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The problem of approximating a given probability distribution using a simpler distribution plays an important role in several areas of machine learning, e.g. variational inference and classification. Within this context, we consider the task of learning a mixture of tree distributions. Although mixtures of trees can be learned by minimizing the KL-divergence using an EM algorithm, its success depends heavily on the initialization. We propose an efficient strategy for obtaining a good initial set of trees that attempts to cover the entire observed distribution by minimizing the $\alpha$-divergence with $\alpha = \infty$. We formulate the problem using the fractional covering framework and present a convergent sequential algorithm that only relies on solving a convex program at each iteration. Compared to previous methods, our approach results in a significantly smaller mixture of trees that provides similar or better accuracies. We demonstrate the usefulness of our approach by learning pictorial structures for face recognition. | CommonCrawl |
Frank den Hollander is professor of mathematics at Leiden University in The Netherlands. His research focuses on probability theory, ergodic theory, statistical physics, population dynamics and complex networks. Frank has supervised 13 PhD students and 33 postdocs, has published 150 papers, and is the author of three monographs. He has served on strategic advisory boards across Europe, and is Fellow of the American Mathematical Society and of the Institute of Mathematical Statistics. Frank is member of the Royal Dutch Academy of Sciences. He has been awarded 25 national and international research grants, including an ERC Advanced Grant and a ten-year consortium grant by the Dutch Ministry of Education, Culture and Science called NETWORKS.
Frank's Medallion Lecture will be delivered at the World Congress in Toronto in July 2016.
Metastability is the phenomenon where a physical system, under the influence of a stochastic dynamics, moves between different subregions of its state space on different time scales. Metastability is encountered in a wide variety of physical systems. The challenge is to devise realistic models and to explain the experimentally observed universality displayed in metastable behaviour, both qualitatively and quantitatively. For an overview, see the monograph by Bovier and den Hollander1.
In statistical physics, metastability is the dynamical manifestation of a first-order phase transition. An example is condensation. When water vapour is cooled down slightly below 100 degrees Celsius, it persists for a very long time in a metastable vapour state before transiting to a stable liquid state under the influence of random fluctuations. The crossover occurs after the system manages to create a critical droplet of liquid inside the vapour, which once present grows and invades the system. While in the metastable vapour state, the system makes many unsuccessful attempts to form a critical droplet, because this requires the system to climb an 'energetic hill'.
In the static Widom–Rowlinson model5, particles are viewed as points carrying disks, and the energy of a particle configuration equals minus the total overlap of the disks (consequently, the interaction between the particles is attractive). It was shown by Ruelle4 and by Chayes, Chayes and Kotecky2 that, on the infinite Euclidean space, this model has a first-order phase transition between a vapour state and a liquid state. The phase transition occurs as the chemical potential, controlling the density of the particles, changes from a subcritical value to a supercritical value. The model is therefore a natural candidate to display metastable behaviour.
with $\beta \in (0,\infty)$ the inverse temperature and $(\Delta H)_+$ the positive part of the change in the Hamiltonian caused by the move ($H$ depends both on the energy and on the chemical potential).
We are interested in the metastable behaviour at low temperature when the chemical potential is supercritical. In particular, we start with the empty torus (which represents the vapour state) and are interested in the first time when we reach the full torus, i.e., the torus is fully covered by disks (which represents the liquid state). In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet of overlapping disks, which plays the role of the critical droplet that triggers the crossover. In the limit as $\beta\to\infty$ (low-temperature limit), we compute the asymptotic scaling of the average crossover time, show that the crossover time divided by its average is exponentially distributed, and identify the size and the shape of the critical droplet.
The leading term corresponds to the classical Arrhenius law, with U(Rc(κ)) the energy of the critical droplet. The correction term, which is large, represents a substantial deviation from this law, with S(Rc(κ)) the entropy of the critical droplet. This correction term comes from the fact that the boundary of the critical droplet is 'bumpy', with many particles sticking out just a little. It turns out that there are ≍β particles inside the critical droplet, ≍β1/3 particles touching the boundary, while the boundary itself fluctuates on scale β−1/2.
Above: The one-species model as the projection of the two-species model.
The Widom–Rowlinson model can be viewed as the projection of a two-species model with hard-core repulsion in which one of the species is not observed. It therefore also serves as a model for phase separation.
1 A. Bovier, and F. den Hollander, Metastability—A Potential–Theoretic Approach, Grundlehren der mathematischen Wissenschaften, Volume 351, Springer, 2015.
2 J.T. Chayes, L. Chayes, and R. Koteck´y, The analysis of the Widom–Rowlinson model by stochastic geometric methods, Commun. Math. Phys. 172, 551–569 (1995).
4 D. Ruelle, Existence of a phase transition in a continuous classical system, Phys. Rev. Lett. 27, 1040–1041 (1971).
5 B. Widom and J.S. Rowlinson, New model for the study of liquid–vapor phase transitions, J. Chem. Phys. 52, 1670–1684 (1970). | CommonCrawl |
Abstract. We study the spectral properties of pseudo-differential operators in the semi-classical limit at energies near a non degenerate minimum of the principal symbol $p$. We give precise asymptotics of the non resonant energy levels for a scalar holomorphic $p$, we get explicit expressions in dimension 2. Precise asymptotics are also derived for the Schr\"odinger and Dirac operator with electro-magnetic field in dimension 2 and 3 (and we give the transport equation of the first term of the $WKB$ expansion of the associated eigenfunction). Then we study the Schr\"odinger and Dirac equation under rotational invariance hypothesis (in dimension 2), and prove that the holomorphic assumption of $p$ can be replaced by $C^\infty $ assumption on the fields. Moreover we prove that there is an associated effective Agmon distance and obtain decay properties of eigenfunctions similar to the case of Schr\"odinger or Dirac operator without magnetic field. | CommonCrawl |
Given a sequence of integers $a_1, a_2, a_3, \ldots , a_ n$, an island in the sequence is a contiguous subsequence for which each element is greater than the elements immediately before and after the subsequence. In the examples below, each island in the sequence has a bracket below it. The bracket for an island contained within another island is below the bracket of the containing island.
Write a program that takes as input a sequence of $12$ non-negative integers and outputs the number of islands in the sequence.
The first line of input contains a single integer $P$, ($1 \le P \le 1000$), which is the number of data sets that follow. Each data set should be processed identically and independently.
Each data set consists of a single line of input. It contains the data set number, $K$, followed by $12$ non-negative integers separated by a single space. The first and last integers in the sequence will be 0.
For each data set there is one line of output. The single output line consists of the data set number, $K$, followed by a single space followed by the number of islands in the sequence. | CommonCrawl |
Nothing in Cantor's proof depends on a "list" however defined.
Well, I would say that both an enumeration and a sequence are types of list and that Cantor uses both, but this is now becoming a semantic argument of little importance to people who understand the proof (among whom I include you).
To me, the most elegant formulation of the proof uses explicit indices in both the enumeration and the sequences and does away entirely with the proof by contradiction which is the root of many misunderstandings in favour of a subsidiary theorem that no countable set of sequences/real numbers contains every such sequence/real number. The contrapositive of this is the statement that the reals are not countable.
You do not believe that Zylo ever has in mind an infinite set, let alone a set of all the real numbers in (0, 1). If you are correct, his argument is infantile because a finite set cannot by definition be put into one-to-one correspondence with an infinite one. On the other hand, I believe that Zylo has in mind (at least some of the time) the set of all real numbers in the interval [0, 1) or (0, 1). He does not, however, even attempt (and would necessarily fail if he did attempt) to show that the set is in one-to-one correspondence with the natural numbers.
There is a difference of opinion here, yes. But you characterise my belief incorrectly. He has in mind the set of reals (sometimes over an interval as in this case). His "proof" then consists of constructing that set in the form of a list. Were he to be successful, this would constitute the required 1-1 mapping with the natural numbers. Of course, he always fails and the reason is that he always works with an countably infinite number of finite subsets of the reals: those consisting of $n \in \mathbb N$ digits after the decimal point. As his most recent post in this thread shows, he is unable to understand that the "limit" as $n$ grows without bound does not exist in his formulation and that his formulation doesn't work for non-terminating decimals and that as a result every one of his real numbers is of the terminating variety.
So his "proof" fails not because there is no mapping with the naturals: because it is a list the mapping exists and the set is countably infinite; but because the set he creates is not the set he believes it to be: the real numbers. $\frac13$, $(\pi-3)$ and $\sqrt2 -1$ are not in it. He is unable or unwilling to see this fact.
Last edited by v8archie; July 19th, 2016 at 06:51 PM. | CommonCrawl |
where $x=(x_1,\dots , x^n)\in\Omega,$ $\Omega$ is a bounded domain in $R^n,$ $t\in(0, T),$ $0 < T <+\infty,$ $f(x, t)$ and $h(x, t)$ are given functions, $\mu$ is a given real, $m$ is a given natural, and $\Delta$ is necessary due to presence of the additional unknown function $q(x)$), the boundary overdetermination condition is used in the article (with $t = 0$ or $t = T$).
For the problems under study, the existence and uniqueness theorems for regular solutions are proved (all derivatives are the Sobolev generalized derivatives).
Fridman A., Partial Differential Equations of Parabolic Type, Prentice Hall, (1964).
Bers L., John F., and Schechter M., Partial Differential Equations, Interscience Publ., New York, London, Sydney (1964).
Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI (1968).
Kostin A. B. and Prilepko A. I., "On certain inverse problems for parabolic equations with final and integral observation," Russ. Acad. Sci., Sb. Math., 75, No. 2, 473–490 (1993).
Prilepko A. I., Orlovsky D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker Inc., New York, Basel (2000).
Isakov V., Inverse Problems for Equations of Parabolic Type, Springer, Berlin (2006).
Kabanikhin S. I., Inverse and Ill-Posed Problems. Theory and Applications, Walter de Gruyter, Berlin, Boston (2012).
Dubinskii Yu. A., "Boundary problems for ellyptic-parabolic equations [in Russian]," Izv. AN Armyan. SSR. Mat., 4, No. 3, 192–214 (1969).
Dubinskii Yu. A., "On some differential-operator equations of arbitrary order," Math. USSR, Sb., 19, No. 1, 1–21 (1973).
Pyatkov S. G., "Solvability of some classes third order equations of mixed-composite type [in Russian]," Preprint, Akad. Nauk SSSR, Sib. Otd., Inst. Mat., Novosibirsk (1980).
Egorov I. E. and Fedorov V. E., Nonclassical Higher Order Equations of Mathematical Physics, Vychisl. Tsentr SO RAN, Novosibirsk (1995).
Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).
Kozhanov A. I., "Questions of posing and solvability of linear inverse problems for elliptic equations," J. Inverse Ill-Posed Probl., 5, No. 4, 337–352 (1997).
Dzhuraev T. D., Boundary Problems for Equations of Mixed and Mixed-Composite Type, Fan, Tashkent (1979).
Trenogin V. A., Functional Analysis, Nauka, Moscow (1980).
Besov O. V., Il'in V. P., and Nikolskii S. M., Integral Representations of Functions and Embedding Theorems, John Wiley and Sons, New York (1978).
Vladimirov V. S., Equations of Mathematical Physics, Marcel Dekker, New York (1971).
Kozhanov A. I. and Pinigina N. R., "Boundary value problems for some higher-order nonclassical differential equations," Math. Notes, 101, No. 3–4, 467–474 (2017).
Akimova, E. and Kozhanov, A. ( ) "Linear inverse problems of spatial type for quasiparabolic equations", Mathematical notes of NEFU, 25(3), pp. 3-17. doi: https://doi.org/10.25587/SVFU.2018.99.16947. | CommonCrawl |
This puzzle replaces all numbers with other symbols.
Your job, as the title suggests, is to find what value fits in the place of $\bigstar$. To get the basic idea down, I recommend you solve Puzzle 1 first.
Each numerical symbol represents integers and only integers. This means fractions and irrational numbers like $\sqrt2$ are not allowed. However, negative numbers and zero are allowed.
Each symbol represents a unique number. This means that for any two symbols $\alpha$ and $\beta$ which are in the same puzzle, $\alpha\neq\beta$.
A solution is a value for $\bigstar$, such that, for the group of symbols in the puzzle $S_1$ there exists a one-to-one function $f:S_1\to\Bbb Z$ which, after replacing all provided symbols using these functions, satisfies all given equations.
What is a Correct Answer?
An answer is considered correct if you can prove that a certain value for $\bigstar$ is a solution. This can be done easily by getting a function from every symbol in the puzzle to the correct values (that is, find an example for $f:S_1\to\Bbb Z$).
A complete answer is a correct answer which also proves that the solution is the only solution. In other words, there is no other possible value for $\bigstar$.
How is an Answer Accepted?
After the puzzle is asked, a one day grace period will be given, in which no answer will be accepted. After that day passes, the complete answer which makes the least assumptions will be accepted. If no complete answer will appear within the grace period, the first complete answer that appears after the grace period will be accepted. | CommonCrawl |
Abstract: This survey is a study of a dynamical system consisting of a massive piston in a cubic container of large size $L$ filled with an ideal gas. The piston has mass $M\sim L^2$ and undergoes elastic collisions with $N\sim L^3$ non-interacting gas particles of mass $m=1$. It is found that under suitable initial conditions there is a scaling regime with time and space scaled by $L$ in which the motion of the piston and the one-particle distribution of the gas satisfy autonomous coupled equations (hydrodynamic equations) such that in the limit $L\to\infty$ the mechanical trajectory of the piston converges in probability to the solution of the hydrodynamic equations for a certain period of time. There is also a heuristic discussion of the dynamics of the system on longer intervals of time.
Balakrishnan V., Van den Broeck C., "Analytic calculation of energy transfer and heat flux in a one-dimensional system", Phys. Rev. E, 72:4 (2005), 046141, 8 pp.
Hurtado P.I., Redner S., "Simplest piston problem. I. Elastic collisions", Phys. Rev. E, 73:1 (2006), 016136, 8 pp.
Hurtado P.I., Redner S., "Simplest piston problem. II. Inelastic collisions", Phys. Rev. E, 73:1 (2006), 016137, 8 pp.
Chernov N., Dolgopyat D., Brownian Brownian motion. I, Mem. Amer. Math. Soc., 198, no. 927, 2009, viii+193 pp. | CommonCrawl |
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First we can tell what the idea of a derivative is. But the issue of computing derivatives is another thing entirely: a person can understand the idea without being able to effectively compute, and vice-versa.
Suppose that $f$ is a function of interest for some reason. We can give $f$ some sort of 'geometric life' by thinking about the set of points $(x,y)$ so that $$f(x)=y$$ We would say that this describes a curve in the $(x,y)$-plane. (And sometimes we think of $x$ as 'moving' from left to right, imparting further intuitive or physical content to the story).
For some particular number $x_o$, let $y_o$ be the value $f(x_o)$ obtained as output by plugging $x_o$ into $f$ as input. Then the point $(x_o,y_o)$ is a point on our curve. The tangent line to the curve at the point $(x_o,y_o)$ is a line passing through $(x_o,y_o)$ and 'flat against' the curve. (As opposed to crossing it at some definite angle).
The idea of the derivative of a function by Paul Garrett is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. For permissions beyond the scope of this license, please contact us. | CommonCrawl |
Share Email Topic 2 error & uncertainty- part 3 byNoel Gallagher 10606views Uncertainty and equipment error byChris Paine 53746views Calculating Uncertainties bymrjdfield 4559views IB Chemistry on uncertainty error c... If we try to read off the numbers on the speedometer and write them down, there'll be a lot of uncertainty in the result. This doesn't affect how we draw the "max" and "min" lines, however. You can change this preference below.
If the latter wildly disagrees with the former, it probably means you made a mistake in doing the digital-numerical calculation. The absolute uncertainty is the actual numerical uncertainty, the percentage uncertainty is the absolute uncertainty as a fraction of the value itself. Hughes and Thomas P.A. If for some reason, however, we want to use the "times" symbol between $X$ and $Y$, the equation is written $Z = X \times Y$.
This is demonstrated in figure 1.2.4 below: Figure 1.2.4 - Intercept uncertainty in a graph Note that in the two figures above the error bars have been exaggerated to improve readability. We hope that these remarks will help to avoid sloppiness when discussing and reporting experimental uncertainties and the inevitable excuse, "Oh, you know what I mean (or meant)." that attends such Note that the previous sentence establishes the length $L$ (actually, its square-root) as the independent variable (what one sets initially) and $T$ as the dependent variable (the quantity that depends on This combination is used so often that a new unit has been derived from it called the watt (symbol: W).
We may summarize this by the simple statement, worth remembering, "You cannot measure zero." What you can say is that if there is a difference between them, it's less than such-and-such Then, using the same menu shown in figure 3, click on the "More error bar options". so use 42s ± 4s. Look at the error bars.
Why not share! time graph with error bars In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Fig. 9: Drag a box over the error on current data. The period of a real (free) pendulum does change as its swings get smaller and smaller from, e.g., air friction.
Hinzufügen Playlists werden geladen... In IB, we do things more precisely. Suppose we want to try to plot a graph of the speed of a car, starting from rest for the first few seconds. The derivation of Eq. (E.9a) uses the assumption that the angle $\theta$ is small.
If you don't check the box, the program will calculate a value for $a$ and its uncertainty $\Delta a$, and it will calculate a value for $b$ and its uncertainty $\Delta Fig. 6: Click the button circled in red. According to the Eq. (E.9c) that we are testing, when $L=0$, $T^2=0$, so you should check the box that asks you if the fit must go through (0,0), viz., "through the Why?
If there is a spread of readings then the uncertainty can be derived from the size of the spread of values. AccuracyA measurement is said to be precise if it has little random errors. In this case, the computer has calculated the gradient for us as well - the acceleration in this case. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000).
Better than nothing is a "guesstimate" for the likely variation based on your experience with the equipment being used for the measurements. Enter the appropriate errors in the +/- boxes and choose "errors in x and y". For example, consider 4.0 ± 0.1 - 3.5 ± 0.1 = 0.5 ± 0.2. To do this, we calculate a result using the given values as normal, with added error margin and subtracted error margin.
Autoplay Wenn Autoplay aktiviert ist, wird die Wiedergabe automatisch mit einem der aktuellen Videovorschläge fortgesetzt. The number of significant digits in a result should not exceed that of the least precise raw value on which it depends. | CommonCrawl |
I found this function, and i have problems with the demonstration of truth or falseness of this afirmation, so some one can help me?
Let S and P be the sum and the product of the divisors of a number T $\in$ $\mathbb Z$ and T different of 1.In the case T is a prime number, the roots are +i, -i. In this last case a think the function prove what a prime number don't have more divisors.
I want to prove when T has an four dividers the roots of the f(x) is the non trivial divisors of T.
I tried prove it some times, but i have problems with the arguments, so some one can give me a hint already help. itried what the product of the roots resolve the problem, i think is correct of this form.
Setting $f(x) = Tx^2 + T^2x + Tx - STx + P$, the roots of $f(x)$ are the values of $x$ that solve the equation $$ Tx^2 + T^2x + Tx - STx + P = 0. \tag1$$ The solutions for $x$ in Equation $(1)$ are also called the roots of the equation.
For $T = 10,$ the divisors are $1,2,5,10,$ so $S = 18$ and $x^2 + (T + 1 - S)x + T = x^2 - 7x + 10.$ The roots of $x^2 -7x + 10 = 0$ are $2$ and $5,$ as predicted.
For $T = 8,$ the divisors are $1,2,4,8,$ so $S = 15$ and $x^2 + (T + 1 - S)x + T = x^2 - 6x + 8.$ The roots of $x^2 - 6x + 8 = 0$ are $2$ and $4,$ as predicted.
So that's one case where $T$ is the product of two distinct primes, and one case where $T$ is the cube of a prime. For an actual proof, you need to show that the claim is true for the product of any two distinct primes and for the cube of any prime. You also need to show that these are the only cases of numbers that can have four divisors; you might consider how we can count the divisors of a number using its prime factorization.
Not the answer you're looking for? Browse other questions tagged number-theory functions divisibility or ask your own question.
product of divisors of n as a power of n.
Find all numbers such that "Product of all divisors=cube of number". | CommonCrawl |
If you are new to Facebook, let alone social media, learning how to use Facebook can be an incredibly difficult and frustrating endeavor. Facebook is a very expansive platform, and is nothing like the chat rooms of old where communicating was easy. Simple tasks on Facebook such as finding friends and adding photos can android how to add drawable resource Industry training, systems and operational training, sales and managing expectations, contracts and agreements, what they mean and why and how they protect the parties involved, how to evaluate the business, preparation of business for sale.
There's no way to keep adding zero until you reach infinity, because you can't reach infinity. It's this inability to "reach" infinity that makes the operations violate your intuition. Traditional algebra/arithmetic doesn't work on infinity. This is why we use the concept of limits, which is well-defined mathematically and allows us to perform algebra on infinities. how to add from xbox to someone on pc fortnite I am getting console logs of two Infinity by adding, subtracting, multiplying and dividing them. While adding/multiplying two infinities, i get inifinity, but while substracting/divinding them, i get NaN(not a number). why this is happening ?
The whole idea of $\infty = \infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number.
What is infinity minus infinity? Other answers provide examples of why (their version of) infinity subtracted from infinity is indeterminate, but the problem with answering the question as stated is the meaning of the terms involved.
Add hair to that section only and complete your infinity braid. Braiding Rhythm Recap: OVER the right section, UNDER the left section, AROUND & OVER the left section, UNDER the right section, AROUND & OVER the right section and REPEAT. | CommonCrawl |
To find a product using the distributive property, you take the coefficent--in this case the 3 outside of the parentheses--and multiply it by each term within the parentheses separately. The first term within the parentheses is $x$, so the first term of the product is $3\times x$, or $3x$. The second term within the parentheses is $5$, so the second term of the product is $3\times5$, or $15$. The terms within the parentheses are connected by a plus sign, so the terms within the product will be too. The final product, therefore, is $3x+15$. | CommonCrawl |
The adsorption of nanoparticles on a surface has interest in fields like heterogeneous catalysts and quantum dots. We simulate the monolayer adsorption of nanoparticles on patterned substrate. We adopted a pattern consisting of equal squares of size $\alpha$ and a distance $\beta$ apart from each other, and characterize the system, by reckoning the mean value and variance of the distance between the nanoparticles and the radial distribution function of their distances. Proper control of $\alpha$ and $\beta$ parameters leads to morphologies range from lattice to homogeneous, with interesting non-trivial behaviors in between. Our study shows the relevance of geometrical constraints to obtain different morphologies of colloidal monolayer films with potential for practical applications. | CommonCrawl |
Abstract: We generalize recent linear lower bounds for Polynomial Calculus based on binomial ideals. We produce a general hardness criterion (that we call immunity), which is satisfied by a random function, and prove linear lower bounds on the degree of PC refutations for a wide class of tautologies based on immune functions. As applications of our techniques, we introduce mod$_p$. Tseitin tautologies in the Boolean case (e.g. in the presence of axioms $x_i^2=x_i$), prove that they are hard for PC over fields with characteristic different from $p$, and generalize them to the flow tautologies that are based on the majority function and are proved to be hard over any field. We also prove the $\Omega(n)$ lower bound for random $k$-CNFs over fields of characteristic 2. | CommonCrawl |
Written by Colin+ in basic maths skills.
A question that frequently comes up in the insalubrious sort of place a mathematician might hang around is, what is that value of $0^0$. We generally sigh and answer that the same way every time.
It was nice, then, to see someone ask a more fundamental one: what is $0 \div 0$?
The short answer is, it's not defined, even though $0 \div a = 0$ pretty much everywhere. But why?
There are probably dozens of explanations for this. My favourite is to look at what division means.
$a \div b$ asks the question "what do you multiply by $b$ to get $a$?" So, $6 \div 2 = 3$ because $3 \times 2 = 6$.
In particular, $0 \div 0$ asks "what do you multiply by $0$ to get $0$?" The answer to that is "anything at all" - which is a Problem.
Basic arithmetic relies on operations giving unique answers to questions - if $2+2$ was both 4 and 5, we'd be in a terrible state. $0 \div 0$ can't possibly be 0 and 7 and $\pi$ and Graham's number and every other value you can multiply by 0 to get 0, for exactly the same reason.
In short, it's undefined because the answer could be anything, and that's not allowed. | CommonCrawl |
Linköping University, Department of Electrical Engineering, Integrated Circuits and Systems.
This report is about control, design and implementation of a low voltage-fed quasi Z-source three-level inverter. The topology has been interesting for photovoltaic-systems due to its ability to boost the incoming voltage without needing an extra switching control. The topology was first simulated in Simulink and later implemented on a full-bridge module to measure the harmonic distortion and estimating the power losses of the inverter. An appropriate control scheme was used to set up a shootthrough and design a three-level inverter. The conclusion for the report is that the quasi Z-source inverter could boost the DC-link voltage in the simulation. But there should be more consideration to the internal resistance of the components for the implementation stage as it gave out a lower output voltage than expected.
Moore's law has until today mostly relied on shrinkage of the size of the devices inintegrated circuits. However, soon the granularity of the atoms will set a limit together with increased error probability of the devices. How can Moore's law continue in thefuture? To overcome the increased error rate, we need to introduce redundancy. Applyingmethods from biology may be a way forward, using some of the strategies that transformsan egg into a fetus, but with electronic cells.
A redundant system is less sensitive to failing components. We define electronic clayas a massive redundancy system of interchangeable and unified subsystems. We show how a mean voter, which is simpler than a majority voter, impact a redundant systemand how optimization can be formalized to minimize the impact of failing subsystems.The performance at given yield can be estimated with a first order model, without the need for Monte-Carlo simulations. The methods are applied and verified on a redundant finite-impulse response filter.
The elementary circuit behavior of the memristor, "the missing circuit element", is investigated for fundamental understanding and how it can be used in applications. Different available simulation models are presented and the linear drift model is simulated with Joglekar-Wolf and Biolek window functions. Driven by a sinusoidal current, the memristor is a frequency dependent component with a cut-off frequency. The memristor can be densely packed and used in structures that both stores and compute in the same circuit, as neurons do. Surrounding circuit has to affect (write) and react (read) to the memristor with the same two terminals.
We looked at artificial neural network for pattern recognition, but also for self organization in electronic cell array. Finally we look at wireless sensor network and how such system can adopt to the environment. This is also a massive redundant clay-like system.
Future electronic systems will be massively redundant and adaptive. Moore's law will continue, not based on shrinking device sizes, but on cheaper, numerous, unified and interchangeable subsystems.
Linköping University, Department of Electrical Engineering, Information Coding. Linköping University, Faculty of Science & Engineering.
The linear-drift memristor model, suggested by HP Labs a few years ago, is used in this work together with two window functions. From the equations describing the memristor model, the transfer characteristics of a memristor is formulated and analyzed. A first-order estimation of the cut-off frequency is shown, that illustrates the bandwidth limitation of the memristor and how it varies with some of its physical parameters. The design space is elaborated upon and it is shown that the state speed, the variation of the doped and undoped regions of the memristor, is inversely proportional to the physical length, and depth of the device. The transfer characteristics is simulated for Joglekar-Wolf, and Biolek window functions and the results are analyzed. The Joglekar-Wolf window function causes a distinct behavior in the tranfer characteristics at cut-off frequency. The Biolek window function on the other hand gives a smooth state transfer function, at the cost of loosing the one-to-one mapping between charge and state. We also elaborate on the design constraints derived from the transfer characteristics.
Operational time is becoming an increasingly important aspect in electronic devices and is also highly relevant in Underwater Acoustic Sensor Networks (UWSN). This thesis contains a study which explores what can be done to de-crease power consumption while maintaining the same functionality of an FPGA inside an underwater sensor-node network. A longer operational time means a more effective system since reconnaissance is one of UWSN's area of application. The thesis will also cover the implementation of a new sensor-node 'mode' which will add new features and increase operational time.
Linköping University, Department of Electrical Engineering, Integrated Circuits and Systems. Linköping University, The Institute of Technology.
A new Vernier time-to-digital converter (TDC) architecture using a delay line and a chain of delay latches is proposed. The delay latches replace the functionality of one delay chain and the sample register commonly found in Vernier converters, hereby enabling power and hardware efficiency improvements. The delay latches can be implemented using either standard or full custom cells, allowing the architecture to be implemented in field-programmable gate arrays, digital synthesized application-specific integrated circuits, or in full custom design flows. To demonstrate the proposed concept, a 7-bit Vernier TDC has been implemented in a standard 65-nm CMOS process with an active core size of 33 mu m x 120 mu m. The time resolution is 5.7 ps with a power consumption of 1.75 mW measured at a conversion rate of 100 MS/s.
Many integrated circuit functional blocks, such as data and power converters, require timing and control signals consisting of complex sequences of pulses. Traditionally, these signals are generated from a clock signal using a combination of flip-flops, latches and delay elements. Due to the large internal switching activity of flips-flops and due to the many, effectively unused, clock cycles, this solution is inefficient from a power consumption point of view and is, therefore, unsuitable for ultralow-power applications. In this paper we present a method to generate non-overlapping control signals without using flip-flops or a clock. We propose to decode and translate the internal states of a ring oscillator into the desired control signal sequence. We show how this can be achieved using a simple combinatorial logic decoder. The proposed architecture significantly reduces the switching activity and the capacitive load, largely reducing the consumed power. We show an example implementation of a 9-bit SAR logic utilizing our proposed method. Furthermore, we show simulation results and compare the power consumption of the example SAR implementation to that of a functionally identical flip-flop-based state-of-the-art ultralow-power SAR. We were able to achieve a 5.8x reduction in consumed power for the complete SAR and 8x for the one-hot generation sub-part.
We are investigating the use of ultrasound in Haptic applications. Initially abrief background of ultrasonic transducers and its characteristics were presented.Then a theoretical research was documented to understand the concepts that govern haptics. This section also discusses the algorithm adopted by various researches to implement haptics in the professional world. Then investigations were made to understand the behavior of ultrasonic transducers and conduct soft-ware simulations to obtain various results. At first simulations were conducted on Field II software. This simulations involved the creation of elements in trans-ducers, transducer's spatial impulse responses, transducer's impulse responsein time and frequency domain, effect of adding apodization to the transducers,pulse echo response of the transducers, beam profile variation along the focallength of the transducers. Then a Matlab based GUI was used to study the relationship between number of elements in transducers, the frequency of the input signal and duty cycle variation of the input wave. A concept of phase shift, which explains the time delay generation was also coded in Matlab.
Modular multiplication (MM) based on the residue number system (RNS) is a widely researched area due to the fast arithmetic operations in the RNS. The major drawback of the RNS based MM architectures is their large area because each arithmetic operation is followed by a modular reduction. In this work, the number of modular reductions is reduced and instead the wordlength of some operations is increased to accommodate the intermediate results. The proposed scheme greatly reduces the number of multipliers and achieves a 55% reduction in the hardware complexity. Moreover the delay of the proposed architecture is also significantly lower than the reference architecture.
A disadvantage with battery powered circuits is the fact that the battery sometimes can run out of power. If a button that can generate energy by applying mechanical work to it was applied instead of batteries, is it possible to enable a transmitter to stay active long enough to transmit data which can later by received and decoded?
This thesis contains a study, in which how to effectively send data wirelessly between a transmitter and receiver module, without the use of any batteries or external power sources, only an energy harvesting push button is constructed and evaluated. There will also be a theoretical comparison between different transmission formats and which is more suitable for a task such as this.
Digital-to-analog (D/A) converters (or DACs) are one the fundamental building blocks of wireless transmitters. In order to support the increasing demand for highdata-ate communication, a large bandwidth is required from the DAC. With the advances in CMOS scaling, there is an increasing trend of moving a large part of the transceiver functionality to the digital domain in order to reduce the analog complexity and allow easy reconguration for multiple radio standards. ΔΣ DACs can t very well into this trend of digital architectures as they contain a large digital signal processing component and oer two advantages over the traditionally used Nyquist DACs. Firstly, the number of DAC unit current cells is reduced which relaxes their matching and output impedance requirements and secondly, the reconstruction lter order is reduced.
Achieving a large bandwidth from ΔΣ DACs requires a very high operating frequency of many-GHz from the digital blocks due to the oversampling involved. This can be very challenging to achieve using conventional ΔΣ DAC architectures, even in nanometer CMOS processes. Time-interleaved ΔΣ (TIDSM) DACs have the potential of improving the bandwidth and sampling rate by relaxing the speed of the individual channels. However, they have received only some attention over the past decade and very few previous works been reported on this topic. Hence, the aim of this dissertation is to investigate architectural and circuit techniques that can further enhance the bandwidth and sampling rate of TIDSM DACs.
The rst work is an 8-GS/s interleaved ΔΣ DAC prototype IC with 200-MHz bandwidth implemented in 65-nm CMOS. The high sampling rate is achieved by a two-channel interleaved MASH 1-1 digital ΔΣ modulator with 3-bit output, resulting in a highly digital DAC with only seven current cells. Two-channel interleaving allows the use of a single clock for both the logic and the nal multiplexing. This requires each channel to operate at half the sampling rate i.e. 4 GHz. This is enabled by a high-speed pipelined MASH structure with robust static logic. Measurement results from the prototype show that the DAC achieves 200-MHz bandwidth, –57-dBc IM3 and 26-dB SNDR, with a power consumption of 68-mW at 1-V digital and 1.2-V analog supplies. This architecture shows good potential for use in the transmitter baseband. While a good linearity is obtained from this DAC, the SNDR is found to be limited by the testing setup for sending high-speed digital data into the prototype.
The performance of a two-channel interleaved ΔΣ DAC is found to be very sensitive to the duty-cycle of the half-rate clock. The second work analyzes this eect mathematically and presents a new closed-form expression for the SNDR loss of two-channel DACs due to the duty cycle error (DCE) for a noise transfer function (NTF) of (1 — z—1)n. It is shown that a low-order FIR lter after the modulator helps to mitigate this problem. A closed-form expression for the SNDR loss in the presence of this lter is also developed. These expressions are useful for choosing a suitable modulator and lter order for an interleaved ΔΣ DAC in the early stage of the design process. A comparison between the FIR lter and compensation techniques for DCE mitigation is also presented.
The nal work is a 11 GS/s 1.1 GHz bandwidth time-interleaved DAC prototype IC in 65-nm CMOS for the 60-GHz radio baseband. The high sampling rate is again achieved by using a two-channel interleaved MASH 1-1 architecture with a 4-bit output i.e only fteen analog current cells. The single clock architecture for the logic and the multiplexing requires each channel to operate at 5.5 GHz. To enable this, a new look-ahead technique is proposed that decouples the two channels within the modulator feedback path thereby improving the speed as compared to conventional loop-unrolling. Full speed DAC testing is enabled by an on-chip 1 Kb memory whose read path also operates at 5.5 GHz. Measurement results from the prototype show that the ΔΣ DAC achieves >53 dB SFDR, < —49 dBc IM3 and 39 dB SNDR within a 1.1 GHz bandwidth while consuming 117 mW from 1 V digital/1.2 V analog supplies. The proposed ΔΣ DAC can satisfy the spectral mask of the 60-GHz radio IEEE 802.11ad WiGig standard with a second order reconstruction lter.
This brief presents an 8-GS/s 12-bit input ΔΣ digital-to-analog converter (DAC) with 200-MHz bandwidth in 65-nm CMOS. The high sampling rate is achieved by a two-channel interleaved MASH 1–1 digital ΔΣ modulator with 3-bit output, resulting in a highly digital DAC with only seven current cells. The two-channel interleaving allows the use of a single clock for both the logic and the final multiplexing. This requires each channel to operate at half the sampling rate, which is enabled by a high-speed pipelined MASH structure with robust static logic. Measurement results show that the DAC achieves 200-MHz bandwidth, 26-dB SNDR, and $-$57-dBc IMD3, with a power consumption of 68 mW at 1-V digital and 1.2-V analog supplies. This architecture shows potential for use in transmitter baseband for wideband wireless communication.
Time-interleaved delta-sigma (Delta Sigma) modulation digital-to-analog converters (TIDSM DACs) have the potential for a wideband operation. The performance of a two-channel interleaved Delta Sigma DAC is very sensitive to the duty cycle of the half-rate clock. This brief presents a closed-form expression for the signal-to-noise-plus-distortion ratio (SNDR) loss of such DACs due to a duty-cycle error for modulators with a noise transfer function of (1 - z(-1))(n). Adding a low-order finite-impulse-response filter after the modulator helps to mitigate this problem. A closed-form expression for the SNDR loss in the presence of this filter is also developed. These expressions are useful for choosing a suitable modulator and filter order for an interleaved Delta Sigma DAC in the early stage of the design process.
This work presents an 11 GS/s 1.1 GHz bandwidth interleaved ΔΣ DAC in 65 nm CMOS for the 60 GHz radio baseband. The high sample rate is achieved by using a two-channel interleaved MASH 1–1 architecture with a 4 bit output resulting in a predominantly digital DAC with only 15 analog current cells. Two-channel interleaving allows the use of a single clock for the logic and the multiplexing which requires each channel to operate at half sampling rate of 5.5 GHz. To enable this, a look-ahead technique is proposed that decouples the two channels within the integrator feedback path thereby improving the speed as compared to conventional loop-unrolling. Measurement results show that the ΔΣ DAC achieves a 53 dB SFDR, -49 dBc IM3 and 39 dB SNDR within a 1.1 GHz bandwidth while consuming 117 mW from 1 V digital/1.2 V analog supplies. Furthermore, the proposed ΔΣ DAC can satisfy the spectral mask of the IEEE 802.11ad WiGig standard with a second order reconstruction filter.
Implementation of wireless wideband transmitters using ΔΣ DACs requires very high speed modulators. Digital MASH ΔΣ modulators are good candidates for speed enhancement using interleaving because they require only adders and can be cascaded. This paper presents an analysis of the integrator critical path of two-channel interleaved ΔΣ modulators. The bottlenecks for a high-speed operation are identified and the performance of different logic styles is compared. Static combinational logic shows the best trade-off and potential for use in such high speed modulators. A prototype 12-bit second order MASH ΔΣ modulator designed in 65 nm CMOS technology based on this study achieves 9 GHz operation at 1 V supply.
Time-interleaved ΔΣ DACs have the potential for wideband and high-speed operation. Their SNR is limited by the timing skew between the output delays of the channels to the output. In a two-channel interleaved ΔΣ DAC, the channel skew arises from the duty cycle error in the half sample rate clock. The effects of timing skew error can be mitigated by hold interleaving, digital pre-filtering or compensation in the form of analog post-correction or digital pre-correction. This paper presents a comparative study of these techniques for two-channel interleaving and the trade-offs are investigated. First order FIR pre-filtering is found to be a suitable solution with a moderate DAC matching penalty of one bit. Higher order pre-filtering achieves a near immunity to timing skew at the cost of higher matching penalty. Correction techniques are found to be less effective than pre-filtering and not well suited for high-speed implementation.
This thesis report presents system specification, such as frequency and output power level, and selection topology of an oscillator circuit suitable for a CMOS Integrated Doppler radar application, in order to facilitate short range target detection within 5-15 m range, using a 0.35 μm CMOS process. With this selected CMOS process, the frequency band at 2.45 GHz or 5 GHz, with a maximum output power level of 25 mW (e.i.r.p), is found to be appropriate for the whole system to obtain a good performance. In this thesis work, a Ring VCO with pseudo-differential architecture has been designed and optimised for 2.45 GHz application. However, for 5 GHz application, a differential cross-coupled LC VCO oscillator topology has been suggested and it is so designed that it can be further scaled down to operate at a frequency of 2.45 GHz. The performance of the oscillator circuits has been tested at circuit level and has been presented as simulation results in this report.
Printed electronics holds the promise of adding intelligence to disposable objects. Low tem- perature additive manufacturing using low-cost substrates, less complex equipment and fewer processing steps allow drastically reduced cost compared to conventional silicon cir- cuits. Ferroelectric memories is a suitable technology for non-volatile storage in printed circuits. Printed organic thin film transistors can be used for logic. Another approach is to reduce the complexity of silicon manufacturing by substituting as many steps as possible for printed alternatives and substitute silicon wafers for cheaper substrates, one such process is printed dopant polysilicon. This thesis explores the possibility of designing circuits using these two transistor technologies for reading and writing ferroelectric memories. Both gen- eration of the voltage pulses necessary for memory operation from a lower supply voltage and the interpretation of the memory response as one of two states is investigated. It is con- cluded, with some reservations, that such circuitry can be implemented using the polysilicon process. Using organic thin film transistors only the latter functionality is shown, generation of the necessary voltage pulses is not achieved but also not completely precluded.
In this thesis the possibility of building an underwater communication system usingelectromagnetic waves has been explored. The focus became designing and testingan antenna even if the entire system has been outlined as well. The conclusion isthat using magnetically linked antennas in the near field it is a very real possibilitybut for long EM waves in the far field more testing needs to be done. This isbecause a lack of equipment and facilitates which made it hard to do the realworld testing for this implementation even if it should work in theory.
Electrical vehicles are getting more popular as the technology around batteries and electrical motors are catching up to the more common combustion engines. Electrical boats are no exception but there are still a lot of boats using old combustion engines that have a big impact on the environment. This study aims to deepen the understanding of the integration of electrical motors into boats by proposing a design of a system using a bidirectional synchronous buckboost converter. This converter is designed to handle the power transfer in a dual battery application, namely consisting of a 12 V battery and a 48 V battery. The converter includes proposed components, a PCB design, as well as the software that is required for the control of the power transfer. The results show that the converter design meets specification and, when using a test-bench, the software is capable of controlling the converter to achieve constant current and constant voltage in both directions.
Wireless sensor networks (WSNs) are employed in many applications, such as for monitoring bio-potential signals and environmental information. These applications require high-resolution (> 12-bit) analog-to-digital converters (ADCs) at low-sampling rates (several kS/s). Such sensor nodes are usually powered by batteries or energy-harvesting sources hence low power consumption is primary for such ADCs. Normally, tens or hundreds of autonomously powered sensor nodes are utilized to capture and transmit data to the central processor. Hence it is profitable to fabricate the relevant electronics, such as the ADCs, in a low-cost standard complementary metal-oxide-semiconductor (CMOS) process. The two-stage pipelined successive approximation register (SAR) ADC has shown to be an energy-efficient architecture for high resolution. This thesis further studies and explores the design limitations of the pipelined SAR ADC for high-resolution and low-speed applications.
The first work is a 15-bit, 1 kS/s two-stage pipelined SAR ADC that has been implemented in 0.35-μm CMOS process. The use of aggressive gain reduction in the residue amplifier combined with a suitable capacitive array digital-to-analog converter (DAC) topology in the second-stage simplifies the design of the operational transconductance amplifier (OTA) while eliminating excessive capacitive load and consequent power consumption. A comprehensive power consumption analysis of the entire ADC is performed to determine the number of bits in each stage of the pipeline. Choice of a segmented capacitive array DAC and attenuation capacitorbased DAC for the first and second stages respectively enable significant reduction in power consumption and area. Fabricated in a low-cost 0.35-μm CMOS process, the prototype ADC achieves a peak signal-to-noise-and-distortion ratio (SNDR) of 78.9 dB corresponding to an effective number of bits (ENOB) of 12.8-bit at a sampling frequency of 1 kS/s and provides a Schreier figure-of-merit (FoM) of 157.6 dB. Without any form of calibration, the ADC maintains an ENOB > 12.1-bit up to the Nyquist bandwidth of 500 Hz while consuming 6.7 μW. Core area of the ADC is 0.679 mm2.
to 260 kHz. The core area occupied by the ADC is 0.589 mm2.
As the low-power sensors might be active only for very short time triggered by an external pulse to acquire the data, the third work is a 14-bit asynchronous two-stage pipelined SAR ADC which has been designed and simulated in 0.18-μm CMOS process. A self-synchronous loop based on an edge detector is utilized to generate an internal clock with variable phase. A tunable delay element enables to allocate the available time for the switch capacitor DACs and the gain-stage. Three separate asynchronous clock generators are implemented to create the control signals for two sub-ADCs and the gain-stage between. Aiming to reduce the power consumption of the gain-stage, simple source followers as the analog buffers are implemented in the 3-stage CCP gain-stage. Post-layout simulation results show that the ADC achieves a SNDR of 83.5 dB while consuming 2.39 μW with a sampling rate of 10 kS/s. The corresponding Schreier FoM is 176.7 dB.
This paper presents a 15-bit, two-stage pipelined successive approximation register analog-to-digital converter (ADC) suitable for low-power, cost-effective sensor readout circuits. The use of aggressive gain reduction in the residue amplifier combined with a suitable capacitive array DAC topology in the second stage simplifies the design of the operational transconductance amplifier while eliminating excessive capacitive load and consequent power consumption. An elaborate power consumption analysis of the entire ADC was performed to determine the number of bits in each stage of the pipeline. Choice of a segmented capacitive array DAC and attenuation capacitor-based DAC for the first and second stages respectively enable significant reduction in power consumption and area. Fabricated in a low-cost 0.35-μm CMOS process, the prototype ADC achieves a peak SNDR of 78.9 dB corresponding to an effective number of bits (ENOB) of 12.8 bits at a sampling frequency of 1 kS/s and provides an FoM of 157.6 dB. Without any form of calibration, the ADC maintains an ENOB >12.1 bits upto the Nyquist bandwidth of 500 Hz while consuming 6.7 μW. Core area of the ADC is 0.679 mm2.
This paper presents a 14-bit, tunable bandwidth two-stage pipelined successive approximation analog to digital converter which is suitable for low-power, cost-effective sensor readout circuits. To overcome the high DC gain requirement of operational transconductance amplifier in the gain-stage, the multi-stage capacitive charge pump (CCP) was utilized to achieve the gain-stage instead of using the switch capacitor integrator. The detailed design considerations are given in this work. Thereafter, the 14-bit ADC was designed and fabricated in a low-cost 0.35-µm CMOS process. The prototype ADC achieves a peak SNDR of 75.6 dB at a sampling rate of 20 kS/s and 76.1 dB at 200 kS/s while consuming 7.68 and 96 µW, respectively. The corresponding FoM are 166.7 and 166.3 dB. Since the bandwidth of CCP is tunable, the ADC maintains a SNDR >75 dB upto 260 kHz. The core area occupied by the ADC is 0.589 mm2.
In this paper, we provide a detailed analysis on the power consumption of two-stage pipeline successive approximation analog-to-digital converter (SAR ADC) and also show the relationship between stage resolution and the total power consumption in 65 nm technology. Thereafter, we evaluate the analysis results with designing a 15-bit pipeline SAR ADC in 65 nm technology and also a power comparison between two-stage pipeline SAR ADC and single SAR ADC is analyzed with the parameters from same technology. The finally results demonstrate that for high resolution ADC design, a particular range is obtained, in which the total power consumption of two-stage pipeline SAR ADC is much lower than single SAR ADC.
This paper presents the design of a multi-stage capacitive charge pump (CCP) as a gain-stage which is used in the two-stage pipelined successive approximation analog-to-digital converter (SAR ADC). The topology of multi-stage CCP and the design considerations are provided. Thereafter, the power comparison between switch capacitor (SC) integrator and multi-stage CCP is analyzed with the parameters from 0.35-mu m CMOS process. The comparison results show that the proposed gain-stage is more power efficient than SC integrator. To verify the analysis, two types of gain-stage, SC integrator and multi-stage CCP, were simulated in 0.35-mu m CMOS process. Simulation results show that the three-stage CCP achieves a gain of 7.9 while only consuming 1.1 mu W with the gain bandwidth of 178.7 kHz. But the SC integrator consumes 1.58 times more power than CCPs to reach the similar gain and gain bandwidth.
Aiming to alleviate operational transconductance amplifiers (OTA), this paper describes the design of a capacitive charge pump (CCP) gain-stage for a two-stage pipelined SAR ADCs suitable for low-power sensors. An analog buffer is inevitable to prevent the charge sharing between the capacitive stages. In this work a simple source follower has been used as the analog buffer, showing sufficient linearity and significant power reduction compared to earlier work where a unity-gain OTA was used. To verify the solution, a CCP gain-stage with source follower has been implemented in design of a 14-bit two-stage pipelined SAR ADC in 0.18 mu m CMOS. Detailed circuit simulations show that the ADC achieves a SNDR of 83.0 dB while consuming 1.8 mu W at a sampling frequency of 10 kHz.
This paper describes the design and implementation of an asynchronous clock generator which has been used in a 14-bit two-stage pipelined SAR ADCs for low-power sensor applications. A self-synchronization loop based on an edge detector was utilized to generate an internal clock with variable phase and frequency. A tunable delay element enables to allocate the available time for the switch capacitor DACs and the gain-stage. Thereafter, three separate asynchronous clock generators were implemented to create the control signals for two sub-ADCs and the gain-stage between. Finally, a 14-bit asynchronous two-stage pipelined SAR ADC was designed and simulated in 0.18 mu m CMOS. Detailed pre-layout circuit simulations show that the ADC achieves a SNDR of 83.5 dB while consuming 2.13 mu W with a sampling rate of 10 kS/s. The corresponding FoM is 177.2 dB.
Oscillators are components providing clock signals. They are widely required by low-cost on-chip applications, such as biometric sensors and SoCs. As part of a sensor, a relaxation oscillator is implemented to provide a clock reference. Limited by the sensor application, a clock reference outside the sensor is not desired. An RC implementation of the oscillator has a balanced accuracy performance with low-cost advantage. Hence an RC relaxation oscillator is chosen to provide the clock inside the sensor.
This thesis proposes a current mode relaxation oscillator to achieve low frequency standard deviation across different supplies, temperatures and process corners. A comparison between a given relaxation oscillator and the proposed design is made as well. All oscillators in this thesis use 0.18 μm technology and 1.8 V nominal supply. The proposed oscillator manages to achieve a frequency standard deviation across all PVT variations less than ±6.5% at 78.4 MHz output frequency with a power dissipation of 461.2 μW. The layout of the oscillator's core area takes up 0.003 mm2.
The proliferation of portable communication devices combined with the relentless demand for higher data rates has spurred the development of wireless communication standards which can support wide signal bandwidths. Benefits of the complementary metal oxide semiconductor (CMOS) process such as high device speeds and low manufacturing cost have rendered it the technology of choice for implementing wideband wireless transceiver integrated circuits (ICs). This dissertation addresses the key challenges encountered in the design of wideband wireless transceiver ICs. It is divided into two parts. Part I describes the design of crucial circuit blocks such as a highly selective wideband radio frequency (RF) front-end and an on-chip test module which are typically found in wireless receivers. The design of high-speed, capacitive DACs for wireless transmitters is included in Part II.
The first work in Part I is the design and implementation of a wideband RF frontend in 65-nm CMOS. To achieve blocker rejection comparable to surface-acousticwave (SAW) filters, the highly selective and tunable RF receiver utilizes impedance transformation filtering along with a two-stage architecture. It is well known that the low-noise amplifier (LNA) which forms the first front-end stage largely decides the receiver performance in terms of noise figure (NF) and linearity (IIP3/P1dB). The proposed LNA uses double cross-coupling technique to reduce NF while complementary derivative superposition (DS) and resistive feedback are employed to achieve high linearity. The resistive feedback also enhances input matching. In measurements, the front-end achieves performance comparable to SAW filters with blocker rejection greater than 38 dB, NF 3.2–5.2 dB, out-of-band IIP3 > +17 dBm and blocker P1dB > +5 dBm over a frequency range of 0.5–3 GHz.
The second work in Part I is the design of an RF amplitude detector for on-chip test. As the complexity of RF ICs continues to grow, the task of testing and debugging them becomes increasingly challenging. The degradation in performance or the drift from the optimal operation points may cause systems to fail. To prevent this effect and ensure acceptable performance in the presence of process, voltage and temperature variations (PVT), test and calibration of the RF ICs become indispensable. A wideband, high dynamic range RF amplitude detector design aimed at on-chip test is proposed. Gain-boosting and sub-ranging techniques are applied to the detection circuit to increase the gain over the full range of input amplitudes without compromising the input impedance. A technique suitable for on-chip third/second-order intercept point (IP3/IP2) test by embedded RF detectors is also introduced.
Part II comprises the design and analysis of high-speed switched-capacitor (SC) DACs for 60-GHz radio transmitters. The digital-to-analog converter (DAC) is one of the fundamental building blocks of transmitters. SC DACs offer several advantages over the current-steering DAC architecture. Specifically, lower capacitor mismatch helps the SC DAC to achieve higher linearity. The switches in the SC DAC are realized by MOS transistors in the triode region which substantially relaxes the voltage headroom requirement. Consequently, SC DACs can be implemented using lower supply voltages in advanced CMOS process nodes compared to their currentsteering counterparts. The first work in Part II analyzes the factors limiting the performance of capacitive pipeline DACs. It is shown that the DAC performance is limited mainly by the clock feed-through and settling effects in the SC arrays while the impact of capacitor mismatch and kT/C noise are found to be negligible. Based on this analysis, the second work in Part II proposes the split-segmented SC array DAC to overcome the clock feed-through problem since this topology eliminates pipelined charge propagation. Implemented in 65-nm CMOS, the 12-bit SC DAC achieves a Spurious Free Dynamic Range (SFDR) greater than 44 dB within the input signal bandwidth (BW) of 1 GHz with on-chip memory embedded for digital data generation. Power dissipation is 50 mW from 1.2 V supply. Similar performance is achieved with a lower supply voltage (0.9 V) which shows the scalability of the SC DAC for more advanced CMOS technologies. Furthermore, the proposed SC DAC satisfies the spectral mask of the IEEE 802.11ad WiGig standard with a second-order reconstruction filter and hence it can be used for the 60-GHz radio baseband.
Catena AB, Stockholm, Sweden .
Analysis and design of a low-noise transconductance amplifier (LNTA) aimed at selective current-mode (SAW-less) wideband receiver front-end is presented. The proposed LNTA uses double cross-coupling technique to reduce noise figure (NF), complementary derivative superposition, and resistive feedback to achieve high linearity and enhance input matching. The analysis of both NF and IIP3 using Volterra series is described in detail and verified by SpectreRF (A (R)) circuit simulation showing NF less than 2 dB and IIP3 = 18 dBm at 3 GHz. The amplifier performance is demonstrated in a two-stage highly selective receiver front-end implemented in 65 nm CMOS technology. In measurements the front-end achieves blocker rejection competitive to SAW filters with noise figure 3.2-5.2 dB, out of band IIP3 greater than+17 dBm and blocker P-1dB greater than+5 dBm over frequency range of 0.5-3 GHz.
In order to achieve blocker rejection comparable to surface acoustic wave (SAW) filters, we propose a two-stage tunable receiver front-end architecture based on impedance frequency transformation and low-noise transconductance amplifier (LNTA) circuits. The filter rejection is captured by a linear periodically varying model that includes band limitation by the LNTA output impedance and the related parasitic capacitances of the impedance transformation circuit. The effect of thermal noise folding on the circuit noise figure, as well as clock phase mismatch on filter gain are also discussed. As a proof of concept, a chip design of a tunable radio-frequency front end using 65-nm CMOS technology is presented. In measurements the circuit achieves blocker rejection competitive to SAW filters with noise figure 3.2-5.2 dB, out of band IIP3 > +17 dBm and blocker P1 dB > +5 dBm over frequency range of 0.5-3 GHz.
A wideband, high dynamic range RF amplitude detector design aimed at on-chip test is presented. Boosting gain and sub-ranging techniques are applied to the detection circuit to increase gain over the full range of input amplitudes without compromising the input impedance. Followed by a variable gain amplifier (VGA) and a 9-bit A/D converter the RF detector system, designed in 65 nm CMOS, achieves in simulation the minimum detectable signal of -58 dBm and 63 dB dynamic range over 0.5 GHz - 9 GHz band with input impedance larger than 4 kΩ. The detector is intended for on-chip calibration and the attained specifications put it among the reported state-of-the-art solutions.
Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, Faculty of Science & Engineering.
In this paper a technique suitable for on-chip IP3/IP2 RF test by embedded RF detectors is presented. A lack of spectral selectivity of the detectors and diverse nonlinearity of the circuit under test (CUT) impose stiff constraints on the respective test measurements for which focused calibration approach and a support by customized models of CUT is necessary. Also cancellation of second-order intermodulation effects produced by the detectors under the two-tone test is required. The test technique is introduced using a polynomial model of the CUT. Simulation example of a practical CMOS LNA under IP3/IP2 RF test with embedded RF detectors is presented showing a good measurement accuracy.
Design of a high speed capacitive digital-to-analog converter (SC DAC) is presented for 65 nm CMOS technology. SC pipeline architecture is used followed by an output driver. For GHz frequency operation with output voltage swing suitable for wireless applications (300 mVpp) the DAC performance is shown to be limited by the capacitor array imperfections. While it is possible to design a highly linear output driver with HD3 < -70 dB and HD2 < -90 dB over 0.55 GHz band as we show, the maximum SFDR of the SC DAC is 45 dB with 8-bit resolution and Nyquist sampling of 3 GHz. The analysis shows the DAC performance is determined by the clock feed-through and settling effects in the SC array and not by the capacitor mismatch or kT/C noise, which appear negligible in this application. The capacitor array is designed based on the DAC design area defined in terms of the switch size and unit capacitance value. A tradeoff between the DAC bandwidth and resolution accompanied by SFDR is demonstrated. The high linearity of the output driver is attained by a combination of two techniques, the derivative superposition (DS) and resistive source degeneration. In simulations the complete Nyquist-rate DAC achieves SFDR of 45 dB with 8-bit resolution for signal bandwidth 1.36 GHz. With 6-bit and 5.5 GHz bandwidth 33 dB SFDR is attained. The total power consumption of the SC DAC is 90 mW with 1.2 V supply and clock frequency of 3 GHz.
A highly selective impedance transformation filtering technique suitable for tunable selective RF receivers is proposed in this paper. To achieve blocker rejection comparable to SAW filters, we use a two stage architecture based on a low noise trans-conductance amplifier (LNTA). The filter rejection is captured by a linear periodically varying (LPV) model that includes band limitation by the LNTA output impedance and the related parasitic capacitances of the impedance transformation circuit. This model is also used to estimate "back folding" by interferers placed at harmonic frequencies. Discussed is also the effect of thermal noise folding and phase noise on the circuit noise figure. As a proof of concept a chip design of a tunable RF front-end using 65 nm CMOS technology is presented. In measurements the circuit achieves blocker rejection competitive to SAW filters with noise figure 3.2-5.2 dB,out of bandIIP3 > +17 dBm and blocker P1dB > +5 dBm over frequency range of 0.5—3 GHz.
A low-noise transconductance amplifier (LNTA) aimed at continuous-time ΣΔ wideband frontend is presented. In this application, the LNTA operates with a capacitive load to provide high linearity and sufficient Gm gain over a wide frequency band. By combination of various circuit techniques the LNTA, which is designed in 65nm CMOS, achieves in simulation the noise figure less than 1.35 dB and linearity of maximum IIP3 = 13.6 dBm over 0.8 - 5 GHz band. The maximum transconductance Gm = 11.6 mS, the return loss S11 <; -14 dB while the total power consumption is 3.9 mW for 1.2 V supply.
Design of a high speed output driver for capacitive digital-to-analog converters (SC DACs) is presented. As the output voltage swing of those DACs is usually greater than 300 mVpp the driver is designed for large signal operation that is a challenge in terms of the DAC linearity. Two non-linearity cancellation techniques are applied to the driver circuit, the derivative superposition (DS) and the resistive source degeneration resulting in HD3 <; -70 dB and HD2 <; -90 dB over the band of 0.5-4 GHz in 65-nm CMOS. For the output swing of 300 mVpp and 1.2 V supply its power consumption is 40 mW. For verification the driver is implemented in a 12-bit pipeline SC DAC. In simulations the complete Nyquist-rate DAC achieves SFDR of 64 dB for signal bandwidth up to 2.2 GHz showing a negligible non-linearity contribution by the designed driver for signal frequencies up to 1.3 GHz and a degradation by 3 dB at 2.2 GHz.
A low-noise transconductance amplifier (LNTA) aimed at current-mode (Saw-less, Software-define radio) wideband receiver frontend is presented. In this application, the LNTA operates with a capacitive load to provide high linearity and sufficient G<;sub>m<;/sub> gain over a wide frequency band. By combination of various circuit techniques the LNTA, which is designed in 65 nm CMOS, achieves in simulation the noise figure in range [1-1.34] dB and linearity of maximum IIP3 = 16.5 dBm over 0.5-6 GHz band. The maximum transconductance G<;sub>m<;/sub> = 12.9 mS, the return loss S11 <; -10 dB while the total power consumption is 4 mW for 1.2 V supply.
In order to achieve high speed and high resolution for switched-capacitor (SC) digital-to-analog converters (DACs), an architecture of split-segmented SC DAC is proposed. The detailed design considerations of kT/C noise, capacitor mismatch, settling time and simultaneous switching noise are mathematically analyzed and modelled. The design area W–Cu is defined based on that analysis. It is used not only to identify the maximum speed and resolution but also to find the design point (W, Cu) for certain speed and resolution of SC DAC topology. The segmentation effects are also considered. An implementation example of this type of DACs is a 12-bit 6-6 split-segmented SC DAC designed in 65 nm CMOS. The linear open-loop output driver utilizing derivation superposition technique for nonlinear cancellation is used to drive off-chip load for the SC array without compromising its performance. The measured results show that the SC DAC achieves a 44 dB spurious free dynamic range within a 1 GHz bandwidth of input signal at 5 GS/s while consuming 50 mW from 1 V digital and 1.2 V analog supplies. The overall performance that was achieved from measurement is poorer than expected due to lower power supply rejection ratio in fabricated chip. This DAC can be used in transmitter baseband for wideband wireless communications.
Energy storage systems is very useful to use in solar panel systems to save money, but also tobe more environment-friendly. The project was given by the solar energy companyPerpetuum Automobile (PPAM) and the project is for their customer, the condominiumcompound Ekoxen. The task is to make a energy regulation for Ekoxen's energy storage sothey can save more money. The energy storage primary task is to shave the top-peaks of theconsumption for Ekoxen. Which means that the battery will supply the household instead forthe three-phase grid. This will make the electric bill for Ekoxen cheaper. Thesimulation/analysis of the energy regulation is done in a spreadsheet tool, where one partworks as a Time-of-Use program and the other work as a modbus feature. Time-of-Use is aweb-based program for PV systems with battery storage, where time-periods can be set toaffect the battery behavior. The modbus feature simulates a system where an algorithm can beimplemented. The results will show that the time-periods for charging the battery with theTime-of-Use program needs to be changed two times per year. One time for the summermonths and a second time for the rest of the months. The results will also show that themodbus feature is better on peak shaving than the time-of-use program.
Målet med projektet var att sammanställa en enkel och optimal modell av ett elsystem för elektrisk kraftförsörjning av olika delsystem och deras apparater genom användande av ett verktyg som kan beräkna spänningsfall i elledningar i flygplan. Modellen ska tjänstgöra som ett verktyg/hjälpmedel vid dimensionering/kontroll av elledningar och säkringar samt snabbt ge besked om huruvida nya laster på kraftbussarna kan ställa till med problem. Detta examensarbete har utförts på Saab Aeronautics och gav ett tillfredställande resultat.
I denna rapport så genomförs en konstruktion av en solpanelkrets. Denna krets kommer att användas i utbildningssyfte så en användare kan skaffa sig en förståelse för hur en solpanel fungerar. Solpanelkretsen seriekopplas för att efterlikna riktiga solpaneler. På kretsen så kan en användare ställa in önskad skuggning som motsvarar olika väderförhållanden som en riktig solpanel kan befinna sig i, samt se hur skugga påverkar en solpanel och seriekopplade solpaneler. Kretsen styrs sedan med någon heter MPPT för att utvinna maximal effekt under alla väderförhållanden som en solpanel kan befinna sig i.
I rapporten så presenteras först väsentlig solteori för att ge upphov till en ökad förståelse för hur solpaneler fungerar. Rapporten bygger mycket på att jämföra simulerade grafer från kretssimuleringsprogrammet Multisim med den fysiska byggda kretsen. Grafer från en solpanel och seriekopplade solpaneler med och utan bypass dioder presenteras. Mätningar från MPPT-styrningen genomförs för att visa vilken maximal effekt som utvanns från den fysiska byggda kretsen. Alla mätningar som genomförts finns i ett resultatkapitel och till sist så diskuteras resultatet och förslag på vidareutvecklingar.
Jitter is generally defined as a time deviation of the clock waveform from its desired position. The deviation which occurs can be on the leading or lagging side and it can be bounded (deterministic) or unbounded (random). Jitter is a critical specification in the digital system design. There are various techniques to measure the jitter. The straightforward approach is based on spectrum analyzer or oscilloscope measurements. In this thesis an on-chip jitter measurement technique is investigated and the respective circuit is designed using 65 nm CMOS technology. The work presents the high level model and transistor level model, both implemented using Cadence software. Based on the Vernier concept the circuit is composed of an edge detector, two oscillators, and a phase detector followed by a binary counter, which provides the measurement result. The designed circuit attains resolution of 10ps and can operate in the range of 100 - 500 MHz Compared to other measurement techniques this design features low power consumption and low chip area overhead that is essential for built-in self-test (BIST) applications.
This thesis work is done for the department of Electronic System at The Institute of Technology at Linköping University (Linköpings Tekniska Högskolan). Study's focus is to design and implement a protocol for smart dust networks to improve the energy consumption algorithm for this kind of network.
Smart dust networks are in category of distributed sensor networks and power consumption is one of the key concerns for this type of network. This work shows that by focusing on improving the algorithmic behavior of power consumption in every network element (so called as mote), we can save a considerable amount of power for the whole network.
Suggested algorithm is examined using Erlang for one mote object and the whole idea has put into test for a small network using SystemC.
This paper presents a novel power amplifier (PA) architecture based on the combination of radio frequency pulse width modulation (RFPWM) and multilevel PWM. The architecture provides better dynamic range at high carrier frequency compared to RFPWM. The benefits of this architecture over multilevel PWM are that it only requires a single PA and no combiner. The average efficiency for an 802.11g baseband signal is better than multilevel PWM. Our results also shows that the proposed technique exhibit a constant dynamic range at carrier frequency of 3, 4 and 5 GHz, in contrast to RFPWM which shows a decrease in dynamic range for increase in carrier frequency.
This paper presents two novel transmitter architectures based on the combination of radio-frequency pulse-width modulation and multiphase pulse-width modulation. The proposed transmitter architectures provide good amplitude resolution and large dynamic range at high carrier frequency, which is problematic with existing radio-frequency pulse-width modulation based transmitters. They also have better power efficiency and smaller chip area compared to multiphase pulse-width modulation based transmitters.
This paper presents a novel pulse-width modulation (PWM) transmitter architecture that compensates for aliasing distortion by combining PWM and outphasing. The proposed transmitter can use either switch-mode PAs (SMPAs) or linear PAs at peak power, ensuring maximum efficiency. The transmitter shows better linearity, improved spectral performance and increased dynamic range compared to other polar PWM transmitters as it does not suffer from AM-AM distortion of the PAs and aliasing distortion due to digital PWM. Measurement results show that the proposed architecture achieves an improvement of 8 dB and 4 dB in the dynamic range compared to the digital polar PWM transmitter (PPWMT) and the aliasing-free PWM transmitter (AF-PWMT), respectively. The proposed architecture also shows better efficiency compared to the AF-PWMT.
Department of Electronic Engineering , NED University of Engineering and Technology, Karachi, Pakistan.
Linearity and efficiency are important parameters in determining the performance of any wireless transmitter. Pulse-width modulation (PWM) based transmitters offer high efficiency but suffer from low linearity due to image and aliasing distortions. Although the problem of linearity can be addressed by using an aliasing-free PWM (AF-PWM), these transmitters have a lower efficiency as they can only use linear power amplifiers (PAs). Moreover, an all-digital implementation of such transmitters is not possible. The aliasing-compensated PWM transmitter (AC-PWMT) has a higher efficiency due to the use of switch-mode PAs (SMPAs) but uses outphasing to eliminate image and aliasing distortions and requires a larger silicon area. In this study, the authors propose a novel transmitter that eliminates both aliasing and image distortions while using a single SMPA. The transmitter can be implemented using all-digital techniques and achieves a higher efficiency as compared to both AF-PWM and AC-PWM based transmitters. Measurement results show an improvement of 11.3, 7.2, and 4.3 dBc in the ACLR as compared to the carrier-based PWM transmitter (C-PWMT), AF-PWMT, and AC-PWMT, respectively. The efficiency of the proposed transmitter is similar to that of C-PWMT, which is an improvement of 5% over AF-PWMT.
Analog-to-digital converters (ADCs) are crucial blocks which form the interface between the physical world and the digital domain. ADCs are indispensable in numerous applications such as wireless sensor networks (WSNs), wireless/wireline communication receivers and data acquisition systems. To achieve long-term, autonomous operation for WSNs, the nodes are powered by harvesting energy from ambient sources such as solar energy, vibrational energy etc. Since the signal frequencies in these distributed WSNs are often low, ultra-low-power ADCs with low sampling rates are required. The advent of new wireless standards with ever-increasing data rates and bandwidth necessitates ADCs capable of meeting the demands. Wireless standards such as GSM, GPRS, LTE and WLAN require ADCs with several tens of MS/s speed and moderate resolution (8-10 bits). Since these ADCs are incorporated into battery-powered portable devices such as cellphones and tablets, low power consumption for the ADCs is essential.
The first contribution is an ultra-low-power 8-bit, 1 kS/s successive approximation register (SAR) ADC that has been designed and fabricated in a 65-nm CMOS process. The target application for the ADC is an autonomously-powered soil-moisture sensor node. At VDD = 0.4 V, the ADC consumes 717 pW and achieves an FoM = 3.19 fJ/conv-step while meeting the targeted dynamic and static performance. The 8-bit ADC features a leakage-suppressed S/H circuit with boosted control voltage which achieves > 9-bit linearity. A binary-weighted capacitive array digital-to-analog converter (DAC) is employed with a very low, custom-designed unit capacitor of 1.9 fF. Consequently the area of the ADC and power consumption are reduced. The ADC achieves an ENOB of 7.81 bits at near-Nyquist input frequency. The core area occupied by the ADC is only 0.0126 mm2.
The second contribution is a 1.2 V, 10 bit, 50 MS/s SAR ADC designed and implemented in 65 nm CMOS aimed at communication applications. For medium-to-high sampling rates, the DAC reference settling poses a speed bottleneck in charge-redistribution SAR ADCs due to the ringing associated with the parasitic inductances. Although SAR ADCs have been the subject of intense research in recent years, scant attention has been laid on the design of high-performance on-chip reference voltage buffers. The estimation of important design parameters of the buffer as well critical specifications such as power-supply sensitivity, output noise, offset, settling time and stability have been elaborated upon in this dissertation. The implemented buffer consists of a two-stage operational transconductance amplifier (OTA) combined with replica source-follower (SF) stages. The 10-bit SAR ADC utilizes split-array capacitive DACs to reduce area and power consumption. In post-layout simulation which includes the entire pad frame and associated parasitics, the ADC achieves an ENOB of 9.25 bits at a supply voltage of 1.2 V, typical process corner and sampling frequency of 50 MS/s for near-Nyquist input. Excluding the reference voltage buffer, the ADC consumes 697 μW and achieves an energy efficiency of 25 fJ/conversion-step while occupying a core area of 0.055 mm2.
The third contribution comprises five disparate works involving the design of key peripheral blocks of the ADC such as reference voltage buffer and programmable gain amplifier (PGA) as well as low-voltage, multi-stage OTAs. These works are a) Design of a 1 V, fully differential OTA which satisfies the demanding specifications of a PGA for a 9-bit SAR ADC in 28 nm UTBB FDSOI CMOS. While consuming 2.9 μW, the PGA meets the various performance specifications over all process corners and a temperature range of [−20◦ C +85◦ C]. b) Since FBB in the 28 nm FDSOI process allows wide tuning of the threshold voltage and substantial boosting of the transconductance, an ultra-low-voltage fully differential OTA with VDD = 0.4 V has been designed to satisfy the comprehensive specifications of a general-purpose OTA while limiting the power consumption to 785 nW. c) Design and implementation of a power-efficient reference voltage buffer in 1.8 V, 180 nm CMOS for a 10-bit, 1 MS/s SAR ADC in an industrial fingerprint sensor SoC. d) Comparison of two previously-published frequency compensation schemes on the basis of unity-gain frequency and phase margin on a three-stage OTA designed in a 1.1 V, 40-nm CMOS process. Simulation results highlight the benefits of split-length indirect compensation over the nested Miller compensation scheme. e) Design of an analog front-end (AFE) satisfying the requirements for a capacitive body-coupled communication receiver in a 1.1 V, 40-nm CMOS process. The AFE consists of a cascade of three amplifiers followed by a Schmitt trigger and digital buffers. Each amplifier utilizes a two-stage OTA with split-length compensation.
This brief presents an 8-bit 1-kS/s successive approximation register (SAR) analog-to-digital converter (ADC), which is targeted at distributed wireless sensor networks powered by energy harvesting. For such energy-constrained applications, it is imperative that the ADC employs ultralow supply voltages and minimizes power consumption. The 8-bit 1-kS/s ADC was designed and fabricated in 65-nm CMOS and uses a supply voltage of 0.4 V. In order to achieve sufficient linearity, a two-stage charge pump was implemented to boost the gate voltage of the sampling switches. A custom-designed unit capacitor of 1.9 fF was used to realize the capacitive digital-to-analog converters. The ADC achieves an effective number of bits of 7.81 bits while consuming 717 pW and attains a figure of merit of 3.19 fJ/conversion-step. The differential nonlinearity and the integral nonlinearity are 0.35 and 0.36 LSB, respectively. The core area occupied by the ADC is only 0.0126 mm2.
This paper presents the design of a 10-bit, 50 MS/s successive approximation register (SAR) analog-to-digital converter (ADC) with an onchip reference voltage buffer implemented in 65 nm CMOS process. The speed limitation on SAR ADCs with off-chip reference voltage and the necessity of a fast-settling reference voltage buffer are elaborated. Design details of a high-speed reference voltage buffer which ensures precise settling of the DAC output voltage in the presence of bondwire inductances are provided. The ADC uses bootstrapped switches for input sampling, a double-tail high-speed dynamic comparator and split binary-weighted capacitive array charge redistribution DACs. The split binary-weighted array DAC topology helps to achieve low area and less capacitive load and thus enhances power efficiency. Top-plate sampling is utilized in the DAC to reduce the number of switches. In post-layout simulation which includes the entire pad frame and associated parasitics, the ADC achieves an ENOB of 9.25 bits at a supply voltage of 1.2 V, typical process corner and sampling frequency of 50 MS/s for near-Nyquist input. Excluding the reference voltage buffer, the ADC consumes 697 μW and achieves an energy efficiency of 25 fJ/conversionstep while occupying a core area of 0.055 mm2.
This paper presents the design of a sampling switch to be used in the input interface to an ultra low-power 8-bit, 1-kS/s SAR ADC in 65 nm CMOS working at a supply voltage of 0.4 V. Important design trade-offs for the sampling switch in this low-voltage and low-power scenario are elaborated upon. The design of a multi-stage charge pump which generates the requisite boosted control voltage is described. A combination of the multi-stage charge pump and a leakage-reduced transmission-gate (TG) switch meets the speed requirement while mitigating leakage without employing additional voltages. Performance of the sampling switch has been characterized over process and temperature (PT) corners. In post-layout simulation, the sampling switch provides a linearity corresponding to 9.42 bits to 13.5 bits over PT corners with a worst-case power consumption of 216 pW while occupying an area of 25.4 μm × 24.7 μm.
This paper presents the design of a fast-settling reference voltage buffer (RVBuffer) which is used to buffer the high reference voltage in a 10-bit, 50 MS/s successive approximation register (SAR) ADC implemented in 65 nm CMOS. Though numerous publications on SAR ADCs have appeared in recent years, the role of RVBuffers in ensuring ADC performance, the associated design challenges and impact on power and FoM of the entire ADC have not been discussed in-depth. In this work, the speed limitation on precise settling of the digital-to-analog converter voltage (DAC) in a SAR ADC imposed by parasitic inductances of the bondwire and PCB trace is explained. The crucial design parameters for the reference voltage buffer in the context of the SAR ADC are derived. Post-layout simulation results for the RVBuffer are provided to verify settling-time, noise and PSRR performance. In post-layout simulation which includes the entire pad frame and associated parasitics, the SAR ADC achieves an ENOB of 9.25 bits at a supply voltage of 1.2 V, typical process corner and sampling frequency of 50 MS/s for near-Nyquist input. Excluding the reference voltage buffer, the ADC consumes 697 ï¿œW and achieves an energy efficiency of 25 fJ/conversion-step while occupying a core area of 0.055 mm2.
This paper presents a fully-differential operational transconductance amplifier (OTA) designed in a 28 nm ultra-thin box and body (UTBB) fully-depleted silicon-on-insulator (FDSOI) CMOS process. An overview of the features of the 28 nm UTBB FDSOI process which are relevant for the design of analog/mixed-signal circuits is provided. The OTA which features continuous-time CMFB circuits will be employed in the programmable gain amplifier (PGA) for a 9-bit, 1 kS/s SAR ADC. The reverse body bias (RBB) feature of the FDSOI process is used to enhance the DC gain by 6 dB. The OTA achieves rail-to-rail output swing and provides DC gain = 70 dB, unity-gain frequency = 4.3 MHz and phase margin = 68ï¿œ while consuming 2.9 μW with a Vdd = 1 V. A high linearity > 12 bits without the use of degeneration resistors and a settling time of 5.8 μs (11-bit accuracy) are obtained under nominal operating conditions. The OTA maintains satisfactory performance over all process corners and a temperature range of [-20oC +85oC].
This paper presents an ultra-low-voltage, sub-μW fully differential operational transconductance amplifier (OTA) designed in 28 nm ultra-thin buried oxide (BOX) and body (UTBB) fully-depleted silicon-on-insulator (FDSOI) CMOS process. In this CMOS process, the BOX isolates the substrate from the drain and source and hence enables a wide range of body bias voltages. Extensive use of forward body biasing has been utilized in this work to reduce the threshold voltage of the devices, boost the device transconductance (gm) and improve the linearity. Under nominal process and temperature conditions at a supply voltage of 0.4 V, the OTA achieves −64 dB of total harmonic distortion (THD) with 75% of the full scale output swing while consuming 785 nW. The two-stage OTA incorporates continuoustime common-mode feedback circuits (CMFB) and achieves DC gain = 72 dB, unity-gain frequency of 2.6 MHz and phase margin of 68o. Sufficient performance is maintained over process, supply voltage and temperature variations.
Linköping University, Department of Electrical Engineering, Integrated Circuits and Systems. Linköping University, The Institute of Technology. AnaCatum AB, Linköping, Sweden.
The paper presents the design of a single-ended amplifier in 1.8~V, 180 nm CMOS process forbuffering the reference voltage in a 10-bit 1-MS/s successive-approximation register (SAR) ADC. The design addresses the comprehensive requirements on the buffersuch as settling time, PSRR, noise, stability, capacitive load variation and power-down features which would be required in a SAR ADC for embedded applications. The buffer is optimized for current consumption and area. Transistor schematic level simulation achieves worst-case settling time of 19.3~ns andcurrent consumption of 66~$\mu$A while occupying an area of (19.2~$\mu$m $\times$ 19.2~$\mu$m). | CommonCrawl |
Ouch! A kitten got stuck on a tree. Fortunately, the tree's branches are numbered. Given a description of a tree and the position of the kitten, can you write a program to help the kitten down?
The input is a description of a single tree. The first line contains an integer $K$, denoting the branch on which the kitten got stuck. The next lines each contain two or more integers $a, b_1, b_2, \ldots $. Each such line denotes a branching: the kitten can reach $a$ from $b_1, b_2, \ldots $ on its way down. Thus, $a$ will be closer to the root than any of the $b_ i$. The description ends with a line containing -1. Each branch $b_ i$ will appear on exactly one line. All branch numbers are in the range $1..100$, though not necessarily contiguous. You are guaranteed that there is a path from every listed branch to the root. The kitten will sit on a branch that has a number that is different than the root.
The illustration above corresponds to the sample input.
Output the path to the ground, starting with the branch on which the kitten sits. | CommonCrawl |
In this problem, we are dividing by a fraction. However, we are never supposed to divide by a fraction. Rather, we must multiply by the reciprocal of the fraction. The reciprocal of a fraction is that fraction with the numerator (top of the fraction) and denominator (bottom of the fraction) switched. Thus, we simplify to find: -5$\div$(-5/3)=-5$\times$(-3/5)=3 Note, the answer is positive, for a negative divided by a negative is a positive. | CommonCrawl |
In neural nets, there is often a weight regularization term in the loss function, which ensures that unnecessarily high weights don't occur. For example, if $C(\theta,x)$ is the baseline loss function, then we have the loss $L(\theta,x)=C(\theta,x)+\sum L(\theta_i)$, (where $L$ is the $L_1$ norm or $L_2$ norm usually).
However, there is nothing in principle that prevents us from choosing a more complicated weight regularization term. we can have a generalized weight regularization term $\mathcal L(\theta)$.
why not tell our neural network that it should "look like a boolean circuit with 2 inputs in each neuron"? We choose sigmoid activation functions, and use weight regularization that penalizes having more than $2$ non-zero weights for each neuron.
why not have a separate neural network look at the original network's weights and its performance, and set weight regularization cost for each weight separately?
Have things in this direction been done? are there benefits? (apart from the obvious computational downsides). | CommonCrawl |
Abstract: Spectral emission lines are local features that represent extra emissions of photons in a narrow band of energy. In a statistical model, it is often appropriate to model the emission lines with a narrow Gaussian function or a delta function. In this article, we show how to identify the location of the narrow line profiles using a model-based Bayesian statistical perspective. Such Bayesian methods are ideally suited to handling the complexity of high-resolution high-energy spectral data such as that obtained with the Chandra X-ray Observatory. van Dyk et al (2001) show how Bayesian methods can account for these complexities of the data generation mechanism as well as the Poisson nature of photon count data. The multimodal nature of the likelihood function poses difficulties for these methods, however, when the location and width of a spectral line are simultaneously fitted or when delta functions are used to model spectral lines. These difficulties necessitate more sophisticated, state-of-the-art statistical computation. We thus develop such methods and illustrate how to detect narrow spectral lines in X-ray spectra using Chandra data sets for the energy spectrum of the high redshift quasar PG 1634+706.
We describe a new spectral classification technique called quantile analysis for X-ray sources with limited statistics. The quantile analysis is superior to the conventional approaches such as X-ray hardness ratios or X-ray color analysis. The median is considered to be an improved substitute for the conventional X-ray hardness ratio and the quantile-based phase diagram is more evenly sensitive over various spectral shapes than the conventional color-color diagrams. We demonstrate the new technique by simulations using Chandra ACIS detector response function and the analysis results from the deep observations at the galactic center.
We will describe the California-Harvard AstroStatistics Collaboration, CHASC. We will provide an introduction to Bayesian methods in the context of some basic X-ray astrophysics problems, such as determining the source strength in the presence of background, and hardness ratios in the regime of (very) low counts. We will also discuss posterior predictive p-values (PPP), which are the preferred alternatives to the often abused F-tests used for model comparisons.
The Markov chain Monte Carlo (MCMC) methods, originating in computational physics about half a century ago, have seen an enormous range of applications in recent statistical literature, due to their ability to simulate from very complex distributions such as the ones needed in realistic statistical models. This talk provides an introductory tutorial of the two most frequently used MCMC algorithms: the Gibbs sampler and the Metropolis-Hastings algorithm. Using simple yet non-trivial examples, we show, step by step, how to implement these two algorithms. The examples involve a family of bivariate distributions whose full conditional distributions are all normal but whose joint densities are not only non-normal, but also bimodal.
Abstract: We explore the convex hull peeling process to develop empirical tools for statistical inferences on multivariate massive data. Convex hull and its peeling process has intuitive appeals for robust location estimation. We define the convex hull peeling depth, which enables to order multivariate data. This ordering process provides ways to obtain multivariate quantiles including median. Based on the generalized quantile process, we define a convex hull peeling central region, a convex hull level set, and a volume functional, which lead us to invent one dimensional mappings, describing shapes of multivariate distributions along data depth. We define empirical skewness and kurtosis measures based on the convex hull peeling process. In addition to these empirical descriptive statistics, we find a few methodologies to separate multivariate outliers in massive data sets. Those outlier detection algorithms are (1) estimating multivariate quantiles up to the level $\alpha$, (2) detecting changes in a measure sequence of convex hull level sets, and (3) constructing a balloon to exclude outliers. The convex hull peeling depth is a robust estimator so that the existence of outliers do not affect properties of inner convex hull level sets. Overall, we illustrate all these characteristics and algorithms of the convex hull peeling process through bivariate synthetic data sets. We show that these empirical procedures are applicable to real massive data set by employing Quasars and galaxies from the Sloan Digital Sky Survey. | CommonCrawl |
Abstract: In a traitor tracing scheme, each user is given a different decryption key. A content distributor can encrypt digital content using a public encryption key and each user in the system can decrypt it using her decryption key. Even if a coalition of users combines their decryption keys and constructs some ``pirate decoder'' that is capable of decrypting the content, there is a public tracing algorithm that is guaranteed to recover the identity of at least one of the users in the coalition given black-box access to such decoder.
In prior solutions, the users are indexed by numbers $1,\ldots,N$ and the tracing algorithm recovers the index $i$ of a user in a coalition. Such solutions implicitly require the content distributor to keep a record that associates each index $i$ with the actual identifying information for the corresponding user (e.g., name, address, etc.) in order to ensure accountability. In this work, we construct traitor tracing schemes where all of the identifying information about the user can be embedded directly into the user's key and recovered by the tracing algorithm. In particular, the content distributor does not need to separately store any records about the users of the system, and honest users can even remain anonymous to the content distributor.
The main technical difficulty comes in designing tracing algorithms that can handle an exponentially large universe of possible identities, rather than just a polynomial set of indices $i \in [N]$. We solve this by abstracting out an interesting algorithmic problem that has surprising connections with seemingly unrelated areas in cryptography. We also extend our solution to a full ``broadcast-trace-and-revoke'' scheme in which the traced users can subsequently be revoked from the system. Depending on parameters, some of our schemes can be based only on the existence of public-key encryption while others rely on indistinguishability obfuscation. | CommonCrawl |
A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poisson-commute and induce an $(\mathbb S^1 \times \mathbb R)$-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo $\&$ Vu Ngoc by means of five invariants.
Three of these five invariants are the so-called Taylor series invariant, the height invariant, and the twisting index. Roughly, the first one describes the behaviour of the system near the focus-focus singular fibre, the second one the position of the focus-focus value, and the third one compares the `distinguished' torus action given near each focus-focus singular fiber to the global toric `background action'.
$\bullet$ Taylor series and twisting index for coupled spin oscillators.
$\bullet$ Taylor series, height invariant, and twisting index for coupled angular momenta.
$\bullet$ Putting the Taylor series and twisting index in relation with wellknown notions from classical dynamical systems like rotation number and rotation vector etc.
$\bullet$ Change of the Taylor series and twisting index when varying the parameters of these systems. | CommonCrawl |
The recursive motif "F +60 F +240 F +60 F" produces the well-known Koch snowflake. Interestingly, for certain choices of angles, this recursive algorithm actually produces an image with finitely many segments and rotational symmetry.
The subdivision of the circle into 360 parts is arbitrary, however. The image above was produced by dividing the circle into 336 parts, and taking 210 and 218 of them, respectively, in the recursive algorithm, which would be described as "F +210 F +218 F +210 F." This is indicated in the caption, where the angles are called $\alpha_0$ and $\alpha_1$ — this is the notation I use in a paper I'm working on which discusses the mathematics of these images in detail. I'll post a link to it when I'm finished.
I stumbled on the white-on-gray color scheme as a result of including some of these images in a paper to be published. We were instructed to make sure that our images, if in color, looked good if printed in black and white. So I decided to use a white-on-gray motif, and was so pleased with the results that I stayed with it.
This is, for me, mathematical art at its purest. Even with the simplest of color schemes, the geometry of the images does all the work in generating the aesthetic appeal of the images. | CommonCrawl |
We reduce a broad class of machine learning problems, usually addressed by EM or sampling, to the problem of finding the $k$ extremal rays spanning the conical hull of a data point set. These $k$ ``anchors'' lead to a global solution and a more interpretable model that can even outperform EM and sampling on generalization error. To find the $k$ anchors, we propose a novel divide-and-conquer learning scheme ``DCA'' that distributes the problem to $\mathcal O(k\log k)$ same-type sub-problems on different low-D random hyperplanes, each can be solved by any solver. For the 2D sub-problem, we present a non-iterative solver that only needs to compute an array of cosine values and its max/min entries. DCA also provides a faster subroutine for other methods to check whether a point is covered in a conical hull, which improves algorithm design in multiple dimensions and brings significant speedup to learning. We apply our method to GMM, HMM, LDA, NMF and subspace clustering, then show its competitive performance and scalability over other methods on rich datasets. | CommonCrawl |
概要:二次元乱流を特徴づける性質として, 高レイノルズ数におけるエネルギー密度スペクトル中のエンストロフィーカスケードとエネルギー逆カスケードに対応する領域の出現が挙げられる. 言い換えれば, 二次元乱流とは粘性ゼロ極限における流体のエネルギー保存とエンストロフィー散逸によって特徴付けられる. よって非粘性流体の運動を記述するEuler方程式の解でこのような性質を持つものが存在すれば, 二次元乱流の数学的構造を解析するうえで重要であると考えられる. しかし, 二次元Euler方程式においてエンストロフィー散逸解を直接構成するには数学的な困難が伴う. 本セミナーでは, Euler方程式の正則化方程式である. Euler-$\alpha$方程式を利用したエンストロフィー散逸解の構成について紹介し, 特に点渦の自己相似3体衝突が生むエンストロフィー散逸に関する数学的結果について説明する予定である.
概要: Mathematical models and statistical arguments play a central role in the assessment of the changes that are observed in Earth's climate system. While much of the discussion of climate change is focused on large-scale computational models, the theory of dynamical systems provides the language to distinguish natural variability from change. In this talk I will discuss some problems of current interest in climate science and indicate how, as mathematicians, we can find inspiration for new applications.
概要: Strong \(A_\infty\) weights are introduced. Then, degenerate elliptic equations with respect to a power of a strong \(A_\infty\) weight are studied. Then, Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions are proved. We stress that regularity results are achieved under minimal assumptions on the coefficients.
概要: In this talk, we will introduce a high-precision numerical method for studying quasicrystals, i.e., the projection method. This method is based on the philosophy that a continuous distributed quasicrystal is a continuous function over a quasilattice. It can be used to study the soft quasicrystals. In particular, the projection method decomposes the quasiperiodic structure by a combination of the almost periodic functions, and provides an efficient algorithm to calculate the combinational coefficients in the higher-dimensional space. At the same time, the projection method provides a unified computational framework for the periodic crystals and quasicrystals. The free energies of the two kinds of ordered structures can be obtained with the same accuracy. Therefore, it can be used to determine the thermodynamic stability of periodic and quasiperiodic crystals in theory. We have applied the algorithm to a series of coarse-grained density functional theories, and obtained 2-dimensional 8-, 10, 12-fold symmetric quasicrystals (computed in the 4-dimensional space), and 3-dimensional icosahedral quasicrystals (calculated in the 6-dimensional space). The corresponding phase diagrams, including periodic crystals and quasicrystals, have been constructed. | CommonCrawl |
Abstract : The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a non-iterative method for computing intersections by solving a matrix singular value decomposition (SVD) and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one-shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton-Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree $q$ of the shape functions and is $2q \times 3q$ for curves and $6 q^2 \times 8 q^2$ for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high-order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/. | CommonCrawl |
The moment problem arises as the result of trying to invert the mapping that takes a measure $μ$ to the sequences of moments, and to resolve the problem of determinacy of such measure.
If all powers of two random variables are uncorrelated, are they independent?
Do orthogonal polynomials determine the moments of their orthogonality measure?
Sample instances of random process given all temporal correlation functions?
What conditions on the moments make a measure a probability measure?
$\int x^k\ d\mu(x)\geq0$ for $k$ even?
About the $k$th moment of the sum of independent r.v.
How are moments used in representing differential equations?
What can be said about the set of distributions with given moment sequence?
Higher moments are minimized around WHAT point?
Which moments identify an absolutely continuous measure on the unit circle?
Does the condition $E[X]=E[X^2]=E[X^3]$ determine the distribution of $X$?
I have read a claim that $E[X^4]<\infty$ and $E[Y^4]<\infty$ imply that $E[(XY)^2]<\infty$ but I cannot prove it. $X$ and $Y$ are not independent and they are correlated.
Why study moment problem in one dimensional case?
Positive definite sequence and its corresponding determinant.
Question about Humburger moment problem and Characteristic function. | CommonCrawl |
Yesterday was Sam's birthday. The most interesting gift was definitely the chessboard. Sam quickly learned the rules of chess and defeated his father, all his friends, his little sister, and now no one wants to play with him any more.
Sam's chessboard has size $N\times N$. A bishop can move to any distance in any of the four diagonal directions. A bishop threatens another bishop if it can move to the other bishop's position. Your task is to compute the maximum number of bishops that can be placed on a chessboard in such a way that no two bishops threaten each other.
The input file consists of several lines. The line number $i$ contains a positive integer representing the size of the $i$-th chessboard. The size of each chessboard is at most $1\, 000\, 000$.
The output file should contain the same number of lines as the input file. The $i$-th line should contain one number – the maximum number of bishops that can be placed on $i$-th chessboard without threatening each other. | CommonCrawl |
I'll give a brief account of Hamiltonian dynamical systems whose degrees of freedom are given by a complex-valued sequence $\alpha_n(t)$, and the Hamiltonian is a quartic combination of $\alpha_n$ and its complex conjugate. Such systems arise naturally as weakly nonlinear approximations to a few interesting equations of mathematical physics, whose linearized perturbations possess a perfectly resonant spectrum of frequencies. A few such equations for sequences display remarkably simple solutions with |$\alpha_n$| being an exactly periodic function of time, while the generating function made of $\alpha_n$ has a simple meromorphic structure in the complex plane. Only the simplest of the equations in this class (the cubic Szego equation) has been thoroughly studied by mathematicians over the course of the last 8 years. The results of that study hint at many more surprising structures waiting to be uncovered for other such systems displaying periodic behaviors. | CommonCrawl |
Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
(where simple means no self intersections and equidecomposable means finitely cut and glued).
For your problem you can take the first polygon to be a unit square and the second to be a sqrt(5) by 1/sqrt(5) rectangle and apply this theorem. Then perform the remaining four cuts.
Also, the generalisation of your question is the 2d analogue of Hilbert's 3rd Problem which asks whether given any two polyhedra with equal volume can one be finitely cut and glued into the other. The answer here, unlike in the 2d case, is "no" which was proved by Dehn using Dehn invariants in 1900.
Cut from (0,0) to (1,1/2), and from (0,1/2) to (1,1). We can glue these three pieces together to get a ring with circumference $\sqrt 5$ and height $\sqrt 5 / 5$. Now it's easy!
Since $1+2i$ has length $\sqrt5$, you can lift a square fundamental domain of $\mathbb C/\mathbb Z[i]$ to $\mathbb C/(1+2i)\mathbb Z[i]$. Overlay a square fundamental domain for the larger torus to get a way to divide a square into 5 smaller squares.
It's pretty easy to decompose any rectangle into a square geometrically, but the general decomposition is not as nice.
Not the answer you're looking for? Browse other questions tagged mg.metric-geometry discrete-geometry tiling polygons or ask your own question.
Planar sets where any line through the center of mass divides the set into two regions of equal area.
If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Can a unit square be cut into rectangles that tile a rectangle with irrational sides?
Could a perfect squared square be split into two perfect squared squares? | CommonCrawl |
How do you canonicalize a matrix over column- and row-swap operations?
Or more specifically, does there exist a function f(M) such that f(M)=f(N) iff there is set of column- and row-swap operations (i.e. a permutation matrix A) on M that would transform M into N (i.e. AMA')?
But how to handle this?
432179865 any of the nine rows, this becomes messy.
896735412 rows or columns must be a square matrix.
Let $M$ be an $m \times n$ matrix. Each row of $M$ may be ordered as a multiset, and one may choose the lexicographically least of these and require that the first row of $f(M)$ should be that multiset (in order). Of course there may be several rows of $M$ whose multisets give this same "lex-least" multiset. One then looks at each such row and asks what the rows below it would look like if that row were placed on top, and column swaps were done in any way preserving the multiset.
Suppose e.g. two rows had lex-least multisets $[1,1,1,2,2,3].$ In $M$ itself these rows might appear as respectively $a=[1,1,2,1,2,3]$ and $b=[2,2,3,1,1,1].$ In either case we can initially do a row swap to put that row on top and then a column swap so the first row is now $[1,1,1,2,2,3].$ The appearance of rows beyond the first will likely be different depending on whether $a$ or $b$ was initially put on top, and we need do decide which to do.
Now note that irrespective to which of $a$ or $b$ was initially placed on top, the first three columns have the same number $1$ on top, and so these first three columns may be permuted among themselves without disturbing the first row $[1,1,1,2,2,3]$. Similarly one may permute the two rows underneath the two $2$'s. Then we can try each of rows $a$ and $b$, with all ways of permuting the columns not disturbing the pattern $[1,1,1,2,2,3]$ of the first row, and see for which one the rest of the matrix is lexicographically least, as a "tie-breaker" on whether to select row $a$ or row $b$ to swap initially to the top row.
What needs work is to choose in which sense the matrix from rows 2 on should be lexicographically ordered. I think it should be based on the fact that row swaps only should be done once row 1 is fixed by the initial single row swap and the column swaps to put it into order. On the other hand maybe one needs something more elaborate such as calling a part of the algorithm recursively.
Added: More detail is needed to see which of the initial rows having the minimal lexicographic multiset one should move to the top via one row swap and a choice of column swaps to put the chosen row 1 into lex order. Suppose the rows of $M$ having the min lex order are put into a collection $R$ of rows. For each $r$ in $R$ we do a subcalculation like the following: We form a copy of $M$ with row $r$ placed at the top via a row switch. We do some column swaps so now row 1 is in nondecreasing order.
Now any columns which are under a block of identical elements of row 1 may be permuted among themselves, still keeping row 1 as it is. Call these "allowable column swaps". Then for each row beyond the first, determine which of the allowable column swaps would put that row in least lex order. Choose the least among these now lex-ordered rows, and call that the "score" of the chosen $r$. We do the above for each candidate $r \in R,$ and see which $r$ gives us the minimal "score". This tells us which $r$ in $R$ we should switch to the top, and also which column swaps to carry out to put the now chosen row 1 into lex order.
I think now that we have row 1 chosen and in order, the same kind of procedure can be used to get row 2, only now we have restricted all column swaps to those keeping row 1 in its (fixed by now) lex order. It may not be the case that say row 2 is in lex order, only that each of its substrings under a constant block from row 1 is in lex order.
Not the answer you're looking for? Browse other questions tagged linear-algebra combinatorics matrices sudoku or ask your own question.
How to find a canonical member of an equivalence class of matrices under row and column swaps? | CommonCrawl |
Molar conductivity increases with decrease in concentration. This is because the total volume, $V$, of solution containing one mole of electrolyte also increases. It has been found that decrease in $\kappa$ on dilution of a solution is more than compensated by increase in its volume. Physically, it means that at a given concentration, $\Lambda_m$ can be defined as the conductance of the electrolytic solution kept between the electrodes of a conductivity cell at unit distance but having area of cross section large enough to accommodate sufficient volume of solution that contains one mole of the electrolyte. When concentration approaches zero, the molar conductivity is known as limiting molar conductivity and is represented by the symbol $E^\circ_m$. The variation in $\Lambda_m$ with concentration is different (Fig. 3.6) for strong and weak electrolytes.
Can anyone explain what does this sentence mean : "it has been found that decrease in k on dilution of a solution is more than compensated by increase in its volume"
Molar conductivity : $\kappa \times V$ ($\kappa$ is specific conductance), $V$ is volume of solution containing 1 mole of electrolyte.
I understand that molar conductivity increase with decrease in concentration. But at the same time this sentence is contradicting that because it says the decrease in "$\kappa$"(specific conductance) is more than increase in volume Which should decrease molar conductivity. I am confused.
I think you missed the word "compensated".
The decrease in specific conductivity is more than compensated by increase in its volume.
This simply answers your question. The reduction in molar conductivity due to the decrease in specific conductivity is overridden by the increase in volume.
It is saying that the latter overrides the former, and not just overrides, but overrides by a lot.
Why does the molar conductivity decrease with increasing charge density? | CommonCrawl |
Induction (recursion actually) will end you up with all finite decimals, but of arbitrary length. This way you'll never end up with infinite decimals.
Induction is how you specify all members of these limits. You can't write them all out.
(c) using a symbol such as "\" or "0" seems to be equivalent to "scratches on a rock".
a) Induction specifies all elements of a list.
If I specify every element of an infinite sequence (calculus), have I specified every element of the sequence?
b) 1,2,3,4,... IS the number system. You can't count something with itself.
Note that the INFINITE harmonic series is specified by 1/1+1/2+1/3+...... Do you have a problem with that?
Last edited by zylo; July 9th, 2018 at 10:46 AM.
OK, this is annoying, please learn the difference between recursion and induction. Cause right now you're just confusing me.
Your response, zylo, to (a) relates to the elements only. Specifying the elements isn't the same as counting the entire endless list.
Your response to (b) is inapplicable, as I didn't ask you to count something with itself. I asked how you would count 1, 2, 3, 4, ... (continued without end) without a number system.
As you didn't respond to (c), I assume you now accept that your suggestions that used repetitions of "\" and "0" violate the criterion that you don't use sticks, stones and the like, because you've already withdrawn that type of response.
Note that the infinite harmonic series has a infinite number of terms but does not have a term $\frac1\infty$. | CommonCrawl |
A tatami tiling is an arrangement of $1 \times 2$ dominoes (or mats) in a rectangle with $m$ rows and $n$ columns, subject to the constraint that no four corners meet at a point. For fixed $m$ we present and use Dean Hickerson's combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when $n \ge m$. Using this decomposition we find the ordinary generating functions of both unrestricted and inequivalent tatami tilings that fit in a rectangle with $m$ rows and $n$ columns, for fixed $m$ and $n \ge m$. This allows us to verify a modified version of a conjecture of Knuth. Finally, we give explicit solutions for the count of tatami tilings, in the form of sums of binomial coefficients. | CommonCrawl |
We consider robust optimization problems, where the goal is to optimize in the worst case over a class of objective functions. We develop a reduction from robust improper optimization to stochastic optimization: given an oracle that returns $\alpha$-approximate solutions for distributions over objectives, we compute a distribution over solutions that is $\alpha$-approximate in the worst case. We show that derandomizing this solution is NP-hard in general, but can be done for a broad class of statistical learning tasks. We apply our results to robust neural network training and submodular optimization. We evaluate our approach experimentally on corrupted character classification and robust influence maximization in networks. | CommonCrawl |
A question was just asked about whether or not we still need model categories of spectra (vs working with the $\infty$-category). To me (and Fernando Muro at least), it is reminiscent of a previous question about whether we still need model categories. Both questions already have answers from experts in both methods. I suspect that which answers people upvote in both threads has more to do with personal preference than anything else. In situations like this, where a user is asking which of two perfectly good methods is better, and everything is opinion based, should the whole thread be Community Wiki?
For non homotopy-theorists, an analogy would be someone asking: "when you do differential geometry, is it better to work in a coordinate free setting or with coordinates." Of course, the answer is: "it depends on what you are trying to do, and your own personal preferences." You can have experts from both sides weigh in, and write answers with +100 upvotes, but it seems strange to me to get so many reputation points for eloquently stating an opinion.
Edit: the question was made Community Wiki on Feb. 10, 2019.
Browse other questions tagged discussion community-wiki . | CommonCrawl |
A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight.
Examples of non-growth tight actions are product groups acting on the $L^1$ products of Cayley graphs of the factors.
In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the $L^p$ metric on the product space, for all $1 < p ≤ \infty$. In particular, the $L^\infty$ metric on a product of Cayley graphs corresponds to a word metric on the product group. This gives the rst examples of groups that are growth tight with respect to an action on one of their Cayley graphs and non-growth tight with respect to an action on another, answering a question of Grigorchuk and de la Harpe. | CommonCrawl |
This tutorial will provide step-by-step guide for building a Recommendation Engine. We will be recommending conference papers based on their title and abstract. There are two major types of Recommendation Engines: Content Based and Collaborative Filtering Engines.
Content Based recommends only using information about the items being recommended. There is no information about the users.
Collaborative Filtering takes advantage of user information. Generally speaking, the data contains likes or dislikes of every item every user has used. The likes and dislikes could be implicit like the fact that a user watched a whole movie or explicit like the user gave the movie a thumbs up or a good star rating.
In general, Recommendation Engines are essentially looking to find items that share similarity. We can think of similarity as finding sets with a relatively large intersection. We can take any collection of items such as documents, movies, web pages, etc., and collect a set of attributes called "shingles".
Shingles are a very basic broad concept. For text, they could be characters, unigrams or bigrams. You could use a set of categories as shingles. Shingles simply reduce our documents to sets of elements that so we can calculate similarity betweens sets.
For our conference papers, we will use unigrams extracted from the paper's title and abstract. If we had data about which papers a set of users liked, we could even use users as shingles. In this way, you could do Item Based Collaborative Filtering where you search a MinHash of items that have been rated positively by the same users.
Likewise, you can flip this idea on its head and make the items the shingles for a User Based Collaborative Filtering. In this case, you would be finding users that had similar positive ratings to another user.
Title 1 = "Reinforcement Learning using Augmented Neural Networks"
Title 2 = "Playing Atari with Deep Reinforcement Learning"
Now, we can find the similarity between these titles by looking at a visual representation of the intersection of shingles between the two sets. In this example, the total number (union) of shingles is 10, and 2 are a part of the intersection. We would measure the similarity as 2/10 = 1/5.
LSH can be considered an algorithm for dimensionality reduction. A problem that arises when we recommend items from large datasets is that there may be too many pairs of items to calculate the similarity of each pair. Also, we likely will have sparse amounts of overlapping data for all items.
To see why this is the case we can consider the matrix of vocabulary that is created when we store all conference papers.
Traditionally, in order to make recommendations we would have a column for every document and a row for every word ever encountered. Since papers can differ widely in their text content, we are going to get a lot of empty rows for each column, hence the sparsity. To make recommendations, we would then compute the similarity between every row by seeing which words are in common.
These concerns motivate the LSH technique. This technique can be used as a foundation for more complex Recommendation Engines by compressing the rows into "signatures", or sequences of integers, that let us compare papers without having to compare the entire sets of words.
LSH primarily speeds up the recommendation process compared to more traditional recommendation engines. These models also scale much better. As such, regularly retraining a recommendation engine on a large dataset is far less computationally intensive than traditional recommendation engines. In this tutorial, we walk through an example of recommending similar NIPS conference papers.
LSH is used to perform Nearest Neighbor Searches based on a simple concept of "similarity".
We say two items are similar if the intersection of their sets is sufficiently large.
This is the exact same notion of Jaccard Similarity of Sets. Recall the picture above of similarity. Our final measure of similarity, 1/5, is Jaccard Similarity.
NOTE Jaccard similarity is defined as the intersection of two sets divided by the union of the two sets.
Note, other metrics for similarity can be used, but we will be strictly using Jaccard Similarity for this tutorial. LSH is a type of Neighborhood Based method like k-nearest neighbors (KNN). As you can see in the table below, methods like KNN scale poorly compared to LSH.
The power of LSH is that it can even scale sub-linearly using a Forest technique as the number of your items grow. As stated before, the goal is to find set(s) similar to a query set.
Hash items such that similar items go into the same bucket with high probability.
Restrict similarity search to the bucket associated with the query item.
Remember that a shingle is an individual element that can be a part of a set like a character, unigram or bigram. In the standard literature there is a concept of shingle size, $k$, where the number of shingles is equal to $20^k$. When you choose what your shingles will be, you are implicitly choosing your shingle size.
Lets use the letters of the alphabet as our example of our shingles. You would have 26 shingles. For our alphabet shingles, k = 1 where $20^k$ is approximately the number of distinct characters in the alphabet that are used frequently.
When shingling documents into unigrams, we'll find there's a very large number of possible words, i.e. we'll be using a large value for $k$. The table below gives a quick view of the number of shingles relative to the shingle size, $k$.
When it comes to choosing your shingles, you must have enough unique shingles so that the probability of a shingle appearing in a given document is low. If a shingle occurs too frequently accross all of your sets, it will not provide significant differentiation between sets. The classic example in Natural Language Processing (NLP) is the word "the". It is too common. Most NLP work would remove the word "the" from all documents being considered because it does not provide any useful information. Likewise, you don't want to work with shingles that will show up in every set. It reduces the effectiveness of the method. You can also see this idea used with TF-IDF. Terms that appear in too many documents get discounted.
Since you want any individual shingle's probability of appearing in any given document to be low, we may even look at shingles that comprise bigrams, trigrams, or greater. If two documents have a large number of bigrams or trigrams in common, it would indicate those documents are quite similar.
When looking for the proper number of shingles, take a look at the average number of characters or tokens that appear in your documents. Make sure that you choose enough shingles such that there will be significantly more shingles than the average number of shingles in your document.
"Self-Organization of Associative Database and Its Applications"
Notice we removed common stop words like "of", "and", and "its" since they provide little value.
The goal of the MinHash is to replace a large set with a smaller "signature" that still preserves the underlying similarity metric.
Randomly permute the rows of the shingle matrix. E.g. Rows 12345 => 35421, so if "reinforcement" was in row 1, it would now be in row 5.
For each set (paper titles in our case), start from the top and find the position of the first shingle that appears in the set, i.e. the first shingle with a 1 in its cell. Use this shingle number to represent the set. This is the "signature".
Repeat as many times as desired, each time appending the result to the set's signature.
Note that in this example we are only using the paper's title for simplicity. In the implementation below we'll also add in the abstract to generate more accurate recommendations.
On the left, we define the matrix with columns as the titles and rows defined as all words encountered in the three titles. If a word appears in a title, we place a 1 next to that word. This will be the input matrix to our hashing function.
Notice our first record in the signature matrix on the right is simply the current row number where we first find a 1. See the next slide for a more detailed explanation.
We perform the first real row permutation. Notice the unigram words have been reordered. Again, we also have the signature matrix on the right where we'll record the result of the permutations.
In each permutation, we are recording the row number of the first word appears in the title, i.e. the row with the first 1. In this permutation, Title 1's and Title 2's first unigram are both in row 1, so a 1 is recorded in row 1 of the signature matrix under Title 1 and 2. For Title 3, the first unigram is in row 5, so 5 is recorded in the first row of the signature matrix under Title 3's column.
We are essentially doing the same operations as Slide 2, but we are creating a new row in the signature matrix for every permutation and putting the result there.
So far we've done three permutations, and we could actually use the signature matrix to calculate the similarity between paper titles. Notice how we have compressed the rows from 15 in the shingle matrix, to 3 in the signature matrix.
After the MinHash procedure, each conference paper will be represented by a MinHash signature where the number of rows is now much less than the number of rows in the original shingle matrix. This is due to us capturing the signatures by performing row permutations. You should need far fewer permutations than actual shingles.
Now if we think about all words that could appear in all paper titles and abstracts, we have potentially millions of rows. By generating signatures through row permutations, we can effectively reduce the rows from millions to hundreds without loss of the ability to calculate similarity scores.
In order to be more efficient, random hash functions are used instead of random permutations of rows when done in practice.
The signature matrix above is now divided into $b$ bands of $r$ rows each, and each band is hashed separately. For this example, we are setting band $b = 2$, which means that we will consider any titles with the same first two rows to be similar. The larger we make b the less likely there will be another Paper that matches all of the same permutations.
For example, notice the first band on the matrix above, conference paper Title 1 and conference paper Title 2 will end up in the same bucket since they both have the same bands, whereas Title 3 is totally different. Even though the ends of Title 1 and 2's signatures differ, they are still bucketed together and considered similar with our band sizing.
Ultimately, the size of the bands control the probability that two items with a given Jaccard similarity end up in the same bucket. If the number of bands is larger, you will end up with much smaller sets. For instance, $b = p$, where p is the number of permutations (i.e. rows in the signature matrix) would almost certainly lead to N buckets of only one item because there would be only one item that was perfect similar across every permutation. What we are really looking for when we select the band size is for our tolerance for false positives (no similar documents ending up in the same bucket) and false negatives (similar documents not ending up in the same bucket).
Convert the query text to shingles (tokens).
Apply MinHash and LSH to the shingle set, which maps it to a specific bucket.
Conduct a similarity search between the query item and the other items in the bucket.
Efficiencies can be achieved by actually using an LSH Forest algorithm for efficient searching. For more info, see M. Bawa, T. Condie and P. Ganesan, "LSH Forest: Self-Tuning Indexes for Similarity Search.
Build an $m \times n$ shingle matrix where every shingle that appears in a set is marked with a 1 otherwise 0.
Permute the rows of the shingle matrix from step 2 and build a new $p \times n$ signature matrix where the number of the row of the first shingle to appear for a set is recorded for the permutation of the signature matrix.
Repeat permuting the rows of the input matrix $p$ times and complete filling in the $p \times n$ signature matrix.
Choose a band size $b$ for the number of rows you will compare between sets in the LSH matrix.
Our goal in this tutorial is to make recommendations on conference papers by using LSH to quickly query all of the known conference papers. As a general rule, you should always examine your data. You need a thorough understanding of the dataset in order to properly pre-process your data and determine the best parameters. We have given some basic guidelines for selecting parameters, and they all require an exploration of your dataset as described above.
For the purposes of this tutorial, we will be working with an easy dataset. Kaggle has the "Neural Information Processing Systems (NIPS) conference papers. You can find them here.
An initial data exploration of these papers can be found here.
Create unigram shingles (tokens) by separating any white space.
For better results, you may try using a natural language processing library like NLTK or spaCy to produce unigrams and bigrams, remove stop words, and perform lemmatization.
#Preprocess will split a string of text into individual tokens/shingles based on whitespace.
You can see a quick example of the preprocessing step below.
To start our example, we will use the standard number of permutations of 128. We will also start by just making one recommendation.
Pass in a dataframe with every string you want to query.
Preprocess a string of text using our preprocessing step above.
Set the number of permutations in your MinHash.
MinHash the string on all of your shingles in the string.
Store the MinHash of the string.
Repeat 2-5 for all strings in your dataframe.
Build a forest of all the MinHashed strings.
Index your forest to make it searchable.
Preprocess your text into shingles.
Set the same number of permutations for your MinHash as was used to build the forest.
Create your MinHash on the text using all your shingles.
Query the forest with your MinHash and return the number of requested recommendations.
Provide the titles of each conference paper recommended.
We will start by loading the CSV containing all the conference papers and creating a new field that combines the title and the abstract into one field, so we can build are shingles using both title and abstract.
Finally, we can query any string of text such as a title or general topic to return a list of recommendations. Note, for our example below, we have actually picked the title of a conference paper. Naturally, we get the exact paper as one of our recommendations.
It took 12.728999853134155 seconds to build forest.
It took 0.006001949310302734 seconds to query forest.
Just as a quick final note, you can build numerous recommendation engines with this method. You do not have to be limited to just Content Based Filtering as we demonstrated. You can also use these techniques for Collaborative Filtered Recommendation Engines.
To summarize, the procedures outlined in this tutorial represent an introduction to Locality-Sensitive Hashing. Materials here can be used as a general guideline. If you are working with a large number of items and your metric for similarity is that of Jaccard similarity, LSH offers a very powerful and scalable way to make recommendations.
Learn more about recommender systems in week five of this course. Gain insight into content-based recommenders and collaborative filtering.
Ray McLendon is a Sole Proprietor, Senior Data Scientist, and Project Consultant at Moxxy Solutions LLC. He has spent 8 years delivering technology and analytics projects in Renewable Energy, Transportation, Consumer Electronics, and Financial Services. | CommonCrawl |
$\alpha$ and $n \alpha$ are two angles and they formed compound angles $(n+1)\alpha$ and $(n-1)\alpha$ by sum and difference. Sine and cosine functions formed a fractional function to represent a quantity in general trigonometric form.
The two sine functions with compound angles are in subtraction form. The subtraction of them can be simplified by the sum to product transformation trigonometric identity.
Similarly, express the sum of cosine functions in terms of product form by the sum to product transformation trigonometric identity.
$\cos n \alpha$ is a common term in expression of the denominator. Take it common from them to simplify it further.
Expand the sine of angle alpha and also express $1+cos \alpha$.
Therefore, it is the required solution for this trigonometric problem. | CommonCrawl |
The aim of this paper is to find cellular automata (CA) rules that are used to describe S-boxes with good cryptographic properties and low implementation cost. Up to now, CA rules have been used in several ciphers to define an S-box, but in all those ciphers, the same CA rule is used. This CA rule is best known as the one defining the Keccak $\chi$ transformation. Since there exists no straightforward method for constructing CA rules that define S-boxes with good cryptographic/implementation properties, we use a special kind of heuristics for that – Genetic Programming (GP). Although it is not possible to theoretically prove the efficiency of such a method, our experimental results show that GP is able to find a large number of CA rules that define good S-boxes in a relatively easy way. We focus on the $4 \times 4$ and $5 \times 5$ sizes and we implement the S-boxes in hardware to examine implementation properties like latency, area, and power. Particularly interesting is the internal encoding of the solutions in the considered heuristics using combinatorial circuits; this makes it easy to approximate S-box implementation properties like latency and area a priori. | CommonCrawl |
Hypoglycemic effect through activity inhibition of $\alpha$-glucosidase and $\alpha$-amylase was evaluated using leaves of Eleutherococcus senticosu, Eleutherococcus gracilistylus, Eleutherococcus sieboldianus and Eleutherococcus sessiliflorus which belong to Acanthopanax sessiliflorus genus. As a result of measuring $\alpha$-glucosidase activity inhibition, extract of Eleutherococcus gracilistylus showed around 43.38% of activity inhibition compared with acarbose and extract of Eleutherococcus senticosu showed 41.24% inhibitory effect. As a result of measuring $\alpha$-amylase activity inhibition, acarbose showed 73.25% of activity inhibition in 10 mg/mL concentration, and the extract of Eleutherococcus senticosu leaves showed 91.90% higher activity inhibition compared with acarbose. Also, after subjects in a model were induced diabetes with streptozotocin (STZ) intake plant extract from Acanthopanax sessiliflorus for 2 weeks, effect of improving blood glucose level and fat was examined. In all groups with specimen, Eleutherococcus senticosu (T1), Eleutherococcus gracilistylus (T2), Eleutherococcus sieboldianus (T3) and Eleutherococcus sessiliflorus (T4), blood glucose level was significantly decreased compared with that in control group (C). In an experiment of examining changes in fat concentration in blood, total cholesterol content increased in a control group compared with a normal, while in T1, T3 and T4, it decreased significantly compared with the control group. As for HDL-cholesterol, it significantly increased in all diabetes induced groups compared with the normal group, and in T3, it increased the most significantly by 55.61% compared with the control group. In case of LDL-cholesterol, specific difference between the normal group and the control group was not found; however, significant increase was detected in T2 and T3, whereas in T1 and T4, it decreased significantly compared with the control group. As for triglyceride, its concentration increased in the control group like total cholesterol. It decreased 60.16% in T3, 60.80% in T4 and 50.16% in T1 compared with the control group. As a result of measuring the concentration of triglyceride in extracted liver, the control group showed significant increase compared with the normal group, whereas T1 and T2 showed significant decrease compared with the normal group. The above results show that extracts from Acanthopanax sessiliflorus genus are effective for hypoglycemic and improving fat metabolism due to diabetes.
Ministry of Health and Welfare. 2005. The Third Korea National Health and Nutrition Examination Survey (KNHANES Ⅲ). Korea. p 193-195.
Zhang W, Xu YC, Guo FJ, Meng Y, Li ML. 2008. Anti-diabetic effects of cinnamaldehyde and berberine and their impacts on retinol-binding protein 4 expression in rats with type 2 diabetes mellitus. Chin Med J 121: 2124-2128.
Won HJ, Lee HS, Kim JT, Hong CO, Koo YC, Lee KW. 2010. The anti-diabetic effects of Kocat-D1 on streptozotocin induced diabetic rats. Korean J Food Sci Technol 42: 204-209.
Han HK, Je HS, Kim GH. 2010. Effects of Cirsium japonicum powder on plasma glucose and lipid level in streptozo tocin induced diabetic rats. Korean J Food Sci Technol 42: 343-349.
Kim YJ, Park MS, Park HK, Kim S, Sung CK. 1996. Eleutheroside E content in Eleutherococcus spp. Korean J Med Crop Sci 4: 333-339.
Kim SK, Kim YG, Lee MK, Han JS, Lee JH, Lee HY. 2000. Comparison of biological activity according to extracting solvents of four Acanthopanax root bark. Korean J Med Crop Sci 8: 21-28.
Jwa CS, Yang YT, Koh JS. 2000. Changes in free sugars, organic acids, free amino acids and minerals by harvest time and parts of Acanthophnax Koreanum. J Korean Soc Agric Chem Biotechnal 43: 106-109.
Lee SE, Son D, Yoon YP, Lee SY, Lee BJ, Lee S. 2006. Protective effects of the methanol extracts of Acanthopanax Koreanum against oxidative stress. Korean J Pharmacogn 37: 16-20.
Yook CS, Lee DH, Seo YK, Ryu KS. 1977. Studies on the constituents in the stem, root bark of Acanthopanax sessiliglorus. Korean J Pharmacogn 8: 31-34.
Ryoo HS, Park SY, Chang SY, Yook CS. 2003. Triterpene components from the leaves of Acanthopanax sessiliglorus Seem. Korean J Pharmacogn 34: 269-273.
Yun YD, Choi CH, Baek JU, Kim HW, Youn DH, Kim KY, Nam KW, Kim GY, Jeong HW. 2007. Effects of Acanthopanacis cortex roots 50% ethyl alcohol extracts on the cerebral bemodynamics and cytokine production in cerebral ischemic rats. Korean J Oriental Physiology & Pathology 21: 891-897.
Kim S, Kim KY, Park MS, Choi SY, Yun SJ. 1998. Intraspecific relationship of E. senticosus Max. by RAPD markers. Korean J Med Crop Sci 6: 165-169.
Park JS, Oh CH, Koh HY, Choi DS. 2002. Anti-mutagenic effect of extract of Eleutherococcus senticosus Maxim. Korean J Food Sci Technol 34: 1110-1114.
Park JH, Lee HS, Mun HC, Kim DH, Seong NS, Jung HG, Bang JK, Lee HY. 2004. Improvement of anti-cancer activation of ultra sonificated extracts from Acanthopanax senticosus Harms, Ephedra sinica Stapf, Rubus coreanus Miq. and Artemisia capillaris Thunb. Korean J Med Crop Sci 12: 273-278.
Jin LH, Han SS, Choi YS. 2002. Anti-oxidant effects of the extracts of Acanthopanax senticosus. Kor J Pharmacogn 33: 359-363.
Brekhman Ⅱ. 1960. A new medicinal plant of the family Araliceae the spiny Eleutherococcus. Izv Sibir Otdel Akad Nauk USSR 9: 113-120.
Shin KH, Lee SH. 2002. The chemistry of secondary products from Acanthopanax species and their pharmacological activities. Natural Product Sciences 8: 111-126.
Kim MJ, Kwon YS, Yu CY. 2005. Anti-oxidative compounds in extracts of Eleutherococcus senticosus Max.plantlets. Korean J Med Crop Sci 13: 194-198.
Park JH, Baek MR, Lee BH, Yon GH, Ryu SY, Kim YS, Park SU, Hong KS. 2009. $\alpha$--Glucosidase and α-amylase inhibitory activity of compounds from roots extract of Pueraria thunbergiana. Korean J Med Crop Sci 17: 357- 362.
Yoon JA, Son YS. 2009. Effects of Opuntia ficus-indica complexes B (OCB) on blood glucose and lipid metabolism in streptozotocin-induced diabetic rats. J Korean Soc Food Sci Nutr 22: 48-56.
Smith EB. 1974. The relationship between plasma and tissue lipid in human atherosclerosis. Adv Lipid Res 11: 1-7. | CommonCrawl |
From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear from the SDE. In fact, if I do simulations using the SDE, x very frequently becomes negative for certain parameter combinations.
The reason SDE may seem to allow a negative value of x is because dW can be a large negative number, and thus can move x from positive to negative values. If that's the confusion, that is justified only in discrete case.
Do a thought experiment. Imagine the transition point of x from positive to negative in discrete case. Now, to move closer to the continuous case, you can decrease dt and imagine x as going from positive to zero first, and then to negative. But your SDE states that dx = x multiplied by something. So when x->0, dx also ->0. This should help you understand that the negative values are only coming from discretization of an otherwise continuous process.
Now, if you fix your formulas in the simulation and make it E3= E2+ D2*E2 (instead of D2+ D2*E2 which you used), you can still get negative values if sigma is really large and dt is not small enough. But try changing dt relative to sigma and you will find it harder to get negative values. Moreover, excel might have a limit of how small you can go.
Not the answer you're looking for? Browse other questions tagged stochastic-processes stochastic-calculus brownian-motion simulations or ask your own question. | CommonCrawl |
) hypothesis while on the other hand very religious people for example have 0 in their priori when it comes to possibility of their religion being made up so they are forced to ignore evidence to the contrary (because bayesian updating breaks for them due to division by zero and mind's way to signal this exception is denial).In general someone with a lot of weight on given hypothesis is "stubborn" or just very convinced and someone with uniform or close distribution just doesn't know anything about given problem.The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution.Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction.Your example is the second one with $\mu_0 = 0$ As a general tip, when doing this type of questions, you should drop the $\frac$, since your expression is only up to a constant of proportionality anyway.(**) You need to expand your expression and write all the exponentials term together, then factorise it as $-\frac$ for some expression $y$. Then you observe, this is proportional the normal distribution with mean and variance given in the wikipedia article.The whole idea is to consider the joint probability of both events, A and B, happening together (a man over 5'10" who plays in the NBA), and then perform some arithmetic on that relationship to provide a updated (posterior) estimate of a prior probability statement.
You will learn to use Bayes' rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm.For $X_, X_,..., X_$ iid $\mathcal(\theta,\sigma^2)$, and a priori distribution $\theta\sim\mathcal(\mu,\tau^2)$, you should obtain posteriori distribution $\mathcal(\mu_,\tau^2_)$, where: $$\mu_=\frac\quad\text\quad\tau^_=\left(\frac \frac\right)^$$ As for the Bayesian estimator - well, I believe that that would depend on your risk function; with a MSE function, you should obtain $\theta^_=\mu_$.Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal(\mu,\sigma^2)$. (Interpretation: we get noisy signals about $\mu$, which are known to be normally distributed with known variance---this is the draw of $X$.Here is a ten-minute overview of the fundamental idea. But there's a catch: Sometimes the arithmetic can be nasty.
On your way to the hotel you discover that the National Basketball Player's Association is having a convention in town and the official hotel is the one where you are to stay, and furthermore, they have reserved all the rooms but yours.
I have to calculate the posteriori distribution on $\theta$ and the Bayes estimator.
Bayesian Updating you are trying to estimate p, the. | CommonCrawl |
Chattopadhyay, K and Mukhopadhyay, NK (1990) On the origin of banded microstructure in rapidly solidified alloys: The case of Al-10%Mn quasicrystalline system. In: Journal of Crystal Growth, 106 (2-3). pp. 387-392.
The formation of banded structure with bands perpendicular to the growth direction is reported in the rapidly solidified $Al$-10at%$Mn$ alloy. The bands consist of fine quasicrystalline phases distributed alternately in the $\alpha-Al$ matrix. It is shown that this banding is different from that observed in the rapidly solidified $Ag-Cu$ alloys. A qualitative explanation based on the interplay of kinetics of solute transport and the nature of free energy curves is proposed to explain the banding in $Al-Mn$ alloys. | CommonCrawl |
Mathematical Circles (Russian Experience); Dmitri Fomin, Sergey Genkin, Ilia Itenberg; Translated from Russian by Mark Saul; Volume 7; American Mathematical Society; ISBN 0-8218-0430-8.
Can we draw a closed path made of 9 line segments, each of which intersects exactly one of the others?
Can a convex 13-gon be divided into parallelograms (not necessarily equal dimensions)?
Can a square checkerboard be covered by dominoes?
Given a convex 101-gon which has an axis of symmetry. Show that the axis of symmetry passes through one of the vertices.
Now for the first problem, if such closed path were possible then we could group the lines into pairs of intersecting line segments but this is not possible as there are odd number of lines. The second one is also not posible, since each domino covers two squares and so the dominoes can cover only an even number of squares. For the third problem, suppose that the convex 13-gon were divided into parallelograms. We choose one of the sides and consider a particular parallelogram it belongs to. The opposite side of this parallelogram is also a side of a second parallelogram. This second parallelogram has another side parallel to the side we started with. We continue this chain of parallelograms until we arrive at another side of the 13-gon, which by our approach is parallel to the first side. Since the 13-gon is convex so this side cannot be parallel to any other side. Thus proceeding this way we could find pairs of parallel sides. But this is not possible as there are 13 sides. Note that the division into parallelograms is not possible even if it is a regular 13-gon. In a similar way, partitioning into pairs solves the other two problems too.
The product of 22 integers is equal to 1. Show that their sum cannot be zero.
Pete bought a notebook containing 96 pages, and numbered them from 1 to 192. Victor tore out 25 pages of Pete's notebook and added the 50 numbers he found on the pages. Could Victor have gotten 1990 as the sum?
Can one form a $latex 6\times 6$ magic square out of the first 36 prime numbers? Magic square means an array (here ) of boxes with a unique number in each box such that the sum of the numbers in every row, column or diagonal is the same.
The numbers 1 through 10 are written in a row. Can the signs + and – be placed in between them, so that the value of the resulting expression is zero?
The numbers are written on a blackboard. We decide to erase from the board any two numbers and include their positive difference in the list. After doing this several times, a single number is left on the board. Could this number be equal to zero?
. Which five of the numbers in L.H.S add up to 50?
The curious readers may try solving these.
Is the product of any five consecutive natural numbers always divisible by 30?
How many zeros are there at the end of the number 100! ? (I mean factorial 100).
For some number , can the number n! have exactly five zeros at the end?
Three prime numbers , $latex q$ and $latex r$, greater than 3, form an arithmetic progression: , and $latex r=p+2d$. Show that must be divisible by 6.
In any group of 13 persons, there exist at least two persons having birthday on the same month.
If a board in the shape of equilateral triangle of side 20 cm is hit with a dart five times, then there exist at least two hits such that the distance between them less than 10cm.
Fifty one points are scattered inside a square of side 1 meter. There exists some set of three of these points can be covered by a square of side 20 cm.
To generalize the above: Given any integers, we can always find two integers out of these such that their difference is divisible by .
A bag contains balls of two colours. What is the minimum number of balls which must be drawn from the bag, of course without looking, so that two balls are of the same colour?
For the 6th problem, we see that the given 8 different numbers can form 28 different pairs (not ordered), the corresponding difference being any of the numbers from 1 to 14. But the difference 14 is possible only in one case ie 15-1=14. Thus remaining 27 pairs are to be fitted in `holes' numbered form 1 to 13. And since by the general pigeon hole principle, at least one such positive difference must correspond to 2+1=3 pairs.
There are various other types of challenging problems in the book I have mentioned. Please do read the book Mathematical Circles for more! | CommonCrawl |
My girlfriend and I have been attending a weekly Dungeons & Dragons night at a local board game cafe, which means that we have been thoroughly immersed in the superstition surrounding the dice. We have heard stories of unlikely strings of bad rolls that ultimately lead to the frustrated fellow adventurer performing the salt water test to see if the dice are balanced poorly. This test is really cool, and it certainly explains the physical mechanism for why a die will come up on some faces more than others, but it doesn't tell you anything about the probabilities of rolling each face. Even with an off balance die, say a d20 that's weighted in favour of rolling a twenty, we shouldn't expect every normal roll on a hard surface to come up twenty. How frequently it comes up will presumably depend on the severity of the balance issue. As an aspiring data scientist I couldn't help but think that a much better approach would be a Bayesian analysis for estimating the probability of rolling each face.
Normally I don't care too much about such superstitions and just take the rolls as they come. I probably never would have implemented this Bayesian analysis, if not for the fact that I have developed a bit of a reputation for rolling high damage numbers on my d10 at our table. The curiosity has now gotten the better of me and I want to know the truth about how my dice really roll.
We will likely need a lot more data to achieve significance using this frequentist approach.
Instead, I think a Bayesian approach will be much more informative. We have a couple options available to us: We can use a numerical approach like MCMC, and compute the posterior distributions for each face that way. Or we can try to find an analytically tractable way of computing the posterior from our prior and our observations. Generally this only occurs when we are lucky enough to have a prior and observations of specific forms that allow to easily compute the posterior, a mathematical feature called conjugacy. Luckily for us, there are two useful cases of conjugacy for us to consider. They are the beta-binomial model, and the more general Dirichlet-multinomial model. Ultimately both will be used in this analysis.
The Dirichlet-multinomial model allows us to model an n-dimensional system of discrete events with varying probabilities that all sum to one. Essentially a dice. The beta-binomial model is a special case of the Dirichlet-multinomial model in which n = 2, and it basically describes a coin. Let's cover the simpler beta distribution and the beta-binomial model before going into the more general Dirichlet distribution and Dirichlet-multinomial model.
This presumes that we already know the true probability for p. In our case we are uncertain about the probability, so how do we deal with that? This is where the beta prior in the beta-binomial model comes in handy. In our Bayesian approach we get to start with a prior belief of what the probability of rolling a specific face of the coin is. Rather than picking a single value like we would in frequentist statistics, we define a probability density function that shows our relative belief in any p being true across all possible values of p. Since we are working with a probability, this is the range 0 to 1. Using observed data of k successes in n trials, we can update the beta prior distribution to find our posterior distribution. This is all a result of conjugacy, which is just a fancy way of saying that if we start with a beta distribution for the probability p, and then gather data that comes from a binomial distribution, we can find a new beta distribution that describes the posterior probability of seeing that face.
Where $\alpha 0$ and $\beta 0$ are parameters of our prior beta distribution, successes is the number of times we saw our chosen face, and failures is the number of times we didn't see that face.
More important than these summary statistics, however, is our ability to calculate a credible interval for the beta distribution. This gives us something similar to a frequentist confidence interval, except that we can actually say there is a certain probability that the interval contains the true value of p given our assumptions (our prior). Much like the median, the credible interval can be found by passing the appropriate quantiles to the qbeta function for our posterior. For a 95% credible interval, these would be 0.025 for the lower bound an 0.975 for the upper bound.
Notice that because we had a prior equivalent to 100 points of data, and collect another 100 points of data to update it, that the new posterior is right in the middle of our prior of p = 0.05 and our observed proportion of 0.20.
The posterior distribution has narrowed, and the mean has moved further towards the right to p = 0.190. This illustrates another useful feature of Bayesian statistics: As we collect more data, our chosen prior has less influence and becomes less important.
This is how the beta-binomial model works when we only care about one face, but what about if we are interested in every face of the dice? To answer that, let's look at the Dirichlet distribution and the Dirichlet-multinomial model. We will consider a made up dice that only has three faces, because it will be very difficult to visualize anything higher.
Unfortunately these visualizations don't work so well for dice with more than three faces since it is hard to visualize probability densities in n-1 dimensions for an n-faced dice.
In fact, if we were to just define $\beta$ as the sum of all the $\theta$s except $\theta_i$, and rename $\theta_i$ to $\alpha$, then the mean posterior probability for $\theta_i$ becomes indistinguishable from a beta distribution defined for $\theta_i$ and $\neg \theta_i$.
If we want the know a 95% credible interval for each face, then we can use this same trick to reduce each face of the dice to it's own beta distribution and find the confidence interval for that. The only caveat to this is that we need to remember that the distributions for each p originally came from a Dirichlet distribution. This means that the condition that the sum of all p equal one still remains, and that if the true value for one of the p is actually lower than our posterior mean probability, then it is necessary that one or more of the other faces has a p higher than it's posterior mean in an amount that cumulatively offsets the difference in the first p.
That's enough theory. Let's start analyzing!
roll_data: An integer vector containing the sequence of roll results.
# Deal with the d100. It has 10 faces starting at 0 instead of 1 and increasing in increments of 10.
It definitely looks like my d10 is a little unbalanced, and in a favourable way for me. But it's no where near as skewed as my reputation was leading me to believe. Based on the 95% credible interval the true probability of rolling a 10 is likely to found somewhere in the interval of approximately 0.105 to 0.180. This is certainly higher than 0.10, but no where near the "trick dice rolls 10 every time" levels I was concerned about. I think I'll keep using the dice and hope that the DM never reads this blog post.
Of all my remaining dice, the only other potential issues are that my d12 may roll a 10 less than expected, and my d100 may roll 90 more that expected. I cannot think of a single time my character has had to roll a d12 so any unfairness in that dice is moot. Furthermore, a 10 is a good roll, so I will probably be disadvantaged by the balance issues in this dice. Similar to my d12, I rarely use my d100, so the benefits of rolling 90 more often than expected are wasted on my character. Overall my dice appear to be less than perfect, but I suppose that is to be expected when you order your dice in bulk from a generic looking amazon seller. | CommonCrawl |
Abstract. We consider the Navier-Stokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number $R$. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as $R\to\infty$. | CommonCrawl |
This notebook proposes a general introduction to goal babbling, and how it differs from motor babbling, in the context of robotics. Goal babbling is a way for robots to discover their body and environment on their own. While representations of those could be pre-programmed, there are many reasons not to do so: environments change, robotic bodies are becoming more complex, and flexible limbs, for instance, are difficult and expensive to simulate. By allowing robots to discover the world by themselves, we use the world itself—the best physic engine we know—for robots to conduct their own experiments, and observe and learn the consequence of their actions, much like infants do on their way to becoming adults.
This notebook requires no previous knowledge beyond some elementary trigonometry and a basic grasp of the Python language. The spirit behind this notebook is to show all the code of the algorithms in a simple manner, without relying on any library beyond numpy (and even, just a very little of it). Only the plotting routines, using the bokeh library, have been abstracted away in the graphs.py file.
The algorithms and results exposed here were originally presented in the chapter 0 and chapter 3 of my Ph.D. thesis. They were implemented using the explorers library then. Here, as explained above, we don't rely on the library, and the code has been exposed in its simplest form. It is entirely available under the Open Science License. A citable version of this notebook is available at figshare. You can contact me for questions or remarks at [email protected].
The core idea behind goal babbling is to explore an unknown environment by deciding which goals to try to pursue in it, rather than deciding which action to try. For robots, it means directing the exploration by choosing which effects to produce in the environments, rather than choosing directly which motor commands to execute. Of course, once a goal has been chosen, the robot must find a motor command to reach it. But the decision on what to explore has been made on which effect to try to produce, not in the motor space, and we will see that it makes an important difference.
The idea of goal babbling was proposed by Oudeyer and Kaplan (2007) (p. 8). Computation implementations were then proposed by Baranes and Oudeyer (2010); Rolf et al. (2011) and Jamone et al. (2011). Formal frameworks were described by Baranes and Oudeyer (2013) and Moulin-Frier and Oudeyer (2013); the work we present here can be understood as an implementation of SAGG-Random.
We consider four different robotic arms, with 2, 7, 20, and 100 joints respectively. All arms measure one meter in length, and all the segments between the joints of the same arm are of equal length. The 7-joint arm, therefore, has segments of 1/7th of a meter, as shown in the figure below. Each joint can move between -150° and +150°. The arm receives a motor commands composed of 2, 7, 20 or 100 angle values, and returns an effect, the position of the tip of the arm after positioning each joint at the required angle, as a cartesian coordinate $x, y$.
Receives an array of angles as a motor commands.
Return the position of the tip of the arm as a 2-tuple.
"""`dim` is the number of joints, which are able to move in (-limit, +limit)"""
self.posture = [(0.0, 0.0)]*dim # hold the last executed posture (x, y position of all joints).
"""Return the position of the end effector. Accepts values in degrees."""
u, v = u + length*np.sin(sum_a), v + length*np.cos(sum_a) # at zero pose, the tip is at x=0,y=1.
Let's create our four arms.
arm2 = RoboticArm(dim=2, limit=150) # you can create new arms here, or modify the parameters of an existing one.
arm7 = RoboticArm(dim=7, limit=150) # for instance, the `limit` parameter can have important consequences.
We desire to explore the possibilities offered by the different arms, that is, understand where we can place the tip of the end-effector.
And we want to do that from an agnostic standpoint: the arm is a black box, receiving inputs (the angles of the joints) and producing outputs (the position of the tip of the arm). We can't assume any knowledge about the arm, nor deduce anything about the order of the inputs (which can be shuffled). In particular, zero can't be used as a special value. This constraint of ignorance allows to reuse the strategies developed on the arm for any other black box (in theory).
We add a second constraint: time. We can only execute motor commands a limited number of times on the arm. Here, we limit ourselves to 5000 executions. 5000 executions may seem a lot, but it is not compared to the size of the motor space: if we wanted sample all the motor commands whose angle values are multiples of 10° (-150°, -140°, ..., 0°, ..., 150° for each joint), we would need $31^7 = 27,512,614,111$ executions, more than 5 million times the budget we have.
So, given these two constraints, what are possible strategies to explore the arm?
N = 5000 # the execution budget.
The simplest strategy is to try random motor commands. We call this strategy random motor babbling (RMB). 'Babbling', here, means that we execute those commands because we don't know—and we want to know—what they will produce.
"""Return a random (and legal) motor command."""
"""Explore the arm using random motor babbling (RMB) during n steps."""
m_command = motor_babbling(arm) # choosing a motor command; does not depend on history.
effect = arm.execute(m_command) # executing the motor command and observing its effect.
history.append((m_command, effect)) # updating history.
This is a simple strategy. By using it for 5000 motor commands, we obtain the following distribution of effects.
history = explore_rmb(arm, N) # running the RMB exploration strategy for N steps.
graphs.show([[figures, figures], # displaying figures.
Each blue dot is the position reached by the tip of the arm during the execution of a motor command.
The random motor babbling strategy does rather well on the 2-joint arm. But as the number of joints increases, the distribution of the effects covers less and less of the approximate reachable area represented by the grey disk. Even after numerous samples of the 7, 20 and 100-joint arms, we don't have a good empirical estimation of the reachable space.
Motor babbling explores the possibilities offered by the arm by exploring the motor space: the choice of which motor command to execute is done by choosing a point in the motor space.
Another strategy is to explore not the motor space but the sensory space, i.e., the space of effects. Goal babbling does exactly this: choose a point in the effect space, consider it as a goal to be reached, and try to find a motor command to reach that goal. We still produce a motor command to execute, but the choice of that motor command is done in the sensory space.
How to find a motor command to reach a specific goal?
The answer to the first question is an inverse model. An inverse model is a process that transforms a goal, i.e., a point in the sensory space, in a motor command, i.e. a point in the motor space. The better an inverse model is, the less distance there is between the goal and the effect obtained when executing the motor command produced by the model.
There are many ways to create an inverse model. Some rely on the arm's schematics. Here, we can't do this, because of our agnostic constraint about not relying on specific knowledge of the arm. Another way is to learn the inverse model during the exploration: each execution of a motor command bring a motor command/effect pair; this data can be exploited to improve our ability to turn effects into motor commands. There are quite sophisticated ways to do that. Here, we choose one of the simplest ways.
When given a goal, our inverse model looks at our history of observations, and select the one that produced the effect closest to the goal. The motor command that produced this effect is retrieved, and we add a small, random perturbation to it. The resulting motor command is the one that is going to be executed.
D = 0.05 # with a range of ±150°, creates a perturbation of ±15° (5% of 300° in each direction).
"""Return the Euclidean distance between a and b"""
"""Return the motor command of the nearest neighbor of the goal"""
nn_command, nn_dist = None, float('inf') # naive nearest neighbor search.
"""Transform a goal into a motor command"""
nn_command = nearest_neighbor(goal, history) # find the nearest neighbor of the goal.
new_command.append(random.uniform(min_i, max_i)) # create a random perturbation inside the legal values.
Now, the nearest neighbor implementation we have is fine, but it is too slow for some of the experiments we will do here. We replace it by a fast implementation from the learners library. If you want to keep the slow but simple implementation, skip the next three code cells.
try: # if learners is not present, the change is not made.
"""Same as nearest_neighbors, using the `learners` library."""
Let's verify that the two nearest neighbor implementations do the same thing.
for i in range(1000): # comparing the results over 1000 random query.
assert nn_a == nn_b # the results should be the same.
Okay, that checks out. Let's override the slow implementation.
Once we have the inverse model, the remaining question is how to choose goals. There are many ways to cleverly select goals, and this is mostly explored in the context of intrinsic motivations (see Oudeyer and Kaplan (2007) for instance). Here, we choose the simplest option: random choice. For each sample, we choose as a goal a random point in the square $[-1, 1]\times[-1,1]$.
goal = [random.uniform(-1, 1), random.uniform(-1, 1)] # goal as random point in [-1, 1]x[-1, 1].
One detail is that the inverse model needs a non-empty history, because it works by creating perturbation of existing motor commands. To solve this problem we begin by doing 10 steps of random motor babbling, and then switch to random goal babbling for the remaining 4990 steps of our 5000-step exploration.
"""Explore the arm using random goal babbling (RGB) during n steps."""
if t < 10: # random motor babbling for the first 10 steps.
else: # then random goal babbling.
m_command = goal_babbling(arm, history) # goal babbling depends on history.
Let's look at how the random goal babbling strategy covers the reachable space for our four arms.
histories_gb.append(history) # we keep the histories for further analysis below.
Why do the 20-joint and 100-joint still don't explore their reachable space fully with goal babbling?
To understand why goal babbling is more efficient than goal babbling, we need to understand how effects are distributed in the sensorimotor space. The sensorimotor space is the space where the mapping between actions and effects can be expressed.
In a typical sensorimotor space, given a random action, all possible effects are not equal in probability. Some effects are more likely to be happening than others. This is due to two main reasons: the non-linear relationship between actions and effects, and sensorimotor redundancy.
The non-linear relationship means that small modifications of motor commands won't have a proportional effect over the whole sensorimotor space. In some parts of the sensorimotor space, a small motor command modification will have a significant impact on the effect produced, while in others it will have little or none at all.
We can demonstrate this with an example. In the code below, we take two motor commands (m_a and m_b), and look at the effects produced when the same 1000 random perturbations are applied to both. The effects stemming from the first motor command, m_a, are spread over a larger area than the effects stemming from the second.
perturbation = [random.uniform(-15, 15) for _ in range(7)] # this is the same perturbation of our inverse model.
The second and by far the most significant reason for the uneven distribution of effect is sensorimotor redundancy. A robotic arm is redundant if there is more than one arm posture that produce the same effect, i.e., the same position of the tip of the arm .
The redundancy of an effect is the number of different motor commands that produce it, which can be infinite. The redundancy is heterogeneously distributed over the sensory space. For instance, there is only one way to reach the point (0, 1) (the motor commands with all zeros), but the point (0, 0) can be reached in an infinite number of ways on an arm with 3 joints or more (the angle of the first joint does not matter).
We can actually show the differences in redundancy on the 2-joint arm. We increase the number of steps to 50000 to get a better picture. This is where it takes a really long time if you have kept the slow implementation of the nearest neighbors algorithm.
history_rmb_50k = explore_rmb(arm2, 50000) # random motor babbling exploration.
history_rgb_50k = explore_rgb(arm2, 50000) # random goal babbling exploration.
In the random motor babbling distribution, there are two areas where the distribution of effect is roughly twice as less dense than the rest.
# separating effects with postures with a positive and negative second joint.
pos_history = [h for h in history_rmb_50k if h > 0] # h is the motor command.
neg_history = [h for h in history_rmb_50k if h < 0] # h is the value of the second joint.
Random motor babbling is as an empirical estimator of the density of the effect distribution. And, because the sensorimotor redundancy has a much larger impact on the effect distribution than other factors, we can consider random motor babbling an approximate estimator of the sensorimotor redundancy.
Therefore, random motor babbling will produce effects preferentially in areas of high redundancy. When the differences in redundancy are high enough, the probability of random motor babbling producing effects in areas of low redundancy becomes too low for any practical purposes. This is what happens with the 20- and 100-joint arm: the center region has a redundancy orders of magnitude higher than the rest: random motor babbling only produces effect clustered in the center.
In contrast, the goal babbling strategy on the 2-joint arm explores different levels of redundancy equally . This is because the exploration is directed uniformly in the sensory space, actively countering the redundancy.
The goal babbling strategy does not explore the entire reachable space in the case of the 20 and 100-joint arm. To understand why, let's look at the posture producing the effect with the lowest x coordinate.
"""Return the posture producing the effect with minimum x"""
There is a loop in the arm starting at the 11th joint. Our inverse model is unable to disentangle that loop through successive perturbation: it tightens it instead. In this posture, the loop is almost completely tightened, as a matter of fact. This limits the span of the arm, and produces a distribution of effects that covers less than the total reachable area. This is a classic example of a local minimum. For another illustrated example of this phenomenon, see Figure 7 (p. 39) of my Ph.D. thesis.
This limitation is created by the interactions of a simple inverse model and a simplistic robotic model. A temptation here is to improve the inverse model, possibly by choosing a more complex learning algorithm. In many cases, however, improving the robot is much cheaper computationally. For instance, real robots usually can't traverse themselves. Adding collision detection to our simulated robot would prevent the apparitions of such loops.
So far, we have seen the difference between motor babbling and goal babbling, and understood that the difference in efficiency between the two is related to how the redundancy is distributed in the motor space. And we have seen that, unsurprisingly, goal babbling is not immune to local minima.
But there's one detail we have overlooked so far. In the goal babbling strategy, goals are chosen in the $[-1, 1]\times[-1, 1]$ square. This is highly suspicious, as this square is almost the axis-aligned bounding box of the reachable space. This is clearly a breach of the agnostic constraint that we imposed at the beginning.
So what would happen if goals were chosen in an overestimated area, or and underestimated area? Let's find out.
"""Goal babbling strategy, with a specific distribution of goals"""
"""Explore the arm using random goal babbling over a specific area."""
We run the strategy for four different cases: for the $[-1, 1]\times[-1, 1]$ square—as before, for the $[-2, 2]\times[-2, 2]$—four times the area of the unit square, for the $[-10, 10]\times[-10, 10]$ square—100 times the area, and $[-0.5, 0.5]\times[-0.5, 0.5]$—an area a quarter of the size of the unit square. We have two overestimated areas and one underestimated.
random.seed(SEED) # same motor babbling phase for all cases.
Here we see that the distribution of effects is strongly correlated with the distribution of goals. This is not surprising, but it illustrates one of the strengths of goal babbling: by manipulating the distribution of goals only, we can manipulate how exploration proceeds. This is exploited by the computational approaches of intrinsic motivations; see Oudeyer and Kaplan (2007) and Baldassarre and Mirolli (2013) for reviews.
"""Update the goal area to be 1.4 times bigger than the current observations"""
if extrema is None: # the first update of the area (based only on motor babbling observations).
else: # we only need to consider the last effect when doing subsequent updates.
history, goals, extrema = , , None # extrema keeps tracks of the min and max values along each dimension.
area, extrema = update_goal_area(history, extrema) # building the goal area from experience.
m_command, goal = goal_babbling_area(arm, history, area) # goal babbling depends on history.
We test the behavior of this algorithm on the 20-joint arm.
This goal babbling algorithm works fine. It still makes a problematic assumption: that the axis-aligned bounding box of the observation is a good approximation of the reachable space. More sophisticated algorithms, such as the Frontiers algorithm we introduced in my Ph.D. thesis, or the direction-sampling algorithm of Rolf (2013) are able to handle more general cases. Baranes and Oudeyer (2013) has also shown that intrinsic motivations could cope with an overestimation of the goal space.
We have shown how goal babbling works on a simple simulation of a 2D arm robot. We have shown that goal babbling, by choosing what to explore in the sensory space, and by relying on a learning algorithm, can explore much better than motor babbling. The next step is to go beyond a random choice of goal, and in particular, to make future goals depend on the history of observations: this is what intrinsic motivations do.
The (0, -1) point is not reachable by the arm, due to the angle constraints: that would require every joint to be at 180 degrees. The unreachable area inside the unit circle decreases with the number of joints; only in the 2-joint arm the area is significant. Additionally, in the 2-joint arm case, the area centered around the base of the arm is not reachable either, for the same reasons.
We could add the orientation to the redundancy criterion. In that context, an arm is redundant only if it can produce the same position and orientation of the tip with different postures. Including the orientation criterion is important in most practical applications. Under that criterion, the 2-joint arm is not redundant.
Some areas near the inner and outer edges of the goal babbling distribution do display higher concentration of effects. This is an effect of our inverse model. For a precise explanation, see chapter 0 (page 38 and Figure 6) of my Ph.D. thesis. | CommonCrawl |
Is there any references on the tensor product of (locally) presentable categories ?
Is there any references that defines it properly and proves the basic properties ?
The canonical reference is Chapter 5 of Greg Bird's thesis.
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Which categories are the categories of models of a Lawvere theory?
Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?
The induced functor in the definition of Deligne's tensor product is exact? | CommonCrawl |
This second version doesn't have to pull in extra packages and doesn't create intermediate lists, but mine is shorter and doesn't have explicit recursion. What do you think of my solution in comparison with the library's implementation?
Data.Bifunctor is part of base since 4.8.0.0, so I'm not sure where you get the "extra packages" from. Yes, Data.Bifunctor was originally in the bifunctors package, but it was transferred into base in 2015.
However, compared to the extras version, your variant is easy to read, as the [x | res] … [x | not res] slightly obfuscates the result. I'd probably bind map fst, but that's personal preference.
It generates a lot of intermediate lists. I haven't checked yet, but as [x|res]++as depends on res cannot get simplified to x:as or as at compile time by rules, so it might create a thunk […]++as. In the end, you have \$\mathcal O(n) \$ additional \$\mathcal O(1)\$-sized lists.
would at least support the optimizer and get rid of those intermediate lists, but is even more verbose. Your variant uses an additional list, sure, but it's easier to read and to understand IMHO.
This version uses only base, doesn't create extra lists and has no explicit recursion. The amount of plumbing this requires disturbs me - there ought to be a library out there that makes these building blocks fit together without naming acc and res. | CommonCrawl |
Let $n$ be a whole number. When is it possible to write $n$ as a sum of two squares, say $n=a^2+b^2$, or as a sum of three squares, say $n=a^2+b^2 +c^2$, or as a sum of four squares, and so on? Of course, $a$, $b$, $c$, $\ldots$ are also meant to be whole numbers here. We can also ask whether there is any convenient test to decide whether $n$ is the sum of one square; that is, whether $n$ is itself a square number, say $n=a^2$. In this article we shall mention some of the interesting answers to these questions. The proofs, which belong to the subject called Number Theory, are too difficult to be given here, but take a look at the article written by Tom Sanders, aged 16.
Although there seems to be no easy test to decide when $n$ is a square number, it is easy to give a test to decide when an even whole number $n$ is not a square number. Suppose that $n$ is even and $n=a^2$. Then $a$ must be even (for if $a$ is odd, then so is $a^2$), so that $a^2$ is divisible by $r$. Thus if $n$ is even and a square number, then $4$ divides $n$ exactly. This shows, for example, that the number $68792734815298359030284382$ (which is too big to put into your calculator) is not a square number. Why does it show this? Well, this number is of the form $100m+82$ and as $100$ is divisible by $4$, we see that $n$ is divisible by $4$ if and only if $82$ is (and it is not). Now you might like to write down some other (very large) numbers that are not square numbers. Can you see why if $n$ is divisible by $3$ but not by $9$ then $n$ is not a square number? (Try some examples of this.) What can you say about numbers divisible by $5$ but not by $25$?
Given a whole number $n$, look at all of the different prime factors of the form $4k+3$ (ignoring the other prime factors) and also the number of times that they occur; if every one of these prime factors occurs an even number of times then $n$ can be written as a sum of two squares; if not, then $n$ cannot be written as the sum of two squares. (Another way of saying this is that $n$ can be written as the sum of two squares if and only if the product of all of its prime factors of the form $4k+3$ is itself a square number.) For example, $490$ and $2450$ can be written as a sum of two squares but $350$ cannot. Try some other examples yourself; for example, $36=4\times 3^2$ so $36$ can be written as a sum of two squares, namely $6^2+0^2$. Try to decide which of the numbers $25$, $37$, $99$ and $245$ can be written as a sum of two squares and when they can, find what the two squares are. For $25$ (and possibly some of the others) there is more than one answer.
Next, let us try to write $n$ as the sum of three squares. It is known that this is possible if and only if $n$ is not of the form $4^m(8k+7)$ (where $m$ can be zero and $4^0=1$); for example, $53=4^0(8\times 6+5)$ can be (try it), but $60=4(8+7)$ cannot (again try it, but not for too long!). What about the numbers $30$, $48$, $77$ and $79$?
You should now choose some whole numbers yourself (not too large, though) and try to express each as a sum of four squares.
One of the important steps in considering sums of two squares is the formula $$(a^2+b^2)(c^2+d^2)=(a c+b d)^2+(a d-b c)^2=(a c-b d)^2+(a d+b c)^2$$ which holds for any whole numbers $a$, $b$, $c$, $d$. This formula shows ust hat if $n$ ($=a^2+b^2$) and $m$ ($=c^2+d^2$) are the sum of two squares then so is their product $m n$. To illustrate this, note that $5=2^2+1^2$ and $13=3^2+2^2$ so we can take $a=2$, $b=1$, $c=3$, $d=2$ and so find $65$ ($=5\times 13$) as the sum of two squares in two different ways. A similar formula to this holds for sums of four squares but sadly not for sums of three squares (and it is the lack of such a formula that makes the problem of sums of three squares more difficult to deal with).
It is much harder to see when a number can be written as a sum of cubes (for example $n=a^3+b^3$, or $n=a^3+b^3+c^3$), but it is known that every whole number can be written as a sum of nine cubes (including, if necessary, $0^3$); for example, $$23=2^3+2^3+1^3+1^3+1^3+1^3+1^3+1^3+1^3$$ $$239=4^3+4^3+3^3+3^3+3^3+3^3+1^3+1^3+1^3$$ Curiously, these two numbers ($23$ and $239$) are the only whole numbers that really do need nine cubes; all other whole numbers need only at most eight cubes. See if you can express the numbers $12$, $21$ and $73$ as the sum of at most eight cubes in as many ways as possible. | CommonCrawl |
Hough transform is a widely used shape detection techniques in computer vision and image processing. Hough transform aims to find imperfect instances of objects of within a certain class of shapes by a voting procedure. Essentially, hough transform transform points in the original space to parametric sapce, in which there is a accumulator to calculate the intersection of line, and the maximum points in the parametric space is the shape represented in the original space by multicollinearity.
For example, hough transform transform points of a line in cartesian coordinate system to polar coordinate system. Why polar coordinate system? The line is represented in cartesian coordinate system: $y=kx+b$, and the parameters are $k$ and $b$, where $k\in [-\infty,+\infty],b\in[-\infty,+\infty]$, it means that when the line with $k\approx \infty$ cannot be represented in the cartesian coordinate system. For each point in the original space, we draw all line across the point in the parametric sapce. We can see that each $\theta$ in parametric space represent a line in the original space. Therefore, the intersection in the parametric space is the line across all points in the original spacel.
Otsu method is a popular method in image segmentation, which aims to automatically perform clustering-based image thresholding, or, the reduction of a graylevel image to a binary image.
Otsu algorithm assumes that image contains two classes of pixels following bi-modal histogram(foreground pixel and background pixels), it then calculates the optimum threshold separating the two classes so that inter-variance is maximized. Consequently, otsu's algorithm is roughly a one-dimensional, discrete analog of Fisher's discriminant analysis.
This equation shows the relationship of class probability $\omega$ and class means $\mu$. | CommonCrawl |
In the first part of this talk I will define a sequence of polynomials resembling the Chebyshev polynomials of the first kind, and present results on their irreducibility and zero distribution. I will also consider $2\times 2$ Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval $(-1,1)$, while those of the other half lie on the unit circle.
If time allows, I will also talk about various other number theoretic polynomials, in particular polynomials with gcd powers a coefficients, and once again consider their irreducibility and zero distribution. | CommonCrawl |
In this paper, H(sub)$\infty$ depth and course controllers of autonomous underwater vehicles using H(sub)$\infty$ servo control are proposed. An H(sub)$\infty$ servo problem is foumulated to design the controllers satisfying a robust tracking property with modeling errors and disturbances. The solution of the H(sub)$\infty$servo problem is as follows; firest, this problem is modified as an H(sub)$\infty$ control problem for the generalized plant that includes a reference input mode, and than a sub-optimal solution that satisfies a given performance criteria is calculated by LMI(Linear Matrix Inequality) approach, The H(sub)$\infty$depth and course controllers are designed to satisfy the robust stability about the modeling error generated from the perturbation of the hydrodynamic coefficients and the robust tracking property under disturbances(was force, wave moment, tide). The performances(the robustness to the uncertainties, depth and course tracking properties) of the designed controlled are evaluated with computer simulations, and finally these simulation results show the usefulness and applicability of the propose H(sub)$\infty$ depth and course control systems. | CommonCrawl |
This is a question I asked on the cognitive science beta, but which never got any answer. I do not know what the policy should be for question migration/reposting (maybe worth discussing in the meta?), but I hoped it might get more answers (i.e. at least one ;)) here.
A rigorous formulation of these axioms can be found on page 8 of Axiomatic Foundations of Expected Utility and Subjective Probability, by Edi Karni, from the Handbook of Economics of risk and uncertainty..
The violations of these axioms I am most interested in are the ones related to the Independence axiom (violations of completeness, transitivity and continuity would probably deserve a separate question. See this question for an example of intransitivity.).
I am looking for situations which cannot be accounted for by the expected utility model. Some well-known examples are the the Allais and Ellsberg paradoxes (although there is still a debate regarding Ellsberg paradox). On the other hand, I do not see the Saint-Peterborough paradox as contradicting expected utility theory, because it can be accounted for by the theory if one assumes an appropriate degree of risk aversion. But you are much welcome to argue against that.
Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory.
A person prefers a trip to Paris to watching a television program about Paris to nothing.
Gamble 1: Win a trip to Paris 99% of the time, the television program 1% of the time.
Gamble 2: Win a trip to Paris 99% of the time, nothing 1% of the time.
It's reasonable to suppose that given the preferences over items, the second gamble might be preferred to the first. Someone who lost the trip to Paris might be so disappointed that they wouldn't be able to stand watching a program about how great it is.
Gamble 1: Win $1 million 99% of the time, win a penny 1% of the time.
Gamble 2: Win $1 million 99% of the time, win nothing 1% of the time.
I suspect that most people prefer winning $1 million for sure to winning a penny for sure to winning nothing for sure, while some people nevertheless prefer gamble 2 to gamble 1.
Kahneman and Tversky's experiments and many in behavioural economics contradict the existence of a utility function (preferences not complete and transitive), therefore also contradict expected utility.
Let me mention another quite well-known one: The calibration theorem by Rabin (2000) and Rabin and Thaler (2002). The idea is that over small stakes individuals must be essentially risk-averse, but in reality they are not.
Only assuming a weakly concave and strictly increasing utility function, Rabin shows that risk aversion on small stakes implies obviously unrealistic risk aversion over large stakes. In other words, under expected-utility theory, a resistance to accept small stake gambles with positive expected value leads to absurd conclusions about individuals' behavior in large stake gambles.
For example, an individual rejecting a coin flip with a gain of USD 125 and a loss of USD 100 would not accept a gain USD $\infty$ and lose USD 600 gamble.
The papers are worth reading, but keep in mind the rebuttals, e.g., by Cox and Sadiraj (2006) or Palacios-Huerta and Serrano (2006).
Picking up my comment under this answer.
One striking issue relevant to decisions not captured by expected utility is the framing effect discussed by Tversky and Kahneman (1981) and others. In their experimental study, they let two different (but with the same characteristics) groups choose between two options. Both groups actually face the same choices, but the wording is different. One group chooses between A and B, and one group between C and D. It is always one safe and one risky choice. While 72 percent picked the save option A vs B, 78 percent picked the risky option D vs C, although in expected-utility terms $A=C$ and $B=D$. So this observation is not compatible with expected utility.
A disease is expected to kill 600 people if no action is taken.
If $A$ is adopted, 200 people will be saved.
If $B$ is adopted, all 600 are saved with probability 1/3 and with probability 2/3 no people are saved.
If $C$ is adopted, 400 people will die.
If $D$ is adopted, nobody dies with probability 1/3 and with probability 2/3 all people die.
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What are different ways of specifying utility and decision making?
Can the Machina Paradox be solved by expanding the choice set?
How can the VNM completeness axiom be derived from the transitivity and continuity axioms?
Which of Anscombe-Aumann's axioms imply the Sure-Thing principle? | CommonCrawl |
In this supplemental set of notes we derive some approximations for , when is large, and in particular Stirling's formula. This formula (and related formulae for binomial coefficients will be useful for estimating a number of combinatorial quantities in this course, and also in allowing one to analyse discrete random walks accurately.
so we see that actually lies roughly at the geometric mean of the two bounds in (3).
thus extracting the factor of that we know from (4) has to be there.
Remark 1 The dominated convergence theorem does not immediately give any effective rate on the decay (though such a rate can eventually be extracted by a quantitative version of the above argument. But one can combine (7) with (4) to show that the error rate is of the form . By using fancier versions of the trapezoid rule (e.g. Simpson's rule) one can obtain an asymptotic expansion of the error term in , but we will not need such an expansion in this course.
Remark 2 The derivation of (7) demonstrates some general principles concerning the estimation of exponential integrals when is large. Firstly, the integral is dominated by the local maxima of . Then, near these maxima, usually behaves like a rescaled Gaussian, as can be seen by Taylor expansion (though more complicated behaviour emerges if the second derivative of degenerates). So one can often understand the asymptotics of such integrals by a change of variables designed to reveal the Gaussian behaviour. A similar set of principles also holds for oscillatory exponential integrals ; these principles are collectively referred to as the method of stationary phase.
when , which suggests that is distributed roughly like the gaussian with mean and variance .
Update, Jan 4: Rafe Mazzeo pointed me to this short article of Joe Keller that gives a heuristic derivation of the full asymptotic expansion of Stirling's formula from a Taylor expansion of the Gamma function.
hi, good lecture notes. There is one typo: "so we heuristically have.. " the formula on right hand side, you miss one negative sign.
this post of yours makes me wonder why the hundreds of books on calculus I have browsed in my life never even came close to your remark at the beginning "so we know already that n! is within an exponential factor of n^n".
There is obviously something flawed in the realm of pedagogy if it takes a mathematician of your caliber to make such childish remarks.
I am using the word "childish" deliberately, because of the the child in Hans Christian Andersen's tale who exclaims that the emperor has no clothes.
"I was like a boy playing on the sea-shore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me".
And Grothendieck likes to compare himself to a child in his autobiography "Récoltes et Semailles".
It might seem strange to compare you to Grothendieck, who in a sense is completely antipodal to you ( I'm not sure he knows Sterling's formula!), but your obvious common love of simplicity and hatred of showing-off might reveal some conceptual similarity, appearances to the contrary notwithstanding.
Greatness lies in knowing its boundaries and genius is just that boundary, I guess. It takes the genius and greatness of Prof. Tao to write such nice and illuminating pots.
I think that in (3) both of the derived bounds are too small by a factor of e. Certainly the upper bound doesn't hold, for instance if n=2.
Compute the antiderivative of , use the fundamental theorem of calculus, then exponentiate all sides of the equation preceding (3).
I see it now. Thank you. It is really a great post.
In the equation before "A classical computation (based for instance on computing", maybe a "dx" is missing.
I kept wondering about the appearance of entropy in the "entropy formula" (exercise 1), and I found the following probabilistic "almost proof" which I decided to share here (I'm hoping this is considered appropriate, if not I apologize). Does anybody know other interesting explanations as to why entropy occurs in the formula?
Take independent random variables with . Define and let for each .
Which is to say that there are a bunch of sequences that satisfy the entropy formula (but sadly, it doesn't show that they all do).
In the second paragraph, a formula appears to be missing after "we obtain a crude lower bound".
I am puzzled by Exercise2. Can you give me some hints? Why k=o(n2/3)?
I have a question about Entropy Formula in Exercise 1.
by Entropy Formula. If it is right, can you tell me how to get it? Some hints will be OK.
(Assuming means ) This formula is only valid for ; for larger values of , the partial sum is of size .
To estimate the order of magnitude of a sum of positive terms, one can often just bound the largest term in the sum, and then multiply by the number of terms, for a crude upper bound. In this case, the largest term will be if .
that is, . Am I right? If it is so, there is a typo in the reply.
This proof using the dominated convergence theorem can be found in a 1989 Monthly article: J. M. Patin, A Very Short Proof of Stirling's Formula, Amer. Math. Monthly 96 (1989), 41–42.
You expanded the argument of the exponential function into a Taylor series. After the term of order $-s^2/2n$ there comes the term $s^3/3n^2$. This terms worries me. Taylor series are local. That is, it should be good for $|s|<<1$. Still the integral is between $-\infty$ to $\infty$. Why does not the error introduce by this next factor $exp(s^3/3n^2)$ count? The integral for that factor alone diverges.
I believe I have an answer for my own question above. As $n$ grows the "Gaussian" shape of the integrand t^n exp(-t) narrows around the "center" or peak which is at t=n. Since we are shifting the function to have the origin at t=n. The Taylor approximation makes sense around that location. Yes, the function extendes between -infty and infty, but the gross of the integral is due to the near to the maxima points since in the limit this "Gaussian" shape becomes an spike or some kind of Dirac Delta.
In the fifth line below (6), it should be "uniform upper bound" (instead of "uniform lower bound"). Also, in the RHS of the second estimate for this bound, " " should be replaced by " ".
It is not difficult to verify (via maximization over ), that the integrand is dominated by – which is best possible for (as it attained by the integrand for ).
which is decreasing in n both for negative and positive x. | CommonCrawl |
We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.
If $(T_t)$ is a semigroup of Markov operators on an $L^1$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t \to \infty$. In this article we generalise and improve this result in several respects.
First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results.
Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.
We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations.
As applications we derive, inter alia, a generalisation of a famous theorem by Doob for operator semigroups on the space of measures and a Tauberian theorem for positive one-parameter semigroups under rather weak continuity assumptions. We also demonstrate how our results are useful to treat semigroups that do not satisfy any irreducibility conditions.
We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell^\infty$ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature.
Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.
We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive c_0-semigroup on an L^p-space is strongly convergent in case that it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.
Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.
We prove that a weakly ergodic, eventually strong Feller semigroup on the space of measures on a Polish space converges strongly to a projection onto its fixed space.
We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.
Given a positive, irreducible and bounded C_0-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator T, we show that the point spectrum of some power T^k intersects the unit circle at most in 1. As a consequence, we obtain a sufficient condition for strong convergence of the C_0-semigroup and for a subsequence of the powers of T, respectively.
We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doob's theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains. The proofs are based on the theory of positive semigroups and do not use probability theory. | CommonCrawl |
Residual intersection theory works well for ideals like the ideal of $p\times p$ minors of a generic $p \times (p+1)$ matrix, but fails for some very nice ideals, such as the ideal of $2 \times 2$ minors of a $2\times n$ matrix for n greater than $3$. Poking around for what might be true, Bernd Ulrich and I stumbled on a new phenomenon that seems to be rather general. We are far from proving all that seems to be true, but I'll describe our conjectures and results, which at least cover the $2 \times n$ matrices. This is joint work of mine with Bernd Ulrich and Craig Huneke. | CommonCrawl |
Das, Apurba Kumar and Haldar, Debasish and Hegde, Raghurama P and Shamala, N and Banerjee, Arindam (2005) X-ray crystallographic signature of supramolecular triple helix formation from a water soluble synthetic tetrapeptide. In: Chemical Communications (14). pp. 1836-1838.
Single crystal X-ray diffraction studies on the water soluble, synthetic tetrapeptide Tyr(1)-Aib(2)-Tyr(3)-Val(4) with a non-coded amino acid residue (Aib: $\alpha$-amino isobutyric acid) reveal that the peptide adopts an "S"-shaped molecular structure which self-assembles to form a supramolecular triple helix using various non-covalent interactions including water mediated hydrogen bonds in the solid state.
The copyright belongs to The Royal Society of Chemistry. | CommonCrawl |
Following on from the previous article on Pricing a Call Option with Multi-Step Binomial Trees, we are now going to discuss what happens as we increase the number of steps, $N$. In particular, we will discuss what happens as $N\rightarrow\infty$. Some care must be taken at this stage, as we will be dealing with infinite series. We must progress in a manner that provides us with a sensible answer.
In order to obtain a meaningful price for our derivative we need to make an assumption about how the underlying asset ("the stock") moves as time increases. Fundamentally, this will involve precisely specifying the random process that governs the asset price.
We denote the spot price of the asset as $S_0$ and the expiry time to be $T$. The mean of the asset at expiry will also equal the spot price, $S_0$, and the variance at expiry will be $\sigma^2 T$. The next step is to split our interval from $0$ to $T$ into $k$ steps, of equal size. For each step we need to ensure that the mean change in the asset price is $0$ and that the variance is $\sigma^2 T/k$. This will guarantee that our overall variance is $\sigma^2 T$, since we are taking the mean and variance of a sequence of independent random variables, which simply sum up.
Because of our assumptions on zero mean growth rate of the asset, as well as zero interest rates, we will utilise our argument of risk neutrality to state that the real world probabilities are equal to the risk neutral probabilities, in this case $p=0.5$. Recall that our investors do not need compensation for the extra risk they are taking on. Also recall that any other probability in this instance will lead to an arbitrage opportunity. If these two concepts are slightly unfamiliar, have a look at our risk neutral discussion for a quick refresher.
So what are the shortcomings of this model in its current state?
It allows a stock price to have a negative value, which is not in line with how a real-stock behaves.
It considers absolute movements in stock prices, rather than relative movements. Share values move in relation to their current price, not in an absolute sense.
We will improve all of these issues in the subsequent articles, eventually leading to a derivation of the famous Black-Scholes equation. | CommonCrawl |
Gavin J Daigle, Karthik Krishnamurthy, Nandini Ramesh, Ian Casci, John Monaghan, Kevin McAvoy, Earl W Godfrey, Dianne C Daniel, Edward M Johnson, Zachary Monahan, Frank Shewmaker, Piera Pasinelli and Udai Bhan Pandey.
Pur-alpha regulates cytoplasmic stress granule dynamics and ameliorates FUS toxicity.. Acta neuropathologica 131(4):605–20, April 2016.
Abstract Amyotrophic lateral sclerosis is characterized by progressive loss of motor neurons in the brain and spinal cord. Mutations in several genes, including FUS, TDP43, Matrin 3, hnRNPA2 and other RNA-binding proteins, have been linked to ALS pathology. Recently, Pur-alpha, a DNA/RNA-binding protein was found to bind to C9orf72 repeat expansions and could possibly play a role in the pathogenesis of ALS. When overexpressed, Pur-alpha mitigates toxicities associated with Fragile X tumor ataxia syndrome (FXTAS) and C9orf72 repeat expansion diseases in Drosophila and mammalian cell culture models. However, the function of Pur-alpha in regulating ALS pathogenesis has not been fully understood. We identified Pur-alpha as a novel component of cytoplasmic stress granules (SGs) in ALS patient cells carrying disease-causing mutations in FUS. When cells were challenged with stress, we observed that Pur-alpha co-localized with mutant FUS in ALS patient cells and became trapped in constitutive SGs. We also found that FUS physically interacted with Pur-alpha in mammalian neuronal cells. Interestingly, shRNA-mediated knock down of endogenous Pur-alpha significantly reduced formation of cytoplasmic stress granules in mammalian cells suggesting that Pur-alpha is essential for the formation of SGs. Furthermore, ectopic expression of Pur-alpha blocked cytoplasmic mislocalization of mutant FUS and strongly suppressed toxicity associated with mutant FUS expression in primary motor neurons. Our data emphasizes the importance of stress granules in ALS pathogenesis and identifies Pur-alpha as a novel regulator of SG dynamics.
Niran Maharjan, Christina Künzli, Kilian Buthey and Smita Saxena.
C9ORF72 Regulates Stress Granule Formation and Its Deficiency Impairs Stress Granule Assembly, Hypersensitizing Cells to Stress.. Molecular neurobiology, April 2016.
Abstract Hexanucleotide repeat expansions in the C9ORF72 gene are causally associated with frontotemporal lobar dementia (FTLD) and/or amyotrophic lateral sclerosis (ALS). The physiological function of the normal C9ORF72 protein remains unclear. In this study, we characterized the subcellular localization of C9ORF72 to processing bodies (P-bodies) and its recruitment to stress granules (SGs) upon stress-related stimuli. Gain of function and loss of function experiments revealed that the long isoform of C9ORF72 protein regulates SG assembly. CRISPR/Cas9-mediated knockdown of C9ORF72 completely abolished SG formation, negatively impacted the expression of SG-associated proteins such as TIA-1 and HuR, and accelerated cell death. Loss of C9ORF72 expression further compromised cellular recovery responses after the removal of stress. Additionally, mimicking the pathogenic condition via the expression of hexanucleotide expansion upstream of C9ORF72 impaired the expression of the C9ORF72 protein, caused an abnormal accumulation of RNA foci, and led to the spontaneous formation of SGs. Our study identifies a novel function for normal C9ORF72 in SG assembly and sheds light into how the mutant expansions might impair SG formation and cellular-stress-related adaptive responses.
Regina Nostramo, Sapna N Varia, Bo Zhang, Megan M Emerson and Paul K Herman.
The Catalytic Activity of the Ubp3 Deubiquitinating Protease Is Required for Efficient Stress Granule Assembly in Saccharomyces cerevisiae.. Molecular and cellular biology 36(1):173–83, January 2016.
Abstract The interior of the eukaryotic cell is a highly compartmentalized space containing both membrane-bound organelles and the recently identified nonmembranous ribonucleoprotein (RNP) granules. This study examines in Saccharomyces cerevisiae the assembly of one conserved type of the latter compartment, known as the stress granule. Stress granules form in response to particular environmental cues and have been linked to a variety of human diseases, including amyotrophic lateral sclerosis. To further our understanding of these structures, a candidate genetic screen was employed to identify regulators of stress granule assembly in quiescent cells. These studies identified a ubiquitin-specific protease, Ubp3, as having an essential role in the assembly of these RNP granules. This function was not shared by other members of the Ubp protease family and required Ubp3 catalytic activity as well as its interaction with the cofactor Bre5. Interestingly, the loss of stress granules was correlated with a decrease in the long-term survival of stationary-phase cells. This phenotype is similar to that observed in mutants defective for the formation of a related RNP complex, the Processing body. Altogether, these observations raise the interesting possibility of a general role for these types of cytoplasmic RNP granules in the survival of G0-like resting cells.
Laura MacNair, Shangxi Xiao, Denise Miletic, Mahdi Ghani, Jean-Pierre Julien, Julia Keith, Lorne Zinman, Ekaterina Rogaeva and Janice Robertson.
MTHFSD and DDX58 are novel RNA-binding proteins abnormally regulated in amyotrophic lateral sclerosis.. Brain : a journal of neurology 139(Pt 1):86–100, January 2016.
Abstract Tar DNA-binding protein 43 (TDP-43) is an RNA-binding protein normally localized to the nucleus of cells, where it elicits functions related to RNA metabolism such as transcriptional regulation and alternative splicing. In amyotrophic lateral sclerosis, TDP-43 is mislocalized from the nucleus to the cytoplasm of diseased motor neurons, forming ubiquitinated inclusions. Although mutations in the gene encoding TDP-43, TARDBP, are found in amyotrophic lateral sclerosis, these are rare. However, TDP-43 pathology is common to over 95% of amyotrophic lateral sclerosis cases, suggesting that abnormalities of TDP-43 play an active role in disease pathogenesis. It is our hypothesis that a loss of TDP-43 from the nucleus of affected motor neurons in amyotrophic lateral sclerosis will lead to changes in RNA processing and expression. Identifying these changes could uncover molecular pathways that underpin motor neuron degeneration. Here we have used translating ribosome affinity purification coupled with microarray analysis to identify the mRNAs being actively translated in motor neurons of mutant TDP-43(A315T) mice compared to age-matched non-transgenic littermates. No significant changes were found at 5 months (presymptomatic) of age, but at 10 months (symptomatic) the translational profile revealed significant changes in genes involved in RNA metabolic process, immune response and cell cycle regulation. Of 28 differentially expressed genes, seven had a ≥ 2-fold change; four were validated by immunofluorescence labelling of motor neurons in TDP-43(A315T) mice, and two of these were confirmed by immunohistochemistry in amyotrophic lateral sclerosis cases. Both of these identified genes, DDX58 and MTHFSD, are RNA-binding proteins, and we show that TDP-43 binds to their respective mRNAs and we identify MTHFSD as a novel component of stress granules. This discovery-based approach has for the first time revealed translational changes in motor neurons of a TDP-43 mouse model, identifying DDX58 and MTHFSD as two TDP-43 targets that are misregulated in amyotrophic lateral sclerosis.
R Nostramo and P K Herman.
Deubiquitination and the regulation of stress granule assembly.. Current genetics, 2016.
Abstract Stress granules (SGs) are evolutionarily conserved ribonucleoprotein (RNP) structures that form in response to a variety of environmental and cellular cues. The presence of these RNP granules has been linked to a number of human diseases, including neurodegenerative disorders like amyotrophic lateral sclerosis (ALS) and spinocerebellar ataxia type 2 (Li et al., J Cell Biol 201:361-372, 2013; Nonhoff et al., Mol Biol Cell 18:1385-1396, 2007). Understanding how the assembly of these granules is controlled could, therefore, suggest possible routes of therapy for patients afflicted with these conditions. Interestingly, several reports have identified a potential role for protein deubiquitination in the assembly of these RNP granules. In particular, recent work has found that a specific deubiquitinase enzyme, Ubp3, is required for efficient SG formation in S. cerevisiae (Nostramo et al., Mol Cell Biol 36:173-183, 2016). This same enzyme has been linked to SGs in other organisms, including humans and the fission yeast, Schizosaccharomyces pombe (Takahashi et al., Mol Cell Biol 33:815-829, 2013; Wang et al., RNA 18:694-703, 2012). At first glance, these observations suggest that a striking degree of conservation exists for a ubiquitin-based mechanism controlling SG assembly. However, the devil is truly in the details here, as the precise nature of the involvement of this deubiquitinating enzyme seems to vary in each organism. Here, we briefly review these differences and attempt to provide an overarching model for the role of ubiquitin in SG formation.
Hilary Bowden and Dorothee Dormann.
Altered mRNP granule dynamics in FTLD pathogenesis.. Journal of neurochemistry, 2016.
Abstract In neurons, RNA-binding proteins (RBPs) play a key role in post-transcriptional gene regulation, e.g. alternative splicing, mRNA localization in neurites and local translation upon synaptic stimulation. There is increasing evidence that defective or mislocalized RBPs - and consequently altered mRNA processing - lead to neuronal dysfunction and cause neurodegeneration, including frontotemporal lobar degeneration (FTLD) and amyotrophic lateral sclerosis (ALS). Cytosolic RBP aggregates containing TDP-43 (TAR DNA binding protein of 43 kDa) or FUS (Fused in sarcoma) are a common hallmark of both disorders. There is mounting evidence that translationally silent mRNP granules, such as stress granules or transport granules, play an important role in the formation of these RBP aggregates. These granules are thought to be "catalytic convertors" of RBP aggregation by providing a high local concentration of RBPs. As recently shown in vitro, RBPs that contain a so-called low complexity domain start to "solidify" and eventually aggregate at high protein concentrations. The same may happen in mRNP granules in vivo, leading to "solidified" granules that lose their dynamic properties and ability to fulfill their physiological functions. This may result in a disturbed stress response, altered mRNA transport and local translation, and formation of pathological TDP-43 or FUS aggregates, all of which may contribute to neuronal dysfunction and neurodegeneration. Here, we discuss the general functional properties of these mRNP granules, how their dynamics may be disrupted in FTLD/ALS, e.g. by loss or gain-of-function of TDP-43 and FUS, and how this may contribute to the development of RBP aggregates and neurotoxicity. This article is protected by copyright. All rights reserved.
Alyssa N Coyne, Shizuka B Yamada, Bhavani Bagevalu Siddegowda, Patricia S Estes, Benjamin L Zaepfel, Jeffrey S Johannesmeyer, Donovan B Lockwood, Linh T Pham, Michael P Hart, Joel A Cassel, Brian Freibaum, Ashley V Boehringer, Paul J Taylor, Allen B Reitz, Aaron D Gitler and Daniela C Zarnescu.
Fragile X protein mitigates TDP-43 toxicity by remodeling RNA granules and restoring translation.. Human molecular genetics 24(24):6886–98, December 2015.
Abstract RNA dysregulation is a newly recognized disease mechanism in amyotrophic lateral sclerosis (ALS). Here we identify Drosophila fragile X mental retardation protein (dFMRP) as a robust genetic modifier of TDP-43-dependent toxicity in a Drosophila model of ALS. We find that dFMRP overexpression (dFMRP OE) mitigates TDP-43 dependent locomotor defects and reduced lifespan in Drosophila. TDP-43 and FMRP form a complex in flies and human cells. In motor neurons, TDP-43 expression increases the association of dFMRP with stress granules and colocalizes with polyA binding protein in a variant-dependent manner. Furthermore, dFMRP dosage modulates TDP-43 solubility and molecular mobility with overexpression of dFMRP resulting in a significant reduction of TDP-43 in the aggregate fraction. Polysome fractionation experiments indicate that dFMRP OE also relieves the translation inhibition of futsch mRNA, a TDP-43 target mRNA, which regulates neuromuscular synapse architecture. Restoration of futsch translation by dFMRP OE mitigates Futsch-dependent morphological phenotypes at the neuromuscular junction including synaptic size and presence of satellite boutons. Our data suggest a model whereby dFMRP is neuroprotective by remodeling TDP-43 containing RNA granules, reducing aggregation and restoring the translation of specific mRNAs in motor neurons.
Ching-Chieh Chou, Olga M Alexeeva, Shizuka Yamada, Amy Pribadi, Yi Zhang, Bi Mo, Kathryn R Williams, Daniela C Zarnescu and Wilfried Rossoll.
PABPN1 suppresses TDP-43 toxicity in ALS disease models.. Human molecular genetics 24(18):5154–73, September 2015.
Abstract TAR DNA-binding protein 43 (TDP-43) is a major disease protein in amyotrophic lateral sclerosis (ALS) and related neurodegenerative diseases. Both the cytoplasmic accumulation of toxic ubiquitinated and hyperphosphorylated TDP-43 fragments and the loss of normal TDP-43 from the nucleus may contribute to the disease progression by impairing normal RNA and protein homeostasis. Therefore, both the removal of pathological protein and the rescue of TDP-43 mislocalization may be critical for halting or reversing TDP-43 proteinopathies. Here, we report poly(A)-binding protein nuclear 1 (PABPN1) as a novel TDP-43 interaction partner that acts as a potent suppressor of TDP-43 toxicity. Overexpression of full-length PABPN1 but not a truncated version lacking the nuclear localization signal protects from pathogenic TDP-43-mediated toxicity, promotes the degradation of pathological TDP-43 and restores normal solubility and nuclear localization of endogenous TDP-43. Reduced levels of PABPN1 enhances the phenotypes in several cell culture and Drosophila models of ALS and results in the cytoplasmic mislocalization of TDP-43. Moreover, PABPN1 rescues the dysregulated stress granule (SG) dynamics and facilitates the removal of persistent SGs in TDP-43-mediated disease conditions. These findings demonstrate a role for PABPN1 in rescuing several cytopathological features of TDP-43 proteinopathy by increasing the turnover of pathologic proteins.
Emma L Scotter, Han-Jou Chen and Christopher E Shaw.
TDP-43 Proteinopathy and ALS: Insights into Disease Mechanisms and Therapeutic Targets.. Neurotherapeutics : the journal of the American Society for Experimental NeuroTherapeutics 12(2):352–63, April 2015.
Abstract Therapeutic options for patients with amyotrophic lateral sclerosis (ALS) are currently limited. However, recent studies show that almost all cases of ALS, as well as tau-negative frontotemporal dementia (FTD), share a common neuropathology characterized by the deposition of TAR-DNA binding protein (TDP)-43-positive protein inclusions, offering an attractive target for the design and testing of novel therapeutics. Here we demonstrate how diverse environmental stressors linked to stress granule formation, as well as mutations in genes encoding RNA processing proteins and protein degradation adaptors, initiate ALS pathogenesis via TDP-43. We review the progressive development of TDP-43 proteinopathy from cytoplasmic mislocalization and misfolding through to macroaggregation and the addition of phosphate and ubiquitin moieties. Drawing from cellular and animal studies, we explore the feasibility of therapeutics that act at each point in pathogenesis, from mitigating genetic risk using antisense oligonucleotides to modulating TDP-43 proteinopathy itself using small molecule activators of autophagy, the ubiquitin-proteasome system, or the chaperone network. We present the case that preventing the misfolding of TDP-43 and/or enhancing its clearance represents the most important target for effectively treating ALS and frontotemporal dementia.
Aditi, Andrew W Folkmann and Susan R Wente.
Cytoplasmic hGle1A regulates stress granules by modulation of translation.. Molecular biology of the cell 26(8):1476–90, April 2015.
Abstract When eukaryotic cells respond to stress, gene expression pathways change to selectively export and translate subsets of mRNAs. Translationally repressed mRNAs accumulate in cytoplasmic foci known as stress granules (SGs). SGs are in dynamic equilibrium with the translational machinery, but mechanisms controlling this are unclear. Gle1 is required for DEAD-box protein function during mRNA export and translation. We document that human Gle1 (hGle1) is a critical regulator of translation during stress. hGle1 is recruited to SGs, and hGLE1 small interfering RNA-mediated knockdown perturbs SG assembly, resulting in increased numbers of smaller SGs. The rate of SG disassembly is also delayed. Furthermore, SG hGle1-depletion defects correlate with translation perturbations, and the hGle1 role in SGs is independent of mRNA export. Interestingly, we observe isoform-specific roles for hGle1 in which SG function requires hGle1A, whereas mRNA export requires hGle1B. We find that the SG defects in hGle1-depleted cells are rescued by puromycin or DDX3 expression. Together with recent links of hGLE1 mutations in amyotrophic lateral sclerosis patients, these results uncover a paradigm for hGle1A modulating the balance between translation and SGs during stress and disease.
Jessica Lenzi, Riccardo De Santis, Valeria Turris, Mariangela Morlando, Pietro Laneve, Andrea Calvo, Virginia Caliendo, Adriano Chiò, Alessandro Rosa and Irene Bozzoni.
ALS mutant FUS proteins are recruited into stress granules in induced pluripotent stem cell-derived motoneurons.. Disease models & mechanisms 8(7):755–66, 2015.
Abstract Patient-derived induced pluripotent stem cells (iPSCs) provide an opportunity to study human diseases mainly in those cases for which no suitable model systems are available. Here, we have taken advantage of in vitro iPSCs derived from patients affected by amyotrophic lateral sclerosis (ALS) and carrying mutations in the RNA-binding protein FUS to study the cellular behavior of the mutant proteins in the appropriate genetic background. Moreover, the ability to differentiate iPSCs into spinal cord neural cells provides an in vitro model mimicking the physiological conditions. iPSCs were derived from FUS(R514S) and FUS(R521C) patient fibroblasts, whereas in the case of the severe FUS(P525L) mutation, in which fibroblasts were not available, a heterozygous and a homozygous iPSC line were raised by TALEN-directed mutagenesis. We show that aberrant localization and recruitment of FUS into stress granules (SGs) is a prerogative of the FUS mutant proteins and occurs only upon induction of stress in both undifferentiated iPSCs and spinal cord neural cells. Moreover, we show that the incorporation into SGs is proportional to the amount of cytoplasmic FUS, strongly correlating with the cytoplasmic delocalization phenotype of the different mutants. Therefore, the available iPSCs represent a very powerful system for understanding the correlation between FUS mutations, the molecular mechanisms of SG formation and ALS ethiopathogenesis.
Anna Emde, Chen Eitan, Lee-Loung Liou, Ryan T Libby, Natali Rivkin, Iddo Magen, Irit Reichenstein, Hagar Oppenheim, Raya Eilam, Aurelio Silvestroni, Betty Alajajian, Iddo Z Ben-Dov, Julianne Aebischer, Alon Savidor, Yishai Levin, Robert Sons, Scott M Hammond, John M Ravits, Thomas Möller and Eran Hornstein.
Dysregulated miRNA biogenesis downstream of cellular stress and ALS-causing mutations: a new mechanism for ALS.. The EMBO journal 34(21):2633–51, 2015.
Abstract Interest in RNA dysfunction in amyotrophic lateral sclerosis (ALS) recently aroused upon discovering causative mutations in RNA-binding protein genes. Here, we show that extensive down-regulation of miRNA levels is a common molecular denominator for multiple forms of human ALS. We further demonstrate that pathogenic ALS-causing mutations are sufficient to inhibit miRNA biogenesis at the Dicing step. Abnormalities of the stress response are involved in the pathogenesis of neurodegeneration, including ALS. Accordingly, we describe a novel mechanism for modulating microRNA biogenesis under stress, involving stress granule formation and re-organization of DICER and AGO2 protein interactions with their partners. In line with this observation, enhancing DICER activity by a small molecule, enoxacin, is beneficial for neuromuscular function in two independent ALS mouse models. Characterizing miRNA biogenesis downstream of the stress response ties seemingly disparate pathways in neurodegeneration and further suggests that DICER and miRNAs affect neuronal integrity and are possible therapeutic targets.
Amandine Molliex, Jamshid Temirov, Jihun Lee, Maura Coughlin, Anderson P Kanagaraj, Hong Joo Kim, Tanja Mittag and Paul J Taylor.
Phase separation by low complexity domains promotes stress granule assembly and drives pathological fibrillization.. Cell 163(1):123–33, 2015.
Abstract Stress granules are membrane-less organelles composed of RNA-binding proteins (RBPs) and RNA. Functional impairment of stress granules has been implicated in amyotrophic lateral sclerosis, frontotemporal dementia, and multisystem proteinopathy-diseases that are characterized by fibrillar inclusions of RBPs. Genetic evidence suggests a link between persistent stress granules and the accumulation of pathological inclusions. Here, we demonstrate that the disease-related RBP hnRNPA1 undergoes liquid-liquid phase separation (LLPS) into protein-rich droplets mediated by a low complexity sequence domain (LCD). While the LCD of hnRNPA1 is sufficient to mediate LLPS, the RNA recognition motifs contribute to LLPS in the presence of RNA, giving rise to several mechanisms for regulating assembly. Importantly, while not required for LLPS, fibrillization is enhanced in protein-rich droplets. We suggest that LCD-mediated LLPS contributes to the assembly of stress granules and their liquid properties and provides a mechanistic link between persistent stress granules and fibrillar protein pathology in disease.
Yang Li, Mahlon Collins, Rachel Geiser, Nadine Bakkar, David Riascos and Robert Bowser.
RBM45 homo-oligomerization mediates association with ALS-linked proteins and stress granules.. Scientific reports 5:14262, January 2015.
Abstract The aggregation of RNA-binding proteins is a pathological hallmark of amyotrophic lateral sclerosis (ALS) and frontotemporal lobar degeneration (FTLD). RBM45 is an RNA-binding protein that forms cytoplasmic inclusions in neurons and glia in ALS and FTLD. To explore the role of RBM45 in ALS and FTLD, we examined the contribution of the protein's domains to its function, subcellular localization, and interaction with itself and ALS-linked proteins. We find that RBM45 forms homo-oligomers and physically associates with the ALS-linked proteins TDP-43 and FUS in the nucleus. Nuclear localization of RBM45 is mediated by a bipartite nuclear-localization sequence (NLS) located at the C-terminus. RBM45 mutants that lack a functional NLS accumulate in the cytoplasm and form TDP-43 positive stress granules. Moreover, we identify a novel structural element, termed the homo-oligomer assembly (HOA) domain, that is highly conserved across species and promote homo-oligomerization of RBM45. RBM45 mutants that fail to form homo-oligomers exhibit significantly reduced association with ALS-linked proteins and inclusion into stress granules. These results show that RMB45 may function as a homo-oligomer and that its oligomerization contributes to ALS/FTLD RNA-binding protein aggregation.
Ana\"ıs Aulas and Christine Vande Velde.
Alterations in stress granule dynamics driven by TDP-43 and FUS: a link to pathological inclusions in ALS?. Frontiers in cellular neuroscience 9:423, January 2015.
Abstract Stress granules (SGs) are RNA-containing cytoplasmic foci formed in response to stress exposure. Since their discovery in 1999, over 120 proteins have been described to be localized to these structures (in 154 publications). Most of these components are RNA binding proteins (RBPs) or are involved in RNA metabolism and translation. SGs have been linked to several pathologies including inflammatory diseases, cancer, viral infection, and neurodegenerative diseases such as amyotrophic lateral sclerosis (ALS) and frontotemporal dementia (FTD). In ALS and FTD, the majority of cases have no known etiology and exposure to external stress is frequently proposed as a contributor to either disease initiation or the rate of disease progression. Of note, both ALS and FTD are characterized by pathological inclusions, where some well-known SG markers localize with the ALS related proteins TDP-43 and FUS. We propose that TDP-43 and FUS serve as an interface between genetic susceptibility and environmental stress exposure in disease pathogenesis. Here, we will discuss the role of TDP-43 and FUS in SG dynamics and how disease-linked mutations affect this process.
Hyun-Hee Ryu, Mi-Hee Jun, Kyung-Jin Min, Deok-Jin Jang, Yong-Seok Lee, Hyong Kyu Kim and Jin-A Lee.
Autophagy regulates amyotrophic lateral sclerosis-linked fused in sarcoma-positive stress granules in neurons.. Neurobiology of aging 35(12):2822–31, December 2014.
Abstract Mutations in fused in sarcoma (FUS), a DNA/RNA binding protein, have been associated with familial amyotrophic lateral sclerosis (fALS), which is a fatal neurodegenerative disease that causes progressive muscular weakness and has overlapping clinical and pathologic characteristics with frontotemporal lobar degeneration. However, the role of autophagy in regulation of FUS-positive stress granules (SGs) and aggregates remains unclear. We found that the ALS-linked FUS(R521C) mutation causes accumulation of FUS-positive SGs under oxidative stress, leading to a disruption in the release of FUS from SGs in cultured neurons. Autophagy controls the quality of proteins or organelles; therefore, we checked whether autophagy regulates FUS(R521C)-positive SGs. Interestingly, FUS(R521C)-positive SGs were colocalized to RFP-LC3-positive autophagosomes. Furthermore, FUS-positive SGs accumulated in atg5(-/-) mouse embryonic fibroblasts (MEFs) and in autophagy-deficient neurons. However, FUS(R521C) expression did not significantly impair autophagic degradation. Moreover, autophagy activation with rapamycin reduced the accumulation of FUS-positive SGs in an autophagy-dependent manner. Rapamycin further reduced neurite fragmentation and cell death in neurons expressing mutant FUS under oxidative stress. Overall, we provide a novel pathogenic mechanism of ALS associated with a FUS mutation under oxidative stress, as well as therapeutic insight regarding FUS pathology associated with excessive SGs.
Physiological protein aggregation run amuck: stress granules and the genesis of neurodegenerative disease.. Discovery medicine 17(91):47–52, 2014.
Abstract Recent advances in neurodegenerative diseases point to novel mechanisms of protein aggregation. RNA binding proteins are abundant in the nucleus, where they carry out processes such as RNA splicing. Neurons also express RNA binding proteins in the cytoplasm and processes to enable functions such as mRNA transport and local protein synthesis. The biology of RNA binding proteins turns out to have important features that appear to promote the pathophysiology of amyotrophic lateral sclerosis and might contribute to other neurodegenerative disease. RNA binding proteins consolidate transcripts to form complexes, termed RNA granules, through a process of physiological aggregation mediated by glycine rich domains that exhibit low protein complexity and in some cases share homology to similar domains in known prion proteins. Under conditions of cell stress these RNA granules expand, leading to form stress granules, which function in part to sequester specialized transcript and promote translation of protective proteins. Studies in humans show that pathological aggregates occurring in ALS, Alzheimer's disease, and other dementias co-localize with stress granules. One increasingly appealing hypothesis is that mutations in RNA binding proteins or prolonged periods of stress cause formation of very stable, pathological stress granules. The consolidation of RNA binding proteins away from the nucleus and neuronal arbors into pathological stress granules might impair the normal physiological activities of these RNA binding proteins causing the neurodegeneration associated with these diseases. Conversely, therapeutic strategies focusing on reducing formation of pathological stress granules might be neuroprotective.
Yun R Li, Oliver D King, James Shorter and Aaron D Gitler.
Stress granules as crucibles of ALS pathogenesis.. The Journal of cell biology 201(3):361–72, 2013.
Abstract Amyotrophic lateral sclerosis (ALS) is a fatal human neurodegenerative disease affecting primarily motor neurons. Two RNA-binding proteins, TDP-43 and FUS, aggregate in the degenerating motor neurons of ALS patients, and mutations in the genes encoding these proteins cause some forms of ALS. TDP-43 and FUS and several related RNA-binding proteins harbor aggregation-promoting prion-like domains that allow them to rapidly self-associate. This property is critical for the formation and dynamics of cellular ribonucleoprotein granules, the crucibles of RNA metabolism and homeostasis. Recent work connecting TDP-43 and FUS to stress granules has suggested how this cellular pathway, which involves protein aggregation as part of its normal function, might be coopted during disease pathogenesis.
Regulated protein aggregation: stress granules and neurodegeneration.. Molecular neurodegeneration 7:56, January 2012.
Abstract The protein aggregation that occurs in neurodegenerative diseases is classically thought to occur as an undesirable, nonfunctional byproduct of protein misfolding. This model contrasts with the biology of RNA binding proteins, many of which are linked to neurodegenerative diseases. RNA binding proteins use protein aggregation as part of a normal regulated, physiological mechanism controlling protein synthesis. The process of regulated protein aggregation is most evident in formation of stress granules. Stress granules assemble when RNA binding proteins aggregate through their glycine rich domains. Stress granules function to sequester, silence and/or degrade RNA transcripts as part of a mechanism that adapts patterns of local RNA translation to facilitate the stress response. Aggregation of RNA binding proteins is reversible and is tightly regulated through pathways, such as phosphorylation of elongation initiation factor 2$\alpha$. Microtubule associated protein tau also appears to regulate stress granule formation. Conversely, stress granule formation stimulates pathological changes associated with tau. In this review, I propose that the aggregation of many pathological, intracellular proteins, including TDP-43, FUS or tau, proceeds through the stress granule pathway. Mutations in genes coding for stress granule associated proteins or prolonged physiological stress, lead to enhanced stress granule formation, which accelerates the pathophysiology of protein aggregation in neurodegenerative diseases. Over-active stress granule formation could act to sequester functional RNA binding proteins and/or interfere with mRNA transport and translation, each of which might potentiate neurodegeneration. The reversibility of the stress granule pathway also offers novel opportunities to stimulate endogenous biochemical pathways to disaggregate these pathological stress granules, and perhaps delay the progression of disease.
Natalie Gilks, Nancy Kedersha, Maranatha Ayodele, Lily Shen, Georg Stoecklin, Laura M Dember and Paul Anderson.
Stress granule assembly is mediated by prion-like aggregation of TIA-1.. Molecular biology of the cell 15(12):5383–98, 2004.
Abstract TIA-1 is an RNA binding protein that promotes the assembly of stress granules (SGs), discrete cytoplasmic inclusions into which stalled translation initiation complexes are dynamically recruited in cells subjected to environmental stress. The RNA recognition motifs of TIA-1 are linked to a glutamine-rich prion-related domain (PRD). Truncation mutants lacking the PRD domain do not induce spontaneous SGs and are not recruited to arsenite-induced SGs, whereas the PRD forms aggregates that are recruited to SGs in low-level-expressing cells but prevent SG assembly in high-level-expressing cells. The PRD of TIA-1 exhibits many characteristics of prions: concentration-dependent aggregation that is inhibited by the molecular chaperone heat shock protein (HSP)70; resistance to protease digestion; sequestration of HSP27, HSP40, and HSP70; and induction of HSP70, a feedback regulator of PRD disaggregation. Substitution of the PRD with the aggregation domain of a yeast prion, SUP35-NM, reconstitutes SG assembly, confirming that a prion domain can mediate the assembly of SGs. Mouse embryomic fibroblasts (MEFs) lacking TIA-1 exhibit impaired ability to form SGs, although they exhibit normal phosphorylation of eukaryotic initiation factor (eIF)2alpha in response to arsenite. Our results reveal that prion-like aggregation of TIA-1 regulates SG formation downstream of eIF2alpha phosphorylation in response to stress. | CommonCrawl |
A set $Q$ is well-quasi-ordered by a relation $\le$ if for every sequence $q_1,q_2,\ldots$ of elements of $Q$ there exist $i<j$ such that $q_i \le q_j$. In their Graph Minors series, Robertson and Seymour prove that graphs are well-quasi-ordered by the minor relation. This result, which is known as the Graph Minor Theorem, is considered one of the deepest results in graph theory and has several algorithmic applications.
Unfortunately, the same is not true for directed graphs and the relation of butterfly minor. In particular, we can easily identify two infinite antichains. We can then ask if classes of graphs that exclude these antichains are well-quasi-ordered by the butterfly minor relation. We prove that this is the case while at the same time providing a structure theorem for these graph classes.
This is joint work with Maria Chudnovsky, S-il Oum, Paul Seymour and Paul Wollan. | CommonCrawl |
Jack and Jill developed a special encryption method, so they can enjoy conversations without worrrying about eavesdroppers. Here is how: let $L$ be the length of the original message, and $M$ be the smallest square number greater than or equal to $L$. Add $(M - L)$ asterisks to the message, giving a padded message with length $M$. Use the padded message to fill a table of size $K \times K$, where $K^2 = M$. Fill the table in row-major order (top to bottom row, left to right column in each row). Rotate the table $90$ degrees clockwise. The encrypted message comes from reading the message in row-major order from the rotated table, omitting any asterisks.
For example, given the original message 'iloveyouJack', the message length is $L=12$. Thus the padded message is 'iloveyouJack****', with length $M=16$. Below are the two tables before and after rotation.
Then we read the secret message as 'Jeiaylcookuv'.
The first line of input is the number of original messages, $1 \le N \le 100$. The following $N$ lines each have a message to encrypt. Each message contains only characters a–z (lower and upper case), and has length $1 \le L \le 10\, 000$.
For each original message, output the secret message. | CommonCrawl |
If $f(x)$ is $U$-almost-periodic and if $f'(x)$ is uniformly continuous on $(-\infty,\infty)$, then $f'(x)$ is $U$-almost-periodic; the indefinite integral $F(x)=\int_0^xf(t)dt$ is $U$-almost-periodic if $F(x)$ is a bounded function.
Bohr's treatise [a1] is a good reference. An up-to-date reference is [a2].
This page was last modified on 24 August 2014, at 12:22. | CommonCrawl |
Department of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box: 6135713895, Ahvaz, Iran.
We introduce and study the concept of $\alpha $-semi short modules.
Using this concept we extend some of the basic results of $\alpha $-short modules to $\alpha $-semi short modules.
We observe that if $M$ is an $\alpha $-semi short module then the dual perfect dimension of $M$ is $\alpha $ or $\alpha +1$.
%In particular, if a semiprime ring $R$ is $\alpha $-semi short as an $R$-module, then its Noetherian dimension either is $\alpha$ or $\alpha +1$. | CommonCrawl |
My current research interests focus on developing theoretical and numerical methods for understanding the phase behavior and structures of block copolymers, polymer brushes and polyelectrolytes both in bulk and under geometrical confinements. During the PhD period, my research focus was conducting both experimental and theoretical studies to elucidate the fundamental mechanism of polymer crystallization, using monolayer crystals of low molecular weight poly(ethylene oxide) (PEO) fractions as a model system.
Block copolymers represent an important class of polymeric materials that can self-assemble into complex microstructures on the scale of 1 to 100 nm. Confining block copolymers in geometrical environments introduces additional parameters, which leads to an even richer set of nanostructures, making it an attractive route for a wide range of applications including nanolithography, nanotemplating, nanoporous membranes, coatings, and biomaterials. I have conducted a comprehensive study of the confining system of block copolymers, including development of highly efficient numerical tools, the surface states of polymer brushes which are often used to modify substrate properties, development of a model to describe the interaction between the modified substrates and the block copolymers, thermodynamic and kinetic aspects of directed self-assembly (DSA).
The surface interaction between substrates and block copolymers is one of the most important factors that control the alignment of self-assembled domains under thin film confinement. Most previous studies simply modeled substrates modified by grafting polymers as a hard wall with a specified surface energy, leading to an incomplete understanding of the role of grafted polymers. We proposed a general model of surface interactions where the role of grafted polymers is decomposed into two independent contributions: the surface softness ($\gamma$) and the surface preference ($\delta$). With this separation, we elucidate the connection among bare hard wall confinement (SST), hard confinement modeled by the "masking" technique, and soft confinement. Soft confinement reduces to the mask hard confinement at large $\gamma$ and to the SST hard confinement in the limit of $\gamma\to\infty$. It is also possible to map the soft confinement model using ideal random copolymer-grafted substrates onto our model. Hence we believe that this separation of surface interactions into two independent parts are universal as long as no internal phase separation occurs across the confining wall itself.
Directed self-assembly (DSA), which utilizes chemoepitaxy or graphoepitaxy to guide the microphase separation of block copolymers (BCPs), is a competitive next-generation lithography candidate for electronic industry. Based on this technique, large scale defect-free ordered nanostructures, have been successfully produced on patterned substrates in experiments. In general, sparsely patterned templates are preferred because the template is produced by photolithography whose cost is proportional to the resolution of the template. However, the directing ability of the template weakens as the pattern in the template becomes sparser. Consequently, it is impossible to reduce the cost by lowering the resolution of the template without limitation.
Inverse design is an emerging concept in materials design that the desired functionality of the new material is declared first and theoretical or numerical calculations are then used to predict which stable compounds/molecules exhibit the required functionality. The common approach for tackling such inverse problems is converting them into optimization problems and optimization methods (such as evolutionary algorithms) are employed to solve these problems. One major issue of this approach is that the optimization method has to be rerun once the desired functionality is changed a little. To settle this issue, we propose a machine learning approach based on deep learning and neural networks for inverse design. One of the advantages of the machine learning approach is that once the model is trained, we can use it again and again. Note that the predicting stage of the machine learning is blazing fast. Thus it can save tremendous amount of computational time in practical applications.
While SCFT achieves remarkable success in modeling self-assembly of block copolymers, it is a mean-field theory which ignores the fluctuating effects. Using beyond mean-field field-theoretic simulations to study fully fluctuating polymer systems remains a big challenge. Current field-based approaches are highly computational demanding which renders it infeasible to tackle large-scale problems. In collaboration with Prof. G. H. Fredrickson at UCSB, we propose a density functional model that can incorporate thermal fluctuations to describe polymer solutions. Meanwhile, it can cut the number of dimensions of the computational space from 4 to 3, leading to a reduction of computational cost for about two magnitude. Using the same strategy for developing the model of polymer solutions, it can be generalized to other polymeric systems such as block copolymers.
to provide new evidence to better understand the nature of polymer crystal-lization.
Based on statistical thermodynamic analysis, we predict that the thickness of amorphous layers of IF(0) crystals increases with temperature, which is confirmed by crystallinity measurements and small angle X-ray scattering (SAXS) experiments.
The melting behavior of PEO monolayer crystals on mica surfaces was studied by AFM. Taking account of the linear decrease of fold surface free energy with temperature, we have shown that the relations between melting points and thickness and/or fold number can be described by a modified Gibbs-Thomson equation. The end group effect on the melting behavior of the monolayer crystals was also analyzed.
As an important part of my research work, I have developed several computational software packages and implemented numerical algorithms to perform computational tasks encountered along research.
Programming languages involved: C/C++, Matlab, and Python. Transition to Julia Planned.
Chebyshev collocation method for SCFT calculations of confined polymers.
Pseudo-spectral method for SCFT calculations.
Multigrid method for SCFT calculations of charged polymers.
Off-lattice Monte-Carlo simulation of polymer crystal growth.
Phase-field simulations of thin film polymer crystal growth. | CommonCrawl |
Effects of unsteady sheared $\bf E \times B$ flow on drift wave turbulence and heat transport driven by slab ion temperature gradient (ITG) instability are investigated by means of Landau fluid simulations. Here, the $\bf E \times B$ flow, which consists of stationary and time-periodic oscillatory parts, is externally applied to the turbulence. The dependence on the amplitude and frequency of $\bf E \times B$ flow are examined in the case where the energy of $\bf E \times B$ flow is the same or larger than the energy of turbulence. In the case above, the transport oscillates with the same period as the $\bf E \times B$ flow and the time-averaged transport coefficient is larger than the coefficient which is evaluated without oscillatory part of $\bf E \times B$ flow. The time-averaged coefficient is maximized at the point where the amplitude of oscillatory part is equal to that of stationary part. As the frequency of $\bf E \times B$ flow increases, the time-averaged coefficient decreases and is close to the coefficient which is evaluated without oscillatory part. These mechanisms are explained. | CommonCrawl |
How can we define the limit of a constant function?
In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.
What if the function was a constant?! A constant function will not approach anything, so, how would we define the limit of a constant function?
Your quote isn't a definition of a limit, but an English language description of what it computes.
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
First let's face the fact there may be more than one limit definition depending on the space in which you operate. For example a limit of a function for a given element of domain where both domain and codomain have some measure you'll likely go with the $\epsilon - \delta$ definition while if you're talking about a limit of an infinite sequence you need to have the sequence definition. They are related but not exactly the same.
The quote you cite from this Wikipedia article that you're referring to uses some textual explanation to make it easier to grasp the idea of a limit but it is not exact in mathematical terms. This is supposed to only make it easier to understand the idea.
$c$ is a $\lim$ of $f$ in (something - either an element in your domain or something that is next to your domain) if regardless how close you want to be to $c$ (by selecting an $\epsilon > 0$ you can define some conditions narrowing which element $x$ you are allowed to pick from your domain - close to where you want to be in your domain so that for any $x$ from your domain selected in that way it is true that you are as close to the $c$ as you wanted to be, i.e.
This is not an actual quote, it's a generalization-rephrasing.
So still using a common English explanation if you get close in your domain to some $x_0$ you will be close to some $c$ in your codomain. If you are in $c$ already you are also very, very close.
Now there is nothing about approaching here, but in most cases most of your values even near your $x_0$ produce a value different than $c$ so using the term approach make it easier for the reader to understand. Yet if you're already in $c$ technically you don't approach it, but it doesn't invalidate the definition. So if you wan't to properly define approach you'd have to say you're either getting very close to the point or you are already there.
Note that your function may jump to $c$ sometimes and then get out of it. As long as you don't get away too far, you're still approaching $c$. But the common sense understanding of approach is different so you need to be careful not to confuse the precise mathematical and common sense meaning.
When trying to describe math in a common language, to make it understandable you use some simplification and loose actual math precision. You need to account that. Especially sources like Wikipedia should be treated with a grain of salt in such informal descriptions.
Terms written in script are not strict mathematical terms. Those terms are replaces with strict definitions/conditions but those definitions/conditions are sometimes difficult to grasp. I deliberately didn't want to use strict math terms.
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How discontinuous can the limit function be?
How can constant functions have limits?
How to approach proving the limit of $[\cos(x) - 1]$ doesn't exist?
What is the limit of this function as x approaches 2? | CommonCrawl |
Many quantum information tasks require measurements to distinguish between different quantum-mechanically entangled states (Bell states) of a particle pair. In practice, measurements are often limited to linear evolution and local measurement (LELM) of the particles. We investigate LELM distinguishability of the Bell states of two qubits (two-state particles) and qutrits (three-state particles), via standard projective measurement and via generalized measurement, which allows detection channels beyond the number of orthogonal single-particle states. Projective LELM can only distinguish 3 of 4 qubit Bell states; we show that generalized measurement does no better. We show that projective LELM can distinguish only 3 of 9 qutrit Bell states that generalized LELM allows at most 5 of 9. We have also made progress on distinguishing qubit $\times$ qutrit hyperentangled Bell states, which are made up of tensor products of the qubit Bell states and the qutrit Bell states, showing that the maximum number distinguishable with projective LELM measurements is between 9 and 11.
Leslie, Nathaniel, "Maximal LELM Distinguishability of Qubit and Qutrit Bell States using Projective and Non-Projective Measurements" (2017). HMC Senior Theses. 108. | CommonCrawl |
Some of the basic probability and statistics concepts. joint probabilities, combinatories; conditional probabilities and independence and their examples; Bayes' rule.
Recall: If A & B are 2 disjoint events, then $P[A\cup B]=P[A]+P[B]$, $P[A\cap B]=P[A]\cdot P[B]$.
If $A\cup B \neq \varnothing$, then $P[A\cap B]=P[A]\cdot P[B] - P[A\cap B]$.
Weather data for 2 consecutive days in some month in some location. We aggregate this data and find: 'R1 is rain on 1st day', 'R2 is rain on 2nd day'. $P[R1]=0.6, P[R2]=0.5, P[R1\cap R2^C]=0.2$. Find $P[R1\cap R2]$ and $P[R1\cup R2]$.
Form a jury with 6 people. We choose from a group with 8 men & 7 women. Find the prob that there are exactly 2 women in the selected jury.
Need an assumption about the relative prob. of people being picked: 6/15. Every one has equal chance of being picked.
Definition: A & B are independence events, if $P[A\cap B] = P[A] \times P[B]$.
The definition comes from the definition of conditional prob.
Definition: Let A & B be two events. The conditional prob of B given A denoted by $P[B|A]$, is $P[A \cap B]/P[A]$.
Note: if $P[B|A] = P[B]$ (given information of A does not affect/influence the chance of B), then the definition is proven.
$P[disease]= 0.006$. Test: positive vs negative. $P[positive|dissease] = 0.98$ (sensitivity). $P[pos|no disease]=0.01$.
then for any $B\in\Omega$, $P[B]=\sum_i P[B|A_i]P[A_i]$.
Try to prove it at home.
$P[A|B]$: B has happend, the prob of A.
$P[A]$ and $P[A^C]$: "The prori" a priori info about the events you are interested in, regard of the observation.
$P[B|A]$ and $P[B|A^C]$: "Likelihood" likelihoods that eventsthat you observe B actually happen given the events of interest to you.
At home, relate disease example problem to Bayes' rule. | CommonCrawl |
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
Can you put the $25$ coloured tiles into the $5\times 5$ square below so that no column, no row and no diagonal line have the same colour in them?
Use the interactivity below to try out your ideas.
Visualising. Combinatorics. Experimental probability. Networks/Graph Theory. Mathematical modelling. Mathematical reasoning & proof. Games. Interactivities. Dynamic geometry. Working systematically. | CommonCrawl |
I am trying to construct an example of a ring satisfying the followings.
I know that a local noetherian ring having a height $1$ principal prime ideal is a domain. Actually I wanted to prove this without the local condition. I couldn't prove this hence I am looking for a counterexample. I need some help. Thanks.
As a hint to what's going on here that can't happen in the local case, notice that $(y-1)x=-x$, so $x$ is divisible by $y-1$ arbitrarily many times. In a local Noetherian ring this would imply $x=0$ by the Krull intersection theorem.
Let $R$ be a noetherian ring with principal prime ideal $P$ of height $1$.
Then $R\times R$ is Noetherian, non local, not a domain, and still has a principal prime ideal of height $1$: $P\times R$.
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$A$ local Noetherian ring with principal maximal ideal implies PIR?
Noetherian domain R is a UFD if every prime ideal of height 1 in R is principal. | CommonCrawl |
$R^\wedge \to S^\wedge $ is formally smooth for the $\mathfrak n^\wedge $-adic topology.
Here $R^\wedge $ and $S^\wedge $ are the $\mathfrak m$-adic and $\mathfrak n$-adic completions of $R$ and $S$.
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What would be the entry for 20 in this list?
20 = ___ . . . ?
Please use and explain the simplest possible rule, not purely mathematical, that accounts for every equivalence from 0 to 99.
Because you gave us a list of equivalences which are more equal than others. So we can assume the remaining numbers are less equal and therefore only equal to themselves.
Because the rules are contrived so I can simply invent whatever I want for the rules that aren't given to me.
However, that did not explain 0 through 10. I could add a 'except for 1 through 10' to my rule, but that wasn't very satisfying.
For a number $n$ made of two digits $a$ and $b$, we have $n = a\cdot 10 + b$. If we say those are equivalent to $m=a\cdot 10 - b$, they are recursively equivalent to a lot of numbers. For example, $$97 = 9\cdot 10 + 7$$ $$9\cdot 10 - 7 = 83 = 8\cdot 10 + 3$$ $$8\cdot10 - 3 = 77 = 7\cdot 10 + 7$$ And so on: $97 \rightarrow 83 \rightarrow 77 \rightarrow 63 \rightarrow 57 \rightarrow 43 \rightarrow 37 \rightarrow \ldots$. This shows that numbers with a zero as second digit do not have any other equivalent numbers, as this process would not lead to any new numbers: $$20 = 2\cdot 10 + 0$$ $$2\cdot10 - 0 = 20$$ $20 \rightarrow 20 \rightarrow \ldots$. The only way I managed to explain $1,\ldots,9$ here is to explain you simply can't apply this process to those numbers, as they do not have two digits.
This doesn't work if you accept that 09 is a perfectly fine way of writing 9. So I'm still not very satisfied with this solution.
The rule (vertically) is: Line 1 + 1, then Line 2 - 1, and so on.
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When are the verbs "divide" and "multiply" synonyms, rather than antonyms?
What are some good resources to practice logical puzzle-solving? | CommonCrawl |
Abstract: I provide updates for the theoretical predictions of the muon and electron anomalous magnetic moments, for the shift in the fine structure constant $\alpha(M_Z)$ and for the weak mixing parameter $\sin^2 \Theta_W(M_Z)$. Phenomenological results for Euclidean time correlators, the key objects in the lattice QCD approach to hadronic vacuum polarization, are briefly considered. Furthermore, I present a list of isospin breaking and electromagnetic corrections for the lepton moments, which may be used to supplement lattice QCD results obtained in the isospin limit and without the e.m. corrections. | CommonCrawl |
Abstract: Solutions of the Polchinski exact renormalization group equation in the scalar O(N) theory are studied. Families of regular solutions are found and their relation with fixed points of the theory is established. Special attention is devoted to the limit $N=\infty$, where many properties can be analyzed analytically. | CommonCrawl |
Abstract: We construct a thermal dark matter model with annihilation mediated by a resonance to explain the positron excess observed by PAMELA, Fermi-LAT and AMS-02, while satisfying constraints from cosmic microwave background (CMB) measurements. The challenging requirement is that the resonance has twice the dark matter mass to one part in a million. We achieve this by introducing an $SU(3)_f$ dark flavor symmetry that is spontaneously broken to $SU(2)_f \times U(1)_f$. The resonance is the heaviest state in the dark matter flavor multiplet and the required mass relation is protected by the vacuum structure and supersymmetry from radiative corrections. The pseudo-Nambu Goldstone Bosons (PNGB's) from the dark flavor symmetry breaking can be slightly lighter than one GeV and dominantly decay into two muons just from kinematics, with subsequent decay into positrons. The PNGB's are produced in resonant dark matter semi-annihilation, where two dark matter particles annihilate into an anti-dark matter particle and a PNGB. The dark matter mass in our model is constrained to be below around 1.9 TeV from fitting thermal relic abundance, AMS-02 data and CMB constraints. The superpartners of Standard Model (SM) particles can cascade decay into a light PNGB along with SM particles, yielding a correlated signal of this model at colliders. One of the interesting signatures is a resonance of a SM Higgs boson plus two collimated muons, which has superb discovery potential at LHC Run 2. | CommonCrawl |
I'm struggling with the concept of the Fatou set because its definition as the largest open set on which the iterates of a map are normal is so abstract. However, on Wikipedia I've caught mentions of "fatou domains", here meaning loosely "largest open set on which the iterates have a certain long term behavior". The Fatou set is then the union of the (disjoint) Fatou domains. For example, in this image, the three colors represent the three Fatou domains, being the basins of attraction of the three attracting fixed points.
I can wrap my head around that concept much better: the complex plane divides up into disjoint open sets with dense union such that the long term dynamics are the same on each set. Their boundary is a nowhere dense set called the Julia set. Behavior at the Julia set is obviously chaotic since by definition, slight perturbations will put you in different Fatou domains, where the long term behavior is different.
A Fatou domain is the basin of attraction of an attracting cycle.
To each $z$ assign $\omega(z)$, the set of limit points of the forward orbit of $z$. Then a Fatou domain is a largest open set on which $\omega$ is constant. The Fatou set is then the union of the Fatou domains.
This definition attempts to generalize the notion of basin of attraction. But I'm not sure if this is correct. Is $\omega$ constant on a Siegel disk, for example? Also, what about the Julia set? How does $\omega$ behave on that? Would it end up being a "Fatou domain" under the above definition?
In summary: Can we define the Fatou set in the following way?
Somehow generalize "basin of attraction" to "largest open set which exhibits a common long term behavior", perhaps using the function $\omega$ above.
Call such sets Fatou domains and prove that they are open and disjoint.
Establish that these sets are dense in the plane.
Call their union the Fatou set and show equivalence with the standard definition.
Browse other questions tagged complex-analysis complex-dynamics or ask your own question.
Do holomorphic functions necessarily blow up at the edge of their maximal domain of definition?
Are limits points continuous on the Fatou set?
Boundary of basin of attraction of $\infty$ = closure of repelling periodic points. | CommonCrawl |
1 Nguyễn Việt Dũng, Nguyen Van Ninh, The Higher Topological Complexity of Complement of Fiber Type Arrangement, Acta Mathematica Vietnamica, 42 (2017), 249–256, Scopus.
2 Nguyễn Việt Dũng, Tran Quoc Cong, The Homotopy Type of the Complement to a System of Complex Lines in $\mathbb C^2$, Vietnam Journal of Mathematics, 42(2014), 365-375, Scopus.
3 Nguyễn Việt Dũng, A model for homotopy type of the complement. Dedicated to the memory of Le Van Thiem (Hanoi, 1998). Acta Math. Vietnam. 27 (2002), 289 - 295.
4 Nguyễn Việt Dũng, Homotopy of configuration spaces, Vietnam J. Math. 30 (2002), 97 - 102.
5 Nguyễn Việt Dũng, Braid monodromy of the complex line arrangements, Kodai Math. J. 22 (1999), 46 - 55.
6 Nguyễn Việt Dũng, The topology of configuration spaces of type B. Ph.D. Dissertation, Hanoi Institute of Mathematics, 1997.
7 Nguyễn Việt Dũng, Hà Huy Vui, The fundamental group of complex hyperplanes arrangements. Acta Math. Vietnam. 20 (1995), 31 - 41.
8 Nguyễn Việt Dũng, On the fundamental group of the complement of arrangements. Kodai Math. J. 17 (1994), 428 - 431.
9 Nguyễn Việt Dũng, The fundamental group of complexified real arrangements. Ann. Sci. Math. Québec 18:2 (1994), 157 - 167.
10 Nguyễn Việt Dũng, The modulo 2 cohomology algebra of wreath products Σ∞≀X, In: Proceedings of Barcelona Algebraic Topology Conference. Springer Lect. notes in Math. 1509 (1990), 115 - 119.
11 Nguyễn Việt Dũng, Note on the structure of cocommutative coalgebras. Acta Math. Vietnam. 17 (1992), 3 - 9.
12 Nguyễn Việt Dũng, The mod 2 equivariant cohomology algebras of finite configuration spaces of type B. Proc. of the 3rd Vietnamese Congress of Mathematicians 2 (1985), 210 - 215.
13 Nguyễn Việt Dũng, The fundamental groups of the spaces of regular orbits of affine Weyl groups. Topology 22:4 (1983), 425 - 435. | CommonCrawl |
Zakaria, Siti Fatimah Binti (2016) Analytic Properties of Potts and Ising Model Partition Functions and the Relationship between Analytic Properties and Phase Transitions in Equilibrium Statistical Mechanics. PhD thesis, University of Leeds.
The Ising and $Q$-state Potts models are statistical mechanical models of spins interaction on crystal lattices. We study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena $-$ how higher level structure is affected by the detailed activity of a very large number of lower level structures. We investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. Our study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. Here we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. The partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. We extend the study of $Q$-state Potts models to a more general case called the $Z_Q$-symmetric model. We evidence the existence of multiple phase transitions for this model in case $Q=5,6,$ and discuss the possible connection to different stages of disordered state. Given sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. Our analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. We evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent $\alpha$. | CommonCrawl |
This page describes how to run constant pH replica exchange MD (pH-REMD) in which replicas are run at the same temperature but different pH values. Note, this tutorial is for implicit (Generalized Born) solvent. After you learn how to run CpHMD in explicit solvent (click here for tutorial), you can set up each replica to run in explicit solvent instead.
Note: You must have at least version 14 of Amber and AmberTools to run pH-REMD.
Creating the input files for pH-REM simulations is identical to creating input files for normal constant pH MD (CpHMD) simulations.
-I: Adding /home/swails/build_amber/amber/dat/leap/prep to search path.
-I: Adding /home/swails/build_amber/amber/dat/leap/lib to search path.
-I: Adding /home/swails/build_amber/amber/dat/leap/parm to search path.
-I: Adding /home/swails/build_amber/amber/dat/leap/cmd to search path.
with the reference energies in the printed cpin file.
evaluate dynamics (or protonation state swaps).
of the GB potential. Default 1.0.
Name of system to titrate. No effect on simulation.
arguments take precedence and are mutually exclusive with each other.
The only required flag is -p prmtop — the rest of the flags are there to give you control over which residues you allow to titrate. You can pick specific residue numbers (in which the first residue is residue 1) using the -resnum flag or specific residue names using the -resname flag. Likewise, -notresnum and -notresname can be used to omit certain residue names and numbers. We can also pre-define the protonation states for each residue using the -states flag.
The original implementation of CpHMD by Mongan, et. al. used the igb=2 Generalized Born model, so that is what we use here.
At this point, we have our topology file (prmtop), input coordinate file (inpcrd) and constant pH input file (cpin), so we can set up our simulation. We will follow the 3 standard steps: minimization, heating, and equilibration (note the apparent misnomer — we're not actually simulating to equilibrium, we're really relaxing our structure).
Note that at this point, we only have a single structure. This will serve as the starting point for each of our replicas next.
At this point, we have a single, 'equilibrated' structure, and we wish to start a pH ladder of replicas. To run replica exchange simulations, we need a groupfile, the -rem and -remlog command-line flags, and the variables nstlim and numexchg in the &cntrl namelist of the input files.
nstlim now means something different. In replica exchange, nstlim is the number of steps between exchange attempts.
numexchg is the number of times to attempt exchanging. Therefore, the total number of steps in a simulation can be found by multiplying nstlim and numexchg together.
icnstph is the flag to turn on constant pH simulations.
ntcnstph is the number of steps between attempting protonation state changes. While it should work for nstlim to be smaller than ntcnstph, I would suggest always having ntcnstph smaller than nstlim.
saltcon is the salt concentration. Because the reference compounds were calculated using a salt concentration of 0.1 M, you should make sure that is present here as well.
In this step, we are attempting replica exchanges every 100 steps (200 fs) and attempting protonation state changes every 5 steps (10 fs). Note this is the sample input file for the replica running with pH set to 2.
Each line should contain the -rem 4 and -remlog rem.log options (where rem.log is the file you want the replica exchange information is stored).
Each line should have a different input file, but the same topology file. The input coordinate file can differ (and will for restarts), or it can be the same.
The ordering of the replicas in the groupfile does not matter. A pH ladder (much like a temperature) ladder is defined, and exchange partners are chosen that way.
Because replicas change pH (rather than changing structure), each output file will contain a combination of snapshots from different pH values, but the pH is automatically reassigned (based on what is in the restart file) at the beginning of the simulation. The pH ladder is automatically generated properly.
Because replicas periodically change their target pH when replica exchange attempts succeed, we need to have some way of extracting snapshots and titration data that correspond to a pH ensemble rather than a replica time series.
I wrote a program in C++ that will parse the constant pH output files from Amber and compute different statistics from these output files. You can download the source code from here: https://github.com/swails/cphmd_tools (there is a "Download ZIP" option that will give you the files in a zipped folder). The README file has instructions for installing the program (it is similar to installing Amber).
-h, --help Print this help and exit.
-V, --version Print the version number and exit.
Allow existing outputs to be overwritten.
being read in and used for the calculations.
This is the time step in ps you used in your simulations.
It will be used to print data as a function of time.
over chunks of the simulation are printed.
printed for every state of every residue.
Output file with requested conditional probabilities.
a trajectory split up into chunks.
(1) Print everything calcpka prints.
like `chunks', `cumulative' data, and running averages.
fraction in time series data (Default behavior).
fraction in time series data.
of fraction (de)protonated. NOT default behavior.
the original, calcpka-style analysis is done.
steps (NOT the number of MC exchange attempts).
This program analyzes constant pH output files (cpout) from Amber.
compression will be detected automatically by the file name extension.
You must have the gzip headers for this functionality to work.
The first thing you must do before doing any kind of analysis is to extract the ensemble of protonation states for a single pH (since the pH is exchanged between replicas, each replica's cpout file has points from every pH we simulated). The --fix-remd flag in cphstats will re-order the cpout files.
The input cpout files should be each cpout file from a single pH-REMD simulation. If you have run 10 ns of simulation in 1 ns chunks, you will need to run this program 10 times! The number of cpout files you provide must be equal to the number of replicas you have. Each pH value will print a cpout file named prefix.pH_2.00 where prefix is the prefix string (it is the argument that follows the --fix-remd flag), and 2.00 is the pH. The pH_# suffix is added for each pH value.
(where each replica has a different number following cpout.md1.).
After you create the pH-based cpout files, you can analyze these pH-specific cpout files using either cphstats or calcpka (the latter is distributed with AmberTools). cphstats is ca. twice as fast as calcpka for the same analyses, and is capable of doing everything calcpka does in addition to computing running averages with a flexible window size, cumulative averages, and 'chunk' averages (the simulation is split up into 'chunks' that are analyzed separately). (Like calcpka, cphstats also performs the same standard analysis and population dumps in exactly the same format that calcpka uses).
Before writing cphstats, I wrote a Python script (remake_cpouts.py) to extract pH-based cpout files from replica-based cpout files. You can download it here. Since implementing CpHMD on GPUs, the cpout files have become so long that processing them with Python was becoming painfully slow. While there is no installation process (just download and run), it is at least 40 times slower than cphstats and does not do any analyses (it is the equivalent of just the --fix-remd flag.
Like the cpout files, trajectory files will be filled with snapshots from each pH value in the ladder (unless of course you get severely limited state space mixing). To account for this, I have 'tricked' the trajectory files into thinking that the pH value is really the temperature. Therefore, you can use ptraj and/or cpptraj the same way you would for T-REMD, but using the pH value where you would normally put the temperature.
This will extract all of the snapshots with pH 2 (which is stored as a temperature of 2) from the trajectory files 1AKI.dry.md1.nc.000, 1AKI.dry.md1.nc.001, … etc.
At this point, if pH-trajectories are generated with a trajout command, then they will align with the reconstructed cpout files created in the previous section as though regular CpHMD simulations were run.
The replica exchange log file has a lot of information regarding the exchange attempts for each replica. A section of a file is shown below.
The columns are the replica number, followed by the number of protons present in the current set of protonation states, followed by the original pH, followed by the "new" pH (if the two pHs are the same, then the exchange attempt failed). The last column is the exchange success rate with the higher-pH replica (notice that the highest pH, 7.5, has a success rate of 0 in the final exchange because its 'higher' pH is the first replica, which has a pH of 1).
This section will be fairly short, as it assumes a basic working knowledge of replica exchange simulations, detailed balance, and Mongan's original constant pH MD method. This will hopefully provide some insight about why things are implemented the way they are (i.e. why replicas change pH rather than change structures).
In Eq.(1), $x_i$ refers to the conformational degrees of freedom (the x-, y-, z-coordinates of each atom), and $N_i$ is the number of titratable protons on the system, which is directly related to the net charge of the system.
An important note — because we are not exchanging either the number of protons or coordinates, all terms involving the energies cancel out, so the energies do not appear in our exchange probability! Therefore, we don't have to calculate any energies, and this exchange attempt is as cheap as exchanges in T-REMD (effectively free). | CommonCrawl |
It was the first meeting of the humans and the aliens, and things were going well: The humans and aliens had learned to understand each other's language and found that both were hoping for a peaceful relationship. In this spirit, the alien and human ambassador decided to play a game.
Yes, yes. We know how queens move on our planet too!
This exchange proved to produce a fateful misunderstanding: The aliens believed queens could make the same moves as a rook or a knight - that is, they would move the piece straight up or left, or go left two and up one, or go up two and left one. The humans believed queens moved in a straight up, diagonal, or leftward line.
Not realizing this, they began playing the game. If we label the top left square $(0,0)$, the square right of it as $(1,0)$ and the square below it as $(0,1)$ and continue with coordinates analogously, then they started by placing the queen at $(2,3)$.
How could you not have seen such a standard move?
The human ambassador immediately understood how the aliens believed the game worked, but politely said nothing. They played again, starting at $(4,2)$ this time. The aliens started, moving to $(3,2)$, believing this to be a winning move. They were very surprised when the humans moved the queen to $(1,0)$, diagonally upwards, forcing the aliens to lose. The aliens were too polite to say anything, but realized how the silly humans must believe queens moved.
Shall we make a bet? Whoever wins the next game will be declared the best species in the universe and will be entitled to ownership of their opponent's planet! We will make this fair: We choose the starting position, you choose which player moves first.
The aliens placed an enormous, $1000000\times 1000000$ chess board on the table. The human ambassador quickly accepted, knowing that a queen moving diagonally was far more powerful than one that could merely make knight leaps. The aliens placed the queen at the position $(170000,170002)$.
What should the human ambassador do?
The human ambassador cannot win.
If either player moves to (0,x) where x>1, they lose because their opponent moves to (0,1). Similarly, (x,0) loses. (1,x) and (x,1) where x>0 also lose for the same reason.
If the human ever moves to (2,2), the alien can move to (1,0), so the human loses. But if the alien moves to (2,2), the human has no move that doesn't lose.
So if the human ever moves to (2,x) or (x,2) where x>2, they lose because the alien can move to (2,2).
If the alien moves to (2,x) or (x,2) where x>3, the human has no move that does not go to one of the previous losing positions. The alien always has such a move unless the human moves to (3,x) or (x,3).
But then the alien can move to (3,3), where the human has no move that does not lose.
So the alien always wins.
The human ambassador should prepare for war, since he will lose this game.
This reasoning is mostly due to user12408's answer, please give credit to them.
First, some observations. When I say a position is losing (winning), I mean that if either player starts there, they will lose (win).
(0,1) and (1,0) are losing. Everything else has one of these as an option, so is winning.
the alien wins. From (2,2), he wins by moving to (1,0). From anything else, he moves to (2,2).
the human wins starting from (3,2) and (2,3) by moving to (1,0) or (0,1), respectively.
the human loses starting from all other squares whose smallest coordinate is 2. If he moves to $(2,y)$ or $(x,2)$, the alien wins [see first bullet in this section]. Else, he must move to somewhere discussed in the first section, which can't be (0,1) or (1,0), so he loses.
the alien loses starting from (3,3). The choices (3,0), (3,1), (0,3), (1,3), (2,1) and (1,2) are bad [see first section]. His other options, (2,3) and (3,2), don't work either [see second section].
otherwise the alien wins. When $x>3$, the winning move is $(x,2)$. When $y>3$, it is $(2,y)$.
We have shown the alien wins starting anywhere, except for the "bad" places (0,1), (1,0) and (3,3). Thus, the humans can only win if they can move to one of these places, showing that (170000,170002) is a loss for the humans if they start there. It is also a loss for the humans if the aliens start there, since it is not one of the three "bad" places. Thus, humanity is doomed by their lack of mathematical rigor.
Not the answer you're looking for? Browse other questions tagged mathematics game nim or ask your own question.
Eat sweets and start your own business! | CommonCrawl |
Is every soft sheaf of countable $\mathbb Q$-vector spaces a direct sum of skyscraper sheaves?
Let $X$ be a finite-dimensional compact metrizable space (these properties might partially be irrelevant; on the other hand, the case $X=[0,1]$ is already interesting to me).
Let $\mathcal F$ be a soft sheaf of $\mathbb Q$-vector spaces on the topology $\mathbb O(X)$ (softness means that restriction of global sections to closed subsets is surjective).
Assume that the dimension of $\mathcal F(X)$ is countable. Question: Is $\mathcal F$ necessarily a direct sum of skyscraper sheaves? If $\mathcal F(X)$ is finite-dimensional then this can be proved using induction on the dimension, I think.
Motivation: I have proved a classification result for certain continuous fields of $C^*$-algebras over $X$. The invariant takes values in soft sheaves (actually in flabby cosheaves, but that is "the same"). Now I am wondering about the range of the invariant.
Let $Y\subseteq X$ be closed and totally disconnected, such as the Cantor set in $[0,1]$. Set $\mathcal F(U)=C(U\cap Y,\mathbb Q)$ -- locally constant functions on $U\cap Y$ with values in (discrete) $\mathbb Q$.
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry gn.general-topology sheaf-theory or ask your own question.
Tensor product of sheaves separated?
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
How to characterize flasque sheaves in more functorial way?
Is the sheaf associated to a differential structure of a specific type?
How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component? | CommonCrawl |
Hannenhalli and Pevzner gave the first polynomial-time algorithm for computing the inversion distance between two signed permutations, as part of the larger task of determining the shortest sequence of inversions needed to transform one permutation into the other. Their algorithm (restricted to distance calculation) proceeds in two stages: in the first stage, the overlap graph induced by the permutation is decomposed into connected components, then in the second stage certain graph structures (hurdles and others) are identified. Berman and Hannenhalli avoided the explicit computation of the overlap graph and gave an $O(n\alpha (n))$ algorithm, based on a Union-Find structure, to find its connected components, where $\alpha$ is the inverse Ackerman function. Since for all practical purposes $\alpha(n)$ is a constant no larger than four, this algorithm has been the fastest practical algorithm to date. In this paper, we present a new linear-time algorithm for computing the connected components, which is more efficient than that of Berman and Hannenhalli in both theory and practice. Our algorithm uses only a stack and is very easy to implement. We give the results of computational experiments over a large range of permutation pairs produced through simulated evolution; our experiments show a speed-up by a factor of 2 to 5 in the computation of the connected components and by a factor of 1.3 to 2 in the overall distance computation.
D.A. Bader, B.M.E. Moret, and M. Yan, ``A Linear-Time Algorithm for Computing Inversion Distance Between Signed Permutations with an Experimental Study,'' Journal of Computational Biology, 8(5):483-491, 2001.
D.A. Bader, B.M.E. Moret, M. Yan, ``A Linear-Time Algorithm for Computing Inversion Distance Between Signed Permutations with an Experimental Study,'' presented at the Seventh International Workshop on Algorithms and Data Structures (WADS2001), F. Dehne, J.-R. Sack, and R. Tamassia (eds.), Springer-Verlag LNCS 2125, 365-376, Brown University, Providence, RI, August 2001. | CommonCrawl |
A financial intermediary is an institution that facilitates the flow of funds between individuals or other economic entities.
A financial intermediary is an institution that facilitates the flow of funds between individuals or other economic entities having a surplus of funds (savers) to those running a deficit of funds (borrowers). Banks are a classic example of financial institutions.
Banks provide a safe and accessible environment for individuals and economic entities to deposit excess funds Additionally, banks also provide a service by packaging deposits into loans that are made available to economic agents (individuals and entities) in need of funds.
Banks are the most common financial intermediaries: Banks convert deposits to loans and thereby increase access to capital by serving as a financial intermediary between savers and borrowers.
Though, perhaps the most well-known of financial intermediaries, banks represent only one intermediary within a larger group. Other financial intermediaries include: credit unions, private equity, venture capital funds, leasing companies, insurance and pension funds, and micro-credit providers.
As noted, financial intermediaries provide access to capital. However, in conjunction with increasing access to funds, through their ability to aggregate funds, intermediaries also reduce the transaction and search costs between lenders and borrowers.
By repurposing funds from savers to borrowers financial intermediaries are able to promote economic growth by providing access to capital. Through diversification of loan risk, financial intermediaries are able to mitigate risk through pooling of a variety of risk profiles and through creating loans of varying lengths from investor monies or demand deposits, these intermediaries are able to convert short-term liabilities to assets of varying maturities.
Returning to the example of a bank used above, banks convert short-term liabilities (demand deposits) into long-term assets by providing loans; thereby transforming maturities. Additionally, through diversified lending practices, banks are able to lend monies to high-risk entities and by pooling with low-risk loans are able to gain in yield while implementing risk management.
Savings are income after-consumption and investment is what is facilitated by saving.
A higher real interest rates increases returns to saving.
Poor expectations for future economic growth, increase households' savings as a precaution.
More disposable income after fixed expenditures (such as mortgage, heating bill, basic goods purchases) have been made increases saving.
Perceived likelihood of reduced return through regulation or taxation on savings will make saving less attractive.
The factors as stated affect the marginal propensity to save (MPS), the percentage of after-tax income that an economic agent will choose to save. The greater the MPS, the more saving households will do as a proportion of each additional increment of income. Stocks and bonds are considered to be important intermediary forms of savings as these get transformed into a capital investment that produces value.
Bonds are a type of savings: Savings are used to fund investments, where investments are defined as expenditures on factory plants, equipment and homes.
Assuming a closed economy, one where there is no export or impart activity to interfere with the domestic savings level, on an aggregate basis individual savings creates the supply of loanable funds available for investment purposes. The amount of savings available in the economy is equal to the amount of funding available for investment activity. The higher the level of savings, typically the lower the relative interest rate, ceteris paribus. On a macroeconomic theory basis, a higher the savings rate promotes business activity my lessening the cost of money and increasing risk taking activities to facilitate growth or production of goods and services.
Financial intermediaries can assist with increasing the incentive to save through developing financial products that offer ease of liquidation but provide a higher return than a savings account. In this manner, financial intermediaries are a significant component to the transformation of savings into investment. Mutual funds, pension obligations, insurance annuities, and other forms of savings marketed by financial intermediaries all consist of stocks, bonds, and cash balances, which in turn pay for the investment capital that increases productivity, efficiency and output of goods and services.
The loanable funds market is a conceptual market where savers (suppliers) and borrowers (demanders) are able to establish a market clearing.
Summarize the mechanics of the loanable funds market.
In economics, the loanable funds market is a conceptual market where savers (suppliers) and borrowers (demanders) are able to establish a market clearing quantity and price (interest rate). In the loanable funds market, market clearing is defined as the interest rate/loanable funds quantity where savings equal investment (the amount of capital needed for property, plant, and equipment based investments). Loanable funds are typically cash, but can also include other financial assets to serve as an intermediary.
Equilibrium in the loanable funds market: When the supply and demand for loanable funds are equal, savings is equal to investment and the loanable funds market is in equilibrium at the prevailing interest rate.
For instance, buying bonds will transfer savers' money to the institution issuing the bond, which can be a firm or government. In return, the borrower's (institution issuing the bond) demand for loanable funds is satisfied when the institution receives cash in exchange for the bond.
Loanable funds are often used to invest in new capital goods. Therefore, the demand and supply of capital is usually discussed in terms of the demand and supply of loanable funds.
The interest rate is the cost of borrowing or demanding loanable funds and is the amount of money paid for the use of a dollar for a year. The interest rate can also describe the rate of return from supplying or lending loanable funds.
As an example, consider this: a firm that borrows $10,000 in funds for one year, at an annual interest rate of 10%, will have to pay the lender $11,000 at the end of the year. This amount includes the original $10,000 borrowed plus $1,000 in interest; in mathematical terms, this can be written as \( \$10,000 \times 1.10 = \$ 11,000\).
Financial intermediaries provide access to capital.
Banks convert short-term liabilities ( demand deposits ) into long-term assets by providing loans; thereby transforming maturities.
Through diversification of loan risk, financial intermediaries are able to mitigate risk through pooling of a variety of risk profiles.
The marginal propensity to save (MPS), the percentage of after-tax income that an economic agent will choose to save.
Savings marketed by financial intermediaries, all consist of stocks, bonds, and cash balances, which in turn pay for the investment capital that increases productivity, efficiency and output of goods and services.
Financial intermediaries are a significant component to the transformation of savings into investment.
In the loanable funds market, market clearing is defined as the interest rate /loanable funds quantity where savings equal investment (the amount of capital needed for property, plant, and equipment based investments).
The interest rate is the cost of borrowing or demanding loanable funds and is the amount of money paid for the use of a dollar for a year.
financial intermediary: A financial institution that connects surplus and deficit agents.
real interest rates: The rate of interest an investor expects to receive after allowing for inflation.
loanable funds: Money available to be issued as debt. | CommonCrawl |
We study principal bundles for strict Lie $n$-groups over simplicial manifolds. Given a Lie group $G$, one can construct a principal $G$-bundle on a manifold $M$ by taking a cover $U$ of $M$, specifying a transition cocycle, and then quotienting $U\times G$ by the equivalence relation generated by the cocycle. We demonstrate the existence of an analogous construction for arbitrary strict Lie $n$-groups. As an application, we show how our construction leads to a simple finite dimensional model of the Lie 2-group String($n$). | CommonCrawl |
Abstract: We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as $ t \to \infty $, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the $ L^2$-norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model. | CommonCrawl |
You are given an array of $n$ integers. You want to modify the array so that it is increasing, i.e., every element is at least as large as the previous element.
On each turn, you may increase the value of any element by one. What is the minimum number of turns?
The first input line contains an integer $n$: the size of the array.
Then, the second line contains $n$ integers $x_1,x_2,\ldots,x_n$: the contents of the array.
Print the minimum number of turns. | CommonCrawl |
Solubility of beryllium in pseudobinary aluminum-magnesium(2)silicide.
The Al-Mg-Si system is the basis of a major class of heat treatable alloys used for both wrought and cast products. Several heat treatable alloys exhibit age hardening at room temperature after a solution treatment (natural aging), and this property is exploited to develop the mechanical properties of the alloy. The active precipitate in the natural age hardening Al-Mg-Si alloys is Mg$\sb2$Si, but the precipitation is sluggish and its acceleration would be beneficial. In a previous investigation, it was shown that Be microadditions to an Al-0.75%Mg-0.50%Si alloy significantly enhances the age hardening response of the alloy, which is associated with a refinement of the Mg$\sb2$Si precipitate. A study of the precipitation kinetics showed this is due to a Be-enhanced nucleation rate. It can be shown that the nucleation rate increases with the nucleation entropy of the precipitating compound (Mg$\sb2$Si), which can be effected by the concentration of the Be in the precipitate compound, or by restricting solubilities of the precipitate's components in the Al solid solution. This study was undertaken to determine the extent to which Be can be incorporated into Mg$\sb2$Si, and its effects on the solubility of Mg and Si in Al($\alpha$). Direct spectroscopic analysis of Be using EDS is not possible due to the low atomic number of Be, and consequently the Be content must be inferred from the combined results using several techniques. In this study, the solubility determination was made using energy dispersive spectroscopy (for Al, Mg, Si), wavelength dispersive spectroscopy (for Be), X-ray diffraction, and optical microscopy coupled with microhardness. From the X-ray diffraction results, it is estimated that Be decreases the solubility of Mg$\sb2$Si in Al($\alpha$) up to about 0.3 at.% at 550$\sp\circ$C, and up to about 4.8 at.% Be may be incorporated into Mg$\sb2$Si compound. Source: Masters Abstracts International, Volume: 32-02, page: 0699. Adviser: W. V. Youdelis. Thesis (M.A.Sc.)--University of Windsor (Canada), 1992.
Hatab, Ali., "Solubility of beryllium in pseudobinary aluminum-magnesium(2)silicide." (1992). Electronic Theses and Dissertations. 3512. | CommonCrawl |
Abstract: A polynomial $f(x_1,\ldots,x_n)$ is said to be an identity for $m \times m$ matrices if $f(M_1,\ldots,M_n) = 0$ for all choices of $m \times m$ matrices for $M_i$s.
In the problem of partial function extension, we are given a partial function consisting of a set of $n$ points in a domain and a function value at each point.
Data assimilation refers to the problem of estimation of state of a high dimensional chaotic system given noisy, partial observations of the system. | CommonCrawl |
One part of making the solver faster were low-level optimizations in the unit-propagation implementation. I won't go into the details of that, but you can look at the new code. The bigger part were several heuristics that play a big role in making CDCL perform well. Below I will explain each implemented in varisat so far.
This is the first post in a series of posts I plan to write while refactoring my SAT solver varisat. In the progress of developing varisat into a SAT solver that can compete with some well known SAT solvers like minisat or glucose, it accumulated quite a bit of technical debt. Varisat is the first larger project I've written in rust and there a lot of things I'd do differently now. Before I can turn varisat into a solver that competes with the fastest solvers out there, I need to do some refactoring.
I'm working on a problem where I want to use a SAT solver to check that a property $P(v_1, \ldots, v_n)$ holds for a bunch of vectors $v_1, \ldots, v_n$, but I don't care about the basis choice. In other words I want to check whether an arbitrary invertible linear transform $T$ exists so that the transformed vectors have a certain property, i.e. $P(T(v_1), \ldots, T(v_n))$. I solved this by finding an encoding for constraining the rank of a matrix. With that I can simply encode $P(M v_1, \ldots, M v_2)$ where $M$ is a square matrix constrained to have full rank and which therefore is invertible.
I've released a new version of my SAT solver Varisat. It is now split across two crates: one for library usage and one for command line usage.
The major new features in this release concern the genration of unsatisfiability proofs. Varisat is now able to directly generate proofs in the LRAT format in addition to the DRAT format. The binary versions of both formats are supported too. Varisat is also able to do on the fly proof trimming now. This is similar to running DRAT-trim but processes the proof while the solver runs.
I've been interested in SAT solvers for quite some time. These are programs that take a boolean formula and either find a variable assignment that makes the formula true or find a proof that this is impossible. As many difficult problems can be rephrased as the satisfiability of a suitable boolean formula, SAT solvers are incredibly versatile und useful. I've recently finished and now released a first version of my SAT solver, Varisat, on crates.io.
Earlier this year, in October, a new widespread cryptography vulnerability was announced. The initial announcement didn't contain details about the vulnerability or much details on how to attack it (updated by now). It did state the affected systems though: RSA keys generated using smartcards and similar devices that use Infineon's RSALib. The announcement came with obfuscated code that would check whether a public key is affected. Also, the name chosen by the researchers was a small hint on how to attack it: "Return of Coppersmith's Attack".
I decided to try and figure out the details before the conference paper describing them would be released. By the time the paper was released, I had reverse engineered the vulnerability and implemented my own attack, which did not use Coppersmith's method at all. This post explains how I figured out what's wrong with the affected RSA-keys and how I used that information to factor affected 512-bit RSA-keys. | CommonCrawl |
Is there any superstable configuration in the game of life?
This question spins off of Gil Kalai's recent question on Conway's game of life for a random initial configuration.
There are numerous configurations in the game of life that are known to be stable---such as blocks, beehives, blinkers and toads---in the sense that if they appear on an otherwise empty board or on part of the board that remains otherwise empty, then they will persevere (or at least reappear on some period) into the indefinite future. All of the common examples of such configurations, however, seem to disintegrate when placed into a hostile environment; when they are hit by a glider or other spaceship, for example, these common stable configurations can be completely ruined.
My question is whether there is any superstable configuration, which can survive even in any hostile environment.
Question 1. Is there any superstable configuration in the game of life?
Specifically, let us define that a finite configuration is superstable, if it can survive in any environment, no matter how hostile, meaning that if it should ever appear on the board, then it will definitely reappear later in exactly that same position, regardless of what else is happening on the board. Perhaps the position is somehow isolated, absorbing whatever is happening around it; or perhaps it is a strong source of some kind, spewing out gliders or other objects, regardless of what else is around it; or perhaps it is some core surrounded by encircling vacuum-cleaners, traveling patterns that sweep up whatever might interfere.
This question is related to Gil Kalai's, in that if there are such superstable configurations, then we will expect that the infinite random position will have them with some (albeit very small) density, which will enable us to prove lower bounds on the density of the expected living infinite random position.
Question 2. Is there any superstable glider?
That is, is there a finite pattern that, regardless of the environment in which it is placed, will repeat itself at some future time with some displacement? A strong form of such a superstable glider would ask also that it be a vacuum cleaner, meaning that it glides around in any given environment while leaving only empty cells in its wake.
Question 2b. Is there a superstable glider vacuum?
I can imagine a small glider that erases everything in its path; or perhaps there is a kind of moving wall, which steadily pushes against whatever it faces, leaving emptiness behind. If there were such a superstable glider vacuum that also moved in a definite direction, then of course there could be no superstable stationary position, since otherwise we could vacuum it up.
Another alternative would seem to be that every finite configuration in the game of life is destructible, in the sense that one can design for it an especially hostile environment, leading to eventual death.
Question 3. Is every finite configuration destructible?
In other words, can every finite configuration in the game of life be extended to a larger configuration whose development leads in finite time to a position with no living cells? A weaker version of this would ask merely that the configuration be extended to a configuration such that eventually, the original configuration does not recur on any subportion of the board.
The existence of a "superstable configuration" is a long-standing open question in the Game of Life community. Years ago I saw Conway ask it as follows: Is there a finite $N$ and a configuration $C$ in an $N \times N$ square that contains some live cell $c$ and guarantees that $c$ will remain live for all time, regardless of what is placed outside the $N \times N$ square in the initial configuration? I think the expectation is that no such $C$ exists but a proof would be very difficult.
I think it could be of interest to mention that something meeting the definition of a finite superstable configuration exists if we add arbitrarily little noise to the original rules of the GoL (Conway's game of life for random initial position). The configuration is trivial – completely empty finite region. Via an argument in the spirit of the second Borel–Cantelli lemma, we see that "if it should ever appear on the board, then it will definitely reappear later in exactly that same position, regardless of what else is happening on the board". Sorry if this is off-topic.
Not the answer you're looking for? Browse other questions tagged co.combinatorics combinatorial-game-theory cellular-automata or ask your own question. | CommonCrawl |
I had a strange correlation test result between two variables ($y$ = residuals of a linear regression, $x$ = dependent variable).
Oh, ok, I have almost zero correlation!
And rho = -0.11. Cool! It seems that $y_0$ and $x_0$ are correlated in some way.
How can I accept null hypothesis if rho ($\rho$) is nearly -1?
That does not make any sense to me.
You have your interpretation of the p-values exactly backwards. Small p-values (<0.05, generally) mean that you should reject the null hypothesis, meaning your correlation is significantly different from 0. The p-value is telling you that it's very unlikely that you'd observe this much correlation if the null hypothesis (no correlation) was true. Large p-values (>0.05) indicate that you don't have sufficient evidence to reject the null hypothesis, meaning that you cannot conclude that you have non-zero correlation.
Your first case with near-zero correlation has a p-value of 0.66, meaning you cannot reject the null hypothesis of 0 correlation. Your second case has a significant p-value (close to 0), meaning you should reject the null hypothesis of no correlation, and accept the alternative hypothesis that the correlation is not equal to 0.
Not the answer you're looking for? Browse other questions tagged hypothesis-testing correlation spearman-rho or ask your own question.
Interpreting the Spearman's Rank Correlation Coefficient output in R. What is 'S'?
How to compare of two Spearman correlations matrices?
How to interpret Wilcoxon test for small difference in location? | CommonCrawl |
Volume 32, Number 1B (2004), 939-995.
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $Dsubsetneqq\C$ is equal to the radial SLE$_2$ path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that $\p D$ is a $C^1$-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A\subset\p D$, is the chordal SLE$_8$ path in $\overline D$ joining the endpoints of A. A by-product of this result is that SLE$_8$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
Ann. Probab., Volume 32, Number 1B (2004), 939-995.
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Let's import the numpy module.
The slowest run took 34.86 times longer than the fastest. This could mean that an intermediate result is being cached.
Matrix multiplication is np.dot(A, B) for two 2D arrays.
The .shape attribute contains the dimensionality array as a tuple. So the tuple (5,4,3) means that we're dealing with a three-dimensional array of size $5 \times 4 \times 3$. | CommonCrawl |
20 Why convolution regularize functions?
17 Suppose that there exist a set $\Gamma$ of positive measure such that $\nabla u=0, a.e.\ x\in\Gamma$.
16 Do diffeomorphisms act transitively on a manifold?
16 Is there no norm in $C^\infty ([a,b])$?
16 How bad can the second derivative of a convex function be? | CommonCrawl |
One special type of real-valued functions that are of interested to study are known as increasing and decreasing (collectively, monotonic) functions which we define below.
Definition: A function $f$ is said to be an Increasing Function on the interval $[a, b]$ if for all $x, y \in [a, b]$ where $x < y$ we have that $f(x) \leq f(y)$. $f$ is said to be a Decreasing Function on the interval $[a, b]$ if for all $x, y \in [a, b]$ where $x < y$ we have that $f(x) \geq f(y)$. $f$ is said to be a Monotonic Function on the interval $[a, b]$ if $f$ either either increasing or decreasing on $[a, b]$.
Furthermore, $f$ is an increasing function on any interval contained in $[0, \infty)$. | CommonCrawl |
We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to Gaussian and $RCD(K, \infty)$ spaces. As a consequence, we obtain quantitative stability for regular Lagrangian flows in Gaussian settings. | CommonCrawl |
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