
6.5-year follow-up
5.5-year follow-up; average earnings by quarter; $USD
Average weekly earnings
Ever enrolled
Self-reported excellent
Ever arrested or charged
Self report
$AUD; 24-month follow-up
Stability: n of moves.
Thirteen of the 17 trials reported employment or NEET status change as an outcome (1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17). Adequate data for meta-analysis (i.e. estimate of difference and standard error) was only available for four samples extracted from three trials (1, 2, 8). Post-intervention, the interventions had a small but significant positive effect on employment compared to control (MD = .04 [0.0–0.7]; see Fig. 2). It should be noted that follow-up periods varied from immediately post-intervention to 48 months. All three trials were multi-component interventions using a mixture of skills/educational training and job-based training.
Meta-analysis of intervention effects on employment
Across all trials that reported employment as an outcome (including those meta-analysed above), nine were experimental and four were quasi-experimental designs, while the interventions used were heterogeneous (see Table 1 for trial characteristics). The majority (nine trials) used a multi-component intervention combining skills/educational training and job-based training. Of these, three had positive effects on employment (1, 8, 16) whereas one had positive effects for women only (2), four had no significant effect (3, 6, 7, 15), and one a negative effect (13). The only other multi-component trial combined one-to-one support with financial incentives (17) and had a significant positive effect.
In terms of the single component interventions, work search verification (5) had a positive impact on employment, whereas case management (4) and individually tailored support (14) had no effect. Across all 13 studies, commonalities of those with significant positive effects were inclusion criteria relating to deprivation indicators (e.g. below poverty line, lowest decile of household income); North or South American-based; post-2000; more likely to use multi-component interventions (e.g. classroom, job-based, and skills) and were for a minimum of 6 months of high intensity contact. Three of these four trials met data reporting requirements and were included in the meta-analysis.
Nine trials reported the effects on actual (2, 3, 6, 7, 8, 15, 16, 18) or expected (13) earnings. Meta-analysis could not be conducted for the outcome of earnings (three samples with sufficient data, three trials: 2, 8, 6), as precision estimates could not be calculated. All but one intervention was multi-component, featuring skills training (e.g. educational, vocational, basic, or social skills) combined with work-based learning (e.g. placement or internship). One involved trialling joined up case management (e.g. employment and housing service providers working cooperatively or collaboratively) (18). Apart from the case management trial, all were intensively delivered (one trial did not report intensity: 13), with a minimum of approximately 2.5 months of daily contact. Given this, analysis of effects by intervention type was not appropriate for this outcome, nor would intervention type explain differences in findings that emerged.
Three reported positive intervention effects on earnings. These were significant for one trial (8), significant for females only in another trial (2), and nonsignificant in one trial (6). Three trials, however, reported a more complex pattern of effects. In these trials, earnings for the intervention groups decreased in the first 2 years of participation but increased beyond the controls in the third and fourth year (7, 15, 16). Despite a common pattern, these differences were statistically significant in only one of the three trials. However, the magnitude of effect was generally small. One trial reported no significant intervention effect on earnings beyond the increase observed with standard provision (18).
There was some evidence suggesting effects on wages might manifest differently in different population subgroups. For example, one trial (3) found no effect on average earnings of female youths and a significant reduction for male youths of approximately $854 over 18 months. Another trial (8) identified stronger impacts on Hispanic participants when compared to whole sample data (U.S. population sample). Lastly, in one trial (7), earnings impact was stronger for those who chose to leave school due to disciplinary problems or dislike, as opposed to those who had left for employment-related reasons. Data for these claims was not included in the paper nor made available for re-examination on request. Of note, the trial that examined wage expectations (wage one expects to receive) and wage reservations (lowest pay one would consider) found a clear pattern of significant increases for men, but not for women.
Six trials reported receipt of welfare outcomes (e.g. income support; child support: 1, 4, 7, 8, 16, 18). Only one trial (8) had adequate data when considering welfare receipt/benefits as the outcome and thus could not be meta-analysed. Two trials found significant intervention effects on receipt of public assistance, with reductions of $84.29/year (8) and $460 ID (16) across a 48-month follow-up period. Both were multi-component interventions, featuring skills training and work-based learning. The most minimal intervention (i.e. a change in case management procedures) had no significant effect on welfare receipt (4, 18).
Two trials reported significant differences in welfare receipt only for specific subgroups. One trial (1) of a multi-component intervention (skills and work-based training) had significant positive effect in credit use for male participants, and a significant negative effect on welfare dependency for females. Another (7) (skills and work-based training) found that the intervention reduced subsequent child-related welfare payments for women who were not custodial mothers on programme entry (relative to the control group), but not for women who were custodial mothers.
Health-related outcomes included general health status (4, 11, 12, 16), health behaviours (16), and psychological health indicators (9, 10). There were no trials with adequate data quality meta-analysis. Two interventions were multi-component (skills and work-based training), and four were single component (three work experience, one case management). The case management approach (4) resulted in no significant difference in health markers post-intervention. Of the work experience-only trials, two resulted in improved general health (11) or self-esteem and distress (10) and one poorer general health relative to the control groups (12). Of the multi-component trials, one had no significant effects on either health or health behaviours (e.g. alcohol, tobacco, or drug use: 16), and the other improved self-esteem but not psychological distress (9).
Four trials reported on either receipt of education or educational attainment (3, 7, 16, 17). There were no trials with adequate data quality for meta-analysis. All were multi-component interventions, and two different approaches were adopted. Three combined skills training and work-based learning, whereas one (17) offered individualised support and advice. Both approaches had positive effects. Three trials reported higher percentage of individuals receiving training for the intervention group compared to the control group (7, 16, 17). Two trials reported a 'significant' or 'highly significant' difference in General Education Development (GED) or High School Diploma (HSD) attainment (3, 7), and one trial a 7% increase in qualification attainment, for intervention groups compared to control groups (17).
This systematic review established the current state of evidence concerning the effectiveness of interventions targeting young people not in education, employment, or training (NEET). Based on the three trials with sufficient data to meta-analyses, the interventions resulted in a small but significant 4% increase in employment. Across the NEET population, this has the potential to enact change for thousands of individuals. Using conservative and somewhat crude estimates of costs and population [4], within the UK, this has the potential to equate to almost £469 million of savings to the public purse.
Successful interventions were high-contact (e.g. 884 h, 6 months, or an 8-month residential programme) and had additional commonalities in terms of inclusion criteria targeting deprivation and using multi-component approaches. Such interventions showed potential to result in small increases in earnings at longer-term follow-up (i.e. over 24 months) and reductions in welfare receipt, particularly for young women and those without children. No consistent effects on participants' health were identified. However, there was evidence of increased educational attainment (for the most part, education would have been a direct consequence of intervention delivery). Although across all trials the majority of effects on employment were nonsignificant, it is important to note that the significant increase above emerged from synthesis of the highest data quality trials.
Taken together, the findings provide promising support for the effectiveness of high-contact multi-component (classroom and work-based) interventions in improving employment prospects for NEET individuals. These share some commonalities with effective practice highlighted in previous reviews. For example, the Department for Education [25] and Public Health England [26] highlighted the importance of work-based placements and basic skills provision and the involvement of local employers and accredited courses, respectively. Other reviews identify perceived important characteristics that are not supported in the current review (e.g. partnership arrangements, effective management and organisation, personalised learning, and clear progression routes: [27]).
We cannot, however, claim that high-contact multi-component interventions were universally effective, and interpreting the data at the intervention level is problematic in this review for multiple reasons. First, the vast majority of interventions were multi-component, combining some form of education or skills-based classroom training with on-the-job training (e.g. internship, work experience, job placements). It was notable that such interventions tended to adopt a pragmatic approach (e.g. classroom and work experience) rather than targeting potentially important psychological barriers to work engagement (e.g. enhancing confidence, reducing distress). Of note, narrative reviews have previously suggested that confidence-enhancing activities are beneficial [25]. Second, findings (both within and across outcomes) are mixed even when the same type of intervention is delivered. Third, there were insufficient number of trials available neither to compare different types of approach, in terms of content or modality (e.g. training versus job search modification), nor to examine required exposure to, or dose of, intervention necessary for a change in outcome.
A repeated finding of differential effectiveness for population sub groups is worthy of consideration here. Differences in intervention effects emerged in some trials dependent on gender, ethnicity, age, and broader circumstances (e.g. prior arrest rate). Sub-group differences were also reported in terms of recruitment to, and engagement with, interventions (e.g. [28]). Whilst the specific sub-groups more or less likely to benefit varied across trials and outcomes, it was notable that sub-groups benefitting less tended to be those that were more disadvantaged at trial commencement (e.g. poorer literacy, higher previous arrest rate, lower socioeconomic status, minority ethnic groups). This raises concerns that, despite often targeting a deprived population, current intervention approaches are not designed to cater for the circumstances and needs of the most disadvantaged, potentially further exacerbating the inequalities experienced by this group.
A more complex interpretation of subgroup effects emerged for gender differences, whereby, trials identified a significant effect on employment [29], a reduction in welfare receipt [30], and no short-term (i.e. <18 months) wage suppression [31] for females only. In these trials, females seemed to benefit more from the intervention, perhaps relating to lower levels of labour market engagement in general for young females relative to males in control populations (thus, improvements were more marked). Individual circumstances also seem to be important, for example, one trial identified reductions in welfare receipt for females who were not custodial mothers at trial commencement, but not for those who were. We tentatively suggest this is because non-custodial mothers were better placed to re-enter the labour market post-intervention, which implies that training alone is not sufficient to improve prospects for custodial mothers and perhaps psychosocial interventions could be beneficial.
Lastly, there was some evidence that contextual factors influenced intervention effectiveness. For example, Cave et al. [28] reported site level differences in effects and problems where different providers were responsible for different services. In trials where different methods of intervention delivery were compared, some reported similar impacts (with altered financial remuneration: [32]), some reported no differences in effect (e.g. between sequential versus simultaneous training delivery: [28]), and some reported different treatment effects (e.g. between variants of the YTS scheme: [33]). It should be noted that the trial locations (six countries, four continents), funders, and delivery partners varied; thus some interventions may have been effective due to the political and economic landscape of location and time of delivery.
Research implications
Overall, the findings from the current review are inconsistent in respect to examining outcomes of interventions for NEET young people. We highlight five main areas for future research to address. First, there is still a need to establish what works to reengage young people. Notably, there is limited delivery and evaluation of interventions based on contemporary behaviour change theory and practice. Second, research is needed to establish what works for whom, particularly in light of interventions not serving some of the most disadvantaged. Third, it is not clear what aspects of interventions work (e.g. education and training, placement, counselling). Indeed, some arguably relevant approaches (e.g. psychological/behaviour change interventions) have not been subject to evaluation, therefore their potential impact is unknown. Fourth, there is a scarcity of research applying theoretically underpinned interventions. Fifth, there is a dearth of research examining physical and mental health outcomes, which is striking given the well established negative impact of unemployment on physical and mental health [6–9].
Previous narrative reviews of supporting young people who are NEET (e.g. [23]) have reported that 'quality of the evidence is high, with most items based upon a strong to moderate evidence base that tends to be qualitative rather than based on statistical measurement'. We disagree. In contrast to this, our review not only found that there exists relevant work research utilising statistical measurement, but that the literature base has substantive issues with quality, methodological rigour, and reporting. For example, of note in the current research is the number of trials that did not provide sufficient data for inclusion in the meta-analysis. We recommend that high quality research is required and that trials evaluating effectiveness of interventions adhere to standardised reporting protocols (e.g. PRISMA) to aid future research examining the effectiveness of interventions with this population.
While methodological rigour is a challenge in terms of controlling for confounds (multiple agencies interacting with the population at any given time) and identification of an appropriate control group, there is a need to stress the importance of implementing randomised controlled trials so as to ascertain evidence for effectiveness and to ensure interventions are not having adverse effects (e.g. loss in earnings). Given that interventions are frequently delivered by commissioned private or voluntary organisations, there is a need for researchers to become involved early in programme development to aid with robust evaluations. Further, there are a broad range of providers and stakeholders working with NEET populations, including multiple local authority departments (e.g. housing, care, health), as well as international, national, and local aid organisations. The literature base reporting on interventions is therefore diverse, and useful information may be difficult to access (e.g. internal local authority project evaluations), incorporate, or control for. There are also systemic, cultural, and economic factors that are likely to impact on NEET status (e.g. recession, deprivation, policy, voting population). These make it difficult to eliminate all confounds when examining intervention effects, but in addition, highlight the importance of attention to these higher-level conditions when seeking to alter NEET population status.
Reporting in terms of cost and cost effectiveness varied, and examining these was beyond the scope of the present review. It is worth noting, however, that intervention costs per recipient are low (e.g. $750 [29]; $1722 [30]). Although it is notoriously difficult to cost up the net social benefit of an individual moving from NEET to non-NEET status, where interventions are simple (e.g. embedded in existing services (5)) cost benefits were demonstrated. This is of mixed value to policymakers given that the strongest effects (i.e. on employment and earnings) emerge for the high-contact interventions.
When considering the commissioning and operation of high-contact schemes, we should be aware that the evidence identifies that the act of participation in such interventions may suppress earnings in the short term (within 24 months). Given this, schemes may need to consider financial incentives or wage replacement to improve recruitment and adherence rates. This may also assist with engagement within the interventions; in one trial [28] that reported effects segregated by contact, participants with low contact levels had poorer outcomes than the control group post-intervention, whereas those with high contact benefited greatly.
Public funders must recognise the need to support and fund rigorous trials as discussed above. Whilst recognising the desire to maximise access to services, this must not be at the expense of determining whether strategies are effective and cost-effective. In addition, limited funding should be allocated to programmes that will not contribute high quality evidence. Without this evidence, policy concerning how best to intervene is speculative. It is worthy of note that of the 18 included trials only three reported to be based on specific theories. One was driven by economic investment framework, and one was designed to increase job search efforts and matching to job vacancies as well as punitive monitoring and motivational feedback. Whilst a theoretical framework was not always explicitly articulated, there is an assumption that behaviourist theories underpinned both of these approaches. Finally, one trial utilised cognitive behavioural therapy-based training; however, this was aimed at improving the mental health of participants and providing them with coping skills to deal better with the negative consequences of prolonged unemployment rather than to reengage them in employment, per se. The limited use of explicitly theory-driven approaches to understanding and driving reengagement may have contributed to the limited variety of approaches utilised, and hence undermines our ability to identify what might work to reengage young people.
Considering most interventions aim to change participant behaviour, it is interesting to note that behavioural change theories were not employed more often. Potentially, this is an artefact of the dominance of economic and policy approaches to NEET interventions. To illustrate, there is a broad range of providers and stakeholders working with NEET populations, including multiple local authority departments as well as international, national, and local aid organisations. It may be that the NEET problem is being tackled by stakeholders focused on economics and social policy as opposed to those best placed to understand human behaviour change, disengagement and reengagement (e.g. psychologists, behaviour change specialists). Policy makers should consider engaging behaviour-change relevant expertise when designing intervention approaches.
As we still do not know how to effectively intervene to reengage NEET individuals, localised innovation should be promoted, accompanied by practice evaluations to identify nuances in delivery between sites and taking into account local contexts. Without effective interventions directly facilitating return to work or education, NEET individuals, and the countries that support them, are left exposed to fluctuations of the macro global economic climate. Good practice in terms of monitoring the NEET population should continue (e.g. within the UK, quarterly statistical releases are provided by the Department of Education), maintaining public and political momentum for tackling the issue. Technological approaches to service delivery and support, as well as monitoring, should be considered in the future as a potentially cost effective and accessible method for engaging this population.
This review included 18 trials and 131,707 participants. While this is not the first review examining the NEET population, other reviews (e.g. [23]) have not been restricted to experimental designs, instead including a broad range of trial methods. As a result, evidence included in these reviews is of limited use in terms of identifying effectiveness. Further, these reviews are prone to selective citation and lack robust quality assessment of included evidence, subsequently examining heterogeneity in a descriptive manner. The current review is the first, we believe, to enforce rigorous inclusion criteria relating to design as well as presenting robust quality appraisal processes.
We do recognise that by constraining the focus of this review to high-quality evidence we omit other work that may be important and useful. The learning from these service evaluations, qualitative trials, case trials, data analyses, models, and philosophical and theoretical texts should be considered holistically when debating the relative merits of different approaches to working with the NEET population.
We reviewed only robust evidence by restricting inclusion to randomised controlled trials and quasi-randomised trials with demonstrable baseline equivalence or a valid matching protocol. Despite this, concerns emerged when critiquing included trials against best scientific practice. All had a high or unclear risk of bias. We cannot know the extent or direction of the influence of bias on trials' findings; however, under- or over-estimation of effects may be present. The ubiquitous nature of the bias risk also prohibited any additional analyses restricted to low risk trials.
As the interventions were all delivered in-service, over multiple sites, fidelity to experimental protocols would have been difficult to identify and were often not reported. We were unable to ascertain whether interventions were delivered as intended, in terms of either contact time or the nature of the provided contact. Where fidelity was reported, findings were not reassuring. For example, one trial [34] reported that 20% of their intervention group never received the intervention, and 50% had only one session in 6 months (as opposed to the targeted fortnightly administration). Concerns over fidelity were exacerbated in trials whereby control or comparison groups were also in receipt of an alternative intervention. For example, in one trial [35], control group members were transferred to intervention groups to compensate for individuals who did not attend the intervention.
In a context where the number of youth classified as NEET is increasing globally and a priority area for labour market policy (International Labour Organisation, 2014; IMPETUS, 2014), identification of effective interventions is important. By considering a broad range of interventions and outcomes, this review has highlighted both gaps in the current evidence base, as well as examples of effective practice. Specifically, we have found that high intensity multi-component interventions, featuring classroom and job-based training, appear to increase employment amongst NEETs by 4% compared to controls. While it is disappointing to find that interventions appear to increase employment prospects by only 4%, it is important to acknowledge that in real terms this could represent a positive difference for thousands of young people. Further, importantly, although employment and earnings were the most commonly measured outcomes, some of the more promising findings emerged for mental health related outcomes. It may be that greater intervention effectiveness would be evident if wellbeing data were routinely monitored; indeed, theoretical questions regarding how we prioritise re-employment as opposed to targeting some of the pathways to re-employment and societal engagement more generally (including improved mental health) need attention from both researchers and policymakers.
However, more needs to be done to effectively meet the growing needs of the NEET population. Furthermore, considering the difficulty and cost of developing and delivering effective for NEET young people, there exists a critical need to do more to prevent individuals becoming NEET in the first place. Restrictions in the amount and quality of evidence leave us in a situation where best practice for changing the lives and prospects of NEET individuals for the better is unclear and robust future research is required. Whilst a key finding of this review was to highlight the need for future research to adopt high-quality evidence methodologies to determine what works best for this population, at present, limited recommendations for policy and practice can be endorsed. This leaves policy makers under-served when designing and implementing new programmes in this area, and a vulnerable population unacceptably neglected.
This research was supported by a grant from the Wolfson Research Institute for Health and Wellbeing, Durham University. The authors have no financial interest or benefit to declare related to the present research.
CBa is an associate director of Fuse, the Centre for Translational Research in Public Health. Funding for Fuse comes from the British Heart Foundation, Cancer Research UK, Economic and Social Research Council, Medical Research Council, the National Institute for Health Research, under the auspices of the UK Clinical Research Collaboration, and is gratefully acknowledged. The views expressed in this paper do not necessarily represent those of the funders or UKCRC. The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
The protocol for this project received funding from the Wolfson Research Institute for Health and Wellbeing, Durham University.
N/A—No primary data collected.
LM and EO are the co-principal investigators of this review and contributed to the review conception, refinement of the design, data analysis, and writing the manuscript. NA contributed to the meta-analysis and writing of the results. CBa participated in the design of the review and protocol, quality assurance, and writing the manuscript. CT participated in the design of the review and protocol, quality assurance, and writing the manuscript. CBr led the design of the protocol and contributed to quality assurance and writing the manuscript. HS assisted in the conception of the study, refinement of the design, quality assurance, and writing the manuscript. All authors read and approved the final manuscript.
Lauren Mawn https://www.researchgate.net/profile/Lauren_Mawn
Emily Oliver https://www.dur.ac.uk/sass/staff/profile/?id=12421
Clare Bambra http://www.ncl.ac.uk/ihs/
Nasima Akhter http://scholar.google.co.uk/citations?user=OYYcbvsAAAAJ&hl=en&oi=ao
Carole Torgerson https://www.dur.ac.uk/directory/profile/?id=10409
Chris Bridle http://staff.lincoln.ac.uk/cbridle
Helen Stain http://research.leedstrinity.ac.uk/en/persons/helen-stain(e08e708a-ff3e-451b-8698-54654505eee4).html
Additional file 1: PRISMA checklist. (DOC 63 kb)
School of Psychology, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK
School of Applied Social Sciences, Durham University, Durham, DH1 3HN, UK
Wolfson Research Institute of Health and Wellbeing, Durham University, Durham, TS17 6BG, UK
Institute of health and Society, Faculty of Medical Sciences, Newcastle University, Newcastle upon Tyne, UK
School of Education, Durham University, Durham, DH1 3LE, UK
Institute of Health, Lincoln University, Lincoln, LN6 7TS, UK
Psychology - School of Social and Health Sciences, Leeds Trinity University, Horsforth, Leeds, LS18 4HD, UK
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CommonCrawl
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Exploring genetic, molecular, mechanical and behavioural methods of sex separation in mosquitoes
Exploring the potential of computer vision analysis of pupae size dimorphism for adaptive sex sorting systems of various vector mosquito species
Mario Zacarés1,
Gustavo Salvador-Herranz2,
David Almenar3,
Carles Tur3,
Rafael Argilés4,
Kostas Bourtzis4,
Hervé Bossin5,6 &
Ignacio Pla3
Several mosquito population suppression strategies based on the rearing and release of sterile males have provided promising results. However, the lack of an efficient male selection method has hampered the expansion of these approaches into large-scale operational programmes. Currently, most of these programmes targeting Aedes mosquitoes rely on sorting methods based on the sexual size dimorphism (SSD) at the pupal stage. The currently available sorting methods have not been developed based on biometric analysis, and there is therefore potential for improvement. We applied an automated pupal size estimator developed by Grupo Tragsa with laboratory samples of Anopheles arabiensis, Aedes albopictus, Ae. polynesiensis, and three strains of Ae. aegypti. The frequency distribution of the pupal size was analyzed. We propose a general model for the analysis of the frequency distribution of mosquito pupae in the context of SSD-sorting methods, which is based on a Gaussian mixture distribution functions, thus making possible the analysis of performance (% males recovery) and purity (% males on the sorted sample).
For the three Aedes species, the distribution of the pupae size can be modeled by a mixture of two Gaussian distribution functions and the proposed model fitted the experimental data. For a given population, each size threshold is linked to a specific outcome of male recovery. Two dimensionless parameters that measure the suitability for SSD-based sorting of a specific batch of pupae are provided. The optimal sorting results are predicted for the highest values of SSD and lowest values of intra-batch variance. Rearing conditions have a strong influence in the performance of the SSD-sorting methods and non-standard rearing can lead to increase pupae size heterogeneity.
Sex sorting of pupae based on size dimorphism can be achieved with a high performance (% males recovery) and a reasonably high purity (% males on the sorted sample) for the different Aedes species and strains. The purity and performance of a sex sorting operation in the tested Aedes species are linked parameters whose relation can be modeled. The conclusions of this analysis are applicable to all the existing SSD-sorting methods. The efficiency of the SSD-sorting methods can be improved by reducing the heterogeneity of pupae size within rearing containers. The heterogeneity between batches does not strongly affect the quality of the sex sorting, as long as a specific separation threshold is not pre-set before the sorting process. For new developments, we recommend using adaptive and precise threshold selection methods applied individually to each batch or to a mix of batches. Adaptive and precise thresholds will allow the sex-sorting of mixed batches in operational conditions maintaining the target purity at the cost of a reduction in performance. We also recommend a strategy whereby an acceptable level of purity is pre-selected and remains constant across the different batches of pupae while the performance varies from batch to batch to fit with the desired purity.
There is a global renewed interest in area-wide integrated mosquito management strategies based on the mass production and release of sterile males to suppress target populations [1,2,3,4]. These techniques are usually referred to as genetic control methods and include, among others, the sterile insect technique (SIT), the incompatible insect technique (IIT) and the release of insects carrying a dominant lethal gene (RIDL) [1, 4,5,6,7]. Several small-scale projects have demonstrated the high potential of these strategies to suppress mosquito populations [6, 8,9,10]. The scaling-up of these projects from pilot to operational has been hampered by several problems, the most significant one being the lack of an efficient sex-sorting method [1, 11, 12]. Given that only the female mosquitoes bite and transmit the human pathogens, those methods must be capable of ensuring a predefined acceptable level of female contamination while maximizing the male pupae recovery.
The successful use of genetic sexing strains (GSS) for the sex sorting of Ceratitis capitata and other fruit fly species [13,14,15,16,17] has encouraged researchers to develop similar GSS strains for mosquitoes. GSS strains that can be sex-sorted at early developmental stages (eggs or L1) are generally accepted to be the optimal solution for mass-scale SIT and related techniques [11, 12]. A genetic sexing strain based on the tolerance to dieldrin has been developed for Anopheles arabiensis; however, this strain presents several problems and has limited potential for SIT applications [18, 19]. In addition, several transgenic genetic sexing strains developed for different mosquito vector species are also of limited applied potential due to either lack of stability, low male performance or subject to extensive regulation [11, 12].
The current lack of a functional GSS has led to the mosquito population suppression projects to use alternative ways in the sex-sorting process. For the mosquito species with strong sexual size dimorphism (SSD), mainly Aedes and Culex species, mechanical methods have been generally adopted for sorting [6, 8, 20]. Although several designs and proposals for sex sorting on a mass scale have been suggested in the past [21, 22], all mosquito genetic control programmes currently use either plate separators [23] or sieves [8] for sex sorting that have been devised for small-scale rearing conditions. The development of new designs with automation capability for unattended sorting would increase the efficiency of those projects and allow their upgrade to large operational programmes [12, 24].
The efficiency of SSD-sorting methods in terms of male recovery, female contamination and speed depends on technical and biological factors. The technical features basically affect the rate of separation per time unit, and differ between methods. The main biological determinant is the size distribution between sexes and their overlap as well as the effect of rearing conditions on this characteristic. All the SSD-sorting methods rely on the same principle: the separation in two samples by a threshold size. It should be noted that an analysis of the biological determinants of the distribution of size will in principle be applicable to all SSD-sorting methods
In order to improve the performance of new designs of sex-sorting methods based on SSD, a previous biometric analysis is required, specifically dealing with the analysis of the frequency distribution of the size of sexes. However, there is scarce information regarding the distribution of size in mosquitoes. Usually, the scope of the biometric studies in mosquitoes has been to find correlations between the body size and other biological traits [25,26,27,28,29], providing only point and variance estimates, and only a limited number of studies have included detailed frequency distributions [30,31,32]. For insects, most of the frequency distributions of the size can fit to normal probability functions. When a strong SSD is present, each sex can fit to an independent normal curve [33]. SSD is generally assumed as a species-specific (or population-specific) trait with a narrow degree of variation caused by complex interactions of factors [34,35,36,37].
Several experiments have shown that variations in the mosquito larval rearing conditions can increase or reduce the average size of the resulting pupae [38,39,40,41]. The SSD is slightly influenced by intraspecific competition [40, 42], but not by food availability [38] or the pollution by conspecifics [43] as size of both sexes is equally affected by these parameters and the difference between the average size of each sex remains constant. The objective of the present study is to optimize the utilization of the SSD-sorting methods through the understanding of the frequency distribution in the pupal size of different mosquito species and strains, with the ultimate goal to: (i) understand the performance of the current sex-sorting methods in different conditions; (ii) assess the relationship between the parameters of importance for sex-sorting devices: female contamination and male recovery; (iii) evaluate the suitability of size-based sex sorting methods for different species and strains of mosquitoes; and (iv) propose features that will optimize the performance of SSD-sorting methods.
To achieve these objectives, we developed a general model for the size distribution of mosquito at the pupal stage. The frequency distribution can be modelled as a mixture of two normal probability density functions. We analyzed the frequency distribution in size of four important mosquito vector species that are currently the target of area-wide integrated vector control projects using SIT-based methods: Aedes aegypti, Ae. albopictus, Ae. polynesiensis and Anopheles arabiensis. The intraspecific variation is also evaluated for Ae. aegypti, since three different laboratory strains were included in the analysis. The use of an automated pupae size estimator system based on artificial vision developed by Grupo Tragsa, Spain, allowed the collection of a large amount of size measurements thus facilitating the achievement of our objective.
Laboratory strains
The Aedes aegypti Sri Lanka strain originates from mosquitoes collected from the Narahenpita area, District of Colombo, Western Province, Sri Lanka. This strain was kindly provided by Ms. Asha Wijegunawardana (University of Kelaniya, Sri Lanka) and has been maintained in the Insect Pest Control Laboratory of the Joint Food and Agriculture Organization and International Atomic Energy Agency (IPCL-Joint FAO/IAEA) laboratories since 2017. F28 mosquitoes from this strain were analyzed in the present study.
The Ae. aegypti GSS has been developed by classical genetic approaches and has been maintained in the IPCL-Joint FAO/IAEA since 2017. F5 mosquitoes from this strain were used in the present study.
The Ae. aegypti WB2 line was recently generated by transfer of Wolbachia wAlbB from Aedes albopictus into Aedes aegypti via embryonic microinjection at Michigan State University (personal communication, Zhiyong Xi), and has been maintained in the IPCL laboratories since 2016. This strain was introgressed into the genomic background of an Ae. aegypti strain from Brazil, provided by Professor Margareth Capurro (University of Sao Paolo, Brazil), through a series of seven backcrosses using in every generation Wolbachia-infected females mated with Ae. aegypti Brazil males. This resulted in the construction of the Ae. aegypti WB2-BRA strain used. F12 mosquitoes from this strain were analyzed in the present study.
The incompatible Ae. polynesiensis "Aito" (BC9) strain carries Wolbachia B from Ae. riversi. This strain was generated through multiple backcrosses between Aedes riversi females and Aedes polynesiensis aposymbiotic males (Hapairai, 2013). This strain which has been maintained at Institut Louis Malardè (ILM), Tahiti since 2010 was recently used in a pilot IIT field study on the atoll of Tetiaroa, French Polynesia (Bossin et al. manuscript in preparation).
The Ae. albopictus Rimini strain was originated from field collections in northern Italy. It has been maintained in the IPCL since 2010.
The An. arabiensis Dongola strain was originated from the Northern State of Sudan. It has been maintained in the IPCL since 2005. It is also available at the Malaria Research and Reference Reagent Resource Center, MR4, as MRA-856.
Mosquito rearing
Standard rearing conditions have been used for the maintenance of experimental colonies, egg collection and hatching of Ae. albopictus and Ae. aegypti [44, 45], Ae. polynesiensis [39], and An. arabiensis [46, 47] colonies.
Pupae production
For each species or strain, three larval containers were prepared for pupae production as described below. These three replicates represented a random sample of the different rearing units found in a mass rearing facility.
For Ae. aegypti and Ae. albopictus, 2000 first-instar larvae were introduced in white acrylonitrile-butadiene-styrene (ABS) plastic trays (41 × 30 × 8 cm) with 1.5 l of deionized water. Since larvae of Ae. polynesiensis must be reared under lower densities [39], 1200 first-instar larvae were introduced in 40 × 60 × 15 cm containers with 4 l of water. The larvae were fed with the standard Aedes IPCL diet [48, 49] at a concentration of 75 g per liter of diet. The diet regime ranged from 0.2 mg of dry weight per larvae on the first day to 0.8 mg on the last days.
Different batches of eggs of Ae. arabiensis were hatched in white plastic trays (41 × 30 × 8 cm). Two days after the hatching, approximately 500-1000 larvae were visually isolated in the same kind of trays with 1.5 l of deionized water. The larvae were fed with the standard Anopheles IPCL diet [46], in a concentration of 10 g/l ranging from 5ml on the first day to 20 ml on the last days.
All pupae in each container were collected on a daily basis starting at 24 hours from the beginning of the pupation. A batch of pupae was defined as all the pupae produced in 24 hours for a specific species/strain and replicate(s). The selected batches were sex-sorted under a binocular microscope. All the pupae in a batch were classified into males and females groups, and the resulting samples are referred to as batch-sex groups. The batch where the proportion of male and female was closer to 50%, usually on the 2nd or 3rd day from the beginning of pupation, was selected for the analysis.
Measurement of pupal size
The lateral profile area of the pupae was automatically measured by means of a computer vision system (Fig. 1), which comprises: (i) a translucent rigid surface with circular uniform movement, acting as conveyor on which the mosquito pupae are arranged; (ii) a uniform and high intensity led white light backlight system and (iii) a high resolution/high speed camera placed in top position. In this way, the mosquito pupae pass continuously under the camera while being backlit by the lighting system. The backlighting of the pupae allows photograph of them with a high contrast, which facilitates their subsequent extraction and isolation from the background (segmentation). In order to increase the precision and accuracy of the measurement of the areas, avoiding errors due to the effects of refraction of light by water droplets, the size of the pupae was measured in dry conditions for each session.
Continuous image capture size
A factor of special importance is that the backlight system must guarantee an illuminated area with a uniform intensity, at least in the interest area of capture of the camera. This is because the light intensity directly affects the size of the areas extracted in the segmentation process, and variations in the intensity could result in errors in the relative measurements. However, even with a uniform backlighting system, minor errors may occur in the measurement due to the position of each pupa with respect to the position of the camera (different projected areas) and electrical noise in the silicone sensor of the camera. To minimise these phenomena, several pictures of each pupa are recorded while in the camera capture area (around 15 shots per pupa), and then the median of all the measurements of each individual is chosen as the value of size. So, each individual has to be identified and its path has to be tracked. For this task, we have developed a predictive tracking algorithm, based on Kalman filters [50], which is able to identify and track individuals in their rotational displacement under the field of view of the camera. Additionally, the algorithm is robust enough to follow the characteristic rapid movements of the pupae in dry conditions, which are quite active.
The median size in square pixels is then transformed to a unidimensional parameter by the square root to linearize the measure of size. The measure of size presented in this study is then the square root of the pixel area. The outcome of the analysis is scale-independent, and the conclusions are valid regardless of the actual value of this magnitude. Since the pupae were sex sorted manually, we assume that a certain degree of identification error occured. Errors in the manual sex identification can affect the estimation of the statistical parameters, especially those pupae with size far larger or smaller than the corresponding sex average value. In order to minimize this effect, we considered each value that exceeded two standard deviations from the average as an error in sex identification. These values were subsequently excluded from the analysis.
Model for the frequency distribution of size
The proposed model relies on the basic assumption that the probability density function for the pupal size of each sex follows a Gaussian distribution, with the mean and standard deviation as the characteristic parameters. The mixture distribution for this situation is:
$$ f(x)={\alpha}_m\mathcal{N}\left(x;{\mu}_m,{\sigma}_m\right)+{\alpha}_f\mathcal{N}\left(x;{\mu}_f,{\sigma}_f\right) $$
where N(x; μi, σi) is the normal probability density function for size (x), with mean μi and standard deviation σi. The scalars αi are the proportions of each sex in the model, being αm + αf = 1. The subscripts m and f denote males and females respectively.
Predictions from the model
One of the goals of our model is to provide a statistical tool to estimate the theoretical outcomes of male recovery and female contamination. In order to quantify them we introduce the performance (PER) and purity (PUR) functions defined as follows:
$$ PER(x)=\frac{recovered\ males\ from\ the\ original\ sample\ for\ a\ given\ x}{males\ in\ the\ original\ sample} \times 100 $$
$$ PUR(x)=\frac{recovered\ males\ from\ the\ original\ sample\ for\ a\ given\ x}{recovered\ pupae\ from\ the\ original\ sample\ for\ a\ given\ x} \times 100 $$
For any given size threshold (x), PER provides the male recovery percentage and PUR the percentage of males on the sorted sample. Assuming the model given by Equation 1, it is easy to show that both functions can be estimated in terms of the normal distribution function:
$$ PER(x)=\varPhi \left(\frac{x-{\mu}_m}{\sigma_m}\right)\times 100 $$
$$ PUR(x)=\frac{\alpha_m\Phi \left(\frac{x-{\mu}_m}{\sigma_m}\right)}{\alpha_m\Phi \left(\frac{x-{\mu}_m}{\sigma_m}\right)+\left(1-{\alpha}_m\right)\Phi \left(\frac{x-{\mu}_f}{\sigma_f}\right)}\times 100 $$
where Φ denotes the standard normal cumulative distribution function. In order to determine both functions, the parameters {αm, μi, σi} of the model must be known. Purity and performance are inversely linked. A decrease in female contamination can be achieved by reducing the value of the threshold, but this unavoidably produces a reduction in the performance (Fig. 2). The features of PER(x) and PUR(x) depend on the chosen set of parameters. However many sets give rise to functions that are related by simple symmetry transformations like translations or scaling. As long as both functions are transformed in the same way, the purity versus performance curve remains invariant (Fig. 2c). Therefore, parameters {αi, μi, σi} are not suitable to classify unequivocally the different samples as they can lead to the same purity-performance curve. In order to find a more appropriate space parameter, we introduce two new dimensionless parameters: the sexual dimorphism index (SDI) and the sexual homoscedasticity index (SHI) defined by
Depiction of the purity-performance relationship under a mixture of two Gaussian distributions applied to the analysis of sex sorting by size. The graphs consider αm = 0.5 and three different sets of parameters s = {μm, σm, μf, σf}, s1 ={10, 1, 11, 1}, s2 = {8, 1, 9, 1} and s3 = {12, 2, 14, 2}. The performance of sets 2 and 3 is obtained by translating and scaling the performance of set 1: PER2(x) = PER1(x + 2), PER3(x) = PER1(x−2/2). The same transformations are applied to purity functions. a Purity versus size (X). b Performance versus size (X). c Purity versus performance
$$ SDI=\frac{\mu_f-{\mu}_m}{2\sqrt{\sigma_m{\sigma}_f}} $$
$$ SHI=\sqrt{\frac{\sigma_m}{\sigma_f}} $$
Combining definitions (6) and (7) with Equations 4 and 5, we can rewrite performance and purity functions as
$$ {\displaystyle \begin{array}{cc} PER(z)& =\Phi \left(\frac{z+ SDI}{\mathrm{SHI}}\right)\times 100\\ {}& \end{array}} $$
$$ PUR(z)\kern0.5em =\frac{\alpha_m\varPhi \left(\frac{z+ SDI}{SHI}\right)}{\alpha_m\varPhi \left(\frac{z+ SDI}{SHI}\right)+\left(1-{\alpha}_m\right)\varPhi \left(\frac{z- SDI}{SHI^{-1}}\right)}\times 100 $$
z being a dimensionless variable defined by \( z=\frac{x-\left(\frac{\mu_m+{\mu}_f}{2}\right)}{\sqrt{\sigma_m{\sigma}_f}} \)
Equations 8 and 9 reveal that PER(z) and PUR(z) only depend on three dimensionless parameters: αm, SDI and SHI. In other words, given αm, all combinations of the original parameters {μm, μf, σm , σf} giving the same pair {SDI, SHI} have exactly the same performance and purity functions in terms of the dimensionless variable z. Consequently, the dimensionless parameters {SDI, SHI} are more suitable than classical measures of center and spread to classify unequivocally different pupae samples with regard to purity and performance.
Basic statistics and model fit
The mean and standard deviation was estimated for every batch-sex group dataset, and its deviation from the normal distribution was tested by means of the Shapiro-Wilk test. The fit of the data to the probability density function of the mixture model was tested by means of the Kolmogornov-Smirnov test. All the statistic computations were done using the base package of R [51]. The significance level was set to α = 0.05.
Partitioning of the variance
A number of pupae equal to the minimal sample size was randomly selected for each batch-sex group per species/strain. This was performed in order to get a balanced factorial design dataset. A linear model was fit for each batch by means of ordinary least squares. The model included sex and batch as fixed factors, and their interaction. The partitioning of variance was assessed through ANOVA. All statistical analyses were done using the base package of R [51].
Table 1 summarizes the statistics of the pupal size of the species and strains used in the present study. In total, 7733 pupae were analyzed. The majority of the batches were not significantly different from a normal distribution, and the mixture of two Gaussian distributions fitted well to all batches of all the species/strains used when male and female pupae data are combined. Table 2 presents the significance of the goodness-of-fit between the models and the data. None of the samples was significantly different from a Gaussian mixture distribution. Only two batches separated by sex showed significant departures from normality: An. arabiensis females of the batch 3, and Ae. albopictus Rimini males of batch 3.
Table 1 Descriptive statistics for the batches of pupae
Table 2 Results for the goodness-of-fit of the data to a probability distribution function. P-values are provided for each test
Figure 3 shows the histograms and fitted models for the three batches of each species/strain studied. After the parameters of the model have been estimated, the purity-performance characteristic curve is computed and the quality of sorting can be analyzed theoretically. Each value of pupal size is linked to a pair of purity and performance values, which are inversely related (Fig. 4). The fitted models allow simulating the output of SSD-sorting methods under different circumstances. Table 3 shows the main descriptors for the predicted output from the fitted model for each batch in the experimental data. These results should not be considered as a general prediction of how a particular strain will perform with SSD-sorting methods, since they are only applicable for the specific rearing conditions of this experiment. However, they show the potential of SSD-based sex-sorting procedures when standard rearing procedures are applied. Table 4 provides examples for one of the strains of Ae. aegypti (GSS) of how the performance-purity output varies under different simulated conditions. The simulation a describes the performance when the three batches are mixed and a threshold size is determined by keeping constant the level of purity of 99.5 %. It is shown how SDI takes lower values than any of the individual batches, and an average reduction of 17 % in performance is predicted for the same level of purity (male recovery of 74.5 % with a female contamination of 0.5 % after mixing the three batches). Simulations b and c assess the effect in the performance of the size heterogeneity and the variations in the SSD respectively. For simulation b the variance of both sexes is scaled by the same factor while the distance between means remains constant. Taking batch 1 as a reference, the standard deviation of both sexes is multiplied by 0.8 and 1.2 respectively. For simulation c, taking again batch 1 as reference, the average size of males and females is increased or decreased by 0.5 √pixels but the standard deviations are not modified in this case. It is worth mentioning that the variation of the statistical parameters in simulations b and c only affect the dimensionless parameter SDI, while SHI remains unaltered. Results show that changes in size heterogeneity and SSD have contrary effects in the performance; an increase in intra-batch variance produces a significant drop in performance whereas an increase in SSD improves the quality of sorting. Simulation d describes how the output of SSD-sorting systems vary with the election of a predefined constant threshold for all the batches.
Frequency distribution of size (√pixel) for pupae of different mosquito species and strains reared under small-scale laboratory conditions. Males are represented in blue and females in red. Three replicates are presented for each mosquito species/strain
Depiction of the SSD-sorting methods functioning simulated by a mixture of two normal distribution functions. Different threshold of sizes separates the sample in two subsamples. The subsample of smaller size has a different male proportion depending on the chosen threshold. Purity = % males on the sorted sample. Performance = % males recovery. The dotted lines depict a value of threshold. a Probability density function of male and female pupal size. b Purity versus performance
Table 3 Predictions of the fitted models with the experimental parameters for different measures of suitability to sexual size dimorphism sorting methods. Performance (% males recovery) at different levels of purity (% males in the sorted sample)
Table 4 Predicted values for the descriptive parameters of the sex sorting of the Ae. aegypti GSS strain simulated under different conditions
The contribution of different factors to the variability in size has been assessed by means of ANOVA. The results for the partitioning of the variance are shown in Table 5. For all the Aedes species, the biggest source of variation is the SSD. For An. arabiensis, there are significant differences in size between sexes, but this factor explained only a relatively small portion of the total variance.
Table 5 ANOVA tables for each species/strain. The factors included are batch (rearing container) and sex (male or female)
For all the species and strains examined in this study, the joint frequency distribution of pupal size included an area of overlap between the individual male and female distributions. This essentially means that a complete separation of sexes according to a given threshold of size is not possible, and every threshold that separates the sample in two will leave a certain proportion of each sex in the batch of the other group: smaller females in the male group and/or bigger males in the female group. This limitation of the SSD-based sorting methods is commonly recognized, altogether with the general observation that rearing conditions have a strong effect over the performance of the methods [8, 20,21,22, 52]. However, the mechanisms under these observations have not been investigated in depth, which may affect the optimization of new sorting methods based on SSD.
For the four species analyzed, including the three Ae. aegypti laboratory strains, the distribution of the size of each sex considered apart followed a normal distribution, as commonly observed in insects [33]. For the three Aedes species studied here, the joint frequency distribution for both sexes is noticeably bimodal, and can be modeled through a mixture of two normal probability density functions. This model is rather simple, with only five parameters that can be easily estimated directly from a population sample. It is likely that this approach can be generalized to other Aedes species as well as to culicine mosquitoes with a marked dimorphism in size [31, 38, 53]. In addition, this method of analysis could be generally applied to all known SSD-based sorting methods, since all of them rely on separating batches of pupae in two groups through the definition of a threshold size.
The features of the distribution in sizes of the individuals determine the differences between samples/ strains/ species in the suitability for any SSD-sorting method. These differences are reflected in two main parameters: the performance (% males recovery) and sample purity (% males on the sorted sample). Both can be predicted from the probability density function. For a given set of the model parameters (αm; μi, σi), each size threshold has a pair of values of performance and purity associated. Under these model assumptions, purity and performance in each sample of pupae are unequivocally linked; each value of purity corresponds to a single value of performance. Since the evaluation of the quality of a given sorting through the predicted values of purity-performance depends on the chosen value of threshold, the relationship of both parameters in a dimensionless space has been analyzed theoretically. This analysis has provided two useful indices that describe the applicability of SSD-sorting methods for a given sample of pupae, i.e. the quality of the biological material and the rearing conditions. As SDI increases, the purity-performance function becomes more optimal (better performance with higher purity). The SDI index has two components, the SSD and the sample variance. Consequently, an increase in SSD and a reduction in variance increase the efficiency of any SSD-sorting method, as will be discussed later. Index SHI modifies the slope of the curve. The higher the SHI value, the more flattened purity-performance curve is obtained. This parameter describes the difference in variance between males and females, which is difficult to control during the rearing process. SDI and SHI can be used for long-term monitoring of the quality control of the sorting process.
The purity and performance are inversely correlated. From the applied point of view, any sorting system must choose a size threshold considering the trade-off between performance and purity. From Equations 8 and 9, it is possible to estimate, for a given sample, the threshold of size needed to obtain a desired value of purity or performance. Unfortunately, it is not possible to calculate a single constant size threshold for sex sorting that keeps constant the purity and performance levels across different batches. It is known that there is heterogeneity in size in the production units (rearing containers) that affects the outcome of the sorting methods [8, 20, 21, 52]. For instance, the three Ae. aegypti GSS SSD batches varied in purity (99.2-100%) and performance (66-94%) when separated by the same threshold value (see Table 4).
Keeping the purity as a constant parameter, and assuming heterogeneity in size, the outcome of the sorting will have a variable percent recovery of males. Since this heterogeneity is important in the output of the sex sorting, we analyzed and quantified the sources of variation in size in the experimental sample. Then, we used the parameters directly estimated from the samples to predict the expected values of purity and performance for each batch of pupae. Changing the value of these parameters in the fitted models allowed us to simulate the outcome of SSD-sorting methods under different scenarios. The partitioning of variance showed two different patterns of relative importance in respect to the source of heterogeneity in size. For the Aedes species, the main source of variation was the sexual difference, followed by the residual, the differences between batches and finally the interaction sex-batch. For Anopheles, the effect of sex was of less importance, and the residual accounted for most of the variation.
The SSD, as the absolute difference between mean size of each sex, is the main factor that explains the interspecific differences in the applicability of SSD-sorting methods. A higher SSD will produce higher performance independently of the scale, and for all the size thresholds considered, yielding a higher SDI. The two species with higher SSD (Ae. aegypti and Ae. polynesiensis) are known to yield better results than Ae. albopictus when separated with plate separators. On the other hand, An. arabiensis, as expected [11, 12], showed a poor suitability for the SSD-sorting methods. The experimental samples of An. arabiensis showed an average SSD of 0.52 √pixels, while the Aedes species ranged from 3.3 to 3.9 √pixels. Likely, even achieving a reduction in the heterogeneity would not be enough to make SSD-sorting methods suitable for An. arabiensis or related species. For example, a SSD-sorting method was used to sort An. albimanus in a trial in El Salvador [54], and resulted in 14% of female contamination in the released mosquitoes which would be currently unacceptable.
The residual variance is the second important source of variation in the Aedes group. It accounts for the unexplained variation due to other factors, mainly the natural heterogeneity in size that can be found in any pupal batch. The heterogeneity in size in a given batch (rearing container) has a strong effect on the performance. As an example, our simulations (Table 4) with the GSS strain predict that a 20 % increase in the standard deviation of the experimental value reduces the performance by about 17 %. Conversely, a reduction of the same magnitude produces an increase in male recovery of 7.6 %. The heterogeneity in size could be due to genetic and/or environmental factors [55, 56]. It is not expected that the genetic heterogeneity of laboratory populations has a major effect given that it has been drastically reduced by the colonization process [55]. On the other hand, it is known that intraspecific asymmetric density-dependent factors can increase the variability in size in other insects which have an aquatic larval developmental stage [57, 58]. Given that the mosquito larvae are usually kept at high densities in the artificial rearing containers, the heterogeneity in size could potentially be reduced by adjusting the larval density.
The variance between rearing containers is also an important factor to consider in SSD-sorting. In a real mass production context, there is variation in size between batches that is present in our experiments as well. This variation affected mainly the average size of the pupae, but also at some extent the absolute SSD magnitude (Table 5, Interaction term Batch:Sex). It has been reported that the food availability or the water pollution by conspecifics does not affect the absolute SSD magnitude in Aedes [38, 43] while larval competition could produce some degree of sexual allometry in size [40, 42]. The intraspecific variation in SSD is a complex issue [35,36,37] out of the scope of this article, but worth to be investigated in mosquitoes in the context of SSD-sorting methods. The variability between rearing containers is of major applied significance because the threshold of size needed to obtain a desired degree of purity is specific for each batch. The use of a common fixed threshold for all the production batches would produce a variable output in respect to purity and performance, and it is therefore not recommended (Table 4). In a mass production context, it can be sometimes useful to mix the pupae production of different rearing containers before the sex sorting, but this would likely increase the size heterogeneity. This is clearly shown in the simulations presented in Tables 3 and 4. For example, mixing the Ae. aegypti GSS pupae production from three rearing trays reduced the performance in about 17 % (with purity level of 99.5 %).
Two main strategies are presently used for SSD sorting methods. First, sieves [8], rows of slots [21] or openings between plates [22], which are based on fixed size thresholds, were developed through a trial and error process. This means that the purity and performance are not controlled and they entirely depend on the rearing conditions. On the other hand, plate separators [23], which rely on a visual adaptive size threshold election system, exhibit better performance [52] at the expense of productivity [21]. Both strategies can be optimized using the appropriate analytical tool. For the fixed threshold methods, a more optimal threshold based on the actual range of variation in size between batches of pupae may be required. The plate separator could be optimized by the determination of less subjective threshold election criteria. Finally, the analytical framework proposed here can be integrated in large scale mechanized sex-sorters of high precision.
The distribution of size in mosquito pupae can be modeled by a mixture of two Gaussian distribution functions. This approach, combined with the parameters obtained from laboratory samples, can be useful to understand and optimize the mechanisms of the SSD-sorting methods. Purity and performance, which are the most relevant features of sex sorting devices, can be directly calculated from the presented model. Two additional dimensionless parameters, SDI and SHI, which are good descriptors of the suitability of a species/strain under given rearing conditions for its sorting with SSD-based methods are proposed. This approach can be applied to all the SSD-sorting methods. The output of the SSD-sorting methods can be improved by reducing the heterogeneity in size within the rearing containers. The heterogeneity between batches can affect the quality of sex sorting when different batches are mixed before the sorting or when a common separation threshold is determined for a series of batches. For new designs of sex-sorting devices based on SSD, we recommend the following: (i) use of an adaptive and precise threshold selection method based on automatic measurement systems and the proposed formulas; and (ii) a specific threshold size for each batch to maintain the purity at a constant level. In this way, the heterogeneity in size will be resulting to a variable male recovery (performance). From the practical point of view, this study shows that enhanced SSD-based sex sorting methods can be applied to Aedes mosquito mass-rearing facilities that depend on lateral area to distinguish sexes to efficiently produce batches of male-only pupae with a male recovery ranging between 70% and 99% and female contamination under 0.5%, with the lower values of male recovery being obtained when different batches are mixed or when larval rearing conditions are not standardized.
GSS:
Genetic sexing strain
IIT:
Incompatible insect technique
RIDL:
Release of insects carrying a dominant lethal
SDI:
Sexual dimorphism index
SHI:
Sexual homoscedasticity index
SIT:
Sexual size dimorphism
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We would like to thank the staff members of the Insect Pest Control Laboratory of the Joint FAO/IAEA Division, Antonios Augustinos, Nanwintoum S. Bimbile-Somda, Danilo Carvalho, Anna Konczal, Hamidou Maïga, Wadaka Mamai, Gulizar Pillwax and Asha Wijegunawardana, for their great support during the preparation of the samples used in this study. We are particularly grateful to Ángela Aguado and José Félix Descalzo for their essential assistance. We thank to Enrique Torrente and Jose Antonio Herrador their indispensable support to carry out this research. This study was supported by the International Atomic Energy research contact no. 17939 as part of the Coordinated Research Project "Exploring genetic, molecular, mechanical and behavioural methods of sex separation in mosquitoes". MZ acknowledges support from Universidad Católica de Valencia "San Vicente Mártir" (grant 2018-253-002)
This research was financially supported by GRUPO TRAGSA. Publication costs for this study was provided by the International Atomic Energy Agency as part of the Coordinated Research Project "Exploring genetic, molecular, mechanical and behavioural methods of sex separation in mosquitoes".
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
About this supplement
This article has been published as part of Parasites & Vectors Volume 11 Supplement 2, 2018: Exploring genetic molecular, mechanical and behavioural methods of sex separation in mosquitoes. The full contents of the supplement are available online at https://parasitesandvectors.biomedcentral.com/articles/supplements/volume-11-supplement-2.
Departamento de Ciencias Experimentales y Matemáticas, Universidad Católica de Valencia "San Vicente Mártir", C/Guillem de Castro 94, 46003, Valencia, Spain
Mario Zacarés
Departamento de Expresión Gráfica, Proyectos y Urbanismo, Universidad CEU Cardenal Herrera, Valencia, Spain
Gustavo Salvador-Herranz
Grupo Tragsa, Avda. de la Industria 26, 46980, Paterna, Valencia, Spain
David Almenar
, Carles Tur
& Ignacio Pla
Insect Pest Control Section, Joint FAO/IAEA Division of Nuclear Techniques in Food and Agriculture, Wagramerstrasse 5, PO Box 100, A-1400, Vienna, Austria
Rafael Argilés
& Kostas Bourtzis
Laboratoire d'Entomologie Médicale, Institut Louis Malardé, BP 30, 98713, Papeete, Tahiti, French Polynesia
Hervé Bossin
IRD, AP-HM, SSA, VITROME, IHU-Méditerranée infection, Univ. Aix Marseille, Marseille, France
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MZ, GS, DA, CT, RA, IP designed the study. HB and KB provided the strains. GS designed and built the acquisition data system based on artificial vision. DA and CT performed the data collection. DA and MZ analyzed the data and wrote the manuscript. MZ proposed the theoretical model for data analysis. GS, CT, RA, IP and KB contributed to the manuscript drafting. All authors read and approved the final manuscript.
Correspondence to Carles Tur.
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution IGO License (https://creativecommons.org/licenses/by/3.0/igo/), which permits unrestricted use, distribution, and reproduction in any medium, provided appropriate credit to the original author(s) and the source is given.
Zacarés, M., Salvador-Herranz, G., Almenar, D. et al. Exploring the potential of computer vision analysis of pupae size dimorphism for adaptive sex sorting systems of various vector mosquito species. Parasites Vectors 11, 656 (2018) doi:10.1186/s13071-018-3221-x
Biometrical analysis
Morphometrics frequency distribution models
Sex sorting methods
Aedes albopictus
Aedes polynesiensis
Anopheles arabiensis
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CommonCrawl
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Geographical model-derived grid-based directional routing for massively dense WSNs
Jing-Ya Li1 &
Ren-Song Ko1
This paper presents the grid-based directional routing algorithms for massively dense wireless sensor networks. These algorithms have their theoretical foundation in numerically solving the minimum routing cost problems, which are formulated as continuous geodesic problems via the geographical model. The numerical solutions provide the routing directions at equally spaced grid points in the region of interest, and then, the directions can be used as guidance to route information. In this paper, we investigate two types of routing costs, position-only-dependent costs (e.g., hops, throughput, or energy) and traffic-proportional costs (which correspond to energy-load-balancing). While position-only-dependent costs can be approached directly from geodesic problems, traffic-proportional costs are more easily tackled by transforming the geodesic problem into a set of equations with regard to the routing vector field. We also investigate two numerical approaches for finding the routing direction, the fast marching method for position-only-dependent costs and the finite element method (and its derived distributed algorithm, Gauss-Seidel iteration with finite element method (DGSI-FEM)) for traffic-proportional costs. Finally, we present the numerical results to demonstrate the quality of the derived routing directions.
With their embedded computation and communication capabilities, wireless sensor networks (WSNs) can extend the senses of human beings to normally inaccessible locations and operate unattended for a long period of time, thus opening up the potential of many new applications [1]. Such applications bring up many challenges in network maintenance since sensors may be unreliable in hazardous situations which prohibit any human intervention to repair or replace malfunctioning sensors. Thus, compared to the cost to access WSNs, advanced developments in manufacturing techniques will make it preferable to deploy a large number of sensors in the region of interest (ROI) in one time, in which sensors can self-organize to operate. However, such a deployment strategy may lead to a massively dense WSN which poses many challenges for efficient algorithm design due to the problem scale and hardware constraints.
For such large-scale networks, the complexity of topological algorithms that model the networks by graphs and then describe network operations by nodes and edges may inevitably increase with the number of nodes and edges, since optimizing, particularly globally, the network performance may require the consideration that all nodes or edges determine the best node or edge to perform a given operation. However, two characteristics of WSNs suggest an alternative approach:
WSN applications are usually spatial-oriented, and spatially close nodes tend to perform the same role in networks.
Extending the working duration of the whole WSN is more important than keeping each sensor node alive. In other words, it may be preferable to exhaust individual nodes in an attempt to achieve better overall performance.
Therefore, rather than optimizing the performance of individual nodes by micro-controlling node operations, the high role substitutability of WSNs allows networks to be managed via geographical parameters, i.e., use the geographical parameters to locate appropriate sensor nodes to perform assigned tasks. Thus, network operations are described by geographical parameters, not node identities, and the complexity, even when considering global optimization, depends on the ROI, not the number of nodes and edges.
Furthermore, one advantage of geographical approaches is that we may use "distributions" or "vector fields" defined in geographical space to describe network states or operations, and these distributions or vector fields have some nice mathematical properties under massively dense networks, such as differentiability or integrability, which allow many techniques developed in classical mathematical analysis to be applicable. For example, several studies [2–6] have used geographical approaches to analyze WSN routing problems from a macroscopic perspective. Without the complexity of detailed descriptions in micromanaging individual nodes, the geographical descriptions can still provide sufficient information to allow meaningful analysis and optimization at the macroscopic level and the derivation of useful insights.
In this paper, we adopt the geographical model to study the minimum routing cost problems for massively dense WSNs in which the problems are formulated as continuous geodesic problems. We use density distributions to describe how nodes are deployed and routing vector fields for how information are transmitted. The relationship between density distributions and various routing costs may be further analyzed, and the equivalence between geodesic problems and optimum routing vector field problems can be established. We investigate two types of routing costs, position-only-dependent costs which are presented in the preliminary work [7] and traffic-proportional costs. Position-only-dependent costs may be the number of hops, throughput, or transmission energy, and traffic-proportional costs correspond to energy-load-balancing. While the routing problems with position-only-dependent costs can be tackled directly from geodesic problems, routing vector field problems provide a better approach to solve the routing problems with traffic-proportional costs.
Numerically solving continuous geodesic problems or routing vector field problems requires discretizing continuous functions involved in problems in a systematic way and then producing solutions (paths or vectors) at finite locations in the ROI, e.g., equally spaced grid points in the ROI. These numerical solutions at grid points provide the directions to the next forwarding nodes, which can be used as guidance to route information. Thus, the resulting routing algorithms, which we call grid-based directional routing algorithms, are actually the natural outcomes of the numerical approaches of these problems and mainly consist of the following two stages:
The ROI is divided into equally spaced grids, and then, each grid point computes its routing direction by numerically solving the continuous geodesic problems or routing vector field problems.
A node may use the routing direction of its closest grid point as guidance to determine its next forwarding node.
In this paper, we mainly focus on two numerical approaches for finding the routing direction of each grid point (i.e., the first stage), namely the fast marching (FM) method [8] for position-only-dependent costs and the finite element method (FEM) [9], including its derived distributed algorithm (namely distributed Gauss-Seidel iteration with FEM, DGSI-FEM), for traffic-proportional costs. We then investigate the quality of the derived routing directions via numerical simulations. Note that though the second stage is needed to completely determine a routing path, the study of the second stage is beyond the scope of this paper and we simply use the mechanism adopted in [10] for the second stage to conduct numerical simulations.
The remainder of this paper is organized as follows. After introducing related work in Section 2, we briefly describe the minimum cost routing problem from a macroscopic perspective and the equivalence between geodesic problems and optimum routing vector field problems in Section 3. The minimum routing cost problems with position-only-dependent costs and traffic-proportional costs, including algorithms and numerical results, are then discussed in Sections 4 and 5, respectively. Finally, conclusions are drawn in Section 6. For the sake of convenience, relevant notations introduced in this paper are listed in Table 1.
Table 1 List of notations introduced in this paper
Mauve et al. [11] argued that, for ad hoc networks, geographical routing scales better than topological routing even given frequently changing network topology. Several approaches are known to be suitable for WSNs, including greedy forwarding (GF) [12], in which each node uses the line segment to the destination to select the optimum forwarding node, and its various remedies [13–15] for the hole problem, in which packets may be trapped in local optima due to the existence of holes. In addition, a global pre-defined trajectory, instead of the local line segment used in GF, may be used to determine the next forwarding node [16].
For massively dense WSNs, several studies have applied analysis techniques developed in the disciplines other than networking to geographical models to analyze the macroscopic behavior of WSNs. For instance, Jacquet [2] analyzed how information traffic may impact the curvature of routing paths from the perspective of geometrical optics. Similarly, Catanuto et al. [5] formulated routing paths as equations of the calculus of variations which state that light follows the path that can be traversed in the least time, i.e., Fermat's principle. Additionally, Kalantari and Shayman [4] formulated the routing problems of WSNs as equations analogous to Maxwell's equations in electrostatic theory.
Jung et al. [17] considered spreading network traffic uniformly throughout the ROI using a potential field-based routing scheme in which the potential field is governed by Poisson's equation via an analogy between physics and network routing problems. Chiasserini et al. [18] used a fluid model to analyze a massively dense WSN in which the media access control and the switch between different operating modes, active and sleep, are considered. Altman et al. [19] analyzed the global optimized routing paths of massively dense networks using the techniques developed in road traffic engineering. Various approaches that work around the scalability problem by creating analogies between various WSN problems and problems in branches of mathematics and physics may be found in [20, 21].
Note that for the approaches mentioned above to be applicable, the massive denseness assumption is required for the validity of some mathematical properties such as continuity or differentiability. In addition to [22] which investigated the relation between the feasibility of such an assumption and node density, Ko [23] provided an operational definition of massively dense networks and then used the definition to derive the upper bound of analysis errors obtained from applying macroscopically derived results to nonmassively dense networks.
Minimum cost routing paths
Typically, a routing algorithm is designed with various optimization goals such as minimum total energy consumption or load-balancing. By introducing the transmission cost function \(\mathcal {C} (x_{v},y_{v})\) (i.e., the cost paid by the node v at (x v ,y v ) to transmit one unit amount of information), a routing problem may be formulated as a geodesic problem which minimizes the route cost \(\sum \limits _{v^{\prime } \in \llbracket {P}\rrbracket }\mathcal {C} (x_{v^{\prime }},y_{v^{\prime }})\). That is, a routing problem is to find a path P ∗ to a sink such that:
$$ \sum\limits_{v^{\prime} \in \llbracket{{P}^{\ast}}\rrbracket}\mathcal{C} (x_{v^{\prime}},y_{v^{\prime}}) \leq \sum\limits_{v^{\prime} \in \llbracket{P}\rrbracket}\mathcal{C} (x_{v^{\prime}},y_{v^{\prime}})\!\!\mid $$
((1))
in which P can be any possible path between a given source node v and any possible sink and ⟦P⟦ denotes the set of nodes on P.
To catch the operations, sensing and networking, we use ρ to represent the amount of information generated by a node located in the ROI (denoted as A) and define the routing vector field, \(\mathbf {D}: A \rightarrow \mathbb {R}^{2}\), in which the direction of D(x,y), called the routing direction and denoted as u f (x,y), points to the next forwarding node of the node at (x,y) and the length |D(x,y)| represents the amount of information transmitted by all nodes at (x,y).
Suppose that the information is conservative; that is, ρ does not consider the information generated and then disappears without being transmitted out, and each node in the ROI relays all the information it received. Thus, for v in A, the net amount of information flowing out of v should be equal to ρ(x v ,y v ). Therefore, we have the following theorem which states that the routing problem may be formulated as a geodesic problem (1) or an optimization problem for the routing vector field incurring the minimum total cost. For the proof please refer to [23].
Theorem 1.
Suppose that the information is conservative for a considered WSN. Hence, ∀v in A, \(\sum \limits _{v^{\prime } \in \llbracket {{P}^{\ast }}\rrbracket }\mathcal {C} (x_{v^{\prime }},y_{v^{\prime }})\) is minimum over all possible paths from v to sinks if and only if \(\sum \limits _{v \,\,\text {in}\, A}\mathcal {C} (x_{v},y_{v})\left |\mathbf {D}(x_{v},y_{v})\right |\) is minimum over all possible vector fields for a given ρ.
In the limit of massively dense networks, routing paths can be considered as continuous lines rather than sequences of discrete nodes [2]. Thus, the geodesic problem (1) may be formulated as the one to find the path P ∗ from (x 0,y 0) to a sink such that:
$$ \int_{{P}^{\ast}}\mathcal{C} (s)\mathrm{d}s \leq \int_{P}\mathcal{C} (s)\mathrm{d}s $$
in which P can be any possible path from (x 0,y 0) to any possible sink and s is the curvilinear coordinate associated with the path P ∗ or P. Similar to Theorem 1, the continuous geodesic problem (2) may be expected to be equivalent to the optimum routing vector field problem for massively dense networks; that is,
Suppose that the information is conservative for a considered WSN. Hence, ∀(x 0,y 0) in A, \(\int _{{P}^{\ast }}\mathcal {C} (x(s),y(s))\mathrm {d}s\) is minimum over all possible paths from (x 0,y 0) to sinks if and only if \(\int _{A}\mathcal {C} (x,y)\left |\mathbf {D}(x,y)\right |\mathrm {d}x\mathrm {d}y\) is minimum over all possible vector fields for a given ρ.
Some routing problems can be tackled via geodesic problems; for example, the cost function \(\mathcal {C} (x,y)\) is isotropic (e.g., sensor nodes with omni-directional antennas) and only depends on position. However, Theorem 2 provides an alternative that allows routing problems to be approached via D. One example is that \(\mathcal {C}(x,y)\) is proportional to |D(x,y)|. We will discuss these two types of \(\mathcal {C} {(x,y)}\), respectively, in Sections 4 and 5.
Position-only-dependent routing cost
Cost function and node density
This section considers the cost functions which are isotropic and only depends on position. Reference [24] discussed the relationship between the transmission energy as the cost and the node density ψ. Note that referring to [25], the energy consumption per unit of information is proportional to \({r}^{\alpha _{\textit {rf}}}\phantom {\dot {i}\!}\) in which r is the distance between the sender and receiver and the RF attenuation exponent α rf is typically in the range of 2 to 5. Additionally, the average inter-distance between nodes is proportional to \(1/\sqrt {\psi }\), which leads to \(\mathcal {C}\propto 1/\psi ^{\alpha _{\textit {rf}}/2}\).
As pointed out in [26], while considering the capacity of wireless communications, the throughput of each node at (x,y) cannot be fully utilized and is only proportional to \(1/\sqrt {\psi (x,y)}\) [3]. Therefore, the optimum total throughput at (x,y) can only be proportional to \(\sqrt {\psi (x,y)}\); that is, \(\mathcal {C} \propto 1/\sqrt {\psi }\) corresponds to a network in which the objective is to maximize the throughput.
Several other possible forms of \(\mathcal {C}\) are also listed in [5]. For example, if the objective is to minimize the number of hops, \(\mathcal {C}\) may be taken to be proportional to 1/r, in which communication is constrained between the nearest neighbors. Thus, \(\mathcal {C} \propto \sqrt {\psi }\). In addition, the case that \(\mathcal {C}\) is a constant corresponds to a setting where routing is equally costly at all parts of the network. Thus, the objective is to minimize the length of routes. The relationships between \(\mathcal {C}\) and ψ for the above objectives are summarized in Table 2.
Table 2 Relationship between \(\mathcal {C}\) and ψ
Grid approximation Dijkstra's method (GADM)
It is infeasible to directly find the minimum cost routing path under massively dense networks. One possible approach to reduce the problem scale is to divide the ROI into equally spaced grids which compose a grid point network, referring to Fig. 1. We then find the minimum cost path between each grid point and sink (e.g., using Dijkstra's method) under the grid point network. The routing direction of a grid point will be the direction pointing to the next grid point on the minimum cost path under the grid point network. For the example of Fig. 1, the direction from to is the routing direction of . Here, we denote as the grid point located at the ith column and the jth row, and say a node belongs to if its closest grid point in the ROI is ; for example, all nodes in the dark gray region belong to . Therefore, a node belonging to may use the direction from to as guidance to determine the next forwarding node [10].
Grid point network of ROI. The ROI is divided into equally spaced grids which compose a grid point network (grid points are connected by dashed lines). is the grid point located at the ith column and the jth row. A node is defined as belonging to if its closest grid point in the ROI is ; for example, all nodes in the dark gray region belong to . The path indicated by the blue solid line is the minimum cost routing path from to the grid point which the sink belongs to (indicated by the red circle). The black region represents the hole (the region without enough working sensors)
Note that the routing direction of derived by grid approximation Dijkstra's method (GADM) always points to one of s four adjacent grid points. Such a restriction is the main reason that GADM cannot approximate continuous paths well (i.e., the minimum cost routing paths under massively dense networks), which thus yields less optimum routing paths.
Fast marching (FM) method
Cost map and eikonal equation
Define the cost map T(x,y) as the minimum total routing cost needed from a node at (x,y) to sinks. Assume that T is differentiable. We then have the following theorem for which proof is given in Appendix 1:
$$ \left|\nabla T\right|=\mathcal{C} $$
In addition,
$$ \frac{\mathrm{d}{P}^{\ast}(s)}{\mathrm{d}s} \parallel -\nabla T $$
in which s is the curvilinear coordinate associated with the minimum cost path P ∗ and ∥ is the symbol for two parallel vectors.
Note that the a priori differentiability requirement of T may not be possible, e.g., existence of multiple sinks, in which case a weak solution may be considered instead. Refer to [27] for details.
Equation (3) is known as the eikonal equation, illustrating how a high-frequency wave front advances; T(x,y) corresponds to the time which the front takes to arrive at (x,y), and \(1/\mathcal {C}(x,y)\) is the speed of the front at (x,y). Theorem 3 indicates that if T may be solved from (3), the minimum cost path may be derived by following the gradient of T.
Geodesic path via eikonal equation
To solve (3), we adopt the FM method proposed by Sethian [8]. We first divide the definition domain of T into equally spaced grids with a gap size h and then approximate the differential terms by differences. Referring to Fig. 2, the definition domain of T should be large enough to cover the ROI. We distinguish the ROI and the definition domain of T to provide a consistent formula of difference approximation at the boundary of the ROI (via δ i,j introduced in (6)).
\(\tilde {f}_{{i},{j}}\) and . The definition domain of f (e.g., T or D) is divided into equally spaced grids with a grid size h. is the grid point located at the ith column and the jth row. \(\tilde {f}_{{i},{j}}\) is the value of f at . The set of grid points, marked by black circles, in A is denoted as . The grid points marked by white circles are not in
Various difference approximations to the length of gradient may be used. In this paper, the following less diffusive difference approximation to |∇T| [28] is chosen; that is, for in ROI, (3) is approximated as:
$$ {\small{\begin{aligned} & \left|\nabla \widetilde{T}_{{i},{j}}\right| = \widetilde{\mathcal{C}}_{{i},{j}}\\ \approx &\sqrt{\max\left(\Delta^{-x}_{i,j}T, -\Delta^{+x}_{i,j}T, 0\right)^{2} + \max\left(\Delta^{-y}_{i,j}T, -\Delta^{+y}_{i,j}T, 0\right)^{2}} \end{aligned}}} $$
in which:
$$\begin{aligned} \Delta^{-x}_{i,j}T=\delta_{i-1,j}\left(\frac{\widetilde{T}_{{i},{j}}-\widetilde{T}_{{i-1},{j}}}{h}\right), \\ \Delta^{+x}_{i,j}T=\delta_{i+1,j}\left(\frac{\widetilde{T}_{{i+1},{j}}-\widetilde{T}_{{i},{j}}}{h}\right), \\ \Delta^{-y}_{i,j}T=\delta_{i,j-1}\left(\frac{\widetilde{T}_{{i},{j}}-\widetilde{T}_{{i},{j-1}}}{h}\right), \\ \Delta^{+y}_{i,j}T=\delta_{i,j+1}\left(\frac{\widetilde{T}_{{i},{j+1}}-\widetilde{T}_{{i},{j}}}{h}\right). \end{aligned} $$
Here, \(\widetilde {T}_{{i},{j}}\) is the value of T at , and:
δ i,j is introduced to ensure a consistent difference formula with the grid points not in the ROI. Note that T is undefined for the grid point not in the ROI; thus, if is not in the ROI, δ i−1,j =0 will force \(\Delta ^{-x}_{i,j}T=0\) which corresponds to no information flow from to .
FM iteratively computes \(\widetilde {T}_{{i},{j}}\) starting from sinks via (5). Conceptually, the iteration of FM works as the wave front advances in the ROI. As the front advances in the ROI, Ts and states of the grid points are determined and updated iteratively as illustrated in Figs. 3 and 4: Upwind side: the zone which has been visited by the wave front. The states of grid points in the upwind zone are marked as accepted, and the values of \(\widetilde {T}\)s at these grid points have been determined. Since \( \mathcal {C}(x,y) > 0\), the front moves outward. Thus, the states of the accepted grid point will not be changed. Narrow band: the zone where the wave front is located. The states of grid points in this zone are marked as trial, and FM is determining the values of \(\widetilde {T}\)s at these grid points. Once finished, the grid point with the smallest \(\widetilde {T}\) in this zone will be included in the upwind side and the wave front expands further. Downwind side: the zone which has not been visited by the wave front. The states of grid points in this zone are marked as far away, and the values of \(\widetilde {T}\)s at these grid points have not been determined.
States of grid points in the process of FM. FM determines the minimum cost routing paths of all grid points to the sink (indicated by a red circle) in the order of wave expansion
Evolution of upwind side, narrow band, and downwind side during the iteration of Algorithm 1. The grid points in the upwind side and narrow band are marked by black circles and cyan circles, respectively
The algorithm of FM is listed in Algorithm 1. Here, , , and are the sets of the grid points in the upwind side, narrow band, and downwind side, respectively, and the neighbor set of , denoted as , is the set of s adjacent grid points in A, i.e., .
Initially, the entire ROI is the downwind side except the sinks which are marked as accepted with \(\widetilde {T}=0\) (Line 2); then, the wave front begins to expand (Line 3). FM uses (5) to compute the \(\widetilde {T}\)s of the grid points in the narrow band (Line 4). Once finished, the grid point with the smallest \(\widetilde {T}\) in the narrow band is marked as accepted (Lines 6–7). The wave front will then keep expanding (Lines 8–13) while updating the \(\widetilde {T}\)s of the grid points in the narrow band (Line 12) for the next iteration until the entire ROI is the upwind side (Lines 5–14). The state changes of grid points are illustrated in Fig. 4.
After determining \(\widetilde {T}\)s at all grid points by Algorithm 1, we may use \(\widetilde {T}\)s and (4) to derive the routing direction, \(\widetilde {\mathbf {u}_{\text {f}}}_{{i,j}}\), which is the unit tangent vector along the geodesic path from to the sink. By (4), the vector V=−∇T is tangent to the geodesic path. We may apply the finite difference method to approximate V: for in the ROI:
$$ \begin{aligned} \widetilde{\mathbf{V}}_{{i},{j_{x}}} & = -\frac{\delta_{i-1,j}\left(\widetilde {T}_{{i},{j}}-\widetilde {T}_{{i-1},{j}}\right)+\delta_{i+1,j}\left(\widetilde {T}_{{i+1},{j}}-\widetilde {T}_{{i},{j}}\right)}{\left(1+\delta_{i-1,j}\delta_{i+1,j}\right)h}\\ \widetilde {\mathbf{V}}_{{i},{j_{y}}} & = -\frac{\delta_{i,j-1}\left(\widetilde{T}_{{i},{j-1}}-\widetilde{T}_{{i},{j}}\right)+\delta_{i,j+1}\left(\widetilde {T}_{{i},{j}}-\widetilde {T}_{{i},{j+1}}\right)}{\left(1+\delta_{i,j-1}\delta_{i,j+1}\right)h} \end{aligned} $$
in which \(\widetilde {\mathbf {V}}_{{i},{j_{x}}}\) and \(\widetilde {\mathbf {V}}_{{i},{j_{y}}}\) are the x and y components of \(\widetilde {\mathbf {V}}_{{i},{j}}\), respectively. Note that it is easy to verify that the formula for \(\widetilde {\mathbf {V}}_{{i},{j}}\) in (7) is consistent with the finite difference approximation of ∇T at . In addition, \(\widetilde {\mathbf {V}}_{{i},{j_{x}}} = 0\) if both and are not in the ROI, which corresponds to zero traffic along the x-direction (the similar reasoning may apply to \(\widetilde {\mathbf {V}}_{{i},{j_{y}}}\)). Once \(\widetilde {\mathbf {V}}_{{i},{j}}\) is computed, \(\widetilde {\mathbf {u}_{\text {f}}}_{i,{j}}\) can be determined by \(\widetilde {\mathbf {u}_{\text {f}}}_{{i,j}} = \widetilde {\mathbf {V}}_{{i},{j}}/\left |\widetilde {\mathbf {V}}_{{i},{j}}\right |\).
Numerical results
We first present numerical results, illustrated in Figs. 5 and 6, to compare the effectiveness of GADM and FM. The settings of both scenarios, as summarized in Table 3, are similar except the number of sinks. Furthermore, the cost function \(\mathcal {C}\) considered is a constant; thus, the minimum cost path is the one with the shortest length.
Routing direction and routing paths. Here, the sink indicated by a circle is located at , and the black regions represent the holes. a Routing direction \(\widetilde {\mathbf {u_{f}}}\). b Routing paths: the source is close to the grid point . c Routing paths: the source is close to the grid point
Routing direction and routing paths. Here, the sinks indicated by circles are located at and , and the black regions represent the holes. a Routing direction \(\widetilde {\mathbf {u_{f}}}\). b Routing paths: the source is close to the grid point . c Routing paths: the source is close to the grid point
Table 3 Simulation settings for the scenarios illustrated in Figs. 5, 6, and 7
If the information is currently routed to a node, denoted as v, belonging to , we use \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}\) and the following mechanism adopted in [10] to determine the next forwarding node (i.e., the second stage of the grid-based directional routing algorithms).
Choose the neighbor nodes within the communication range R c of v which can make positive progress to sink. The progress of the neighbor node v ′ is defined as the inner product of \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}\) and the vector from v to v ′. If there are multiple candidates, choose the one which makes the greatest progress.
If no nodes are making positive progress, increase R c by ΔR c .
Note that due to the characteristics of wireless communication [3], it is preferred to use multiple short-range transmissions for optimal power consumption and communication capacity. Therefore, we gradually increase the communication range R c of v in searching for the next forwarding nodes to avoid long distance transmissions. The values of R c and ΔR c are also listed in Table 3.
Figure 5 a depicts the routing directions derived by FM. Figure 5 b, c illustrate the routing paths via the routing directions derived by GADM and FM. The route lengths listed in Table 4 show that FM may derive shorter routing paths than GADM. Note that GADM and FM may result in different routes to bypass the hole for the same source node, as illustrated in Fig. 5 c. Similar results may be found for the second scenario, referring to Fig. 6 and Table 4. In addition, GADM and FM may result in routing to different sinks for the same source node, as illustrated in Fig. 6 c.
Table 4 Length of routing path
The reason that FM outperforms GADM is that the minimum cost path derived under the grid point network may not approximate the actual minimum cost path well. In addition, the routing direction of a grid point always points to the neighbors of (that is, the four adjacent grid points of in our simulations). Though this problem may be alleviated by extending the neighbor set (for example, adding the diagonal grid points to the neighbor set), the direction restriction (the routing direction always points to one of the neighbors) cannot be removed. On the other hand, (5) used in Algorithm 1 approximates |∇T| well, and the routing direction (via using \(\widetilde {T}\)s and (7)) has no such direction restriction.
Figure 7 illustrates how node density may affect routing paths for the optimization objectives listed in Table 2 with α rf =4. Twenty thousand nodes are randomly deployed according to ψ(x,y)∝(3.5×10−5 y 2+0.02). Note that routing directions are solved (i.e., the first stage of the grid-based directional routing algorithms) using only the macroscopic parameter, ψ, but not the detailed position of each node. Thus, FM derives the same routing directions under the same density distribution regardless of the node positions. The node positions are merely used to determine the next forwarding node (i.e., the second stage) from the routing directions using the approach described earlier in this section.
Minimum cost paths for the optimization objectives listed in Table 2 with α rf =4. The scenario settings are listed in Table 3
The results show that routing should utilize the nodes in the sparse area to minimize the number of hops and use the nodes in the dense area to increase the throughput and to avoid long distance transmissions for less energy consumption. Of course, routing should use the straight line to the sink for minimizing the route length.
We also conducted simulations to compare the routing cost of \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}\) obtained from FM and GADM with the optimum routing cost determined by applying a shortest path algorithm to the connectivity graph of the WSN shown in Fig. 7, which basically is a microscopic routing approach. The results illustrated in Fig. 8 reveal that \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}\) obtained from FM may lead to a reasonable routing cost; the average cost is 5 % more than the average optimum cost. On the other hand, \(\widetilde {\mathbf {u_{f}}}_{i,j}\) obtained from GADM may have a routing cost up to 28 % more than the optimum cost. In Fig. 8, the mean of the routing cost is the average cost of all nodes to the sink. The relative mean of FM (or GADM) is defined as the mean of the routing cost of FM (or GADM) divided by the mean of the optimum routing cost.
The relative mean of routing cost of the scenario in Fig. 7: the mean of the routing cost of FM (or GADM) is the average cost of all nodes to the sink using the routing directions derived by FM (or GADM). The relative mean of FM (or GADM) is defined as the mean of the routing cost of FM (or GADM) divided by the mean of the optimum routing cost (OPT)
Traffic-proportional routing cost
Load-balancing routing
This section considers the case in which \(\mathcal {C}(x,y)= \lambda (x,y)^{2}\left |\mathbf {D}(x,y)\right |\); here, λ is the energy cost e, normalized to the initial energy E, for transmitting one unit of information, i.e., λ=e/E. As pointed out in [2], in the context of a massively dense network, routing paths can be considered as continuous lines, instead of sequences of discrete nodes, and D may be considered differentiable. Thus, the fact that information is conservative (i.e., at each location, the net amount of traffic is equal to the amount of information generated) can be formulated as [29]:
$$ \nabla \cdot \mathbf{D}(x,y) - \rho (x,y) = 0. $$
Thus, from Theorem 2, if (8) holds, the geodesic problem (2) with \(\mathcal {C}(x,y)=\lambda (x,y)^{2}\left |\mathbf {D}(x,y)\right |\) is equivalent to the optimization problem which finds the vector field D(x,y) to minimize:
$$ \int_{A}\lambda (x,y)^{2}\left|\mathbf{D}(x,y)\right|^{2}\mathrm{d}x\mathrm{d}y. $$
Note that the variance of λ|D|, \(\int _{A}\left (\lambda (x,y)\left |\mathbf {D}(x,y)\right |-\overline {\lambda \left |\mathbf {D}\right |}\right)^{2}\mathrm {d}x\mathrm {d}y\), is positive; here, \(\overline {\lambda \left |\mathbf {D}\right |}\) is the average of λ(x,y)|D(x,y)|. Since:
$$\begin{aligned} &\int_{A}\left(\lambda (x,y)\left|\mathbf{D}(x,y)\right|-\overline{\lambda \left|\mathbf{D}\right|}\right)^{2}\mathrm{d}x\mathrm{d}y \\ = &\int_{A}\lambda(x,y)^{2}\left|\mathbf{D}(x,y)\right|^{2}\mathrm{d}x\mathrm{d}y-\overline{\lambda\left|\mathbf{D}\right|}^{2}\cdot\text{area(}{A}\text{)} \end{aligned} $$
in which area(A) is the area of A; minimizing (9) not only minimizes the difference of each location's λ|D| but also inherently reduces \(\overline {\lambda \left |\mathbf {D}\right |}\).
Since λ is the normalized communication energy cost per unit of information, λ(x,y)|D(x,y)| is the normalized total communication energy consumption. Thus, it is not difficult to reason that keeping λ|D| the same everywhere in A is equivalent to exhausting the energy of each location in A simultaneously. In other words, the objective of the geodesic problem (2) with \( \mathcal {C}(x,y)=\lambda (x,y)^{2}\left |\mathbf {D}(x,y)\right |\) is to achieve global load-balancing (by minimizing the difference of each location's λ|D|, i.e., the variance) and reduce the total communication energy consumption (by reducing \(\overline {\lambda \left |\mathbf {D}\right |}\)).
As mentioned in [30], the necessary condition for deriving the minimum value of (9) is the existence of a scalar function Φ called potential that satisfies:
$$ \mathbf{D}=J\nabla \varPhi $$
in which J=1/λ 2. In addition, there is no information flow from the outside of A; that is, there is no traffic along the inward pointing normal direction at the boundary of A, denoted as ∂A, which leads to the following boundary condition:
$$ \mathbf{D}(x,y) \cdot \hat{\mathbf{n}}(x,y) = 0, \forall (x,y) \in \partial A $$
in which \(\hat {\mathbf {n}}\) is the unit inward pointing normal vector to ∂A.
Therefore, the minimum cost routing problem with the cost \( \mathcal {C} (x,y)=\lambda (x,y)^{2}\left |\mathbf {D}(x,y)\right |\) can be transformed into a set of partial differential equations that we call load-balancing routing equations, (8), (10), and (11). We may combine these equations into the following single equation called the weak formulation of the load-balancing routing equations:
$$ \underset{A}{\int }J\nabla \varPhi \cdotp \nabla \nu \mathrm{d}y\mathrm{d}x=-\underset{A}{\int}\rho \nu \mathrm{d}y\mathrm{d}x $$
in which ν is an arbitrary smooth scalar valued function.
Note that there is no differential term of D in (12), and the a priori differentiability requirement of D is weakened. Thus, the weak formulation allows us to consider irregular problems in which true solutions cannot be continuously differentiable [9], e.g., the problems in which ψ or ρ are jump functions in A. For the sake of brevity, the derivation of (12) is given in Appendix 2.
The relationship between J and the node density distribution ψ may be further established if the transmission energy consumption model is given. For example, we may adopt the energy consumption model in [25], in which the energy consumption per unit of information (denoted as e) is proportional to \(r^{\alpha _{\textit {rf}}}\phantom {\dot {i}\!}\). Here, r is the sender-to-receiver distance and the RF attenuation exponent α rf is typically in the range of 2 to 5. Since the average inter-distance between nodes is proportional to \(1/\sqrt {\psi }\), \(r \propto 1/\sqrt {\psi }\) and hence \( e\propto {\psi}^{-{\alpha}_{rf}/2} \). In addition, suppose that the nodes have an equal amount of initial energy; thus, the initial energy E is proportional to ψ, which leads to:
$$ J=1/{\lambda}^2={E}^2/{e}^2\propto {\psi}^{2+{\alpha}_{rf}}. $$
Finite element method (FEM) and DGSI-FEM algorithm
Equation (12) can be solved numerically by FEM in which (12) is locally approximated (posed over small partitions called elements of the entire ROI) and a global solution is built by combining the local solutions over these elements [9]. Similarly, referring to Fig. 2, we may divide the ROI into equally spaced grids and then use these grid points to form the elements (i.e., the gray hexagon on the x–y plane illustrated in Fig. 9).
A piecewise-linear finite element basis function. The linear basis function μ i,j is a pyramid with the peak at and is nonzero only within the element centered at (i.e., the gray hexagon). In addition, (x i ,y j ) is the position of
Consider the set of basis functions, μ i,j with , defined on the A such that μ i,j has the following properties:
$$\mu_{i,j}(x_{i^{\prime}}, y_{j^{\prime}})=\left\{ \begin{array}{ll} 1, & \text{if}\,\, i^{\prime}=i\,\, \text{and}\,\, j^{\prime}=j\\ 0, & \text{otherwise} \end{array} \right. $$
Here, \((x_{i^{\prime }}, y_{j^{\prime }})\) is the position of We then approximate Φ, J, and ρ, respectively, by:
in which \(\widetilde {\Phi }_{{i},{j}}=\Phi (x_{i}, y_{j})\), \( {\overset{\sim }{J}}_{i,j}=J\left({x}_i,{y}_j\right) \), and \(\widetilde {\rho }_{{i},{j}}=\rho (x_{i}, y_{j})\) (i.e., the values of Φ, J, and ρ at the grid point , respectively). By substituting (15), (16), and (17) into (12), we obtain the following set of linear equations:
One possible set of candidate functions satisfying (14) are pyramids with peaks at grid points as illustrated in Fig. 9. That is:
$$ {}\mu_{i,j}(x,y) =\left\{ \begin{array}{lll} -\frac{(x-x_{i})}{h}+1 & \text{if}\, (x,y)\, \text{is in}\, {_{i,j}}\triangle{^{i+1,j}_{i+1,j-1}}\\ \frac{(y-y_{j})}{h}+1 & \text{if}\, (x,y)\, \text{is in}\, {_{i,j}}\triangle{^{i+1,j-1}_{i,j-1}}\\ \frac{(x-x_{i})+(y-y_{j})}{h}+1 & \text{if}\, (x,y)\, \text{is in}\, {_{i,j}}\triangle{^{i,j-1}_{i-1,j}}\\ \frac{(x-x_{i})}{h}+1 & \text{if}\, (x,y)\, \text{is in}\, {_{i,j}}\triangle{^{i-1,j}_{i-1,j+1}}\\ -\frac{(y-y_{j})}{h}+1 & \text{if}\,(x,y)\, \text{is in}\, {_{i,j}}\triangle{^{i-1,j+1}_{i,j+1}} \\ -\frac{(x-x_{i})+(y-y_{j})}{h}+1 & \text{if}\, (x,y) \text{is in}\, {_{i,j}}\triangle{^{i,j+1}_{i+1,j}} \\ 0 & \text{otherwise.} \end{array} \right. $$
Here, \({_{i_{1},j_{1}}}\triangle {^{i_{2},j_{2}}_{i_{3},j_{3}}}\) is the triangle formed by and if all , and are in the ROI. If any of these three grid points is not in the ROI, \({_{i_{1},j_{1}}}\triangle {^{i_{2},j_{2}}_{i_{3},j_{3}}}\) is an empty region. That is:
With the linear basis functions (19), the coefficients, \(K^{i^{\prime },j^{\prime }}_{i,j}\) and g i,j , in (18) can be derived:
$$\begin{aligned} & g_{i,j} = -h^{2}/24\left(\mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i+1,j}_{i+1,j-1}} +\, \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i+1,j-1}_{i,j-1}}\right.\\ & + \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i,j-1}_{i-1,j}} +\, \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i-1,j}_{i-1,j+1}} \\ & + \left.\mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i-1,j+1}_{i,j+1}} +\, \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i,j+1}_{i+1,j}}\right), \end{aligned} $$
in which δ i,j is defined in (6) and:
$$\begin{aligned} &\mathcal{B}^{0}\left[{f}\right]_{{i,j}\triangle^{i_{1},j_{1}}_{i_{2},j_{2}}} = \delta_{i_{1},j_{1}}\delta_{i_{2},j_{2}}\left(\,\,\widetilde{f}_{{i},{j}} + \widetilde{f}_{{i_{1}},{j_{1}}} + \widetilde{f}_{{{i_{2}},{j_{2}}}}\right), \\ &\mathcal{B}^{1}\left[{f}\right]_{{i,j}\triangle^{i_{1},j_{1}}_{i_{2},j_{2}}} = \delta_{i_{1},j_{1}}\delta_{i_{2},j_{2}}\left(2\widetilde{f}_{{i},{j}} + \widetilde{f}_{{i_{1}},{j_{1}}} + \widetilde{f}_{{i_{2}},{j_{2}}}\right). \end{aligned} $$
Note that it is not difficult to verify that \(K^{i^{\prime },j^{\prime }}_{i,j}=K^{i,j}_{i^{\prime },j^{\prime }}\). For the sake of brevity, the detailed computation of \(K^{i^{\prime },j^{\prime }}_{i,j}\) and g i,j is given in Appendix 3.
The Gauss-Seidel iteration (GSI) may solve (18) for \(\widetilde {\Phi }_{{i},{j}}\) via iteratively updating each \(\widetilde {\Phi }_{{i},{j}}\) in lexicographical order from the most updated \(\widetilde {\Phi }\) value at other grid points until the update change \(\left |\widetilde {\Phi }_{{i},{j}}^{(k)}-\widetilde {\Phi }_{{i},{j}}^{(k-1)}\right | \leq \varepsilon \) for all That is, \(\widetilde {\Phi }_{{i},{j}}^{(k)}\) are computed sequentially by:
$$ \begin{aligned} &\widetilde {\Phi}_{{i},{j}}^{(k)} \leftarrow\\ &\frac{1}{K_{i,j}^{i,j}}\left(g_{i,j} - \sum\limits_{\substack{ \mathcal{O}_{L}(i^{\prime}, j^{\prime}) \\ < \mathcal{O}_{L}(i,j)}}K_{i,j}^{i^{\prime}, j^{\prime}}\widetilde {\Phi}_{{i^{\prime}},{j^{\prime}}}^{(k)}- \sum\limits_{\substack{\mathcal{O}_{L}(i^{\prime}, j^{\prime}) \\ > \mathcal{O}_{L}(i,j)}}K_{i,j}^{i^{\prime}, j^{\prime}}\widetilde{\Phi}_{{i^{\prime}},{j^{\prime}}}^{(k-1)}\right), \end{aligned} $$
in which \( \mathcal {O}_{L}(i,j)\) defines the lexicographical order; that is:
$$\mathcal{O}_{L}(i_{1},j_{1}) < \mathcal{O}_{L}(i_{2},j_{2})\,\text{if} \left\{ \begin{array}{ll} i_{1} < i_{2}, \text{or} & \\[2ex] i_{1} = i_{2}\, \text{and}\, j_{1} < j_{2}. \end{array} \right. $$
In GSI, only one \(\widetilde {\Phi }\) is updated in one iteration (21). We say GSI has gone through one sweep when each \(\widetilde {\Phi }\) has been updated once. \(\widetilde {\Phi }_{{i},{j}}^{(k)}\) is the value of \(\widetilde {\Phi }_{{i},{j}}\) after the kth sweep.
Note that if and . Thus, only \(\widetilde {\Phi }_{{i},{j-1}}^{(k)}\), \(\widetilde {\Phi }_{{i-1},{j}}^{(k)}\), \(\widetilde {\Phi }_{{i+1},{j}}^{(k-1)}\), and \(\widetilde {\Phi }_{{i},{j+1}}^{(k-1)}\) are needed to compute \(\widetilde {\Phi }_{{i},{j}}^{(k)}\) via (21). In other words, as long as \(\widetilde {\Phi }_{{i},{j-1}}^{(k)}\) and \(\widetilde {\Phi }_{{i-1},{j}}^{(k)}\) are computed, \(\widetilde {\Phi }_{{i},{j}}^{(k)}\) can be computed.
Accordingly, the distributed routing algorithm, DGSI-FEM, is proposed to coordinate sensors to solve \(\widetilde {\Phi }\)s from (18) in parallel using (21). In DGSI-FEM, a nearby node is selected as the grid head for each grid point to compute the value of \(\widetilde {\Phi }\). The grid head of may update \(\widetilde {\Phi }_{{i},{j}}\) as long as the most updated \(\widetilde {\Phi }_{{i},{j-1}}\) and \(\widetilde {\Phi }_{{i-1},{j}}\) are known; it does not need to wait for the grid heads of all the grid points with lexicographical order less than \(\mathcal {O}_{L}(i,j)\). Note that only these grid heads are involved in the computation of \(\widetilde {\Phi }\)s, resulting in low overhead for a massively dense network. For the sake of brevity, we simply describe the operations of grid points without explicitly mentioning that the operations are actually executed by grid heads.
Since the termination condition is that all the changes made by a sweep fall below a size threshold ε, each grid point needs to know all these changes. To achieve this, DGSI-FEM uses two state packets, PRECISE and DONE, for each grid point, which represent the convergence status and the termination decision, i.e., whether the update changes are small enough and whether the iteration should terminate, respectively. In addition, DGSI-FEM uses two phases (namely, a forward sweep followed by a backward sweep) to propagate the termination decision (via the state packet DONE) and collect the convergence status (via the state packet PRECISE) of all \(\widetilde {\Phi }\)s. Detailed DGSI-FEM is illustrated in Algorithm 2. Note that \(K_{i,j}^{i^{\prime },j^{\prime }}\) and \(\delta _{i^{\prime },j^{\prime }}\) for all and are known in advance. This may be done by letting each grid point discover its adjacent grid points and, once found, exchange \( {\overset{\sim }{J}}_{i,j} \) with them. Additionally, the algorithms for sending and waiting for messages are depicted in Algorithms 3 and 4, respectively. Both algorithms will check whether the communication counterpart is in the ROI and wait will return 〈0,true〉 if not.
After initialization (Lines 1–2), the iteration for will proceed as follows, referring to Fig. 10 for the sequence diagram of DGSI-FEM iteration:
Sequence diagram for in the iteration of DGSI-FEM. For example, in the forward sweep, and update and then send their \(\widetilde {\Phi }\)s and DONEs to After updating \(\widetilde {\Phi }_{{i},{j}}\) and DONE i,j , sends \(\widetilde {\Phi }_{{i},{j}}\) and DONE i,j to and . The dark rectangles drawn on top of the lifelines indicate that grid points are updating \(\widetilde {\Phi }\)s and DONEs (or PRECISEs in the backward sweep)
Forward sweep: iteration direction begins from bottom-left (the grid point with the smallest \(\mathcal {O}_{L}\)) to top-right (the grid point with the largest \(\mathcal {O}_{L}\)).
waits for \(\widetilde {\Phi }\)s and DONEs, respectively, from the down and left adjacent grid points, and , which have smaller \(\mathcal {O}_{L}\) values. (Lines 5–9)
updates \(\widetilde {\Phi }_{{i},{j}}\) by (21). (Line 10)
updates DONE i,j . (Lines 11–19)
sends \(\widetilde {\Phi }_{{i},{j}}\) and DONE i,j to the right and top adjacent grid points, and , respectively, which have larger \(\mathcal {O}_{L}\) values. (Lines 20–20)
Backward sweep: iteration direction moves from top-right to bottom-left.
waits for \(\widetilde {\Phi }\)s and PRECISEs from the top and right adjacent grid points, respectively. (Lines 27–29)
updates PRECISE i,j . (Lines 31–33)
sends \(\widetilde {\Phi }_{{i},{j}}\) and PRECISE i,j to the down and left adjacent grid points, respectively. (Lines 34–36)
Note that will set PRECISE i,j as true if the update made by itself is small enough and the PRECISEs of its top and right grid points are true (Lines 31–33). The PRECISEs collected by the \(\mathcal {O}_{L}\)-initiator, which has no bottom and left adjacent grid points (e.g., the grid point with the smallest lexicographical order), indicate whether the update changes of all \(\widetilde {\Phi }\)s are small enough and are used to determine the termination by the \(\mathcal {O}_{L}\)-initiator.
In addition, will set DONE i,j as true based on the following rules:
is an \(\mathcal {O}_{L}\)-initiator, and the update changes in the last backward sweep are small enough, i.e., PRECISE i,j is true (Lines 11–14), or
is not an \(\mathcal {O}_{L}\)-initiator, but DONE i,j−1=true and DONE i−1,j =true (Lines 15–19).
Note that DONEs are propagated from bottom-left to top-right in the forward sweep, so the termination proceeds from bottom-left to top-right. Thus, the if statement that checks whether the down or left adjacent grid points have terminated is added (Line 6) to avoid infinite waiting when waiting for messages from the down or left adjacent grid points (Line 7).
After \(\widetilde {\Phi }_{{i},{j}}\)s are solved from (18), \(\widetilde {\mathbf {D}}_{{i},{j}}\) may be approximated by the following formulae, which are derived by using (15) to approximate (10):
$$ \begin{array}{cc}{\overset{\sim }{\mathbf{D}}}_{i,{j}_x}=& \frac{\delta_{i+1,j}{\delta}_{i-1,j}{\overset{\sim }{J}}_{i,j}}{2h}\left({\overset{\sim }{\varPhi}}_{i+1,j}-{\overset{\sim }{\varPhi}}_{i-1,j}\right)\\ {}{\overset{\sim }{\mathbf{D}}}_{i,{j}_y}=& \frac{\delta_{i,j+1}{\delta}_{i,j-1}{\overset{\sim }{J}}_{i,j}}{2h}\left({\overset{\sim }{\varPhi}}_{i,j+1}-{\overset{\sim }{\varPhi}}_{i,j-1}\right)\end{array} $$
in which \(\widetilde {\mathbf {D}}_{{i},{j_{x}}}\) and \(\widetilde {\mathbf {D}}_{{i},{j_{y}}}\) are, respectively, the x and y components of \(\widetilde {\mathbf {D}}_{{i},{j}}\). For the sake of brevity, the derivation is given in Appendix 4. Once \(\widetilde {\mathbf {D}}_{{i},{j}}\) is computed, \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}\) can be determined by \(\widetilde {\mathbf {u_{f}}}_{{i},{j}}=\widetilde {\mathbf {D}}_{{i},{j}}/\left |\widetilde {\mathbf {D}}_{{i},{j}}\right |\) and then used as guidance to find the next forwarding nodes for routing information.
We present several numerical results to demonstrate the effectiveness of DGSI-FEM for different scenarios, namely the ROI with holes, the ROI with a nonuniform information generation rate, and the ROI with a nonuniform density. The simulation settings for these scenarios are listed in Table 5. Twenty thousand sensors are randomly deployed based on the density distribution ψ and generate information based on ρ except for the sink which will consume all the information generated. Similarly, routing directions are solved using only the macroscopic parameters, ψ and ρ, but not the detailed position of each node. The node positions are merely used to determine the next forwarding node from the routing directions. In addition, the energy consumption per unit of information is proportional to \(r^{\alpha _{\textit {rf}}}\phantom {\dot {i}\!}\) with α rf =2 and thus J=ψ 4 as indicated in (13).
Table 5 Simulation settings for the scenarios illustrated in Fig. 11
The routing directions obtained by DGSI-FEM are depicted as arrows in Fig. 11. The arrows provide the routing guidance for load-balancing. For example, Fig. 11 a reveals that information may be forwarded in a direction which deviates from a straight line to the sink to bypass the holes in advance. Thus, unlike many hole-bypassing algorithms [15, 31–33], using routing directions may alleviate the excess energy consumption of the boundary sensors.
Routing direction. All ROIs are square regions divided into 37×37 grids. Sinks which will consume all the information generated are marked as circles, and the arrows represent the routing directions, \(\widetilde {\mathbf {u_{f}}}\)s. a Uniform ψ and ρ: the black regions represent the holes. b Uniform ψ and nonuniform ρ: the sensors in the gray region generate ten times more information than other sensors. c Nonuniform ψ and uniform ρ: the gray region has 50 % higher sensor density than the white region. d Nonuniform density and uniform ρ: the gray region has 30 % lower sensor density than the white region
The routing directions shown in Fig. 11 b indicate that, to achieve load-balancing, information may be forwarded to the sensors outside the high- ρ region and then to the sink, instead of being delivered straight to the sink by the sensors in the high- ρ region. Particularly, some nodes around the bottom-right corner of the high- ρ region may forward packets in the opposite direction to the sink in order to avoid using nodes in the high- ρ region. Note that the sensors in the high- ρ region generate more events and potentially have more loading.
The routing directions shown in Fig. 11 c, d indicate that the information traffic tends to flow into the high-density regions and bypass the low-density regions to achieve load-balancing. Similar to Fig. 11 b, Fig. 11 d depicts that some nodes along the bottom-left boundary of the low-density region may forward packets in the opposite direction to the sink in order to avoid using nodes in the low-density region. In the last two scenarios, the ρs are uniform; thus, the sensors in the high-density (or low-density) region generate fewer (or more) events and potentially have less (or more) loading.
We then conducted simulations to compare the energy consumption of the routing directions obtained from DGSI-FEM with that of a microscopic routing algorithm, namely greedy perimeter stateless routing (GPSR) [31]. We used the approach described in Section 4.4 to route the information via the routing directions. Note that GPSR normally works as GF [12]; that is, the next forwarding node will be the one closest to the destination among the current sender's neighbors. However, if GF fails to find the node making any progress in delivering information, the node to the left in a planar subgraph of the connectivity graph of the WSN will be selected as the next forwarding node until GF recovers. The planarization used here is RNG [34].
We also conducted comparative simulations for another microscopic routing algorithm, namely geographical and energy aware routing (GEAR) [35], which attempts to achieve load-balancing by considering both the distance to the sink and the energy consumption. If a node has neighbors closer to the sink, GEAR will choose among these neighbors the one with the smallest weighted sum of the distance to the sink and the energy consumed as the next forwarding node; otherwise, the neighbor with the smallest weighted sum is the next forwarding node.
Figure 12 depicts the means and standard deviations of the energy consumption of DGSI-FEM and GEAR, normalized to the energy consumption of GPSR. Referring to Fig. 11 a, DGSI-FEM may forward information in a direction which deviates from a straight line to the sink to bypass the holes in advance, while our GPSR implementation uses the left-hand rule to forward information, thus resulting in excess energy consumption along the holes. Hence, DGSI-FEM may achieve better load-balancing with less energy consumption than GPSR for the ROI with holes.
The relative statistics of energy consumption of the scenarios in Fig. 11: the relative mean (or standard deviation) of DGSI-FEM (or GEAR) is defined as the mean (or standard deviation) of the routing energy consumption of DGSI-FEM (or GEAR) divided by the mean (or standard deviation) of the routing energy consumption of GPSR. a The relative mean of the energy consumption. b The relative standard deviation of the energy consumption
Furthermore, GPSR is degenerated to GF for the ROIs without holes and thus exhibits a lower average energy consumption for the scenarios shown in Fig. 11 b–d. On the other hand, DGSI-FEM will avoid the nodes in the high- ρ and low-density region and utilize the nodes in the high-density region for load-balancing. Thus, the routing paths will bypass the high- ρ and low-density regions or bend into the high-density region.
In addition, though GEAR strives to achieve load-balancing by considering the distance and energy factors, the best next forwarding node is still a local optimum; thus, GEAR provides less effective load-balancing than does DGSI-FEM. The standard deviation results in Fig. 12 b indicate that the routing directions solved by DGSI-FEM can effectively achieve load-balancing, particularly for the ROIs having holes.
This paper studies the minimum routing cost problems for massively dense WSNs via the geographical model, which leads to the grid-based directional routing algorithms. The minimum routing cost problems are formulated as continuous geodesic problems under massively dense WSNs, and the grid-based directional routing algorithms are the natural outcomes of numerically solving these problems; numerical solutions of the geodesic problems provide the directions to the next forwarding nodes at equally spaced grid points in the ROI, and these directions can be used as guidance to route information.
We first consider the position-only-dependent costs (e.g., hops, throughput, or energy) and investigate two numerical approaches, GADM and FM. GADM uses Dijkstra's method to determine the minimum cost path (under the grid point network). However, GADM may yield less optimum routing paths due to the direction restriction. On the other hand, by introducing the cost map T, the geodesic problem can be transformed into the eikonal equation and then solved by FM. Note that the geodesic problem considered here is to find a continuous curve which has the minimum cost from a given source to a sink. Our numerical results show that FM is more suitable than GADM for approximating the continuous curves and thus yields a path with less routing cost. The routing cost comparison results shows that FM has a routing cost 5 % more than the optimum cost, and GADM may have a routing cost 28 % more than the optimum cost.
We then consider the traffic-proportional costs which correspond to energy-load-balancing. By the equivalence between geodesic problems and optimum routing vector field problems, we transform the geodesic problem into a set of equations with regard to the routing vector field, which can be more easily tackled. We propose a distributed algorithm, i.e., DGSI-FEM, for solving the routing vector field via FEM and present the numerical results to demonstrate the quality of the derived routing directions. The routing energy consumption results show that routing directions may effectively achieve load-balancing than the microscopic routing algorithms, GPSR, and GEAR, particularly for the ROIs with holes.
Many aspects of this paper, the problems studied and the approaches taken, are the application of the existing work, e.g., minimum cost routing paths [23], cost function and node density [5, 24, 26], geodesic path via eikonal equation [27], fast marching method [8, 28], load-balancing routing equations [2, 24, 29, 30], and finite element method [9]. However, these works either do not specifically focus on the network routing problems or only theoretically analyze the routing problems without providing routing algorithms. The main contribution of this paper is to propose a systematic framework to develop low overhead routing algorithms for massively dense WSNs, i.e., coordinate sensor nodes themselves to solve the routing directions using these existing techniques and then route the information with the routing directions. In addition, there have existed numerous strategies for solving geodesic paths and PDEs. We believe this paper will open up a potential research direction toward the development of routing algorithms via investigation of the appropriateness of these strategies for implementation on WSNs.
Appendix 1: proof of theorem 3
For the sake of convenience, we use the position vector x to represent the position (x,y). Consider T(x) and for any dx, i.e., a small change of x:
$$ T(\mathbf{x} + \mathrm{d}\mathbf{x}) = T(\mathbf{x}) + \nabla T \cdot \mathrm{d}\mathbf{x} + \mathrm{O}\left({\left|\mathrm{d}\mathbf{x}\right|^{2}}\right) $$
by the Taylor expansion [29]. Let the cost of the straight line from x to x+dx be ΔT ′; then:
$$\Delta T^{\prime} = \mathcal{C}\left|\mathrm{d}\mathbf{x}\right| + \mathrm{O}\left({\left|\mathrm{d}\mathbf{x}\right|^{2}}\right). $$
Since T is the minimum cost:
$$T(\mathbf{x} + \mathrm{d}\mathbf{x}) \leq T(\mathbf{x}) + \Delta T^{\prime}. $$
Thus, choosing dx = small multiple of ∇T:
$$\left|\nabla T\right| \leq \mathcal{C}. $$
On the other hand, consider x and x+dx along a minimum cost path. We have:
$$T(\mathbf{x} + \mathrm{d}\mathbf{x})-T(\mathbf{x})=\mathcal{C}\left|\mathrm{d}\mathbf{x}\right| + \mathrm{O}\left({\left|\mathrm{d}\mathbf{x}\right|^{2}}\right), $$
and then by (22):
$$ \mathcal{C}\left|\mathrm{d}\mathbf{x}\right| + \mathrm{O}\left({\left|\mathrm{d}\mathbf{x}\right|^{2}}\right)=\nabla T \cdot \mathrm{d}\mathbf{x} = \left|\nabla T\right|\left|\mathrm{d}\mathbf{x}\right|\cos{\theta} $$
in which θ is the angle between ∇T and dx. Therefore, \(\left |\nabla T\right | \geq \mathcal {C}\), and (3) is proved.
Furthermore, consider x and x+dx along a minimum cost path. It is also clear from (3) and (23) that θ=0. Since both x and x+dx are on a minimum cost path, dx and thus ∇T are tangent to the minimum cost path.
Appendix 2: weak formulation of the load-balancing routing equations, (8), (10), and (11)
Multiply (8) by an arbitrary smooth scalar valued function ν and integrate it over the ROI; then:
$$\int_{A}\left(\nabla \cdot \mathbf{D} - \rho\right)\nu\mathrm{d}y\mathrm{d}x = 0. $$
By the product rule of a scalar valued function and a vector field:
$$\nabla\cdot\nu\mathbf{D} = \nu\nabla\cdot\mathbf{D} + \mathbf{D}\cdot\nabla\nu, $$
$$\int_{A}\left(\nabla\cdot\nu\mathbf{D}-\mathbf{D}\cdot\nabla\nu\right)\mathrm{d}y\mathrm{d}x - \int_{A}\rho\nu\mathrm{d}y\mathrm{d}x = 0, $$
and hence:
$$\int_{A}\mathbf{D}\cdot\nabla\nu\mathrm{d}y\mathrm{d}x = -\int_{A}\rho\nu\mathrm{d}y\mathrm{d}x + \int_{A}\nabla\cdot\nu\mathbf{D}\mathrm{d}y\mathrm{d}x. $$
By divergence theorem, we obtain:
$$\int_{A}\mathbf{D}\cdot\nabla\nu\mathrm{d}y\mathrm{d}x = -\int_{A}\rho\nu\mathrm{d}y\mathrm{d}x + \int_{\partial A}\nu\mathbf{D}\cdot \hat{\mathbf{n}}\mathrm{d}y\mathrm{d}x. $$
By substituting (10) and the boundary condition (11) into the above equation, we have the weak formulation (12).
Appendix 3: values of \(K^{i',j'}_{i,j}\) and g i,j
Referring to Fig. 9, for the element centered at (i.e., the gray hexagon), denoted H i,j as the set of vertices, that is:
and T i,j as the set of the triangles forming the element, that is:
$$ \begin{array}{cc}{T}_{i,j}=& \left\{{\kern1em }_{i,j}{\triangle}_{i+1,j-1}^{i+1,j},\kern0.3em {\kern1em }_{i,j}{\triangle}_{i,j-1}^{i+1,j-1},\kern0.60em {\kern1em }_{i,j}{\triangle}_{i-1,j}^{i,j-1},\kern0.60em {\kern1em }_{i,j}{\triangle}_{i-1,j+1}^{i-1,j},\right.\\ {}\left.{\kern1em }_{i,j}\triangle {\kern1em }_{i,j+1}^{i-1,j+1},\kern0.60em {\kern1em }_{i,j}{\triangle}_{i+1,j}^{i,j+1}\right\}.\end{array} $$
Similar to (15), we approximate ν by:
By substituting (15) and the above equation into (12), (12) becomes:
By reordering the summation and integral of the above equation, we then have:
Since ν is arbitrary, \(\widetilde {\nu }_{{i,j}}\) are arbitrary. Therefore, the above equation leads to:
By reordering the summation and integral of the above equation, we obtain:
Define:
$$ \begin{array}{cc}{K}_{i,j}^{i^{\prime },{j}^{\prime }}=& \underset{A}{\int }J\nabla {\mu}_{i^{\prime },{j}^{\prime }}\cdotp \nabla {\mu}_{i,j}\mathrm{d}y\mathrm{d}x\\ {}=& \underset{A}{\int }J\left(\frac{\partial {\mu}_{i^{\prime },{j}^{\prime }}}{\partial x}\frac{\partial {\mu}_{i,j}}{\partial x}+\frac{\partial {\mu}_{i^{\prime },{j}^{\prime }}}{\partial y}\frac{\partial {\mu}_{i,j}}{\partial y}\right)\mathrm{d}y\mathrm{d}x\end{array} $$
$$g_{i,j} = -\int_{A}\rho \mu_{i,j}\mathrm{d}y\mathrm{d}x. $$
Then (24) can be written as (18).
From (19):
$$ \frac{\partial \mu_{i,j}(x,y)}{\partial x} =\left\{ \begin{array}{lll} -1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i+1,j}_{i+1,j-1}\\ 0 & \text{if}\, \,(x,y)\, \, \text{is in}_{i,j}{\triangle}^{i+1,j-1}_{i,j-1} \\ 1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i,j-1}_{i-1,j}\\ 1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i-1,j}_{i-1,j+1} \\ 0 & \text{if} \, \,(x,y)\,\, \text{is in}_{i,j}{\triangle}^{i-1,j+1}_{i,j+1} \\ -1/h & \text{if}\, \, (x,y) \text{is in}_{i,j}{\triangle}^{i,j+1}_{i+1,j} \\ 0 & \text{otherwise} \end{array} \right. $$
$$ \frac{\partial \mu_{i,j}(x,y)}{\partial y} =\left\{ \begin{array}{lll} 0 & \text{if}\, \,(x,y)\,\, \text{is in}_{i,j}{\triangle}^{i+1,j}_{i+1,j-1} \\ 1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i+1,j-1}_{i,j-1} \\ 1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i,j-1}_{i-1,j} \\ 0 & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i-1,j}_{i-1,j+1}\\ -1/h & \text{if}\, \,(x,y)\, \, \text{is in}_{i,j}{\triangle}^{i-1,j+1}_{i,j+1} \\ -1/h & \text{if}\, \, (x,y)\, \, \text{is in}_{i,j}{\triangle}^{i,j+1}_{i+1,j} \\ 0 & \text{otherwise} \end{array} \right. $$
Note that if and the element centered at and the element centered at do not overlap; therefore, it is not difficult to verify that:
$$\frac{\partial \mu_{i^{\prime},j^{\prime}}}{\partial x}\frac{\partial \mu_{i,j}}{\partial x} + \frac{\partial \mu_{i^{\prime},j^{\prime}}}{\partial y}\frac{\partial \mu_{i,j}}{\partial y} = 0\; \text{and hence}\; K^{i^{\prime},j^{\prime}}_{i,j} = 0. $$
In addition, if (x,y) in \( {}_{i_1}{,}_{j_1}{\triangle}_{i_3,{j}_3}^{i_2,{j}_2}\in {T}_{i_1,{j}_1} \), (x,y) is located within the elements centered at and and \( J\left(x,y\right)=\sum_{k=1}^3{\overset{\sim }{J}}_{i_k,{j}_k}{\mu}_{i_k,{j}_k}\left(x,y\right) \) by (16) and (19). For example, for (x,y) in \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\):
$$ \begin{array}{cc}J\left(x,y\right)=& {\overset{\sim }{J}}_{i,j}{\mu}_{i,j}\left(x,y\right)+{\overset{\sim }{J}}_{i+1,j}{\mu}_{i+1,j}\left(x,y\right)\\ {}+{\overset{\sim }{J}}_{i+1,j-1}{\mu}_{i+1,j-1}\left(x,y\right).\end{array} $$
Thus:
$$ \begin{array}{cc}\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }J\mathrm{d}y\mathrm{d}x& ={\overset{\sim }{J}}_{i,j}\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }{\mu}_{i,j}\mathrm{d}y\mathrm{d}x\\ {}\kern1em +{\overset{\sim }{J}}_{i+1,j}\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }{\mu}_{i+1,j}\mathrm{d}y\mathrm{d}x\\ {}\kern1em +{\overset{\sim }{J}}_{i+1,j-1}\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }{\mu}_{i+1,j-1}\mathrm{d}y\mathrm{d}x.\end{array} $$
Note that referring to Fig. 9, \( \int {_{_{i,j}\triangle ^{i+1,j}_{i+1,j-1}}}\mu _{i,j}\mathrm {d}y\mathrm {d}x\) is the volume of the triangular pyramid formed by the vertex (x i ,y j ,μ i,j (x i ,y j )) and \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\); here, the volume is h 2/6, since μ i,j (x i ,y j )=1 and the area of \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\) is h 2/2. The same argument can be used to compute the rest of two integrals in the above equation. Thus, referring to (20) and (6):
$$ \begin{array}{c}\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }J\mathrm{d}y\mathrm{d}x=\\ {}{\delta}_{i+1,j}{\delta}_{i+1,j-1}{h}^2/6\left({\overset{\sim }{J}}_{i,j}+{\overset{\sim }{J}}_{i+1,j}+{\overset{\sim }{J}}_{i+1,j-1}\right).\end{array} $$
Here, δs are added to check whether the vertices of \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\) are in the ROI.
Similarly, this integral computation can apply to other \(\vartriangle \)s in T i,j , and we have:
$$ \underset{i,j{\triangle}_{i_2,{j}_2}^{i_1,{j}_1}}{\int }J\mathrm{d}y\mathrm{d}x={\delta}_{i_1,{j}_1}{\delta}_{i_2,{j}_2}{h}^2/6\left({\overset{\sim }{J}}_{i,j}+{\overset{\sim }{J}}_{i_1,{j}_1}+{\overset{\sim }{J}}_{i_2,{j}_2}\right), $$
for i,j △i 2,j 2 i 1,j 1∈T i,j . By denoting:
$$\mathcal{B}^{0}\left[{f}\right]_{{i,j}\triangle^{i_{1},j_{1}}_{i_{2},j_{2}}} = \delta_{i_{1},j_{1}}\delta_{i_{2},j_{2}}\left(\,\,\widetilde{f}_{{i},{j}} + \widetilde{f}_{{i_{1}},{j_{1}}} + \widetilde{f}_{{i_{2}},{j_{2}}}\right), $$
we then have:
$$ \underset{i,j{\triangle}_{i_2,{j}_2}^{i_1,{j}_1}}{\int }J\mathrm{d}y\mathrm{d}x=\frac{h^2}{6}{\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j}{\triangle}_{i_1,{j}_2}^{i_1,{j}_1}. $$
Thus, by (25) and (26), for
$$ \begin{array}{cc}{K}_{i,j}^{i,j}=& 1/{h}^2\left(\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }J\mathrm{d}y\mathrm{d}x+\underset{{}_{i,j}{\triangle}_{i,j-1}^{i+1,j-1}}{\int }J\mathrm{d}y\mathrm{d}x\kern0.3em +\right.\\ {}2\underset{{}_{i,j}{\triangle}_{i-1,j}^{i,j-1}}{\int }J\mathrm{d}y\mathrm{d}x+\underset{{}_{i,j}{\triangle}_{i-1,j+1}^{i-1,j}}{\int }J\mathrm{d}y\mathrm{d}x\kern0.3em +\\ {}\left.\underset{{}_{i,j}{\triangle}_{i,j+1}^{i-1,j+1}}{\int }J\mathrm{d}y\mathrm{d}x+2\underset{{}_{i,j}{\triangle}_{i+1,j}^{i,j+1}}{\int }J\mathrm{d}y\mathrm{d}x\right)\\ {}=& 1/6\left({\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i+1,j-1}^{i+1,j}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i,j-1}^{i+1,j-1}}+\kern0.3em 2{\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i-1,j}^{i,j-1}}\right.\\ {}+& \left.{\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i-1,j+1}^{i-1,j}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i,j+1}^{i-1,j+1}}+\kern0.3em 2{\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i+1,j}^{i,j+1}}\right).\end{array} $$
Now we compute \(K^{i^{\prime },j^{\prime }}_{i,j}\)s for We first consider Referring to (25), the only \(\vartriangle \)s for both ∂μ i,j /∂x≠0 and ∂μ i+1,j /∂x≠0 are \(_{i,j}\triangle ^{i,j+1}_{i+1,j}\) and \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\), and there is no \(\vartriangle \) for both ∂μ i,j /∂y≠0 and ∂μ i+1,j /∂y≠0. Therefore:
$$ \begin{array}{cc}{K}_{i,j}^{i+1,j}=& -1/{h}^2\left(\underset{{}_{i,j}{\triangle}_{i+1,j}^{i,j+1}}{\int }J\mathrm{d}y\mathrm{d}x+\underset{{}_{i,j}{\triangle}_{i+1,j-1}^{i+1,j}}{\int }J\mathrm{d}y\mathrm{d}x\right)\\ {}=& -1/6\left({\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i+1,j}^{i,j+1}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i+1,j-1}^{i+1,j}}\right).\end{array} $$
The same argument can be used to compute \(K^{i^{\prime },j^{\prime }}_{i,j}\)s for the rest of in H i,j , and we have:
$$ \begin{array}{c}{K}_{i,j}^{i+1,j-1}=0,\\ {}{K}_{i,j}^{i,j-1}=-1/6\left({\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i,j-1}^{i+1,j-1}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i-1,j}^{i,j-1}}\right),\\ {}{K}_{i,j}^{i-1,j}=-1/6\left({\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i-1,j}^{i,j-1}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i-1,j+1}^{i-1,j}}\right),\\ {}{K}_{i,j}^{i-1,j+1}=0,\\ {}{K}_{i,j}^{i,j+1}=-1/6\left({\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i,j+1}^{i-1,j+1}}+\kern0.3em {\mathcal{B}}^0{\left[\kern0.3em J\right]}_{i,j{\triangle}_{i+1,j}^{i,j+1}}\right).\end{array} $$
Note that as mentioned earlier, \(K^{i^{\prime },j^{\prime }}_{i,j} = 0\) if and Thus, together with \(K^{i+1,j-1}_{i,j}= 0\) and \(K^{i-1,j+1}_{i,j}= 0\), we have:
To compute g i,j , we use (17) to expand g i,j as follows:
If and the element centered at and the element centered at do not overlap; therefore, it is not difficult to verify from (19) that \(\mu _{i^{\prime },j^{\prime }} \mu _{i,j} = 0\). Hence:
In addition, it is obvious that if \( \vartriangle \notin {T}_{i,j} \), \(\int _{\vartriangle }\mu _{i^{\prime },j^{\prime }} \mu _{i,j}\mathrm {d}y\mathrm {d}x=0\). We only need to compute the integral over the region \( \vartriangle \in {T}_{i,j} \).
We first consider the integral over \(_{i,j}\triangle ^{i+1,j}_{i+1,j-1}\):
The computation of each integral of the above equation is carried out as follows:
$${\small{\begin{aligned} & \int_{_{i,j}\triangle^{i+1,j}_{i+1,j-1}}\mu_{i,j} \mu_{i,j}\mathrm{d}y\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}\int_{x_{i}+y_{j}-x}^{y_{j}}\left(-(x-x_{i})+h\right)^{2}\mathrm{d}y\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}\left(-(x-x_{i})+h\right)^{2}y|_{x_{i}+y_{j}-x}^{y_{j}}\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}\left(-(x-x_{i})+h\right)^{2}(x-x_{i})\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}h\left(-(x-x_{i})+h\right)^{2} \\ & \quad-\left(-(x-x_{i})+h\right)^{3}\mathrm{d}x\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\left(-h/3\left(-(x-x_{i})+h\right)^{3}\right.\\ & \left.\quad+\,1/4\left(-(x-x_{i})+h\right)^{4}\right)|_{x_{i}}^{x_{i}+h}\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2} \left(h^{4}/3-h^{4}/4\right) \\ & = \delta_{i+1,j}\delta_{i+1,j-1}h^{2}/12, \end{aligned}}} $$
$${\small{\begin{aligned} & \int_{_{i,j}\triangle^{i+1,j}_{i+1,j-1}}\mu_{i+1,j} \mu_{i,j}\mathrm{d}y\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}\int_{x_{i}+y_{j}-x}^{y_{j}}\left((x-x_{i})+(y-y_{j})\right)\\ &\qquad\qquad\qquad\qquad\qquad\left(-(x-x_{i})+h\right)\mathrm{d}y\mathrm{d}x\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\int_{x_{i}}^{x_{i}+h}\left(-(x-x_{i})+h\right)\\ &\qquad\qquad\qquad\qquad\qquad\left((x-x_{i})+(y-y_{j})\right)^{2}|_{x_{i}+y_{j}-x}^{y_{j}}\mathrm{d}x\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\int_{x_{i}}^{x_{i}+h}\left(-(x-x_{i})+h\right)(x-x_{i})^{2}\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\int_{x_{i}}^{x_{i}+h}h(x-x_{i})^{2}-(x-x_{i})^{3}\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\left(h(x-x_{i})^{3}/3-(x-x_{i})^{4}/4\right)|_{x_{i}}^{x_{i}+h} \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\left(h^{4}/3-h^{4}/4\right) \\ & = \delta_{i+1,j}\delta_{i+1,j-1}h^{2}/24, \end{aligned}}} $$
$${\small{\begin{aligned} & \int_{_{i,j}\triangle^{i+1,j}_{i+1,j-1}}\mu_{i+1,j-1} \mu_{i,j}\mathrm{d}y\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/h^{2}\int_{x_{i}}^{x_{i}+h}\int_{x_{i}+y_{j}-x}^{y_{j}}\left(-(y-y_{j})\right)\\ &\qquad\qquad\qquad\qquad\qquad\left(-(x-x_{i})+h\right)\mathrm{d}y\mathrm{d}x\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\int_{x_{i}}^{x_{i}+h}-(y-y_{j})^{2}\\ &\qquad\qquad\qquad\qquad\qquad\left(-(x-x_{i})+h\right)|_{x_{i}+y_{j}-x}^{y_{j}}\mathrm{d}x\\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\int_{x_{i}}^{x_{i}+h}(x-x_{i})^{2}\left(-(x-x_{i})+h\right)\mathrm{d}x \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\left(h(x-x_{i})^{3}/3-(x-x_{i})^{4}/4\right)|_{x_{i}}^{x_{i}+h} \\ & = \delta_{i+1,j}\delta_{i+1,j-1}/2h^{2}\left(h^{4}/3-h^{4}/4\right) \\ & = \delta_{i+1,j}\delta_{i+1,j-1}h^{2}/24. \end{aligned}}} $$
By denoting:
$$\mathcal{B}^{1}\left[{f}\right]_{{i,j}\triangle^{i_{1},j_{1}}_{i_{2},j_{2}}} = \delta_{i_{1},j_{1}}\delta_{i_{2},j_{2}}\left(2\widetilde{f}_{{i},{j}} + \widetilde{f}_{{i_{1}},{j_{1}}} + \widetilde{f}_{{i_{2}},{j_{2}}}\right), $$
The same computation can be carried out for the rest of △s in T i,j , and we have:
$$\begin{aligned} g_{i,j} &= -h^{2}/24\left(\mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i+1,j}_{i+1,j-1}} + \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i+1,j-1}_{i,j-1}}\right. \\ & \quad+ \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i,j-1}_{i-1,j}} + \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i-1,j}_{i-1,j+1}} \\ & \quad+ \left.\mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i-1,j+1}_{i,j+1}} + \mathcal{B}^{1}[\!{\rho}]_{{i,j}\triangle^{i,j+1}_{i+1,j}}\right). \end{aligned} $$
Appendix 4: values of \(\widetilde {\mathbf {D}}_{{i},{j}}\)
By (10) and (15) together with the boundary condition (11):
$$ \begin{array}{cc}{\overset{\sim }{\mathbf{D}}}_{i,{j}_x}& ={\mathbf{D}}_x\left({x}_i,{y}_j\right)={\delta}_{i+1,j}{\delta}_{i-1,j}J\left({x}_i,{y}_j\right)\frac{\partial \varPhi }{\partial x}\left({x}_i,{y}_j\right)\\ {}={\delta}_{i+1,j}{\delta}_{i-1,j}{\overset{\sim }{J}}_{i,j}{ \lim}_{\varDelta x\to 0}\frac{\varPhi \left({x}_i+\varDelta x,{y}_j\right)-\varPhi \left({x}_i-\varDelta x,{y}_j\right)}{2\varDelta x}.\end{array} $$
Here, D x is the x component of D and (x i ,y j ) is the position of
Note that (x i +Δx,y j ) is located within the elements centered at and . Thus:
$$\begin{aligned} \Phi(x_{i}+\Delta x, y_{j}) = & \widetilde{\Phi}_{{i},{j}}\mu_{i,j}(x_{i}+\Delta x, y_{j})\\ & + \widetilde{\Phi}_{{i+1},{j}}\mu_{i+1,j}(x_{i}+\Delta x, y_{j}) \\ & + \widetilde{\Phi}_{{i},{j+1}}\mu_{i,j+1}(x_{i}+\Delta x, y_{j}) \\ & + \widetilde{\Phi}_{{i+1},{j-1}}\mu_{i+1,j-1}(x_{i}+\Delta x, y_{j}). \end{aligned} $$
From (19), μ i,j+1(x i +Δx,y j )=0 and μ i+1,j−1(x i +Δx,y j )=0. Thus:
$${\small{\begin{aligned} {}\Phi(x_{i}+\Delta x, y_{j}) \\ = & \widetilde{\Phi}_{{i},{j}}\mu_{i,j}(x_{i}+\Delta x, y_{j}) + \widetilde{\Phi}_{{i+1},{j}}\mu_{i+1,j}(x_{i}+\Delta x, y_{j}). \end{aligned}}} $$
$$\mu_{i,j}(x_{i}+\Delta x, y_{j}) = -\Delta x/h + 1, $$
$$\mu_{i+1,j}(x_{i}+\Delta x, y_{j}) = \Delta x/h. $$
$$\Phi(x_{i}+\Delta x, y_{j}) = \left(-\Delta x/h + 1\right)\widetilde{\Phi}_{{i},{j}} + \left(\Delta x/h\right)\widetilde{\Phi}_{{i+1},{j}}. $$
Similarly:
$$\Phi(x_{i}-\Delta x, y_{j}) = \left(-\Delta x/h + 1\right)\widetilde{\Phi}_{{i},{j}} + \left(\Delta x/h\right)\widetilde{\Phi}_{{i-1},{j}}. $$
$${}{\lim}_{\Delta x \to 0}\frac{\Phi(x_{i}+\Delta x, y_{j})-\Phi(x_{i}-\Delta x, y_{j})}{2\Delta x} = \frac{\widetilde{\Phi}_{{i+1},{j}} - \widetilde{\Phi}_{{i-1},{j}}}{2h}. $$
Therefore:
$$ {\overset{\sim }{\mathbf{D}}}_{i,{j}_x}=\frac{\delta_{i+1,j}{\delta}_{i-1,j}{\overset{\sim }{J}}_{i,j}}{2h}\left({\overset{\sim }{\varPhi}}_{i+1,j}-{\overset{\sim }{\varPhi}}_{i-1,j}\right). $$
$$ {\overset{\sim }{\mathbf{D}}}_{i,{j}_y}=\frac{\delta_{i,j+1}{\delta}_{i,j-1}{\overset{\sim }{J}}_{i,j}}{2h}\left({\overset{\sim }{\varPhi}}_{i,j+1}-{\overset{\sim }{\varPhi}}_{i,j-1}\right). $$
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This research was supported by Ministry of Science and Technology of Taiwan, under Grant MOST 104-2221-E-194-008. The authors gratefully acknowledge this support.
Department of Computer Science and Information Engineering, National Chung Cheng University, No. 168, Sec. 1, University Road, Chia-Yi, 621, Taiwan
Jing-Ya Li
& Ren-Song Ko
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Correspondence to Ren-Song Ko.
Li, J., Ko, R. Geographical model-derived grid-based directional routing for massively dense WSNs. J Wireless Com Network 2016, 17 (2016). https://doi.org/10.1186/s13638-015-0492-1
Geographical routing algorithm
Geodesic problem
Fast marching method
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CommonCrawl
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\begin{definition}[Definition:Differentiable Mapping/Real Function/Real Number Line]
Let $f$ be a real function defined on $\R$.
By definition, $\R$ is an (unbounded) open interval.
Let $f$ be differentiable on the open interval $\R$.
That is, let $f$ be differentiable at every point of $\R$.
Then $f$ is '''differentiable everywhere (on $\R$)'''.
Category:Definitions/Differentiable Real Functions
\end{definition}
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ProofWiki
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