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The embodiment of the invention discloses a rate sampling method and device, and rate sampling equipment, and the method comprises the steps: determining the number x of equally divided pulse signalsthrough the total number m of pulse signals transmitted by an encoder through each path S and the number n of equally divided pulse signals transmitted by the encoder through each path S; acquiring astarting time point T0 when the encoder sends a starting pulse signal, and sequentially acquiring an equant time point Ti of each x pulse signals from the starting pulse signal sent by the starting time point, so as to calculate the time difference of the equant pulse signals at an interval of n; and calculating a rate sampling value according to the distance S and the time difference. The embodiment of the invention has higher sampling efficiency and higher refreshing frequency. | |
Research into grassland ecology started at the Northeast Normal University (NENU) in the 1950s when Tingcheng Zhu, founder of NENU’s Institute of Grassland Science (IGS), surveyed the vegetation of marsh areas in northeastern China. Since then, ecological research programmes have developed quickly at NENU. Through cross-disciplinary integration and international exchange, IGS has emerged as a key contributor to diverse research, from species interactions in ecological communities, the impacts of livestock grazing, to climate change.
Rethinking interactions
Grassland studies by IGS are linked with research on the interactions between species, the focus of an IGS professor, Deli Wang. Specializing in ecosystem ecology, Wang’s team explores how human activities (such as grazing, mowing, and nitrogen deposition) affect interactions between species of different trophic levels in a food web. It also considers their impacts on biotic communities, including biodiversity, and ecosystem processes and functions, such as productivity, and carbon and nitrogen cycles.
A highlight of Wang’s work is finding that livestock grazing can change habitat structure, regulating the trophic interactions between predators and prey in the arthropod food web, and between plant-eating insects and plants. Livestock can also have inter-species facilitation with their co-existing insects, contributing to plant biodiversity and productivity.
Grazing and grassland ecology
Livestock overgrazing is widely considered a threat to grassland ecosystems, as selective grazing and trampling may harm biodiversity and damage plants. However, when managed well, research shows, selective grazing can also bring conservation benefits.
By investigating the effects of livestock grazing on grassland ecological structures, processes and functions, along with the regulatory mechanisms, an IGS team led by Ling Wanghas been exploring ways to optimize grazing management, preserving multiple ecosystem functions.
Based on manipulative field experiments to compare grazing variables at northeast China’s Songnen Plain, Wang’s team found that the way to do mixed grazing of different livestock species, such as cattle and sheep, plays a significant role in grazing effects on plant diversity, soil health (including nitrogen cycling, heterogeneity, and carbon flux), and soil’s capacity in storing carbon.
The researchers found that mixed grazing of cattle, sheep, and other livestock can increase grassland biodiversity and versatility. Preserving the biodiversity of insects and plants, as well as underground micro-organisms and nematodes is important for maintaining the multiple functions of grassland ecosystems affected by grazing.
Studying climate factors
Investigating how grassland responds to global changes, particularly, to changed rainfall patterns and their interactions with other global change factors (GCF) is the focus of an IGS team led by Wei Sun, a specialist in plant eco-physiology. “The exploration will provide insights to adaptive management of grassland,” said Sun.
Sun’s analysis of long-term observational data and field experiment results suggests that rainfall patterns are a controlling factor in grassland ecosystems, as they determine the effects of other GCFs, including temperature increase and nitrogen deposition. After prolonged adaptation to interannual changes in rainfall, meadows can develop resistance to short-term drought. Nitrogen deposition can also affect key vegetation features, including the diversity of plant functional groups, root-to-shoot ratio, and leaf area index, which determine how sensitive ecosystem carbon cycles are in response to changes in precipitation patterns. | https://www.nature.com/articles/d42473-021-00060-3?error=cookies_not_supported&code=490e4c92-d9c5-4e09-8c0b-27e04035543f |
Q:
Space of linear and bounded operators is Banach
Show that if $(X,\|\cdot\|_X)$ is a normed space and $(Y,\|\cdot\|_Y)$ is a Banach space then $(B(X,Y),\|\cdot\|_{op})$ is a Banach space.
$B(X,Y) = \{T\colon X\to Y : T$ is bounded and linear$\}$
What I did :
Let $\|x\|_X \le 1$ and $\|T\|_{op} = \sup\{\|T(x)\|_Y\ : \|x\|_X \le 1\}$ and $\sum_{n=1}^\infty \|T_n\|_{op}$ is convergent i.e. $\sum_{n=1}^\infty T_n$ is absolute convergent series in $B(X,Y)$
We have $\|T_n(x)\|_Y \le \|T_n\|_{op}$ from definition
Since $\sum_{n=1}^\infty \|T_n\|_{op}$ is convergent $\sum_{n=1}^\infty \|T_n(x)\|_Y$ is convergent too.
Since $(Y,\|\cdot \|_Y)$ is a Banach space $\sum_{n=1}^\infty T_n(x)$ is convergent.
Let $S_k= \sum_{n=1}^k T_n(x)$
$\exists y \in Y$ such that $S_k \to y$
I've stuck in that point. How must I pursue? I think there are some mistakes or unnecessary things. Could you please fix me? How can I show $\sum_{n=1}^\infty T_n$ is convergent in $B(X,Y)$?
Thanks
A:
We have established that $\sum_{n=1}^\infty T_nx$ converges for all $x \in X$. Define a linear map $T : X \to Y$ with $$Tx = \sum_{n=1}^\infty T_nx$$
For $N \in \mathbb{N}$ and $x \in X$ we have
$$\left\|Tx - \sum_{n=1}^N T_nx\right\| = \left\|\sum_{n=N+1}^\infty T_nx\right\| \le \sum_{n=N+1}^\infty \|T_nx\| \le \left(\sum_{n=N+1}^\infty \|T_n\|\right)\|x\|$$
taking the supremum over $\|x\| = 1$ we get
$$\left\|T - \sum_{n=1}^N T_n\right\| \le \sum_{n=N+1}^\infty \|T_n\| \xrightarrow{N\to\infty} 0$$
so $\sum_{n=1}^\infty T_n = T$ with respect to the operator norm.
Furthermore, let $N \in \mathbb{N}$ such that $\left\|T - \sum_{n=1}^N T_n\right\| \le 1$. We have
$$\|T\| \le \left\|T - \sum_{n=1}^N T_n\right\| + \left\|\sum_{n=1}^N T_n\right\| \le 1 + \sum_{n=1}^N \|T_n\| < +\infty$$
so $T$ is bounded.
We conclude that $\sum_{n=1}^\infty T_n$ converges to an element of $B(X,Y)$ so $B(X,Y)$ is a Banach space.
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Q:
How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
I was asked to prove an inequality:
For any $n$ positive numbers $\{a_i\}$ with $a_{1}a_{2}\cdots a_{n} = 1$ and $m \geq n-1$ be a non-negative integer,
$a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
After several attempts, I think I had to use induction on both $m$ and $n$. In particular, I fix an $n$ and claim that the inequality holds for every $m$, and use induction on $n$.
The case where $n = 2$ could be done by rewriting the inequality as $a_1^2 + a_2^2 \geq a_1 + a_2$ and assume that $a_1 \leq 1 \leq a_2$.
In the inductive step, given $a_1^k + a_2^k \geq a_1 + a_2$ , by writing $a_1^{k+1} + a_2^{k+1} = (a_1^{k} + a_2^{k})(a_1 + a_2) - a_1^{k}a_2^{k} (a_1 + a_2)$, the result follows.
However, this method seems to fail for $n>2$, as the right-hand side of the inequality becomes more complicated. Should I try not to prove it by induction?
A:
We will prove that
$$a_1^m + a_2^m + \cdots + a_n^m \ge a_1^{n-1} + a_2^{n-1} + \cdots + a_n^{n-1} \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}.$$
Left-hand side inequality:
Obviously we have $(x^{n-1}-1)(x^{m-n+1}-1) \ge 0$ for any $x>0$ (note that $m\ge n-1$). Expanding we get
$$x^m\ge x^{n-1} + (x^{m-n+1}-1).$$
Applying this inequality for $x=a_1,\ldots,a_n$ and taking the sum we get
$$\sum_{i=1}^n a_i^m \ge \sum_{i=1}^n a_i^{n-1} + \left(\sum_{i=1}^n a_i^{m-n+1}- n \right).$$
But by AM-GM inequality
$$\sum_{i=1}^n a_i^{m-n+1} \ge n\times (a_1a_2\cdots a_n)^{(m-n+1)/n} = n.$$
Thus we have
$$\sum_{i=1}^n a_i^m \ge \sum_{i=1}^n a_i^{n-1}.$$
Right-hand side inequality:
Denote $S=a_1^{n-1} + a_2^{n-1} + \cdots + a_n^{n-1}$. Using AM-GM inequality for $n-1$ numbers we have
$$S-a_1^{n-1} = a_2^{n-1} + \cdots + a_n^{n-1} \ge (n-1)a_2\cdots a_n = \frac{n-1}{a_1},$$
i.e.
$$S-a_1^{n-1} \ge \frac{n-1}{a_1}.$$
Similarly, we have the other inequalities for $a_2,\ldots,a_n$. Taking the sum of these $n$ inequalities we get
$$nS-(a_1^{n-1} + a_2^{n-1} + \cdots + a_n^{n-1}) \ge (n-1)\left(\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}\right).$$
The left-hand side of the last inequality is just $(n-1)S$. Divided both sides by $(n-1)$ we get the desired inequality.
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Q:
A function for calculating velocity at several distances as you fall towards the earth's center
Assuming there is no drag, no friction and no other objects affecting you. If you drop into the earth (through a tube). Your velocity will be 7900 m/s at the center of the earth according to http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html
I assume that was the summation of all the velocity gained as gravitational force changes/decreases such as v= (9.8* first nano-meter traveled) + (9.7...*next nano-meter traveled)+...+(0*last nano-meters traveled)
I am sure there is a way to write a function using calculus (integration) that calculates your velocity at every point as you get closer. Lets approximate the earth's radius is 6,371 km and the acceleration due to gravity on the earth's surface is 9.807 m/s^2.
I will like a function that calculates your velocity at every given distance from the earth center as you fall.
A:
This is a surprisingly simple thing to calculate.
It is a well known result that a consequence of the inverse square law is that there is no force inside a symmetrical hollow shell. This means that as the object falls into the hole, it will appear to be attracted by a sphere of decreasing radius - the mass outside "doesn't count."
The acceleration of gravity at the surface of a sphere of radius R (assuming uniform density $\rho$) is given by
$$\begin{align}
a &= \frac{GM}{R^2} \\
&= \frac{4G\pi R^3\rho}{3R^2} \\
&= \frac43\pi \rho G R\\
\end{align}$$
Where $G$ is the gravitational constant, and $R$ is the distance to the center of the earth. In other words - the acceleration is proportional to the distance to the center. The corollary is that an object dropped into a hole through the center of the earth will exhibit simple harmonic motion.
Let's do the math in more detail. Put the distance from the center as $r$, then the acceleration $\frac{d^2r}{dt^2}$ is given by
$$\frac{d^2r}{dt^2}=-\frac43 \pi \rho G r$$
(since the acceleration is pointing towards the center). This looks like the differential equation for simple harmonic motion:
$$\frac{d^2x}{dt^2} = -\omega^2 x$$
for which the solution (if velocity is zero at t=0, and amplitude is $x_0$) is
$$x(t) = x_0 \cos\omega t$$
and the velocity is
$$v(t) = -\omega x_0 \sin\omega t$$
It follows that we can write the expression for the velocity as a function of position by eliminating time:
$$\begin{align}\\
v(x) &= -\omega x_0 \sin \cos^{-1}\left(\frac{x}{x_0}\right)\\
&= -\omega x_0\sqrt{1-\left(\frac{x}{x_0}\right)^2}\end{align}$$
If we now substitute $\omega^2 = \frac43 \pi \rho G = \frac{g}{R}$ where $g$ is the gravitational acceleration at the surface of the earth, we get
$$\begin{align}\\
v(r) &= -\sqrt{\frac{g}{R}} R \sqrt{1-\left(\frac{r}{R}\right)^2}\\
&=\sqrt{\frac{g}{R}\left(R^2-r^2\right)}
\end{align}$$
If we substitute $r=R$, we get $v=0$ as expected; and when we put in $r=0$, we get $v = \sqrt{gR}$ = 7904 m/s using your values for $R$ and $g$. I think that's pretty close...
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Customer Analysis The total industrial consumption of cyano-acrylates which the new Bond-A-Matic 2000 would dispense was 265,000 pounds in FY 1978, expected to grow to about 335,000 pounds in FY 1979. Across 16 SIC categories, approximately 174,909 firms currently used cyano-acrylates (at a 15.5% market penetration.) 11% of CA users, i.e. approximately 19,240 firms used over 10 pounds of CAs per year, comprising at-least 75% of total current market. Assuming that growth in the CA segment stagnates, and that only heavy CA applicators would be interested in dispensing equipment the total market is still estimated at 19,240 users.
Question 1. a) How many parts should you purchase each time you place an order? R=800- ORDERING COST Annual carrying cost of capital= 20% .20 lost per dollar R= 50000 H= .80 r= .20 cost per dollar C= unit cost H=rC Qo= √2RS/H= √(2(50,000)(800)/.8) = 10,000 10,000 parts should be purchased each time an order is placed. (b) To satisfy annual demand, how many times per year will you place orders for this part? R/Qo= 50,000/10,000= 5 times per year Question 2: (a) Determine BIM’s total annual cost of production and inventory control. Q= 4 weeks supply = 1600 units R= 400 units a week= 20000 units/ year C= purchase cost per unit= $1250 X (1-.20)= 1,000 H= holding cost= rC= $200 per unit / year S= setup cost= 2000 + 93.75 = $2,093.75 Setups per year= R/Q= 20000/1600= 12.5 Annual setup cost= (R/Q)(S)=12.5X $2,093.75= $26,172 Annual Holding cost= (q/2)(H)= (1600 /2)X $200= $160,000 Total Annual Cost= Annual Setup Cost+ Annual Holding Cost Total Annual Cost= 26,172+160,000 BIM’S Total Annual Cost= $186,172 (b) Compute the economic batch size and the resulting cost savings.
At 10 cents each, the expected revenue of $500 per day, and the amount will be lost while the copier is broken. The standard number of breakdowns per year to be 12.255 (dividing 52 weeks/year by 4.243 weeks/breakdown), the profit lost per breakdown to be $1,125 (days lost/breakdown * revenue lost/day), and finally the profit lost per year due to breakdowns to be $13,787 (breakdowns/year * profit lost/breakdown). 6. The profits lost per year due to breakdowns is $13,787, which is bigger than $12,000, they should procure a backup copier. A high
According to the text, the exception amount for active owners is phased out as income increases: the $25,000 maximum exception amount is phased out by 50 cents for every dollar the taxpayer’s adjusted gross income (before considering the rental loss) exceeds $100,000. 120,000-100,000*.5= 10,000 phase out 25,000-10,000= 15,000 total deductible loss; Alexa can still deduct the $2,400 because her loss is less than the 15,000. New AGI 120,000-2,400= $117,600 d. Assume that Alexa’s AGI from other sources is $200,000. This consists of $150,000 salary, $10,000 of dividends, and $25,000 of
Capacity Next, we assume that Ryanair plans to operate at a higher capacity than BA, which will have a direct impact on the spread of fixed costs per unit. In the case, airport authorities claimed a volume of 500,000 passengers per year, which has remained stagnant over the last decade. Additionally, 750,000 round-trip travelers used rail and sea ferries rather than aircraft (which takes 9 hours to travel versus 1 hour for air travel). Together, this makes approximately 1.25 million round trip travelers per year on the DUB-LHR route. Ryanair intends to enter the market with a 44 seat turboprop.
How much does a worker make a month making 1.00 per hour (40 hour work weeks)? _____________________________________________________________ 2. How much would a 1920 Ford Touring cost per month if bought with an installment plan of 12 installments (one every month) in one year?_______ 3. What happened in the 1920s that greatly lowered the price of cars (automobiles) and other consumer
The project would shut down the production for 45 days while the renovations are implemented. The price of polypropylene is GBP675 per ton. The tax rate required for this analysis is 30%. New assets can be depreciated over 15 years on an accelerated basis. Due to the increased production the work-in-process inventory (WIP-inventory) would need to increase by 3.0% of cost of goods. | https://www.antiessays.com/free-essays/Break-Even-Point-Analysis-682926.html |
Q:
Optimization-like question
Let's say I have a formula like $ax + by + cz = N$. $a, b, c$, and $N$ are known and cannot be changed. $x, y$, and $z$ are known and can be changed.
The problem is that the equation is not true! My problem (for a program I'm writing) is: how can $x, y$, and $z$ be changed enough that they equal $N$ while differing from their previous values as little as possible?
A:
IN CASE YOU WANT THE VARIABLES TO BE INTEGERS:
This is called the "closest vector problem". Also the "nearest lattice point." Given one solution $X_0 = (x_0, y_0,z_0)$ to $ax+by+cz = N,$ you get a "lattice" by taking $X-X_0$ for every solution $X.$
You have a nonsolution, call it $Y = (x_1, y_1,z_1).$ You want the closest lattice point $X-X_0$ to $Y-X_0,$ so that $X-Y$ is shortest.
There is plenty of research on this, and probably some textbooks have good methods in dimensions 2 and 3 for the shortest vector problem and the related closest vector problem. https://en.wikipedia.org/wiki/Lattice_problem
EDIT: the lattice given is two dimensional. This means that finding a reduced basis is just Gauss reduction for (positive) a binary quadratic form. Once you have a reduced basis, it becomes straightforward to produce a finite list of lattice vectors that must contain the best solution, then check them.
A:
The solution space of the equation $ax+by+cz=N$ is a plane in three dimensions, thus for any given point, say, $(x_0,y_0,z_0)$, you may easily calculate the distance between this point and the plane of actual solutions (this is equivalent to the possibly-familiar "least-squares method", which I think is the answer you're after?)
A:
Let $x_0, y_0, z_0$ be the current values of $x, y, z$.
An equation of the form $ax + by + cz = N$ specifies a plane in space. Let your plane be $P$. So your problem is equivalent to the following one: in space, given a point $(x_0, y_0, z_0)$ and a plane $P$, find the point on $P$ that is closest to $(x_0, y_0, z_0)$.
The closest point will be the point you get by dropping a perpendicular line from $(x_0, y_0, z_0)$ to $P$ and seeing where it lands. (This must be the closest because a straight line is the shortest distance between any two points!)
What I just described is also called the orthogonal projection of a point onto a plane. It's solved e.g. here.
As charlestoncrabb mentions, this is equivalent to the method of least squares because we measure distance in space by $\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2}$. So that's the "default" meaning of "close to" in this context, oddly enough. If want to measure "close to" some other way, you can use more general optimization methods (like Lagrange multipliers).
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Conventional information retrieval systems (also known as text retrieval systems or text search engines) view document collections as standalone text corpora with little or no structured information associated with them. However, there are two primary reasons why such a view is no longer tenable. First, modern enterprise applications for customer relationship management, collaboration, technical support, etc., regularly create, manipulate, and process data that contains a mix of structured and unstructured information. In such applications, there is inherently a fair amount of structured information associated with every document. Second, advances in natural language processing techniques has led to the increased availability of powerful and accurate text analysis engines. These text analysis engines are capable of extracting structured semantic information from text. Such semantic information, usually extracted in the form of semantic annotations, has the potential to significantly improve the quality of free text search and retrieval.
However, the architectures of conventional information retrieval systems are not explicitly designed to take advantage of semantic annotations. In particular, semantic annotations provide the capability for describing content in terms of types and relationships, that is concepts that are not intrinsic to conventional information retrieval systems. For example, a particular document in a corpus may contain a person name “John” and a telephone number for John: “555-1234”, but not the actual word “telephone”. A person may search on that corpus using the keyword phrase “John telephone”. However, a conventional retrieval system does not find the document since the keyword “telephone” is not present. In essence, conventional information retrieval systems merely recognize keywords but not the types into which a word or phrase may be categorized or the relationships between such types.
A conventional information retrieval system is typically designed to return a ranked list of matching documents in response to a keyword search query comprising search words or tokens. In a standard implementation of such a system, an entire corpus of documents is processed in advance to build an inverted index. This inverted index maps each token to a list of occurrences of that token. A token is usually a word or a phrase; however, a token can also be a more complex entity.
Upon receiving a keyword query, the inverted index is used to compute a list of candidate documents that are potentially relevant to the query. Each of these candidate documents is assigned a rank, using a pre-designed ranking formula. The rank ordered list of candidate documents is then presented to the user. Although this technology has proven to be useful, it would be desirable to present additional improvements. The tightly integrated architecture of conventional information retrieval systems directly maps a query to storage and index structures. Consequently, it becomes difficult to exploit available semantic annotations. In conventional information retrieval systems, the available semantic annotations can only be exploited in an ad-hoc fashion by hand crafting specialized ranking formulae. Such ad-hoc ranking formulae are difficult to construct and are very often not portable across document collections. As a result, every time an information retrieval system is deployed over a new document collection, a significant amount of time and effort is required to craft a ranking formulae appropriate to that collection.
What is therefore needed is a system, a computer program product, and an associated method for exploiting semantic annotations in executing keyword queries over a collection of text documents, allowing a user to search on a corpus and locate information based on types and relationships found in the corpus by, for example, a text analysis engine. The need for such a solution has heretofore remained unsatisfied.
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Three Basic Economics Questions
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
***This is not a request for a written paper, but points to expound on.***
Write a 2-3 page thorough analysis of the three basic economic questions along with an analysis of the centralized command and control system, the price system, and the mixed economic system. Remember to cite your research and do not plagiarize.
https://brainmass.com/economics/applied-economics/three-basic-economics-questions-628105
Solution Preview
Step 1
The first basic economic question is "what to produce?". This is concerned with how scarce resources are allocated. This allocation question is answered by the economic system. The economic system allocates productive resources in the form of land, labor, and capital. Land includes natural resources, real estate, and raw materials. Labor determines the work that is done by the people and the their earnings in the form of pay. Capital includes physical capital by way of buildings, equipment, and tools. It also includes human resources, including the quality of education, and training on the job (a). In most economic systems the production of some items is determined by supply and demand, whereas some essential commodities such as food items are subsidized and controlled by the government.
In a centralized command and control system the question of what to produce is determined by the government. In a price system what is produced is determined by the market forces. In a mixed system, some aspects of what is produced is determined by the ...
Solution Summary
This posting gives you a step-by-step explanation of three basic economics questions. The response also contains the sources used. | https://brainmass.com/economics/applied-economics/three-basic-economics-questions-628105 |
Q:
Can you find the quadratic coefficient of quadratic equation?
How do you find the quadratic coefficient a, b, and c, given some values of x and y?
For example: Suppose we have
f(0.1524)=0.9961
f(0.8258)=0.0782
f(0.5383)=0.4427
and given the quadratic equation: $y = ax^2 + bx + c$, then can you find values of x and y?
Is there a formula to find these a, b, and c? when I think about the general formula, but in this case y=0. So, it does't work. When I think about substitute by the formula: $y = ax^2 + bx + c$, I cannot get go long; because we have three unknown in one equation. So, what you suggest as next step.
Thank you!
A:
If you are given $3$ points with distinct $x$ values, by substituting those values inside $y=ax^2+bx+c$, we can obtain $3$ linear equations in terms of $a, b$, and $c$.
Hence, the problem reduces to the problem of solving a linear system of equations, which can be solved, for example, by Gaussian elimination.
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Solution: Let the line graph obtained by joining the points (-2, 5) and (1, -4) be the graph of y = mx c.
So, the given pairs of values of (x, y) obey the relation y = mx c. (ii)Subtracting (ii) from (i), we get: 5 4 = -2m – m⟹ 9 = -3m⟹ -3m = 9⟹ m = \(\frac\)⟹ m = -3 Putting m = -3 in (ii), we have: -4 = -3 c ⟹ c = -1.
In this slope intercept calculator, we will focus only on the straight line, but those interested in knowing more about the parabolic function should not worry.
We have two special calculators dedicated to such an equation, namely the parabola calculator and the quadratic formula calculator.
Here we will learn how to solve different types of problems on slope and y-intercept.1.
(i) Determine the slope and y-intercept of the line 4x 7y 5 = 0Solution: Here, 4x 7y 5 = 0 ⟹ 7y = -4x – 5⟹ y = -\(\frac\)x - \(\frac\).There you can find a full description of these types of functions!Linear equations, or straight line equations, can be quickly recognized as they have no terms with exponents in them.Any line on a flat plane can be described mathematically as a relationship between the vertical (y-axis) and horizontal (x-axis) positions of each of the points that contribute to the line.This relation can be written as representing any real numbers) is the relationship of a straight line.Draw the graph of 3x - √3y = 2√3 using its slope and y-intercept. Solution: Here, 3x - √3y = 2√3 ⟹ - √3y = -3x 2√3⟹ √3y = 3x - 2√3y = √3x – 2Comparing with y = mx c, we find the slope m = √3 and y-intercept = -2. wiki How's Content Management Team carefully monitors the work from our editorial staff to ensure that each article meets our high quality standards. The slope intercept form calculator will teach you how to find the equation of a line from any two points that this line passes through.Solution: Remember, all you have to do to transform an equation into slope-intercept form is to solve it for y.To isolate the y, subtract x from both sides of the equation and then divide everything by the coefficient of - 4: © 2004 by W.
Comments How To Solve Slope And Y-Intercept Problems
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Linear Equations - CSUN.edu
Algebra - Linear Equations & Inequalities. T-37/H-37. Find the slope, m, of the line whose graph contains the. number b is called the y-intercept of the line.…
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How to Solve Slope Intercept Form With Two Points - Sciencing
Solving for Slope-Intercept Form From Two Points. Imagine that you've been. you have y = -2_x_ + b. You're almost there, but you still need to find the y-_intercept that _b represents. How to Convert Graphs to Equations.…
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Find the Equation of a Line Given That You Know Its Slope. | https://vizitt.ru/how-to-solve-slope-and-y-intercept-problems-525.html |
There Are Over 300 Undiscovered Meteorite Impact Craters on Earth, Geologists Claim
It is essentially common knowledge in this day and age that phenomena in space can play a significant role in shaping life on earth. Meteorites have perhaps played the most crucial role, as they have struck the earth many times over the course of the planet's existence. Most notably, an enormous collision of a meteorite with Earth at the end of the Cretaceous period is thought to have caused the extinction of the dinosaurs. Now, geologists are attempting to calculate just how many meteorite impact sites are present on the surface of the earth that may yet be undiscovered.
Geologists Dr. Stefan Hergarten and Dr. Thomas Kenkmann from the University of Freiburg have published the first study that estimates just how many meteorite craters there should be on Earth's surface. According to the results of a probability calculation, published in Earth and Planetary Science Letters, a total of 188 craters have been confirmed so far, and 340 more are still awaiting discovery.
This may seem like quite a lot, but compared to the 300,000-plus craters present on Mars, 340 is almost underwhelming. Although meteorites are not less likely to crash on Earth than on Mars, Earth's surface changes much more rapidly, so it is much more likely that a crater will be eroded to the point that it's undetectable. Hergarten told Science Daily, "The main challenge of the study was to estimate the long-term effect of erosion, which causes craters to disappear over time."
Aside from erosion, the other fact that determines a crater's lifespan, and therefore its probability of detection, is its size. Obviously, large craters are much easier to spot than smaller ones, but they are also much less likely to exist, as large impacts are much rarer than small impacts. "A surprising, initially sobering finding we made was that there are not many craters of above six kilometers in diameter left to discover on the Earth's surface," Hergarten stated. It is also much more difficult to find large craters, as they are more likely to be covered in deep sediment.
Although they may be difficult to find, the researchers estimate that 90 craters with a diameter of one to six kilometers and a further 250 with a diameter of 250 to 1000 meters are yet to be discovered by scientists. Large craters, in particular, will most likely be hidden from detection, but at least we now know what exactly is buried underneath Earth's changing surface. | https://www.outerplaces.com/science/item/9264-there-are-over-300-undiscovered-meteorite-impact-craters-on-earth-geologists-claim |
Essentials of understanding psychology/ Robert S Feldman.
By: Feldman, Robert S.Material type: BookPublisher: New York, NY : McGraw-Hill Education ; 2015Edition: Eleventh edition.Description: 1 volume (various pagings) ; color illustrations , 28 cm.ISBN: 9781259255786; 1259255786.Subject(s): PsychologyDDC classification: 150
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|149.97 NOE கோட்பாட்டு விமர்சன யுகம் விமர்சனக் கோட்பாட்டு யுகம்:||150 BUR Psychology,||150 CIC Psychology||150 FEL Essentials of understanding psychology/||150 RAT தூக்கம் ஒரு பிரச்சினையா?:||150.15195 BEL Statistics for psychology using R /||150.1950 SHA Psychoanalysis, culture, and religion : essays in honour of Sudhir Kakar|
1: Introduction to psychology 2: Neuroscience and behavior 3: Sensation and perception 4: States of consciousness 5: Learning 6: Memory 7: Thinking, language, and intelligence 8: Motivation and emotion 9: Development 10: Personality 11: Health psychology: stress, coping, and well-being 12: Psychological disorders 13: Treatment of psychological disorders 14: Social psychology
Your students are individuals. Do your teaching materials treat them that way? Using a revolutionary revision process, this is an integrated learning system that brings the "Students First" goal to a new level.
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---
abstract: 'The distance of a binary operation from being associative can be measured by its *associative spectrum*, an appropriate sequence of positive integers. Particular instances and general properties of associative spectra are studied.'
address:
- 'Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary'
- 'Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary'
author:
- Béla Csákány
- Tamás Waldhauser
title: Associative Spectra of Binary Operations
---
[^1]
Introduction
============
Let $n$ be a positive integer. We call a string consisting of symbols $x,(,$ and $)$ a *bracketing of size* $n$ if it contains $n$ symbols $x$, and $n-1$ symbols $($ (left parentheses) as well as $)$ (right parentheses) so that they are properly placed to determine a product of $n$ factors $x$ (see, *e.g.* \[1,15\]). More formally,
1\. $x$ is the unique bracketing of size 1,
2\. the bracketings of size $n$ are exactly the strings of form $(PQ)$ where $P$ and $Q$ are bracketings of size $k$ resp. $l$ with $k+l=n$.
*E.g.* $(xx)$ is the only bracketing of size 2, and $((x(xx))(xx))$ is a bracketing of size 5. Note that we always use an outermost pair of parentheses whenever $n>1$, in contrary to the everyday usage of parentheses. We shall denote bracketings by capital letters, and $|B|$ stands for the size of $B$.
Bracketings are, in fact, the elements of the free groupoid[^2] with one free generator $x$ (*cf.* \[1\], p. 133), or, equivalently, they are the unary groupoid terms. The corresponding unary term operations on special groupoids were investigated by several authors (see, *e.g.* \[5,7\]). In any bracketing of size $n$ we can indicate the position of symbols $x$ by subscripts $1,\ldots,n$, *e.g.* $(x_{1}x_{2}),((x_{1}(x_{2}x_{3}))(x_{4}x_{5}))$. Thus, a bracketing of size $n$ provides also an element of the free groupoid with free generators $x_{1},\ldots,x_{n}$, *i.e.*, an $n$-ary groupoid term (although, of course, not all $n$-ary groupoid terms originate from bracketings in such a way). Here we always study bracketings considered as $n$-ary groupoid terms, even if in some cases we omit the subscripts $1,\ldots,n$. On every groupoid $G$, these terms give rise to $n$-ary term operations. We call them *regular* $n$*-ary operations of* $G$ (or, *regular over the operation of* $G$), and, in concrete cases, *operations induced by given bracketings*. For notation of the regular operation induced by the bracketing $B,P,Q,$ *etc.* we use the corresponding lowercase letters $b,p,q,$ *etc*.
If $G$ is associative, then by the generalized associative law there is exactly one regular $n$-ary operation for each $n$. In the general case, we have a sequence $${s_{{}_{G}}}(1),{s_{{}_{G}}}(2),\ldots,{s_{{}_{G}}}(n),\ldots$$ of positive integers with $s_{{}_{G}}(n)$ denoting the number of distinct $n$-ary regular operations of $G$. *E.g.*, $s_{{}_{G}}(1)=s_{{}_{G}}(2)=1 $ for every groupoid $G$, and $s_{{}_{G}}(3)=2$ if and only if $G$ is nonassociative, as then the two possible bracketings of size 3, $(x_{1}(x_{2}x_{3}))$ and $((x_{1}x_{2})x_{3})$ induce different ternary term operations.
The sequence $$\{s_{{}_{G}}(n)\}=(s_{{}_{G}}(1),s_{{}_{G}}(2),\ldots,s_{{}_{G}}(n),\ldots)$$ measures, in some sense, the distance of $G$ from associativity: the smaller its entries are, the closer the operation of $G$ is to being associative. Hence we call this sequence *the associative spectrum of* $G$ (or, *of the operation of* $G$). Instead of $s_{{}_{G}}(n)$ we write $s(n)$ if this cannot cause misunderstanding. Usually we also omit $s(1)$ and $s(2)$, bearing no information about $G$.
In this paper we study the introduced notion from several points of view. The next section contains some well-known facts, simple observations, and auxiliary results on bracketings and associative spectra; there and later, the routine inductive proofs will often be omitted. Most frequently we use induction on size; we leave out the words on size in these cases. The third section contains samples of determining associative spectra of some familiar nonassociative operations. The problem of characterizing all associative spectra of operations on a set with a given power seems to be hard. However, the case of the two-element set is, as it might be expected, easy (Section 4), and a lot of three-element groupoids are accessible (Section 5). In the final section we present some facts on the general behavior of associative spectra, and formulate several problems.
Further on, we write simply *spectrum* for associative spectrum.
Properties of bracketings and spectra
=====================================
For any bracketing $B$ of size $n(>1)$, we can *pair* its left and right parentheses in a natural way (\[9,15\]). Induction shows that we can always choose a consecutive quadruple $(xx)$ in $B$; its left and right parentheses will be associated to form a pair. Replacing then $(xx)$ with $x$ we obtain a bracketing $B^{\prime}$ of size $n-1$, for which the preceding process can be repeated until no unpaired parentheses remain. This way of forming pairs involves that any pair together with the symbols between them is also a bracketing. It is called a *subbracketing of* $B$; *e.g.*, if $B=(PQ)$, then $P$ and $Q$ are subbracketings of $B$, as outermost parentheses of any bracketing are paired. We call $P$ and $Q$ the (*left* resp. *right*) *factors* of $B$. The symbols $x$ are considered as subbracketings of size 1, too. Observe that pairing is unique, and if a parenthesis lies between a pair then its associate also lies between them. Hence the representation of bracketings of size $>1$ in form $(PQ)$ is unique, too.
Substituting $x$ for one or several disjoint subbracketings in $B$ we obtain *a quotient bracketing* of $B$. *E.g.* $(x(xx))$ and $((xx)(xx))$ are (disjoint) subbracketings of $B=(((x(xx))x)((xx)(xx)))$, and replacing them with $x$ provides the quotient bracketing $((xx)x)$ of $B$. A bracketing is a *nest* if it is either of size 1 (*a trivial nest*) or one out of its factors is $x$, and the other one is a nest (\[5,7\]). *E.g.*, all bracketings of size 4 save $(xx)(xx)$ are nests. Given a bracketing $B$, there are subbracketings of $B$ which are nests; in particular, each $x_{i}$ is contained in a unique maximal nest. We call these maximal nests simply *the nests of* $B$. A nontrivial nest has a unique subbracketing of form $(x_{i}x_{i+1})$; we say that $x_{i},x_{i+1}$ are *the eggs of the nest*.
The *Catalan numbers* $C_{n}$ are defined recursively by
$\left( 1\right) $ $C_{0}=1$,
$\left( 2\right) $ $C_{n}=C_{0}C_{n-1}+C_{1}C_{n-2}+\cdots
+C_{n-2}C_{1}+C_{n-1}C_{0}\quad(n>0)$,
or, equivalently, by the formula$$C_{n}={\frac{1}{{n+1}}}{\binom{{2n}}{n}}.$$ Compare (1) and (2) with the formal definition of bracketings in the introduction, and take into account the unicity of the representation of bracketings in form $(PQ)$. Then the following standard result follows:
2.1.* The number of bracketings of size* $n$* equals* $C_{n-1}$ (see, *e.g.* \[8\]).
Hence we infer:
2.2.* For any spectrum* $\{s(n)\}$*,*$$1\leq s(n)\leq C_{n-1}$$ *holds for every* $n$*.*
If $s_{{}_{G}}(n)=C_{n-1}$ for every $n$, then the groupoid $G$ and its operation are said to be *Catalan*. *E.g.*, free groupoids are Catalan. Another inequality also follows from the definition of bracketings:
2.3.* For any spectrum* $\{s(n)\}$*,* $$s(n)\leq s(1)s(n-1)+s(2)s(n-2)+\cdots+s(n-2)s(2)+s(n-1)s(1)$$ *holds for every* $n(\geq2)$*.*
Hence if $s_{{}_{G}}(n_{0})<C_{{n_{0}}-1}$ then $s_{{}_{G}}(n)<C_{n-1}$ for every $n>n_{0}$. The following trivial observations are useful, too:
2.4.* If the groupoids* $G$* and* $H$* are isomorphic or antiisomorphic, then their spectra coincide.*
2.5.* If the groupoid* $H$* is a subgroupoid or a factorgroupoid of* $G$*, then* $$s_{{}_{H}}(n)\leq s_{{}_{G}}(n)$$ *holds for every* $n$*.*
By 2.5, in order to show that $G$ is Catalan it is sufficient to find a Catalan subgroupoid or factorgroupoid of $G$. The next fact goes back to Łukasiewicz (for a proof, see \[3\], Ch. 3.2, or \[11\], Exercise 1.38):
2.6.* Bracketings are uniquely determined by the places of their right (or left) parentheses between the symbols* $x_{1},\ldots,x_{n}$*.*
Next we introduce sequences of nonnegative integers which arise naturally from bracketings, and also contain full information on them. Consider the free monoid $F_{2}$ with unit element $e$, generated by symbols 0 and 1. A subset $M$ of $F_{2}$ is *prefix-free* if no word in $M$ is a prefix (*i.e.*, a left segment) of another word in $M$. There exist finite maximal prefix-free sets (*FMPF-sets* in short) in $F_{2}$, *e.g.*, the set containing the empty word $e$ only, the sets $\{0,1\},\{00,010,011,10,11\},$ *etc*. Assign to each bracketing an ordered sequence of words in $F_{2}$ inductively by the rule:
\(a) $x\mapsto(e)$,
\(b) if$\hfill P\,\mapsto\,(w_{1},\ldots,w_{k})\hfill$and$\hfill
Q\,\mapsto\,(w_{k+1},\ldots,w_{k+l})\hfill$then$\hfill(PQ)\mapsto
(0w_{1},\ldots,0w_{k},$
$~1w_{k+1},\ldots,1w_{k+l}).$
It is a routine to check that, in this way, a unique, lexicographically listed FMPF-set of $n$ words is assigned to every bracketing of size $n$. Now we can use the defining properties (1),(2) of Catalan numbers to show that the number of distinct FMPF-sets of $n$ elements equals $C_{n-1}$. Therefore, (a) and (b) provide a 1-1 correspondence between bracketings and lexicographically ordered FMPF-sets.
Consider a bracketing $B$ of size $n$ viewed with subscripts, *i.e.*, as an $n$-ary groupoid term. Let $({w_{1}}(B),\ldots,{w_{n}}(B))$ be the lexicographically ordered FMPF-set corresponding to $B$. Call the length of ${w_{i}}(B)$ *the depth of* $x_{i}$ *in* $B$, and the number of $0$’s (resp. of $1$’s) in ${w_{i}}(B)$ *the left depth* (resp. *the right depth*) *of* $x_{i}$ *in* $B$.
Inspecting (a) and (b) we get the intuitive meaning of depth of $x_{i}$: the number of pairs of parentheses (or, equivalently, of the subbracketings of size at least 2) containing $x_{i}$. Similarly, *e.g.* the right depth of $x_{i}$ in $B$ is the number of those subbracketings in which $x_{i}$ is contained in the right factor. The sequence consisting of the depths of $x_{1},\ldots,x_{n}$ in $B$ will be called *the depth sequence of* $B$. *Left* and *right depth sequences of* $B$ are defined analogously. *E.g.*, the depth sequence of $((x(xx))(xx))$ is $(2,3,3,2,2)$, and its right depth sequence is $(0,1,2,1,2)$.
\[ptb\]
[rtree2.eps]{}
FMPF-sets — and thus also bracketings — can be imagined as such minimal sets of vertices in the infinite binary rooted tree that separate the top of the tree from its bottom. See the figure where the sets of vertices corresponding to $(x_{1}x_{2})$ and $(x_{1}(x_{2}x_{3}))(x_{4}x_{5})$ are marked by squares, resp. circles; correspondence between vertices and binary strings is indicated, too. In this representation, the depth of $x_{i}$ is the number of edges in the path $p$ connecting $e$ with $x_{i}$. Similarly, the left (right) depth is the number of left(right) edges in $p$.
2.7.* Bracketings are uniquely determined by their depth sequences.*
This is clearly true for bracketings of size $\leq3$. Suppose the bracketings $(P_{1}Q_{1})$ and $(P_{2}Q_{2})$ of size $n(>3)$ have the same depth sequence $(d_{1},\ldots,d_{n})$. From the definition, the equality$$\sum_{i=1}^{n}{\frac{1}{2^{e_{i}}}}=1 \tag{1}$$ follows for every depth sequence $(e_{1},\ldots,e_{n})$. If $|P_{1}|=j,|P_{2}|=k$, then, in view of (a) and (b), the depth sequences of $P_{1}$ and $P_{2}$ are of form $(d_{1}-1,\ldots,d_{j}-1)$ and $(d_{1}-1,\ldots
,d_{k}-1)$, respectively. Therefore,$$\sum_{i=1}^{j}{\frac{1}{2^{d_{i}}}}=\sum_{i=1}^{k}{\frac{1}{2^{d_{i}}}}=1/2.$$ Hence the sizes of $P_{1}$ and $P_{2}$ are equal. Now the proposition follows by induction.
2.8.* Bracketings are uniquely determined by their right (or left) depth sequences.*
Let $B=(PQ)$ be a bracketing with right depth sequence (in short, *RD-sequence*)$$(d_{1},\ldots,d_{n}). \tag{2}$$ Then there is a $k$ between $1$ and $n$ such that the RD-sequence of $P$ is $(d_{1},\ldots,d_{k})$, and that of $Q$ is $(d_{k+1}-1,\ldots,d_{n}-1)$. Induction shows that always$$d_{1}=0,\quad d_{2}=1, \tag{3}$$ and, for $i=1,\ldots,n-1$, $$1\leq d_{i+1}\leq d_{i}+1. \tag{4}$$ Call a sequence (2) of nonnegative integers a *zag sequence* (*cf.* \[6\], Ch. 1.2, where *zig* is defined) if it has the properties (3) and (4). We use induction to prove that for any zag sequence (2) there exists at most one bracketing with RD-sequence (2). This is clearly true for $n\leq2$. As $(d_{k+1}-1,\ldots,d_{n}-1)$ is a zag sequence, we have $d_{k+1}=1$, and $d_{j}\geq2$ for $j=k+2,\ldots,n$. It follows that if the size of the first factor of $B$ is $k$, then the last $1$ in the RD-sequence of $B$ appears on the $(k+1)$st place. Hence if the RD-sequences of $B=(PQ)$ and $B^{\prime}=(P^{\prime}Q^{\prime})$ are the same, then $|P|=|P^{\prime}|$. Thus the RD-sequences of $P$ and $P^{\prime}$ coincide, and, by induction, $P=P^{\prime}$. Similarly we obtain $Q=Q^{\prime}$, completing the proof.
An analogous straightforward induction shows that every zag sequence is the RD-sequence of some bracketing. Consequently, the number of zag sequences of length $n$ equals that of the bracketings of size $n$, *i.e.,* $C_{n-1}$ (*cf.* \[14\], Ch. 5, Exercise 19(u)).
Examples
========
In this section we determine spectra of several common operations. Given a particular operation, we denote the members of its spectrum by $s(n)$ (without subscript), and we write $s(n)=f(n)$ to indicate that this equality holds for $n\geq3$.
3.1.* For the subtraction of numbers,* $s(n)=2^{n-2}$*.*
Induction shows that any regular operation $b(x_{1},x_{2},\ldots,x_{n})$ over the subtraction is of form $x_{1}-x_{2}\pm x_{3}\pm\cdots\pm x_{n}$. It is enough to prove that actually every possible sequence of the $+$ and $- $ signs occurs. This is true for $n\leq3$; suppose $n>3$, and apply induction. If $b(x_{1},x_{2},\ldots,x_{n})=x_{1}-x_{2}-\cdots-x_{n}$, then $b$ is induced by $((\ldots((x_{1}x_{2})x_{3})\ldots)x_{n})$. Otherwise there exists a first $+$ sign in $f$, say $b(x_{1},x_{2},\ldots,x_{n})=x_{1}-x_{2}-\cdots
-x_{k+1}+x_{k+2}\pm\cdots\pm x_{k+l}~(k+l=n)$. Then $b(x_{1},\ldots
,x_{n})=(x_{1}-x_{2}-\cdots-x_{k})-(x_{k+1}-x_{k+2}\mp\cdots\mp x_{k+l})$, and this is induced by $B=(PQ)$, where $P=((\ldots((x_{1}x_{2})x_{3})\ldots
x_{k})$, and $Q$ is the bracketing that induces the subtrahend (such a $Q$ exists by induction). In fact, this reasoning is valid for subtraction in arbitrary Abelian groups except those of exponent 2.
3.2.* The arithmetic mean as a binary operation on numbers is Catalan.*
We prove that distinct bracketings induce distinct regular operations over the arithmetic mean. Induction shows that a bracketing $B$ of size $n$ induces$b(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}{2^{-d_{i}}x_{i}}$ over the arithmetic mean, where $d_{i}$ is the depth of $x_{i}$ in $B$. Let $B^{\prime}\,(\neq B)$ be another bracketing of size $n$ which induces $b^{\prime}(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}{2^{{-d_{i}}^{\prime}}x_{i}}$. In virtue of 2.7, there exists a $j~(1\leq j\leq n)$ such that $d_{j}\neq{d_{j}}^{\prime}$. Then $b(\delta_{1}^{j},\ldots,\delta_{n}^{j})=2^{-d_{j}}\neq{2^{{-d_{j}}^{\prime}}}=b^{\prime}(\delta_{1}^{j},\ldots,\delta_{n}^{j})$, *i.e.*, $b$ and $b^{\prime}$ are distinct operations, as required. This holds for an arbitrary set of numbers closed under arithmetic mean, containing more than one element.
3.3.* The geometric mean and the harmonic mean as binary operations on positive real numbers are Catalan.*
This follows from 3.2 and 2.4, as the groupoids $(\mathbf{R},\left(
{x+y}\right) /2)$ and $(\mathbf{R}_{+},\sqrt{xy})$ are isomorphic, as well as $(\mathbf{R}_{+},\left( x+y\right) /2)$ and $({\mathbf{R}_{+}},{2xy/}\left(
x+y\right) )$.
3.4.* The exponentiation as a binary operation* $(a,b)\mapsto
a^{b}$* on numbers is Catalan.*
Let $p_{1},\ldots,p_{n}$ be distinct prime numbers. Consider bracketings $B,\,B^{\prime}(\neq B)$ and the regular operations $b,b^{\prime}$ they induce over the exponentiation. We show that $b\neq b^{\prime}$. Making use of the law $(r^{s})^{t}=r^{st}$, and the usual convention of writing $r^{s^{t}}$ instead of $r^{(s^{t})}$, we can write expressions of form $b(p_{1},\ldots,p_{n})$ without parentheses, *e.g.*, if $B=\left( \left(
x_{1}\left( x_{2}x_{3}\right) \right) \left( x_{4}x_{5}\right) \right) $ and $p_{i}$ are the first primes, we have $b(2,3,5,7,11)=2^{{3^{5}}{7^{11}}}$. Here the exponents are at different levels: say, 2 is at the zeroth, 3 and 7 are at the first level, *etc*. The key observation is that the height of the level of $p_{i}$ in $b$ always equals the right depth of $x_{i}$ in $B$; this can be verified using induction. As $B\neq B^{\prime}$, by 2.8. there exists a $j$ such that the right depth of $x_{j}$ in $B$ is different from that of $x_{j}$ in $B^{\prime}$. Then the fundamental theorem of arithmetic implies $b(p_{1},\dots,p_{n})\neq b^{\prime}(p_{1},\ldots,p_{n})$.
3.5.* The cross product of vectors is Catalan.*
Consider three pairwise perpendicular unit vectors, their additive inverses, and the zero vector. They form a groupoid under cross product, and, if we identify the unit vectors with their negatives, we obtain a four-element factorgroupoid $C$ with Cayley operation table $$\renewcommand{\arraystretch}{1.4} \setlength{\tabcolsep}{0.2cm}\begin{tabular}
[c]{c|c|c|c|c|}$\times$ & $0$ & $u$ & $v$ & $w$\\\hline
$0$ & $0$ & $0$ & $0$ & $0$\\\hline
$u$ & $0$ & $0$ & $w$ & $v$\\\hline
$v$ & $0$ & $w$ & $0$ & $u$\\\hline
$w$ & $0$ & $v$ & $u$ & $0$\\\hline
\end{tabular}$$
In virtue of 2.5., it is enough to prove that this operation is Catalan. Let $B,B^{\prime},b,b^{\prime}$ be as in 3.4. We shall find nonzero elements $c_{1},\ldots,c_{n}\in C$ such that $b(c_{1},\ldots,c_{n})=0\neq b^{\prime
}(c_{1},\ldots,c_{n})$. The case $n=3$ is obvious. The general case needs some preparations:
3.5.1. Let $F$ be a nontrivial nest of size $k$ which induces the regular operation $f$ on $C$. Given $i~(1\leq i\leq k)$, and $c,d\in C$ with $d\notin\{0,c\}$, we can choose elements $c_{1},\ldots,c_{i-1},c_{i+1},\ldots,c_{k}\in C$ so that $f(c_{1},\ldots,c_{i-1},c,c_{i+1},\ldots,c_{k})=d$.
This is valid also for any bracketing $B$ and its induced regular operation $b$ instead of $F$ and $f$. Indeed, apply 3.5.1 to the nest of $B$ containing $x_{i}$, if this nest is nontrivial, and replace this nest by $x$; while if $x_{i}$ is a trivial nest, replace the eggs of another nest by $x$. Then, in both cases, use induction for the quotient bracketing. We remark that this generalized form of 3.5.1 implies that any regular operation over the cross product is surjective (*i.e.*, it maps $C^{n}$ onto $C$; in fact, this is the case for all surjective binary operations, *cf.* 4.2.1).
3.5.2. If $x_{j},x_{j+1}$ are no eggs of any nest of a bracketing $B$, we can choose $d_{1},\ldots,d_{j-1},\allowbreak d,d_{j+2},\ldots,d_{k}$ in $C$ such that $f(d_{1},\ldots,d_{j-1},d,d,d_{j+2},\ldots,d_{k})\neq0$.
If $B=(PQ)$ with $|P|=k$ and $j+1\leq k$, then for suitable elements $d,d_{i}\in C$ by induction we have $p(d_{1},\ldots,d_{j-1},d,d,d_{j+2},\ldots,d_{k})=e\neq0$. Now by 3.5.1 there are $d_{k+1},\ldots,d_{n}\in C$ such that $q(d_{k+1},\ldots,d_{n})=f\neq0,e$. Then $b(d_{1},\ldots
,d,d,\ldots,d_{n})=e\times f\neq0$. The case $k<j$ can be treated in a similar way. Finally, suppose $k=j$. Let us fix $d\neq0$, and apply 3.5.1 to $P$ and $Q$ with $i=k$ and $i=k+1$, respectively. Then we have elements $d_{1},\ldots,d_{k-1},d_{k+2},\ldots,d_{n}\in C$ such that $p(d_{1},\ldots
,d_{k-1},d)=e$ and $q(d,d_{k+2},\ldots,d_{n})=f$, where $C=\{0,d,e,f\}$. Thus $b(d_{1},\ldots,d,d,\ldots,d_{n})=e\times f=d\neq0$, completing the proof of 3.5.2.
In order to prove 3.5., first suppose that there is an $i~(1\leq i\leq n)$ such that $x_{i}$ and $x_{i+1}$ are the eggs of a nest of $B$ as well as of $B^{\prime}$. Replacing $(x_{i}x_{i+1})$ by $x$ in $B$ and $B^{\prime}$, we obtain quotient bracketings $B_{1}$ resp. ${B_{1}}^{\prime}$ of size $n-1$ with induced regular operations $b_{1}$ and ${b_{1}}^{\prime}$. By induction, there exist nonzero elements $e_{1},\ldots,e_{n-1}\in C$ such that ${b_{1}}(e_{1},\ldots,e_{i},\ldots,e_{n-1})=0\neq{b_{1}}^{\prime}(e_{1},\ldots
,e_{i},\ldots,e_{n-1})$. Let $e^{\prime},e^{\prime\prime}\in C$ be distinct, and different from $0$ and $e_{i}$. Then $e^{\prime}\times e^{\prime\prime
}=e_{i}$ , and $b(e_{1},\ldots,e_{i-1},e^{\prime},e^{\prime\prime},e_{i+1},\ldots,e_{n-1})=b_{1}(e_{1},\ldots,e_{i},\ldots,e_{n-1})=0\neq
b^{\prime}(e_{1},\ldots,e_{i-1},\allowbreak e^{\prime},e^{\prime\prime
},e_{i+1},\ldots,e_{n-1})$.
Now suppose that no nests of $B$ and $B^{\prime}$ have a common pair of eggs. Let $x_{j}$ and $x_{j+1}$ be the eggs of a nest of $B$. Then $b(d_{1},\ldots,d_{j-1},d,d,d_{j+1},\ldots,d_{n})=0$ for any choice of $d_{1},\ldots,d_{j-1},d,d_{j+2},\ldots,d_{n}\in C$. However, as $x_{j}$ and $x_{j+1}$ are eggs of no nest in $B^{\prime}$, from 3.5.2 it follows that there is a choice of $d_{1},\ldots,d_{j-1},d_{j+2},\ldots,d_{n}$ such that $b^{\prime}(d_{1},\ldots,d_{j-1},d,d,d_{j+2},\ldots,d_{n})\neq0$.
Groupoids on two-element sets
=============================
In what follows we consider operations on finite sets. For uniform treatment, we study groupoids of form $(\mathbf{n},\circ)$, where $\mathbf{n}$ stands for the set $\{0,1,\ldots,n-1\}$. Each two-element groupoid is isomorphic or antiisomorphic with $(\mathbf{2},\circ)$, where $x\circ y$ is one of the following seven Boolean functions:
\(1) the constant 1 operation; (2) $x$ (the first projection); (3) $x\wedge y$ (*i.e.*, $\min(x,y)$); (4) $x+y$$\operatorname{mod}$ 2; (5) $x+1$$\operatorname{mod}$ 2; (6) $x|y$ (the Sheffer function: neither $x$, nor $y$); (7) $x\rightarrow
y$ (implication).
Here (1) — (4) are associative. We determine the spectra of (5) — (7).
4.1.* For the operation* $x+1\,\operatorname{mod}2$,$~s(n)=2$*.*
Indeed, induction shows that for an arbitrary bracketing $B$ of size $n$ and $\allowbreak c_{1},\ldots,c_{n}\in\mathbf{2}$, $b(c_{1},\ldots,c_{n})=c_{1}+d\,\operatorname{mod}2$, where $d$ is the left depth of $x_{1}$ in $B$.
4.2.* The Sheffer function is Catalan.*
Recall, that $0|0=1$ and $x|y=0$ otherwise. We shall need some preliminaries.
4.2.1. Regular operations over a surjective operation are surjective (*i.e.*, they take on all elements of their base sets; the inductive proof is trivial).
4.2.2. If the Cayley table of a surjective operation $\circ$ has neither two identical columns nor two identical rows, then each variable of any regular operation over $\circ$ is essential.
This is obvious for at most binary regular operations. Let $B=(PQ)$, $|B|=n\geq3,|P|=k$. Take a variable $x_{i}$ of $b$. We have to prove that there are elements $c_{1},\ldots,c_{i-1},u,v,c_{i+1},\ldots,c_{n}$ in the base set $M$ of $\circ$ such that $b(c_{1},\ldots,c_{i-1},u,c_{i+1},\allowbreak
\ldots,c_{n})\neq b(c_{1},\ldots,c_{i-1},v,c_{i+1},\ldots,c_{n})$. Without loss of generality, suppose $i\leq k$. Then, by induction there exist $c_{1},\ldots,c_{i-1},u,v,c_{i+1},\ldots,c_{k}\in M$ such that $g=p(c_{1},\ldots,c_{i-1},u,c_{i+1},\ldots,c_{n})\neq p(c_{1},\ldots,c_{i-1},v,c_{i+1},\ldots,c_{n})=h$. The rows of $g$ and $h$ in the Cayley table of $\circ$ are not identical, *i.e.*, there is a $d\in M$ such that $g\circ d\neq h\circ d$. Further, by 4.2.1, there are $c_{k+1},\ldots,c_{n}\in
M$ with $q(c_{k+1},\ldots,c_{n})=d$. Then $b(c_{1},\ldots,c_{i-1},u,c_{i+1},\ldots,c_{n})=g\circ d\neq h\circ d=b(c_{1},\ldots,c_{i-1},v,c_{i+1},\ldots,c_{n})$, which was needed.
4.2.3. If $\circ$ fulfils the conditions of 4.2.2, then regular operations of distinct arities over $\circ$ cannot be identically equal.
Indeed, otherwise the last variable of the regular operation of greater arity could not be essential.
We see that 4.2.1—3 apply to the Sheffer function. Let $B_{1},B_{2}$ be bracketings of size $n\,(\geq3),B_{1}=(P_{1}Q_{1}),B_{2}=(P_{2}Q_{2})$, and suppose that their induced operations $b_{1}$ and $b_{2}$ coincide. We have to prove $B_{1}=B_{2}$. This is true for $n=3$, as $(0|0)|1=0\neq1=0|(0|1)$. Let $n>3$, and assume $k=|P_{1}|\leq|P_{2}|=l$. First we show that, for arbitrary $c_{1},\ldots,c_{k},\ldots,c_{l}\in\mathbf{2},p_{1}(c_{1},\ldots,c_{k})=0$ if and only if $p_{2}(c_{1},\ldots,c_{l})=0$. Let $p_{1}(c_{1},\ldots,c_{k})=0$. By 4.2.1, there exist $c_{k+1},\ldots,c_{n}\in\mathbf{2}$ with $q_{1}(c_{k+1},\ldots,c_{n})=0$. Hence it follows $$\begin{gathered}
b_{1}(c_{1},\ldots,c_{k},c_{k+1},\dots,c_{n})=p_{1}(c_{1},\ldots
,c_{k})\,|\,q_{1}(c_{k+1},\ldots,c_{n})=1=\\
=b_{2}(c_{1},\ldots,c_{l},c_{l+1},\ldots,c_{n})=p_{2}(c_{1},\ldots
,c_{l})\,|\,q_{2}(c_{l+1},\ldots,c_{n}),\end{gathered}$$ implying $p_{2}(c_{1},\ldots,c_{l})=0$. This reasoning is valid in the opposite direction, too, showing that $p_{1}$ identically equals $p_{2}$. Now from 4.2.3 we infer $k=l$ and, by induction, $P_{1}=P_{2}$. It remains to establish $Q_{1}=Q_{2}$. Let, once more, $p_{1}(c_{1},\ldots,c_{k})=0$. If $Q_{1}\neq Q_{2}$, then, again by induction, there are $c_{k+1},\ldots
,c_{n}\in\mathbf{2}$ such that $q_{1}(c_{k+1},\ldots,c_{n})\neq q_{2}(c_{k+1},\ldots,c_{n})$. Then $$b_{1}(c_{1},\ldots,c_{n})=0\,|\,q_{1}(c_{k+1},\ldots,c_{n})\neq0\,|\,q_{2}(c_{k+1},\ldots,c_{n})=b_{2}(c_{1},\ldots,c_{n}),$$ a contradiction. Thus $Q_{1}=Q_{2}$, as required.
4.3.* Implication is Catalan.*
In view of 2.4., instead of implication we can consider the operation $x\ast
y$, defined by $0\ast1=1$ and $x\ast y=0$ otherwise, as $(\mathbf{2},\rightarrow)$ and $(\mathbf{2},\ast)$ are isomorphic. For $\ast$, the proof of 4.2. can be literally adapted.
Groupoids on three-element sets
===============================
There are 3330 essentially distinct three-element groupoids in the sense that each three-element groupoid is isomorphic with exactly one of them (see the Siena Catalog \[2\], in which code numbers from $1$ to $3330$ are given to each of these representatives), therefore a plain survey of their spectra such as in the two-element case seems to be impossible. In this section we determine the spectra of all groupoids on $\mathbf{3}$ with minimal clones of term operations, and give examples for further spectra.
There exist 12 essentially distinct groupoids on $\mathbf{3}$ with minimal clones, and each of them is idempotent (see \[4\]). The operations of an *idempotent* groupoid on $\mathbf{3}$ may be encoded by the numbers $0,1,\ldots,728$ in the following transparent way: let the code of $\circ$ be$$(0\circ1)\cdot3^{5}+(0\circ2)\cdot3^{4}+(1\circ0)\cdot3^{3}+(1\circ
2)\cdot3^{2}+(2\circ0)\cdot3+(2\circ1)$$ (see the examples below). The operations of the groupoids on $\mathbf{3}$ with minimal clones are (or, more exactly, may be chosen as) $0,8,10,11,16,17,26,33,35,68,178,624$ (their codes in the Siena Catalog are $80,102,105,106,122,125,147,267,271,356,\allowbreak1108,2346$ respectively). It is easy to check that $0,8,10,11$ and $26$ are associative. Here we display the Cayley tables of the remaining seven operations: $$\begin{gathered}
\renewcommand{\arraystretch}{1.4}\setlength{\tabcolsep}{0.2cm}\begin{tabular}
[c]{cccc}$\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$0$ & $1$ & $1$\\
$2$ & $1$ & $2$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$0$ & $1$ & $1$\\
$2$ & $2$ & $2$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$1$ & $1$ & $0$\\
$2$ & $0$ & $2$\end{tabular}
$2$ & $2$ & $2$\end{tabular}
$\\
$16$ & $17$ & $33$ & $35$\end{tabular}
\\
\renewcommand{\arraystretch}{1.4}\setlength{\tabcolsep}{0.2cm}\begin{tabular}
[c]{ccc}$\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$2$ & $1$ & $1$\\
$1$ & $2$ & $2$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $2$\\
$0$ & $2$ & $1$\\
$2$ & $1$ & $0$\\
$1$ & $0$ & $2$\end{tabular}
$\\
$68$ & $178$ & $624$\end{tabular}\end{gathered}$$ As we apply three different approaches, we parcel our task into three parts.
5.1.* The operations* 16, 17* and* 178* are Catalan.*
Observe that $\mathbf{3}$ with each of the operations 16, 17 and 178 is a groupoid in which $\{0,1\}$ is a subgroupoid with two-sided zero element 0, while $\{1,2\}$ and $\{2,0\}$ are subgroupoids with left zero elements 1 and 2, respectively. Here and in what follows, the just considered operations will be denoted by circle.
Let $B_{i}=(P_{i}Q_{i})~(i=1,2)$ be distinct bracketings of size $n\,(\geq3)
$. For $n=3$, $1\circ(2\circ0)=1\circ2=1\neq0=1\circ0=(1\circ2)\circ0$, *i.e.*, $b_{1}\neq b_{2}$. To prove the same for $n>3$, first suppose $|P_{1}|=k<l=|P_{2}|$. Then$$\begin{aligned}
b_{1}(1,\ldots,1,2,\ldots,2,0,\ldots,0) & =p_{1}(1,\ldots,1)\circ
q_{1}(2,\ldots,2,0,\ldots,0)=1\circ2=1,\\
b_{2}(1,\ldots,1,2,\ldots,2,0,\ldots,0) & =p_{2}(1,\ldots,1,2,\ldots,2)\circ
q_{2}(0,\ldots,0)=1\circ0=0.\end{aligned}$$
Thus, we can assume $|P_{1}|=|P_{2}|=k$. If $P_{1}\neq P_{2}$, by induction there exist elements $c_{1},\ldots,c_{k}\in\mathbf{3}$ with $g_{1}=p_{1}(c_{1},\ldots,c_{k})\neq p_{2}(c_{1},\ldots,c_{k})=g_{2}$. Let $d$ be the element of $\mathbf{3}$ that is different from $g_{1}$ and $g_{2}$. Then $g_{1}\circ d\neq g_{2}\circ d$ (see the Cayley tables), and hence $b_{1}(c_{1},\ldots,c_{k},d,\ldots,d)=g_{1}\circ d$ differs from $b_{2}(c_{1},\ldots,c_{k},d,\ldots,d)=g_{2}\circ d$. It remains to settle the case $Q_{1}\neq Q_{2}$. Again, we can choose elements $c_{k+1},\ldots,c_{n}\in\mathbf{3}$ with $h_{1}=q_{1}(c_{k+1},\ldots,c_{n})\neq q_{2}(c_{k+1},\ldots,c_{n})=h_{2}$.
*Case* 17*:* 0 and 2 are left zero elements, whence $c_{k+1}=1$, and we can assume $h_{1}=0,\,h_{2}=1$. Now $b_{1}(1,\ldots
,1,c_{k+1},\ldots,c_{n})=1\circ0=0\neq1=1\circ1=b_{2}(1,\ldots,1,c_{k+1},\ldots,c_{n})$.
*Cases* 16* and* 178*:* for distinct elements $h_{1},h_{2}\in\mathbf{3}$ there exists $e\in\mathbf{3}$ with $e\circ
h_{1}\neq e\circ h_{2}$. Hence it follows $b_{1}(e,\ldots,e,c_{k+1},\ldots,c_{n})\neq b_{2}(e,\ldots,e,c_{k+1},\ldots,c_{n})$, concluding the proof.
5.2.* The operation* 33* is Catalan. For* 35* and* 68*,* $s(n)=2^{n-2}$*.*
Consider a groupoid $(G,\circ)$ with idempotent elements $d,e(\neq d),f$ such that
$\left( \alpha\right) $ in the Cayley table of $\circ$, $d$ occurs only in its own row;
$\left( \beta\right) $ in the row of $e$, $e\circ d$ occurs only once;
$\left( \gamma\right) $ $f$ is a right unit element.
First check that $\mathbf{3}$ with 33, 35, or 68 satisfies these conditions. Now let $B_{1}=(P_{1}Q_{1})$ and $B_{2}=(P_{2}Q_{2})$ be bracketings of size $n$ such that their induced operations over $\circ$ coincide. We prove $p_{1}=p_{2}$. Suppose $k=|P_{1}|<|P_{2}|=l$. Then $$\begin{aligned}
b_{2}(e,\ldots,e,d,\ldots,d) & =p_{2}(e,\ldots,e)\circ q_{2}(d,\ldots
,d)=e\circ d,\\
b_{1}(e,\ldots,e,d,\ldots,d) & =p_{1}(e,\ldots,e)\circ q_{1}(e,\ldots
,e,d,\ldots,d).\end{aligned}$$ As by $\left( \alpha\right) $ we have $q_{1}(e,\ldots,e,d,\ldots,d)\neq d$, from $\left( \beta\right) $ it follows that $b_{1}(e,\ldots,e,\allowbreak
d,\ldots,d)\neq b_{2}(e,\ldots,e,d,\ldots,d)$. Thus $|P_{1}|=|P_{2}|$, and , by $\left( \gamma\right) $, for arbitrary $c_{1},\ldots,c_{k}\in G$ it holds $p_{1}(c_{1},\ldots,c_{k})=b_{1}(c_{1},\ldots,c_{k},f,\ldots,f)=b_{2}(c_{1},\ldots,c_{k},\allowbreak f,\ldots,f)=p_{2}(c_{1},\ldots,c_{k})$, which was needed.
Take into account that 33 is surjective, and its Cayley table has no two identical columns. We show that in the case of 33 if $b_{1}=b_{2}$, then $q_{1}=q_{2}$, which together with $p_{1}=p_{2}$ implies *via* induction that 33 is Catalan. Indeed, suppose that, although $b_{1}=b_{2}$, there exist $c_{k+1},\ldots,c_{n}\in\mathbf{3}$ such that $q_{1}(c_{k+1},\ldots,c_{n})\neq q_{2}(c_{k+1},\ldots,c_{n})$. Then the columns of these two elements are also distinct, *i.e.* $c\circ q_{1}(c_{k+1},\ldots,c_{n})\neq c\circ q_{2}(c_{k+1},\ldots,c_{n})$ for some $c\in\mathbf{3}$. In virtue of 4.2.1 we can choose $c_{1},\ldots,c_{k}\in\mathbf{3}$ so that $p_{1}(c_{1},\ldots,c_{k})=c$. Now $b_{1}(c_{1},\ldots,c_{n})=p_{1}(c_{1},\ldots,c_{k})\circ q_{1}(c_{k+1},\ldots,c_{n})\neq
p_{1}(c_{1},\ldots,c_{k})\circ q_{2}(c_{k+1},\ldots,c_{n})=b_{2}(c_{1},\ldots,c_{n})$, a contradiction.
Concerning 35 and 68, observe that in these cases if $u\circ v\neq u\circ w$ then at least one of $v$ and $w$ is a left zero which satisfies $\left(
\alpha\right) $. We have seen that $b_{1}=b_{2}$ implies $p_{1}=p_{2}$; now we prove that the converse implication also holds. Suppose not, *i.e.*, there are $c_{1},\ldots,c_{n}\in\mathbf{3}$ such that $b_{1}(c_{1},\ldots,c_{n})=p_{1}(c_{1},\ldots,c_{k})\circ q_{1}(c_{k+1},\ldots,c_{n})\neq
p_{1}(c_{1},\ldots,c_{k})\circ q_{2}(c_{k+1},\ldots,c_{n})=b_{2}(c_{1},\ldots,c_{n})$. Hence, without loss of generality, the element $d=q_{1}(c_{k+1},\ldots,c_{n})$ is a left zero, and $d$ does not occur in other rows. We infer that $c_{k+1}=d$, and, as a consequence, $q_{2}(c_{k+1},\ldots
,c_{n})=d=q_{1}(c_{k+1},\ldots,c_{n})$, whence $b_{1}(c_{1},\ldots
,c_{n})=b_{2}(c_{1},\ldots,c_{n})$, a contradiction.
This shows that, for 35 and 68, $s(n)=s(n-1)+\cdots+s(2)+s(1)$, and this means $s(n)=2^{n-2}$, as stated.
5.3.* For the operation* 624*,* $s(n)=\lfloor
2^{n}/3\rfloor$*.*
624 is actually $2x+2y\,\operatorname{mod}3$ on $\mathbf{3}$. We shall write it in form $-x-y$; our considerations are valid for this operation on numbers, too. An $n$-ary regular operation (over $-x-y$) is always of form $t(x_{1},\ldots,x_{n})=\pm x_{1}\pm\cdots\pm x_{n}$. We call such operations *complete linear*. As $x_{1}-x_{2}+x_{3}$ shows, not every complete linear operation is regular. Denote by $\pi(t)$ the number of $+$ signs in a complete linear operation $t=t(x_{1},\ldots,x_{n})$, and call a complete linear $t$ *subregular*, if $\pi(t)\equiv2n-1\,(\operatorname{mod}3)$. The following assertion can be checked immediately:
5.3.1. If $t,t_{1},t_{2}$ are complete linear operations such that the equality $t(x_{1},\ldots,x_{n})=-t_{1}(x_{1},\ldots,x_{k})-t_{2}(x_{k+1},\ldots,x_{n})$ holds, then every one of $t,t_{1},t_{2} $ is subregular provided the other two of them are subregular.
Next we characterize the regular operations over $-x-y$.
5.3.2. A complete linear operation $t(x_{1},\ldots,x_{n})$ is regular over $-x-y$ if and only if it is subregular but not of form $$x_{1}-x_{2}+x_{3}-\cdots+x_{n} \tag{5}$$ (*i.e.*, not of odd arity with alternating signs and beginning with a $+$ sign).
Clearly, this is true for $n\leq3$. Suppose that $t$ is regular. Then $t(x_{1},\ldots,x_{n})=-t_{1}(x_{1},\ldots,x_{k})-t_{2}(x_{k+1},\ldots,x_{n})$ with $t_{1}$ and $t_{2}$ regular. By induction, $t_{1}$ and $t_{2}$ are subregular, and 5.3.1 implies that $t$ is subregular. If $t$ is regular and it is of form (5), then one of $t_{1}$ and $t_{2}$ — say, $t_{1}$ — must be of even arity with alternating signs. However, a complete linear operation $t$ of arity $2m$ with alternating signs cannot be subregular, as $\pi
(t)=m\not \equiv 2\cdot2m-1\,(\operatorname{mod}3)$. Hence $t_{1}$ is not subregular, a contradiction.
Conversely, assume that $t$ is subregular but not regular. We have to prove that $t$ is of form (5). We show that the first sign in $t$ is $+$ . If not, then $t(x_{1},\ldots,x_{n})=-x_{1}\pm x_{2}\pm\cdots\pm x_{n}=-x_{1}-(\mp
x_{2}\mp\cdots\mp x_{n})=-x_{1}-t_{2}(x_{2},\ldots,x_{n})$, and from 5.3.1 it follows that $t_{2}$ is subregular. If, in addition, $t_{2}$ is not of form (5), then by induction $t_{2}$ is regular, hence $t$ is regular, in contrary to the assumption. However, if $t_{2}$ is of form (5), then$$\begin{aligned}
t(x_{1},\ldots,x_{n}) & =-x_{1}-x_{2}+x_{3}-\cdots+x_{n-1}-x_{n}=\\
& =-(x_{1}+x_{2}-x_{3}+\cdots-x_{n-1})-x_{n}=\\
& =-t_{1}(x_{1},\ldots,x_{n-1})-x_{n},\end{aligned}$$ and here $t_{1}$ is regular, implying again the regularity of $t$.
Thus, $t$ starts with a $+$ sign, and it is enough to prove that the signs alternate in $t$. If not, consider the first two consecutive identical signs in $t$. Suppose they are $+$ ; the other case can be treated analogously. Then$$\begin{aligned}
t(x_{1},\ldots,x_{n})= & \ x_{1}-x_{2}+\cdots-x_{2k-2}+x_{2k-1}+x_{2k}\pm\\
& \pm x_{2k+1}\pm\cdots\pm x_{n}=\\
= & -(-x_{1}+x_{2}-\cdots+x_{2k-2}-x_{2k-1}-x_{2k})-\\
& -(\mp x_{2k+1}\mp\cdots\mp x_{n})=\\
= & -t_{1}(x_{1},\ldots,x_{2k})-t_{2}(x_{2k+1},\ldots,x_{n}).\end{aligned}$$ We can check that $t_{1}$ is subregular and not of form (5), hence regular; further, $t_{2}$ is subregular by 5.3.1. As above, supposing that $t_{2}$ is not of form (5) leads to a contradiction. Hence $t_{2}(x_{2k+1},\ldots
,x_{n})=x_{2k+1}-x_{2k+2}+\cdots-x_{n-1}+x_{n}$, and $$\begin{aligned}
t(x_{1},\ldots,x_{n})= & \ x_{1}-x_{2}+x_{3}-\cdots+x_{2k-1}+x_{2k}-x_{2k+1}+\\
& +x_{2k+2}-\cdots+x_{n-1}-x_{n}=\\
= & -(-x_{1}+x_{2}-x_{3}+\cdots-x_{2k-1}-x_{2k}+x_{2k+1}-\\
& -x_{2k+2}+\cdots-x_{n-1})-x_{n}=\\
= & -{t_{1}}^{\prime}(x_{1},\ldots,x_{n-1})-x_{n}.\end{aligned}$$ Here ${t_{1}}^{\prime}$ is subregular and not of form (5), so it is regular by induction, whence we obtain that $t$ is regular, and this final contradiction proves that a subregular but not regular complete linear operation is of form (5).
From 5.3.2 it follows that the number $s(n)$ of the $n$-ary regular operations over $-x-y$ equals $\sum_{k}{\binom{n}{{3k+i}}}-(n\,\operatorname{mod}2)$, if $n\equiv2-i\,(\operatorname{mod}3)~(i=0,1,2)$. It is known that each of these numbers is equal to $\lfloor{2}^{n}/3\rfloor$ (see \[6\], Ch. 5, Exercise 75). This completes the description of spectra of three-element groupoids with minimal clones.
The next seven operations are of some interest from various reasons. The first two pairs have the same spectra but with different coincidences of induced regular operations. Fibonacci numbers appear at the fifth one. Nest structure is exploited in the next example, and the last one is related to the Sheffer operation on $\mathbf{2}$. These operations are numbered by their codes in the Siena Catalog \[2\]: $$\begin{gathered}
\renewcommand{\arraystretch}{1.4}\setlength{\tabcolsep}{0.2cm}\begin{tabular}
[c]{cccc}$\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $2$\\
$0$ & $0$ & $2$\\
$2$ & $2$ & $1$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$0$ & $0$ & $0$\\
$1$ & $0$ & $0$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $1$\\
$0$ & $0$ & $1$\\
$1$ & $1$ & $0$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$1$ & $1$ & $1$\\
$2$ & $2$ & $2$\\
$0$ & $0$ & $0$\end{tabular}
$\\
$1066$ & $10$ & $405$ & $3242$\end{tabular}
\\
$0$ & $1$ & $0$\\
$0$ & $0$ & $1$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$0$ & $0$ & $0$\\
$0$ & $1$ & $0$\\
$0$ & $1$ & $2$\end{tabular}
$ & $\begin{tabular}
[c]{|ccc}\hline
$1$ & $0$ & $0$\\
$0$ & $2$ & $0$\\
$0$ & $0$ & $0$\end{tabular}
$\\
$79$ & $82$ & $2407$\end{tabular}\end{gathered}$$
5.4.* For the operations* 1066* and* 10*,* $s(n)=n-1$*.*
Denote by $t(c_{1},\ldots,c_{n})$ the number of occurrences of $2$ among $c_{1},\ldots,c_{n}$. Concerning 1066, induction shows that, for arbitrary bracketing $B=(PQ)$ with $|B|=n,\,|P|=k$, and $c_{1},\ldots,c_{n}\in
\mathbf{3}$, $$b(c_{1},\ldots,c_{n})=2\text{ if and only if }t(c_{1},\ldots,c_{n})\text{ is
odd,}$$ and $$b(c_{1},\ldots,c_{n})=1\text{ if and only if both }t(c_{1},\ldots,c_{k})\text{
and }t(c_{k+1},\ldots,c_{n})\text{ are odd.}$$ As a consequence, $b(c_{1},\ldots,c_{n})=0$ if and only if both $t(c_{1},\ldots,c_{k})$ and $\linebreak t(c_{k+1},\ldots,c_{n})$ are even. Hence it follows that two bracketings of equal size induce the same operation if and only if the sizes of their left factors are equal.
In order to manage 10 (which, for this once, will be written as multiplication), we introduce the *priority of a bracketing* $B$ ($\mathrm{pr}(B)$ in sign) for $|B|>2$ as follows: If $B=(PQ)$ and $|P|>1$, then $\mathrm{pr}(B)=0$; if $B=(x_{1}(x_{2}(\ldots(x_{k}(R))\ldots)))$, and $\mathrm{pr}(R)=0$ or $|R|=2$, then $\mathrm{pr}(B)=k$. We call the bracketing $R$ *the core of* $B$. Clearly, if $n>2$, for every $k=0,1,\ldots,n-2$ there exist bracketings of size $n$ with priority $k$. Hence it is sufficient to prove that two bracketings of size $n$ induce the same regular operation over 10 if and only if they are of the same priority.
If: $\mathrm{pr}(B)=0$ implies that $b$ is the constant $0$ operation. If $k=n-2$ or $k=n-3$, then there is only one bracketing $B$ with $\mathrm{pr}(B)=k$. Suppose $B_{1}$ and $B_{2}$ are of size $n$ with cores $R_{1}$, resp. $R_{2}$, and $\mathrm{pr}(B_{1})=\mathrm{pr}(B_{2})=k<n-3$. Then$$\begin{aligned}
b_{1}(c_{1},\ldots,c_{n}) & =(c_{1}(\ldots(c_{k}\cdot r_{1}(c_{k+1},\ldots,c_{n}))\ldots))=(c_{1}(\ldots(c_{k}\cdot0)\ldots))=\\
& =(c_{1}(\ldots(c_{k}\cdot r_{2}(c_{k+1},\ldots,c_{n}))\ldots))=b_{2}(c_{1},\ldots,c_{n})\end{aligned}$$ for arbitrary $c_{1},\ldots,c_{n}\in\mathbf{3}$.
Only if: Let again $B_{1}$ and $B_{2}$ be bracketings with cores as above, and let $\mathrm{pr}(B_{1})=k<l=\mathrm{pr}(B_{2})$. Induction on priority shows that bracketings with positive priority induce nonconstant operations over 10. Hence there are $c_{k+1},\ldots
,c_{l},\allowbreak c_{l+1},\ldots,c_{n}\in\mathbf{3}$ such that $(c_{k+1}(\ldots(c_{l}\cdot r_{2}(c_{l+1},\ldots,c_{n}))\ldots))=1$. For $i=0,1$, check the equality $(2(2(\ldots(2\cdot i)\ldots)))=(k-i)\,\operatorname{mod}2$, where $k$ is the number of occurrences of 2 in the left side, and choose $c_{1}=\cdots=c_{k}=2$. It follows$$\begin{aligned}
b_{1}(c_{1},\ldots,c_{n}) & =(c_{1}(\ldots(c_{k}\cdot r_{1}(c_{k+1},\ldots,c_{n}))\ldots))=(c_{1}(\ldots(c_{k}\cdot0)\ldots))=\\
& =k\,\operatorname{mod}2\neq(k-1)\,\operatorname{mod}2=(c_{1}(\ldots
(c_{k}\cdot1)\ldots))=\\
& =(c_{1}(\ldots(c_{k}(c_{k+1}(\ldots(c_{l}\cdot r_{2}(c_{l+1},\ldots
,c_{n}))\ldots)))\ldots))=\\
& =b_{2}(c_{1},\ldots,c_{n}).\end{aligned}$$
5.5.* For the operations* 405* and* 3242*,* $s(n)=3$* if* $n>3$*.*
Let $B_{1},B_{2}$ be bracketings of size $n$, $B_{i}=(P_{i}Q_{i})$. We show that the induced regular operations $b_{1},b_{2}$ over 405 coincide if and only if one of the following conditions is satisfied:
$\left( 1\right) $ $|P_{1}|=|P_{2}|=1$;
$\left( 2\right) $ $1<|P_{1}|,|P_{2}|<n-1$;
$\left( 3\right) $ $|P_{1}|=|P_{2}|=n-1$.
Indeed, in the case $\left( 1\right) $ the first variable, and in the case $\left( 3\right) $ the last variable determines the value of $b_{i}$. In the case $\left( 2\right) $ $b_{i}$ is the constant zero operation. Finally, if $B_{1}=(x_{1}Q_{1}),\,B_{2}=(P_{2}x_{n})$, then $b_{1}(0,\ldots,2)=0\neq
1=b_{2}(0,\ldots,2)$.
3242 is $x+1\,\operatorname{mod}3$. Similarly to 4.1, for any bracketing $B$ and its induced operation $b$ over 3242 we have $b_{1}(c_{1},\ldots
,c_{n})=c_{1}+d\,\operatorname{mod}3$, where $d$ is the left depth of $x_{1}$ in $B$.
5.6.* For the operation* 79*,* $s(n)=F_{n+1}-1$*, where* $F_{k}$* is the* $k\hspace{1pt}$*th Fibonacci number.*
First we show that, for bracketings $B_{1},B_{2}$ of equal size, $b_{1}$ coincides with $b_{2}$ if and only if the eggs of nests of $B_{1}$ are the same as the eggs of nests of $B_{2}$. Suppose that $x_{i},x_{i+1}$ are the eggs of a nest of $B_{1}$ but of no nest of $B_{2}$. Put $c_{j}=2$, if $j=i$ or $j=i+1$, and $c_{j}=1$ otherwise. Then $b_{1}(c_{1},\ldots,c_{n})=1\neq0=b_{2}(c_{1},\ldots,c_{n})$. On the other hand, if the eggs of nests of $B_{1}$ and $B_{2}$ are the same, induction on the number of nests proves $b_{1}=b_{2}$. Note that this number is 1 exactly when $B_{1}$ and $B_{2}$ are nests, and for nests we can apply the usual induction on size.
Choose several non-overlapping pairs $(i,i+1)$ in the sequence $1,\ldots,n$. The number of such choices (including the empty choice) is $F_{n+1}$. Induction shows that for every such nonempty choice $C$ there exists a bracketing $B$ such that $x_{i},x_{i+1}$ are the eggs of a nest of $B$ if and only if $(i,i+1)$ occurs in the choice $C$. This proves our proposition.
5.7.* The operation* 82* is Catalan.*
Induction shows that the first (*i.e.*, leftmost) right parenthesis in $B$ together with its left pair encloses just the eggs of the leftmost nontrivial (maximal) nest of $B$. Let $|B_{1}|=|B_{2}|=n,~b_{1}=b_{2}$, and let the eggs in question of $B_{1}$ and $B_{2}$ consist of $x_{k},x_{k+1}$ and $x_{l},x_{l+1}~(k<l)$, respectively. For $c_{1}=\cdots=c_{k}=c_{k+2}=\cdots=c_{n}=1,~c_{k+1}=2$ we get $b_{1}(c_{1},\ldots,c_{n})=0\neq
b_{2}(c_{1},\ldots,c_{n})$. Thus, the first right parentheses in $B_{1}$ and $B_{2}$ cannot be in different positions. Collapsing $x_{k}$ and $x_{k+1}$ we obtain quotient bracketings $B_{1}^{\prime}$ and $B_{2}^{\prime}$ of size $n-1$. Remark that, for arbitrary $c_{1},\ldots,c_{k-1},c_{k+1},\ldots
,c_{n}\in\mathbf{3}$,$~b_{i}^{\prime}(c_{1},\ldots,c_{k-1},c_{k+1},\ldots,c_{n})=b_{i}(c_{1},\ldots,c_{k-1},2,c_{k+1},\ldots,c_{n})$ holds, as $2$ is a left unit for 82. In such a way, $b_{i}$ determines $b_{i}^{\prime}$, and the latter determines the place of the first right parenthesis in $B_{i}^{\prime}$, which is the second right parenthesis in $B_{i}$; *etc*. We see that the induced operation determines the positions of all right parentheses in its parent bracketing. Now 5.7 follows from 2.6.
5.8.* The operation* 2407* is Catalan.*
The proof consists of a suitable adaptation of 4.2. The observations 4.2.1, 4.2.2, and 4.2.3 apply to 2407. Now, from $B_{1}=(P_{1}Q_{1}),~B_{2}=(P_{2}Q_{2})$, and $b_{1}=b_{2}$ we can deduce not only the equivalence of $p_{1}(c_{1},\ldots,c_{k})=0$ and $p_{2}(c_{1},\ldots,c_{l})=0$ but also that of $p_{1}(c_{1},\ldots,c_{k})=1$ and $p_{2}(c_{1},\ldots,c_{l})=1$. Thus, again we have $p_{1}=p_{2}$, and, by induction, $P_{1}=P_{2}$. In order to refute $Q_{1}\neq Q_{2}$, assume that there exist $c_{k+1},\ldots,c_{n}\in\mathbf{3}$ with $q_{1}(c_{k+1},\ldots,c_{n})=i\neq j=q_{2}(c_{k+1},\ldots,c_{n})$; here we can suppose $i\neq2$. There are $c_{1},\ldots
,c_{k}\in\mathbf{3}$ with $p_{1}(c_{1},\ldots,c_{k})=i$. Then $b_{1}(c_{1},\ldots,c_{n})=i\circ i=i+1\,\operatorname{mod}3\neq i\circ
j=b_{2}(c_{1},\ldots,c_{n})$.
The Sheffer function on $\mathbf{2}$ and 2407 on $\mathbf{3}$ are the smallest instances of groupoids $(\mathbf{n},\circ)$ with operations$$i\circ j=\begin{cases}
i+1, & \text{if }i=j\\
0, & \text{otherwise.}\end{cases}
\tag{6}$$ All these groupoids are *primal*; *i.e.*, all possible operations on $\mathbf{n}$ are term operations of such a groupoid. The proof of 5.8 can be generalized for them without trouble. Hence we could (in fact, we did) conjecture for a minute that primality implies a Catalan spectrum; however, operation 3233 testifies that this is not the case. Its Cayley table comes from that of 3242 by writing $1\circ2=0$ instead of $1\circ2=2$. For 3233 we have $s_{6}=41<C_{5}(=42)$. Actually,$$x_{1}\circ\left( \left( x_{2}\circ\left( x_{3}\circ\left( x_{4}\circ
x_{5}\right) \right) \right) \circ x_{6}\right) =x_{1}\circ\left( \left(
x_{2}\circ\left( \left( x_{3}\circ x_{4}\right) \circ x_{5}\right)
\right) \circ x_{6}\right)$$ identically holds for 3233 on $\mathbf{3}$ (but no other regular operations over 3233 induced by distinct bracketings of size $\leq6$ are equal). On the other hand, the primality of $\mathbf{3}$ with 3233 as well as of $\mathbf{n}$ with operation (6) follows, *e.g.*, from Rousseau’s criterion: a finite algebra with a single operation is primal if and only if it has neither proper subalgebras, nor congruences, nor automorphisms \[12\].
We have checked all the 3330 entries of the Siena Catalog by computer for the five initial elements of their spectra, *i.e.* $(s(3),s(4),s(5),s(6),s(7))$. It is known that there are 24 nonisomorphic three-element semigroups. The table below shows the number of essentially distinct three-element nonassociative groupoids with a given initial segment of spectrum: $$\renewcommand{\arraystretch}{1.2} \setlength{\tabcolsep}{0.2cm}\begin{tabular}
[c]{rrrrrrr}$2$ & $2$ & $2$ & $2$ & $2$ & & $16$\\
$2$ & $3$ & $3$ & $3$ & $3$ & & $4$\\
$2$ & $3$ & $4$ & $5$ & $6$ & & $15$\\
$2$ & $4$ & $4$ & $4$ & $4$ & & $2$\\
$2$ & $4$ & $5$ & $6$ & $7$ & & $6$\\
$2$ & $4$ & $6$ & $8$ & $10$ & & $4$\\
$2$ & $4$ & $7$ & $12$ & $20$ & & $4$\\
$2$ & $4$ & $7$ & $12$ & $21$ & & $12$\\
$2$ & $4$ & $8$ & $15$ & $27$ & & $12$\\
$2$ & $4$ & $8$ & $16$ & $32$ & & $62$\\
$2$ & $5$ & $8$ & $12$ & $16$ & & $2$\\
$2$ & $5$ & $10$ & $18$ & $31$ & & $4$\\
$2$ & $5$ & $10$ & $20$ & $40$ & & $4$\end{tabular}
\qquad\qquad\qquad\begin{tabular}
[c]{rrrrrrr}$2$ & $5$ & $10$ & $21$ & $42$ & & $5$\\
$2$ & $5$ & $11$ & $23$ & $47$ & & $2$\\
$2$ & $5$ & $11$ & $24$ & $53$ & & $4$\\
$2$ & $5$ & $12$ & $28$ & $65$ & & $12$\\
$2$ & $5$ & $13$ & $34$ & $87$ & & $12$\\
$2$ & $5$ & $13$ & $34$ & $89$ & & $2$\\
$2$ & $5$ & $13$ & $34$ & $90$ & & $4$\\
$2$ & $5$ & $13$ & $34$ & $91$ & & $24$\\
$2$ & $5$ & $13$ & $35$ & $96$ & & $2$\\
$2$ & $5$ & $13$ & $35$ & $97$ & & $32$\\
$2$ & $5$ & $14$ & $41$ & $123$ & & $6$\\
$2$ & $5$ & $14$ & $41$ & $124$ & & $16$\\
$2$ & $5$ & $14$ & $42$ & $132$ & & $3038$\end{tabular}$$
Several sequences beginning with some quintuples above, *e.g.* $(2,5,10,21,42)$ (*cf.* 5.3) and $(2,5,14,41,123)$, are recently missing in the Encyclopedia \[13\].
General remarks and problems
============================
All the spectra considered up to now are monotonic. Groups with the commutator operation provide examples of nonmonotonic spectra: if a group $G$ is nilpotent then there exists an $n$ such that all $n$-ary regular term operations over the commutator of $G$ are equal (to the constant unit operation), hence $s(n)=1$, and if $G$ is not nilpotent of class 2 then the commutator is not associative (see, *e.g.* \[10\] ). The spectrum always stabilizes in these examples: $s(n)=1$ implies $s(m)=1
$ for every $m>n$. In fact, this is a common property of all spectra, which generalizes the generalized associative law:
6.1.* For an arbitrary spectrum* $s$*, *$s(n)=1$* for some* $n\,(\geq3)$* implies* $s(m)=1$* for every* $m>n$*.*
Call two bracketings of size $m$ *adjacent* if there exists a $j$ such that $x_{j},x_{j+1}$ are eggs of nests for each of these bracketings. It is easy to see that the transitive closure of the adjacency relation is the trivial equivalence if $m\geq5$.
Let $n\,(\geq3)$ be a number such that $s(n)=1$ for an operation $\circ$ on a set $M$. Consider bracketings $B,B^{\ast}$ of size $n+1$. We have to prove $b=b^{\ast}$. For $n=3$ this is the generalized associative law. Assume $n>3$. Then $n+1\geq5$, hence there exist bracketings $B_{0}=B,B_{1},\ldots
,B_{k}=B^{\ast}$ such that, for $i=0,1,\ldots,k-1$,$~B_{i}$ is adjacent to $B_{i+1}$. Let $x_{j},x_{j+1}$ be common eggs of a nest of $B_{i}$ and a nest of $B_{i+1}$. Replacing $(x_{j}x_{j+1})$ by $x_{j} $ in both of them, we obtain quotient bracketings $B_{i}^{\prime},B_{i+1}^{\prime}$ of size $n$. As $s(n)=1$, we have $b_{i}^{\prime}=b_{i+1}^{\prime}$, and thus$$\begin{aligned}
b_{i}(c_{1},\ldots,c_{n+1}) & =b_{i}^{\prime}(c_{1},\ldots,c_{j-1},c_{j}\circ c_{j+1},c_{j+2},\ldots,c_{n+1})=\\
& =b_{i+1}^{\prime}(c_{1},\ldots,c_{j-1},c_{j}\circ c_{j+1},c_{j+2},\ldots,c_{n+1})=\\
& =b_{i+1}(c_{1},\ldots,c_{n+1})\end{aligned}$$ for arbitrary $c_{1},\ldots,c_{n+1}\in M$.
Groups provide also examples showing that the difference $s(n)-s(n-1)$ of consecutive entries of a spectrum can be arbitrarily large:
6.2*. The spectrum of the commutator operation on the dihedral group of degree* $2^{t}~(t\geq3)$* is*$$s(n)=\begin{cases}
2, & \text{if }n=3\\
n, & \text{if }3<n\leq t\\
1, & \text{if }n>t.
\end{cases}$$
$D_{m}$, the dihedral group of degree $m$ is generated by a rotation $\alpha$ of order $m$ and a reflection $\rho$. We write $i$ for $\alpha^{i}$ and $j^{\prime}$ for $\alpha^{j}\rho$. Here is the concise Cayley table of the commutator on $D_{m}$:
$$\renewcommand{\arraystretch}{1.4}\setlength{\tabcolsep}{0.2cm}\begin{tabular}
[t]{c|c|c}
& $j$ & $j^{\prime}$\\\hline
$i$ & $0$ & $-2i\,\operatorname{mod}m$\\\hline
$i^{\prime}$ & $2j\,\operatorname{mod}m$ & $2\left( i-j\right)
\,\operatorname{mod}m$\end{tabular}$$
The following observations are immediate: If a bracketing $B$ over the commutator on $D_{n}$ has at least two nests, then it induces the constant zero operation. Further, if $B$ is a nest with eggs $x_{k},\,x_{k+1}$, then $b(c_{1},\ldots,c_{n})\neq0$ only if all $c_{i}\,(\in D_{m})$ but at most one of $c_{k},c_{k+1}$ are of form $i^{\prime}$ (*i.e.,* $\alpha^{i}\rho$). From the Cayley table we learn that for such a nest $B$ and such elements $c_{1},\ldots,c_{n}$$$b(c_{1},\ldots,c_{n})=[c_{k},c_{k+1}]\,{2^{k-1}}(-2)^{n-k-1}\,\operatorname{mod}m \tag{7}$$ holds. From (7) we infer that the position of eggs of $B$ determines the induced operation $b$. As all commutators are of form $2u\,\operatorname{mod}m$, (7) shows also that always $b(c_{1},\ldots,c_{n})={2^{n-1}}\cdot
v\,\operatorname{mod}m$ with suitable integers $v$. This means that $b$ is the zero operation if $m=2^{t}$ and $n>t$.
It remains to show that nests of equal size $n\,(\leq t)$ but with distinct eggs induce distinct operations. In fact, besides $B$ consider another nest $B^{\prime}$ with eggs $x_{l},x_{l+1}~(l>k)$. Let $c_{k}=1,\,c_{k+1}=2^{\prime}$, and choose elements $c_{i}~(i\neq k,k+1)$ of form $i^{\prime}$ arbitrarily. Then $[1,2^{\prime}]=-2\,\operatorname{mod}2^{t}$, and, by (7), $b(c_{1},\ldots,c_{n})=(-1)^{n-k}2^{n-1}\,\operatorname{mod}2^{t}\neq0$. On the other hand, $l>k$ implies $b^{\prime}(c_{1},\ldots,c_{n})=0$ because $c_{k}=1$, and $x_{k}$ is out of the egg of $B^{\prime}$.
The same reasoning shows that the commutator on $D_{1},\,D_{2}$ and $D_{4}$ is associative, and if $m$ is not a power of $2$ (*e.g.*, in the case of $D_{3}=S_{3}$) the spectrum of the commutator on $D_{m}$ is $s(n)=n$ for $n>3$.
The next example leads to groupoids whose spectra begin with arbitrarily many Catalan numbers and still reach 1.
6.3.* The following operation on the nonnegative integers is Catalan:*$$a\circ b=\begin{cases}
\min(a,b)-1, & \text{if }a,b>0\\
0, & \text{otherwise.}\end{cases}$$
For the proof, denote by $d_{{}_{B}}(x_{i})$ the depth of $x_{i}$ in the bracketing $B$. Consider an arbitrary bracketing $B=(PQ)$ with $|B|=n,~|P|=k$. First we show that$$b(d_{{}_{B}}(x_{1})+1,\ldots,d_{{}_{B}}(x_{n})+1)=1.$$ Note that, for any $B$, $b(c_{1},\ldots,c_{n})>0$ implies $b(c_{1}+1,\ldots,c_{n}+1)=b(c_{1},\ldots,c_{n})+1$. By induction we have $p(d_{{}_{B}}(x_{1}),\ldots,d_{{}_{B}}(x_{k}))=p(d_{{}_{P}}(x_{1})+1,\ldots,d_{{}_{P}}(x_{k})+1)=1$, and similarly $q(d_{{}_{B}}(x_{k+1}),\ldots,d_{{}_{B}}(x_{n}))=1$, whence it follows$$\begin{aligned}
b(d_{{}_{B}}(x_{1})+1,\ldots,d_{{}_{B}}(x_{n})+1)= & \ p(d_{{}_{B}}(x_{1})+1,\ldots,d_{{}_{B}}(x_{k})+1)\circ\\
& \circ q(d_{{}_{B}}(x_{k+1})+1,\ldots,d_{{}_{B}}(x_{n})+1)=\\
= & \ (1+1)\circ(1+1)=1,\end{aligned}$$ as needed.
Next we show that for any other $B^{\prime}$ of size $n$, $b^{\prime}(d_{{}_{B}}(x_{1})+1,\ldots,d_{{}_{B}}(x_{n})+1)=0$. Again, induction shows that for arbitrary $B$, nonnegative integers $c_{1},\ldots,c_{n}$, and $i~(1\leq i\leq n)$$$b(c_{1},\ldots,c_{n})\leq\max(c_{i}-d_{{}_{B}}(x_{i}),0) \tag{8}$$ holds; we omit the details. As $B^{\prime}\neq B$, 2.7 implies that there exists an $i$ such that $d_{{}_{B^{\prime}}}(x_{i})\neq d_{{}_{B}}(x_{i})$, and in view of (1) we can suppose even $d_{{}_{B^{\prime}}}(x_{i})>d_{{}_{B}}(x_{i})$. Then applying (8) to $B^{\prime}$ we obtain$$b^{\prime}(d_{{}_{B}}(x_{1})+1,\ldots,d_{{}_{B}}(x_{n})+1)\leq\max(d_{{}_{B}}(x_{i})+1-d_{{}_{B^{\prime}}}(x_{i}),0)=0,$$ concluding the proof.
For any bracketing $B$ with $|B|=k<n$, and for every $i\,(=1,\ldots,k)$, we have $d_{{}_{B}}(x_{i})<k$, hence $d_{{}_{B}}(x_{i})+1\in\mathbf{n}$. Therefore the above reasoning shows that in $(\mathbf{n},\circ)$, which is a subgroupoid of $(\mathcal{N}_{0},\circ)$, distinct bracketings of size $k\,(<n)$ induce different regular operations. On the other hand, every bracketing $B$ whose size exceeds $2^{n-2}$ has a symbol $x_{j}$ with $d_{{}_{B}}(x_{j})\geq n-1$. Applying (8) to the regular operation $b$ of $(\mathbf{n},\circ)$ we obtain$$b(c_{1},\ldots,c_{n})\leq\max(c_{j}-d_{{}_{B}}(x_{j}),0)=0,$$ as $c_{j}\leq n-1$. Hence any bracketing of size $2^{n-2}+1$ induces the constant zero operation of $(\mathbf{n},\circ)$. Thus, for the spectrum of $(\mathbf{n},\circ)$, $s(k)=C_{k-1}$ if $k<n$, and $s(k)=1$ if $k>2^{n-2}$.
The study of spectra of linear operations $px+qy$ (and $px+qy+r$) on numbers (or, more generally, on modules over rings) also offers remarkable facts. As a specimen, we prove the following generalization of 3.2.
6.4.* The linear operations* $px+py$* and* $x+py$* on the complex numbers are not Catalan if and only if* $p$* is a root of unity.*
Concerning $px+py$, induction shows that for any bracketing $B$ of size $n$, the induced operation over $px+py$ is$$b(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}p^{d_{i}}x_{i}, \tag{9}$$ where $d_{i}$ is the depth of $x_{i}$ in $B$. From 2.7 it follows that if $p$ is not a root of unity then $px+py$ is Catalan. Suppose $p^{k}=1$. Define the bracketings $B_{i}$ by $B_{1}=(xx)$, and $B_{n+1}=(B_{n}B_{n})$ for $n>0$. The depth sequences of $B^{\prime}=(xB_{k})$ and $B^{\prime\prime}=(B_{k}x)$ are $(1,k+1,\ldots,k+1)$ and $(k+1,\ldots,k+1,1)$, respectively. Now (9) implies $b^{\prime}=b^{\prime\prime}$. Hence, for $m=2^{k}+1$, $s(m)<C_{m-1}$.
Analogous considerations apply to $x+py$: (9) remains valid for this case with *right* depths instead of depths. If $p$ is not a root of unity, 2.8 guarantees that $x+py$ is Catalan. Suppose again $p^{k}=1$, and redefine $B_{i}$ by $B_{1}=(xx)$, and $B_{n+1}=(xB_{n})$ for $n>0$. The RD-sequences of $B^{\prime}=(B_{k}x)$ and $B^{\prime\prime}=B_{k+1}$ are $(0,1,2,\ldots,k,1)$ and $(0,1,2,\ldots,k,k+1)$, respectively, implying $b^{\prime}=b^{\prime
\prime}$, and, for $m=k+2$, $s(m)<C_{m-1}$.
In conclusion, we formulate a few problems:
1\. For every positive integer $n$ there exists a minimal $f(n)$ with the property that, if for two spectra $s_{1},s_{2}$ of $n$-element groupoids $s_{1}(i)=s_{2}(i)$ holds whenever $i\leq f(n)$, then these spectra coincide. Propositions 4.1—3 imply $f(2)=4$, and the table at the end of Section 5 shows that $f(3)\geq7$. What is the actual value of $f(3)$ (and that of $f(4)$, *etc.*)?
2\. We gave a rough estimation for the subsequent entries of a spectrum with a given initial segment in 2.3 which *e.g.,* for $s(3)=2
$ and $s(4)=4$ provides $s(5)\leq12$. However, a case-by-case analysis shows that $s(3)=2$ and $s(4)=4$ actually imply $s(5)\leq8$. Do they imply $s(n)\leq2^{n-2}$ for all $n\,(>1)$? If so, call $s(n)=2^{n-2}$ a *maximal extension* of the initial segment $(2,4)$. Prove or disprove that the maximal extension of $(2,3)$ is $s(n)=n-1$, and that of $(2,2)$ is $s(n)=2$.
3\. All nonconstant spectra we exhibited above are ultimately constant or monotonic. In the latter case their growth rates are either linear or exponential. Is there any other possibility? More concretely: find, *e.g.*, a spectrum with quadratic growth rate.
4\. The statistics of the three-element groupoids and the abundance of appropriate examples leave such an impression that a huge majority of binary operations is Catalan. Is it true that, in some sense, almost all operations are Catalan (or almost Catalan)?
[99]{}
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[^1]: This research was supported by Hungarian National Foundation for Scientific Research (OTKA) grants no. T022867 and T026243.
[^2]: [Here by a groupoid we mean a nonempty set with one binary operation.]{}
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This invention relates to hydroponic crop production.
Horticulture and agriculture increasingly rely on advanced production technology to maintain their competitiveness. The use of hydroponic crop production methods has led to the production of high quality crops at high yields. The crops so produced include tomatoes and cucumbers.
Hydroponic crop production normally involves the use of a substrate, for example of rockwool or perlite, on which the crop is grown. Some methods do not involve the use of a substrate however and rely on a thin film of nutrient liquid being guided by a trough or other conduit to flow over the plant roots and provide nutrition for the plant.
In both the production methods which involve substrates and those which do not, it is important that the substrate or fluid conduit is free at the beginning of each production cycle from any residues or contaminants harmful to the plants to be grown. This necessity has led to hydroponic substrates being discarded after a single growing cycle, with the resultant considerable wastage of materials, or to the substrates being removed from the locations they occupy during crop production to be sterilised by, for example, steam treatment or microwave or radio- frequency radiation treatment. All these treatments involve handling and transportation of the substrates and are energy- and labour-intensive. As an alternative, treatment of the substrates with a fungicide or soil sterilant might be considered. These materials are however themselves subject to considerable disadvantages: fungicides, which usually produce undesirable residues in effluent wash water, are moreover usually specific in their action (and would therefore not be effective against a wide range of contaminants unless a large number of fungicides were used simultaneously); soil sterilants such as methyl bromide (a toxic gas requiring careful, specialist handling) usually leave behind toxic residues which would render the substrate unsuitable for further use. Corresponding disadvantages arise when it is desired to remove harmful residues from the hydroponic fluid conduits used in hydroponic production not involving a substrate.
It has now been found that when a hydroponic substrate is treated after use with sufficient of a percarboxylic acid to reduce the microbial population without totally sterilising the medium, improved plant growth is obtained in a subsequent growth cycle compared even with the fresh medium. This surprising result is apparently due to the ability of benign microflora to recolonise the substrate after percarboxylic acid treatment, and the absence of toxic residues from the percarboxylic acid itself.
The result is particularly surprising since percarboxylic acids have not been found very satisfactory as soil sterilants. It is believed that this may be due to the relatively high concentration of organic matter in soil which may inactivate percarboxylic acid at all but very high levels thereof. Hydroponic substrates are generally characterised by relatively low organic contents, typically less than 5% by volume e.g. less than 3% preferably less than 2% less than 1% or less than 0.5% by volume.
The present invention accordingly provides a method of hydroponic crop production in which a hydroponic substrate and/or the liquid-supply means of a hydroponic system which has been infected with a microbial population is treated after a crop-production cycle and prior to a subsequent crop-production cycle with sufficient of a dilute aqueous solution containing at least one peracid having a ##STR2## group substantially to reduce but not eliminate the microbial population. Preferably the dilute aqueous solutions also contain hydrogen peroxide.
With such a method, in situ disinfection of the hydroponic substrate or liquid-supply means can be readily carried out and the need for time- and energy-consuming removal and transportation of the substrate avoided. The dilute treatment solution can be conveniently supplied to the substrate through the same irrigation system as is used for supply of liquid nutrients to the plants during the growing cycle. Any trace of peracid remaining after treatment can be removed by flushing through the same irrigation system. In any event, the use of a peracid as mentioned above, for example peracetic acid, will normally produce no harmful residue because of its breakdown to the corresponding acid, for example acetic acid, and hydrogen peroxide which itself decomposes to water and oxygen. Disinfection of the substrate is achieved, giving a significant reduction in the microbial population but leaving a residual community and a dynamic environment in which recolonisation may readily occur. Recolonisation with a low level of microorganisms is believed to promote higher plant growth yields. In particular, a balanced microbial population inhibits the growth of any pathogens that may reinfect the medium. Moreover, with treatment processes according to the invention, no phytotoxic effects have been observed. Treatment according to the invention can be carried out quickly, with consequent reduction of the downtime of the hydroponic system, and is effective against a broad spectrum of contaminants. Furthermore, no deterioration in the hydroponic substrate as a result of treatment according to the invention has been observed.
The peracid used is preferably a percarboxylic acid, more preferably peracetic or perpropionic acid.
The treatment solution will usually be prepared by dilution of a more concentrated solution of the peracid. Such solutions of the peracid may be unstable on prolonged storage and the treatment solution will preferably be analysed prior to the treatment process in order to establish the concentration of the peracid in the treatment solution.
The solutions will thereby preferably not contain any additional ingredient which may tend to destabilise the peracid. In particular, the presence of significant quantities of dissolved salts is to be avoided.
The concentrates may contain a wetting agent. A preferred group of wetting agents is those which exhibit some bactericidal activity, the most preferred wetting agent being aromatic alkyl and especially benzene sulphonic acids. Examples of such acids are the alkyl aryl sulphonic acids which have from 6 to 18 carbon atoms in the alkyl substituent of the molecule, especially those which have from 9 to 15 carbon atoms in the alkyl substituent. The sulphonic acid may take the form of a mixture of alkyl aryl sulphonic acids wherein the number of carbon atoms in the alkyl substituent varies but has an average value of from 6 to 18 carbon atoms. Examples of alkyl aryl sulphonic acids which are useful include acids derived from aromatic nuclei other than benzene, such as toluene and xylene, as well as those derived from benzene itself. Examples of useful sulphonic acids include decyl toluene, dodecyl xylene, octyl benzene, nonyl benzene, decyl benzene, tridecyl benzene, tetradecyl benzene, pentadecyl benzene, dodecyl benzene and hexadecyl benzene sulphonic acids. The preferred sulphonic acid is dodecyl benzene sulphonic acid. The concentrates may also comprise any of the conventional stabilisers for percarboxylic acids such as 2,6 pyridine dicarboxylic acid and phytic acid.
The concentrates may be produced by mixing the components thereof in any order. The peracid may be added as a solution thereof or may be generated in situ by the reaction of hydrogen peroxide with the corresponding acid or acid anhydride. The product of these mixing procedures is an aqueous solution comprising hydrogen peroxide, peracid and acid in dynamic equilibrium. The rate of reaction between hydrogen peroxide and the acid is usually relatively slow at ambient temperatures and the mixture may not reach equilibrium for a considerable period in some circumstances. These processes represent the preferred method for the production of the concentrates and the conditions under which the mixing of the appropriate quantities of peroxide and the acid or peracid is carried out may be adjusted to ensure that a predetermined minimum concentration of peracid is produced.
The preferred concentrates for use in this invention contain from 0.5 to 20%, preferably from 1.0 to 10.0% and more preferably from 2. 0 to 7. 0% by weight of peracid and from 10 to 50%, preferably 10 to 35%, and most preferably 15 to 25% by weight of hydrogen peroxide, the ratio of the weight of hydrogen peroxide to peracid preferably being in the range 2:1 to 10:1, more preferably in the range 4:1 to 7:1. In general the stability of the concentrates increases as the concentration of the peracid increases and should be at least 3.0% by weight and more preferably at least 3.5% by weight of the concentrate. Any hydroponic substrate, including rockwool, perlite, silica, gels and plastics foam substrates, may be treated in accordance with the invention, as may all troughs and other conduits used. Substrates high in organic matter, for example soil and peat, cause breakdown of the concentrates. Preferably, the substrates to be treated in the present invention contain less than 2% by volume of organic matter when first used. Crop residues derived from plant growth do not inactivate the concentrates.
The stability of the concentrates decreases as the amount of water present increases. The amount of water present in the concentrates is preferably in the range 40 to 75% by weight and more preferably in the range 50 to 70%. The ratio of the amount of water to the amount of peracid (at equilibrium) is preferably in the range 10:1 to 20:1, more preferably in the range 12:1 to 18:1. The maximum concentration of peracid is preferably not greater than 7.0% and more preferably not greater than 5.0% by weight because greater concentrations of peracid present difficulties in transporting and handling the concentrate. Where the peracid is to be generated in situ by the reaction of hydrogen peroxide with the corresponding acid or acid anhydride, the quantity of hydrogen peroxide which is used will be adjusted so as to take into account the quantity of hydrogen peroxide which is liable to be consumed in this reaction.
A concentrate containing peracid is preferably produced by adding glacial acetic acid to a solution of hydrogen peroxide. The formulation of peracetic acid may be accelerated by the addition of a catalytic quantity of a mineral acid, but this is less preferred since it adds to the corrosive nature of the product. Alternatively the concentrate can be produced by the addition of hydrogen peroxide solution to commercially available peracetic acid products.
The concentrates mentioned may be used at dilution rates of, say, 25, 50, 75, 100 or 200 v/v. The concentration of peracid in the treatment solution being preferably from 0.001 to 0.05 mol dm.sup.-3, more preferably from 0.0025 to 0.02 mol dm.sup.-3 and most preferably from 0. 005 to 0.01 mol dm.sup.-3 .
The application rate of treatment solution to hydroponic substrate is preferably 0.1:1 to 5.0:1 v/v (i.e., from one-tenth to five volumes of treatment solution applied per volume of substrate), more preferably from 0.5:1 to 1.0:1 and most preferably from 0.66:1 to 0.83:1.
The present invention is applicable equally to hydroponic substrates used for the production of any crop, including edible crops, such as tomatoes and cucumbers, ornamental plants and flowers for cutting.
The invention will now be described further with reference to the following example and test results.
EXAMPLE 1
An aqueous concentrate comprising peracetic acid was made by mixing the following ingredients.
______________________________________
Percent m/m
______________________________________
Acetic acid 10.3
Hydrogen peroxide (35% solution)
70.9
2,6-pyridine dicarboxylic acid
0.015
Dodecyl benzene sulphonic acid
1.2
Water 17.585
______________________________________
The composition (composition "A") was mixed, left to stand at 50. degree. C. for 19 hours, cooled to ambient temperature and sampled prior to use to determine the peracetic content.
TEST RESULTS
A range of hydroponics substrate samples was treated with diluted composition A and assayed for populations of bacteria and fungi. Controls were of untreated material.
Four substrate samples were used:
(1) Rockwool (Grodania A/S) previously used for cucumber production.
(2) New rockwool (Grodania A/S).
(3) Horticultural grade perlite (Tilcon Ltd) previously used for tomato production.
(4) New horticultural grade perlite (Tilcon Ltd).
Composition A was diluted at 1:50 and 1:100 v/v and the diluted solution applied to the substrate at a rate of 0.66 to 0.83:1 v/v (20 to 25 dm.sup.3 solution applied to 30 dm.sup.3 of substrate). Variable grading and settlement of materials, in particular perlite, preclude accurate determination of volume.
The substrate samples were assayed as follows. Samples of approximately 30 dm.sup.3 of each substrate were treated with a dilute solution of composition A and with equivalent volumes of clean mains water. Representative sub-samples, each weighing 5 g, were ground in 15 cm.sup.3 sterile distilled water and 0.1 cm.sup.3 of serial dilutions plated onto each of two plates of Czapex-Dox plus yeast extract agar (CD+ ye). CD+ye is a specific medium, pH 4, for promoting growth of fungi. In addition, two plates containing Nutrient agar (NA) were inoculated with sub-samples from each serial dilution. The plates were incubated for 3 days at 20° C. prior to counting of bacterial colonies on NA, and for 4 days at 20° C. prior to counting of fungal colonies on CD+ ye. The moisture contents of the substrate samples were determined and results expressed as colony forming units (cfus) or bacteria per gram (dr wt).
Application of composition A to both used and new rockwool and perlite at a dilution of 1:50 eliminated all fungal and bacterial populations. Reference should now be made to Table 1.
TABLE 1
______________________________________
Fungal and bacterial populations in rockwool
and perlite treated or untreated with
composition A (g.sup.-1 dr wt)
Treatment/ Fungal
sample cfus Bacteria
______________________________________
Composition A @ 1:50
Used rockwool 0 0
0 0
New rockwool 0 0
0 0
Used perlite 0 0
0 0
New perlite 0 0
0 0
Water controls
Used rockwool 2.0 × 10.sup.5
2.6 × 10.sup.5
3.8 × 10.sup.5
2.6 × 10.sup.5
New rockwool 1.2 × 10.sup.2
3.5 × 10.sup.7
1.2 × 10.sup.2
1.4 × 10.sup.7
Used perlite 0 2.7 × 10.sup.6
0 4.7 × 10.sup.6
New perlite 1.7 × 10.sup.2
1.1 × 10.sup.7
1.7 × 10.sup.2
6.9 × 10.sup.6
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Growing and phytotoxicity trials were carried out to identify the level of phytotoxicity of composition A when used as a pre-planting drench to once-used hydroponic substrates. Phytotoxicity was tested using rapid bioassay techniques. The use of treated material for the production of salad crops was then investigated.
In order to determine the presence or absence of any phytotoxic residues in hydroponic substrates previously treated with composition A, a rapid assay method based on standard procedures was adopted. The assay method for Dazomet soil residue determination using cress germination and growth (BASF UK Ltd) was utilised.
Rockwool and perlite samples were treated with composition A at a dilution of 1:50 or 1:100, or with clean water. Test solutions were applied to approximately 0.5 dm.sup.3 samples of used rockwool (Grodania A/S). Samples of a similar volume of used perlite (Tilcon Ltd) were also used.
Assessment of phytotoxicity was carried out by treating substrate samples with composition A and subsequently flushing twice, to run off, with clean water prior to bioassay.
Substrate samples were treated with composition A solutions or water controls and allowed to soak for one hour. The samples were then drained overnight and placed in plastic trays in a glasshouse before being surface-seeded with cress and maintained in the glasshouse with a heating thermostat setpoint of 18° C. and a ventilation setpoint 24° C. Germination and subsequent growth were assessed on a subjective scale of 0 to 5. (0=no germination or seedling death; 5=good germination and growth). The samples were kept moist with nutrient solution as required. The results obtained showed that germination was good in all cases.
TABLE 2
______________________________________
Mean growth score for cress sown onto flushed rockwool
and perlite previously treated with composition A
0 = poor; 5 = good
Treatment/sample Mean growth score after 6 days
______________________________________
Composition A @ 1:100
Used rockwool 4
Used perlite 4.5
Composition A @ 1:50
Used rockwool 4
Used perlite 4.5
Water controls
Used rockwool 5
Used perlite 5
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On the basis of results from rapid bioassay of compound A residues and efficacy data indicating activity of product against natural populations of bacteria and fungi, compound A was tested on crop plants. Tomatoes were grown in treated and untreated perlite and cucumbers grown in treated and untreated rockwool and maintained under standard commercial conditions prior to assessment of plant establishment and growth.
Materials and methods
Substrate and crop combinations:
i) Cucumber grown in rockwool.
ii) Tomato grown in perlite.
Substrates:
i) Rockwool
Propagation: New cubes (Capogro Ltd) approximately 7 cm×7 cm. times.7 cm.
Growing on: Once-used slabs (Grodania A/S) (Glass Glover Ltd) approximately 90 cm×27 cm×6 cm.
ii) Perlite (Tilcon Ltd)
Propagation: New perlite in 13 cm mesh based pots (Plantpak Ltd) .
Growing on: Once-used perlite (Tilcon Ltd) 25 dm.sup.3 growing modules.
Crop species:
i) Cucumber cv Tinda (Enza Zaden BV).
ii) Tomato cv 663 RZ F1 (Rijk Zwaan BV).
`Planting` was carried out using current commercial practices over a six-day period. Cucumbers were `planted` by placing young plants directly onto growing slab surfaces. Tomato plants were `planted` by partial plunging of the propagation pot into the top 2 to 3 cm of the perlite growing modules.
Each 25 dm.sup.3 perlite growing module was planted with three tomato plants. The modules were positioned end to end in rows within a glasshouse. Each rockwool slab was planted with two cucumber plants. The modules were positioned end-to-end in rows within a glasshouse. Both species were trained up vertical strings and side shoots removed at an early stage to maintain a single plant stem in each case. Irrigation water and nutrients were applied manually on a twice-daily basis, or more frequently as required under bright conditions. Nutrition was supplied according to current recommendations. Environmental control was provided by a Van Vliet CR-12 environmental computer. The heating thermostat setpoint was 18° C. with a ventilation set point of 24. degree. C. The plants were destructively harvested five weeks after planting and assessed for establishment and growth in the various treated substrates.
The plants were visually assessed for plant establishment and root and shoot development after 5 weeks growth under standard commercial conditions. Within the course of the trial, plant growth was sufficiently good to ensure flower set and fruit development in all treatments. This formed the basis of records of plant productivity in view of the commercial relevance of fruit, rather than foliage, production.
All tomato and cucumber plants established well both in untreated perlite and rockwool and in material previously treated with compound A and flushed out with water prior to planting. All plants scored on the basis of a subjective scale (1=poor; 5=good) scored 5 for both root production and healthy foliage production. No evidence of root death or scorching of foliage was noted for any treatment. Differences between treated plots and untreated controls (Tables 3 and 4) were slight in all cases. No evidence of phytotoxic residues in either rockwool or perlite was detected.
TABLE 3
______________________________________
Plant growth and fruit development of cucumbers
grown in substrate treated or untreated with
composition A (mean of 6 plants)
Height Fruit wt.
Total fresh
Crop (cm) No. of fruit
(g) weight (g)
______________________________________
Cucumber (i)
Rockwool 112.17 1 14.80 519.86
treated with
composition A
@ 1:50
S.E.M. 2.06 0 2.53 7.96
(standard
error of
mean)
Cucumber (i)
Rockwool 115.33 1.17 13.50 538.74
untreated
control
S.E.M. 6.31 0.41 8.56 61.40
Cucumber (ii)
Rockwool 139.83 6.00 1208.20 1610.80
treated with
composition A
@ 1:50
S.E.M. 13.60 0.89 254.00 266.00
Cucumber (ii)
Rockwool 158.20 6.40 1467.80 2003.50
untreated
control
S.E.M. 21.50 0.55 703.00 773.00
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TABLE 4
______________________________________
Plant growth and development of tomatoes grown
in substrate treated or untreated with
composition A (mean of 12 plants)
Height No. of fruit
Fruit wt.
Total fresh
Crop (cm) (Truss No)
(g) weight (g)
______________________________________
Tomato
Perlite 130.50 22.50 124.29 602.12
composition A (3.75)
@ 1:100
S.E.M. 1.66 1.17 7.56 10.47
(0.13)
Tomato
Perlite 124.50 20.25 137.49 595.14
untreated (3.17)
controls
S.E.M. 3.29 0.91 9.87 18.60
(0.16)
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Composition A at a dilution of 1:50 or 1:100 may also be used to prepare the liquid-supply troughs of a no-substrate hydroponics system for a fresh crop production cycle. The composition can be supplied through the irrigation system normally used for the nutrient liquid, the irrigation system and liquid supply troughs being thoroughly flushed afterwards with clean water.
EXAMPLE 2
Laboratory scale investigations were performed to investigate the effect of sample preparation on recovery of micro-organisms from treated and untreated perlite and the influence of organic matter on the effectiveness of composition A as a sterilant.
Method
1. Six or eight flasks were inoculated with 100 g perlite to which 3 or 4 had 66 cm.sup.3 water added and 3 or 4 had 66 cm.sup.3 of 1:100 composition A.
2. Flasks were shaken to mix the sample and were then incubated statically for 24 hours at 30° C.
3. The samples were then removed, the composition A inactivated, by either filter washing or sodium thiosulphate, and microbial counts performed.
Two procedures for inactivating composition A were tested. The first was the filter wash where composition B was diluted and then removed by washing the perlite with water, trapping bacteria in the wash water on a membrane filter, followed by homogenisation of both the perlite and the filters.
The other procedure was chemical inactivation with sodium thiosulphate.
There was no significant differences between the counts obtained using either method and it was concluded that sodium thiosulphate is a satisfactory inactivator to use in this application.
The first laboratory scale test using perlite taken from recently used bags in-situ in the greenhouse gave a 2.5 log reduction in bacterial counts and approximately 2 log reductions in yeast and fungal counts. There was some peaty organic matter in the perlite and laboratory trails were carried out to determine the impact of this organic material on the effectiveness of composition A.
Two samples of perlite were compared. A stored used perlite, low in organic matter provided the low organic matter samples. High organic matter perlite was prepared by the addition of the peaty residues of propagation pot compost, which were obtained from used perlite bags.
In the low organic matter perlite the bacterial counts were reduced by 4.5 logs while bacterial spore counts, yeasts and moulds were reduced by around 2 logs. The presence of the peaty material almost completely inactivated composition A. There was no significant reduction in bacterial or yeast numbers and there was only a 30 to 50% reduction in the numbers of bacterial spores and moulds.
Reference should now be made to Tables 5 and 6.
TABLE 5
______________________________________
Counts per gram dry weight of treated and untreated
low organic content perlite (lab scale)
N.B. Counts are exponents,
i.e. 2.74 10E7 = 2.74 × 10.sup.7
Treat-
ment Time Bacteria Spores Yeast Moulds
______________________________________
A X 2.74 10E7 1.08 10E4
1.22 10E5
9.46 10E4
A X 1.57 10E7 1.04 10E4
4.05 10E4
8.11 10E4
A X 1.84 10E7 1.19 10E4
2.0 10E4 6.76 10E4
B X 1.57 10E7 8.78 10E3
2.0 10E4 2.7 10E4
B X 2.19 10E7 1.36 10E4
4.05 10E4
8.11 10E4
B X 2.38 10E7 1.84 10E4
2.0 10E4 5.4 10E4
C X 2.0 10E2 2.0 10E2
2.0 10E1 2.0 10E2*
C X 2.0 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
C X 2.0 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
D X 2.0 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
D X 4.05 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
D X 1.62 10E3 2.0 10E2
2.0 10E2 2.0 10E2*
A Y 9.86 10E6 1.32 10E4
2.0 10E4 2.0 10E4*
A Y 1.42 10E7 2.76 10E4
2.0 10E4 2.0 10E4*
A Y 1.40 10E7 8.65 10E3
1.22 10E4
8.11 10E3
B Y 1.00 10E7 2.85 10E4
2.0 10E4 2.0 10E4*
B Y 1.70 10E7 1.00 10E4
2.0 10E4 2.70 10E4
B Y 8.65 10E6 7.43 10E4
1.22 10E4
1.08 10E4
C Y 2.0 10E2 2.7 10E2
4.05 10E2
4.05 10E2
C Y 2.0 10E2 5.4 10E2
2.0 10E2 2.0 10E2*
C Y 2.0 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
D Y 0.0 10E0 5.40 10E2
2.0 10E3 3.51 10E3
D Y 4.05 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
D Y 2.0 10E2 2.0 10E2
2.0 10E2 2.0 10E2*
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Key:
Treatment:
A = Before application and filter wash
B = Before application and thiosulphate inactivation
C = After filter wash
D = After thiosulphate
Time:
X = After 24 hours treatment
Y = After 48 hours treatment
*Inaccurate due to plate overgrowth
TABLE 6
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Counts per gram dry weight of treated and untreated
high organic content perlite (lab scale)
N.B. Counts are exponents,
i.e. 1.61 10E8 = 1.61 × 10.sup.8
Treatment
Bacteria Spores Yeast Moulds
______________________________________
A 1.61 10E8
4.05 10E6 8.95 10E5
2.04 10E6
A 1.94 10E8
5.38 10E6 1.31 10E6
4.20 10E6
A 1.30 10E8
5.04 10E6 7.82 10E5
2.87 10E6
B 1.33 10E8
3.28 10E6 1.02 10E6
1.91 10E6
B 2.21 10E8
3.15 10E6 1.18 10E6
3.28 10E6
B 1.25 10E8
4.52 10E6 7.82 10E5
3.26 10E6
C 1.31 10E8
1.67 10E6 1.09 10E6
7.71 10E5
C 1.06 10E8
1.95 10E6 1.17 10E6
8.71 10E5
C 1.35 10E8
1.46 10E6 9.37 10E5
8.86 10E5
D 1.26 10E8
1.52 10E6 1.29 10E6
1.08 10E6
D 1.15 10E8
1.38 10E6 1.34 10E6
8.96 10E5
D 1.42 10E8
1.26 10E6 8.86 10E5
7.59 10E5
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Key: Treatment
A = Before application and filter wash
B = Before application and thiosulphate inactivation
C = After filter wash
D = After thiosulphate
In vivo application
Perlite, separate from that used for crop production, was treated with composition A and assessed for microbiological `kill`. This was carried out according to the following schedule.
1. One or two plastic `tubes` containing ten perlite bags were chosen for sampling.
2. The 10 perlite bags were labelled as appropriate.
3. Each perlite bag contained 3 inlets for composition A, which were sampled.
4. The samples were removed from the whole depth of each inlet using a scoop sterilised with alcohol between samples. Samples remote from the inlet were taken by cutting the plastic bag and sampling to approximately 10 cm depth.
5. Once sampled, the hole left by sampling was caved in using adjacent perlite.
6. Samples were put into stomacher bags and transferred to the lab for testing.
N.B. The method of sampling was randomised using 10 petri dishes with 3 labels in each, i.e. left, right and middle. A label was chosen from each dish in turn, so as to create a sampling plan.
No attempt was made to remove any peat, algae, roots etc, which were present in the samples.
Two series of tests were run using perlite.
In test series 1, the perlite around the inlets was contaminated with peaty material derived, presumably, from the rooting compost in which the plants were grown before planting out. The perlite in test series 2 was free from such extraneous material. Perlite bags were tested in the glass house before and directly after treatment with composition A.
TEST SERIES 1
The composition A reduced the bacterial count by 2.5 log, roughly 300- fold. After one week the numbers of bacteria had recovered to about 30% of the original count. Immediately after treatment yeast numbers were only reduced by 1.2 logs, about 15-fold, and the moulds by only 0.7 logs, or about 5-fold. Numbers of yeasts recovered to their original level after 1 week but mould numbers did not increase.
TEST SERIES 2
There was roughly a 3 log reduction in bacterial counts. The bacterial spore count was reduced by only 1.2 logs while the yeasts and moulds were reduced by 2 logs. The bacterial counts recovered after 1 week while the small increases in spore and mould numbers were not statistically significant.
There is a greater reduction in microbial counts in series 2 compared to series 1. This confirms the laboratory scale findings that composition A is more effective in disinfecting perlite containing no peat compost around the planting sites.
Reference should be made to Tables 7 and 8.
TABLE 7
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Series 1
Counts per gram dry weight of perlite before, during
and after treatment.
N.B. Counts are exponents,
i.e. 3.31 10E8 = 3.31 × 10.sup.8
Type
Point (a) Bacteria Yeast Moulds
______________________________________
1 A 3.31 10E8 1.58 10E5
4.74 10E6
2 A 6.23 10E8 7.62 10E6
1.63 10E7
3 A 4.03 10E8 1.65 10E4
1.58 10E7
4 A 1.21 10E9 3.27 10E4
5.81 10E5
5 A 5.55 10E8 2.67 10E9
3.75 10E7
6 A 9.97 10E8 1.68 10E6
1.68 10E6
7 A 7.35 10E8 1.7 10E6
9.62 10E6
8 A 1.29 10E8 1.18 10E6
2.24 10E6
9 A 2.5 10E8 6.13 10E5
1.66 10E6
10 A 2.97 10E8 1.51 10E6
1.92 10E6
11 B 3.42 10E6 4.69 10E5
1.78 10E5
12 B 2.46 10E6 1.64 10E5
1.38 10E6
13 B 1.08 10E6 6.68 10E5
4.01 10E6
14 B 2.79 10E6 1.72 10E5
3.21 10E7
15 B 1.81 10E5 4.29 10E6
9.15 10E6
16 B 3.57 10E6 1.6 10E4
2.17 10E5
17 B 9.40 10E5 6.42 10E3
4.28 10E4
18 B 5.39 10E4 1.88 10E3
2.92 10E5
19 B 1.07 10E6 4.63 10E4
8.91 10E5
20 B 1.43 10E6 1.05 10E4
2.02 10E5
21 C 9.39 10E7 8.36 10E5
1.91 10E6
22 C 2.96 10E8 1.44 10E6
8.21 10E5
23 C 1.19 10E8 1.38 10E6
1.36 10E6
24 C 4.06 10E8 9.30 10E5
3.97 10E5
25 C 2.84 10E8 4.20 10E5
1.09 10E6
26 C 2.81 10E7 5.83 10E5
9.87 10E5
27 C 7.17 10E7 1.62 10E6
1.90 10E5
28 C 4.88 10E6 1.39 10E5
1.50 10E5
29 C 1.71 10E8 6.72 10E6
2.56 10E6
30 C 4.14 10E8 6.64 10E5
6.35 10E5
______________________________________
Key: (a)
A = Before treatment
B = Immediately after application
C = After flushing
TABLE 8
______________________________________
Series 2
Counts per gram dry weight in treated and untreated
low organic content perlite in a glasshouse.
N.B. Counts are exponents,
i.e. 5.08 10E7 = 5.08 × 10.sup.7
Treat-
ment Type Bacteria Spore Yeast Moulds
______________________________________
A X 5.08 10E7 4.41 10E4
1.17 10E6
3.0 10E7*
A Y 7.80 10E7 2.15 10E4
2.43 10E6
2.43 10E5
A X 9.18 10E7 9.23 10E4
2.18 10E5
1.30 10E6
A Y 7.85 10E7 2.00 10E5
4.45 10E5
5.0 10E7*
A X 1.38 10E7 8.65 10E6
1.65 10E7
1.41 10E7
A Y 2.38 10E8 1.83 10E5
4.46 10E6
1.15 10E6
A X 1.50 10E8 5.35 10E4
7.45 10E5
4.00 10E7*
A Y 7.57 10E7 7.35 10E4
7.00 10E4
4.00 10E7*
A X 7.61 10E7 1.31 10E5
1.20 10E6
2.00 10E7*
A Y 1.54 10E8 1.83 10E4
6.24 10E4
1.25 10E5
A X 1.32 10E7 6.27 10E4
9.00 10E4
1.91 10E5
A Y 6.36 10E7 4.57 10E4
8.00 10E4
2.24 10E5
A X 8.02 10E7 1.50 10E5
8.58 10E5
2.20 10E7*
A Y 2.11 10E7 3.76 10E4
9.01 10E5
5.00 10E5
A X 6.87 10E7 1.80 10E4
2.43 10E6
9.59 10E5
A Y 3.47 10E7 2.56 10E4
2.00 10E4
2.00 10E7
A X 9.38 10E7 3.33 10E5
9.96 10E5
1.42 10E7
A Y 1.53 10E8 1.01 10E5
1.53 10E6
7.64 10E4
A X 2.46 10E7 3.38 10E4
9.00 10E4
4.56 10E7
A Y 1.44 10E7 2.06 10E4
9.00 10E4
4.78 10E7
B X 8.00 10E2 8.00 10E2
8.00 10E2
1.30 10E3
B Y 8.89 10E6 6.80 10E3
1.46 10E4
4.39 10E4
B X 5.56 10E8 2.04 10E3
2.80 10E6
8.99 10E6
B Y 4.08 10E5 6.43 10E4
1.00 10E3
7.00 10E6*
B X 3.18 10E3 1.59 10E3
1.00 10E3
1.06 10E3
B Y 5.36 10E5 3.22 10E3
1.00 10E3
7.50 10E5
B X 3.20 10E5 6.80 10E4
1.00 10E3
3.05 10E4
B Y 2.68 10E4 3.35 10E4
1.91 10E3
1.44 10E3
B X 1.83 10E3 2.44 10E3
1.22 10E3
3.06 10E3
B Y 5.36 10E5 7.49 10E3
3.12 10E4
4.37 10E4
B X 5.63 10E3 9.39 10E2
9.00 10E2
9.00 10E2*
B Y 1.15 10E5 2.19 10E3
8.00 10E2
4.60 10E4
B X 1.31 10E3 1.00 10E3
1.00 10E3
1.31 10E3
B Y 1.18 10E4 1.31 10E3
1.00 10E3
6.54 10E4
B X 2.68 10E3 1.00 10E3
1.00 10E3
6.71 10E2
B Y 5.63 10E5 2.64 10E3
1.00 10E3
9.10 10E4
B X 1.66 10E4 5.53 10E2
1.00 10E3
2.21 10E5
B Y 1.17 10E7 2.74 10E4
1.00 10E3
2.28 10E5
B X 3.14 10E3 6.28 10E2
1.00 10E3
1.26 10E3
B Y 1.15 10E7 1.10 10E5
1.00 10E3
7.00 10E6*
C X 5.43 10E6 2.22 10E3
1.00 10E4
1.11 10E4
C Y 5.69 10E7 2.00 10E4
3.01 10E6
5.57 10E4
C X 1.38 10E7 2.59 10E3
1.00 10E4
1.00 10E4*
C Y 1.69 10E8 1.75 10E5
1.15 10E7
6.00 10E7*
C X 2.07 10E6 9.74 10E3
1.00 10E4
5.93 10E6
C Y 1.09 10E6 2.16 10E3
1.08 10E5
4.31 10E5
C X 1.52 10E7 5.99 10E4
1.00 10E4
1.16 10E5
C Y 3.12 10E7 2.48 10E4
1.00 10E4
5.74 10E4
C X 1.00 10E7 5.00 10E4
1.00 10E4
2.08 10E6
C Y 2.20 10E7 4.77 10E3
7.94 10E5
1.06 10E5
C X 6.00 10E6 6.58 10E2
1.00 10E4
1.00 10E4*
C Y 1.63 10E7 1.24 10E4
1.55 10E4
3.09 10E4
C X 4.96 10E7 2.89 10E3
1.15 10E4
2.31 10E4
C Y 5.29 10E7 5.71 10E4
1.00 10E4
6.91 10E5
C X 1.62 10E7 1.40 10E3
1.00 10E4
1.00 10E4*
C Y 1.20 10E7 2.08 10E4
1.00 10E4
4.02 10E6
C X 4.25 10E8 2.83 10E3
1.00 10E4
1.00 10E4*
C Y 6.24 10E7 2.11 10E4
1.00 10E4
6.14 10E6
C X 1.41 10E7 6.40 10E2
1.00 10E4
1.28 10E4
C Y 1.56 10E7 2.51 10E4
5.61 10E5
1.00 10E4*
______________________________________
Key:
Treatment:
A = Before application
B = Immediately after application
C = 1 week after application/flushing
Location:
X = Below inlet
Y = Horizontally remote from inlet
= *Inaccurate due to overgrown plates. Plant growth under simulated
commercial conditions following substrate treatment were investigated:
TOMATO PRODUCTION
Procedure
Tomato seed, cv. Shirley, was sown into fine grade perlite, and propagated under standard commercial conditions. Seedlings were pricked out into horticultural grade perlite in 13 cm mesh-based pots. These were maintained in a heated glasshouse until the first flower on the majority of plants was open, at which point they were planted by plunging the pots into the perlite contained in bags, arranged as previously described. Planting was carried out after 45 days.
On the basis of previous trials results application of composition A was at a dilution rate of 1:100 with mains water. This was applied by commercially available irrigation harness as previously described.
Experimental design was modified because of the complexity of the experimental procedure adopted for microbiological assay and the need for re-installation of treated and untreated substrates for the procedures described previously. Plots were of previously used perlite only for crop trials.
Crop yields were recorded over the period 101/2 to 12 weeks after plating. Yield was only recorded in terms of gross marketable class I yield irrespective of colour. This was done in order to include green and blotchy red fruit, produced solely as a result of low autumn/early winter light, in the data. Fruit of this type would mature normally for crops grown in spring/summer as is generally the case in the U.K.
Nutrients were applied according to current WSC recommendations.
No evidence of phytotoxic residues were detected in crops grown in perlite previously treated with composition A and subsequently flushed with water. No scorching of leaf margins, as noted in some previous, preliminary trials work occurred, nor were any chlorotic symptoms, typical of low dose phytotoxicity, observed. All foliage was of a uniform dark green colour typically associated with healthyplants.
Early yield of fruit from plots previously treated with product was greater than that from untreated substrates, (Table 9). This trend continued at the second pick but the situation was subsequently reversed. In each case differences between treatments were small. In economic terms early plant establishment and yield is more important than development later in the cropping sequence. These results support the view that composition A can have a beneficial effect on plant establishment in comparison to untreated plots, even in the absence of specific disease organisms.
TABLE 9
______________________________________
Cumulative yield of tomato fruit from plots
treated with or without composition A
Yield (kg/plot)
Pick No.
Treatment 1 2 3
______________________________________
Treated 0.29 0.77 1.53
Untreated 0.27 0.65 1.57
______________________________________
CUCUMBER PRODUCTION
Procedure
Cucumber seed (cv. Birgit) was surface sown onto moist perlite.
Chitted seeds were transferred to horticultural grade perlite contained in 13 cm mesh-based pots and propagated under standard commercial conditions.
Planting, after 45 days, was by plunging the pots directly into planting holes of the previous crop in perlite. Treatments were of samples previously treated or untreated with composition A.
The crop was trained vertically, and side shoots removed until plants reached the glasshouse eaves height.
Picking commenced 31/2 weeks after planting, and continued for seven weeks. Fruit was graded for quality according to EEC regulations. Class I fruit only was graded for weight of individual fruit.
Cucumber plants, which are widely acknowledged as being very sensitive to changes both in aerial and root environments, all established well. No evidence of phytotoxic residues, either as necrosis or chlorosis, was observed. Foliage was all a uniform dark green colour.
Total yield of fruit was better in plots previously treated with composition A (Table 10).
Differences in yield were most marked for early picks, indicating a beneficial effect of product on plant establishment and young plant development.
In terms of fruit grade and quality, treatments were not significantly different. Fruit was predominantly small (250-400 g) with a high proportion of Class II fruit, resulting from reducing light levels over the trial period (Table 11).
TABLE 10
______________________________________
Mean cumulative yield of cucumber fruit, in plots
treated with composition A and in untreated controls.
Yield (fruit/plant)
Picking Date
Treatment
24th Oct 31st Oct 7th Nov
21st Nov
12th Dec
______________________________________
Treated 0.61 1.00 1.45 1.91 3.44
Untreated
0.50 0.80 1.39 1.89 3.38
______________________________________
TABLE 11
______________________________________
Comparative quality and numbers of fruit grown in
treated and untreated perlite.
Mean numbers of fruit/plot
Class I Total
250-400 g
400-500 g
Class I Class II
Total
______________________________________
Treated 68 2 70 33 103
Untreated
66 1 67 35 102
______________________________________
EXAMPLE 3
An aqueous concentrate comprising peracetic acid was made by mixing the following ingredients.
______________________________________
Percent m/m
______________________________________
Acetic acid (80% solution)
12.8
Hydrogen peroxide (50% solution)
51.5
2,6-pyridine dicarboxylic acid
0.015
Dodecylbenzene sulphonic acid
1.2
Acetodiphosphonic acid
2.0
Deionised water 32.485
______________________________________
The composition (composition "B") was mixed, left to stand at 50. degree. C. for 19 hours, cooled to ambient temperature and sampled prior to use to determine the peracetic acid content.
The effect of changes in the concentration of composition B and the effect of increased rate of application and efficacy in two different growing substrates were investigated.
MATERIALS
Corp: Tomato cultivar was MARATHON
Substrates: Perlite (Tilcon Ltd.) Rockwool (Capogro [Pilkington Ltd])
Composition B application rates for each substrate were as follows.
______________________________________
1:25 1:50 1:75 1:100
Control
______________________________________
Plot size was 16 plants/plot, with three replicates per treatment.
Application method
Bulk tank of diluted composition B was pumped directly via standard irrigation harness to crop plant `stations`.
Mean application rate was 14.84 dm.sup.3 per hour (10.975 dm. sup. 3).
Substrates were treated to excess run-off, (2.5 dm.sup.3 /station for rockwool, 5.3 dm.sup.3 /station for perlite).
The crop was arranged in a statistically randomised layout of three replicates of each treatment.
Crop substrates were sampled for populations of fungi, bacteria and yeasts before, immediately after application of composition B and after subsequent rinsing.
No evidence of phytotoxicity was noted in plants established by normal commercial procedures into Crop substrates treated with composition B and subsequently flushed with clean water.
With regard to efficacy, composition B's ability to reduce populations of fungi, bacteria and yeasts was again demonstrated. Results are summarised in Tables 12 to 17.
In the case of Rockwool substrates application of composition B at 1:50 or above to markedly reduce the population of fungi, bacteria and yeasts. In the case of perlite a similar effect was achieved at a lower concentration;-1:75. It is a feature of this technique that recolonisation of the material by naturally occurring organisms takes place within a short time. This feature of low persistence/low toxicity is compatible with the theoretical reintroduction of beneficial organisms in an integrated biological control programme.
TABLE 12
______________________________________
Rockwool Bacterial Counts
Counts per gram dry weight
Before After After
disinfection
disinfection
washing
______________________________________
Control 4.70 × 10.sup.8
6.19 × 10.sup.7
4.90 × 10.sup.8
(water) 4.32 × 10.sup.6
9.23 × 10.sup.8
8.30 × 10.sup.8
2.88 × 10.sup.7
1.47 × 10.sup.9
1.98 × 10.sup.8
1:100 1.35 × 10.sup.9
5.76 × 10.sup.5
1.30 × 10.sup.9
dilution 7.47 × 10.sup.7
1.53 × 10.sup.8
2.15 × 10.sup.9
1.63 × 10.sup.7
9.36 × 10.sup.6
4.20 × 10.sup.7
1:75 1.22 × 10.sup.7
1.81 × 10.sup.8
2.68 × 10.sup.7
dilution 2.70 × 10.sup.5
1.42 × 10.sup.8
1.66 × 10.sup.9
3.20 × 10.sup.8
3.93 × 10.sup.8
1.96 × 10.sup.4
1:50 1.72 × 10.sup.8
1.78 × 10.sup.4
1.84 × 10.sup.9
dilution 3.24 × 10.sup.8
< 2.70 × 10.sup.2
2.60 × 10.sup.9
1.22 × 10.sup.6
< 2.70 × 10.sup.2
1.23 × 10.sup.4
1:25 7.57 × 10.sup.6
< 2.70 × 10.sup.2
1.49 × 10.sup.9
dilution 5.27 × 10.sup.6
3.57 × 10.sup.6
1.39 × 10.sup.7
2.83 × 10.sup.8
< 2.70 × 10.sup.2
3.32 × 10.sup.6
______________________________________
TABLE 13
______________________________________
Rockwool Yeast Counts
Counts per gram dry weight
Before After After
disinfection disinfection
washing
______________________________________
Control
2.83 × 10.sup.6
3.43 × 10.sup.3
2.24 × 10.sup.6
(water)
2.70 × 10.sup.4
3.93 × 10.sup.3
4.03 × 10.sup.4
2.30 × 10.sup.4
3.70 × 10.sup.5
2.26 × 10.sup.4
1:100 < 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
1.05 × 10.sup.4
6.40 × 10.sup.3
1.20 × 10.sup.5
5.44 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:75 5.73 × 10.sup.3
6.27 × 10.sup.5
6.35 × 10.sup.5
dilution
< 2.70 × 10.sup.3
4.16 × 10.sup.3
5.42 × 10.sup.4
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:50 < 2.70 × 10.sup.3
< 2.70 × 10.sup.2
4.36 × 10.sup.5
dilution
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
4.05 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:25 < 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
9.46 × 10.sup.5
2.69 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
______________________________________
TABLE 14
______________________________________
Rockwool Mould Counts
Counts per gram dry weight
Before After After
disinfection disinfection
washing
______________________________________
Control
2.53 × 10.sup.6
1.49 × 10.sup.3
9.47 × 10.sup.6
(water)
< 2.70 × 10.sup.3
6.05 × 10.sup.5
2.78 × 10.sup.5
1.01 × 10.sup.5
5.86 × 10.sup.5
2.90 × 10.sup.5
1:100 1.51 × 10.sup.7
1.46 × 10.sup.5
6.05 × 10.sup.4
dilution
4.21 × 10.sup.5
5.22 × 10.sup.5
6.84 × 10.sup.6
1.75 × 10.sup.5
7.52 × 10.sup.3
3.80 × 10.sup.4
1:75 3.64 × 10.sup.5
7.09 × 10.sup.5
7.47 × 10.sup.6
dilution
1.62 × 10.sup.4
1.52 × 10.sup.5
4.69 × 10.sup.6
5.59 × 10.sup.5
1.89 × 10.sup.6
< 2.70 × 10.sup.2
1:50 1.06 × 10.sup.6
2.27 × 10.sup.3
6.28 × 10.sup.6
dilution
1.54 × 10.sup.6
< 2.70 × 10.sup.2
2.55 × 10.sup.7
1.49 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:25 1.22 × 10.sup.6
3.08 × 10.sup.3
> 2.00 × 10.sup.7
dilution
5.14 × 10.sup.6
3.11 × 10.sup.4
> 2.00 × 10.sup.5
2.62 × 10.sup.6
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
______________________________________
TABLE 15
______________________________________
Perlite Bacterial Counts
Counts per gram dry weight
Before After After
disinfection disinfection
washing
______________________________________
Control
1.33 × 10.sup.8
9.67 × 10.sup.7
2.18 × 10.sup.8
(water)
9.71 × 10.sup.7
1.05 × 10.sup.8
8.63 × 10.sup.6
3.44 × 10.sup.7
7.02 × 10.sup.5
1.85 × 10.sup.5
1:100 1.70 × 10.sup.7
2.68 × 10.sup.6
3.01 × 10.sup.6
dilution
4.60 × 10.sup.7
< 2.70 × 10.sup.2
1.83 × 10.sup.6
1.10 × 10.sup.7
3.36 × 10.sup.6
4.20 × 10.sup.7
1:75 1.22 × 10.sup.7
< 2.70 × 10.sup.2
1.58 × 10.sup.7
dilution
1.11 × 10.sup.8
< 2.70 × 10.sup.2
2.93 × 10.sup.6
1.80 × 10.sup.7
< 2.70 × 10.sup.2
1.06 × 10.sup.4
1:50 1.50 × 10.sup.8
< 2.70 × 10.sup.2
1.00 × 10.sup.7
dilution
8.53 × 10.sup.6
< 2.70 × 10.sup.2
1.22 × 10.sup.8
4.97 × 10.sup.6
2.84 × 10.sup.3
< 2.70 × 10.sup.2
1:25 3.20 × 10.sup.7
< 2.70 × 10.sup.2
9.91 × 10.sup.7
dilution
1.40 × 10.sup.8
4.05 × 10.sup.2
4.55 × 10.sup.8
2.32 × 10.sup.6
< 2.70 × 10.sup.2
6.64 × 10.sup.4
______________________________________
TABLE 16
______________________________________
Perlite Yeast Counts
Counts per gram dry weight
Before After After
disinfection disinfection
washing
______________________________________
Control
2.22 × 10.sup.5
< 2.70 × 10.sup.2
2.24 × 10.sup.5
(water)
5.74 × 10.sup.4
4.20 × 10.sup.4
< 2.70 × 10.sup.2
2.15 × 10.sup.4
< 2.70 × 10.sup.2
3.12 × 10.sup.3
1:100 1.97 × 10.sup.5
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
6.43 × 10.sup.4
< 2.70 × 10.sup.2
9.94 × 10.sup.4
3.21 × 10.sup.5
4.11 × 10.sup.4
4.48 × 10.sup.4
1:75 1.36 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
4.30 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1.37 × 10.sup.6
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:50 < 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
< 2.70 × 10.sup.3
3.35 × 10.sup.3
< 2.70 × 10.sup.2
1:25 1.38 × 10.sup.5
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
dilution
3.28 × 10.sup.4
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
______________________________________
TABLE 17
______________________________________
Perlite Mould Counts
Counts per gram dry weight
Before After After
disinfection disinfection
washing
______________________________________
Control
2.57 × 10.sup.5
< 2.70 × 10.sup.2
5.59 × 10.sup.5
(water)
2.68 × 10.sup.4
5.08 × 10.sup.4
1.36 × 10.sup.5
2.23 × 10.sup.5
7.34 × 10.sup.3
7.28 × 10.sup.5
1:100 > 2.00 × 10.sup.7
1.82 × 10.sup.6
1.72 × 10.sup.5
dilution
1.37 × 10.sup.5
< 2.70 × 10.sup.2
3.05 × 10.sup.4
> 2.00 × 10.sup.7
7.80 × 10.sup.4
2.56 × 10.sup.5
1:75 5.45 × 10.sup.4
< 2.70 × 10.sup.2
> 2.00 × 10.sup.7
dilution
6.62 × 10.sup.5
< 2.70 × 10.sup.2
3.86 × 10.sup.5
1.68 × 10.sup.5
< 2.70 × 10.sup.2
4.89 × 10.sup. 3
1:50 > 2.00 × 10.sup.7
< 2.70 × 10.sup.2
4.08 × 10.sup.6
dilution
> 2.00 × 10.sup.5
3.30 × 10.sup.3
< 2.70 × 10.sup.2
> 2.00 × 10.sup.5
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
1:25 2.29 × 10.sup.6
< 2.70 × 10.sup.2
2.33 × 10.sup.3
dilution
7.55 × 10.sup.4
< 2.70 × 10.sup.2
1.26 × 10.sup.7
< 2.70 × 10.sup.3
< 2.70 × 10.sup.2
< 2.70 × 10.sup.2
______________________________________ | |
As individual creative professionals, we as designers need to find our processes and ways of gaining inspiration as everyone is different in their approach when commissioned for a design project they then need to find inspiration for.
In this community-based article, we discuss where to find logo design inspiration, with insights from designers we have interviewed, and other designers from the design community who answer the question Where do you find logo design inspiration?
I personally have a set process when designing that I trust and follow, but I don’t have a set routine to find inspiration this varies from project to project. It can be a quick or really slow process and I think most designers will agree.
I try not to force ideas they come to me once I have done the research stage and I’m fully aware of the client and scope of work, their competitors and the goals the company is aiming to achieve and the problems they are facing.
One of my biggest inspirations to me is my dad and his approach to life, he loved it and he lived it and let nothing stand in his way. He was a thinker and a doer he had a strategic and creative mindset, and planned everything before taking action. He was logical in his approach. He wasn’t afraid to fail and focused to achieve his goal no matter what!.
I’m a firm believer in trusting your process as inspiration is in the process and inspiration can come to you at any point during the process, often at times unexpected.
The simplest solutions are never the easiest to develop by trying to take the most relevant and simplest elements of a brand and communicating it in the simplest visual form is always challenging. All logo designers should be aiming to create meaning in its simplest form that communicates visually to the brand’s audience.
In a recent designer interview, we did with Lance Wyman a legend in logo design most well known for creating the Mexico 68 logo for the 1968 Olympic games in Mexico city.
When asked
“What is your favourite style of logo design? And why?”
he answered:
“I seem to migrate to simple with meaning”
in another question about what his advice would be for new logo designers, he went on to say:
“Make an effort to really understand what you are out to represent” “See clearly as you can what you are dealing with”
When it came to gaining inspiration in another designer interview, we asked Stefan Sagmeister another legend in the design world.
“What is your daily inspiration when you design?”
Stefan answered:
“One of my most frequent sources of inspiration is a newly occupied hotel room. I find it easy to work in a place far away from the studio, where thoughts about the implementation of an idea don’t come to mind immediately but I can dream a bit more freely.
Ideas can’t be forced they come at any time and more than normally a time that’s is not appropriate. I had some really strange timings like one time I was driving down the motorway on the way to a client meeting and suddenly an idea came into my head for another identity I was working on for another client I pulled into the next service station whipped out my notebook and pencil and started sketching away.
I also like to go to the coast a lot as I love the sea air and walking on the beach. I have had times where ideas have come to me while taking a walk on the beach and drawn the idea in the sand then took a pic on my phone for later.
Ideas come to us when we allow our minds to empty and we forget and stop forcing the focus on the ideation stage of a project. It’s like Jacob Boghosian says in his designer interview
“Inspiration happens around us so I try to keep myself open to ideas wherever I go.”
and Mark Richardson in his designer interview mentions
“Anything can be inspiring and ideas usually evolve when my mind is elsewhere.”
What other designers in the community are saying
I recently reached out to some other creative and logo designers to ask them the question “Where do you find logo design inspiration?” and here’s what they said.
“I’m sure for a lot of logo designers out there this is quite an easy question to answer. People have set routines and procedures in place to help. Whether it’s creating a mood board, brainstorming or simply browsing logo galleries, there is a “go-to” that helps them get their creative juices flowing.
I don’t really do any of the above, not for inspiration anyway. For me, I’ve always just allowed ideas and concepts to just pop into my head out of nowhere. Sitting with the specific task of coming up with ideas just don’t work for me. I ended up just sitting wasting time.
My method of finding logo inspiration is to distract myself. If I let myself almost forget about the logo project at hand, it’s almost guaranteed that at some point an idea will randomly come to mind when I’m doing something else.
Personally, I don’t have the ability to sit and force my creative juices to flow. It’s just going to spontaneously happen, and when it does that’s the time to sit down and focus. Obviously, the main downside to this is when you’re out and about with friends and family, and all of a sudden you’re scribbling a design down on a bar menu to make sure you don’t forget it!”
Mark Sims – The Logo Mark
“When looking for design inspiration, I always tend to search for photos first. I search for photo’s that fits the target audience but also the feeling I want to express in my design. In this stage I don’t look for logo designs or branding projects. I use photos a lot because they help me to set the mood/feeling for the project. Using photos in this first stage often lets me really connect with the audience.
After receiving the client brief I started to search the internet for photos and create some kind of mood board with al the photo’s I have collected. In my first sketching face I just use my mood board and start sketching. I try to find some usable shapes in the photos. It can be the shape of a building, a flower, a shoe, a person, or just a face or arm. After creating this ‘mood board’ It’s time to search for logos and symbols that fit my mood board. Most of my inspiration I get from the internet. I spend a lot of time on sites like Behance and Dribble, twitter and Pinterest for logo designs. But I also still use my design books.
Why don’t I start with searching for logo designs?
When I start with searching for logo designs it kind of feels like I limit myself and only focus on the symbol and not on the target audience or the feeling I want to express. When I do start with logo designs I often pick out designs that I like, but this does not always fit the brand I’m designing for. So I personally need a visual image of the brand and its target audience to come up with a logo design that fits.”
I find photos on different sites like:
Some examples of the mood boards I have done:
Kevin van Eijk – dipx Design
“My way of work is quite simple. I collect information about the company, it’s products, the history and philosophy. Anything that helps you find a concept.
I never try to be overly innovative and keep my designs as minimal as possible. I will reach out to create a memorable design and keep it very minimal. The downside of this approach is the time I invest.
It can take weeks to find “that one” design. But if someone is willing to wait, the reward is much greater.”
Alen Pajazetovic Helvetiphant™
“I largely use the usual tools for inspiration such as feature sites, Google and Pinterest. In doing this, I look for both other people’s work and general images that give me the right vibe for a certain project. However, when I am really stuck for what to do, I most often let the inspiration come to me rather than the other way around. I find that personally when I try to force ideas they rarely end up in a quality design.
For this reason, I’ll do the usual routine of research, exploring, mind maps, mood boards and initial sketches to make sure that the overall feel of the project is clear in my mind and that the vision is solid. Then I simply leave the project for 2 days, try to work in a different place, go for a walk and tackle other things.
I was a scientist before I was a designer so I am a bit of nerd when it comes to how the brain works and how we can get the most out of it. I like to have a change of environment or go for a stroll because everything we see is made up of shapes and patterns and all of them are subconsciously stored in our brains. When we then relax, our brain is able to make connections between the things we’ve seen and the problems we need to solve.
Thomas Edison came up with the concept of the light bulb just as he was drifting off to sleep rather than whilst researching and pouring over literature and blueprints. I therefore like to tackle creative block with a few days of focusing on other things, after which I’ll usually have a list of ideas to explore that have come to me without forcing them.
In the past I’ve found inspiration for a logo in the negative space of some flowery curtains or the shapes made up on some frosted glass. I don’t see these as random sparks of inspiration but rather associations my brain has made between a creative problem I have been storing and a solution it’s seen.”
Aimee Castle – Clarity Zest Creative
Experience as a designer
There really is no magic process or way of finding inspiration, it comes with experience. As designers we all learn from past experiences and the things we learn are normally things that cannot be taught and we self-develop over time we change our style, our process and way of thinking, we are always improving our mindset to work on bigger and more demanding projects over time.
I myself have been designing logos for 20 years and learn a new thing with each project I work on which has given me the experience to develop simple and meaningful logos that are fit for purpose and effectiveness of the clients brand.
I have always had a talent for coming up with ideas right from being at school studying A Level Art and Graphic Design through to my university years when studying for my design diplomas at first yes I admit it was more challenging for me to develop simple and effective concepts but as the years have passed I have gained more experience running my design studio and working with clients internationally that it does come more natural to me now but it has taken years of study and research and practice and the overall commitment to doing what I love tot do has allowed me to train my brain to work the way it does and visualise this concepts how I do.
Research is key
A lot of designers overlook research and think they can just jump in and start designing and maybe you do but you need to ask yourself is this the right way to go about doing this? Your client is expecting you to solve a problem, not just design a logo.
In order to solve a problem, you need to determine how this will be achieved. Knowledge of the client and the brand is crucial there are other key aspects also such as the target market and business competitors and how my client wants their brand to be positioned into the market so it’s not just about designing a logo for the client but designing a logo for the consumer and this takes thorough research which is a very time-consuming process but a very effective stage that is vital when designing a logo and without research an effective solution is not possible to achieve.
The solution is unique to every client
We have all heard the common, cliché phrase “there is no one size fits all approach” when designing a logo and brand identity. When conducting this vital research stage in the process, the findings will prove that there is no fit for any instance magic solution as every client and their brand is unique and with unique comes a bespoke solution that’s going to be appropriate and fit for purpose for the brand. It will solve a problem and communicate the brand to the consumer, it will gain trust in an instance and be a unique visual identifier for the brand.
Think simple, not complex, try to take the most basic elements that will communicate to the consumer, be clever and think differently but above all else know what you are dealing with and solve the problem. It’s not about style or trends, it’s about communicating visually and solving the problem.
In another recent interview, we had with Brand Strategist and author of Creative Strategy and Business of Design Douglas Davis when asked
“What makes a great logo? and an excellent identity system?”
Douglas sums it up quite simply with his answer
“Clients approach us with various forms of the exact same request: Increase my sales, reposition my brand, expand my customer base. It’s all the same ask: Solve my problem. An excellent identity system or great logo does that.”
Another example is from Alina Wheeler who we recently interviewed and when we asked her.
“What is your favourite style of logo design? And why?”
she replied:
“I have no favourite style. I like what solves the problem best and fills the need across communications channels. I like logos that give the client latitude to grow, are dramatically differentiated within their sector, and work well across touchpoints—from an app icon to a digital billboard on Times Square.”
When your next commissioned to design a logo or full brand identity system remember this is not a fashion contest about who can design the fanciest logo with all the latest trends remember the saying “KISS” Keep it simple stupid! Our job as designers is to solve problems and communicate visually. So do the research, know what you’re dealing with think like the consumer, think – Simple that’s fit for purpose, adaptable, memorable, communicate visually and gain trust across all brand touch points.
Don’t try to force inspiration, it’s all around us sometimes we need to open our eyes and look around once in a while, trust your process as the inspiration is in the process!
Be sure to check out our Designer Interview section for some great interviews full of inspiration and designer insights. Below are some other links to various sources that you can find some inspiration from to get the creative juices flowing.
Places on the internet to find creative inspiration
- Dribbble — Is a well-known creative social network in the creative community. Dribbble is a site where creative professionals can post their work known as “shots”, comment on other creatives work and find paid projects. A site I recommend that every creative to your bookmark list.
- Behance — A similar concept to Dribbble but owned by Adobe. Behance is another great place to find inspiration and recent work from designers around the world. This website is more for viewing and adding full projects with descriptions and the creative process as opposed to dribbble which are single image shots. Behance is a great place to find designers you like and follow them so you can follow their work over time and stay inspired.
- Twitter — the creative community are incredibly active on Twitter so it’s the perfect place to find inspiration. Do searches for “graphic design” “logo design” etc (or any other discipline) and filter by designers, you can even create lists and add your favorite creative agencies or designers to keep an up to date news feed or follow them with check to be notified whenever they tweet so you’re alerted instantly. It’s also a great place to engage and interact with fellow designers.
- Instagram —The place to be for visually creative people around the world to start browsing by hashtags like #graphicdesign, #logodesign, #monogram, #iconography, typography I could go on for ages, but I will leave it there just to search and find super talented designers to follow on Instagram.
- Pinterest — Possibly even more search-friendly than Instagram. Pinterest is an incredible place to look for inspiration in all forms. It’s also handy to use to gather your inspiration findings, you can create a private board or share with your client and pin your findings to create somewhat of a mood board.
Finding Logo, Branding, and Identity Design Inspiration
- Brand New — Resource for Creative in the branding + identity design industry. They aggregate collections of corporate branding work, the only downside to brand new and I’m not bashing anything here but this is why I’m not a personal reader or fan of the site is I’m dead against the rating system, in my opinion, this is not fair, I totally respect the work that goes into the articles thumbs up here! and I respect that everyone has an opinion but this just turns into a bashing system when something kicks off and this encourages others to just follow suit and the comments can be quite nasty at times. It would be great without ratings and comments when it’s other people’s work that’s centre stage.
- Identity Designed — This site by David Airey who does a deeper dive into specific branding work for individual brands. Rather than just showing the logo design, they show the entire branding process with a description and lots of photos. Even though the site is not updated much possibly once a month as opposed to brandnew which seems to aim for quantity, identity design aims for quality over quantity and all the posts are worth checking out for inspiration.
- Inspiration Grid — Fantastic site for all areas of design inspiration, but they do a particularly great job organising branding and logo design inspiration.
- The Logo Creative – Designer Spotlight – We share out identity design projects on Twitter under the #hashtag #designerspotlight with every post we spotlight the designer and company and also credit the work by providing a link to the projects page.
Finding Lettering & Typography Inspiration
Inspiration grid mentioned above has a nice section for typography that is organised and nicely laid out like the branding and logo sections.
Typo / graphic posters are a platform for inspiration and promotion of good design through the poster culture.
They focus exclusively on typographical and graphical posters, the once that challenge type, colours and shapes to express a message.
It is a curated gallery with a graphic design point of view. each poster is reviewed to meet a standard in visual quality and functional effort.
Behance is a great place for designers to display their work and gain client projects. It’s also a fantastic inspiration resource at your fingertips.
It’s a great site for searching for typography inspiration for your logo design projects. You can search certain typefaces by name, the format such as books or packaging and the industry such as technology or education and you can really narrow down your search to the type of projects you’re gaining inspiration for. As you can see from the example pic above i was searching by industry for education for an educational project i am currently working on.
Started as a personal archive for designer Jonathan Lawrence as a source of inspiration for himself and is an ongoing archive of found typography photographed and curated by Jonathan himself.
Was set up to be a good resource for choosing fonts for design projects and focused on highlighting what’s trending and popular in type.
Creative Education for Inspiration
If your getting started or perhaps your already a logo designer you will find this course very handy and useful, a very cost effective online course that will teach you about efficiently using Adobe Illustrator to design visual identities and implementing design strategy. Through each lesson, you will be taught a specific design theory, provide interactive assignments for practice, and then let you tackle realistic client projects to improve your visual problem-solving. Highly recommended by us here at The Logo Creative
Skillshare
Probably the best known creative education site out there today, Skillshare paved the way for incredibly high-quality creative classes from well-known and accomplished professionals. It’s well worth £10 a month same as a Netflix subscription plus you get 2 months free of charge you have nothing to lose and everything to gain!
Udemy – Logo | Branding
Udemy is broader in approach, but you can find amazing creative courses there to supplement your creative education. And unlike Skillshare, a good number of them are
1,500+ Life changing classes. 650+ industry-leading experts. 10+ million students. A host of creative classes full of inspiration and tips to grow as a professional creative. It also includes classes from the likes of Debbie Millman who has also taken part in out designer interviews.
Below is a very interesting video presented by Ivan Chermayeff, Tom Geismar & Sagi Haviv who give insights into the process and design thinking that went into some of the worlds famous and timeless logos they have created.
Hope this article will be of value to you and hope it will help you move forward in your career as a designer, if you have anything to add then please feel free to drop a comment in the comments section below.
If you would like to stay up to date with our articles in our blog section then don’t forget to subscribe and you will only get an email once a week with new articles.
Find more logo design inspiration on medium.com/theymakedesign/logo-design-inspiration-d6c0199dd2f9. | https://www.thelogocreative.co.uk/where-do-you-find-logo-design-inspiration/ |
Birds that build cup nests use a mixture of organic debris and saliva or mud to build bowl-shaped structures in trees, shrubs and beneath rocky outcroppings. According to the Cornell University Laboratory of Ornithology, cup nests are the preferred nest type for hummingbirds, swifts, flycatchers, blackbirds and other small birds.
9/26/2011 · As birds gained more experience, they dropped blades of grass less often. "If birds built their nests according to a genetic template, you would expect all birds to build their nests the same way ...
Birds' nests are amazing structures that come in a wide range of sizes and styles. Understanding more about why and how birds build nests gives birders even better insights into the amazing lives and reproductive habits of their favorite bird species.
The smallest bird nests are those of some hummingbirds, tiny cups which can be a mere 2 cm (0.79 in) across and 2–3 cm (0.79–1.18 in) high. At the other extreme, some nest mounds built by the dusky scrubfowl measure more than 11 m (36 ft) in diameter and stand nearly 5 m (16 ft) tall. Not all bird species build nests.
4/20/2015 · Q: How do birds learn to build their nests–especially the intricate ones? Milensky: There has been some recent debate about whether nest building is a learned or instinctive behavior. It is primarily instinctive, but it has been clearly shown that birds that build intricate nests, like the group called weavers, learn and become better nest builders over time.
Bird nests are extremely diverse, although each species typically has a characteristic nest style. Some birds do not make nests at all and instead lay their eggs in a simple scrape in the ground. Other birds construct nests from natural materials, such as grass, leaves, mud, lichen, and fur, or from man-made materials like paper, plastic, and yarn.
8 Different Kinds of Bird Nests and How to Spot Them ... water. Others, such as loons, grebes, coots and gallinules, nest directly on top of the water. Eggs will sink, so the birds build floating platform nests out of cattails, reeds, other aquatic vegetation, or mud. ... Thank you for sending such great examples and beautiful pictures of so ...
Eastern Birds’ Nests or Western Birds’ Nests (Peterson Field Guides series), by Hal H. Harrison Nests, Eggs, and Nestlings of North American Birds , by Paul Baicich and Colin Harrison Remember not to take nests from the wild; it is always best to leave them where …
2/17/2014 · Animals and their habitats: Junior kids, toddlers, preschoolers and primary school children can learn a lot about animals and their homes and names of the homes . This education video is ...
Challenge kids with this fun "Make A Birds Nest" STEAM project and get them testing out their ideas and problem solving. ... It’s amazing to see all the different types of nests that birds can build just using materials found around them and no tools. ... Not only do their nests look fabulous but this Make a Birds Nest STEAM Project gave the ...
bird nests such as, Where do birds build nests?, What time of year do birds usually build their nests?, and What kind of materi-als do you think birds would use to build Building Bird Nests Activity 4 Objective: To teach students about songbird nests, the different types, placement and uses. Materials: Nesting material
No, birds don’t destroy their own nests. But if extreme weather destroys a nest, they quite often use the same material from the old nest to re-build nearby. Finding material is a lot of work! Some birds, particularly raptors, use the same nest fo...
Cowbirds do not build their own nest - they lay eggs in the nests of other birds, depending on the host to incubate and raise their young. Eggs are oval with variable shape, with a moderately glossy, granulated shell, white or grayish white, evenly dotted with brown/ reddish brown/gray, sometimes with heavier markings at the large end.
1/23/2014 · OK, so birds don’t sleep in their nests. Let’s get more specific than that. There are a lot of different kinds of birds, and they roost in different ways.
Basic Hummingbird Nest Facts. Females build their nests 10 to 90 feet high, generally in trees or shrubs (with a few exceptions). Hummingbirds build velvety, compact cups with spongy floors and elastic sides that stretch as the young grow.
7/4/2014 · Here are amazing 10 facts about birds' nests. Not all birds go through the trouble of building a nest, but among those that do some really extraordinary efforts are put forth. Here are 10 facts ...
Build or purchase a nestbox designed specifically for bluebirds. These boxes are made of unpainted, untreated 3/4" - 1" wood or PVC, have an overhanging slanted roof (2-5", with a shallow saw kerf (groove) to keep rain from entering the box), no perch, a round 1.5" …
What can we learn about squirrels to help us understand them better? What other tools do we need as we try to coexist with them as people who also love to feed birds? If you’ve observed these creatures darting up a tree, you may have asked yourself “Where do …
4/9/2013 · My older son is quite the birding enthusiast and this backyard bird nesting ball was a great craft for him and a good addition to our homemade bird feeders.. Birds start to build nests in spring and our urban feathered friends appreciate a little help finding good nesting material.
You might think that all birds live in a nest that looks like a small bowl. That's because we see those nests most often in pictures, but actually there are many different kinds of birds' nests.
Use this mini-lesson to introduce students to the different kinds of nests birds build, and the variety of material they use in building them. Excerpted from Eyewitness: Bird David Burnie An in-depth, comprehensive look at the world of birds with a unique integration of words and pictures.
f ree hummingbird pictures licensed under the Creative Commons License. 1 week old baby hummingbird Chlorostilbon mellisugus (Common Emerald) Hummingbird nests- photos by Supersnail. licensed under the Creative Commons License. Click the link what do baby hummingbird look like to see more baby hummingbird videos.
If you’ve been hearing an endless string of 10 or 15 different birds singing outside your house, you might have a Northern Mockingbird in your yard. These slender-bodied gray birds apparently pour all their color into their personalities. They sing almost endlessly, even sometimes at night, and they flagrantly harass birds that intrude on their territories, flying slowly around them or ...
When you come across a helpless-looking baby bird out of its nest, it’s hard to resist the overpowering urge to come to the rescue. But most of the time it’s best to do nothing. Many birds that people try to rescue are still being cared for by their parents and should be left alone. Here’s a primer on when to take action for songbirds (baby ducks or geese require a different approach ...
Golden eagle nests, while large, are generally smaller and flatter than bald eagle nests. Visit the National Eagle Center to see our life-size replica bald eagle nest – climb in and see just how big it is! When do eagles build the nest? Nest building may begin 1-3 months prior to mating and is considered part of the breeding process.
Photos by Earle Robinson, April 1997, South Central Arizona. Verdins are small, industrious birds that build nests throughout the year. Usually when a bird is observed carrying nesting material, like the Verdin to the left, it is a sure sign that soon eggs will be laid in the nest. However, some birds build "roost" nests and "brood" nests.
Unique among North American raptors for its diet of live fish and ability to dive into water to catch them, Ospreys are common sights soaring over shorelines, patrolling waterways, and standing on their huge stick nests, white heads gleaming. These large, rangy hawks do well around humans and have rebounded in numbers following the ban on the pesticide DDT.
Bird nests come in many shapes and sizes. Each species has a unique method used to choose suitable locations and construct nests to house their young. Many birds build their nests among tree branches or shrubs, while others prefer tree cavities, on the ground, in …
5/21/2018 · What Time of Year Do Wild Birds Lay Eggs? By J. Dianne Dotson; Updated May 21, 2018 . ... Wild birds begin making nests in trees, on the ground or even in human-made structures. Once mated, females can produce one egg per day. The iconic American robin, often considered a harbinger of spring, may make four or five nests per season. ...
You will learn about 1) where birds build nests, 2) the materials they use, 3) different nest characteristics and 4) how to observe nesting activity. Introduction Birds hatch their young in nests. Nests can vary from a smoothed place on the ground to the elaborately woven nests of the oriole. A nest is not a permanent “house” for a bird.
What Kind of Birds Build Nest Out of Mud? ... Both birds build gourd-shaped nests about the size of a grapefruit. Each mud nest has a small opening to allow for entry and exit. These birds build their nests by collecting mud in their bills and mixing it together with grass stems to create pellets. ... Learn More About Alexa Hands-free voice ...
9/23/2018 · Platform nests are large, flat nests built in trees, on the ground, on the tops of vegetation, or even on debris in shallow water. Many platform nests are reused year after year by the same birds, with additional materials added to the nest with each use. This practice can create huge nests that damage trees—especially in bad weather.
3/25/2018 · Paper wasps build umbrella-shaped wasp nests suspended underneath eaves and overhangs. Bald-faced hornets construct large, football-shaped nests. Yellowjackets make their nests underground. Regardless of where a wasp builds its nest or what shape the nest is, the process wasps use to construct their nests is generally the same.
The children fill up a brown lunch bag with whatever they would like to use to help birds build their nests. Then, I staple the bag shut and put on one of these labels. ... The children love seeing pictures of the different types of nests. ... The students can record what they learn about their favorite bird. | http://mangup-kale.ru/How-do-birds-learn-to-build-nests-pictures.html |
Q:
Gson - ignore json fields when parsing JSON to Object
There is a question here that is similar to my question but not exactly what I'm looking for.
I've a JSON response from a webservice, let's say this JSON response:
{
"routes" : [
{
"bounds" : {
"northeast" : {
"lat" : 45.5017123,
"lng" : -73.5672184
},
"southwest" : {
"lat" : 43.6533103,
"lng" : -79.3827675
}
},
"copyrights" : "Dados do mapa ©2015 Google",
"legs" : [
{
"distance" : {
"text" : "541 km",
"value" : 540536
},
"duration" : {
"text" : "5 horas 18 min.",
"value" : 19058
},
"end_address" : "Montreal, QC, Canada",
"end_location" : {
"lat" : 45.5017123,
"lng" : -73.5672184
},
"start_address" : "Toronto, ON, Canada",
"start_location" : {
"lat" : 43.6533103,
"lng" : -79.3827675
},
(...)
In this JSON I' just interested on the distance object. My question is, how can I ignore all the other fields?
I tried to build my object starting from legs, as it is the first non repetitive object name from distance to the root.
Here's my object:
public class MyObject {
public ArrayList<Distance> legs;
public static class Distance {
public String text;
public String value;
}
}
but the ArrayList legs is always null.
How can I accomplish this? Ignore fields to the left from the pretended json field.
A:
I think that the philosophy of Gson is to map the Json structure to an objects graph. So in your case I would probably create all the needed java objects to map correctly the json structure. In addition of that, maybe one day you will need some other information of the response, so it will be easier to do the evolution. Something like that (the proper way I think):
class RouteResponse {
private List<Route> routes;
}
class Route {
private List<Bound> bounds;
private String copyrights;
private List<Leg> legs;
}
class Leg {
private Distance distance;
private Duration duration;
private String endAddress;
...
}
class TextValue {
private String text;
private String value;
}
class Distance extends TextValue {
}
// And so on
And I would use an ExclusionStrategy to have light objects and to have only the fields I'm interested in. It sounds to me like the proper way to do that.
Now if you really want to retrieve only the list of distances, I'm sure you can do that with a custom TypeAdapter and TypeAdapterFactory.
Something like that (the bad way :-)):
The objects to map the response:
public class RouteResponse {
private List<Distance> distances;
// add getters / setters
}
public class Distance {
private String text;
private String value;
// add getters / setters
}
The factory that instantiates our adapter (With a reference to the Gson object, so the adapter can retrieve delegates) :
public class RouteResponseTypeAdapterFactory implements TypeAdapterFactory {
@Override
public <T> TypeAdapter<T> create(Gson gson, TypeToken<T> type) {
if (type.getRawType() == RouteResponse.class) {
return (TypeAdapter<T>)new RouteResponseTypeAdapter(gson);
}
return null;
}
}
And the type adapter: this implementation will first unmarshall the Json document to a JsonElements tree and then will retrieve the needed JsonObjects to create through the delegate the Distance objects (sorry for the bad code, quickly written).
public class RouteResponseTypeAdapter extends TypeAdapter<RouteResponse> {
private final TypeAdapter<JsonElement> jsonElementTypeAdapter;
private final TypeAdapter<Distance> distanceTypeAdapter;
public RouteResponseTypeAdapter(Gson gson) {
this.jsonElementTypeAdapter = gson.getAdapter(JsonElement.class);
this.distanceTypeAdapter = gson.getAdapter(Distance.class);
}
@Override
public void write(JsonWriter out, RouteResponse value) throws IOException {
throw new UnsupportedOperationException("Not implemented");
}
@Override
public RouteResponse read(JsonReader jsonReader) throws IOException {
RouteResponse result = new RouteResponse();
List<Distance> distances = new ArrayList<>();
result.setDistances(distances);
if (jsonReader.peek() == JsonToken.BEGIN_OBJECT) {
JsonObject responseObject = (JsonObject) jsonElementTypeAdapter.read(jsonReader);
JsonArray routes = responseObject.getAsJsonArray("routes");
if (routes != null) {
for (JsonElement element:routes) {
JsonObject route = element.getAsJsonObject();
JsonArray legs = route.getAsJsonArray("legs");
if (legs != null) {
for (JsonElement legElement:legs) {
JsonObject leg = legElement.getAsJsonObject();
JsonElement distanceElement = leg.get("distance");
if (distanceElement != null) {
distances.add(distanceTypeAdapter.fromJsonTree(distanceElement));
}
}
}
}
}
}
return result;
}
}
Finally, you can parse your json document:
String json = "{ routes: [ ....."; // Json document
Gson gson = new GsonBuilder().registerTypeAdapterFactory(new RouteResponseTypeAdapterFactory()).create();
RouteResponse response = gson.fromJson(json, RouteResponse.class);
//response.getDistances() should contain the distances
Hope it helps.
| |
---
abstract: 'In the present paper, a class of stochastic Runge–Kutta methods containing the second order stochastic Runge–Kutta scheme due to E. Platen for the weak approximation of Itô stochastic differential equation systems with a multi–dimensional Wiener process is considered. Order one and order two conditions for the coefficients of explicit stochastic Runge–Kutta methods are solved and the solution space of the possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge–Kutta schemes of order one and two with minimal stage numbers is calculated. Further, within the considered class of stochastic Runge–Kutta schemes coefficients for optimal schemes in the sense that additionally some higher order conditions are fulfilled are presented.'
address: 'Technische Universität Darmstadt, Fachbereich Mathematik, Schlo[ß]{}gartenstr.7, D-64289 Darmstadt, Germany'
author:
- Kristian Debrabant
- 'Andreas Rö[ß]{}ler'
title: 'Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations'
---
and
Stochastic Runge–Kutta method ,stochastic differential equation ,classification ,weak approximation ,optimal scheme\
MSC 2000: 65C30 ,60H35 ,65C20 ,68U20
Introduction {#Introduction}
============
Recently, the development of numerical schemes for strong as well as weak approximation of stochastic differential equations (SDEs) has focused amongst others on Runge–Kutta type schemes [@BuBu00a; @BuBu96; @KP99; @KoMiSu97; @MacNav01; @Mil95; @Mil04; @Roe06a; @Roe06b; @Roe03; @ToVA02]. This is due to the increasing complexity of stochastic Taylor expansions and the desire to avoid derivatives in higher order approximation schemes. In section \[Sec:SRK-methods\], a class of stochastic Runge–Kutta (SRK) methods due to Rö[ß]{}ler [@Roe06a; @Roe06b; @Roe03] for the weak approximation of Itô SDE systems with a multi–dimensional Wiener process is considered. This class contains as a special case the second order SRK scheme proposed by Platen [@KP99] as well as the class of SRK methods proposed by Tocino and Vigo-Aguiar [@ToVA02]. Order conditions for coefficients of the SRK methods have been calculated by applying the multi-colored rooted tree analysis due to Rö[ß]{}ler [@Roe06a; @Roe06b; @Roe03]. In contrast to earlier work on this topic, the aim of the present paper is to analyze these order conditions with the objective to determine a full classification of the coefficients for this class of SRK methods. A full classification for order one SRK schemes with $s=1$ stage and for order one SRK schemes with deterministic order two for $s=2$ stages as well as for second order SRK schemes with $s=3$ stages is calculated in section \[Sec:Parameter-families\]. Further, some optimal schemes are derived from this classification in section \[Sec:Optimal-schemes\] by taking into account additional higher order conditions. Their performance is studied by some numerical examples in section \[Sec:Numerical-Example\].\
\
We denote by $(X_t)_{t \in I}$ the solution of the $d$-dimensional Itô SDE defined by $$\label{St-lg-sde-ito-1}
{\mathrm{d}} X_t = a(t,X_t) \, {\mathrm{d}}t + b(t,X_t) \,
{\mathrm{d}}W_t, \qquad X_{t_0} = x_{0},$$ with an $m$-dimensional Wiener process $(W_t)_{t \geq 0}$ and $I=[t_0,T]$. We assume that the Borel-measurable coefficients $a : I \times
\mathbb{R}^d \rightarrow \mathbb{R}^d$ and $b : I \times
\mathbb{R}^d \rightarrow \mathbb{R}^{d \times m}$ satisfy a Lipschitz and a linear growth condition such that the Existence and Uniqueness Theorem [@KP99] applies. In the following, let $b^j(t,x) = (b^{i,j}(t,x))_{1 \leq i \leq d} \in \mathbb{R}^d$ denote the $j$th column of the diffusion matrix $b(t,x)$ for $j=1,
\ldots, m$.\
\
Let a discretization $I_h = \{t_0, t_1, \ldots, t_N\}$ with $t_0 <
t_1 < \ldots < t_N =T$ of the time interval $I=[t_0,T]$ with step sizes $h_n = t_{n+1}-t_n$ for $n=0,1, \ldots, N-1$ be given. Further, define $h = \max_{0 \leq n < N} h_n$ as the maximum step size. Let $C_P^l(\mathbb{R}^d, \mathbb{R})$ denote the space of all $g
\in C^l(\mathbb{R}^d,\mathbb{R})$ fulfilling a polynomial growth condition and let $g \in C_P^{k,l}(I \times \mathbb{R}^d,
\mathbb{R})$ if $g(\cdot,x) \in C^{k}(I,\mathbb{R})$ and $g(t,\cdot) \in C_P^l(\mathbb{R}^d, \mathbb{R})$ for all $t \in I$ and $x \in \mathbb{R}^d$ [@KP99].
A time discrete approximation $Y=(Y_t)_{t \in I_h}$ converges weakly with order $p$ to $X$ as $h \rightarrow 0$ at time $t \in I_h$ if for each $f \in C_P^{2(p+1)}(\mathbb{R}^d, \mathbb{R})$ exist a constant $C_f$ and a finite $\delta_0 > 0$ such that $$| {\operatorname{E}}(f(X_t)) - {\operatorname{E}}(f(Y_t)) | \leq C_f \, h^p$$ holds for each $h \in \, ]0,\delta_0[\,$.
Stochastic Runge–Kutta Methods {#Sec:SRK-methods}
==============================
We consider stochastic Runge–Kutta methods as proposed in [@Roe06a; @Roe06b; @Roe03] for the weak approximation of SDE (\[St-lg-sde-ito-1\]). Therefore, the $d$-dimensional approximation process $Y$ of an explicit $s$-stage SRK method is defined by $Y_{t_0} = x_0$ and $$\label{SRK-method-Ito-Wm-allg01}
\begin{split}
Y_{t_{n+1}} = Y_{t_n} & + \sum_{i=1}^s
\alpha_i \, a(t_n+c_i^{(0)} h_n, H_i^{(0)}) \, h_n \\
& + \sum_{i=1}^s
\sum_{k=1}^m
\beta_i^{(1)} \, b^{k}(t_n+c_i^{(1)} h_n, H_i^{(k)}) \, \hat{I}_{(k)} \\
& +
\sum_{i=1}^s \sum_{k=1}^m
\beta_i^{(2)} \, b^{k}(t_n+c_i^{(1)} h_n, H_i^{(k)}) \,
\tfrac{\hat{I}_{(k,k)}}{\sqrt{h_n}} \\
& + \sum_{i=1}^s
\sum_{\substack{k,l=1 \\ k \neq l}}^m
\beta_i^{(3)} \, b^{k}(t_n+c_i^{(2)} h_n, \hat{H}_i^{(l)}) \, \hat{I}_{(k)} \\
& +
\sum_{i=1}^s \sum_{\substack{k,l=1 \\ k \neq l}}^m
\beta_i^{(4)} \, b^{k}(t_n+c_i^{(2)} h_n, \hat{H}_i^{(l)}) \,
\tfrac{\hat{I}_{(k,l)}}{\sqrt{h_n}}
\end{split}$$ for $n=0,1, \ldots, N-1$ with stage values $$\begin{aligned}
{5}
H_i^{(0)} &=&\,\, Y_{t_n} &+ \sum_{j=1}^{i-1} A_{ij}^{(0)}
\, a(t_n+c_j^{(0)} h_n, H_j^{(0)}) \, h_n \\
&& &+ \sum_{j=1}^{i-1} \sum_{r=1}^m
B_{ij}^{(0)} \, b^r(t_n+c_j^{(1)} h_n, H_j^{(r)}) \, \hat{I}_{(r)} \\
H_i^{(k)} &=&\,\, Y_{t_n} &+ \sum_{j=1}^{i-1} A_{ij}^{(1)}
\, a(t_n+c_j^{(0)} h_n, H_j^{(0)}) \, h_n \\
&& &+ \sum_{j=1}^{i-1}
B_{ij}^{(1)} \, b^k(t_n+c_j^{(1)} h_n, H_j^{(k)}) \,
\sqrt{h_n} \\
\hat{H}_i^{(k)} &=&\,\, Y_{t_n} &+ \sum_{j=1}^{i-1} A_{ij}^{(2)}
\, a(t_n+c_j^{(0)} h_n, H_j^{(0)}) \, h_n \\
&& &+ \sum_{j=1}^{i-1}
B_{ij}^{(2)} \, b^k(t_n+c_j^{(1)} h_n, H_j^{(k)}) \, \sqrt{h_n}\end{aligned}$$ for $i=1, \ldots, s$ and $k=1, \ldots, m$. Here, $\alpha,
\beta^{(1)},\dots,\beta^{(4)},c^{(q)}\in \mathbb{R}^s$ and $A^{(q)}$, $B^{(q)} \in \mathbb{R}^{s \times s}$ for $0\leq q \leq
2$ with $A_{ij}^{(q)} = B_{ij}^{(q)} = 0$ for $j \geq i$ are the vectors and matrices of coefficients of the SRK method. We choose $c^{(q)}=A^{(q)} e$ for $0 \leq q \leq 2$ with a vector $e=(1,
\ldots, 1)^T$ [@Roe06a]. In the following, the product of column vectors is defined component-wise. The coefficients of the SRK method (\[SRK-method-Ito-Wm-allg01\]) are determined by the following Butcher tableau:
----------- ------------- ------------------- -------------------
$c^{(0)}$ ${A}^{(0)}$ $B^{(0)}$
$c^{(1)}$ ${A}^{(1)}$ $B^{(1)}$
$c^{(2)}$ ${A}^{(2)}$ $B^{(2)}$
$\alpha^T$ ${\beta^{(1)}}^T$ ${\beta^{(2)}}^T$
${\beta^{(3)}}^T$ ${\beta^{(4)}}^T$
----------- ------------- ------------------- -------------------
The random variables of the SRK method are defined by three-point distributed random variables with ${\operatorname{P}}(\hat{I}_{(r)} = \pm
\sqrt{3 \, h_n} ) = \frac{1}{6}$ and ${\operatorname{P}}(\hat{I}_{(r)} = 0 ) =
\frac{2}{3}$. Further, $ \hat{I}_{(k,l)} = \frac{1}{2} (
\hat{I}_{(k)} \, \hat{I}_{(l)} + V^{k,l} )$. The $V^{k,l}$ are independent two-point distributed random variables with ${\operatorname{P}}(V^{k,l} = \pm h_n) = \frac{1}{2}$ for $l=1, \ldots, k-1$, $V^{k,k} = -h_n$ and $V^{k,l} = -V^{l,k}$ for $l=k + 1, \ldots, m$ and $k=1, \ldots, m$ [@KP99].\
\
By the application of the multi–colored rooted tree analysis [@Roe06a; @Roe03], order conditions for the coefficients of the SRK method (\[SRK-method-Ito-Wm-allg01\]) can be easily determined. As a result of this, the following Theorem \[SRK-theorem-ito-ord2-Wm-main1\] due to Rö[ß]{}ler [@Roe03] gives order conditions for the SRK method (\[SRK-method-Ito-Wm-allg01\]) up to order two.
\[SRK-theorem-ito-ord2-Wm-main1\] Let $a^i, b^{ij} \in C_P^{2,4}(I \times
\mathbb{R}^d,\mathbb{R})$ for $1 \leq i \leq d$, $1 \leq j \leq m$. If the coefficients of the SRK method (\[SRK-method-Ito-Wm-allg01\]) fulfill the equations $$\begin{aligned}
{5}
1&. \quad \alpha^T e = 1 \qquad \qquad
&2. \quad &{\beta^{(4)}}^T e = 0 \qquad \qquad
&3. \quad &{\beta^{(3)}}^T e = 0 \\
4&. \quad ({\beta^{(1)}}^T e)^2 = 1 \qquad \qquad
&5. \quad &{\beta^{(2)}}^T e = 0 \qquad \qquad
&6. \quad &{\beta^{(1)}}^T {B^{(1)}} e = 0 \\
7&. \quad {\beta^{(3)}}^T
{B^{(2)}} e = 0
\end{aligned}$$ then the SRK method converges with order 1 in the weak sense. In addition, if $a^i, b^{ij} \in C_P^{3,6}(I \times
\mathbb{R}^d,\mathbb{R})$ for $1 \leq i \leq d$, $1 \leq j \leq m$ and if the equations [ $$\begin{aligned}
{3}
8&. \quad \alpha^T A^{(0)} e = \tfrac{1}{2}
\qquad \qquad
&9. \quad &\alpha^T (B^{(0)} e)^2 = \tfrac{1}{2} \\
10&. \quad ({\beta^{(1)}}^T e) (\alpha^T B^{(0)} e) =
\tfrac{1}{2}
\qquad \qquad
&11. \quad &({\beta^{(1)}}^T e) ({\beta^{(1)}}^T A^{(1)} e) = \tfrac{1}{2} \\
12&. \quad {\beta^{(3)}}^T A^{(2)} e = 0
\qquad \qquad
&13. \quad &{\beta^{(2)}}^T B^{(1)} e = 1 \\
14&. \quad {\beta^{(4)}}^T B^{(2)} e = 1
\qquad \qquad
&15. \quad &({\beta^{(1)}}^T e) ({\beta^{(1)}}^T
(B^{(1)} e)^2) = \tfrac{1}{2} \\
16&. \quad ({\beta^{(1)}}^T e) ({\beta^{(3)}}^T
(B^{(2)} e)^2) = \tfrac{1}{2} \qquad
&17. \quad &{\beta^{(1)}}^T (B^{(1)} (B^{(1)} e)) =
0 \\
18&. \quad {\beta^{(3)}}^T (B^{(2)}
(B^{(1)}e)) = 0 \qquad
&19. \quad &{\beta^{(3)}}^T (A^{(2)} (B^{(0)}
e)) = 0 \\
20&. \quad {\beta^{(1)}}^T (A^{(1)} (B^{(0)} e)) =
0 \qquad
&21. \quad &\alpha^T (B^{(0)} (B^{(1)} e)) = 0 \\
22&. \quad {\beta^{(2)}}^T A^{(1)} e = 0 \qquad
&23. \quad &{\beta^{(4)}}^T A^{(2)} e = 0 \\
24&. \quad {\beta^{(1)}}^T ((A^{(1)} e) (B^{(1)} e)) = 0
\qquad
&25. \quad &{\beta^{(3)}}^T ((A^{(2)} e)
(B^{(2)} e)) = 0 \\
26&. \quad {\beta^{(4)}}^T (A^{(2)} (B^{(0)}
e)) = 0 \qquad
&27. \quad &{\beta^{(2)}}^T (A^{(1)} (B^{(0)} e))
= 0 \\
28&. \quad {\beta^{(2)}}^T (A^{(1)} (B^{(0)} e)^2) = 0
\qquad
&29. \quad &{\beta^{(4)}}^T (A^{(2)} (B^{(0)}
e)^2) = 0 \\
30&. \quad {\beta^{(3)}}^T (B^{(2)} (A^{(1)}
e)) = 0 \qquad
&31. \quad &{\beta^{(1)}}^T (B^{(1)} ( A^{(1)} e)) =
0 \\
32&. \quad {\beta^{(2)}}^T (B^{(1)} e)^2 = 0 \qquad
&33. \quad &{\beta^{(4)}}^T (B^{(2)} e)^2 = 0 \\
34&. \quad {\beta^{(4)}}^T (B^{(2)} (B^{(1)}
e)) = 0 \qquad
&35. \quad &{\beta^{(2)}}^T (B^{(1)} (B^{(1)} e))
= 0 \\
36&. \quad {\beta^{(1)}}^T (B^{(1)} e)^3 = 0 \qquad
&37. \quad &{\beta^{(3)}}^T (B^{(2)} e)^3 = 0 \\
38&. \quad {\beta^{(1)}}^T (B^{(1)} (B^{(1)}
e)^2) = 0 \qquad
&39. \quad &{\beta^{(3)}}^T (B^{(2)} (B^{(1)}
e)^2) = 0
\end{aligned}$$ ]{} $$\begin{aligned}
{3}
40&. \quad \alpha^T ((B^{(0)} e) (B^{(0)}
(B^{(1)} e))) = 0 \\
41&. \quad {\beta^{(1)}}^T ((A^{(1)} (B^{(0)}
e)) (B^{(1)} e)) = 0 \\
42&. \quad {\beta^{(3)}}^T ((A^{(2)}
(B^{(0)} e)) (B^{(2)} e)) = 0 \\
43&. \quad {\beta^{(1)}}^T (A^{(1)} (B^{(0)} (B^{(1)} e))) = 0 \\
44&. \quad {\beta^{(3)}}^T (A^{(2)} (B^{(0)}
(B^{(1)} e))) = 0 \\
45&. \quad {\beta^{(1)}}^T (B^{(1)} (A^{(1)}
(B^{(0)} e))) = 0 \\
46&. \quad {\beta^{(3)}}^T (B^{(2)} (A^{(1)}
(B^{(0)} e))) = 0 \\
47&. \quad {\beta^{(1)}}^T ((B^{(1)} e) (B^{(1)}
(B^{(1)} e))) = 0 \\
48&. \quad {\beta^{(3)}}^T ((B^{(2)} e)
(B^{(2)} (B^{(1)} e))) = 0 \\
49&. \quad {\beta^{(1)}}^T (B^{(1)} (B^{(1)}
(B^{(1)} e))) = 0 \\
50&. \quad {\beta^{(3)}}^T (B^{(2)} (B^{(1)}
(B^{(1)} e))) = 0
\end{aligned}$$ are fulfilled then the stochastic Runge–Kutta method (\[SRK-method-Ito-Wm-allg01\]) converges with order 2 in the weak sense.
In the case of $m>1$ one has to solve 50 non-linear equations in order to calculate coefficients for an order two SRK method (\[SRK-method-Ito-Wm-allg01\]). However, in the case of $m=1$ these conditions are reduced to 28 equations which have to be solved [@Roe06b]. Thus, the analysis of the space of all admissible coefficients is not an easy task. It turns out that explicit order one SRK methods need at least $s=1$ stage while order two SRK methods need $s \geq 3$ stages. This is due to e.g.conditions 6. and 15., which can not be fulfilled in the case of $s \leq 2$ stages for explicit order two SRK methods. In the following, we distinguish between the stochastic and the deterministic order of convergence. Let $p_S=p$ denote the order of convergence of the SRK method if it is applied to an SDE and let $p_D$ with $p_D \geq p_S$ denote the order of convergence of the SRK method if it is applied to a deterministic ordinary differential equation (ODE), i.e., SDE (\[St-lg-sde-ito-1\]) with $b \equiv 0$. We also write $(p_D,p_S)$ in the following [@Roe06b; @Roe03].
Parameter Families for SRK Methods {#Sec:Parameter-families}
==================================
Coefficients for SRK Methods of Order (1,1) {#Sec:Coeff-SRK-Ord-1-1}
-------------------------------------------
First, we analyze explicit SRK methods (\[SRK-method-Ito-Wm-allg01\]) of order $p_D=p_S=1$ with $s=1$ stage. Considering the order one conditions 1.–7. in Theorem \[SRK-theorem-ito-ord2-Wm-main1\], the corresponding coefficients are uniquely determined for $c_1 \in \{-1,1\}$ by $$\label{Parameter-Ord11-all}
\alpha_1 = 1, \qquad \beta_1^{(1)} = c_1,
\qquad \beta_1^{(2)} = 0, \qquad
\beta_1^{(3)} = 0, \qquad
\beta_1^{(4)} = 0.$$ The resulting class of SRK schemes coincides with the well-known Euler-Maruyama scheme.
Coefficients for SRK Methods of Order (2,1) {#Sec:Coeff-SRK-Ord-2-1}
-------------------------------------------
Next, we consider the case of $s=2$ stage explicit SRK methods (\[SRK-method-Ito-Wm-allg01\]). As already mentioned in Section \[Sec:SRK-methods\], it is not possible to attain order $p_S=2$. However, we can find some SRK methods of order $p_D=2$ and $p_S=1$ corresponding to the following parameter family: From condition 1. of Theorem \[SRK-theorem-ito-ord2-Wm-main1\] follows $\alpha_1 = 1 -\alpha_2$ and taking into account the order 2 condition 8. we obtain $\alpha_2 = \frac{1}{2 A^{(0)}_{21}}$ for $A^{(0)}_{21} \neq 0$. Further, condition 2. yields $\beta^{(4)}_1
= -\beta^{(4)}_2$, condition 3. results in $\beta^{(3)}_1 =
-\beta^{(3)}_2$ and condition 5. is fulfilled if $\beta^{(2)}_1 =
-\beta^{(2)}_2$ while condition 4. holds for $\beta^{(1)}_1 = c_1 -
\beta^{(1)}_2$ with $c_1 \in \{-1,1\}$. Finally, considering condition 6. we need that $\beta^{(1)}_2 = 0$ or $B^{(1)}_{21}=0$ and for condition 7. analogously that $\beta^{(3)}_2 = 0$ or $B^{(2)}_{21}=0$ hold. Thus, this class of SRK methods is determined by $$\begin{aligned}
{7}
\alpha^T &= \begin{bmatrix} 1-\frac{1}{2 c_2} && \frac{1}{2
c_2} \end{bmatrix} , &\quad \quad
{\beta^{(1)}}^T &= \begin{bmatrix} c_1-c_4 && c_4 \end{bmatrix},
&\quad \quad
{\beta^{(2)}}^T &= \begin{bmatrix} c_5 && -c_5 \end{bmatrix}, \notag \\
&& {\beta^{(3)}}^T &= \begin{bmatrix} c_6 && -c_6 \end{bmatrix},
&\quad
{\beta^{(4)}}^T &= \begin{bmatrix} c_7 && -c_7 \end{bmatrix}, \notag \\
A^{(0)} &= \begin{bmatrix} 0 && 0 \\ c_2 && 0 \end{bmatrix}, &\quad
A^{(1)} &= \begin{bmatrix} 0 && 0 \\ c_8 && 0 \end{bmatrix}, &\quad
A^{(2)} &= \begin{bmatrix} 0 && 0 \\ c_{9} && 0 \end{bmatrix},
\notag \\
B^{(0)} &= \begin{bmatrix} 0 && 0 \\ c_3 && 0 \end{bmatrix},
&\quad
B^{(1)} &= \begin{bmatrix} 0 && 0 \\ c_{10} && 0 \end{bmatrix}, &\quad
B^{(2)} &= \begin{bmatrix} 0 && 0 \\ c_{11} && 0 \label{Parameter-Ord21-all}
\end{bmatrix},\end{aligned}$$ for $c_1 \in \{-1,1\}$ and $c_2, \ldots, c_{11} \in \mathbb{R}$ with $c_2 \neq 0$, $c_4 \, c_{10} =0$ and $c_6 \, c_{11} = 0$.
Coefficients for SRK Methods of Order (2,2) {#Sec:Coeff-SRK-Ord-2-2}
-------------------------------------------
Now, we consider explicit SRK methods (\[SRK-method-Ito-Wm-allg01\]) of order $p_D=p_S=2$ with $s=3$ stages. Then, the SRK schemes of the class under consideration are completely characterized by the following families of coefficients which follow from the order conditions in Theorem \[SRK-theorem-ito-ord2-Wm-main1\]: Due to conditions 13.and 32. we need $B^{(1)}_{21} \neq 0$ and from conditions 15. and 36. follows $\beta^{(1)}_3 \neq 0$. Thus, there exist no SRK schemes of the considered class attaining order $p_D=p_S=2$ with less than 3 stages. Now, by condition 17. follows that $B^{(1)}_{32}=0$ and we deduce from 6., 15. and 36. that $B^{(1)}_{31} = -B^{(1)}_{21}\neq0$. Analyzing the weights, we calculate from conditions 5., 13. and 32. that $\beta^{(2)}_2 = \tfrac{1}{2
B^{(1)}_{21}}$, $\beta^{(2)}_3 = -\tfrac{1}{2 B^{(1)}_{21}}$ and $\beta^{(2)}_1 = 0$. For $c_1 \in \{-1,1\}$ we obtain from conditions 4., 6. and 15. the weights $\beta^{(1)}_1 = c_1 - \tfrac{c_1}{2
(B^{(1)}_{21})^2}$ and $\beta^{(1)}_2 = \beta^{(1)}_3 =
\tfrac{c_1}{4 (B^{(1)}_{21})^2}$. Now, due to 24. and 11. we need that $A^{(1)}_{21} = (B^{(1)}_{21})^2$ and $A^{(1)}_{31} =
(B^{(1)}_{21})^2 - A^{(1)}_{32}$. Applying now conditions 3., 7., 16. and 37. we conclude that $B^{(2)}_{21} + B^{(2)}_{31} +
B^{(2)}_{32} = 0$, $B^{(2)}_{21} \neq 0$, $B^{(2)}_{21} \neq
B^{(2)}_{31} + B^{(2)}_{32}$ and that $\beta^{(3)}_1 =
-\tfrac{c_1}{2 (B^{(2)}_{21})^2}$, $\beta^{(3)}_2 = \tfrac{c_1}{4
(B^{(2)}_{21})^2}$ and $\beta^{(3)}_3 = \tfrac{c_1}{4
(B^{(2)}_{21})^2}$. Further, we can now determine the remaining weights as $\beta^{(4)}_1 = 0$, $\beta^{(4)}_2 = \tfrac{1}{2
B^{(2)}_{21}}$ and $\beta^{(4)}_3 = -\tfrac{1}{2 B^{(2)}_{21}}$ from conditions 2., 14. and 33., and we have $\alpha_1 = 1 - \alpha_2 -
\alpha_3$ due to condition 1. Then, we can consider condition 18.which needs $B^{(2)}_{32} = 0$ and we thus get with the previous considerations that finally $B^{(2)}_{21} = -B^{(2)}_{31}$ has to be fulfilled. Now, we obtain from conditions 12. and 23. that $A^{(2)}_{21} = 0$ and that $A^{(2)}_{32} = -A^{(2)}_{31}$ has to be fulfilled. Continuing in this manner, we have to distinguish the following cases:
(A) \[Parameter-Fall1\] For $\alpha_3 = 0$, the parameter family is given by $\alpha_1 = \alpha_2 = \tfrac{1}{2}$ and $B^{(0)}_{21}
= c_1$, which follows from conditions 1., 9. and 10. Further, we calculate from condition 8. that $A^{(0)}_{21} = 1$, from 20. that $A^{(1)}_{32} = 0$ and from condition 19. that $A^{(2)}_{32} = 0$.
(B) \[Parameter-Fall2\] For $\alpha_3 \neq 0$, condition 21. yields now that $B^{(0)}_{32} = 0$ and we have to consider the following cases:
1. \[Parameter-Fall2.1\] For $B^{(0)}_{21} = 0$ it follows from conditions 9. and 10. that $B^{(0)}_{31} = c_1$ and $\alpha_3 = \tfrac{1}{2}$. Thus, by condition 1. it follows immediately that $\alpha_1 =
\tfrac{1}{2} - \alpha_2$.
1. \[Parameter-Fall2.1.1\] If $A^{(0)}_{21} =
0$ then condition 8. implies that in this case $A^{(0)}_{32} = 1 - A^{(0)}_{31}$ has to be fulfilled.
2. \[Parameter-Fall2.1.2\] If $A^{(0)}_{21}
\neq 0$ then condition 8. yields that $\alpha_1 =
\tfrac{1}{2} - \tfrac{1-A^{(0)}_{31}-A^{(0)}_{32}}{2
A^{(0)}_{21}}$ and $\alpha_2 = \tfrac{1-A^{(0)}_{31} -
A^{(0)}_{32}}{2 A^{(0)}_{21}}$.
2. \[Parameter-Fall2.2\] For $B^{(0)}_{21}
\neq 0$, we calculate from condition 20. that $A^{(1)}_{32}
= 0$ and from condition 19. that $A^{(2)}_{32} = 0$ which implies that also $A^{(2)}_{31} = 0$ due to $A^{(2)}_{32} =
-A^{(2)}_{31}$. Now, with $\kappa = \alpha_2 \, \alpha_3 (2
\alpha_2 + 2 \alpha_3 - 1)$ and conditions 9. and 10. we have to consider the following cases:
1. \[Parameter-Fall2.2.1\] In the case of $\kappa \geq 0$, $\alpha_2 \neq -\alpha_3$, $\alpha_2
\neq 0$ and $\alpha_2 \neq \pm \sqrt{\kappa}$ it follows that $B^{(0)}_{21} = \tfrac{c_1 (\alpha_2 \mp \sqrt{\kappa})}{2
\alpha_2 (\alpha_2+\alpha_3)}$ and $B^{(0)}_{31} =
\tfrac{c_1 (\alpha_3 \pm \sqrt{\kappa})}{2 \alpha_3
(\alpha_2 + \alpha_3)}$.
2. \[Parameter-Fall2.2.2\] If $\alpha_2 =
0$ and $\alpha_3 = \tfrac{1}{2}$ then we can conclude that $B^{(0)}_{31} = c_1$ has to hold.
3. \[Parameter-Fall2.2.3\] For $\alpha_2 =
-\alpha_3$ and $\alpha_2 \neq -\tfrac{1}{2}$ it follows that $B^{(0)}_{21} = c_1 ( \tfrac{1}{2}+\tfrac{1}{4 \alpha_2})$ and $B^{(0)}_{31} = c_1 (\tfrac{1}{2} -
\tfrac{1}{4 \alpha_2})$ has to be fulfilled.
Due to condition 8. it follows that $A^{(0)}_{31} = \tfrac{1 - 2 \alpha_2 \, A^{(0)}_{21}}{2
\alpha_3} - A^{(0)}_{32}$.
Finally, one can easily check that all remaining conditions of Theorem \[SRK-theorem-ito-ord2-Wm-main1\], which have not been mentioned explicitly in our analysis, are fulfilled by each parameter family and thus do not contribute any further restrictions for the coefficients.\
\
Summarizing our results, we have the following classification for the SRK schemes of order $p_D=p_S=2$ for the considered class with $s=3$ stages: For $c_1 \in \{-1,1\}$ and $c_2, c_3, c_4, c_5 \in
\mathbb{R}$ with $c_3 \neq 0$ and $c_4 \neq 0$ it holds $$\begin{aligned}
{5}
{\beta^{(1)}}^T &= \begin{bmatrix} c_1-\frac{c_1}{2 c_3^2} & &
\frac{c_1}{4 c_3^2} & & \frac{c_1}{4 c_3^2} \end{bmatrix},
&\quad \quad
{\beta^{(2)}}^T &= \begin{bmatrix} 0 & & \frac{1}{2 c_3} & &
-\frac{1}{2 c_3} \end{bmatrix}, \\
{\beta^{(3)}}^T &= \begin{bmatrix} -\frac{c_1}{2 c_4^2} & &
\frac{c_1}{4 c_4^2} & & \frac{c_1}{4 c_4^2} \end{bmatrix},
&\quad \quad
{\beta^{(4)}}^T &= \begin{bmatrix} 0 & & \frac{1}{2 c_4} & &
-\frac{1}{2 c_4} \end{bmatrix}, \\
A^{(1)} &= \begin{bmatrix} 0 && 0 && 0 \\ c_3^2 && 0 && 0 \\
c_3^2-c_2 && c_2 && 0 \end{bmatrix}, &\quad \quad B^{(1)} &=
\begin{bmatrix} 0 && 0 && 0 \\ c_3 && 0 && 0 \\ -c_3 && 0 && 0
\end{bmatrix}, \label{Parameter-Ord22-A1-B1} \\
A^{(2)} &= \begin{bmatrix} 0 && 0 && 0 \\ 0 && 0 && 0 \\
c_5 && -c_5 && 0 \end{bmatrix}, &\quad \quad B^{(2)} &=
\begin{bmatrix} 0 && 0 && 0 \\ c_4 && 0 && 0 \\ -c_4 && 0 && 0
\end{bmatrix} . \label{Parameter-Ord22-A2-B2}\end{aligned}$$ Now, the following cases are possible:\
\
In the case (\[Parameter-Fall1\]), we get with $c_2=c_5=0$ in (\[Parameter-Ord22-A1-B1\])–(\[Parameter-Ord22-A2-B2\]) that $$\label{Parameter-Coeff-1}
\alpha^T = \begin{bmatrix} \frac12 & &
\frac12 & & 0 \end{bmatrix} ,
\quad
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ 1 && 0 && 0 \\
0&& 0 && 0 \end{bmatrix}, \quad B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ c_1 && 0 && 0 \\0 &&0 && 0
\end{bmatrix} ,$$ with $A^{(0)}_{31}=A^{(0)}_{32}=B^{(0)}_{31}=B^{(0)}_{32}=0$ because these coefficients are not relevant for the scheme due to $\alpha_3=0$.\
\
For the case (\[Parameter-Fall2.1.1\]) we get with $c_6, c_7 \in
\mathbb{R}$ the coefficients $$\label{Parameter-Coeff-2.1.1}
\alpha^T = \begin{bmatrix} \frac{1}{2}-c_6 &&
c_6 && \frac{1}{2} \end{bmatrix},
\quad
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ 0 && 0 && 0 \\
c_7 && 1-c_7 && 0 \end{bmatrix}, \quad B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ 0 && 0 && 0 \\ c_1 && 0 && 0
\end{bmatrix} .$$\
Considering the case (\[Parameter-Fall2.1.2\]) we obtain for $c_6, c_7, c_8 \in \mathbb{R}$ with $c_6 \neq 0$ that $$\label{Parameter-Coeff-2.1.2}
\alpha^T = \begin{bmatrix} \frac{1}{2}-\frac{1-c_7-c_8}{2 c_6}
&& \frac{1-c_7-c_8}{2 c_6} && \frac{1}{2} \end{bmatrix},
\,\,
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ c_6 && 0 && 0 \\
c_7 && c_8 && 0 \end{bmatrix},
\,\,
B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ 0 && 0 && 0 \\ c_1 && 0 && 0
\end{bmatrix} .$$\
Next, we have the case (\[Parameter-Fall2.2.1\]) with $c_2=c_5=0$ in (\[Parameter-Ord22-A1-B1\])–(\[Parameter-Ord22-A2-B2\]), $c_6, c_7, c_8,c_9 \in \mathbb{R}$ and with $c_6 \neq 0$ and $c_6 \neq
-c_7 \neq 0$. Then, it holds with $\kappa = c_6 c_7 (2 c_6 + 2 c_7
-1)$ and $\lambda = \frac{1-2 c_6 c_8}{2 c_7}$ for $c_6 \neq \pm
\sqrt{\kappa}$ and $\kappa \geq 0$ that $$\label{Parameter-Coeff-2.2.1}
\alpha^T = \begin{bmatrix} 1-c_6-c_7 && c_6 && c_7
\end{bmatrix}, \,\,
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ c_8 && 0 && 0 \\
\lambda-c_9 && c_9 && 0 \end{bmatrix}, \,\, B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ \frac{c_1}{2}
\frac{c_6 \mp \sqrt{\kappa}}{c_6 (c_6 + c_7)} && 0 && 0 \\
\frac{c_1}{2} \frac{c_7 \pm \sqrt{\kappa}}{c_7 (c_6 + c_7)} && 0 && 0
\end{bmatrix} .$$\
The case (\[Parameter-Fall2.2.2\]) yields for $c_6, c_7, c_8 \in
\mathbb{R}$ with $c_8 \neq 0$ and $c_2=c_5=0$ in (\[Parameter-Ord22-A1-B1\])–(\[Parameter-Ord22-A2-B2\]) the coefficients $$\label{Parameter-Coeff-2.2.2}
\alpha^T = \begin{bmatrix} \frac{1}{2}
&& 0 && \frac{1}{2} \end{bmatrix},
\quad
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ c_6 && 0 && 0 \\
1-c_7 && c_7 && 0 \end{bmatrix}, \quad B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ c_8 && 0 && 0 \\ c_1 && 0 && 0
\end{bmatrix} .$$\
Finally, we have the case (\[Parameter-Fall2.2.3\]) for $c_6, c_7,
c_8 \in \mathbb{R}$ with $c_6 \notin \{-\tfrac{1}{2},0\}$ and $c_2=c_5=0$ in (\[Parameter-Ord22-A1-B1\])–(\[Parameter-Ord22-A2-B2\]) which leads to $$\label{Parameter-Coeff-2.2.3}
\alpha^T = \begin{bmatrix} 1 && c_6 && -c_6 \end{bmatrix},
\quad
A^{(0)} = \begin{bmatrix} 0 && 0 && 0 \\ c_7 && 0 && 0 \\
\frac{1-2 c_6 c_7}{-2 c_6}-c_8 && c_8 && 0 \end{bmatrix},
\quad
B^{(0)} =
\begin{bmatrix} 0 && 0 && 0 \\ \frac{c_1}{2}(1+\frac{1}{2 c_6}) && 0 && 0 \\
\frac{c_1}{2}(1-\frac{1}{2 c_6}) && 0 && 0 \end{bmatrix} .$$
Coefficients for SRK Methods of Order (3,2) {#Sec:Coeff-SRK-Ord-3-2}
-------------------------------------------
If we consider the classification of the coefficients for explicit SRK methods, we can see from Section \[Sec:Coeff-SRK-Ord-2-2\] that in some of the resulting cases there are still degrees of freedom in choosing the coefficients for $\alpha$ and $A^{(0)}$. Therefore, we analyze now the classification for explicit SRK methods (\[SRK-method-Ito-Wm-allg01\]) with $s=3$ stages of order $p_D=3$ and $p_S=2$. Thus, we additionally have to take into account the well known deterministic order 3 conditions [@Butcher87; @HNW93] $$\begin{aligned}
{3}
\alpha^T (A^{(0)} e)^2 &= \frac{1}{3} \, ,
\label{cond-det-ord-3a} \\
\alpha^T (A^{(0)} (A^{(0)} e)) &= \frac{1}{6} \, .
\label{cond-det-ord-3b}\end{aligned}$$ Clearly, these conditions can not be fulfilled in case (\[Parameter-Fall1\]) where $\alpha_3=0$ as well as in case (\[Parameter-Fall2.1.1\]) due to $A^{(0)}_{21}=0$ and in case (\[Parameter-Fall2.2.2\]) due to the restrictions for $\alpha$ and $A^{(0)}$. However, in the case of parameter family (\[Parameter-Fall2.1.2\]) we obtain from (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) an SRK method of order (3,2) if in (\[Parameter-Coeff-2.1.2\]) it holds $$c_7 = \frac{1}{2} c_6 \pm \frac{1}{6} \sqrt{9 c_6^2-36 c_6
+24} - \frac{1}{3 c_6} \, , \quad \quad c_8 = \frac{1}{3 c_6}
\, .$$\
For the parameter families in case (\[Parameter-Fall2.2.1\]) and (\[Parameter-Fall2.2.3\]) we have to distinguish the following three possibilities due to condition 8. of Theorem \[SRK-theorem-ito-ord2-Wm-main1\] and due to (\[cond-det-ord-3a\]):
a) \[case-a)\] $\alpha_2 = \frac{3}{4}$, $A^{(0)}_{21}=\frac{2}{3}$, $A^{(0)}_{31} = -A^{(0)}_{32}$.\
b) \[case-b)\] $\alpha_3=\frac{3}{4}-\alpha_2$, $A^{(0)}_{21}=\frac{2}{3}$, $A^{(0)}_{31}=\frac{2}{3}-A^{(0)}_{32}$.\
c) \[case-c)\] $\alpha_2 = \frac{1}{6} \frac{2-3(A^{(0)}_{31} +
A^{(0)}_{32})}{A^{(0)}_{21} (A^{(0)}_{21}-A^{(0)}_{31}
-A^{(0)}_{32})}$, $\alpha_3 = \frac{1}{6} \frac{3
A^{(0)}_{21}-2}{(A^{(0)}_{31}+A^{(0)}_{32}) (A^{(0)}_{21} -
A^{(0)}_{31}-A^{(0)}_{32})}$ if $A^{(0)}_{31}+A^{(0)}_{32} \neq
A^{(0)}_{21} \neq 0$ and $A^{(0)}_{31}+A^{(0)}_{32} \neq 0$.
If we consider now the case of the parameter family (\[Parameter-Fall2.2.1\]) then the conditions (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) are fulfilled for \[case-a)\]) if $$c_6 = \frac{3}{4} \, , \quad \quad c_7 = \frac{1}{4 c_9} \, ,
\quad \quad c_8 = \frac{2}{3} \, ,$$ with $c_7 \notin \{-\tfrac{3}{4}, 0,\tfrac12
\} \cup \, ]-\tfrac{1}{4},0[ \,$ in (\[Parameter-Coeff-2.2.1\]). Further, the conditions (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) are also fulfilled in the case (\[Parameter-Fall2.2.1\]) combined with \[case-b)\]) if $$c_6 = \frac{3}{4} - \frac{1}{4 c_9} \, , \quad \quad c_7 =
\frac{1}{4 c_9} \, , \quad \quad c_8 = \frac{2}{3} \, ,$$ with $c_9 \neq 0$ and for $c_6 \in \, ]0,\tfrac{1}{4}[ \, \cup \,
]\tfrac{1}{4},\tfrac{3}{4}[\,$ in (\[Parameter-Coeff-2.2.1\]). Finally, the considered parameter family (\[Parameter-Fall2.2.1\]) fulfills the conditions (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) also for the case \[case-c)\]) if $$\label{2.2.1.c}
c_6 = \frac{1}{6} \frac{2-3 \lambda}{c_8 (c_8- \lambda)} \, , \quad
\quad c_7 = \frac{1}{6} \frac{3 c_8 -2}{\lambda (c_8-\lambda)} \, ,\quad
\quad c_9 = \frac{\lambda(c_8-\lambda)}{(3c_8-2)c_8}$$ in (\[Parameter-Coeff-2.2.1\]) for $\lambda \in \mathbb{R}$ with $\lambda \notin \{0, \tfrac{2}{3}, c_8, \tfrac{2}{3}-c_8\}$, $(\lambda-1)c_8\neq\lambda^2-\frac23$, $c_8 \notin \{0,
\tfrac{2}{3}\}$ and with $\lambda<\frac23$ if $c_8=1$; $\tfrac{3c_8-2}{3(c_8-1)}\leq\lambda<\frac23$ if $\tfrac{2}{3} < c_8 < 1$ holds; $\frac23<\lambda$ or $\lambda\leq\tfrac{3c_8-2}{3(c_8-1)}$ if $0 < c_8 <
\tfrac{2}{3}$ holds and with $\lambda<\frac23$ or $\lambda\geq\tfrac{3c_8-2}{3(c_8-1)}$ if $c_8 < 0$ or $c_8 > 1$ holds. Note that $\lambda = \frac{1-2 c_6
c_8}{2 c_7}$ is thus automatically fulfilled in (\[Parameter-Coeff-2.2.1\]).\
\
Finally, we consider the case of parameter family (\[Parameter-Fall2.2.3\]) which fulfills the additional order three conditions (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) in case \[case-a)\]) for $c_6 = \frac{3}{4}$ and $c_7 =
\frac{2}{3}$ as well as in case \[case-c)\]) with $c_6 =
\frac{1}{4 c_7 - \frac{4}{3}}$ if $c_7 \notin \{-\frac{1}{6},0,
\frac{1}{3}\}$ in (\[Parameter-Coeff-2.2.3\]). For case \[case-b)\]) there exists no solution.
Optimal SRK Schemes {#Sec:Optimal-schemes}
===================
In the present section, coefficients for the SRK method (\[SRK-method-Ito-Wm-allg01\]) of different orders of convergence are presented. Due to some degrees of freedom in choosing the coefficients, we consider additional conditions in order to specify the free parameters of the SRK scheme. Clearly, these additional conditions need not necessarily be fulfilled for the desired order of convergence. However, coefficients fulfilling also higher order conditions yield SRK schemes with the objective to obtain smaller error constants and we call them optimal SRK schemes in the following.
Coefficients for Optimal SRK Schemes of Order (2,1)
---------------------------------------------------
For SRK methods of order $p_D=2$ and $p_S=1$, we need 2 stages for the drift part, however only one stage is needed for the diffusion. Therefore, we consider only the case of $c_4=
\ldots=c_{11}=0$ in (\[Parameter-Ord21-all\]). Next, we want to specify $c_2$ and $c_3$. Therefore, we consider additional order conditions which need not necessarily be fulfilled for order $(2,1)$ schemes. Taking into account condition 9.yields $c_2 = c_3^2$. From condition 10. it follows that $c_2 = c_1 \, c_3$. Further, one can consider the deterministic order 3 conditions (\[cond-det-ord-3a\]) and (\[cond-det-ord-3b\]) [@Butcher87; @HNW93], whereas only (\[cond-det-ord-3a\]) can be fulfilled which yields $c_2 =
\frac{2}{3}$. However, one can only combine two of the mentioned additional conditions. Condition 9. together with 10. yields $c_2=1$ and $c_3=c_1$, condition 9. together with (\[cond-det-ord-3a\]) yields $c_3 = \pm \sqrt{\frac{2}{3}}$ while condition 10. together with (\[cond-det-ord-3a\]) results in $c_3=c_1 \frac{2}{3}$. One can easily verify that all the order 2 conditions 8.–50. are fulfilled with the exception of conditions 11. and 13.–16. and condition 9.or 10. if (\[cond-det-ord-3a\]) is fulfilled in combination with only one of them. Therefore, we consider the additional condition (\[cond-det-ord-3a\]) which is fulfilled for $c_1 = 1$ and $c_2 = c_3 = \frac{2}{3}$. This leads to the SRK scheme RDI1WM presented in Table \[Table-Coeff-RDI1\], which is an improved Euler-Maruyama scheme with two evaluations of the drift and one of the diffusion coefficients for each step. Thus, it may be superior to the widely used Euler-Maruyama scheme, especially in practical applications where small noise is inherent to the system.
$$\begin{array}{r|ccc|ccc|ccc}
0 &&&&&&&&& \\
\frac{2}{3} & \frac{2}{3} & & & \frac{2}{3} &&&&& \\
\hline
0 &&&&&&&&& \\
0 & 0 & & & 0 &&&&& \\
& \frac{1}{4} & & \frac{3}{4} &
1 & & 0 & 0 & \quad & 0 \\
\cline{2-10}
& & & & 0 & & 0 & 0 & & 0
\end{array}$$
Coefficients for Optimal SRK Schemes of Order (2,2) and (3,2)
-------------------------------------------------------------
If we consider the order three tree $(\sigma_{j_1}, \sigma_{j_2},
\{\tau, \sigma_{j_4} \}_{j_3})$ (see [@Roe06a; @Roe03] for details) in the case of $j_1=j_2=j_3=j_4$, then we obtain the corresponding order condition $$\label{tree-cond-1}
{\beta^{(2)}}^T
((A^{(1)} e) (B^{(1)} e))
({\beta^{(1)}}^T e)^2 = \frac{2}{3} .$$ For the coefficient families (\[Parameter-Coeff-1\])-(\[Parameter-Coeff-2.2.3\]), this order condition is fulfilled if $c_3 = \pm \sqrt{\frac{2}{3}}$. For the tree $(\sigma_{j_1},\{\sigma_{j_2},\sigma_{j_2},
\sigma_{j_3}, \sigma_{j_3} \}_{j_1})$ (see [@Roe06a; @Roe03]) we calculate in the case of $j_1 \neq j_2$ and $j_2=j_3$ the following order three condition $$\label{tree-cond-2}
({\beta^{(1)}}^T e) ({\beta^{(3)}}^T
( B^{(2)} e )^4) = 1$$ which is fulfilled if $c_4 = \pm \sqrt{2}$. Due to some symmetry in the SRK schemes, we obtain always the same SRK schemes regardless what sign we choose for $c_3$, $c_4$ and $c_1$. In the following, we choose $c_3 = \sqrt{\frac{2}{3}}$, $c_4
= \sqrt{2}$ and $c_1=1$.\
\
Then, the parameter family (\[Parameter-Coeff-1\]) definitely provides the optimal SRK scheme
RDI2WM of order $p_D=p_S=2$ presented in Table \[Table-Coeff-RDI2\].
$$\begin{array}{r|ccccc|ccccc|cccccc}
0 & & & && &&&&& &&& \\
1 & 1 & &&& & 1 & && &&& \\
0 & 0 & & 0 && & 0 & & 0 & &&& \\
\hline
0 &&& &&&&& &&& \\
\frac{2}{3} & \frac{2}{3} & & & & & \sqrt{\frac{2}{3}} &&&& &&& \\
\frac{2}{3} & \frac{2}{3} & & 0 & & & -\sqrt{\frac{2}{3}} && 0 && &&& \\
\hline
0 & & &&& &&&&& &&& \\
0 & 0 & &&& & \sqrt{2} &&&& &&& \\
0 & 0 & & 0 & & & -\sqrt{2} & & 0 & & &&& \\
\hline
& \frac{1}{2} & & \frac{1}{2} & & 0 &
\frac{1}{4} & & \frac{3}{8} & & \frac{3}{8} & & 0 & &
\frac{\sqrt{6}}{4} & & -\frac{\sqrt{6}}{4} \\
\cline{2-17}
& & & & & & -\frac{1}{4} & & \frac{1}{8} & & \frac{1}{8}
& & 0 & & \frac{\sqrt{2}}{4} & & -\frac{\sqrt{2}}{4}
\end{array}$$
\
\
Next, we calculate SRK schemes of order $p_D=3$ and $p_S=2$ for the family (\[Parameter-Coeff-2.2.1\]) in the case (\[2.2.1.c\]). Again, we try to specify the remaining coefficients in the deterministic part of the scheme by additionally considering the order four conditions [@Butcher87; @HNW93] $$\begin{aligned}
{3}
\alpha^T (A^{(0)} (A^{(0)}e)^2) &=
\frac{1}{12} \, , \label{Zusatz-Gl-001a} \\
\alpha^T ((A^{(0)}e)(A^{(0)}
(A^{(0)}e) ) ) &= \frac{1}{8} \, . \label{Zusatz-Gl-001}\end{aligned}$$ These conditions are fulfilled if $\lambda=\tfrac{3}{4}$ and $c_8=\tfrac{1}{2}$. As a result of this, we obtain the coefficients of the SRK scheme RDI3WM presented in Table \[Table-Coeff-RDI3\].
$$\begin{array}{r|ccccc|ccccc|cccccc}
0 & & & && &&&&& &&& \\
\frac{1}{2} & \frac{1}{2} & &&& & \frac{9-2\sqrt{15}}{14} & && &&& \\
\frac{3}{4} & 0 & & \frac{3}{4} && & \frac{18+3 \sqrt{15}}{28} & & 0 & &&& \\
& \frac{2}{9} & & \frac{1}{3} & & \frac{4}{9} &
\frac{1}{4} & & \frac{3}{8} & & \frac{3}{8}
& & 0 & & \frac{\sqrt{6}}{4} & & -\frac{\sqrt{6}}{4} \\
\
\
However, if we claim for the family (\[Parameter-Coeff-2.2.1\]) in the case of (\[2.2.1.c\]) that the order four conditions (\[Zusatz-Gl-001a\]) and $$\label{Zusatz-Gl-002}
\begin{split}
\alpha^T (A^{(0)}e)^3 &= \frac{1}{4}
\end{split}$$ are fulfilled instead of (\[Zusatz-Gl-001\]), then we get the coefficients $\lambda=1$ and $c_8=\frac{1}{2}$. As a result of this, we obtain the coefficients of the SRK scheme RDI4WM presented in Table \[Table-Coeff-RDI4\]. Here, the deterministic part of scheme RDI4WM coincides with the well known Simpson scheme for ODEs [@HNW93].
$$\begin{array}{r|ccccc|ccccc|cccccc}
0 & & & && &&&&& &&& \\
\frac{1}{2} & \frac{1}{2} & &&& & \frac{6-\sqrt{6}}{10} & && &&& \\
1 & -1 & 2 & && & \frac{3+2\sqrt{6}}{5} & & 0 & &&& \\
& \frac{1}{6} & & \frac{2}{3} & & \frac{1}{6} &
\frac{1}{4} & & \frac{3}{8} & & \frac{3}{8}
& & 0 & & \frac{\sqrt{6}}{4} & & -\frac{\sqrt{6}}{4} \\
\cline{2-17}
& & & & & & -\frac{1}{4} & & \frac{1}{8}
& & \frac{1}{8} & & 0 & & \frac{\sqrt{2}}{4} & & -\frac{\sqrt{2}}{4}
\end{array}$$
Numerical example {#Sec:Numerical-Example}
=================
In the following, some of the SRK schemes presented in Section \[Sec:Optimal-schemes\] are applied to test equations in order to analyze their order of convergence in comparison to some well known schemes. Therefore, the functional $u = {\operatorname{E}}(f(X_t))$ is approximated by a Monte Carlo simulation based on the optimal SRK schemes RDI1WM of order 1 and RDI3WM and RDI4WM of order 2. The optimal SRK schemes are compared to the second order SRK scheme PL1WM due to Platen [@KP99], the Euler–Maruyama scheme EM of order 1 and the extrapolated Euler-Maruyama scheme EXEM [@KP99] also attaining order 2. The SRK scheme PL1WM is contained in the class of SRK schemes (\[SRK-method-Ito-Wm-allg01\]) with coefficients $c_1
= c_3 = c_4 = 1$ in (\[Parameter-Coeff-1\]) due to case (\[Parameter-Fall1\]). The extrapolated Euler-Maruyama approximation is given by $2 {\operatorname{E}}(f(Z_t^{h/2}))- {\operatorname{E}}(f(Z_t^{h}))$ based on the Euler-Maruyama approximations $Z_t^{h/2}$ and $Z_t^h$ calculated with step sizes $h$ and $h/2$.
The sample average $u_{M,h} = \frac{1}{M} \sum_{k=1}^M f(Y_t(\omega_k))$, $\omega_k \in
\Omega$, of $M$ independent simulated realizations of the considered approximation $Y_t$ is calculated in order to estimate the expectation. In the following, we denote by $\hat{\mu} = u_{M,h} - {\operatorname{E}}(f(X_t))$ the mean error and by $\hat{\sigma}^2_{\mu}$ the empirical variance of the mean error. Further, we calculate the confidence interval with boundaries $a$ and $b$ to the level of 90% for the estimated error $\hat{\mu}$ (see [@KP99; @Roe03] for details).\
\
As the first example, we consider the non-linear SDE [@KP99; @MacNav01; @Roe06b] $$\label{Simu:nonlinear-SDE2}
dX_t = \left( \tfrac{1}{2} X_t + \sqrt{X_t^2 + 1} \right) \,
dt + \sqrt{X_t^2 + 1} \, dW_t, \qquad X_0=0,$$ on the time interval $I=[0,2]$ with the solution $X_t = \sinh (t +
W_t)$. Here, we choose $f(x)=p({\operatorname{arsinh}}(x))$, where $p(z) = z^3 -
6z^2 + 8z$ is a polynomial. Then the expectation of the solution can be calculated as $${\operatorname{E}}(f(X_t)) = t^3 - 3t^2 + 2t \,\,.$$ The solution ${\operatorname{E}}(f(X_t))$ is approximated with step sizes $2^{-1},
\ldots, 2^{-4}$ and $M=10^9$ simulations are performed in order to determine the systematic error of the considered schemes at time $t=2$. The results for the applied schemes are presented in Table \[Table1\]. The orders of convergence correspond to the slope of the regression lines plotted in Figure \[Bild001\] where we get the order $0.58$ for the EM scheme, order $1.11$ for RDI1WM, order $1.80$ for EXEM, order $1.81$ for PL1WM, order $1.93$ for RDI3WM and order $2.01$ for the scheme RDI4WM.
$h$ $|\hat{\mu}|$ $\hat{\sigma}_{\mu}^2$ $a$ $b$
-- ---------- --------------- ------------------------ ------------ ------------
$2^{-1}$ 8.797E-01 6.534E-07 -8.799E-01 -8.795E-01
$2^{-2}$ 7.705E-01 1.592E-06 -7.708E-01 -7.702E-01
$2^{-3}$ 4.825E-01 1.599E-06 -4.828E-01 -4.822E-01
$2^{-4}$ 2.691E-01 1.754E-06 -2.694E-01 -2.688E-01
$2^{-1}$ 1.101E-00 1.381E-06 -1.101E-00 -1.100E-00
$2^{-2}$ 5.342E-01 2.080E-06 -5.346E-01 -5.339E-01
$2^{-3}$ 2.390E-01 3.297E-06 -2.394E-01 -2.386E-01
$2^{-4}$ 1.112E-01 2.984E-06 -1.116E-01 -1.107E-01
$2^{-1}$ 1.359E-00 2.990E-06 -1.359E-00 -1.359E-00
$2^{-2}$ 6.614E-01 7.315E-06 -6.620E-01 -6.607E-01
$2^{-3}$ 1.945E-01 8.629E-06 -1.952E-01 -1.938E-01
$2^{-4}$ 5.570E-02 9.014E-06 -5.641E-02 -5.499E-02
$2^{-1}$ 3.837E-01 1.885E-06 -3.841E-01 -3.834E-01
$2^{-2}$ 1.165E-01 3.207E-06 -1.169E-01 -1.161E-01
$2^{-3}$ 3.348E-02 2.475E-06 -3.386E-02 -3.311E-02
$2^{-4}$ 8.949E-03 3.447E-06 -9.390E-03 -8.509E-03
$2^{-1}$ 3.926E-01 1.400E-06 -3.929E-01 -3.923E-01
$2^{-2}$ 1.041E-01 2.787E-06 -1.045E-01 -1.037E-01
$2^{-3}$ 2.748E-02 2.427E-06 -2.785E-02 -2.711E-02
$2^{-4}$ 7.054E-03 1.813E-06 -7.373E-03 -6.734E-03
$2^{-1}$ 3.760E-01 1.488E-06 -3.762E-01 -3.757E-01
$2^{-2}$ 9.454E-02 2.823E-06 -9.494E-02 -9.414E-02
$2^{-3}$ 2.318E-02 2.441E-06 -2.355E-02 -2.281E-02
$2^{-4}$ 5.816E-03 1.816E-06 -6.135E-03 -5.496E-03
: Mean errors, empirical variances and confidence intervals for SDE (\[Simu:nonlinear-SDE2\]).[]{data-label="Table1"}
\
\
As a second example, a multi-dimensional SDE with a $2$-dimensional driving Wiener process is considered: $$\label{Simu:dm-SDE}
\begin{split}
d \begin{pmatrix} X_t^1 \\ X_t^2 \end{pmatrix}
&= \begin{pmatrix} -\frac{273}{512} & 0 \\
-\frac{1}{160} & -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix} \,
\begin{pmatrix} X_t^1 \\ X_t^2 \end{pmatrix} \, dt +
\begin{pmatrix} \frac{1}{4} X_t^1 & \frac{1}{16} X_t^1 \\
\frac{1-2\sqrt{2}}{4} X_t^2 & \frac{1}{10} X_t^1 + \frac{1}{16} X_t^2
\end{pmatrix} \, d \begin{pmatrix} W_t^1 \\ W_t^2
\end{pmatrix}
\end{split}$$ with initial value $X_0=(1,1)^T$. This SDE system is of special interest due to the fact that it has no commutative noise. Here, we are interested in the second moments which depend on both, the drift and the diffusion function (see [@KP99] for details). Therefore, we choose $f(x) = (x^1)^2$ and obtain $$\begin{split}
{\operatorname{E}}(f(X_t)) = \exp(- t) \, .
\end{split}$$ We approximate ${\operatorname{E}}(f(X_t))$ at $t=4$ by $M=5\cdot 10^8$ simulated trajectories with step sizes $2^{-0}, \ldots, 2^{-3}$. The results for the considered schemes are presented in Table \[Table2\] and Figure \[Bild001\]. Here, the order of convergence is $0.88$ for the Euler-Maruyama scheme, $1.53$ for RDI1WM, $2.22$ for PL1WM, $2.18$ for EXEM, $2.24$ for RDI3WM and order $2.28$ for the optimal SRK scheme RDI4WM.
$h$ $|\hat{\mu}|$ $\hat{\sigma}_{\mu}^2$ $a$ $b$
-- ---------- --------------- ------------------------ ------------ ------------
$2^{-0}$ 1.178E-02 3.946E-11 -1.178E-02 -1.178E-02
$2^{-1}$ 7.002E-03 6.669E-11 -7.004E-03 -7.000E-03
$2^{-2}$ 3.738E-03 5.799E-11 -3.740E-03 -3.736E-03
$2^{-3}$ 1.922E-03 8.614E-11 -1.925E-03 -1.920E-03
$2^{-0}$ 9.000E-03 1.275E-10 8.998E-03 9.004E-03
$2^{-1}$ 2.472E-03 1.127E-10 2.470E-03 2.475E-03
$2^{-2}$ 8.870E-04 8.278E-11 8.848E-04 8.891E-04
$2^{-3}$ 3.714E-04 8.926E-11 3.691E-04 3.736E-04
$2^{-0}$ 2.223E-03 2.871E-10 -2.227E-03 -2.219E-03
$2^{-1}$ 4.733E-04 2.500E-10 -4.771E-04 -4.696E-04
$2^{-2}$ 1.071E-04 3.585E-10 -1.116E-04 -1.026E-04
$2^{-3}$ 2.348E-05 3.919E-10 -2.818E-05 -1.879E-05
$2^{-0}$ 4.230E-03 6.967E-11 4.228E-03 4.232E-03
$2^{-1}$ 7.736E-04 8.594E-11 7.714E-04 7.758E-04
$2^{-2}$ 1.728E-04 8.412E-11 1.706E-04 1.750E-04
$2^{-3}$ 4.148E-05 8.356E-11 3.932E-05 4.365E-05
$2^{-0}$ 1.909E-03 3.700E-11 -1.910E-03 -1.907E-03
$2^{-1}$ 3.822E-04 6.597E-11 -3.841E-04 -3.803E-04
$2^{-2}$ 8.282E-05 6.356E-11 -8.471E-05 -8.093E-05
$2^{-3}$ 1.797E-05 8.787E-11 -2.019E-05 -1.574E-05
$2^{-0}$ 1.608E-03 4.285E-11 -1.609E-03 -1.606E-03
$2^{-1}$ 3.089E-04 6.812E-11 -3.108E-04 -3.069E-04
$2^{-2}$ 6.583E-05 6.403E-11 -6.773E-05 -6.394E-05
$2^{-3}$ 1.392E-05 8.803E-11 -1.615E-05 -1.170E-05
: Mean errors, empirical variances and confidence intervals for SDE (\[Simu:dm-SDE\]).[]{data-label="Table2"}
![Orders of convergence for SDE (\[Simu:nonlinear-SDE2\]) and SDE (\[Simu:dm-SDE\]).[]{data-label="Bild001"}](Bsp12_1 "fig:"){width="6.8cm"} ![Orders of convergence for SDE (\[Simu:nonlinear-SDE2\]) and SDE (\[Simu:dm-SDE\]).[]{data-label="Bild001"}](Bsp13_1 "fig:"){width="6.8cm"}
![Computational effort per simulation path versus precision for SDE (\[Simu:nonlinear-SDE2\]) and SDE (\[Simu:dm-SDE\]).[]{data-label="Bild002"}](Bsp12_2 "fig:"){width="6.8cm"} ![Computational effort per simulation path versus precision for SDE (\[Simu:nonlinear-SDE2\]) and SDE (\[Simu:dm-SDE\]).[]{data-label="Bild002"}](Bsp13_2 "fig:"){width="6.8cm"}
\
\
Due to the results in Figure \[Bild001\], we can see that for both test equations the so called optimal SRK scheme RDI1WM attains much better orders of convergence than the well known order one EM scheme. The same holds for the optimal SRK schemes RDI3WM and RDI4WM compared to the order two schemes EXEM and PL1WM. Clearly, the optimal SRK schemes RDI1WM, RDI3WM and RDI4WM need some additional computational effort compared to the schemes EM, EXEM and PL1WM, respectively. Therefore, we take the number of evaluations of the drift function $a$ and of each diffusion function $b^j$, $1 \leq
j \leq m$, as well as the number of random variables that have to be simulated as a measure for the computational effort. Then we can compare the computational effort versus the errors of the analyzed schemes. The results are presented in Figure \[Bild002\], and again, RDI1WM performs much better than the scheme EM for both test equations. Further, RDI3WM yields similar results like RDI4WM which is for higher precisions slightly better than PL1WM and significantly better than the scheme EXEM for the test equation (\[Simu:nonlinear-SDE2\]). Considering the multi-dimensional test equation (\[Simu:dm-SDE\]), the scheme RDI3WM is again close to RDI4WM which performs for higher precisions slightly better than EXEM. However both optimal SRK schemes RDI3WM and RDI4WM are significantly better than the SRK scheme PL1WM.
Conclusion {#Sec:Conclusion}
==========
In the present work, a full classification of the coefficients for a class of explicit SRK methods of order $(1,1)$ for $s=1$ and order $(2,1)$ for $s=2$ stages as well as for the orders $(2,2)$ and $(3,2)$ with $s=3$ stages is calculated. Based on this classification, coefficients for so called optimal SRK schemes are determined by considering additional higher order conditions. Optimal coefficients for SRK methods of order $(2,1)$, $(2,2)$ and $(3,2)$ are calculated and similarly to the deterministic setting [@HNW93], better convergence results are expected for these schemes in general. Finally, the SRK schemes based on optimal coefficients are applied to some test equations. Here, it turned out that the proposed optimal SRK schemes attain higher orders of convergence than the well known schemes under consideration and they also perform very well if the computational effort is taken into account.\
\
For future research, it would be interesting to extend the presented classification to diagonal or fully implicit SRK methods. Further, the given classification may be applied in order to determine coefficients for SRK methods with optimal stability properties.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are very grateful to the unknown referees for their fruitful comments and suggestions.
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Q: I just finished reading your October ’08 column highlighting workouts for beginning bodybuilders. I found your advice timely, as I’m starting that journey myself, although with a big difference—I’m 52. I’ve lifted weights off and on over the years but have never been consistently serious until almost six years ago. At that time, I was 47, 6’5” and almost 250 pounds with 20-plus percent bodyfat. I started doing full-body workouts and slow-motion cardio. After almost three years and a couple of injuries that helped me learn how to be more careful, I was doing very well. My weight was down to 210 with about 12 percent bodyfat, according to my online trainer. I’d gone from a 90- to a 170-pound bench press for three sets of 12 reps, deadlifting 350, doing 24 to 30 sets per workout using a split routine, and although I never saw my abs and never really put on much mass, my waist was down to 36 inches. That was three years ago, November, 2005.
My daughter’s health took me away from training for almost three years, but she’s now on the mend. I’ve started training again and getting refocused. I can still bench-press 145 pounds for three sets of eight and deadlift 250 pounds. I’ve never made the commitment to be a real bodybuilder, but I’d like to do so now.
I’ve returned to an old full-body routine similar to one you suggested in your column. I’m doing three sets of 12 reps for a dozen exercises. My plan was to do that for two to three months while doing HIT cardio on off-days, then switch to the split routine you suggested.
I was amazed at the volume in your split routine. Is that really what I need to put on solid mass? What level of intensity is needed with that volume? Later in the same issue, C.S. Sloan, a former powerlifter, recommends lots of volume, staying away from failure and moderate reps (six to eight), while Steve Holman’s approach with X Reps and POF suggests less volume with extreme intensity, even for an older guy like me.
My question comes down to this: Can I apply the general outline of your article to my own training needs as a 52-year-old relative newbie? What general modifications, if any, would you recommend to ensure that it’s effective? Or would you recommend an alternative approach such as Sloan’s or Holman’s for someone in my situation?
A: I think it’s great that you’re making the commitment to be a “real bodybuilder.” It’s never too late, and I’m happy that you’re ready to take on the challenge and achieve your best condition ever.
The routine I listed in the October issue is an intermediate program designed for younger bodybuilders who are trying to add size and gain weight. That was the routine I used to bulk up when I was in my early 20s.
That program is centered on the basic exercises, using heavy resistance for six to eight reps. I trained each bodypart twice a week, but I trained only four days a week so I’d have three full days to recover. The volume is relatively high because I trained four muscle groups at each workout with an average of 10 to 12 sets per bodypart. Here’s the routine again:
Monday
Bench presses 4 x 10, 8, 6, 6
Incline dumbbell presses 3 x 8, 6, 6
Flyes 3 x 8, 6, 6
Dumbbell pullovers 3 x 10, 8, 8
Seated military presses 4 x 10, 8, 6, 6
Lateral raises 4 x 10, 8, 6, 6
Bent-over lateral raises 3 x 8, 6, 6
Barbell shrugs 4 x 10, 8, 6, 6
Pushdowns 4 x 10, 8, 6, 6
Lying triceps extensions 3 x 8, 6, 6
Weighted dips 3 x 8, 6, 6
Standing calf raises 4 x 12, 10, 8, 6
Seated calf raises 3 x 12, 10, 8
Tuesday
Incline situps 3 x max
Incline knee raises 3 x max
Squats 5 x 10, 8, 6, 6, 6
Leg presses 4 x 12, 10, 8, 6
Leg curls 4 x 10, 8, 6, 6
Stiff-legged deadlifts 3 x 10, 8, 6
Wide-grip chins 4 x 10, 8, 6, 6
Barbell rows 4 x 10, 8, 6, 6
Seated cable rows 3 x 10, 8, 6
Incline curls 3 x 10, 8, 6
Barbell curls 3 x 8, 6, 6
Wrist curls 4 x 12, 10, 8, 8
Wednesday
Rest
Thursday
Bench presses 4 x 10, 8, 6, 6
Incline barbell presses 3 x 8, 6, 6
Incline flyes 3 x 10, 8, 6
Weighted dips 3 x 8, 6, 6
Seated dumbbell presses 4 x 10, 8, 6, 6
Seated lateral raises 4 x 10, 8, 6, 6
Upright rows 3 x 10, 8, 6
Power cleans 3 x 8, 6, 6
Close-grip bench presses 4 x 10, 8, 6, 6
Seated barbell extensions 3 x 8, 6, 6
Donkey calf raises 4 x 20
Leg press calf raises 3 x 15, 12, 10
Friday
Incline situps 3 x max
Incline knee raises 3 x max
Squats 4 x 12, 10, 8, 8
Front squats 3 x 10, 8, 6
Leg curls 4 x 10, 8, 6, 6
Stiff-legged deadlifts 3 x 10, 8, 6
Wide-grip chins 4 x 12, 10, 8, 6
One-arm dumbbell rows 3 x 8, 6, 6
T-bar rows 4 x 10, 8, 6, 6
Hyperextensions 3 x 15-20
Seated dumbbell curls 3 x 10, 8, 6
Preacher curls 3 x 8, 6, 6
Wrist curls 4 x 12, 10, 8, 6
As we get older, recuperation plays a bigger role in our workouts. I’m in my mid-40s now, and I never train more than four days a week. I’ve modified the intermediate routine by training each bodypart only once instead of two times a week. That seems to help recuperation, not only for my muscles but also for my joints and tendons—a big consideration as we get older.
By training only two muscle groups per workout instead of four, I keep my total volume much lower. That makes it easier to train harder at each workout, and it gives me more days off before I train the same bodyparts again. I can no longer train each muscle group twice a week because I don’t recuperate as fast as I used to.
I think a good routine for you at your level would be to split your body up over three workouts. Don’t train three days in a row—you’ll need a day off after two consecutive days of workouts. Instead, train two days on/one day off. That will give you five days of rest for each muscle group and your body two to three complete days of rest per week. Here’s an example:
Day 1: Chest, arms, calves
Dumbbell bench presses 4 x 12, 10, 8, 6
Incline presses 3 x 10, 8, 6
Incline flyes 3 x 10, 8, 8
Close-grip bench presses 3 x 10, 8, 8
Lying extensions 3 x 10, 8, 6
Incline curls 3 x 10, 8, 8
Barbell curls 3 x 10, 8, 6
Seated calf raises 4 x 20, 15, 12, 10
Day 2: Abs, legs
Hanging knee raises 3 x 30-40
Crunches 3 x 30-40
Leg extensions 3 x 15, 12, 10
Leg presses 3 x 12, 10, 8
Squats 3 x 10, 8, 8
Leg curls 3 x 12, 10, 8
Stiff-legged deadlifts 3 x 10, 8, 8
Day 3: Off
Day 4: Delts, back, calves
Seated dumbbell presses 3 x 10, 8, 6
Lateral raises 3 x 10, 8, 8
Bent-over lateral raises 3 x 10, 8, 8
Shrugs 3 x 10, 8, 6
Wide-grip chins 3 x 12, 10, 8
Barbell rows 3 x 10, 8, 6
Deadlifts 3 x 10, 8, 6
Leg press calf raises 4 x 12, 10, 8, 8
Day 5: Off
Day 6: Repeat cycle
The program focuses on the basic exercises, but by training fewer bodyparts per workout, you cut way back on the volume. It should be a good routine for you. Keep training hard by making the workouts progressive—training heavier or employing techniques such as supersets or drop sets to increase the intensity—and you should be able to achieve your goal of getting into the best shape of your life.
Q: My calves, and a lot of other people’s, are extremely stubborn. I’ve noticed improvement recently, but not much. As the calves are fast-recovering muscles, would it be all right to hit them with four sets four times a week? I want to do a soleus-prominent day, then a gastrocnemius-prominent day, alternating between standing and seated calf raises. Any thoughts on that?
A: The calves are notorious for being the bodybuilder’s most challenging bodypart. The bodybuilders with the best calves usually owe much of their development to heredity. Chris Dickerson, Ken Waller, Mike Matarazzo and Mike Mentzer always credited their parents for their incredible calf development; however, many bodybuilders don’t put the effort into developing their calves that they do other muscles, like the chest, arms or even thighs. Another problem is the overwhelming opinion that if your calves are weak, it’s all genetics and there’s nothing you can do about it.
My calves were very skinny when I started bodybuilding as a teenager. My legs were a weak point, and I always worked hard on them to bring them into balance with my upper body. My calves eventually developed into one of my stronger bodyparts.
Calves seem to recuperate much more quickly than most muscle groups, so they need to be trained more than once a week. I work my calves with two exercises for three to four sets each, twice a week. I usually train them on Monday and Thursday, giving them three to four days of rest between workouts.
I think the best exercise for calves is the donkey calf raise. I love doing donkeys because the weight is distributed directly over the hips—unique for a calf exercise. In addition, donkey calf raises give you heavy resistance and higher repetitions.
The key to building the calves is to train them heavy and use high repetitions to really pump the blood into the muscle. If you think about it, the calf muscles are the farthest from the heart, so you have to do more repetitions to get the blood into the calves. Even so, you can’t train the calves with light weights and expect them to grow. The muscle fibers in the calves are accustomed to walking every day, so doing high reps with a light-to-moderate resistance won’t do the trick.
I do donkey calf raises with someone sitting on my back instead of using a machine. Most of the machines I’ve seen don’t have enough resistance to really develop the calves. In addition to the weight of my partner, I use a belt to strap on more weight.
At one point in my career, when I made the best gains in calf development, I had two guys sit on my back along with another 100 pounds in resistance hanging from the belt. I could force out 15 reps with that substantial weight—about 500 pounds—before one of the guys would jump off. I’d continue doing the exercise with just one guy on my back and the extra 100 pounds. I could usually get another 10 reps with that resistance before the other guy would jump off. Then, with just the weight from the belt, I’d force out another 10 reps. That amounted to a total of 35 reps.
By doing drop sets that way, I trained my calves with very heavy weight but still did a lot of repetitions to force the blood into the muscle. The combination of high resistance and high reps really made the difference in building up my calves.
My current program has me working calves on Monday, doing seated calf raises—the best exercise for developing the soleus—for four sets of 10 to 20 reps followed by standing calf raises for the gastrocs, three to four sets of eight to 12 reps.
I use high reps on the seated calf raises because the soleus is composed mostly of slow-twitch fibers, and it responds best to high repetitions—15 to 20. After I have the blood in the muscle, I move to standing calf raises. I use heavier weights for eight to 12 reps to develop the fast-twitch muscle fibers of the gastrocnemius.
On Thursday I begin with donkey calf raises for four sets of 20 to 30 reps. That blows up my calves and gives me a great pump. When I finish the four sets, I can hardly walk. I finish off with another three sets of calf raises on the leg press machine, keeping my knees straight and getting a full range of motion—three sets of 10 to 12 reps.
You can see the pattern here. Do high reps with the first exercise to get a great pump in your calves and then follow up with another exercise that uses heavier resistance. The combination is the best for developing size in the calves. | https://www.ironmanmagazine.com/best-workout-to-grow/ |
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Monday October 9th:
Round 2 of the women’s championships. The players faced a windy day but luckily the rain held off. A grade winner with a 93-20-73 was Sandra Kent and runner up was Sandra Toone with a score of 102-26-76 while Kerryn Wagstaff won the putting with 29 putts. B grade was won by Jean Hansen with a score of 120-41-79 nett with Faith Morrissy close behind with a 119-39-80, whilst Valerie Johnson won the putting with 32 putts.
Sunday October 8th:
Round 1 of the Men’s Championships: 24 Men headed out in quite a heavy shower of rain to start their 1st round of Championships.
A grade was a close encounter with Andrew Hill wining with a 74-8-66 on a countback from Bob Bedson 76-10-66 nett. B grade also ran close with Rocky Dimieri winning with 81-15-66 from Brian McKenzie 84-17-67.
C grade saw John Dwyer win with 89-19-70 by one shot from Bill Buckley with a score of 89-18-71 nett.
Nearest the pin on 1st/10th A grade Craig McKenzie
B grade Brian McKenzie and C grade Ken Johnson. On the 9th/18th only A grade player Craig McKenzie managed to get on the green.
Saturday October 7th:
Mixed 4 person Ambrose Texas Scramble. The winners of this competition were Carmel and Rob Knowles partnered by Joycie Henderson and Rocky Dimieri with a score of 75-15- 59. Runners up by the merest of margins were Jeff and Jenny McKeon along with Noel and Kaylene Axelson 72-12-59.
Thursday October 5th:
17 women competed in the first round of the club championship. In the 0-26 handicap range, Sandra Kent was the winner with 88-21-67 and Sandra Toone was runner-up with 101-26-75. In the 27-45 category Valerie Johnson won with 101-30-79. Runner-up was Faith Morrissy with 119-39-80 on a count-back from Sue McKenzie and Caroline Wren. Sandra Kent won the Juice Lab trophy and also the A grade putting with 28 putts. Caroline Wren won the B grade putting with 30 putts.
Wednesday October 4th:
Men’s 4BBB stableford was won by Richard Stone and Darren Gill with 42 points. Runners-up were David Hand and Rick Johnston with 41 points in a 4-way count-back from Bruce Thomas and Graham McKenzie, 41, Doug Honey and Dick McGrath 41, and Don Wagstaff and Peter Tovey 41. David Hand had nearest the pins on 1st/10th and 9th/18th holes, and Ray Matthews was nearest the pin 1st/10th for B Grade.
Monday October 2nd:
Women’s 9 holes Stroke was
Won by Vicki Hannah 43-7 ½ – 35 ½ Sally Stone was runner-up on a c/b from Kerryn Wagstaff with 55-16-39. Putting 0-26 was won by Kerryn Wagstaff with 14 putts and 27-45 by Cheryle Polgeest with 16 putts.
Sunday October 1st:
Men’s Par Championship was won by Don Wagstaff +1 and Ray Matthews was runner-up with -5
Thursday September 28th:
13 women competed in an 18 hole stableford event with Janice McKenzie the A Grade winner with 31 points and runner-up Vicki Hannah with 30 points. In B Grade Faith Morrissy won with 28 points and runner-up Jean Hansen had 26 points. Jean also netted a nearest the pin on 1st/10th.
Sunday 17th September:
Men’s Foursomes Championship was won by Rick Johnston and Kurt Rixon with 73. Runner’s up were Craig McKenzie and Don Wagstaff with 74.
Wednesday September 27th:
In the final round of the 4
week stableford competition Lance Thomas was the winner with 35 points on a count-back from Runner-up Don Wagstaff, Ian Robinson and David Hand who also had 35 points. Nearest the pin 1st/10 in A Grade was Bob Bedson, and in B Grade Lance Thomas. On 9th/18th Brian McKenzie won the trophy. The overall 4 week winner was Don Wagstaff with 71 points and runner-up was Brian McKenzie with 68 points. | http://apollobaygolfclub.org.au/category/results/ |
In order for an amino acid sequence alignment to be as realistic as possible we need to understand the processes which control amino acid substitutions, as well as the effects of insertions and deletions. As mentioned in the chapter on protein structures, the degree of conservation of a protein three-dimensional structure is much higher than the degree of sequence conservation. And the reason is simply that the tertiary structure to a large degree determines protein function. Now, imagine a deletion of a couple of residues, for example, in an alpha-helix. What will happen to that helix? There is a good chance that it will collapse, since we may introduce a distortion in the hydrogen bonding network, in the packing of side chains, in the mutual adjustment of the torsion angles along the helix, etc. This in turn, may modify the overall 3D structure of the protein, which will affect its function, or probably result in denaturation and total loss of function, loss of the proteins ability to interact with its partners cell, etc. I think that one of the consequences of the principle of preservation of protein tertiary structure is the observation that insertions and deletions are most often found in regions between secondary structure elements - loop regions. Insertions and deletions within secondary structure elements may simply affect protein structure and function in some undesirable way. At the same time, the core region of proteins normally has a higher degree of sequence conservation and smaller number of insertions an deletions, which may be recognized by a smaller number of gaps in a sequence alignment.
As discussed in the previous page, in a computer-generated alignment insertions and deletions show up as gaps. If we count the score of the alignment taking into account the number of aligned identical and similar residues, we also need to find a way to limit the number of gaps in the alignment while maximizing the alignment score. Common sense would tell us that the number of insertions and deletions must be limited. To control the number of gaps, their size, position, etc., gap penalties are introduced. Then, the score used to judge the correctness of a given alignment is modified accordingly and gaps are allowed only if they increase the score (despite the negative effect of the penalty):
S= Sum of costs (identities, replacements) - Sum of penalties (number of gaps x gap creation penalties)
The values of identities and amino acid replacements in the expression are pre-calculated numbers, which are usually presented in a form of a 20 x 20 matrix (20 is the number of the most common amino acids). In total there are 210 possible pairs of amino acids, which includes 190 pairs of different amino acids + 20 pairs of identical substitutions. An example is presented below, the Gonnet matrix:
The first matrix of this type was developed in the 1970s by Margaret Dayhoff and co-workers, who pioneered the field of protein sequence analysis, databases and bioinformatics. Their scoring model was based on observed frequencies of substitutions of each of the 20 amino acids, derived from alignment of closely related sequences. In the resulting mutation data (or probability) matrix Mij each element provides an estimate of the probability of an amino acid in column i to be mutated to the amino acid in row j after certain evolutionary time. An evolutionary unit of 100 million years was adapted, resulting in the PAM (percentage accepted mutations / 100 million years) matrix. 1 PAM corresponds to an average amino acid substitution in 1% of all positions. Although 100 PAM does not mean that all the amino acids in the sequence have been changed since many of them will be mutated back to their original type. This is logical to assume since preservation of structure and function always has higher priority in selection.
Different versions of the amino acid substitution matrix can be used for different purposes. For example, low PAM (20, 40, 60) may be preferred in database scanning, which usually outputs short alignments of the closest-related sequence segments. The higher the number associated with PAM the longer the evolutionary distance. Thus, high PAM will be suitable for aligning more distant proteins, and if used for database scanning, it will find weaker and longer alignments. It has been shown that at 256 PAM 80 % of all amino acids will be substituted, although to various degrees: 48% of Trp, 41% of Cys and 20% of His would be unchanged, but only 7% of Ser will remain (More details). By other words different amino acids have different propensities for change, presumably due to both structural and functional reasons. For example, as mentioned earlier, tryptophane has a large side chain, and if located within the core of the structure, it would not be easy to replace it by some other amino acid. This may leave a cavity inside the structure, which may destabilize the protein structure as a whole.
Since the Dayhoff matrix was based on a limited set of protein sequences known at that time, no statistical data could be collected for many of the possible 190 substitutions. This was corrected for in a more recent PET91 substitution matrix, essentially an updated Dayhoff matrix (Jones et al., 1992). PET91 was constructed based on a study which included 2,621 protein families from the SwissProt database (part of the Expasy server). Meanwhile, other types of substitution matrices were developed, based on slightly different principles. One of the most popular is the BLOSUM matrix (BLOcks of Amino Acid SUbstitution Matrix, Henikoff S, Henikoff JG. 1992). BLOSUM scores amino acid replacements based on the frequencies of amino acid substitutions in un-gaped aligned blocks of sequences with a certain percentage sequence identity. This constitutes a major difference between PAM and BLOSUM matrices, since PAM matrices are based on mutations observed throughout a global alignment, which includes highly conserved regions as well as low-conservation regions with gaped alignment. The numbers associated with each matrix (e.g. BLOSUM62, BLOSUM80, etc) refer to the minimum percentage sequence identity of the sequences group within a certain block. Thus, higher numbers correspond to shorter evolutionary distance. By other words, BLOSUM with high numbers will be used for highly related sequences, while low BLOSUM numbers are used for distantly related proteins, for example is screening databases.
A number of substitution matrices have also been developed based on the comparison of three-dimensional structures (structure-based alignment). The 3D structure provides information on the position and length of secondary structure elements as well as loop regions, allowing a more precise positioning of gaps. To generate a structure-based sequence alignment it is possible to use a superposition of the 3D structures of the proteins in question (if structures are available for both) or to use the 3D structure of one member of the protein family to guide and correct placement of gaps in a multiple sequence alignment. Many graphics programs include superposition of 3D structures as an option.
Correct sequence alignment is crucial, e.g. for successful homology modeling. In the next sections we will consider some examples of sequence alignment (Tutorial 1 and Tutorial 2). The results of these tutorials will be used later in the tutorials on homology modeling. | http://proteinstructures.com/Sequence/Sequence/amino-acid-substitution.html |
Now that the programme for FantasyCon 2017 has been finalized, here’s what my schedule will look like:
Historical Fantasy, Historical Fiction, on Friday 29 September, at 4:30pm, in Panel Room 3
“The fantastic realms we write our stories in are often based on some real historical context. Join our panel in exploring the different historical research that can be necessary to create robust fantasy fiction, whether set in the primary world or in an invented one.”
Participants: Sandra Unerman (mod), Daniel Godfrey, Irene Soldatos, Peter McLean, Andrew Knighton, David Stokes
and
Writing Research, on Sunday 1 October, at 1:30pm, in Panel Room 1
“They always say a writer’s internet history would be alarming. There are stories about ‘ride alongs’ and all sorts of adventures in the name of authenticity. What do our panel research, how do they research it and how does it help their writing?”
Participants: Iain Grant (mod), Irene Soldatos, Joely Black, Tiffani Angus, Jacey Bedford
I will also be doing a READING, on Saturday 30 September, at 6:00pm.
I hope to see you there! Come say hello. Don’t be shy!
Hannah’s Bookshelf
On Saturday, 9 September 2017, you can catch me chatting with Hannah on North Manchester FM. We’ll be talking about fiction, history and research, books in general, and even musicology. And I’ll be telling her which three books I’ll save when the apocalypse comes. There’ll also be music, so you don’t get bored.
Here you can listen to previous shows on Hannah’s Bookshelf, and see what’s coming up.
Fiction Fix in Peterborough
I’ll be reading at Fiction Fix in Peterborough on 1 October 2017, right after this year’s FantasyCon (where I will also be attending. More info to follow).
The event will take place at the Draper’s Arms. I’ll post more information nearer the time. Do come by, if you’re free. There’ll be beer. And fiction. Can’t get better than that!
FantasyCon by the Sea
On 23 September 2017, I’ll be on the panel It’s a Kind of Magic, talking about magic in Fantasy fiction, at FantasyCon by the Sea, with four other great writers. | https://irenesoldatos.eu/news/ |
The typical age for the early onset menopause is 51 years old. The average is between the ages 45-55, but it commencing during 30’s or 40’s is not uncommon. Apart from the physical symptoms, women experiencing an early start to menopause may find it more difficult to cope because of the emotional implications. Menopause is the start of decline in reproductive functions, so women who still wish to have children it is a difficulty reality to reconcile.
Signs of Early Onset Menopause
There is little difference in menopause symptoms of whether it is early or not. These symptoms include:
- Hot flashes
- Irregular or missed periods
- Heavy or light periods
- Decreased libido
- Emotional changes
- Vaginal dryness
- Night sweats and excessive sweating
Your primary health care provider can help diagnose whether you are experiencing an early onset of menopause or not. This can be done through an early menopause test (blood test) to measure specific estradiol levels that may indicate the hormonal changes in women. When this level falls below 30, it can be an indicator of early onset menopause. Other blood test include testing for FSH, which accounts for the functions of ovaries. A decline in production shows that you may be in menopause.
What Causes Early Menopause
- Surgical removal of the ovaries
In cases where ovaries are needed to be removed, the result is induced menopause. Ovaries are responsible for the release of eggs that dictate the fertility of a woman, and to have them abruptly stop releasing the eggs altogether leads to early menopause. Oftentimes, surgically induced menopause can cause severe symptoms due to the abrupt decline in hormone levels. Ovarian function ceases which cause estrogen levels to drop significantly, forcing the woman to induced menopause.
Women who suffer from endometriosis, ovarian cancer, or polyps may require the removal of the ovaries or uterus.
- Premature ovarian failure
In about 1% of women, premature ovarian failure may occur. Although it is not fully understood, premature ovarian failure leads to gradual decline of the production of hormones and release of eggs. This can then turn into early onset menopause since it commonly occurs in women before the age 40.
- Chemotherapy and radiotherapy
The results of chemotherapy and radiation therapy can have adverse effects on the reproductive function of women. Depending on the proximity to the site and on the amount of therapy administered, these treatments can cause damage to the ovaries. Since chemotherapy kills not only cancer, but also healthy cells, it can affect the production of egg cells. Those who are taking Tamoxifen in order to reduce their symptoms may also develop early onset menopause as its side effect.
- Infection
Although evidence is still inconclusive, infections especially those in viral form could possibly trigger early menopause in women. This includes conditions such as cytomegalovirus or mumps. Tuberculosis can also infect the ovaries, causing effects in hormonal balance.
- Other Factors
In a recent study done by Gold, EB, et. al., an early commencement of menopause has been associated with different demographic factors such as; prior contraceptive use, alcohol consumption, smoking, physical activity, baseline weight, and even educational level.
Early Onset Menopause Depends on the Individual
Menopause is an individual experience and there are many predisposing factors to its occurrence. In order to find specific approaches to you own care, a proper understanding of your condition is essential. Further information on how to manage your symptoms of menopause can be found here. | http://www.menopausesymptomsfacts.com/tag/early-onset-menopause/ |
Any example where the divisor is less than 1 .
It's easier to visualize with smaller numbers. 18 divided by 3 = 6 18 divided by 6 = 3 If the dividend is the same, the smaller the divisor, the larger the quotient.
Dividing a number by a decimal always gives a number greater than the dividend. Some decimal numbers are bigger than 1, eg 506.23 , so the answer is less. The statement "a number greater than you" has no meaning so the question cannot be properly understood.
Fccrucgcfgfthfyft vfgygcfcvgvyfcfyff
Greater the divisor is less than 1,so the quotient is greater than the dividend
True.
I have no idea about the quotation, but the quotient is less than the divisor.
The quotient for whole numbers will always be less than or equal to the dividend. It will never be more.
Because you can take a piece of an apple out of a bag of apples more times than the number of whole apples in the bag.
The answer depends on the sign of the numbers.(1/4) / 2 = 1/8, which is smaller.(-1/4) / 2 = -1/8, which is greater.
1
There are usually more zeros in dividends because it is more preferible that the larger number is in the dividends section
Yes, but this is true of not just unit fractions but any positive number.
most definitely, especially if the divisor is a negative number
There is more than one answer to your question. One answer is a divisor of 99 and a dividend of 30492. A divisor of 33 and a dividend of 10164 would do as well. In fact you can use any number between 33 and 99 as the divisor if you adjust the dividend accordingly.
24/6 = 4 and 24/4 = 6, therefore 24 divided by 4 is the bigger quotient.
Divisor must be greater than 10000/308 ie 33 or more
yes
There are four parts to a division equation:1. Dividend (number being divided up into groups)2. Divisor (how many groups the dividend is bring broken into)3. Quotient (how many in each group from the dividend)4. Remainder (the leftover that is less than the divisor)Another way to locate them in an expression using the little division house:Dividend is inside the house. Divisor is outside at the door. Quotient is on top of the house and the Remainder are close by their friends, the quotient.Read more: http://wiki.answers.com/Parts_of_division#ixzz18rDC1tVp
Generally, the quotient of a whole number divided by a fraction will be greater than that whole number, because division is simply multiplying the dividend by the reciprocal of the divisor. For instance: 2 / (1/2) = 2 * (2/1)
If you think about it, it makes sense. when you divide a whole number (call that number X) by any number less than it(call that Z), the quotient (answer) is smaller than the first number. For example, 10/2=5, 12/4=3. Also, the lower the Z number is (ignoring if it were 1), the higher the quotient is. Therefore, since fractions are lower than 0, the quotient will be higher than the X number. | https://math.answers.com/Q/What_is_an_equation_when_the_quotient_is_bigger_than_the_dividend |
How Difficult Is Overbite Correction for Adults in 2022
Perfectly aligned teeth are something that we all want, but only a certain number of people can boast of it. Most people have a deep bite or, in other words, overbite. Overbite is a type of malocclusion, which means that it is a form of deviation. Like many other malocclusions, a deep bite can be treated and corrected. However, there are different types of overbite and not all of them need to be treated. In reality, it differs depending on many different factors. If you’re an adult, and you’re thinking of correcting an overbite, you’re probably wondering if this is possible and what kind of results you can expect. Here are some more details about it.
What is an overbite and do I need to correct it?
An overbite means that your upper teeth overlap lower ones. It is pretty common and most people have it. A regular deep bite is somewhere between two and five millimeters. If it is significantly over this range, teeth will likely look less aesthetically pleasing.
An overbite usually doesn’t mean that you need to do any kind of procedure to correct it. However, sometimes the overbite is too large, so it causes some health problems. If that is the case, it is recommended to correct it and prevent future complications. On the other hand, people sometimes do not like how their teeth look due to deep bite, so they are determined to correct it. Corrective procedures can be applied in both scenarios.
Why do I have a deep bite?
The causes of a deep bite are different. Of course, it can be genetically determined when genes are the ones that determine the shape and size of the jaw that leads to an overbite. It can also be caused (or exacerbated) at an early age by sucking a thumb or overuse of a bottle. This is why parents need to be extra careful and prevent this as soon as possible. An overbite can even occur in teenagers due to a bad habit of biting nails, pencils, etc. So all of this information leads to a conclusion that you don’t always need to be born with an overbite to have it as an adult. It can be a consequence of various external factors.
What can happen over time if I don’t treat my overbite?
If your overbite is very pronounced, it can lead to medical problems and complications. For example, you may feel discomfort while eating, pain in the jaw, headache, and others. A deep bite can lead to gum damage and having trouble with chewing food. It can also affect the way you speak and pronounce certain words. After some longer time, an overbite can cause teeth to wear and tear. Over time, teeth can also become less aesthetically appealing, which is a huge concern to some people and affects their self-esteem. That is why most adults are looking for a way to solve this problem and get a nice smile if possible. Most people that are working on correcting their overbite doesn’t experience any health issues. They just want to look better.
If you have some medical problems due to a deep bite, or you just don’t like how your smile looks, you should visit https://mynazarethdentist.com/.
On their website, you can request an appointment and consult their experts to hear their professional opinion. A nice smile is just one request away.
How can I treat my overbite?
There are a few corrective procedures you can apply to treat an overbite in adults. There are several different approaches to this condition, depending on how pronounced the overbite is.
1. Braces
Braces correct an overbite by moving the teeth and can be very effective. They can realign the jaw and help to reduce a deep bite that way. Braces can be a pretty quick solution, but the thing with them is they are an aesthetic problem for most people. If this is not big deal for you, it can be an easy, practical choice. In some cases, a dentist can recommend teeth extraction in combination with braces if they find that it would be a better solution for you. Sometimes by reducing teeth number, the jaw goes back into its natural position, and overbite can be corrected. You may need to wear braces for up to three years, so make sure you collect all relevant information about this treatment before making your final decision.
2. Clear braces
Clear braces were introduced to the world a long time ago and for a long time, they weren’t as effective as metal ones. Today, you can achieve pretty similar results with both types of braces, and the difference is that clear braces offer you a more aesthetically pleasing option. At least most people prefer clear braces compared to metal ones that stand out much more. A bad thing is that not every dentist offers this kind of treatment, so you might have trouble finding someone who will be able to do this. But if you prefer this option much more, we recommend searching around and maybe set aside a little bit more time to find what you’re looking for. It will surely pay off eventually.
3. Invisalign
Another great option for treating deep bites in adults is Invisalign. It is a pretty old treatment that has been improved over time, and nowadays it provides results pretty similar to ones you can achieve with braces. A dentist can also add elastics or other attachments that can help achieve results faster.
Conclusion: Having a beautiful smile with perfectly aligned teeth is something we’d all like to have. But most people have a malocclusion called deep bite or an overbite. It means that upper teeth overlap bottom ones to a certain extent. Most people don’t need to do anything about it since it doesn’t cause any medical concerns. However, other ones would like to correct a deep bite to make their smile more aesthetically pleasing. You can fix an overbite by wearing metal or clear braces, Invisalign, or applying a few other treatments. If having beautiful teeth will make you happier, you should consult with a dentist and see what can you do about it to achieve desired results. | https://theeventchronicle.com/overbite-correction-for-adults/ |
Q:
How to prove that: $19.999<e^\pi-\pi<20$?
I would like to know how to prove
$$e^\pi-\pi\sim 20.$$
More precisely, I want to show by using only mathematical tools that,
$$19.999<e^\pi-\pi<20$$
I have checked with online calculator and I got
$$e^\pi-\pi\approx19.9990999792\sim 20.$$
I tried to use the Tyalor expansion for exponential
$$ e^\pi =\sum_{n=0}^{\infty} \frac{\pi^n}{n!} = \pi +1+\sum_{n=2}^{\infty} \frac{\pi^n}{n!}$$
then, $$e^\pi -\pi =1+\sum_{n=2}^{\infty} \frac{\pi^n}{n!}$$
which is not easy to continue from here, since the factor $\pi^n$ is involved. Any idea?
A:
Use the iteration to calculate Gelfond's constant:
$$k_0 = 1/\sqrt{2},\quad k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}
{1+{\sqrt {1-k_{n}^{2}}}}}$$
$$e^\pi =\lim_{n\to\infty} \left(\frac{4}{k_{n+1}}\right)^{2^{-n}}$$
Within two iterations you should have (assuming you know $\pi$ to 5 or more digits and that you're willing to compute three square roots), that $e^\pi-\pi\approx 19.999$:
$$k_1=3-2\sqrt{2}$$
$$k_2=33+24\sqrt{2}-4\sqrt{140+99\sqrt{2}}$$
And:
$$\sqrt{4\over k_2} \approx 19.99926 + \pi$$
| |
For Toronto Maple Leafs’ head coach Mike Babcock, the NHL season is broken up into five-game segments in which his team needs to accumulate six points per segment. So far, in 24 games, the Leafs have collected 25 points.
On Wednesday, the Leafs will look to add to that total and finish their fifth five-game segment of the season averaging 5.4 points per segment. It’s a feat they will have to put to the test against a surging Minnesota Wild team in a game that Babcock has tapped as a must-win game for the young Leafs squad.
Part of grabbing the two big points from the Wild will be convincing the team that it’s a must-win matchup – something it seems the Leafs young players have already bought into.
The second meeting between these two clubs this season, the Maple Leafs will look to avenge a 3-2 loss in Minnesota back on October 20. In that game, the Leafs saw a 2-1 second period lead get swallowed up by two third period goals by the Wild – including Eric Staal’s second of the game.
The Leafs will look, once again, to their young goal to provide some offence for the club and will likely have Andersen back between the pipes after the team’s three-day layoff. On the other side of the puck, the Wild will rely heavily on experience veterans as they look to climb their way up the Western Conference standings.
Minnesota Wild at Toronto Maple Leafs
Air Canada Centre – 7:30 p.m. EST
Broadcast channels – SN and FS-N
2016-17 Season Series: October 20 – Wild 3 – Maple Leafs 2
Minnesota Wild – 12-8-4 – 28 Points
Road Record: 5-5-4
Hot Players: Charlie Coyle, Nino Niederreiter and Devan Dubnyk
Key Injuries: Zac Dalpe
Projected Lines:
Forwards
Nino Niederreiter-Eric Staal-Charlie Coyle
Jason Zucker-Mikko Koivu-Mikael Granlund
Zach Parise-Erik Haula-Jason Pominville
Chris Stewart-Tyler Graovac-Kurtis Gabriel
Defense
Ryan Suter-Jared Spurgeon
Jonas Brodin-Mathew Dumba
Marco Scandella-Nate Prosser
Goaltender
Devan Dubnyk
[irp]
Toronto Maple Leafs – 10-9-5 – 25 Points
Home Record: 8-3-0
Hot Players: James van Riemsdyk, Auston Matthews and Nazem Kadri
Projected Lines:
Forwards
Leo Komarov-Nazem Kadri-Nikita Soshnikov
James van Riemsdyk-Tyler Bozak-Mitch Marner
Zach Hyman-Auston Matthews-Connor Brown
Matt Martin-William Nylander-Ben Smith
Defense
Morgan Rielly-Nikita Zaitsev
Jake Gardiner-Martin Marincin
Matt Hunwick-Roman Polak
Goaltender
Frederik Andersen
Game Notes
[miptheme_dropcap style=”normal” color=”#222222″ background=””]1)[/miptheme_dropcap] Leafs’ Frederik Andersen has played three career games against the Minnesota Wild with a record of 2-1-0 with a 2.67 goals against average and .904 save percentage over that span.
[miptheme_dropcap style=”normal” color=”#222222″ background=””]2)[/miptheme_dropcap] Wild goaltender, Dubnyk, has played in seven games against the Maple Leafs over his career. He holds a 4-1-1 record with 16 goals against, a 2.53 goals against average, a .918 save percentage and one shutout over that time.
[miptheme_dropcap style=”normal” color=”#222222″ background=””]3)[/miptheme_dropcap] The Wild and Leafs have faced off 17 times since Minnesota came into the league in 2000. The two teams have split the head-to-head match-ups with the Leafs holding an 8-8-1 record. While they’ve split the games in terms of win-loss record, they’ve combined for an average of 4.5 goals per game with the Leafs outscoring the Wild 39-37.
[miptheme_dropcap style=”normal” color=”#222222″ background=””]4)[/miptheme_dropcap] The Leafs will have Antoine Bibeau on the bench – backing up Andersen – after the team placed Jhonas Enroth on waivers earlier this week. Bibeau has yet to suit up in an NHL regular season game, but has 49 wins in 85 AHL games with the Marlies.
[miptheme_dropcap style=”normal” color=”#222222″ background=””]5)[/miptheme_dropcap] Staal has 17 points (5g-12a) in 24 games in his first season with the Wild. He’s on pace for a 58 point season which will be his most since he recorded 61 with the Hurricanes in 2013-14.
[miptheme_dropcap style=”normal” color=”#222222″ background=””]6)[/miptheme_dropcap] The Maple Leafs could be in tough if they can’t score early and often against Dubnyk. While he’s 10-6-3 in 19 games so far this season, the Wild goaltender ranks first in the NHL in goals against average (1.63), first in save percentage (.946) and first in shutouts (4). He’s also perfect in the shootout so far this season having stopped all six shots he’s seen so far.
The Leafs are coming off a shootout loss to the Vancouver Canucks in their last outing while the Wild will look for their second win in a row after a 2-1 victory over the Edmonton Oilers on Sunday. With a win, not only will the Leafs meet coach Babcock’s five-game point challenge, but the Leafs will jump back to within four points of a playoff spot. The Leafs should be able to use their speed and rest to jump on the Wild early, but they will need to find a way to protect any lead they might grab in this contest.
Andrew is in his 8th year reporting for The Hockey Writers covering the Toronto Maple Leafs. He began his broadcasting with CBC’s Hockey Night in Canada team as well as being part of their coverage of the 2014 Winter Olympic Games in Sochi. He’s the former play-by-play voice of the London Jr. Knights for Rogers TV and currently hosts the Sticks in the 6ix podcast. You can follow him on Twitter at @AndrewGForbes. | https://thehockeywriters.com/preview-maple-leafs-must-win-versus-minnesota/ |
Served over french green beans, cherry tomatoes, fingerling potatoes, kalamata olives, horseradish aioli, balsamic reduction.
Spicy marinara, olive oil, garlic, anchovy, white wine, fresh basil, butter, Romano cheese.
Acorn squash roasted with fresh sage and pure maple syrup, prosciutto, roasted hazelnuts, evoo, arugula.
French green beans tossed in a warm bacon Sherry Dijon dressing, toasted almonds, dried cranberries, crumbled goat cheese, shaved red onion, over a bed of spinach.
Sauteed leeks, garlic, olive oil, white wine, butter, jalapenos, fresh thyme.
Sauteed with white wine, olive oil, marinara, basil, crushed red pepper, Romano cheese.
Stuffed with thyme and romano and drizzled with balsamic reduction.
Slow cooked in tomato vodka cream sauce, melted mozzarella.
Herbed chevre, pomodoro sauce, basil toast.
Flash fried, tossed with fresh lemon juice, salt, black pepper and romano.
4 garlic marinated, grilled shrimp, sauteed spinach, diced red peppers and lemon butter sauce.
Cherry tomatoes, spinach, cannellini beans, white wine, chili flakes.
Panko breaded, Calabrian chili and roasted red pepper cream sauce and arugula.
Panko breaded, calabrian chili & roasted red pepper cream sauce, arugula.
Five garlic marinated, grilled shrimp, sauteed spinach, diced red peppers, lemon butter sauce.
Browned in butter, garlic, white wine, vegetable stock, wilted spinach, Parmesan cheese.
Tossed with salt, black pepper, fresh rosemary, Romano cheese, and served with a side of chive creme fresh.
Sweet fennel sausage, button mushrooms, Spanish onions, peas, creamy tomato sauce, Romano cheese, fresh basil.
panko crusted, pan fried, filled with prosciutto, Romano, asiago, garlic, served with a Calabrian chili aioli.
Heirloom cherry tomatoes, balsamic vinaigrette, fresh basil, salt & pepper, crostini, balsamic reduction.
Savory Pecorino custard, marinara, cracked black pepper, fresh basil, rustic toast tips.
Red onion, toasted hazelnuts, pecorino romano, evoo, balsamic vinegar, lemon juice.
Field greens, pears, strawberries, candied walnuts, gorgonzola and balsamic reduction.
Field greens, goat cheese, almonds, oranges and orange mint dressing.
Romaine, tomatoes, cucumbers, belgian endive, radicchio, balsamic vinaigrette.
Romaine, herbed croutons, sun-dried tomatoes, shaved Parmesan cheese.
French green beans, bacon, cranberries, almonds, goat cheese, warm Dijon-sherry vinaigrette.
Arugula, radicchio, shaved fennel, toasted hazelnuts, medjool dates, ricotta salata, apple cider vinaigrette.
Chopped Romaine, fresh grilled sweet corn, cherry tomatoes, radishes, scallion, ricotta salata, passion fruit vinaigrette.
Arugula, diced Granny Smith apples, pancetta lardons, goat cheese, pine nuts, lemon balsamic vinaigrette.
Braised with red wine and mirepoix vegetables until fork tender, served with sauteed spinach and polenta.
R ed wine, mirepoix vegetables, marinara, capers.
Fresh egg pappardelle ribbon pasta, zucchini, yellow squash, cherry tomatoes, capers, basil, shrimp, Romano cheese, tomato white wine sauce.
Pan seared, oven roasted filet, nestled atop sauteed spinach and classic Puttanesca sauce of San Marzano tomatoes, fresh basil, garlic, anchovy, capers, caperberries, kalamata olives and white wine.
Beef tenderloin tips braised with mirepoix vegetables, red wine, marinara, capers, tossed with pappardelle pasta, topped with shaved parmesan.
Three-two oz beef tenderloin medallions, grilled medium rare, cracked black pepper port reduction, sauteed broccolini, garlic mashed potatoes.
Fresh egg pasta ribbons tossed with braised Beef Short ribs in red wine, mirepoix, marinara, hint of chili flakes, capers.
Fresh egg pasta filled with braised beef short rib, asiago, parmesan cheese, grilled onions, tossed in a Chianti cream sauce, shiitake and porcini mushrooms, sweet peas, thyme, Romano cheese.
Pan seared, lemon butter white wine sauce, caper berries, Calabrian chilis, roasted cherry tomatoes, capers, fresh garlic, sauteed spinach.
Pan seared, baked, served over sweet corn relish, chopped asparagus, black beans, red bell peppers, onions, garlic, honey, balsamic.
Boneless short rib, mirepoix vegetables, red wine chocolate reduction, creamy polenta, spinach.
Saffron egg pasta stuffed with Maine lobster claw meat, sherry wine cream sauce, fresh chives.
Olive oil, white wine, marinara, garlic, peas, capers, crushed red pepper, fresh basil.
Pounded thin with prosciutto and sage, pan seared, melted provolone, tomato lemon caper sauce, garlic mashed potatoes and grilled asparagus.
Pan-fried with pistachio-breadcrumb crust, red pepper-cream balsamic sauce, creamy polenta and grilled asparagus.
Pan seared, blistered cherry tomatoes, capers, Kalamata olives, caper-berries, white wine lemon butter and sauteed spinach.
Hammered thin, pan seared, Marsala wine cream sauce with shiitake, button, porcini mushrooms, Gorgonzola, garlic mashed potatoes, grilled asparagus.
Grilled and served with a butter white wine sauce with fresh garlic, caper berries, Calabrian chilis, oven roasted cherry tomatoes and capers with spinach.
Pork medallions pounded thin with prosciutto, provolone, fresh sage, pan seared, lemon tomato sauce, capers, garlic mashed potatoes, grilled asparagus.
Pan seared, roasted almonds, lemon zest, fresh herbs, haricot verts, lemon beurre blanc.
Pan seared with sage, prosciutto and provolone, lemon tomato caper sauce, grilled asparagus, garlic mashed potatoes.
Pan seared, baked, served over French green beans with pancetta lardons, mango Beurre blanc.
Tossed with artichokes, kalamata olives, oven cured tomatoes, marinara, romano, basil, chili flakes.
Rigatoni, homemade meat sauce, tomatoes, cream, onions and fresh mozzarella.
Grilled Italian sausage, roasted red peppers, herbed ricotta, marinara and romano cheese.
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Oreo crumb crust, dark chocolate ganache, whipped topping. | https://www.allmenus.com/il/oak-park/237477-cucina-paradiso/menu/ |
---
abstract: 'We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geq\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. [In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.]{}'
address:
- ' School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287-1804, USA'
- ' Department of Computer Science, University of Toronto, Toronto ON M5S 2E4, Canada'
- ' Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada'
author:
- Samantha Dahlberg
- Adrian She
- Stephanie van Willigenburg
title: 'Schur and $e$-positivity of trees and cut vertices'
---
[^1]
Introduction {#sec:intro}
============
The generalization of the chromatic polynomial, known as the chromatic symmetric function, was introduced by Stanley in 1995 [@Stan95] and has seen a resurgence of interest and activity recently. Much of this has centred around trying to resolve the 1995 conjecture of Stanley [@Stan95 Conjecture 5.1] and its equivalent incarnation [@StanStem Conjecture 5.5], which states that if a poset is $(3+1)$-free, then its incomparability graph is a nonnegative linear combination of elementary symmetric functions, that is, $e$-positive. The study of chromatic symmetric function $e$-positivity [@ChoHuh; @Dahl; @lollipop; @Foley; @FoleyKin; @Gash; @GebSag; @GP; @Hamel; @MM; @HuhNamYoo; @Wolfe], and related Schur-positivity [@Gasharov; @Paw; @SW; @Stanley2], is also an active area due to connections to the representation theory of the symmetric and general linear group.
Many partial results regarding chromatic symmetric functions have been obtained such as when the graph involved is the path or the cycle [@lollipop; @Stan95; @Wolfe], when the graph is formed from complete graphs [@ChoHuh; @GebSag; @MM], or when a graph avoids another [@Foley; @Gash; @Hamel; @Tsujie]. These proofs have not always worked directly with the chromatic symmetric function. Instead, sometimes generalizations of the chromatic symmetric function have been employed such as to quasisymmetric functions [@ChoHuh; @MM; @SW] and noncommutative symmetric functions [@GebSag].
Another research avenue that has seen activity is to determine whether two nonisomorphic trees can have the same chromatic symmetric function [@Jose2+1; @Jose2; @HeilJi; @Loebl; @MMW; @Orellana]. The data for up to 29 vertices [@HeilJi] shows that two trees $T_1, T_2$ have the same chromatic symmetric function if and only if $T_1$ and $T_2$ are isomorphic. Further evidence towards this includes that for $T_1$ and $T_2$ to have the same chromatic symmetric function they must have the same number of vertices, edges and matchings, and many of these results have been collected together in [@Orellana].
In this paper, we meld these two avenues and discover criteria on trees and graphs with cut vertices that ensure they are not $e$-positive or not Schur-positive. In particular, we discover a trove of trees that are not $e$-positive, supporting Stanley’s observation from 1995 [@Stan95 p 187] that a tree is likely only to be $e$-positive “by accident”. More precisely, this paper is structured as follows.
In Section \[sec:background\] we review the necessary notions before reducing the graphs we need to study to spiders in Subsection \[subsec:redspiders\]. [We also prove the following in Theorem \[the:perfect\_matching\] and relate it to the claw in Corollary \[cor:clawsandmatching\].]{}
[Let $G$ be an $n$-vertex connected graph. If $G$ has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then $G$ is not $e$-positive.]{}
In Section \[sec:spiders\] we study the $e$-positivity of spiders including showing that a spider with at least three legs of odd length is not $e$-positive in Corollary \[cor:matching\_cor\]. We also show that if the length of each spider leg is less than half the total number of vertices, then the spider is not $e$-positive in Lemma \[lem:short\_legs\] and generalize this to trees and graphs in Theorem \[the:gen\_short\_legs\]. In Lemma \[lem:induction\_lem\], Theorem \[the:induction\_short\_res\_1\] and Theorem \[the:induction\_short\_res\_2\], we show that if a spider is not $e$-positive, then we can create infinitely many more spiders from it that are not $e$-positive. Meanwhile Lemmas \[lem:quotient\_construction\], \[lem:quotient\_construction\_2\] and \[lem:quotient\_construction\_3\] give divisibility criteria on the total number of vertices, which ensure in Proposition \[prop:all\_e\_positive\_spiders\] that the spider is not $e$-positive. Applying these results on spiders yields our most general result, the following, given in Theorem \[the:gen\_partial\_e\_thm\].
If $G$ is an $n$-vertex connected graph with a cut vertex whose deletion produces a graph with $d\geq 3$ connected components such that $$d \geq \log_2 n + 1$$then $G$ is not $e$-positive.
We show the utility of our results in Example \[ex:gen\_short\_legs\], where we easily classify when a windmill graph $W^d_n$ for $d\geq1$, $n\geq 1$ is $e$-positive. In Section \[sec:bipartite\] we turn our attention to Schur-positivity, proving the following in Theorem \[the:bipartite\_s\_pos\].
If $G$ is an $n$-vertex bipartite graph with a vertex of degree greater than $\lceil \frac{n}{2} \rceil$, then $G$ is not Schur-positive.
Finally, in Section \[sec:further\] we conclude with two captivating conjectures on the $e$-positivity of trees.
Background {#sec:background}
==========
In order to describe our results, let us first recall the necessary combinatorics and algebra. We say a *partition* $\lambda = (\lambda _1, \ldots , \lambda _{\ell(\lambda)})$ of $N$, denoted by $\lambda \vdash N$, is a list of positive integers whose *parts* $\lambda _i$ satisfy $\lambda _1 \geq \cdots \geq \lambda _{\ell(\lambda)}>0$ and [$\sum _{i=1} ^{\ell(\lambda)} \lambda _i=N$]{}. If we have $j$ parts equal to $i$ then we often denote this by $i^j$. Related to every partition $\lambda$ is its *transpose*, $\lambda ^t = (\lambda _1^t, \ldots , \lambda ^t _{\lambda_1})$, which is the partition of $N$ obtained from $\lambda$ by setting $$\lambda _i^t = \mbox{ number of parts of }\lambda \geq i.$$For example, if $\lambda =(2,2,1)$ then $\lambda ^t = (3,2)$.
Given a graph $G$ with vertex set $V_G$ and edge set $E_G$, we say that $G$ is an *$n$-vertex graph*, or has *size* $n$, if $|V_G| =n$. We say that a connected graph $G$ contains a *cut vertex* if there exists a vertex $v\in V_G$ such that the deletion of $v$ and its incident edges yields a graph $G'$ with more than one connected component. A *connected partition* $C$ of an $n$-vertex graph $G$ is a partitioning of its vertex set $V_G$ into $\{V_1, \dots, V_k\}$ such that each induced subgraph formed by the vertices in each subset $V_i$ only is a connected graph. The *type* of a connected partition $C$ is the partition of $n$ formed from sorting the sizes of each set $V_i$ in decreasing order. We say $G$ *has a connected partition* of type $\lambda$ if and only if there exists a connected partition of $G$ of type $\lambda$, and is *missing a connected partition* of type $\lambda$ otherwise.
\[ex:connpartition\] Consider the $n$-vertex star $S_n$ for $n\geq 4$, consisting of a single vertex connected to $n-1$ vertices of degree 1. The star $S_4$ is below. $$\centering
\begin{tikzpicture}[scale=0.5]
\coordinate (A) at (0,0);
\coordinate (B) at (1.5,0);
\coordinate (C) at (3,0);
\coordinate (D) at (1.5,1);
\draw[thick] (A)--(D);
\draw[thick] (B)--(D);
\draw[thick] (C)--(D);
\filldraw (A) circle (7pt);
\filldraw (B) circle (7pt);
\filldraw (C) circle (7pt);
\filldraw (D) circle (7pt);
\end{tikzpicture}$$ The graph $S_n$ has a connected partition of type $\lambda$ if and only if $\lambda = (k, 1^{n-k})$ for some $1 \leq k \leq n$. Examples of connected partitions for $S_4$ of type $(4), (3,1), (2,1^2)$ and $(1^4)$ are below. $$\centering
\end{tikzpicture}
\hspace{1cm}
\end{tikzpicture}$$ Thus $S_n$ is missing a connected partition of type $(n-2, 2)$ for $n\geq 4$. For example, $S_4$ is missing a connected partition of type $(2,2)$.
\[rem:claw\] The star $S_4$ is also known as the claw. It is intimately connected to the aforementioned 1995 conjecture of Stanley [@Stan95 Conjecture 5.1] since if a poset is $(3+1)$-free, then its incomparability graph is claw-free. In contrast, the graphs we will study are mostly not claw-free.
We say that an $n$-vertex graph $G$ has a *perfect matching* if it has a connected partition of type $(2^{\frac{n}{2}})$ and an *almost* perfect matching if it has a connected partition of type $(2^{\frac{n-1}{2}},1)$. Classically stated, a graph $G$ has a perfect matching if there exists a subset of its edges $M\subseteq E_G$, such that every vertex in the graph is incident to exactly one edge in $M$. Similarly $G$ has an almost perfect matching if there exists a vertex $v\in V_G$ whose deletion, along with its incident edges, yields a graph $G'$ that has a perfect matching.
Graphs that will be of particular interest to us will be *trees*, namely connected graphs with no cycles. Recall that degree 1 vertices in trees are called *leaves*, and that a disjoint union of trees is called a *forest*. Two types of tree that will be crucial to our results are paths and spiders. Recall that the *path* $P_n$ of *length* $n$ where $n\geq1$ is the $n$-vertex tree with $n-2$ vertices of degree 2, and 2 leaves, for $n\geq 2$ or a single vertex for $n=1$. Meanwhile, given a partition $\lambda = (\lambda _1, \ldots , \lambda _d) \vdash n-1$ where $d\geq 3$, the *spider* $$S(\lambda) = S(\lambda _1, \ldots , \lambda _d)$$is the $n$-vertex tree consisting of $d$ disjoint paths $P_{\lambda _1}, \ldots , P_{\lambda _d}$ (each respectively called a *leg* of *length* $\lambda _i$ for $1\leq i \leq d$) and a vertex (called the *centre*) joined to a leaf in each path. Extending this notation, $S(i,\lambda)$ is the $(n+i)$-vertex spider with legs of length $i, \lambda _1, \ldots , \lambda _d$.
\[ex:spider\] The $n$-vertex star $S_n$ for $n\geq4$ is also the spider $S(1^{n-1})$. The $7$-vertex spider $S(4,1,1)$ is below.
(0,0) circle (4pt); (1,1) circle (4pt); (0,2) circle (4pt); (2.5,1) circle (4pt); (4,1) circle (4pt); (5.5,1) circle (4pt); (7,1) circle (4pt); (0,0)–(1,1); (0,2)–(1,1); (1,1)–(7,1);
We now turn to the algebra we will need. The algebra of symmetric functions is a subalgebra of ${\mathbb{Q}}[[ x_1, x_2, \ldots ]]$ that can be defined as follows. The *$i$-th elementary symmetric function* $e_i$ for $i\geq 1$ is given by $$e_i = \sum _{j_1<\cdots < j_i} x_{j_1}\cdots x_{j_i}$$and given a partition $\lambda = (\lambda _1, \ldots , \lambda _{\ell(\lambda)})$ the *elementary symmetric function* $e_\lambda$ is given by $$e_\lambda = e_{\lambda _1} \cdots e_{\lambda _{\ell(\lambda)}}.$$The *algebra of symmetric functions*, $\Lambda$, is then the graded algebra $$\Lambda = \Lambda ^0 \oplus \Lambda ^1 \oplus \cdots$$where $\Lambda ^0 = {\operatorname{span}}\{1\} = {\mathbb{Q}}$ and for $N\geq 1$ $$\Lambda ^N = {\operatorname{span}}\{e_\lambda {\;|\;}\lambda \vdash N\}.$$Moreover, the elementary symmetric functions form a basis for $\Lambda$. Perhaps the most studied basis of $\Lambda$ is the basis of Schur functions. For a partition $\lambda = (\lambda _1, \ldots, \lambda _{\ell(\lambda)})$, the *Schur function* $s_\lambda $ is given by $$\label{eq:JT}
s_{\lambda }=\det \left( e_{\lambda ^t_i -i +j}\right) _{1\leq i,j \leq \lambda_1}$$where if $\lambda ^t_{i}-i+j <0$ then $e _{\lambda ^t_{i}-i+j}=0$.
If a symmetric function can be written as a nonnegative linear combination of elementary symmetric functions then we say it is *$e$-positive*, and likewise if a symmetric function can be written as a nonnegative linear combination of Schur functions then we say it is *Schur-positive*. Although not clear from , it is a classical result that any $e$-positive symmetric function is Schur-positive, however Example \[ex:S411\] shows that the converse does not hold.
However, the symmetric functions that we will focus on will be the chromatic symmetric function of a graph, which is reliant on a graph that is *finite* and *simple* and we will assume that our graphs satisfy these properties from now on. Given a graph, $G$, with vertex set $V_G$ a *proper colouring* $\kappa$ of $G$ is a function $$\kappa : V_G\rightarrow \{1,2,\ldots\}$$such that if $v_1, v_2 \in V_G$ are adjacent, then $\kappa(v_1)\neq \kappa(v_2)$. Then the chromatic symmetric function is defined as follows.
[@Stan95 Definition 2.1]\[def:chromsym\] For an $n$-vertex graph $G$ with vertex set $V_G=\{v_1, \ldots, v_n\}$, the *chromatic symmetric function* is defined to be $$X_G = \sum _\kappa x_{\kappa(v_1)}\cdots x_{\kappa(v_n)}$$ where the sum is over all proper colourings $\kappa$ of $G$.
For succinctness, if we say that a graph $G$ is $e$-positive or Schur-positive, then we mean that $X_G$ is $e$-positive or Schur-positive, respectively.
\[ex:S411\] The spider $S(4,1,1)$ from Example \[ex:spider\] is not $e$-positive, but is Schur-positive since $$\begin{aligned}
X_{S(4,1,1)}&=e_{(2^3,1)} + 4e_{(3,2,1^2)} - 3e_{(3,2^2)} + 10e_{(3^2,1)} \\
&+ 10e_{(4,2,1)}+ 17e_{(4, 3)} + 4e_{(5, 1^2)} + 3e_{(5, 2)} + 11e_{(6, 1)} + 7e_{(7)}\\
&=64s_{(1^7)} + 88s_{(2,1^5)} + 76s_{(2^2, 1^3)} + 57s_{(2^3,1)} + 36s_{(3,1^4)} \\
&+ 36s_{(3,2,1^2)} + 18s_{(3,2^2)} + 4s_{(3^2,1)} + 5s_{(4,1^3)} + 6s_{(4,2,1)} + s_{(4,3)}.\end{aligned}$$
The following result gives us one way to test whether a graph is $e$-positive, and will be vital in many of our proofs.
[@Wolfgang Proposition 1.3.3] \[the:e\_positivity\_crit\] If a connected $n$-vertex graph $G$ is $e$-positive, then $G$ has a connected partition of type $\mu$ for every partition $\mu \vdash n$.
Hence, to prove that a graph $G$ is not $e$-positive, it suffices to find a partition $\mu$ such that $G$ is missing a partition of type $\mu$. In particular, we get the following.
\[the:perfect\_matching\] [Let $G$ be an $n$-vertex connected graph. If $G$ has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then $G$ is not $e$-positive. In particular, let $T$ be an $n$-vertex tree. If $T$ has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then $T$ is not $e$-positive.]{}
[It is a well-known result that if a connected graph with an even number of vertices is claw-free, then it has a perfect matching. This immediately yields the following corollary to Theorem \[the:perfect\_matching\].]{}
\[cor:clawsandmatching\] [Let $G$ be a connected graph with an even number of vertices and no perfect matching. Then $G$ contains the claw and is not $e$-positive.]{}
\[rem:clawsandmatching\] [Note that Corollary \[cor:clawsandmatching\] cannot be strengthened further regarding graphs that contain the claw since $S(2,1,1)$ and $S(6,2,1)$ both contain the claw and are $e$-positive. However, $S(2,1,1)$ has an odd number of vertices, and $S(6,2,1)$ has a perfect matching.]{}
It is also known when a tree has a perfect matching by the following specialization of Tutte’s Theorem on graphs with perfect matchings [@Tutte].
\[lem:perfect\_matching\] Let $T$ be a tree. Then $T$ has a perfect matching if and only if for every vertex $v$, the deletion of $v$ and its incident edges produces a forest with exactly one connected component with an odd number of vertices.
As an example, we use the two theorems above to test the star to see in two ways that it is not $e$-positive.
\[ex:perfect\_matching\] The $n$-vertex star $S_n$ for $n\geq4$ is not $e$-positive since it is missing a connected partition of type $(n-2,2)$ by Example \[ex:connpartition\]. It is also not $e$-positive since it is missing a connected partition of type $(2^{\frac{n}{2}})$ for $n$ even and $(2^{\frac{n-1}{2}},1)$ for $n$ odd.
Note, however, that the converse of Theorem \[the:e\_positivity\_crit\] and of Theorem \[the:perfect\_matching\] is false since the spider $S(4,1,1)$ is not $e$-positive by Example \[ex:S411\] and yet has a connected partition of every type.
The reduction to spiders {#subsec:redspiders}
------------------------
The next two lemmas allow us to reduce our study of connected graphs to the study of spiders. For ease of notation in the proof of the next lemma, in the $i^{th}$ leg of a spider, which has $\lambda_i$ vertices, label the vertices by $\{s_{i,1}, \dots, s_{i,\lambda_i}\}$ where $s_{i,1}$ is the vertex connected to the centre, $s_{i,\lambda_i}$ is a leaf, and there are edges between each $s_{i,j}$ and $s_{i,j+1}$ for $1\leq j \leq \lambda _i -1$.
\[lem:spider\_lem\] Let $T$ be a tree with a vertex of degree $d \geq 3$, and let $v$ be any such vertex. Let $(t_1, \dots, t_d)$ be the partition whose parts denote the sizes of the subtrees $(T_1, \dots, T_d)$ rooted at each vertex adjacent to $v$. If $T$ has a connected partition of type $\mu$, then the spider $S = S(t_1, \dots, t_d)$ has a connected partition of type $\mu$ as well.
Let $C = \{V_1, \dots, V_k\}$ be a connected partition of $T$ of type $\mu$. [We will work towards constructing a connected partition of $S$ of type $\mu$.]{} Without loss of generality, suppose $v \in V_1$. Since $T$ is a tree, no subset in $C$ contains vertices from two different subtrees unless vertex $v$ is also included. Therefore, all subsets in $C$ except possibly $V_1$ contain vertices from one subtree only. Hence, let $V_1$ contains $n_i$ vertices from subtree $T_i$ for $1 \leq i \leq d$, and let $\mathcal{T}_i \subset C$ denote the sets in the partition with vertices in subtree $T_i$ only. Notice that each $\mathcal{T}_i$ contains $t_i - n_i$ vertices.
A connected partition $\{W_1, \dots, W_k\}$ of the same type $\mu$ in $S$ may now be formed as follows. Let $W_1 = \{v\} \cup \{s_{i,j}: 1 \leq i \leq d, 1 \leq j \leq n_i\}$ where $v$ is the centre of $S$. Note that [$|W_1|=|V_1|$ and $W_1$ is connected.]{} Now notice that $S$ with deletion of all vertices in $W_1$ and their incident edges is now a collection of $d$ disjoint paths of length $t_i - n_i$ for $1 \leq i \leq d$. For each $\mathcal{T}_i$, let $\nu_i \vdash (t_i - n_i)$ be the partition formed from the size of each set in $\mathcal{T}_i$. Since a path may be decomposed into a connected partition of any type, in particular, a connected partition of type $\nu_i$ can be formed from a path of length $t_i - n_i$. Hence, the result follows.
In fact, the above argument can be generalized as follows.
\[lem:gen\_spider\_lem\] Let $G$ be a connected graph with a cut vertex $v$ whose deletion produces a graph with connected components $(C_1, \dots, C_d)$ with $d \geq 3$. Let $(c_1, \dots, c_d)$ be the partition whose parts denote the sizes of each of these connected components. If $G$ has a connected partition of type $\mu$, then the spider $S = S(c_1, \dots, c_d)$ has a connected partition of type $\mu$ as well.
Suppose $C$ is a connected partition of $G$ of type $\mu$. By assumption that $v$ is a cut vertex, any path between $w, u$ in distinct connected components $C_i$ and $C_j$ passes through $v$, for otherwise the deletion of $v$ would not leave $C_i$ and $C_j$ as distinct connected components. Hence, there are no edges between any distinct $C_i$ and $C_j$. The proof then proceeds as before in Lemma \[lem:spider\_lem\] as $C$ contains exactly one set $V_1$ including the vertex $v$ and possibly some other vertices in $(C_1, \dots, C_d)$, and every other set $V_i$ in $C$ contains vertices from exactly one connected component $C_j$.
The $e$-positivity of spiders {#sec:spiders}
=============================
We now work towards our first result on spiders by classifying when they have perfect and almost perfect matchings, for which we will need the following straightforward observation.
\[obs:matchedpaths\] A path, $P_n$, has a perfect matching if and only if $n$ is even.
With this observation we can now classify when a spider has a perfect or almost perfect matching.
\[lem:spider\_matching\] We have the following.
(a) A spider has a perfect matching if and only if it has exactly one leg of odd length.
(b) A spider has an almost perfect matching if and only if it has zero or two legs of odd length.
Suppose a spider $S$ has a perfect matching. The deletion of the centre of $S$ produces a union of disjoint paths, and exactly one of these paths must have odd length by Lemma \[lem:perfect\_matching\]. Conversely, if $S$ has exactly one leg of odd length, by matching the centre to its neighbour in that leg, a perfect matching exists by Observation \[obs:matchedpaths\] since the deletion of these vertices and their incident edges then produces a set of disjoint paths, each of even length.
Now suppose a spider $S$ has an almost perfect matching [so that there exists a vertex $v'$ whose deletion results in a perfect matching]{}. If the deletion of the centre produces a graph with a perfect matching, all legs in $S$ have even length, by Observation \[obs:matchedpaths\]. Otherwise, [$v'$]{} was in one of the legs $L$. Hence, unless the centre had degree 3 and $v'$ was adjacent to it, the deletion of $v'$ produces a spider $S'$ with an even number of vertices and a path of even length, which may [have 0 vertices]{}. Notice that an odd number of vertices in total was deleted from $S$ to form $S'$.
Hence, if the leg $L$ in $S'$ now has odd length, [then]{} every other leg in $S$ had even length by the first part. Therefore, all legs in $S$ had even length as an odd number of vertices was deleted from $L$ to form $S'$. Otherwise, if the leg $L$ in $S'$ now has even length, then some other leg in $S'$ has odd length by the first part, and so as an odd number of vertices was deleted from $L$ to form $S'$, $S$ originally had 2 legs of odd length.
In the case where $v'$ is adjacent to a degree 3 centre, the deletion of $v'$ produces two disjoint paths of even length. Hence, in this case, $S$ had exactly 2 legs of odd length. [This is because one of the paths of even length must consist of 2 legs and the centre, so one of the legs must be of odd length. Furthermore, the other path of even length must be a path of length one less than the remaining third leg, due to the deletion of $v'$, so the third leg must be of odd length.]{}
For the converse, if $S$ has no legs of odd length, then the deletion of the centre produces a disjoint union of paths with even length, which has a perfect matching by Observation \[obs:matchedpaths\]. If $S$ has 2 legs of odd length, the the deletion of the leaf in exactly one of those legs produces a spider with exactly one leg of odd length, which has a perfect matching by the first part.
\[cor:matching\_cor\] Every spider with at least 3 legs of odd length is not $e$-positive.
This follows immediately from Lemma \[lem:spider\_matching\] and Theorem \[the:perfect\_matching\]. In particular, if we have an $n$-vertex spider, then a connected partition of type $(2^{\frac{n}{2}})$ for $n$ even or $(2^{\frac{n-1}{2}},1)$ for $n$ odd, is missing.
\[ex:threeoddlegs\] All spiders $S(\lambda _1, \lambda _2, \lambda _3, 1,1,1)$ are not $e$-positive.
However, it is not only the parity of the legs that can determine the $e$-positivity of a spider, but also the length of the legs.
\[lem:short\_legs\] Let $\lambda \vdash (n-1)$ have at least 3 parts and $m$ be the maximum of the parts of $\lambda$. If $m < \lfloor \frac{n}{2} \rfloor$, then the $n$-vertex spider $S(\lambda)$ is missing a connected partition of type $$(n-m-1, m+1)$$and hence is not $e$-positive.
Consider the types of connected partitions that can be formed by the deletion of one edge from $S(\lambda)$. It is straightforward to see that the only connected partitions of type $(n-i, i)$ can be formed where $1\leq i \leq m$. Since $m < \lfloor \frac{n}{2} \rfloor$ we have that $n-(m+1)\geq m+1$ and so $S(\lambda)$ is missing a connected partition of type $$(n-(m+1), m+1)=(n-m-1, m+1).$$Hence, $S(\lambda)$ is not $e$-positive by Theorem \[the:e\_positivity\_crit\].
\[rem:short\_legs\] If $\lambda$ and $m$ are as in Lemma \[lem:short\_legs\] and $\mu = (n-m-1, m+1)$, then using the Newton-Girard Identities and [@Stan95 Theorem 2.5] one can compute directly that $$[e_\mu] X_{S(\lambda)} = -n \mbox{ if } m+1\neq n-m-1$$and $$[e_\mu] X_{S(\lambda)} = -\frac{n}{2} \mbox{ if } m+1= n-m-1$$where $[e_\mu] X_{S(\lambda)}$ denotes the coefficient of $e_\mu$ in $X_{S(\lambda)}$ when expanded as a linear combination of elementary symmetric functions.
\[ex:short\_legs\] If $\lambda = (2,2,1,1)\vdash 6$ then $m=2$ and $S(2,2,1,1)$, below, is missing a connected partition of type $(4,3)$ so is not $e$-positive by Lemma \[lem:short\_legs\]. More precisely, it is not $e$-positive since $X_{S(2,2,1,1)}$ contains the term $-7e_{(4,3)}$ by the above remark.
(0,0) circle (4pt); (0,2) circle (4pt); (1,1) circle (4pt); (2,0) circle (4pt); (2,2) circle (4pt); (3.5,0) circle (4pt); (3.5,2) circle (4pt); (0,0)–(1,1); (0,2)–(1,1); (1,1)–(2,0); (1,1)–(2,2); (2,2)–(3.5,2); (2,0)–(3.5,0);
For our next three results we construct larger spiders that are not $e$-positive and have been created from smaller ones. Hence, our focus will temporarily not be on the total number of vertices, $n$, but on the number of vertices in an initial set of legs.
\[lem:induction\_lem\] Let $i \geq 0$, $\lambda \vdash N$ have at least two parts, and $m$ be the maximum of the parts of $\lambda$. Suppose the spider $S(i,\lambda)$ is missing a connected partition of type $\mu \vdash (i + N+1)$ where $m + 1 \leq \mu_k \leq N$ for each part $\mu_k$ of $\mu$. Then the spider $S(i+N, \lambda)$ is missing a connected partition of type $$(N, \mu)$$ and hence is not $e$-positive.
To aid comprehension, we give an example before the proof.
\[ex:induction\_lem\] If $\lambda = (2,1,1)$ then $N=4$ and $m=2$. Setting $i=2$, we know from Example \[ex:short\_legs\] that $S(2,2,1,1)$ is missing a connected partition of type $\mu = (4,3)\vdash (2+4+1)$ and its parts satisfy $m+1=3\leq 4,3 \leq 4 = N$. Hence, $S(6,2,1,1)$ is missing a connected partition of type $(4,4,3)$.
[We will try to construct a connected partition of type $(N, \mu)$ and find this is impossible.]{} Suppose $V_1$ is a connected component in a connected partition of $S = S(i+N,\lambda)$ with $N$ vertices. Let $L'$ denote the vertices that are part of the leg of length $i + N$ in $S$ and [$L_k$]{} denote the vertices that are part of the leg of length [$\lambda_k$]{} in $S$.
If $V_1$ does not contain any vertex from $L'$, $V_1$ must contain the centre as each set [$L_k$]{} by assumption has less than or equal to $N-1$ vertices. As $V_1$ must contain the centre and furthermore cannot contain all $N+1$ vertices in the subtree $S(\lambda)$ in $S$, the deletion of $V_1$ and its incident edges produces an isolated vertex. Therefore, a connected partition of type $\mu$ cannot be formed from $S$ with all vertices in $V_1$ deleted, as each part of $\mu$ has size greater than or equal to 2 by assumption.
Hence, $V_1$ must contain some vertices from $L'$. If $V_1$ contained some vertex in some [$L_k$]{} as well, it must contain the centre since $V_1$ is connected. Once again, since the subtree $S(\lambda)$ has $N+1$ vertices and $V_1$ contains the centre, $V_1$ cannot contain the subtree $S(\lambda)$ entirely and the deletion of $V_1$ from $S$ then produces at least two connected components where one of them is a path with less than or equal to $m$ vertices. Hence, a connected partition of type $\mu$ cannot be formed from the remaining graph as each part has size greater than or equal to $m+1$.
[Thus, $V_1$ must contain either the centre and vertices from $L'$ only, or vertices from $L'$ only. If $V_1$ contains the centre and vertices from $L'$ only, then, as in the previous paragraph, the deletion of $V_1$ from $S$ then produces at least two connected components where one of them is a path with less than or equal to $m$ vertices. Hence, a connected partition of type $\mu$ cannot be formed from the remaining graph as each part has size greater than or equal to $m+1$.]{}
Therefore, $V_1$ must contain vertices from $L'$ only, [which]{} form an induced path of length $N$. Upon deleting $V_1$ and its incident edges, the graph $G_j$ formed is the disjoint union of $S(j,\lambda)$ and a path of length $i - j$ for some $0 \leq j \leq i$. However, if $G_j$ had a connected partition of type $\mu$ for some $j$, this contradicts the assumption that $S(i,\lambda)$ was missing a connected partition of type $\mu$, since any $G_j$ can be formed by first deleting an edge from the length $i$ path in $S(i,\lambda)$.
As we have exhausted all possibilities for $V_1$ and $|V_1| = N$, then $S(i+N,\lambda)$ is missing a connected partition of type $(N, \mu)$, and hence it is not $e$-positive by Theorem \[the:e\_positivity\_crit\].
We now come to a substantial way to create families of spiders that are missing a connected partition.
\[the:induction\_short\_res\_1\] Let $\lambda \vdash N$ and $m$ be the maximum of the parts of $\lambda$. If $\max(2m-N+1, 0) \leq i < N$, then the spider $S(i + Na, \lambda)$ is not $e$-positive for every integer $a \geq 0$. In particular, the spider $S(i+Na,\lambda)$ is missing a connected partition of type $\mu$ where $$\mu = \begin{cases} (N^{a+1}, i+1) & m \leq i < N \\ (N^a, N+i-m, m+1) & i < m \end{cases}.$$
Note that if a spider is missing a connected partition then it is not $e$-positive by Theorem \[the:e\_positivity\_crit\]. Hence, we now proceed to find a missing connected partition in all cases.
Before we do this, observe that if $\lambda = (N)$, so $m=N$, then no such $i$ exists. Thus assume that $\lambda \vdash N$ with at least 2 parts. We will now study two cases, $m\leq i$ and $i<m$. In each case we will study the spider $S(i,\lambda)$ before drawing our desired conclusions about $S(i+Na, \lambda)$.
For the first case, if $1\leq m \leq i <N$, then $2m-N+1 \leq m$ since $m\leq N-1$, so our condition on $i$ is trivially satisfied. Also note that all legs in the spider $S(i,\lambda)$ have length less than $\lfloor \frac{i+N+1}{2} \rfloor$. This is because there are $i+N+1$ vertices in total and $$\left\lfloor \frac{i + N + 1}{2} \right\rfloor > {\left\lfloor\frac{2i}{2} \right\rfloor}= i \geq m.$$Hence, by Lemma \[lem:short\_legs\], since $i$ is the maximum of the parts of $\lambda$ [and $i$]{}, the spider $S(i,\lambda)$ is missing a connected partition of type $(N, i+1)$. Hence, by Lemma \[lem:induction\_lem\] since $m+1\leq i+1\leq N$, by assumption, we have that $S(i+N, \lambda)$ is missing a connected partition of type $(N,N,i+1)$.
Consequently, $S(i+Na, \lambda)$ is missing a connected partition of type $(N^{a+1}, i+1)$, since if not then $(a-1)$ connected components with $N$ vertices would have to be contained in the leg of length $i+Na$ (this is because otherwise one of these connected components with $N$ vertices would consist of the centre vertex connected to $N-1$ other vertices, which could not yield a connected partition of type $(N^{a+1}, i+1)$ since $i\geq m$). However, this would imply that $S(i+N, \lambda)$ is not missing a connected partition of type $(N,N,i+1)$, a contradiction.
For the second case, first suppose $0 \leq 2m-N+1 \leq i < m$. Since, therefore, $2m+1\leq i+N$, $$\left\lfloor \frac{i+N+1}{2} \right\rfloor \geq \left\lfloor \frac{2m+2}{2} \right\rfloor = m+1 > m > i$$the spider $S(i, \lambda)$ is missing a connected partition of type $(N+i-m, m+1)$ by Lemma \[lem:short\_legs\] since $m$ is the length of the longest leg in $S(i, \lambda)$. Hence, by Lemma \[lem:induction\_lem\], since $m+1=N+(2m-N+1)-m \leq N+i-m<N$, because $i<m$ by assumption, we have that $S(i+N, \lambda)$ is missing a connected partition of type $(N, N+i-m, m+1)$.
Alternatively suppose $2m-N+1<0\leq i <m$. The first inequality implies that $m<\frac{N-1}{2}$ so [$$\left\lfloor \frac{i+N+1}{2} \right\rfloor > m$$]{}and hence the spider $S(i, \lambda)$ is missing a connected partition of type $(N+i-m, m+1)$ by Lemma \[lem:short\_legs\] since $m$ is the length of the longest leg in $S(i, \lambda)$. Hence, by Lemma \[lem:induction\_lem\], since $m+1\leq N+i-m<N$, because $i<m$ by assumption, we have that $S(i+N, \lambda)$ is missing a connected partition of type $(N, N+i-m, m+1)$.
Consequently, in both subcases of the second case $S(i+Na, \lambda)$ is missing a connected partition of type $(N^a, N+i-m, m+1)$ since if not then $(a-1)$ connected components with $N$ vertices would have to be contained in the leg of length $i+Na$ (this is because otherwise one of these connected components with $N$ vertices would consist of the centre vertex connected to $N-1$ vertices, which could not yield a connected partition of type $(N,N+i-m,m+1)$). However, this would imply that $S(i+N, \lambda)$ is not missing a connected partition of type $(N,N+i-m,m+1)$, a contradiction.
\[ex:induction\_short\_res\_1\] If $\lambda = (2,1,1)$, then $N=4$ and $m=2$. Since $\max(2m-N+1, 0)=1 \leq 2 < 4=N$ we can set $i=2$ and hence, by Theorem \[the:induction\_short\_res\_1\], every spider $S(2+4a, 2, 1,1)$ is missing a connected partition of type $(4^{a+1}, 3)$ for every integer $a \geq 0$.
Applying the above theorem now leads us to a surfeit of spiders that are not $e$-positive.
\[the:induction\_short\_res\_2\] Let $\lambda \vdash N$ and $m$ be the maximum of the parts of $\lambda$. If $m < \lfloor \frac{N}{2} \rfloor$ and $i \geq 0$ is an integer, then every spider $S(i,\lambda)$ is not $e$-positive.
By Theorem \[the:induction\_short\_res\_1\], it suffices to show that if $m < \left \lfloor \frac{N}{2} \right \rfloor$, then $2m-N+1 \leq 0$. If $N$ is even, then $2m < N$, so $2m - N < 0$ and $2m - N + 1 \leq 0$. Otherwise if $N$ is odd, then $2m < N-1$, so once again $2m - N + 1 \leq 0$.
An alternative approach for finding a missing connected partition is given in the following lemma.
\[lem:quotient\_construction\] Suppose $S=S(\lambda_1, \dots, \lambda_d)$ is an $n$-vertex spider with $(\lambda_1, \dots, \lambda_d)$ a partition, $\lambda _1 \geq \lambda _2 +\cdots + \lambda _d$, $\lambda_2 \leq \lambda_3 + \dots + \lambda_d$, and with $\lambda_2 \geq 2$. Let $n = q(\lambda_2 + 1) + r$ [where $0\leq r <\lambda _2 +1$]{}, and $r = qd' + r'$ [where $0\leq r' <q$]{}. If $\lambda_2 \geq 3$, or if $\lambda_2 = 2$ and $q \geq 3$, then $S$ is missing a connected partition of type $$(\lambda_2 + d' + 2)^{r'} (\lambda_2 + d' + 1)^{q - r'}.$$
Suppose $S$ has a connected partition $C$ of type $(\lambda_2 + d' + 2)^{r'} (\lambda_2 + d' + 1)^{q - r'}$. Consider the set $V_1\in C$ containing the leaf on a leg of length $\lambda _2$. Since every set [in $C$]{} must contain at least $\lambda_2 + 1$ vertices, $V_1$ contains the centre, and hence also all legs of length less than or equal to $\lambda_2$ since all sets in the connected partition have size greater than $\lambda_2$.
Hence, since $\lambda_2 \leq \lambda_3 + \dots + \lambda_d$, $$|V_1| \geq 1 + \lambda_2 + \dots + \lambda_d \geq 1 + 2 \lambda_2.$$
However, we claim that $1 + 2 \lambda_2 > \lambda_2 + d' + 2$. This is because $$n = 1 + \lambda_1 + \dots + \lambda_d \geq 1 + 2(\lambda_2 + \dots + \lambda_d) > 2(\lambda_2 + 1)$$ by assumption that $\lambda _1 \geq \lambda _2 +\cdots + \lambda _d$. So $q \geq 2$. Hence, if $\lambda_2 \geq 3$, then $$\frac{\lambda_2}{\lambda_2 - 1} = 1 + \frac{1}{\lambda_2 - 1} < 2 \leq q.$$Otherwise if $\lambda_2 = 2$, then $\frac{\lambda_2}{\lambda_2 - 1} = 2 < q$ when $q \geq 3$. Therefore, in either case,
$$r - r' = qd' \leq r \leq \lambda_2 < q(\lambda_2 - 1),$$which implies that $d' < \lambda_2 - 1$ by dividing both sides by $q$. Adding $\lambda_2 + 2$ to both sides shows that $1 + 2\lambda_2 > \lambda_2 + d' + 2$, which contradicts that $C$ is a connected partition of the desired type as $V_1$ does not contain $\lambda_2 + d' + 2$ or $\lambda_2 + d' + 1$ vertices.
\[ex:quotient\_construction\] Let $S = S(8, 2, 2, 1)$. Since $8\geq 2+2+1$ and $2\leq 2+1$, and $n = 14 = 4(3) + 2$, the conditions of Lemma \[lem:quotient\_construction\] are met. Since $2 = 4(0) + 2$, $S$ is missing a connected partition of type $(\lambda_2 + 2)^2 (\lambda_2 + 1)^2 = (4,4,3,3)$.
We now will generalize Lemma \[lem:quotient\_construction\] and then bound the number of vertices before giving our key result on the $e$-positivity of a spider.
\[lem:quotient\_construction\_2\] Let $i \geq 3$ be an integer. Suppose $S(\lambda_1, \dots, \lambda_d)$ is an $n$-vertex spider with $(\lambda_1, \dots, \lambda_d)$ a partition, with a $\lambda_i \geq 2$ satisfying $\lambda_i \leq \lambda_{i+1} + \dots + \lambda_d$ [where $i<d$]{}, and $\lambda_j > \lambda_{j+1} + \dots + \lambda_d$ for all $j < i$. Let $n = q(\lambda_i + 1) + r$ [where $0\leq r <\lambda _i +1$]{}, and $r = qd' + r'$ [where $0\leq r' <q$]{}. Then $S$ is missing a connected partition of type $$(\lambda_i + d' + 2)^{r'} (\lambda_i + d' + 1)^{q - r'}.$$
Suppose $S$ has a connected partition $C$ of the desired type. Consider the set $V_1 \in C$ containing the leaf on a leg of length $\lambda_i$. Since every set [in $C$]{} must contain at least $\lambda_i + 1$ vertices, $V_1$ contains the centre, and hence also all legs of length less than or equal to $\lambda_i$ since all sets in the connected partition have size greater than $\lambda_i$. Hence, since $\lambda_i \leq \lambda_{i+1} + \dots + \lambda_d$,
$$|V_1| \geq 1 + \lambda_i + \dots + \lambda_d \geq 1 + 2 \lambda_i.$$
However, we claim that $\frac{\lambda_i}{\lambda_i - 1} \leq 2 < q$. This is because $\lambda_i \geq 2$ so $\frac{\lambda_i}{\lambda_i - 1} \leq 2$, and repeatedly using the condition that $\lambda_j > \lambda_{j+1} + \dots + \lambda_d$, for all $j < i$ gives
[$$\begin{aligned}
n&=1+\lambda _1 + \cdots + \lambda _d\\
&>1+2(\lambda _2 + \cdots + \lambda _d)\\
&>1+4(\lambda _3 + \cdots + \lambda _d)\\
&\vdots\\
&> 1 + 2^{i-1} (\lambda_i + \dots + \lambda_d) \geq 2^{i-1} (\lambda_i + 1).\end{aligned}$$Hence, $$(q+1)(\lambda _i +1) > q(\lambda_i + 1) + r = n > 2^{i-1} (\lambda_i + 1)$$and so]{} $q \geq 2^{i-1} \geq 4>2$ since $i \geq 3$. This implies that $1 + 2 \lambda_i > \lambda_i + d' + 2$ as in the proof of Lemma \[lem:quotient\_construction\], which contradicts that $C$ is a connected partition of the desired type as $V_1$ does not contain [$\lambda_i + d' + 2$ or $\lambda_i + d' + 1$]{} vertices.
\[lem:count\_vertices\] Let $S(\lambda_1, \dots, \lambda_d)$ be an $n$-vertex spider, with $(\lambda_1, \dots, \lambda_d)$ a partition.
(a) If $\lambda_1 \geq \lambda_2 + \dots + \lambda_d$ and $\lambda_i > \lambda_{i+1} + \dots + \lambda_d$ for $2 \leq i \leq d - 1$, then $n > 2^{d-1}$.
(b) If $\lambda_1 \geq \lambda_2 + \dots + \lambda_d$, $\lambda_i > \lambda_{i+1} + \dots + \lambda_d$ for [$d\geq 4$ and ]{}$2 \leq i \leq d-2$, and $\lambda_{d-1} = \lambda_d = 1$, then $n > 2^{d-1}$.
For (a), if the partition $(\lambda_1, \dots, \lambda_d)$ satisfies the conditions stated, then we claim that $\lambda_{d-i} > 2^{i-1} \lambda_d$ for $1 \leq i \leq d-1$. This is true for $i = 1$ by assumption. Next, assuming by induction that $\lambda_{d-j} > 2^{j-1} \lambda_d$ for all $1 \leq j < i$,
$$\lambda_{d-i} \geq \sum_{j = 0}^{i-1} \lambda_{d-j} > \lambda_d + \sum_{j=1}^{i-1} 2^{j-1} \lambda_d = \lambda_d + (2^{i-1} - 1) \lambda_d = 2^{i-1} \lambda_d.$$
Hence, the total number of vertices $n$ satisfies
$$n = \lambda_1 + \dots + \lambda_d + 1 > \lambda_d + \sum_{i=1}^{d-1} 2^{i-1} \lambda_d = {\lambda_d + (2^{d-1} - 1) \lambda_d} = 2^{d-1} \lambda_d.$$
Since $\lambda_d \geq 1$, then [$n > 2^{d-1} \lambda_d\geq2^{d-1}$.]{}
For (b), we claim that $\lambda_{d-i} \geq 3 (2^{i-2})$ for $2 \leq i \leq d-2$. This is true for $i=2$ since $\lambda_{d-2} \geq 3$ from the conditions given. Next, assuming by induction that this holds for all $2 \leq j < i$,
$$\lambda_{d-i} \geq \sum_{j=2}^{i-1} \lambda_{d-j} + \lambda_{d-1} + \lambda_d + 1 \geq 3 + \sum_{j=2}^{i-1} 3 (2^{j-2}) = 3 + 3 (2^{i-2} - 1) = 3 (2^{i-2}).$$
Thus, $\lambda_1 \geq \lambda_2 + \dots + \lambda_d \geq \sum_{i=2}^{d-2} 3 (2^{i-2}) + 2 = 3 (2^{d-3}) - 1.$
Hence, the total number of vertices $n$, by considering the centre and lengths of each leg, satisfies
$$n={1+\lambda_1+\cdots + \lambda _d} \geq \sum_{j=0}^{d-3} 3 (2^j) - 1 + 3 = 3 (2^{d-2} - 1) + 2 = 3 (2^{d-2}) - 1 > 2^{d-1}$$ since $d \geq 3$ for a spider.
We now arrive at our most general result on the $e$-positivity of spiders.
\[the:partial\_e\_thm\] Suppose $S=S(\lambda_1, \dots, \lambda_d)$ is an $n$-vertex spider with a vertex of degree $$d \geq \log_2 n + 1.$$ Then $S$ is missing a connected partition of some type $\mu$, and hence is not $e$-positive.
By Lemma \[lem:count\_vertices\](a), either $\lambda_1 < \lambda_2 + \dots + \lambda_d$ or $\lambda_i \leq \lambda_{i+1} + \dots + \lambda_d$ for some $i \geq 2$ [and $i<d$]{}, since if not, $d < \log_2 n + 1$.
[For the first case, if]{} $\lambda_1 < \lambda_2 + \dots + \lambda_d$, then Lemma \[lem:short\_legs\] provides a missing connected partition. [For the second case]{}, if $\lambda_1 \geq \lambda_2 + \dots + \lambda_d$ but $\lambda_2 \leq \lambda_3 + \dots + \lambda_d$ and $\lambda _2\geq 3$, then Lemma \[lem:quotient\_construction\] provides a missing connected partition. If $\lambda _2 =2$ then since $d\geq3$, $\lambda _3$ exists and $\lambda _3 \leq 2$. Hence, $n\geq 9$ so the conditions of Lemma \[lem:quotient\_construction\] are satisfied and it provides a missing connected partition. Meanwhile, if $\lambda _2=1$ then since $d\geq 3$, $\lambda _3$ exists and [$\lambda _3 = 1$.]{} Hence, $n\geq 5$ and $d\geq 4$ since $d \geq \log_2 n + 1$ [so $\lambda _4$ exists and $\lambda _4 = 1$.]{} Thus by Lemma \[lem:spider\_matching\] the spider does not have a perfect [or almost perfect]{} matching and hence is missing a connected partition as well.
Otherwise, if the spider does not fall into the [second case subcases]{} above, [then]{} let $i \geq 3$ be the least index for which $\lambda_i \leq \lambda_{i+1} + \dots + \lambda_d$ [and $i<d$ is true. Then $d\geq 4$.]{} If $\lambda_i \geq 2$, Lemma \[lem:quotient\_construction\_2\] provides a missing connected partition. Otherwise, $\lambda_i = 1$. By Lemma \[lem:count\_vertices\](b), $i < d-1$ since if $i = d-1$, then $d < \log_2 n + 1$. Hence, the centre of the spider is attached to at least three leaves, and by Lemma \[lem:spider\_matching\], the spider does not have a perfect or almost perfect matching, and hence is missing a connected partition.
We are now left with spiders $S(\lambda _1, \ldots , \lambda _d)$ with $(\lambda _1, \ldots , \lambda _d)$ a partition, $\lambda_1 \geq \lambda_2 + \dots + \lambda_d$, and either every other leg $\lambda_i$ satisfies $\lambda_i > \lambda_{i+1} + \dots + \lambda_d$, or $\lambda _{d-1}=\lambda _d = 1$ and all other $\lambda _i$ satisfy this inequality. In this case, a spider may not be $e$-positive but still have a connected partition of every type. The spider $S(6,4,1,1)$ is one such example. Otherwise, we can get a missing partition for “sufficiently large” spiders by a method similar to Lemma \[lem:quotient\_construction\_2\].
\[lem:quotient\_construction\_3\] Suppose $S=S(\lambda_1, \dots, \lambda_d)$ is an $n$-vertex spider with $(\lambda_1, \dots, \lambda_d)$ a partition. Pick some $\lambda_i$ with $i \geq 2$ and let $n = q (\lambda_i + 1) + r$ [where $0\leq r <\lambda _i +1$]{}, $r = qd' + r'$ [where $0\leq r' <q$]{}, and $t = \lambda_{i+1} + \dots + \lambda_d$ [where $i<d$ and $t>1$]{}. If $q \geq \frac{\lambda_i + 1}{t - 1}$, then $S$ is missing a connected partition of type $$(\lambda_i + d' + 2)^{r'} (\lambda_i + d' + 1)^{q-r'}.$$
Suppose $S$ has a connected partition $C$ of the desired type. Consider the set $V_1\in C$ containing the leaf on a leg of length $\lambda _i$. Since every set [in $C$]{} must contain at least $\lambda_i + 1$ vertices, $V_1$ contains the centre, and hence also all legs of length less than or equal to $\lambda_i$ since all sets in the connected partition have size greater than $\lambda_i$. Hence, $|V_1| \geq 1 + \lambda_i + \dots + \lambda_d = 1 + \lambda_i + t$, but we claim that this is greater than $\lambda_i + d' + 2$. It suffices to show that $d' < t - 1$. This follows since,
$$d' = \frac{r-r'}{q} \leq \frac{\lambda_i}{q} \leq \frac{\lambda_i}{\lambda_i + 1} (t-1) < t-1$$by the assumption on the value of the quotient $q$. Hence, $1+\lambda _i+t>\lambda _i+d'+2$, which contradicts that $C$ is a connected partition of the desired type as $V_1$ does not contain $\lambda_i + d' + 2$ or $\lambda_i + d' + 1$ vertices.
Hence, if $\lambda_{i+1}, \dots, \lambda_d$ are fixed, then for sufficiently large values of the sum $\lambda_1 + \dots + \lambda_i$, the spider $S(\lambda_1, \dots, \lambda_d)$ will be missing a connected partition of some type.
\[ex:136411\] If $\lambda = (13, 6,4,1,1)$ and let $i=2$ so that $\lambda _i = 6$, then $26 = 3(7)+5$, $5=3(1) +2$ and $6=4+1+1$. Hence, $q=3$, $\lambda _i +1 = 7$, and $t=6$. Since $3\geq \frac{7}{5}$, we have that $S(\lambda)$ is missing a connected partition of type $$(6+1+2)^2(6+1+1)=(9,9,8).$$
To end this section, we collect together the various latter lemmas on missing partitions of various types and draw the following conclusion on $e$-positivity, which is immediate by Theorem \[the:e\_positivity\_crit\].
\[prop:all\_e\_positive\_spiders\] If a spider satisfies the criteria of Lemmas \[lem:quotient\_construction\], \[lem:quotient\_construction\_2\] or \[lem:quotient\_construction\_3\], then it is not $e$-positive.
The $e$-positivity of trees and cut vertices {#sec:trees}
============================================
We can now use our results from the previous section in conjunction with Lemmas \[lem:spider\_lem\] and \[lem:gen\_spider\_lem\] to deduce criteria for $e$-positivity of trees and graphs in general.
\[the:gen\_partial\_e\_thm\] If $G$ is an $n$-vertex connected graph with a cut vertex whose deletion produces a graph with $d\geq 3$ connected components such that $$d \geq \log_2 n + 1$$then $G$ is not $e$-positive.
In particular, if $T$ is an $n$-vertex tree with a vertex of degree $d\geq 3$ such that $$d \geq \log_2 n + 1$$then $T$ is not $e$-positive.
For the first part, by Lemma \[lem:gen\_spider\_lem\] and Theorem \[the:partial\_e\_thm\] every such $n$-vertex graph is missing a connected partition of some type, and hence is not $e$-positive by Theorem \[the:e\_positivity\_crit\]. For the second part we can either repeat the above argument but this time using Lemma \[lem:spider\_lem\] instead of Lemma \[lem:gen\_spider\_lem\], or we can note that if we delete a vertex of degree $d$ from a tree, then $d$ connected components remain.
As a simple example, every tree with 1000 vertices that contains a vertex of degree 11 or more is not $e$-positive. In fact, we can use every result from the previous section on spiders that involves a missing connected partition to obtain a result on trees, cut vertices and $e$-positivity. We illustrate this using Lemma \[lem:short\_legs\].
\[the:gen\_short\_legs\] If $G$ is an $n$-vertex connected graph with a cut vertex whose deletion produces a graph with connected components $C_1, \ldots , C_d$ such that $d\geq 3$ and $|V_{C_i}|< \lfloor \frac{n}{2}\rfloor$ for all $1\leq i \leq d$, then $G$ is not $e$-positive.
In particular, if $T$ is an $n$-vertex tree with a vertex for degree $d\geq3$ whose deletion produces subtrees $T_1, \ldots , T_d$ and $|V_{T_i}|< \lfloor \frac{n}{2}\rfloor$ for all $1\leq i \leq d$, then $T$ is not $e$-positive.
For the first part, by Lemma \[lem:gen\_spider\_lem\] and Lemma \[lem:short\_legs\] every such $n$-vertex graph is missing a connected partition type $(n-m-1, m+1)$ where $$m=\max \{|V_{C_1}|, \ldots, |V_{C_d}|\}$$and hence is not $e$-positive by Theorem \[the:e\_positivity\_crit\]. For the second part we can either repeat the above argument but this time using Lemma \[lem:spider\_lem\] instead of Lemma \[lem:gen\_spider\_lem\], or we can note that if we delete a vertex of degree $d$ from a tree, then $d$ connected components remain.
As a more meaningful example, we will now classify when a windmill graph is $e$-positive.
\[ex:gen\_short\_legs\] Let $K_n$ be the *complete graph* on $n$-vertices, namely the $n$-vertex graph in which every two vertices are adjacent. Let $W^d_n$ for $d\geq 1, n\geq 1$ be the *windmill graph* in which $d$ copies of $K_n$ all have one common vertex $c$. For example, $W^4_3$ is below.
(0,0) circle (4pt); (-2,1) circle (4pt); (-2,-1) circle (4pt); (2,1) circle (4pt); (2,-1) circle (4pt); (1,2) circle (4pt); (-1,2) circle (4pt); (1,-2) circle (4pt); (-1,-2) circle (4pt); (0,0)–(-2,1)–(-2,-1)–(0,0); (0,0)–(2,1)–(2,-1)–(0,0); (0,0)–(1,2)–(-1,2)–(0,0); (0,0)–(1,-2)–(-1,-2)–(0,0);
Note that $W^d_n$ has $d(n-1)+1$ vertices. Also note that [for $n>1$]{} the deletion of $c$ produces $d$ connected components, more precisely $d$ copies of $K_{n-1}$ each with $n-1$ vertices. Hence, for $d\geq 3$ since $$(n-1)<\left\lfloor \frac{d(n-1)+1}{2}\right\rfloor$$every $W^d_n$ for $d\geq 3, {n>1}$ is not $e$-positive by Theorem \[the:gen\_short\_legs\]. In contrast, by say [@ChovW Theorem 8] and [@Stan95 Corollary 3.6] respectively, every $W^1_n=K_n$ for [$n>1$]{} and $W^2_n$ for [$n>1$]{} is $e$-positive. [Lastly, note that $W^d_1$ for $d\geq 1$ is $K_1$ and so, by say [@ChovW Theorem 8], is $e$-positive.]{}
The Schur-positivity of bipartite graphs {#sec:bipartite}
========================================
While $e$-positivity implies Schur-positivity, it is possible for a graph to be Schur-positive but not $e$-positive, for example the spider $S(4,1,1)$ from Examples \[ex:spider\] and \[ex:S411\].
(0,0) circle (4pt); (1,1) circle (4pt); (0,2) circle (4pt); (2.5,1) circle (4pt); (4,1) circle (4pt); (5.5,1) circle (4pt); (7,1) circle (4pt); (0,0)–(1,1); (0,2)–(1,1); (1,1)–(7,1);
Again we can determine whether trees or certain graphs are not Schur-positive using a vertex degree criterion, but this time we will need the dominance order on partitions, bipartite graphs, and stable partitions.
For the first of these, recall that given two partitions of $N$, $\lambda = (\lambda _1, \ldots , \lambda _{\ell(\lambda)})$ and $\mu = (\mu _1, \ldots , \mu _{\ell(\mu)})$, we say that $\lambda$ *dominates* $\mu$, denoted by $\lambda \geq _{dom} \mu$ if $$\lambda _1 + \cdots + \lambda _i \geq \mu _1 + \cdots + \mu _i$$for all $1\leq i \leq \min\{\ell(\lambda), \ell(\mu)\}$. For the second of these, recall that a graph $G$ is *bipartite* if there exists a proper colouring of $G$ with 2 colours. For the third of these, we say a *stable* partition of an $n$-vertex graph $G$ is a partitioning of its vertex set $V$ into sets $\{V_1, \dots, V_k\}$ such that every set $V_i$ in the partitioning is an independent set, namely no edge $e\in E_G$ exists between any $v_1, v_2 \in V_i$. The *type* of a stable partition is the partition of $n$ formed by sorting the sizes of each set $V_i$ in decreasing order. We say $G$ *has a stable partition* of type $\lambda$ if and only if there exists a stable partition of $G$ of type $\lambda$, and is *missing a stable partition* of type $\lambda$ otherwise.
\[ex:stablepartition\] Consider again the $n$-vertex star $S_n$ for $n\geq 4$. The graph $S_n$ has a stable partition of type $(n-1, 1)$ but is missing a stable partition of type $(n-2,2)$.
Stable partitions are intimately related to the Schur-positivity of a graph via the following result.
[@Stanley2 Proposition 1.5] \[thm:s\_positivity\_crit\] Suppose an $n$-vertex graph $G$ has a stable partition of type $\lambda \vdash n$. If $G$ is Schur-positive, then $G$ has a stable partition of type $\mu$ for every $\mu \leq _{dom}\lambda$.
This in turn yields a criterion for when a graph is not Schur-positive that is dependent on vertex degrees.
\[the:bipartite\_s\_pos\] If $G$ is an $n$-vertex bipartite graph with a vertex of degree greater than $\lceil \frac{n}{2} \rceil$, then $G$ is not Schur-positive.
In particular, if $T$ is an $n$-vertex tree with a vertex of degree greater than $\lceil \frac{n}{2} \rceil$, then $T$ is not Schur-positive.
For the first part, let $G$ be an $n$-vertex bipartite graph with a vertex $v$ of degree $d > \lceil \frac{n}{2} \rceil$. By assumption that $G$ is bipartite, there is a proper colouring of $G$ with two colours. Call these colours red and black, and note that $(V_1, V_2)$ where $V_1$ is the set of the vertices coloured red and $V_2$ is the set of vertices coloured black is a stable partition of $V_G$. The type of this stable partition will be a partition $(m, n - m)$ where $m > \lceil \frac{n}{2} \rceil$, since the $d > \lceil \frac{n}{2} \rceil$ vertices adjacent to $v$ must be assigned a different colour from the colour of $v$ by assumption.
We claim now that $G$ does not have a stable partition of type $(\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor)$. Suppose $G$ did have such a partitioning of its vertices into $(V_1, V_2)$ with $|V_1| = \lceil \frac{n}{2} \rceil$ and $|V_2| = \lfloor \frac{n}{2} \rfloor$. If $v \in V_1$, then its neighbours must be in $V_2$, which is impossible since $v$ has degree $d > \lceil \frac{n}{2} \rceil \geq \lfloor \frac{n}{2} \rfloor$. Similarly, if $v \in V_2$, then its neighbours must be in $V_1$, which is impossible since $v$ has degree $d > \lceil \frac{n}{2} \rceil$.
Since $(\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor) < _{dom}(m, n - m)$, and $G$ has a stable partition of type $(m, n-m)$, but $G$ is missing a stable partition of type $(\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor)$, then by Theorem \[thm:s\_positivity\_crit\], $G$ is not Schur-positive.
For the second part, note that all trees are bipartite.
\[ex:bipartite\_s\_pos\] Since the star $S_n$ for $n\geq 4$ is a tree and has one vertex of degree $(n-1)>\lceil \frac{n}{2} \rceil$, it is not Schur-positive, and hence again not $e$-positive, since $e$-positivity implies Schur-positivity.
Further avenues {#sec:further}
===============
A natural avenue to pursue is to tighten the bound in Theorem \[the:partial\_e\_thm\], and to this end we conjecture the following, which has been checked for all trees with up to 12 vertices.
\[con:4vertices\] If $T$ is an $n$-vertex tree with a vertex of degree $d\geq 4$, then $T$ is not $e$-positive.
Towards this one can use our techniques to check that certain families of spiders are missing a particular type of partition. However, so far these results have been local to the family of spiders being studied, and there does not seem to exist a natural global type of partition that is missing. Moreover, as noted just before Lemma \[lem:quotient\_construction\_3\], there exist spiders that may not be $e$-positive but still have a connected partition of every type, such as $S(6,4,1,1)$.
In this case we can prove that the family of spiders $S(r,s,1,1)$ is not $e$-positive by first noting that if $r$ or $s$ is odd then $S(r,s,1,1)$ is not $e$-positive by Corollary \[cor:matching\_cor\]. If $r$ and $s$ are even then we can prove this by using the triple-deletion rule of Orellana-Scott [@Orellana Theorem 3.1], and generalized to $k$-deletion by the first and third authors [@lollipop Proposition 5], to express $X_{S(r,s,1,1)}$ as a linear combination of chromatic symmetric functions of unions of paths. From here, by using the formula of Wolfe [@Wolfe Theorem 3.2] for expressing $X_{P_n}$ as a linear combination of elementary symmetric functions we can show that if $r=2k$ and $s=2\ell$ then $$[e_{(3,2^{k+\ell})}]X_{S(r,s,1,1)} = -2(r+s)+7,$$which is negative when $r\geq 2$ and $s\geq 2$. This technique can also be used to show that $S(r, 1, 1)$ for $r\geq 3$ has $$[e_{(r-1,2^2)}]X_{S(r,1,1)} = -(r-1).$$For example, returning to Example \[ex:S411\], note the term $-3e_{(3,2^2)}$ in $X_{S(4,1,1)}$. Direct calculation yields that $S(1,1,1)$ is not $e$-positive but $S(2,1,1)$ is $e$-positive, and hence deducing that $S(r, 1, 1)$ for $r\geq 3$ is not $e$-positive supports Stanley’s statement [@Stan95 p 187] that $S(2,1,1)$ is $e$-positive “by accident”. [Meanwhile, regarding Schur-positivity, we believe that the bound in Theorem \[the:bipartite\_s\_pos\] cannot be improved. This is implied by the following conjecture, which has been checked for all trees with up to 19 vertices, and with which we end.]{}
\[con:Spositivetrees\] For all $n\geq 2$, there exists an $n$-vertex tree with a vertex of degree [$\lfloor \frac{n}{2} \rfloor$]{} that is Schur-positive.
Acknowledgements {#sec:acknow .unnumbered}
================
The authors would like to thank John Shareshian for fruitful conversations, [and the referee for drawing our attention to the connection between claw-free graphs and perfect matchings that then gives Corollary \[cor:clawsandmatching\].]{}
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[^1]: All authors were supported in part by the National Sciences and Engineering Research Council of Canada.
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I grew up in Belfast, Maine, and received a BFA from The Massachusetts College of Art in Boston in 2000. It was here that I studied a multitude of disciplines involving glass as a medium.
Torch work, which was not covered at Mass Art, had certain qualities that were desirable for the creations that I had dreamed of making.
Immediately following graduation I spent the summer beginning to learn, from instructors Sally Prasch and Robert Mickelsen, the torch working techniques that I employ today.
My floral visions investigating adaptation, that had not until then been able to be produced with the detail and delicacy I had so desired, were on their way to being realized. I currently spend most of my time behind the torch instructing others about this process that, after eighteen years, still creates much joy and wonder for me.
This is my first time entering an ISGB challenge. I have fallen head over heels for silver plum, I made my first few beads with Ivory and silver plum but decided that the reactions were too much.
So I started again with white and silver plum. I played with bending the colors and disrupting patterns we all know and love. I of course wondered does clear count as a color?
(always a trouble maker) with the last bead I created windows for the pattern with clear, and filled in around them to block the reminder out. I have also been playing with egg shapes.
Making an egg that is pleasing is incredibly difficult there are inherent proportions that have to be conveyed to be appreciated. Unfortunately the egg shape for the window is not ideal but I was too afraid to close the windows. I learned that I have to give myself more time to do this it is of course coming in last minute.
Artist Statement
My workstation is always full of colored rods ready for my imaginative and creative juices. So, when I was challenged with 2 colors I looked to nature, who better can show us the possibilities than Mother Earth. The ocean has always brought me peace and calm with its sounds and beauty of endless combinations of colors and marine life. I choose white and Elektra blue to create my Ocean beads collection. For my shapes I chose a starfish, shell, a wave, and the beluga whale. Glass, gravity, fire, scissors, tweezers, a small knife, and graphite/brass paddle tools brought them to life. This challenge pushed me to think more like an artist, to create a cohesive collection that conveys a message and displays beauty in the simple things using a small amount of color and tools. I enjoyed moving the glass in a playful manner to create the majestic objects and shapes I was looking for.
Artist Statement
I found that this challenge was so fun....very open ended...gave the artist something to think about....it reminded me of many of the assignments that I would give to my high school art students. It was good to have some parameters but still have room to develop an individual artistic representation in glass.
Thank you for the challenge.
Artist Statement
As soon as I heard about this challenge, I knew I was going to use black and ivory as my 2 colors. The contrast of these colors really highlights the precise application of décor: simple dots and raking in this case. There is so much you can do with a dot! I have been experimenting with various masking patterns, so that was a logical choice, once I picked the color. Of course, I am obsessed with birds, so they had to be included. I really enjoyed this project and made several other shapes and designs as well. I chose these four for the variety of shapes and masking patterns. Now I am looking forward to the October challenge. Keep 'em coming!
Artist Statement
I have chosen two types of glass - black and transparent - to create two colors: black and silver bubbles that result from brass sublimation. When creating my beads I focuse on making unique peaces which can be worn as solitaires. I have chosen two shapes to be worn as rings (a flat bead with a hole and a triangle flat one with a screw) and two shapes to be worn as solitaire pendants - one big hole bead and one donut. The challenge was: how to create a glamorous bead for my customers, using minimum colors.
Best regards
Artist Statement
The September Challenge to choose two colors and use them to make and decorate beads of various shapes and sizes was an invitation to play. My approach was to choose the two colors and just have fun. I decided that I wanted to use transparent colors with an emphasis of the darker transparent to primarily work like a border to the lighter color, so I used aqua and ink blue glass by Effetre.
I wanted to experiment a bit. In the first bead on the left, I layered the glass to see what light looked like with the bordered layers. (to me, it was great to see, but convinced me that this technique didn't appeal to my aesthetic. It didn't look clean enough as a single layer of glass did.) The second bead (simple tree), I wanted to use the border and change the shape once it was applied to see how it looked. I would do that again. The third bead (bird with hole) was an experiment to see what it would be like to make the shape and also whether I could make a hole (to possibly dangle something in jewelry) and have the hole with the second color as a border around it. It worked, but I would create it in a different way next time. The fourth bead was to see whether stretching then tightly curving the glass would make a difference. It reminded me of rams' horns or an architectural piece. Short answer: yes it does, but it's a matter of heat control. I will try this again but in different combinations of the two colors to attempt additional results.
What a fun excuse to play with glass and make some beads I hadn't before. Thank you for this challenge.
Artist Statement
I recently started carving beads after an inspirational class this past summer with Holly Cooper, and so for this challenge I decided to incorporate carved design to enhance shape. I chose grey glass with yellow enamel to highlight contrast. In considering shape, I went with a drop shape because that provides a lot of surface to carve. I have an unusual heavy brass lamp that could double as a weapon in any martial art movie, and I thought it would be a challenge to recreate it in glass. And as it’s nearly impossible to get glass into a cube shape, I was curious if I could make beads close to a cube shape, and enhance their corners with a carved design. I really enjoyed this challenge!
Artist Statement
This challenge correspond to another one I am doing as part of the Facebook group 'Limited Pallet Challenge'. We are using Thai Orchid, Opal Yellow, Emerald Green, Hades, and White for the month, so I picked two of those to do the ISGB creativity challenge. My favorite thing to make are hollow beads. I decided to try out making some of the basic bead shapes we all start out with making solid beads with on hollow beads instead. The dot one is the normal doughnut shape that most of my hollows end up with. Next I did the square one, squares are always more challenging in hollows because as you flatten the sides you have to keep blowing them out to keep them from becoming concave. The same is true with the lentil shape, and I think this one is my favorite. The last one is more of an egg shape and it started small so the stringer really stretched out and I love that look.
The challenge was fun, hope that we keep doing them!
Artist Statement
I used Ivory EFF 264 and EFF 036 Aqua. One of my favourite colours. I live on an island and these colours represent the sand and sea.
Next challenge was thinking of shapes... but there I would have loved to have more time to come up with lots more shapes. But the greatest challenge was making animal shapes with only 2 colours.
I am so looking forward to the next challenge. I love it when my customers make requests to push me out of my comfort zone. I can usually hardly wait to get to my torch.
What I learned... to limit myself sometimes, having fewer choices to come up with something new. Also how wide the colour shade ranges when working with only two colours and one being transparent.
Artist Statement
For this learning experience, I used Effetre light Ivory and Transparent Black. I wanted to get away from the traditional silver foil / ivory / intense black and see what I could do with a simple basic black and white. I learned that with enough heat, even transparent black will web and shadow on the ivory. I am showing 4 beads, and at least 3 different shapes, (lentil, barrel, round hollow, and also a bi-cone), using 2 colors, and the reactions I got with varying applications of heat to the bead. The barrel with stacked dots are crisp and defined separation of colors, the lentil is showing the effects of more applied heat and the beginnings of webbing, the round/hollow is showing definite webbing and the beginnings of shadowing, with the bi-cone showing webbing and definite shadowing after heating (technical term coming up) “the snot out of it”. I used a Nortel Mega Minor with 2 5lpm OxyCons for this experience. All beads were hand shaped using only a marver and were annealed directly from the flame in a digital kiln, and learning definitely took place.
Artist Statement
Colors: Dark Ivory and Test color Double Helix OSLT.
I struggled with color choice. I wanted to maximize the results with reaction and I intended to use Dark multi color or Raku, but when I sat down at the torch, the rich gold just felt like the color of Fall and the reaction would represent the changing colors of the season.
Shapes:
Straight sided Lentil; I always love this shape for designing a story, and I intended to work with the concept of trees and fall, but then a mountain appeared. I often sit at the torch with one vision and create something entirely different, letting the glass or my subconscious lead the way.
The Heart represents my love of the Mountains and I am always practicing stringer with a love of metal work and scrolls.
Two small Rounds; a pair of earrings of course.
Thanks for letting me play!
Artist Statement
After reading the terms of the contest, I started thinking - what two colors would I use? Contrasting colors or restrained soft tones? How will they look in different forms and under different lighting?
Good challenge! I immediately dismissed all the glass that needs a transparent coat - as “transparent” is also a color!
I thought about silver-containing glass, but it can produce many shades, which would violate the two-color requirement. Hence, I decided that colors should be simple, and therefore contrasting.
In addition to color, I thought I could also take advantage of the contrast between transparent and matte glass for an added effect.
Additionally, the glass must be able to form different shapes, as indicated by the requirements of the competition to create at least three different shapes.
As a result, I opted for the Effetre Transparent 076 Red (a transparent bright glass) and Effetre Pastel 275M Silver Plum (a soft matte dark glass).
I chose different shapes for the beads to highlight the contrast of these two glasses. These two glasses provide endless opportunities for creativity.
As a result, participating in this competition led me to create several beads that are somewhat unusual for my style.
Thank you for a great challenge! | https://isgb.org/creativity-challenge-september-2019/ |
Q:
Linear Algebra Proof of Nonsingularity
Let $A\in \mathbb R^{n×n}$ , $B\in \mathbb R^{n×m}$, and $C\in \mathbb R^{m×n}$.
If $A$ and $I − CA^{-1}B$ are nonsingular, show that $A − BC$ is nonsingular
A:
If $A-BC$ is singular, i.e. if there is a non-zero vector $v$ s.t.
$$Av=BCv$$ then $Cv\neq 0$ since $Av\neq 0$ ($A$ is non-singular), and we get, by multiplying with $CA^{-1}$,
$$Cv=CA^{-1}BCv$$ i.e. $(I-CA^{-1}B)Cv=0$, but since $Cv\neq 0$, this means that $I-CA^{-1}B$ is singular
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Q:
Proving that a function is 1-1 to show that it is invertible
I want to prove that $h(x)=x^3 +2x+1$ is a $1-1$ function to show that it is invertible on all of $\mathbb{R}$.
This my attempt: Let $x_1,x_2\in \mathbb{R}$ where $x_1\neq x_2$.
Suppose for contradiction $h(x_1)=h(x_2)$.
Then $h(x_1)=x_1^3 +2x_1+1$ and $h(x_2)=x_2^3 +2x_2+1$.
$x_1^3 +2x=x_2^3 +2x_2$
Then where do I go from here to show that $x_1=x_2$?
A:
An easier way: The derivative is
$$h'(x) = 3x^2 + 2$$
is strictly positive for all $x$, and so the function is strictly increasing. Formal justification could be made by, say, Rolle's theorem.
Alternatively, given $$x_1^3 + 2x_1 = x_2^3 + 2x_2$$
this can be rearranged as
$$0 = x_1^3 - x_2^3 + 2x_1 - 2x_2 = (x_1 - x_2)(x_1^2 + x_1 x_2 + x_2^2) + 2(x_1 - x_2)$$
Dividing by the non-zero quantity $x_1 - x_2$, this shows that
$$0 = x_1^2 + x_1 x_2 + x_2^2 + 2$$
This is a quadratic equation in $x_1$, whose discriminant is
$$x_2^2 - 4 (x_2^2 + 2)(1) = -3x_2^2 - 8$$
As this is strictly negative, there are no real solutions.
| |
1. A school has eight teachers. If two new teachers start working at the school, how many teachers will there be in total?
There will be ten teachers in total.
2. Luke had scored seven goals. After his last game his total moved up to ten. How many goals did he score in his last game?
He scored three goals in his last game.
3. Sol has six cousins. If two of his aunts each give birth to a new baby, how many cousins will he now have?
He will now have eight cousins.
4. In a soccer game one team had two players sent off. If they started with eleven, how many players were left on the pitch from that team?
There were nine players left on the pitch from that team.
5. Joel is playing pool and has potted five balls in a row. How many more does he have to pot to reach a total of eight balls potted?
He has to pot three more.
6. Sean is on a fourteen day holiday. He has eight days left. For how many days has he been on holiday?
He has been on holiday for six days.
7. There are four chairs around the kitchen table. If four more chairs can be fitted around the table, how many people will be able to sit around the table?
Eight people will be able to sit around the table.
8. Mel wants to use the express checkout but is allowed only twelve items. She has to put three items back to leave her with twelve. How many items did she have before she put some back?
She had fifteen items before she put some back.
9. Three children start playing basketball at recess. By the end of recess there are a total of twelve children playing basketball. How many children joined the three who were playing at the start?
Nine children joined the three who were playing at the start.
10. Jack had twelve friends for a big sleep over. Three of them had to leave early because they had too much ice cream. How many friends were left for the sleep over.
There were nine friends left for the sleep over.
11. Bronwen buys herself two new console games. This means she now has a total of eight games. How many games did she have before buying the two new console games?
She had six console games before buying the two new ones.
12. Maria has fourteen friends on her social network. How many friends will she have if she adds five new ones?
She will have nineteen friends.
13. When Zach started swimming lessons there were five students in the group. When he finished his lessons the number of students had increased to nine. How many students joined the group after the start?
Four students joined the group after the start.
14. Will is playing an online game. When he starts there are fifteen players. After twenty minutes there are only nine still playing. How many players must have left the game?
Six players must have left the game.
15. One class at school has sixteen students. After the holidays two new students join the class. How many students are now in the class?
There are now eighteen students in the class.
16. Shelley is helping her Dad wash the windows. It takes a long time. They have cleaned eight and their are still six more left to do. How many windows needed cleaned when they started?
Fourteen windows needed cleaned when they started.
17. Sam’s team had seven games left until the end of the season. They then played two games. How many games did they then have left?
They then have five games left to play.
18. Ben was asked put some chairs around a table before dinner. There were already six chairs but there needed to be a total of ten. How many more chairs does Ben need to put around the table?
Ben needs to put another four chairs around the table.
19. Melissa posts a photo on a website. Right away thirteen people add a comment to it. The next day three people comment on it. How many people will then have made a comment?
Sixteen people will then have made a comment.
20. Joshua had eight pairs of shorts but he gave four pairs away because they did not fit him anymore. How many pairs did he have left?
He had four pairs left .
21. Kelvin is given eight questions for homework. He does some of them before dinner. After dinner he has five left to do. How many questions did he do before dinner?
He did three questions before dinner.
22. Joshua’s team has scored fourteen goals this year. In the last game they scored two. How many goals had they scored before the last game?
They scored twelve goals before the last game.
23. Sophie’s family is extending their house. It used to have two bedrooms but after the building work it will have five bedrooms. How many bedrooms are being added?
24. Sara has scored six goals for her team this year. She scores three in her next game. How many goals has she now scored?
She has now scored nine goals.
25. Leanne had seven books from the library. She takes three of them back. How many books does she have left?
She has four books left.
26. Shaun is having a party. The last three people arrived late and made a total of fourteen at the party. How many people were at the party before the three that arrived late?
There were eleven people at the party before the three arrived late.
27. Emma posts of photograph onto a website one evening. When she goes to sleep the photograph has eight comments. When she wakes up in the morning she checks and finds that it has seventeen comments. How many comments were added while she was sleeping?
Nine comments were added while she was sleeping.
28. Brad has taken seven books out of the library to help with a project. He goes back the next day and takes four more books out. How many books does he now have out of the library?
He has eleven books out of the library .
29. Olav’s dog has given birth to some puppies. Three have been given away and there are two left. How many puppies did Olav’s dog give birth to?
Olav's dog had five puppies.
30. Andy loves playing tennis. At the start of the year he gets a present of twelve tennis balls. By the end of the year he has lost most of them and only has three left. How many tennis balls did he lose?
He lost nine tennis balls.
31. Jenny’s team has just won four games in a row. This means they have now won a total of eighteen games. How many game must they have won before they won four in a row.
They must have won fourteen games before they won four in a row.
32. Vince has twenty minutes of talk time left on his phone. If he talks for eight minutes, how much time will have left?
He will have twelve minutes left.
33. In a project at school Michele has to write about seven different countries. She has already written about four. How many more countries does she have to write about?
She has to write about three more countries.
34. Nicole has eight candles on her birthday cake. She blows six of them out. How many candles are still to be blown out?
There are still two candles to be blown out.
35. Joel is learning to play golf. He played on Sunday and lost four golf balls. He now has eight golf balls left. How many golf balls did he have when he started playing on Sunday?
They had twelve golf balls when he started on Sunday.
36. There a sixteen children swimming in a pool. Three children dive in and start swimming too. What is the total number of children swimming in the pool?
There are nineteen children swimming in the pool.
37. Ashley is generous and likes giving her toys away. On Saturday she counts that she has fourteen dolls. She gives some to her best friend and she then has only eight left. How many dolls did she give away?
She gave away six dolls.
38. Lindsay’s class finishes the year with twenty students. Three new students had joined the class that year. How many students were in the class at the start of the year?
There were seventeen students at the start of the year.
39. Sven is allowed to invite eight friends to his birthday. He invites six. How many friends can he still invite?
He can invite two more friends.
40. Jessica’s family was having a fun holiday. They decided to add some extra days and turn their seven day holiday into a ten day holiday. How many days did they add?
41. Selene’s family is having a get together at a campsite. Seventeen people are there for lunch and then three people arrive late. How many are at the campsite now?
There are then twenty people at the campsite.
42. The Carter family have just bought a pack of six toothbrushes. They take four of them out the pack to use. How many are left in the pack?
There are two left in the pack.
43. Jordan has already taken seven books out of the library. She takes some more books out on Friday and reaches her limit of fifteen. How many more books must she have taken out on Friday?
She must have taken out eight more books.
44. A hockey team has fourteen jerseys but that never seems to be enough so the coach buys three more. What is the total number of jerseys that the team then has?
The team then has seventeen jerseys .
45. Sue is doing a charity walk with her friends. They must walk for six miles. Eight of them finish the walk but four of them dropped out and did not finish. How many of them started the walk?
Twelve of them started the walk.
46. Eight boys are hoping to find some more players so they can have a team of eleven to play soccer. How many more players do they need?
They need three more players.
47. Jerome has been on holiday for seven days. He has seven more days left. How many days is his holidays in total?
48. A class at school started the year with twelve students. At the end of the year there were fifteen students in the class. How many students joined the class during the year?
Three students joined the class. | https://www.helpingwithmath.com/printables/worksheets/word_problems/1oa1-word-problems01.htm |
Dexamethasone for prevention and treatment of acute mountain sickness.
We wished to determine in a field study the effectiveness of dexamethasone for prevention and treatment of acute mountain sickness (AMS). Prevention Trial: We transported 15 subjects from sea level to 4,400 m (PB = 400 mm Hg) on Denali (Mt. McKinley) by means of a 1-h helicopter flight. In a randomized, double-blind fashion we gave eight subjects a placebo and seven subjects 2 mg dexamethasone orally every 6 h, starting 1 h before take-off. The entire placebo group and five of the dexamethasone group developed AMS within 5 h, and became progressively more ill until 12 h when the trial was terminated. We concluded that 2 mg of dexamethasone every 6 h did not prevent AMS in active soldiers rapidly transported to high altitude. Treatment Trial: We treated 11 of those with moderate to severe AMS (symptom score 4.5 +/- 0.7, range 3 to 11) with 4 mg of dexamethasone every 6 h orally or intramuscularly for 24 h. All were markedly improved at 12 h (symptom score 1.0 +/- 0.3, p less than 0.001, range 0 to 3), but symptoms increased after the drug was discontinued at 24 h (symptom score = 2.4 +/- 0.5). We conclude that dexamethasone in a dosage of 4 mg PO or IM every 6 h is an effective treatment for AMS, but that illness may recur with abrupt discontinuation of the drug.
| |
---
abstract: |
This paper devotes to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements are established for the $H^1$ and $H(\rm rot)$ elliptic problems, respectively. The two finite element spaces on general quadrilateral grids, together with the space of piecewise constant functions, formulate a discretised de Rham complex. First order convergence rate can be proved for both of them under the asymptotic-parallelogram assumption on the grids.
The local shape functions of the two spaces are piecewise polynomials, and the global finite element functions are nonconforming. A rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element defined as Ciarlet’s triple whose shape functions are always piecewise polynomials and which can form conforming subspaces on a grid that consists of arbitrary quadrilaterals.
address:
- 'University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China'
- 'LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China'
- 'LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China'
author:
- 'QimengQuan, Xia Ji, Shuo Zhang'
bibliography:
- 'reference.bib'
title: 'Lowest-degree polynomial de Rham complex on general quadrilateral grids'
---
[^1]
Introduction
============
There has been a long history on the study of finite element methods on general quadrilateral grids. Many conforming and nonconforming finite elements are established for various model problems. The classical strategy for constructing quadrilateral elements is to utilize isoparametric technique (cf., e.g., [@ciarlet2002finite]). With this strategy, one begins with a given shape function space on a reference square and a class of bilinear transforms (or Piola transforms, etc.), the finite element on any convex quadrilateral cell can be constructed correspondingly. Great success has been achieved via this approach, particularly in constructing conforming finite element spaces; we refer to [@arnold2002approximation; @arnold2005quadrilateral; @falk2011hexahedral] for more details. On the other hand, a solid difficulty of these methods, as discussed in, e.g., Zhang [@zhang2004nested], is that one will encounter the problem of rational function integration in practical numerical computation due to non-constant Jacobian determinants and inverse Jacobian matrices. Besides, when we take the discretized differential complexes, which is one of the most fundamental structural feature of the finite element schemes and has been a central topic of the finite element methods during the passed decades, into consideration, we have to pay extra attention to the different but relevant Jacobian matrices for the spaces, respectively. We are thus motivated to study finite element schemes with piecewise polynomials on every cell that possess clearer structure and more friendly implementation. In this paper, we present three finite element spaces on grids that consist of general quadrilaterals, and they form a discrete analogue of the de Rham complex which reads in 2D $$\label{eq:dr2d}
\mathbb{R}\xrightarrow{\rm inclusion} H^1 \xrightarrow{\nabla} H(\rot) \xrightarrow{\rot} L^2\xrightarrow{\rm integration}\mathbb{R}.$$
There have been various finite element schemes on general quadrilateral grids with piecewise polynomials; we refer to [@rannacher1992simple; @jeon2013class; @park2003p; @zhang2016stable] for example. Also, the first finite element complex with piecewise polynomials on general quadrilateral grids can be found in [@zhang2016stable]. It is worthy of attention that all these aforementioned finite element spaces are nonconforming, except those for the space $L^2$. Indeed, as will be proved in Section \[sec:geo\], it is impossible to construct a practically useful finite element whose shape functions are always piecewise polynomials and which can form conforming subspaces on a grid that consists of arbitrary quadrilaterals rather than parallelograms only. We thus do not seek to construct conforming elements; in view of , we impose the continuity of evaluation on vertices for the approximation of $H^1$ functions and the continuity of the average of the tangential component along the edges for the approximation of $H(\rot)$ functions, and these spaces can be called quasi-conforming ones. For low-degree polynomials, the continuity can guarantee the conformity of the respective finite element spaces on parallelograms, while on general quadrilaterals, the continuity does not even guarantee the functions to pass the patch test. In the analysis and the implementation, we will then adopt the general $\mathcal{O}(h^2)$ asymptotic-parallelogram regularity assumption for the convergence. This assumption is normally used for the analysis of nonconforming finite element schemes. Further, due to the lack of full elliptic regularity, for the nonconforming finite element scheme of the $H(\curl)$ problem, higher than general regularity assumption may in general be needed for the same convergence rate. In contrast, by the aid of this assumption, we can prove the optimal convergence rate for the proposed elements with the proper regularity assumptions of the exact solutions. Though is crucial for our lowest-degree case, the assumption can be weakened for higher degree schemes, such as [@arbogast2016two; @dubach2009pseudo].
The remaining of the paper is organized as follows. In the remaining of this section, we introduce some necessary notations. In Section \[sec:geo\], we introduce some geometrical features of the grids. We will particularly prove that, shortly speaking, no practically useful conforming element with piecewise polynomials can be constructed for general quadrilateral grids. In Section \[sec:a sequence FEs\], three finite elements are introduced with a commutative diagram. The approximation error is analyzed by the technique of combining the classical Taylor expansion procedure and a specific commutative property. In Section \[sec:NFEs and mc\], the modulus of the continuity of the finite element functions are given. Then in Section \[sec:FE schemes for model problems\], the performance of the finite elements are studied for the $H^1$ and $H(\rot)$ problems; both theoretical analysis and numerical verifications are given. Some conclusions and comments are given in Section \[sec:concluding remarks\]. \
#### **Notations**
In this paper, conventional notations for the Sobolev spaces and grid-related quantities will be used. Let $\Omega\subset \mathbb{R}^2$ be a simple connected Lipschitz domain and $\Gamma = \partial\Omega$ be the piecewise boundary with $\un$ the outward unit normal vector and $\ut$ the counterclockwise unit tangential vector, “$\undertilde{\cdot}$" representing the vector valued quantities. Denote by $H^{m}(\Omega)$ and $H^{m}_{0}(\Omega)$ the standard Sobolev spaces equipped with the norm $\|\cdot\|_{m,\Omega}$ and seminorm $|\cdot|_{m,\Omega}$ as usual, and $L^2_0(\Omega):=\{q\in L^2(\Omega):\int_{\Omega} q \ \mathrm{d}x=0\}$. We also denote by $\uL^2(\Omega)=(L^{2}(\Omega))^2$ , $\uH^m(\Omega)=(H^{m}(\Omega))^2$ and $\uH{}_{0}^{m}(\Omega)=(H^{m}_0(\Omega))^2$. The inner product of $L^2$ and $\uL^2$ is denoted by $(\cdot,\cdot)$ on the domain $\Omega$. We define two forms of rotation operator in two-dimensional case by $$\begin{aligned}
& \text{Given a vector}\
\usigma(x_{1},x_{2})
=(\sigma_{1},\sigma_{2})^{T} &
\mathrm{rot}\usigma&=
\frac{\partial \sigma_{2}}{\partial x_{1}}-
\frac{\partial \sigma_{1}}{\partial x_{2}} \\
& \text{Given a scalar function}\
\sigma=\sigma(x_{1},x_{2}) &
\undertilde{\mathrm{curl}}\sigma&=
(\frac{\partial\sigma}{\partial x_{2}},
-\frac{\partial\sigma}{\partial x_{1}})^{T}.\end{aligned}$$ Superscript $T$ indicates transposition of vector or matrix as usual. We also use these notations to denote Sobolev spaces $H(\mathrm{rot},\Omega)=
\{\usigma|\usigma\in \uL^2(\Omega),\mathrm{rot}\usigma\in L^{2}(\Omega)\}
$ and $
H_{0}(\mathrm{rot},\Omega)=\{\usigma|\usigma\in H(\mathrm{rot},\Omega), \usigma\cdot\ut=0 \ \text{on}\ \Gamma\}$ equipped with standard norm $\|\cdot\|^2_{\rot,\Omega}=\|\cdot\|^2_{0,\Omega}+|\cdot|^2_{{\rm rot},\Omega}$. Specially, a new notation is used for the space $
H^{1}(\mathrm{rot},\Omega)
\triangleq
\{
\usigma|\usigma\in \uH^1(\Omega), \mathrm{rot}\usigma\in H^{1}(\Omega)
\}$.
Let $\mathcal{J}_{h}$ be a regular subdivision of domain $\Omega$, with the elements being convex quadrilaterals, i.e., $\Omega=\cup_{K\in\mathcal{J}_{h}} K$. And any two distinct quadrilaterals $K_{1}$ and $K_{2}$ in $\mathcal{J}_{h}$ with $\bar{K_{1}}\cap \bar{{K}_{2}}\neq\emptyset$, share exactly one vertex or have one edge in common. Denote a finite element $(K,P_K,D_K)$ by Ciarlet’s triple [@ciarlet2002finite], the subscription $K$ implying the dependence of the quadrilateral $K$. Let $\mathcal{N}_{h}$ denote the set of all the vertexes, $\mathcal{N}_{h}=
\mathcal{N}_{h}^{i}\cup
\mathcal{N}_{h}^{b}$, with $\mathcal{N}_{h}^{i}$ and $\mathcal{N}_{h}^{b}$ consisting of the interior vertexes and the boundary vertexes, respectively. Similarly, let $\mathcal{E}_{h}=
\mathcal{E}_{h}^{i}\cup
\mathcal{E}_{h}^{b}$ denote the set of all the edges, with $\mathcal{E}_{h}^{i}$ and $\mathcal{E}_{h}^{b}$ consisting of the interior edges and the boundary edges, respectively. The subscript $h$ in various notations implies the dependence of the subdivision. Denote by $h_{K}$ the diameter of each quadrilateral $K$ and the grid size $h\triangleq\max_{K\in\mathcal{J}_{h}}{h_{K}}$. On the edge $e$, we use $[\![]\!]_{e}$ for the jump across $e$.
Throughout the paper we denote by $C$ a positive constant not necessarily the same at each occurrence but always independent of the diameter $h_K$ or the grid size $h$. Denote $\lambda_{F,G}$ by the generalized eigenvalue of matrix pair $(F,G)$, i.e., $F\ux=\lambda_{F,G}G\ux$. We use notations $P_e$ and $P_{K}$ to denote the average of the integral on the edge $e$ and quadrilateral $K$, respectively.
Geometry of the quadrilaterals {#sec:geo}
==============================
Quadrilateral and functions {#subsec:quad and fun}
---------------------------
Let $K$ be a convex quadrilateral with $A_{i}$ the vertices and $e_{i}$ the edges, $i = 1:4$, see Figure \[fig:convquad\]. Let $m_{i}$ be the mid-point of $e_{i}$, then the quadrilateral $\square m_{1}m_{2}m_{3}m_{4}$ is a parallelogram. The cross point of $m_{1}m_{3}$ and $m_{2}m_{4}$, which is labelled as $O$, is the midpoint of both $m_{1}m_{3}$ and $m_{2}m_{4}$. Denote $\undertilde{r}=\overrightarrow{Om_4}$ and $\undertilde{s}=\overrightarrow{Om_1}$. Then, the coordinates of the vertices in the coordinate system $\undertilde{r}O\undertilde{s}$ are $A_{1}(1+\alpha, 1+\beta)$, $A_{2}(-1-\alpha, 1-\beta)$, $A_{3}(-1+\alpha, -1+\beta)$, $A_{4}(1-\alpha, -1-\beta)$ and for some $\alpha, \beta$. Since $K$ is convex, $|\alpha|+|\beta|<1$. Without loss of generality, we assume $\alpha>0$, $\beta>0$ and $\undertilde{r}\times\undertilde{s}>0$. Here and after, we call $\alpha$, $\beta$ local shape parameters.
Define the shape regularity indicator of the quadrilateral $K$ by $\mathcal{R}_K=\max\{\frac{|r||s|}{\ur\times\us},\frac{|r|}{|s|},\frac{|s|}{|r|}\}$. Evidently $\mathcal{R}_K\geqslant 1$, and $\mathcal{R}_K=1$ if and only if $K$ is a square. A given family of quadrilateral subdivisions $\{\mathcal{J}_h\}$ of $\Omega$ is called regular, if all the shape regularity indicators of the quadrilaterals of all the subdivisions are uniformly bounded.
(0.05, 0.26)node[$O$]{};
(-2, 2)–(3, 2); (-3.5, -2)–(1.5, -2); (-3.5, -2)–(-2, 2); (1.5, -2)–(3, 2);
(0.5, 2)–(-1, -2); (2.25, 0)–(-2.75, 0);
(-2.345, 0.275)node[$m_2$]{}; (-0.75, -1.9)node[$m_3$]{}; (2.1, 0.35)node[$m_4$]{}; (0.85, 1.75)node[$m_1$]{};
(-3.4, 0.175)node[$e_2$]{}; (-0.95, -2.4)node[$e_3$]{}; (2.7, 0.05)node[$e_4$]{}; (0.4, 2.35)node[$e_1$]{};
(-0.25, 0)– node\[auto\] [$\undertilde{r}$]{} (2.25, 0); (-0.25, 0)– node\[auto\] [$\undertilde{s}$]{} (0.5, 2);
(3.725, 2.6)–(-2.725, 1.4); (3.725, 2.6)–(0.775, -2.6); (-2.775, -1.4)–(0.775, -2.6); (-2.775, -1.4)– (-2.725, 1.4);
(-2.9, 1.7)node[$A_{2}$]{}; (-2.8, -1.7)node[$A_{3}$]{}; (0.9, -2.85)node[$A_{4}$]{}; (3.85, 2.8)node[$A_{1}$]{};
(0.5, 2)–(2.25, 0); (2.25, 0)–(-1, -2); (-1, -2)–(-2.75, 0); (-2.75, 0)–(0.5, 2);
Define two linear functions $\xi$ and $\eta$ by $\xi(a\undertilde{r}+b\undertilde{s})=a$ and $\eta(a\undertilde{r}+b\undertilde{s})=b$. The two functions play the same role on quadrilaterals as that played by barycentric coordinates on triangles. Additionally we also define two functions $\hat{\xi}$ and $\hat{\eta}$ by $\hat{\xi}=\xi-\dashint_{K}\xi\ \mathrm{d}x$, $\hat{\eta}=\eta-\dashint_{K}\eta\ \mathrm{d}x$ for convenience of calculation in Section \[sec:NFEs and mc\]. Technically, we construct two tables about the evaluation of some functions which will be useful in theoretical analysis and numerical computation.
Function $\xi^{2}$ $\xi\eta$ $\eta^{2}$
---------------- ----------------------------------- ------------------------------------- -----------------------------------
$\int_{e_{1}}$ $\frac{(1+\alpha)^{2}|e_{1}|}{3}$ $\frac{(1+\alpha)\beta|e_{1}|}{3}$ $\frac{(3+\beta^{2})|e_{1}|}{3}$
$\int_{e_{2}}$ $\frac{(3+\alpha^{2})|e_{2}|}{3}$ $\frac{\alpha(-1+\beta)|e_{2}|}{3}$ $\frac{(1-\beta)^{2}|e_{2}|}{3}$
$\int_{e_{3}}$ $\frac{(1-\alpha)^{2}|e_{3}|}{3}$ $\frac{(-1+\alpha)\beta|e_{3}|}{3}$ $\frac{(3+\beta^{2})|e_{3}|)}{3}$
$\int_{e_{4}}$ $\frac{(3+\alpha^{2})|e_{4}|}{3}$ $\frac{\alpha(1+\beta)|e_{4}|}{3}$ $\frac{(1+\beta)^{2}|e_{4}|}{3}$
: boundary integral evaluation of some functions[]{data-label="tab:tab1"}
Function $1$ $\xi$ $\eta$ $\xi^{2}$ $\xi\eta$ $\eta^{2}$
------------ ----------------- -------------------------------- --------------------------------- -------------------------------------------------- -------------------------------------- --------------------------------------------------
$\int_{K}$ $4\ur\times\us$ $\frac{4\beta}{3}\ur\times\us$ $\frac{4\alpha}{3}\ur\times\us$ $\frac{4}{3}(1+\alpha^{2})\ur\times\us$ $\frac{4}{3}\alpha\beta\ur\times\us$ $\frac{4}{3}(1+\beta^{2})\ur\times\us$
Function $1$ $\hat{\xi}$ $\hat{\eta}$ $\hat{\xi}^{2}$ $\hat{\xi}\hat{\eta}$ $\hat{\eta}^{2}$
$\int_{K}$ $4\ur\times\us$ $0$ $0$ $\frac{4}{9}(3+3\alpha^{2}-\beta^2)\ur\times\us$ $\frac{8}{9}\alpha\beta\ur\times\us$ $\frac{4}{9}(3+3\beta^{2}-\alpha^2)\ur\times\us$
: integral evaluation of some functions in domain K[]{data-label="tab:tab2"}
Grid refinement {#subsec:mesh refinement}
---------------
Denote by $d_{K}$ the distance between the midpoints of the diagonals of the quadrilateral $K$, then we introduce a lemma.
All refined quadrilaterals produced by a bisection scheme of grid subdivisions have the property $d_{K}=\mathcal{O}(h^{2}_{K})$. \[lemma:mesh refinement\]
The proof can be found in [[@shi1984convergence; @zhang2004polynomial]]{}.
Here and after, we call the grid generated by the bisection scheme as an asymptotically parallelogram grid. We notice that the quantity $\max_{K\in\mathcal{J}_{h}}\{\alpha_K,\beta_K\}$ is of order $\mathcal{O}(h)$ uniformly for asymptotically regular parallelogram grid by Lemma \[lemma:mesh refinement\]. This proposition will be used frequently in the Section \[sec:NFEs and mc\].
On the construction of conforming element with piecewise polynomials {#subsec:mcpp on quads}
--------------------------------------------------------------------
Hereby, we prove that, as rigorously presented in the below theorem, for general quadrilateral grids, no practically useful conforming elements can be constructed with piecewise polynomials.
\[thm:nocon\] Let ${\rm FEM_{pq}}=(K, P_K, N_K)$ be a finite element defined by Ciarlet’s triple, with $K$ being any quadrilateral, and $P_K$ being a space of polynomials on $K$. If the finite element space generated by ${\rm FEM_{pq}}$ by the continuity of nodal parameters is an $H^1$ subspace on any grid that consists of arbitrary quadrilaterals, then $P_K$ only contains polynomials that vanish on the boundary of $K$.
Shortly speaking, if a finite element ${\rm FEM_{pq}}$, the subscripts “p" for [*polynomial*]{} and “q" for [*quadrilateral*]{}, can formulate continuous piecewise polynomial space on general quadrilateral grids, then the shape function space of the finite element consists of bubble functions on $K$ only. This theorem shows the non-existence of practically useful conforming finite element defined by Ciarlet’s triple on general quadrilateral grids. However, we emphasize that it does not exclude the possibility that, on a given quadrilateral grid, a subspace of $H^1_0(\Omega)$ that consists of piecewise $k$-th degree polynomials can contain more than cell bubbles.
#### **Proof of Theorem \[thm:nocon\]**
By the assumption that ${\rm FEM_{pq}}$ can formulate an $H^1$ subspace on a grid that consists of arbitrary quadrilaterals, given a quadrilateral $K_1$ with $D$ being one of its vertices, without loss of generality, we assume there is a shape function (polynomial) $q$ defined on $K_1$, such that $q=0$ along the two edges of which $D$ is [**not**]{} an endpoint and $q$ is nontrivial along one edge $e\ni D$ of $K_1$. By the continuity of the finite element space generated from ${\rm FEM_{pq}}$ by the continuity of nodal parameters and by the arbitrariness of choosing the evaluation of nodal parameters, for any groups of quadrilaterals (including $K_1$) which can form a patch $\omega_D$ centered at $D$ (see Figure \[fig:patch for no CFE space\] for a reference), there exists a finite element function $r\in H^1_0(\omega_D)$, such that $r|_K=q$. Again, we emphasize that the evaluation on these common nodal parameters do not inflect the evaluation of the finite element function on the edges which do not intersect with $K_1$, and this is why $r$ can be chosen in $H^1_0(\omega_D)$.
Now, since $r|_f=0$, we can rewrite $r|_{K_2}=r_{-1}\cdot l_f$, where $l_f$ is a first degree polynomial which vanishes on $f$ and $r_{-1}$ is a polynomial with one degree lower than $r$. Without loss of generality, we assume $f$ is not parallel to $e$. By the continuity of $r$ on $e$, we can rewrite $q|_e=q_{-1}l_f|_e=q_{-1}(t-\theta)$, where $t$ is the length parameter of $e$, $q_{-1}$ is a polynomial on $e$ with one degree lower than $q$ and $\theta$ varies as the angle between $e$ and $f$ varies. Recall that $K_1$ and $q_{K_1}$ are fixed, but $f$ can be arbitrary. Change $f$ to another direction $f'$, by elementary calculation, it follows that $q|_e=q_{-2}(t-\theta)(t-\theta')$. This way, by repeating the procedure, we can see that $q|_e$ contains a polynomial factor with growing degree and thus can not be a nontrivial polynomial. This leads to a contradiction to the assumption that $q|_e\not\equiv 0$ and completes the proof.
(-1.5,-1.25)–(0,-2); (0,-2)–(2.25,-1.8); (-1.5,0)–(0.5,0); (0.5, 0)–(2.5, 0); (-0.7, 1.8)–(1,2); (1,2)–(2.6, 1.4);
(-1.5,-1.25)–(-1.5,-0); (-1.5,-0)–(-0.7, 1.8); (0,-2)–(0.5,0); (0.5, 0)–(1, 2); (2.25,-1.8)–(2.5,-0); (2.5, -0)–(2.6, 1.4);
(0.38,0.17)node[$D$]{}; (-0.6,-1)node[$K_{1}$]{}; (1.15,-1)node[$K_{2}$]{}; (-0.35,1)node[$K_{4}$]{}; (1.65,1)node[$K_{3}$]{}; (0.05,-1)node[$e$]{}; (2.15,-1)node[$f$]{}; (0.5,0) circle(1.5pt);
\[fig:patch for no CFE space\]
\[rem:nocon\] Similarly, conforming finite elements can not be defined for $H(\rot)$ with piecewise polynomials for general quadrilateral grids. Indeed, the assertion can be generalized to general Sobolev spaces.
A sequence of lowest-degree finite elements {#sec:a sequence FEs}
===========================================
Definitions of the finite elements {#subsec:def of FEs}
----------------------------------
In the subsection we introduce three types of finite elements.
The quadrilateral finite element presented below is similar to the bilinear element on rectangle, and we call it the quadrilateral bilinear (${\rm QBL}$) element.
${\rm QBL}$ element defined above is unisolvent. Indeed, define $$\left\{
\begin{aligned}
\phi_{1} & =
\frac{\alpha+\beta-1}{4(\alpha^{2}+\beta^{2}-1)}\xi\eta+ \frac{(\beta-1)(-\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\xi+ \frac{(\alpha-1)(\alpha-\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\eta+
\frac{-(\alpha-1)(\beta-1)(\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)} \\
\phi_{2} & =
\frac{-\alpha+\beta+1}{4(\alpha^{2}+\beta^{2}-1)}\xi\eta+ \frac{-(\beta+1)(\alpha+\beta-1)}{4(\alpha^{2}+\beta^{2}-1)}\xi+ \frac{(\alpha-1)(\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\eta+
\frac{(\alpha-1)(\beta+1)(\alpha-\beta+1)}{4(\alpha^{2}+\beta^{2}-1)} \\
\phi_{3} & =
\frac{-(\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\xi\eta+ \frac{(\beta+1)(\alpha-\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\xi+ \frac{(\alpha+1)(-\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\eta+
\frac{(\alpha+1)(\beta+1)(\alpha+\beta-1)}{4(\alpha^{2}+\beta^{2}-1)} \\
\phi_{4} & =
\frac{\alpha-\beta+1}{4(\alpha^{2}+\beta^{2}-1)}\xi\eta+ \frac{(\beta-1)(\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}\xi+ \frac{-(\alpha+ 1)(\alpha+\beta-1)}{4(\alpha^{2}+\beta^{2}-1)}\eta+
\frac{(\alpha+1)(\beta-1)(-\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}.
\end{aligned}
\right.
\label{equ:local dual basis of QBL}$$ then we can verify directly $\phi_{i}(A_{j})=\delta_{ij}$, $i,j=1:4$.
Here and after, the functions $\{\phi_{i}\}_{i=1:4}$ are called local basis of $P_{K}^{\rm QBL}$. We use the notation $\phi_i^{(j)}$ to denote $j$-th coefficient of $i$-th basis, for example, $\phi_{1}^{(2)} = \frac{(\beta-1)(-\alpha+\beta+1)}{4(\alpha^{2}+\beta^{2}-1)}$.
Given a ${\rm QBL}$ element $(K,P_{K}^{\rm QBL},D_{K}^{\rm QBL})$, define the local interpolation operator $J_{K}$ by $$J_{K}u=\sum_{i=1}^{4}u(A_{i})\phi_{i},
\quad
\forall
u \in H^2(K).
\label{def:J_K}$$ Furthermore, given a family of ${\rm QBL}$ elements $(K_{i},P_{K_{i}}^{\rm QBL},D_{K_{i}}^{\rm QBL})$ in a subdivision $\mathcal{J}_h$, define the global interpolation operator $J_{h}$ by $$J_{h}u|_{K_{i}}=
J_{K_{i}}u
\quad
\forall K_{i}\in\mathcal{J}_{h}.
\label{def:J_h}$$
The quadrilateral finite element presented below is similar to the Raviart-Thomas element on rectangle, and we call it the quadrilateral Raviart-Thomas (${\rm QRT}$) element.
The ${\rm QRT}$ element as above is unisolvent. Indeed, define $$\left\{
\begin{aligned}
\uphi{}_{1} & =
\frac{(1-\alpha)(1-\beta^2)|e_1|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\xi+ \frac{\alpha(1-\alpha)\beta|e_1|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\eta+ \frac{-\alpha(1-\alpha)|e_1|}{4(\alpha^{2}+\beta^{2}-1)}\xi\nabla\eta+
\frac{(1-\alpha-\beta^2)|e_1|}{4(\alpha^{2}+\beta^{2}-1)}\eta\nabla\xi \\
\uphi{}_{2} & =
\frac{\alpha\beta(1+\beta)|e_2|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\xi+ \frac{(1-\alpha^2)(1+\beta)|e_2|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\eta+ \frac{-(1-\alpha^2+\beta)|e_2|}{4(\alpha^{2}+\beta^{2}-1)}\xi\nabla\eta+
\frac{-\beta(1+\beta)|e_2|}{4(\alpha^{2}+\beta^{2}-1)}\eta\nabla\xi \\
\uphi{}_{3} & =
\frac{-(1+\alpha)(1-\beta^2)|e_3|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\xi+ \frac{-\alpha(1+\alpha)\beta|e_3|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\eta+ \frac{\alpha(1+\alpha)|e_3|}{4(\alpha^{2}+\beta^{2}-1)}\xi\nabla\eta+
\frac{(1+\alpha-\beta^2)|e_3|}{4(\alpha^{2}+\beta^{2}-1)}\eta\nabla\xi \\
\uphi{}_{4} & =
\frac{-\alpha\beta(1-\beta)|e_4|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\xi+ \frac{-(1-\alpha^2)(1-\beta)|e_4|}{4(\alpha^{2}+\beta^{2}-1)}\nabla\eta+ \frac{-(1-\alpha^2-\beta)|e_4|}{4(\alpha^{2}+\beta^{2}-1)}\xi\nabla\eta+
\frac{\beta(1-\beta)|e_4|}{4(\alpha^{2}+\beta^{2}-1)}\eta\nabla\xi.
\end{aligned}
\right.
\label{equ:local dual basis of QRT}$$ Denote $\{D_i^{\rm QRT}\}_{i=1:4}$ by the components of $D_K^{\rm QRT}$, then we can verify directly $D_i^{\rm QRT}(\uphi{}_j)=\delta_{ij}$, $i,j=1:4$.
Here and after, the functions $\{\uphi{}_i\}_{i=1:4}$ are called local basis of $P_K^{\rm QRT}$.
Given the ${\rm QRT}$ element $(K,P_{K}^{\rm QRT},D_{K}^{\rm QRT})$, define the local interpolation operator $\sqcap_{K}$ by $$\sqcap_{K}
\undertilde{\sigma} =
\sum_{i=1}^{4}
D_i^{\rm QRT}(\usigma)\uphi{}_i
\quad
\forall
\usigma \in \uH^1(K).
\label{def:local interpolation of rot}$$ Furthermore, given a family of ${\rm QRT}$ elements $(K_i,P_{K_i}^{\rm QRT},D_{K_i}^{\rm QRT})$ in a subdivision $\mathcal{J}_{h}$, define the global interpolation operator $\sqcap_{h}$ by $$\sqcap_{h}\undertilde{\sigma}|_{K_{i}}=
\sqcap_{K_{i}}\undertilde{\sigma}
\quad
\forall K_{i}\in\mathcal{J}_{h}.
\label{def:global rot interpolation}$$
Finally, for any $q\in L^2(\Omega)$ define the interpolation operator $P_h$ by $P_hq|_{K_i}= P_{K_{i}}q, \forall K_{i}\in \mathcal{J}_h$.
### Exact sequences on a quadrilateral {#subsubsec:exact sequence on a quad}
The commutative diagram holds as below: $$\begin{array}{ccccccccc}
\mathbb{R} & ~~~\longrightarrow~~~ & H^2(K) & ~~~\xrightarrow{{\boldsymbol}{{\mathrm}{\nabla}}}~~~ & \uH^1(K) & ~~~\xrightarrow{{\mathrm}{rot}}~~~ & L^2(K) & ~~~\xrightarrow{\int_K\cdot}~~~ & \mathbb{R} \\
& & \downarrow J_K & & \downarrow \sqcap_{K} & & \downarrow P_K & & \\
\mathbb{R} & \longrightarrow & P_K^{\rm QBL} & \xrightarrow{{\boldsymbol}{{\mathrm}{\nabla}}} & P_K^{\rm QRT} & \xrightarrow{{\mathrm}{rot}} & \mathbb{R} & \xrightarrow{\int_K\cdot} & \mathbb{R}.
\end{array}$$ \[thm:commutativity on a cell\]
We first prove the discretized de Rham complex. Evidently ${\rm ker}(\nabla)=\mathbb{R}$ and $\nabla P^{\rm QBL}_K\subset P_{K}^{\rm QRT}$. On the other hand, $\rot P_{K}^{\rm QRT}=\mathbb{R}$. It remains to prove that ${\rm ker}({\rm rot})=\nabla P^{\rm QBL}_K$. Given a $\utau\in P_K^{\rm QRT}$, such that ${\rm rot}\utau=0$. Since $\utau= d_1\nabla\xi+d_2\nabla\eta+d_3\xi\nabla\eta+d_4\eta\nabla\xi$, then we have $d_3=d_4$ and $\utau \in \nabla P^{\rm QBL}_K$.
Then we are going to show that $\nabla J_K=\sqcap_{K}\nabla$ on $H^2(K)$, and $\mathrm{rot}\sqcap_{K}=P_K\mathrm{rot}$ on $\uH^1(K)$. We first prove the former. Given a $\usigma \in \uH^1(K)$, let $\sqcap_{K}\usigma=g_{1}\cdot\nabla\xi+g_{2}\cdot\nabla\eta+g_{3}\cdot\xi\nabla\eta+g_{4}\cdot\eta\nabla\xi$. By definition, we have $$\begin{aligned}
g_{1}&=
\frac{
(1-\alpha)(1-\beta^2)\undertilde{e}{}_{1}\cdot P_{e_{1}}(\usigma)+
\alpha\beta(1+\beta)\undertilde{e}{}_{2}\cdot P_{e_{2}}(\usigma)-
(1+\alpha)(1-\beta^{2})\undertilde{e}{}_{3}\cdot P_{e_{3}}(\usigma)-
\alpha\beta(1-\beta)\undertilde{e}{}_{4}\cdot P_{e_{4}}(\usigma)
}
{4(\alpha^2+\beta^{2}-1)}\\
g_{2}&=
\frac{
\alpha(1-\alpha)\beta\undertilde{e}{}_{1}\cdot P_{e_{1}}(\usigma)+
(1-\alpha^{2})(1+\beta)\undertilde{e}{}_{2}\cdot P_{e_{2}}(\usigma)-
\alpha(1+\alpha)\beta\undertilde{e}{}_{3}\cdot P_{e_{3}}(\usigma)-
(1-\alpha^{2})(1-\beta)\undertilde{e}{}_{4}\cdot P_{e_{4}}(\usigma)
}
{4(\alpha^2+\beta^{2}-1)}\\
g_{3}&=
\frac{
-\alpha(1-\alpha)\undertilde{e}{}_{1}\cdot P_{e_{1}}(\usigma)
-(1-\alpha^{2}+\beta)\undertilde{e}{}_{2}\cdot P_{e_{2}}(\usigma)
+\alpha(1+\alpha)\undertilde{e}{}_{3}\cdot P_{e_{3}}(\usigma)
-(1-\alpha^{2}-\beta)\undertilde{e}{}_{4}\cdot P_{e_{4}}(\usigma)
}
{4(\alpha^2+\beta^{2}-1)}
\\
g_{4}&=
\frac{
(1-\alpha-\beta^2)\undertilde{e}{}_{1}\cdot P_{e_{1}}(\usigma)
-\beta(1+\beta)\undertilde{e}{}_{2}\cdot P_{e_{2}}(\usigma)
+(1+\alpha-\beta^{2})\undertilde{e}{}_{3}\cdot P_{e_{3}}(\usigma)
+\beta(1-\beta)\undertilde{e}{}_{4}\cdot P_{e_{4}}(\usigma)
}
{4(\alpha^2+\beta^{2}-1)}.\\
\end{aligned}$$ Now given a $u\in H^2(K)$, we take $\usigma=\nabla u\in \uH^1(K)$ and the former follows by simple calculation.
It remains to prove the latter. Since $\nabla\xi=\frac{s^{\bot}}{\ur\times\us}$, $\nabla\eta=\frac{r^{\bot}}{\ur\times\us}$, then for $\usigma\in\uH^1(K)$ it holds $$\mathrm{rot}(\sqcap_{K}\usigma)=
(g_{3}-g_{4})\nabla\xi\times\nabla\eta=
\frac{g_3-g_4}{\ur\times\us}=
\frac{1}{4\ur\times\us}\int_{\partial K}
\undertilde{\sigma}
\cdot
\undertilde{t}\ \mathrm{d}s=
P_K({\rm rot}\usigma).$$ The proof is completed.
Interpolation error estimation {#subsec:interpolation error}
------------------------------
### Interpolation error estimations in $L^2$ norm {#subsubsec:interpolation error in L2 norm}
Let $K$ be a convex quadrilateral, then it holds $$\| u-J_{K}u\|_{0, K}
\leqslant
Ch^{2}_{K}|u|_{2, K}
\quad
\forall u\in H^2(K).
$$
By density, it suffices to consider $u\in C^{2}(\bar{K})$. Let $A$ be any point in the quadrilateral $K$ with vertexes $\{A_i\}_{i=1:4}$. Using Taylor expansion with integral remainder, we have $$u(A_{i})=u(A)+\nabla u(A)\cdot(A_{i}-A)+R_{i}(A),
\quad
R_{i}(A)=\int_{0}^{1}
(1-t)
\frac{\mathrm{d}^{2}u}
{\mathrm{d} t^{2}}
(\xi_{i},\eta_{i})
\ \mathrm{d}t.$$ Here $\xi_{i}=tx_{i}+(1-t)x$, $\eta_{i}=ty_{i}+(1-t)y$.
Since $J_{K}$ preserves linear polynomial $$J_{K}u(A)
=\sum_{i=1}^{4}
u(A_{i})\phi_{i}(A)
=u(A)+\sum_{i=1}^{4}R_{i}(A)\phi_{i}(A).$$ Then we obtain $$u(A)-J_{K}u(A)=-\sum_{i=1}^{4}R_{i}(A)\phi_{i}(A).$$
Evidently $$\begin{aligned}
|R_{i}(A)|^{2}
&=|\int_{0}^{1}(1-t)
(\frac{\partial^{2}u}{\partial \xi_{i}^{2}}(x_{i}-x)^{2}+
2\frac{\partial^{2}u}{\partial\xi_{i}\partial\eta_{i}}(x_{i}-x)(y_{i}-y)+\frac{\partial^{2}u}{\partial \eta_{i}^{2}}(y_{i}-y)^{2})
\ \mathrm{d}t|^{2} \\
&\leqslant
4h_{K}^{4}\int_{0}^{1}(1-t)^{2}
\sum_{|m|=2}
|\partial^{m}u(\xi_{i}, \eta_{i})|^2
\ \mathrm{d}t \\
\| u-J_{K}u\|_{0, K}^{2}
&\leqslant
Ch_{K}^{4}\sum_{i=1}^{4}\sum_{|m|=2}
\int_{0}^{1}(1-t)^2\int_{K}|\partial^{m}u(\xi_{i}, \eta_{i})|^2\ \mathrm{d}x\mathrm{d}y\mathrm{d}t.
\end{aligned}$$
Take integral variable substitution: $\mathrm{d}\xi_{i}=(1-t)\mathrm{d}x$, $\mathrm{d}\eta_{i}=(1-t)\mathrm{d}y$, then we have $$\| u-J_{K}u\|_{0, K}^{2}
\leqslant
Ch_{K}^{4}
\sum_{i=1}^{4}\sum_{|m|=2}
\int_{0}^{1}\int_{K}
|\partial^{m}u(\xi_{i}, \eta_{i})|^2
\ \mathrm{d}\xi_{i}\mathrm{d}\eta_{i}\mathrm{d}t
=Ch_{K}^{4}|u|_{2, K}^2.$$ The proof is completed.
Let $K$ be a convex quadrilateral, then it holds $$\|\usigma-\sqcap_{K}\usigma\|_{0, K}
\leqslant Ch_{K}|\usigma|_{1, K}
\qquad\forall\usigma\in\uH^1(K).
\label{inequ:L2 norm for local estimation in H(rot)}$$ \[thm:L2 norm for interpolation estimation in H(rot)\]
We postpone the proof of the theorem after a technical lemma. Define a new interpolation operator $\sqcap_{Q}: \uH^1(K)\to \undertilde{\rm{span}}\{1,\xi,\eta,\xi^{2}-\eta^{2}\}$ by $\dashint_{e_{i}}\usigma\ \mathrm{d}s=
\dashint_{e_{i}}\sqcap_{Q}\usigma\ \mathrm{d}s, i=1:4$. Evidently $\sqcap_{Q}$ is well-defined.
The local interpolation operator $\sqcap_{K}$ is $H^{1}$ stable, namely $$|\sqcap_{K}\usigma|_{1, K}\leqslant C|\usigma|_{1, K}
\qquad\forall \usigma\in\uH^1(K).$$ \[lemma:H\^[1]{} stable on interpolation associated with QRT\]
Let $
\sqcap_{Q}\usigma=
\undertilde{d}{}_{1}\cdot 1+
\undertilde{d}{}_{2}\cdot\xi+
\undertilde{d}{}_{3}\cdot\eta+
\undertilde{d}{}_{4}\cdot(\xi^{2}-\eta^{2})
$, then by definition we have $$\begin{aligned}
\undertilde{d}{}_{1}&=
\frac{\alpha^{2}-\beta^{2}+2}{8}P_{e_{1}}(\usigma)+
\frac{-\alpha^{2}+\beta^{2}+2}{8}P_{e_{2}}(\usigma)+
\frac{\alpha^{2}-\beta^{2}+2}{8}P_{e_{3}}(\usigma)+
\frac{-\alpha^{2}+\beta^{2}+2}{8}P_{e_{4}}(\usigma)\\
\undertilde{d}{}_{2}&=
-\frac{\beta}{4}P_{e_{1}}(\usigma)+
\frac{\beta-2}{4}P_{e_{2}}(\usigma)-
\frac{\beta}{4}P_{e_{3}}(\usigma)+
\frac{\beta+2}{4}P_{e_{4}}(\usigma)\\
\undertilde{d}{}_{3}&=
\frac{\alpha+2}{4}P_{e_{1}}(\usigma)-
\frac{\alpha}{4}P_{e_{2}}(\usigma)+
\frac{\alpha-2}{4}P_{e_{3}}(\usigma)-
\frac{\alpha}{4}P_{e_{4}}(\usigma)\\
\undertilde{d}{}_{4}&=
-\frac{3}{8}P_{e_{1}}(\usigma)+
\frac{3}{8}P_{e_{2}}(\usigma)-
\frac{3}{8}P_{e_{3}}(\usigma)+
\frac{3}{8}P_{e_{4}}(\usigma).
\end{aligned}$$
Denote by row vector $\usigma=(\sigma_{1},\sigma_{2})$, $\undertilde{p}{}_{1}=(P_{e_{i}}(\sigma_{1}))_{i=1:4}$, $\undertilde{p}{}_{2}=(P_{e_{i}}(\sigma_{2}))_{i=1:4}$, and column vector $\undertilde{p}=
(\undertilde{p}{}_{1},\undertilde{p}{}_{2})^{T}$. Denote $\undertilde{=}$ by $=$ up to a constant, independent of diameter $h_{K}$, then $$\begin{aligned}
\int_{K}|\nabla(\sqcap_{Q}\usigma)|^{2}\ \mathrm{d}x
&\undertilde{=}
h_{K}^{-2}\int_{K}
|\partial_{\xi}(\sqcap_{Q}\usigma)|^{2}+
|\partial_{\eta}(\sqcap_{Q}\usigma)|^{2}
\ \mathrm{d}x
=h_{K}^{-2}\int_{K}|\undertilde{d}{}_{2}+2\undertilde{d}{}_{4}\xi|^{2}+|\undertilde{d}{}_{3}-2\undertilde{d}{}_{4}\eta|^{2}\ \mathrm{d}x \\
&\undertilde{=}
|\undertilde{d}{}_{2}|^{2}+
|\undertilde{d}{}_{3}|^{2}+
|\undertilde{d}{}_{4}|^{2}
=\undertilde{p}{}_{1}B_{1}^{T}B_{1}\undertilde{p}{}_{1}^{T}+
\undertilde{p}{}_{2}B_{1}^{T}B_{1}\undertilde{p}{}_{2}^{T}
=\undertilde{p}^{T}B\undertilde{p}.
\end{aligned}$$ Here $\undertilde{p}\in \mathbb{R}^{8}$ and
$$B_{1}=
\begin{bmatrix}
-\frac{\beta}{4} & -\frac{\beta-2}{4} & -\frac{\beta}{4} & \frac{\beta+2}{4} \\
\frac{\alpha+2}{4} & -\frac{\alpha}{4} & \frac{\alpha-2}{4} & -\frac{\alpha}{4} \\
-\frac{3}{8} & \frac{3}{8} & -\frac{3}{8} & \frac{3}{8} \\
\end{bmatrix},
\quad
B=
\begin{bmatrix}
B_{1}^{T}B_{1} & \Large{0} \\
\Large{0} & B_{1}^{T}B_{1} \\
\end{bmatrix}.$$
Similarly, let $\sqcap_{K}\usigma=
g_{1}\cdot\nabla\xi+
g_{2}\cdot\nabla\eta+
g_{3}\cdot\xi\nabla\eta+
g_{4}\cdot\eta\nabla\xi$, recalling $\{g_i\}_{i=1:4}$ in Theorem \[thm:commutativity on a cell\], then there exists a semi-positive matrix $D$ with $\mathcal{O}(1)$ elements such that $
|\sqcap_{K}\undertilde{\sigma}|_{1, K}^{2}=
\undertilde{p}^{T}D\undertilde{p}
$.
Since $\sqcap_{Q}$ is $H^{1}$ stable (see [@zhang2016stable]), it remains to show that $
|\sqcap_{K}\undertilde{\sigma}|_{1, K}
\leqslant
C
|\sqcap_{Q}\undertilde{\sigma}|_{1, K}
$.
We first show that $\ker B\subset\ \ker D$. Given a $\undertilde{p} \in \ker B$, then $|\sqcap_{Q}\usigma|_{1, K}^{2}=0$ and $\sqcap_{Q}\usigma$ is a constant vector. Since $\sqcap_{K}\usigma=\sqcap_{K}(\sqcap_{Q}\usigma)$, thus $\sqcap_{K}\usigma$ is also a constant vector and $|\sqcap_{K}\usigma|_{1, K}^{2}=0$, i.e. $\undertilde{p}\in \ker D$.
Subsequently, we calculate all the eigenvalues of $B^{T}_{1}B_{1}$, (below we use $\lambda_{i}=\lambda_{i}(B_{1}^{T}B_{1})$, $i=1:4$) $$\begin{aligned}
\lambda_{1}&=0
&
\lambda_{3}&=
\frac{17}{32}+\frac{\alpha^{2}+\beta^{2}}{8}-
\frac
{
\sqrt{
16(\alpha^{4}+\beta^{4})+
32\alpha^{2}\beta^{2}+
136(\alpha^{2}+\beta^{2})+1
}
}
{32} \\
\lambda_{2}&=\frac{1}{2}
&
\lambda_{4}&=
{
}
}
\end{aligned}$$ The eigenvalues $\lambda(B)=
\{\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}\}$ of matrix $B$ are all double eigenvalues. Let $\{\undertilde{\nu}{}_{1}$, $\undertilde{\nu}{}_{2}\}$ be two eigenvectors subordinating to the eigenvalue $\lambda_{1}=0$. Then we can decompose $
\mathbb{R}^8=
\mathrm{span}
\{\undertilde{\nu}{}_{1}, \undertilde{\nu}{}_{2}\}
\oplus
(\mathrm{span}
\{\undertilde{\nu}{}_{1}, \undertilde{\nu}{}_{2}\})^{\bot}
$. Suppose $
\undertilde{p}=
\undertilde{\psi}{}_{1}+
\undertilde{\psi}{}_{2}
$, $\undertilde{\psi}{}_{1}
\in
\mathrm{span}\{\undertilde{\nu}{}_{1}
$, $\undertilde{\nu}{}_{2}\}$, $\undertilde{\psi}{}_{2}
\in
(\mathrm{span}\{\undertilde{\nu}{}_{1}
,\undertilde{\nu}{}_{2}\})^{\bot}$. Rayleigh quotient theorem reads $$\begin{aligned}
|\sqcap_{K}\usigma|_{1, K}^{2}
&=
\up^{T}D\up=\upsi{}_{2}^{T}D\upsi{}_{2}\leqslant
\lambda_{max}(D)|\upsi{}_{2}|^{2} \\
|\sqcap_{Q}\usigma|_{1, K}^{2}
& \undertilde{=}
\up^{T}B\up=\upsi{}_{2}^{T}B\upsi{}_{2}\geqslant
\text{min}
\{\lambda_{2}, \lambda_{3}, \lambda_{4}\}|\upsi{}_{2}|^{2}.
\end{aligned}$$ This finishes the proof.
#### **Proof of Theorem \[thm:L2 norm for interpolation estimation in H(rot)\]**
Dividing a quadrilateral $K$ along the diameter into two triangles, we have the following estimation from the similar argument referred to Theorem $5.1$ in [@carstensen2012explicit]. $$\| \usigma-\sqcap_{K} \usigma\|_{0, K}
\leqslant Ch_{K}\|\nabla(\usigma-\sqcap_{K}\usigma)\|_{0, K}.$$ This proves by Lemma \[lemma:H\^[1]{} stable on interpolation associated with QRT\].
Let $K$ be a convex quadrilateral, then it holds $$\| q-P_{K}q\|_{0, K}
\leqslant
Ch_{K}|q|_{1, K}
\quad
\forall
q\in H^1(K).
$$
The proof can be found in [@payne1960optimal].
### Interpolation error estimations in energy norms {#subsubsec:interpolation error in energy norms}
Let $K$ be a convex quadrilateral, then it holds $$|u-J_{K}u|_{1,K}
\leqslant
Ch_{K}|u|_{2, K}
\quad
\forall u\in H^2(K),
$$ and $$|\usigma-\sqcap_{K}\usigma|_{\mathrm{rot},K}
\leqslant Ch_{K}|\mathrm{rot}\usigma|_{1,K}
\quad\forall\usigma\in H^1({\rm rot},K).
$$ \[thm:inequ:energy norm for interpolation estimation in H1\]
We prove the estimations by the commutative diagrams. Since $\nabla (J_K)=\sqcap_{K}(\nabla)$ on $H^2(K)$, we have $$|u-J_Ku|_{1,K} =
\|\nabla u -\sqcap_{K}(\nabla u)\|_{0,K}
\leqslant
Ch_K|u|_{2,K}.$$ Similarly, since ${\rm rot}\sqcap_{K}=P_K{\rm rot}$ on $\uH^1(K)$, we have $$\|\mathrm{rot} \usigma-\mathrm{rot}(\sqcap_{K}\usigma)\|_{0,K}=
\|\mathrm{rot} \usigma-P_K(\mathrm{rot}\usigma)\|_{0,K}\leqslant
Ch_K|\mathrm{rot} \usigma|_{1,K}.$$ The proof is completed.
Finite element spaces on a grid $\mathcal{J}_h$ {#subsec:FE sapces on a grid}
-----------------------------------------------
Associated with the ${\rm QBL}$ element, define the finite element spaces $V_{h}^{\rm QBL}$ and $V_{h0}^{\rm QBL}$ by $$V_{h}^{\rm QBL}
\triangleq
\{
v_{h}\in L^{2}(\Omega):
v_{h}|_{K}\in P_{K}^{\rm QBL},
v_{h}\ \text{is continuous at two endpoints of edge}\ e\in\mathcal{E}^{i}_{h}
\},$$ $$\mbox{and}\ \ \
V_{h0}^{\rm QBL}
\triangleq
\{
v_{h}\in V_{h}^{\rm QBL}:
v_{h}=0\ \text{at two endpoints of edge}\ e \in\mathcal{E}^{b}_{h}
\}.$$
Associated with the ${\rm QRT}$ element, define the finite element spaces $V_{h}^{\rm QRT}$ and $V_{h0}^{\rm QRT}$ by $$V_{h}^{\rm QRT}
\triangleq
\{
\utau{}_{h}
\in \uL^2(\Omega):
\utau{}_{h}|_{K}
\in P_{K}^{\rm QRT},
\dashint_{e}
\utau{}_{h}
\cdot
\ut{}_{e}
\ \mathrm{d}s
\ \mbox{is continuous at the edge}\ e\in\mathcal{E}^{i}_{h}
\},$$ \[def:P\_h\^rot,Omega\] $$\mbox{and}\ \ \ V_{h0}^{\rm QRT}
\triangleq
\{
\undertilde{\tau}{}_{h}
\in V_{h}^{\rm QRT}:
\dashint_{e}
\undertilde{\tau}{}_{h}\cdot\undertilde{t}{}_{e}
\ \mathrm{d}s=0
\ \text{at the edge}\ e \in\mathcal{E}^{b}_{h}
\}.$$
Define the piecewise constant finite element spaces $W_h$ and $W_{h0}$ by $$W_h\triangleq
\{q_h\in L^2(\Omega):q_h|_K=P_Kq, q\in L^2(\Omega)\},
\quad \mbox{and}\ \ W_{h0}\triangleq
\{q_h\in W_h:\int_{\Omega}q_h\ \mathrm{d}x = 0\}.$$
The properties on a single cell can be generalized on a grid. We firstly adopt a lemma below.
It holds on the subdivision that $$\label{eq:infsup1}
\sup_{\utau{}_h\in V^{\rm QRT}_h}(\rot_h\utau{}_h,q_h)\geqslant C\|\utau{}_h\|_{\rot,h}\|q_h\|_{0,\Omega},\ \ \mbox{for\ any}\ q_h\in W_h,$$ and $$\label{eq:infsup2}
\sup_{\utau{}_h\in V^{\rm QRT}_{h0}}(\rot_h\utau{}_h,q_h)\geqslant C\|\utau{}_h\|_{\rot,h}\|q_h\|_{0,\Omega},\ \ \mbox{for\ any}\ q_h\in W_{h0}.$$
Given $q_h\in W_h$, there exists a $\utau\in \uH{}^1(\Omega)$, such that $\rot \utau=q_h$, and $\|\utau\|_{1,\Omega}\leqslant C\|q_h\|_{0,\Omega}$. Set $\utau{}_h:=\sqcap_h\utau$, then $\rot_h\utau{}_h=q_h$, and $\|\utau{}_h\|_{\rot,h}\leqslant C\|\utau\|_{1,\Omega}$. This proves . Similarly is proved.
The commutative diagrams hold as below: $$\begin{array}{ccccccccc}
\mathbb{R} & ~~~\longrightarrow~~~ & H^2(\Omega) & ~~~\xrightarrow{{\boldsymbol}{{\mathrm}{\nabla}}}~~~ & \uH^1(\Omega) & ~~~\xrightarrow{{\mathrm}{rot}}~~~ & L^2(\Omega) & ~~~\xrightarrow{\int_\Omega\cdot}~~~ & \mathbb{R} \\
& & \downarrow J_h & & \downarrow \sqcap_{h} & & \downarrow P_h & & \\
\mathbb{R} & \longrightarrow & V^{\rm QBL}_h & \xrightarrow{{\boldsymbol}{{\mathrm}{\nabla_h}}} & V^{\rm QRT}_h & \xrightarrow{{\mathrm}{rot_h}} & W_h & \xrightarrow{\int_\Omega\cdot} & \mathbb{R},
\end{array}
\label{eq:ddrc1}$$ and $$\begin{array}{ccccccccc}
\{0\} & ~~~\longrightarrow~~~ & H^2(\Omega)\cap H^1_0(\Omega) & ~~~\xrightarrow{{\boldsymbol}{{\mathrm}{\nabla}}}~~~ & \uH^1(\Omega)\cap H_0(\rot,\Omega) & ~~~\xrightarrow{{\mathrm}{rot}}~~~ & L_0^2(\Omega) & ~~~\xrightarrow{\int_\Omega\cdot}~~~ & \{0\} \\
& & \downarrow J_h & & \downarrow \sqcap_{h} & & \downarrow P_h & & \\
\{0\} & \longrightarrow & V^{\rm QBL}_{h0} & \xrightarrow{{\boldsymbol}{{\mathrm}{\nabla_h}}} & V^{\rm QRT}_{h0} & \xrightarrow{{\mathrm}{rot_h}} & W_{h0} & \xrightarrow{\int_\Omega\cdot} & \{0\}.
\end{array}
\label{eq:ddrc2}$$
We first consider . The commutativity is trivial by Theorem \[thm:commutativity on a cell\] and it remains us to verify the discretized de Rham complex by the standard dimension counting technique.
Evidently ${\rm ker}(\nabla_h)=\mathbb{R}$ and $\nabla_h V^{\rm QBL}_h\subset V_{h}^{\rm QRT}$. On the other hand, by , $\rot_h V_{h}^{\rm QRT}=W_h$. This way, follows by noting that $\dim(V^{\rm QBL}_h)=\#(\mathcal{N}_h)$ and $\dim(V_{h}^{\rm QRT})=\#(\mathcal{E}_h)$, and that $\dim(V_{h}^{\rm QRT})=\dim(V^{\rm QBL}_h)+\dim(W_h)-1$ by the Euler formula.
Similarly is proved. The proof is completed.
The error estimation of the global interpolator is the same as that of the respective local ones.
There exists a constant $C$ depending on the shape regularity of $\mathcal{J}_h$ only, such that
1. $\displaystyle\|u-J_{h}u\|_{0,\Omega}+h|u-J_hu|_{1,h}
\leqslant
Ch^2|u|_{2,\Omega}
\quad\forall u\in H^2(\Omega);$
2. $\displaystyle \|\usigma-\sqcap_{h}\usigma\|_{{\rm rot},h}
\leqslant
Ch(|\usigma|_{1,\Omega}+|{\rm rot}\usigma|_{1,\Omega})
\quad\forall \usigma\in H^1({\rm rot},\Omega);$
3. $ \displaystyle\| q-P_{h}q\|_{0,\Omega}
\leqslant
Ch|q|_{1,\Omega}
\quad\forall q\in H^1(\Omega).$
Nonconforming finite element spaces and their modulus of continuity {#sec:NFEs and mc}
===================================================================
In this section, we show that on a grid that consists of arbitrary quadrilaterals and satisfies the condition that the cells are asymptotically parallelograms, the spaces $V^{\rm QBL}_h$ and $V^{\rm QRT}_h$, though not subspaces of $H^1$ and $H(\rot)$ respectively, the consistency can be controlled well. We begin with an analysis that $V^{\rm QBL}_h$ is in general not continuous.
Continuity and non-continuity of $V_{h0}^{\rm QBL}$ {#subsec:cnc of QBL}
---------------------------------------------------
Let $\mathcal{G}_D$ be a patch with the center $D$ and four cells $K_{1},K_{2},K_{3},K_{4}$, see Figure \[fig:patch for QBL FE space\]. Let $V_{h0}^{\rm QBL}(\mathcal{G}_D)$ be ${\rm QBL}$ finite element space defined on $\mathcal{G}_D$ with zero boundary condition. Denote by local shape parameters $\alpha_i,\beta_i$ of $K_i$, $i=1:4$. Here $\alpha_i=\beta_i=0$ for $i=2:4$.
(-1.5,-1.25)–(0,-2); (0,-2)–(2,-2); (-1.5,0)–(0.5,0); (0.5, 0)–(2.5, 0); (-1, 2)–(1,2); (1,2)–(3, 2);
(-1.5,-1.25)–(-1.5,-0); (-1.5,-0)–(-1, 2); (0,-2)–(0.5,0); (0.5, 0)–(1, 2); (2,-2)–(2.5,-0); (2.5, -0)–(3, 2);
(0.38,0.17)node[$D$]{}; (-0.6,-1)node[$K_{1}$]{}; (1.15,-1)node[$K_{2}$]{}; (-0.35,1)node[$K_{4}$]{}; (1.65,1)node[$K_{3}$]{}; (0.5,0) circle(1.5pt);
For $\alpha_1,\beta_1\neq 0$ and $v_h\neq 0\in V_{h0}^{\rm QBL}(\mathcal{G}_D)$, there exists a function $\varphi\in C^{\infty}(\mathcal{G}_D)$ such that $$(\nabla\varphi,\nabla_{h} v_h)+(\Delta \varphi,v_h)\neq 0$$
Since $\alpha_i,\beta_i=0$ for $i=2:4$, then $$(\nabla \varphi,\nabla_h v_h)+(\Delta \varphi,v_h)=
\int_{\partial K_1}\frac{\partial \varphi}{\partial n}(v_h-q)\ \mathrm{d}s.$$ Here $q$ is the linear interpolation of $v_h$ on $e$ with respect to endpoints. Without loss of generality, we assume $\alpha_1\neq 0$. Noticing that $\dim V_{h0}^{\rm QBL}(\mathcal{G}_D)=1$, we denote by $\phi_{D}$ the basis of $V_{h0}^{\rm QBL}(\mathcal{G}_D)$ and $\phi_{D}|_{K_1}=\phi_{1}$ (see ), then $v_h|_{K_1}=v_h(D)\phi_{1}$. Let $\varphi$ be a linear polynomial whose gradient is $\ur{}_{K_1}/\ur{}_{K_1}\times\us{}_{K_1}$, defined on the patch $\mathcal{G}_D$. Then the proof is completed by simple calculation.
Modulus of continuity of $V_{h0}^{\rm QBL}$ {#subsec:mc of QBL}
-------------------------------------------
Define consistency functional $E(\uzeta,v_{h})$ by $$\begin{aligned}
E(\uzeta, v_{h})&=(\uzeta,\nabla_{h}v_{h})+(\dv\uzeta,v_{h})
\quad
\mbox{for}\
\uzeta\in \uH^{1}(\Omega),
v_{h}\in V_{h0}^{\rm QBL} \\
E(\uzeta, v_{h})&=(\uzeta,\nabla_{h}v_{h})+(\dv\uzeta,v_{h})
\quad
\mbox{for}\
\uzeta\in \uH{}_{0}^{1}(\Omega),
v_{h}\in V_{h}^{\rm QBL} \cap L^2_0(\Omega).\end{aligned}$$
Let $\{\phi_{i}\}_{i=1:4}$ be local basis of $P_{K}^{\rm QBL}$, then define by $$\iota_{1}=\sum_{i=1}^{4}\phi_{i}^{(1)}v_{h}(A_{i})\quad
\iota_{2}=\sum_{i=1}^{4}\phi_{i}^{(2)}v_{h}(A_{i})\quad
\iota_{3}=\sum_{i=1}^{4}\phi_{i}^{(3)}v_{h}(A_{i}).$$
For $\uzeta\in \uH^{1}(\Omega)$, $v_{h}\in V_{h0}^{\rm QBL}$ or $\uzeta\in \uH{}_{0}^{1}(\Omega)$, $v_{h}\in V_{h}^{\rm QBL}$, it holds $$E(\uzeta,v_{h})
\leqslant
Ch\|\uzeta\|_{1,\Omega}|v_{h}|_{1,h}.$$ \[thm:global estimate of E(u,v\_h)\]
Evidently the consistency functional can be decomposed into $$E(\uzeta, v_{h})=
\sum_{e\in\mathcal{E}_{h}}
\int_{e}\uzeta\cdot\un{}_{e}[\![v_{h}-q]\!]_{e}\ \mathrm{d}s=
\sum_{K\in\mathcal{J}_{h}}\sum_{e\subset\partial K}
\int_{e}\uzeta\cdot\un{}_{e}(v_{h}-q)\ \mathrm{d}s.$$ Here $q$ is the linear interpolation of $v_h$ on $e$ with respect to endpoints. Then by direct calculation, we have $$\begin{aligned}
\int_{\partial K}\uzeta\cdot\un(v_{h}-q)\ \mathrm{d}s
&=
\beta_K\iota_{1}\int_{e_{1}}\uzeta\cdot\un{}_{1}
(\frac{\xi^2}{1+\alpha_K}-(1+\alpha_K))\ \mathrm{d}s+
\alpha_K\iota_{1}\int_{e_{2}}\uzeta\cdot\un{}_{2}
(\frac{\eta^2}{-1+\beta_K}-(-1+\beta_K))\ \mathrm{d}s \\
&+
\beta_K\iota_{1}\int_{e_{3}}\uzeta\cdot\un{}_{3}
(\frac{\xi^2}{-1+\alpha_K}-(-1+\alpha_K))\ \mathrm{d}s+
\alpha_K\iota_{1}\int_{e_{4}}\uzeta\cdot\un{}_{4}
(\frac{\eta^2}{1+\beta_K}-(1+\beta_K))\ \mathrm{d}s \\
&\leqslant
Ch\iota_{1}\|\uzeta\|_{1,K}
\label{equ:consistency functional}.
\end{aligned}$$ On the other hand, setting $
\undertilde{\iota}=
(\iota_{1},\iota_{2},\iota_{3})^{T}
$ and $|v_{h}|^{2}_{1, K}=\undertilde{\iota}^{T}G\undertilde{\iota}$, then $$G=
\begin{bmatrix}
\frac
{4(1+\beta_K^{2}) \undertilde{s}{}_K\cdot\undertilde{s}{}_{K}
-8\alpha_K\beta_K \undertilde{r}{}_{K}\cdot\undertilde{s}{}_{K}
+4(1+\alpha_K^{2})\undertilde{r}{}_{K}\cdot\undertilde{r}{}_{K}}
{3 \undertilde{r}{}_{K}\times\undertilde{s}{}_{K}} &
\frac
{4\alpha_K \undertilde{s}{}_{K}\cdot\undertilde{s}{}_{K}
-4\beta_K \undertilde{r}{}_{K}\cdot\undertilde{s}{}_{K}}
{3 \undertilde{r}{}_{K}\times\undertilde{s}{}_{K}} &
\frac
{-4\alpha_K \undertilde{r}{}_{K}\cdot\undertilde{s}{}_{K}
+4\beta_K \undertilde{r}{}_{K}\cdot\undertilde{r}{}_{K}}
{4\undertilde{s}{}_{K}\cdot\undertilde{s}{}_{K}}
{ \undertilde{r}{}_{K}\times\undertilde{s}{}_{K}} &
\frac
{-4\undertilde{r}{}_{K}\cdot\undertilde{s}{}_{K}}
{ \undertilde{r}{}_{K}\times\undertilde{s}{}_{K}} \\
{4 \undertilde{r}{}_{K}\cdot\undertilde{r}{}_{K}}
{ \undertilde{r}{}_{K}\times\undertilde{s}{}_{K}} \\
\end{bmatrix}.$$ By the generalized Rayleigh quotient theorem, then for $
F=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
$ and all $\undertilde{\iota}\in R^3$ $$\undertilde{\iota}^{T}F\undertilde{\iota}
\leqslant
\lambda_{F,G}\undertilde{\iota}^{T}G\undertilde{\iota},
\quad
\lambda_{F,G}=
\frac
{9\ur{}_K\times\us{}_K}
{4((3-\alpha_K^2+3\beta_K^2)\us{}_{k}\cdot\us{}_{k}
-4\alpha_K\beta_K\ur{}_{K}\cdot\us{}_{K}
+(3+3\alpha_K^2-\beta_K^2)\ur{}_{K}\cdot\ur{}_{K})}.$$
In summary, we have $$E(\uzeta,v_h)\leqslant
Ch\sum_{K\in\mathcal{J}{}_{h}}\|\uzeta\|_{1,K}|v_h|_{1,K}
\leqslant Ch\|\uzeta\|_{1,\Omega}|v_h|_{1,h}.$$ The proof is completed.
Modulus of continuity of $V_{h0}^{\rm QRT}$ {#subsec:mc of QRT}
-------------------------------------------
Define consistency functional $E(w,\undertilde{\tau}{}_{h})$ by $$\begin{aligned}
E(w,\undertilde{\tau}{}_{h})&=
(w,\mathrm{rot}_{h}\undertilde{\tau}{}_{h})-
(\undertilde{\mathrm{curl}}\ w,\undertilde{\tau}{}_{h})
\quad
\mbox{for}\
w\in H^{1}(\Omega),
\forall
\undertilde{\tau}{}_{h}\in V_{h0}^{\rm QRT} \\
w\in H^{1}_0(\Omega),
\forall
\undertilde{\tau}{}_{h}\in V_{h}^{\rm QRT}.\end{aligned}$$ Evidently the consistency functional can be decomposed into $$E(w,\undertilde{\tau}{}_{h})=
\sum_{K\in\mathcal{J}{}_{h}}\sum_{e\subset\partial K}
\int_{e}
(w-c_K)
(\undertilde{\tau}{}_{h}
\cdot
\ut{}_{e}-
P_{e}
\label{equ:consistency functional decomposition}$$ Here $c_K$ is an arbitrary constant.
For $w\in H^{1}(\Omega)$, $\utau{}_{h}
\in V_{h0}^{\rm QRT}$ or $w\in H_0^{1}(\Omega)$, $\utau{}_{h}
\in V_{h}^{\rm QRT}$, it holds $$E(w,\utau{}_{h})\leqslant Ch
|w|_{1,\Omega}\|\utau{}_{h}\|_{{\rm rot},h}.
\label{inequ: rot consistency error}$$ \[thm: rot consistency error\]
Evidently, we have $\inf_{c_{K}\in \mathbb{R}}\|w-c_K\|_{0,\partial K}^{2}\leqslant Ch_{K}|w|_{1,K}^{2}$. Let $
\utau{}_{h}=
\gamma_{1}\nabla\xi+
\gamma_{2}\nabla\eta+
\gamma_{3}\hat{\xi}\nabla\eta+
\gamma_{4}\hat{\eta}\nabla\xi
$ and $\|\utau{}_{h}\|_{{\rm rot},K}^2=\ugamma^T V\ugamma$, with $\ugamma=(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})^T$ and $$V=
\begin{bmatrix}
\frac
{4 \us{}_K\cdot\us{}_K}
{ \ur{}_K\times\us{}_K} &
\frac
{-4\ur{}_K\cdot\us{}_K}
{ \ur{}_K\times\us{}_K} &
0 &
0 \\
\frac
{-4\ur{}_K\cdot\us{}_K}
{ \ur{}_K\times\us{}_K} &
\frac
{4 \ur{}_K\cdot\ur{}_K}
{ \ur{}_K\times\us{}_K} &
0 &
0 \\
0 &
0 &
\frac
{4(3+3\alpha_K^2-\beta_K^2)\ur{}_K\cdot\ur{}_{K}+36}
{9 \ur{}_K\times\us{}_K} &
\frac
{-8\alpha_K\beta_K \ur{}_K\cdot\us{}_K-36}
{9 \ur{}_K\times\us{}_K} \\
0 &
0 &
\frac
{-8\alpha_K\beta_K \ur{}_K\cdot\us{}_K-36}
{9 \ur{}_K\times\us{}_K} &
\frac
{4(3+3\beta_K^2-\alpha_K^2)\us{}_K\cdot\us{}_K+36}
{9 \ur{}_K\times\us{}_K}
\end{bmatrix}.
\label{matrix:V}$$
On the other hand $$\|
\utau{}_{h}
\|_{0,e}^{2}
=
\int_{e}
\ut{}_{e}))^2
\ \mathrm{d}s
=
\ \mathrm{d}s,$$ then by simple calculation $$\begin{aligned}
&\int_{e_{1}}
\undertilde{\tau}{}_{h}
\cdot
\undertilde{t}{}_{1}
(\undertilde{\tau}{}_{h}
\cdot
\undertilde{t}{}_{1}-
P_{e_{1}}
(\undertilde{\tau}{}_{h}
\cdot
\undertilde{t}{}_{1}))
\ \mathrm{d}s
\\
=
&\int_{e_{1}}
(
\gamma_{3}\nabla\eta\cdot
\undertilde{t}{}_{1}\xi+
\gamma_{4}\nabla\xi\cdot
\undertilde{t}{}_{1}\eta
)
\cdot
(
\gamma_{3}\nabla\eta\cdot
\undertilde{t}{}_{1}
(\xi-P_{e_{1}}(\xi))+
\gamma_{4}\nabla\xi \cdot
\undertilde{t}{}_{1}
(\eta-P_{e_{1}}(\eta))
)
\ \mathrm{d}s \\
=
&\int_{e_{1}}
(\gamma_{3}\nabla\eta\cdot
\undertilde{t}{}_{1})^{2}\xi^{2}+
2(\gamma_{3}\nabla\eta\cdot
\undertilde{t}{}_{1})
(\gamma_{4}\nabla\xi\cdot
\undertilde{t}{}_{1})\xi\eta+
(\gamma_{4}\nabla\xi\cdot
\undertilde{t}{}_{1})^{2}\eta(\eta-1)
\ \mathrm{d}s =
\frac{4(1+\alpha_K)^{2}\beta_K^{2}}{3|e_{1}|}
\undertilde{\gamma}^{T}U\undertilde{\gamma}.
\end{aligned}$$ Here $U=
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
\end{bmatrix}$. Similarly $$\begin{aligned}
&
\|\utau{}_{h}\cdot\ut{}_{2}-
P_{e_{2}}({\utau}{}_{h}\cdot\ut{}_{2})\|_{0,e_{2}}^{2}
=
\frac{4\alpha_K^{2}(-1+\beta_K)^{2}}{3|e_{2}|}
\ugamma^{T}U\ugamma \\
&\|
\utau{}_{h}\cdot\ut{}_{3}-
P_{e_{3}}(\utau{}_{h}\cdot\ut{}_{3})\|_{0,e_{3}}^{2}
=
\frac{4(-1+\alpha_K)^{2}\beta_K^{2}}{3|e_{3}|}
\ugamma^{T}U\ugamma \\
&\|
\utau{}_{h}\cdot\ut{}_{4}-
P_{e_{4}}(\utau{}_{h}\cdot\ut{}_{4})\|_{0,e_{4}}^{2}
=
\frac{4\alpha_K^{2}(1+\beta_K)^{2}}{3|e_{4}|}
\ugamma^{T}U\ugamma.
\end{aligned}$$ By the generalized Rayleigh quotient theorem, then for all $\ugamma \in \mathbb{R}^4$ $$\|
\utau{}_{h}\cdot\ut{}_{e}-
P_{e}(\utau{}_{h}\cdot\ut{}_{e})\|_{0,\partial {K}}^{2}
\leqslant
C\lambda_{U,V}h_K\|\utau{}_h\|_{{\rm rot},K}^2.$$ Here $\lambda_{U,V}=p(\alpha_K,\beta_K,\ur{}_K,\us{}_K)/q(\alpha_K,\beta_K,\ur{}_K,\us{}_K)$ with $$p(\alpha_K,\beta_K,\ur{}_K,\us{}_K) =
9((3+3\alpha_K^{2}-\beta_K^2)\ur{}_K\cdot\ur{}_K+
4\alpha_K\beta_K \ur{}_K\cdot\us{}_K+
(3+3\beta_K^{2}-\alpha_K^2)\us{}_K\cdot\us{}_K+
36)\ur{}_K\times\us{}_K$$ and $$\begin{gathered}
q(\alpha_K,\beta_K,\ur{}_K,\us{}_K) =
4((3+3\alpha_K^{2}-\beta_K^2)(3+3\beta_K^{2}-\alpha_K^2)
(\ur{}_K\cdot\ur{}_K)(\us{}_K\cdot\us{}_K)
-4\alpha_K^{2}\beta_K^{2}(\ur{}_K\cdot\us{}_K)^2
\\+(27-9\alpha_K^2+27\beta_K^2)\us{}_K\cdot\us{}_K-36\alpha_K\beta_K\ur{}_K\cdot\us{}_K
+(27+27\alpha_K^2-9\beta_K^2)\ur{}_K\cdot\ur{}_K).
\end{gathered}$$ follows from the Cauchy-Schwarz inequality, the proof is completed.
Finite element schemes for respective model problems {#sec:FE schemes for model problems}
====================================================
A finite element scheme for the Poisson equation {#subsec:a FE scheme for the Poisson equation}
------------------------------------------------
We consider the Poisson problem with homogeneous boundary condition $$\left\{
\begin{aligned}
-\Delta u & =f & \quad & \mbox{in} \ \Omega \\
u & =0 & \quad & \mbox{on} \ \Gamma
\end{aligned}
\right.
\label{equ:Poisson problem in numerical experiment}$$
The variational formulation is to find $u\in H^1_0(\Omega)$, such that $$\int_\Omega \nabla u\nabla v \ \mathrm{d}x=\int_\Omega fv \ \mathrm{d}x,\quad\forall\,v\in H^1_0(\Omega).
\label{equ:variational problem on Poisson }$$ The finite element problem is to find $u_h\in V_{h0}^{\rm QBL}$, such that $$\sum_{K\in\mathcal{J}_h}\int_K\nabla u_h\nabla v_h\ \mathrm{d}x=\int_\Omega fv_h \ \mathrm{d}x,\quad\forall\,v_h\in V_{h0}^{\rm QBL}.
\label{equ:discrete problem on Poisson}$$
Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ and $u_h$ be the solutions of , and , respectively. Then $$|u-u_h|_{1,h}\leqslant Ch\|u\|_{2,\Omega}.$$
The theorem is proved by the standard technique.
### Numerical verification
We choose the computational domain to be the quadrilateral with vertexes $(0,0)$, $(1,0)$, $(2,2)$, $(-1,1)$. The data $f$ is chosen such that the exact solution is the polynomial $u = y(x+y)(x-3y+4)(2x-y-2)$. We subdivide the domain with quadrilateral grids and triangle grids, respectively, and numerical solutions are computed on both grids. To generate quadrilateral grids, we use bisection strategy. To generate triangle grids, we firstly subdivide the domain with quadrilateral grids, then bisect all of them each to two triangles, see Figure \[fig:Poissonshowmesh\]. We first test the performance of the ${\rm QBL}$ element on the quadrilateral grids, then we test Courant element on the triangle grids as a comparison. The results are recorded in Table \[tab:tab6\].
\[fig:Poissonshowmesh\]
------------------------- ------------------- -------------------------- -- ------------------- --------------------------
Size($\mathcal{J}_{h}$)
$|u-u_{h}|_{1,h}$ $\|u-u_{h}\|_{0,\Omega}$ $|u-u_{h}|_{1,h}$ $\|u-u_{h}\|_{0,\Omega}$
$8\times8$ 1.67E0 1.39E-1 3.59E0 2.57E-1
$16\times16$ 8.35E-1 3.52E-2 1.83E0 6.73E-2
$32\times32$ 4.18E-1 9.69E-3 9.23E-1 1.70E-2
$64\times64$ 2.09E-1 2.42E-3 4.62E-1 4.57E-3
Convergence order 1 2 1 2
------------------------- ------------------- -------------------------- -- ------------------- --------------------------
: Numerical results for Poisson problem[]{data-label="tab:tab6"}
Figure \[fig:convergence order on Poisson problem\] reports on approximation results of ${\rm QBL}$ and Courant elements for Poisson equation. The $x$-axis and the $y$-axis represent the logarithm of grid size $h$ and of the error, respectively. The dashed line and the solid line represent the error associated with the norm $|\cdot|_{1,h}$ and $\|\cdot\|_{0,\Omega}$, respectively. The results confirm our conclusion: a clear first-order of convergence is observed with $|\cdot|_{1,h}$.
![The log-log plot of the error of ${\rm QBL}$ and Courant elements for Poisson equation[]{data-label="fig:convergence order on Poisson problem"}](Poissonconverorder.pdf){width="1\linewidth"}
Application on Laplace eigenvalue equation {#subsec:application on Laplace eigenvalue equation}
------------------------------------------
We consider the Laplace eigenvalue problem with homogeneous boundary condition $$\label{eq:evpoisson}
\left\{
\begin{aligned}
-\Delta u & =\lambda u & \quad & \mbox{in} \ \Omega \\
u & =0 & \quad & \mbox{on} \ \Gamma.
\end{aligned}
\right.
$$
The variational formulation is to find $(\lambda,u)\in \mathbb{R}\times H^1_0(\Omega)$, such that $$\int_\Omega \nabla u\nabla v \ \mathrm{d}x=\lambda\int_\Omega uv \ \mathrm{d}x,\quad\forall\,v\in H^1_0(\Omega).
\label{eq:variational problem on evpoisson}$$ The finite element problem is to find $(\lambda_h,u_h)\in \mathbb{R}\times V_{h0}^{\rm QBL}$, such that $$\sum_{K\in\mathcal{J}_h}\int_K\nabla u_h\nabla v_h \ \mathrm{d}x=\lambda_h\int_\Omega u_hv_h \ \mathrm{d}x,\quad\forall\,v_h\in V_{h0}^{\rm QBL}.
\label{eq:dicrete on evpoisson}$$
Let the eigenvalues of the problem and be sorted from small to big. Let $(\lambda, u)$ and $(\lambda_h,u_h)$ be the $k$-th eigenpair of and , respectively. Then for $h$ small enough, $$|\lambda-\lambda_h|=\mathcal{O}(h^2)\ \ \ \mbox{and}\ \ \ |u-u_h|_{1,h}=\mathcal{O}(h).$$
The theorem is proved by the standard technique.
### Numerical verification
We choose the computational domain to be the unit square $\Omega = (0,1)\times(0,1)$. The eigenvalue $\lambda$ is chosen such that the exact solution is the function $u = \sin(\pi x)\sin(\pi y)$. We first divide the computational domain into four trapezoids, then use the same strategy as Subsection \[subsec:a FE scheme for the Poisson equation\] to generate the grids, see Figure \[fig:Laplaeigshowmesh\], and repeat numerical test by same elements as before. The results are recorded in Table \[tab:tab7\].
\[fig:Laplaeigshowmesh\]
------------------------- --------------- ------------------------- -- --------------- ------------------------- -----------
Size($\mathcal{J}_{h}$)
$\lambda_{h}$ $|\lambda-\lambda_{h}|$ $\lambda_{h}$ $|\lambda-\lambda_{h}|$ $\lambda$
$8\times8$ 2.035E1 6.090E-1 2.074E1 9.995E-1 2$\pi^2$
$16\times16$ 1.988E1 1.359E-1 1.998E1 2.424E-1 2$\pi^2$
$32\times32$ 1.977E1 3.210E-2 1.980E1 6.120E-2 2$\pi^2$
$64\times64$ 1.9747E1 7.800E-3 1.9754E-1 1.480E-2 2$\pi^2$
Convergence order 2 2
------------------------- --------------- ------------------------- -- --------------- ------------------------- -----------
: Numerical results for Laplace eigenvalue problem[]{data-label="tab:tab7"}
Figure \[fig:convergence order on Laplace eigenvalue problem\] reports on approximation results of ${\rm QBL}$ and Courant elements for Laplace eigenvalue equation. The $x$-axis and the $y$-axis represent the logarithm of grid size $h$ and of the error, respectively. In this numerical experiment, a clear second-order of convergence is observed and the numerical performance of ${\rm QBL}$ element is better than that of Courant element.
![The log-log plot of the error of ${\rm QBL}$ and Courant elements for Laplace eigenvalue equation[]{data-label="fig:convergence order on Laplace eigenvalue problem"}](Laplaeigconverorder.pdf){width="1\linewidth"}
Application on $H(\mathrm{rot})$ equation {#subsec:application on rot equation}
-----------------------------------------
We consider the problem with homogeneous boundary condition
$$\left\{
\begin{aligned}
\undertilde{\mathrm{curl}}
&(\mathrm{rot}\usigma)+\usigma=\uf &
\quad & \mbox{in} \ \Omega \\
&\usigma\ut=0 &
\quad & \mbox{on} \ \Gamma.
\end{aligned}
\right.$$
The variational formulation is to find $\usigma\in H_0({\rm rot},\Omega)$, such that $$\int_\Omega {\rm rot}\usigma{\rm rot}\utau+\usigma\utau \ \mathrm{d}x=\int_\Omega \uf\utau\ \mathrm{d}x,\quad\forall\,v\in H_0({\rm rot},\Omega).
\label{equ:variational problem on rot}$$ The finite element problem is to find $\usigma{}_h\in V_{h0}^{\rm QRT}$, such that $$\sum_{K\in\mathcal{J}_h}\int_K{\rm rot}\usigma{}_h{\rm rot}\utau{}_h+\usigma{}_h\utau{}_h\ \mathrm{d}x=\int_\Omega \uf\utau{}_h \ \mathrm{d}x,\quad\forall\,\utau{}_h\in V_{h0}^{\rm QRT}.
\label{equ:discrete problem on rot}$$
Let $\usigma\in H^1({\rm rot},\Omega)\cap H_0({\rm rot},\Omega)$ and $\usigma{}_h$ be the solutions of , and , respectively. Then $$\|\usigma-\usigma{}_h\|_{{\rm rot},h}\leqslant Ch(|\usigma|_{1,\Omega}+|{\rm rot}\usigma|_{1,\Omega}).$$
The theorem is proved by the standard technique.
### Numerical verification
We choose the computational domain to be the unit square $\Omega = (0,1)\times(0,1)$. The source term $\uf$ is chosen such that the exact solution is given by $\usigma=(xy^2-xy,x^2y-xy)^{T}$. Then we test the performance of the ${\rm QRT}$ element on the quadrilateral grids as Subsection \[subsec:application on Laplace eigenvalue equation\] and the results are recorded in Table \[tab:tab8\].
Size($\mathcal{J}_{h}$) $\|\usigma-\usigma{}_{h}\|_{0,\Omega}$ $|\usigma-\usigma{}_{h}|_{\mathrm{rot},h}$ $\|\usigma-\usigma{}_{h}\|_{\mathrm{rot},h}$
------------------------- ---------------------------------------- -------------------------------------------- ----------------------------------------------
$8\times8$ 1.13E-1 5.95E-2 1.28E-1
$16\times16$ 5.53E-2 2.98E-2 6.28E-2
$32\times32$ 2.78E-2 1.49E-2 3.15E-2
$64\times64$ 1.40E-2 7.45E-3 1.58E-2
Convergence order 1 1 1
: Numerical results for $H(\mathrm{rot})$ problem[]{data-label="tab:tab8"}
Figure \[fig:convergence order on rot problem\] reports on approximation results of ${\rm QRT}$ element for $H(\mathrm{rot})$ equation. The $x$-axis and the $y$-axis represent the logarithm of grid size $h$ and of the error, respectively. The error associated with $\|\cdot\|_{0,\Omega}$, $|\cdot|_{\mathrm{rot},h}$ and $\|\cdot\|_{\mathrm{rot},h}$ are plotted by dotted line, dashed line and solid line, respectively. The results confirm our conclusion: a clear first-order of convergence is observed.
![The log-log plot of the error of ${\rm QRT}$ for $H(\mathrm{rot})$ equation[]{data-label="fig:convergence order on rot problem"}](rotconverorder.pdf){width="1\linewidth"}
Concluding remarks {#sec:concluding remarks}
==================
In this paper, we present a polynomial de Rham complex on a grid that consists of arbitrary quadrilaterals by constructing two nonconforming finite elements for the $H^1$ and $H(\rot)$ problems, respectively. As is proved in the present paper, cf. Theorem \[thm:nocon\] and Remark \[rem:nocon\], we can not theoretically expect the finite element be conforming anyway; however, the two spaces are both quasi-conforming and are consistent to the requirement of the differential complex. Moreover, with respect to the $\mathcal{O}(h^2)$ asymptotic parallelogram assumption, the scheme for $H(\rot)$ problem is $\mathcal{O}(h)$ convergent for $H^1(\rot)$ exact solution; namely, this element does not suffer from the extra requirement on the regularity for general nonconforming $H(\curl)$ element (cf. [@shi2009low]). For the Poisson equation, numerical experiments show that the [QBL]{} element plays superior to the triangular linear element with the same amount of unknowns for both source and eigenvalue problems, which confirms the need of quadrilateral elements.
It follows immediately that a piecewise polynomial complex can be constructed by rotation according to $$\mathbb{R}\xrightarrow{\rm inclusion} H^1 \xrightarrow{\rm curl} H(\dv) \xrightarrow{\dv} L^2\xrightarrow{\rm integration}\mathbb{R}$$ by simply a rotation. Further, the methodology which use the same shape functions as that on a framework parallelogram and the nodal parameters on a physical cells can be generalized to more complicated cases, such as higher order schemes and higher dimension problems. These would be discussed in future.
[^1]: The research of X. Ji is supported by the National Natural Science Foundation of China under grants 91630313 and 11971468, and National Centre for Mathematics and Interdisciplinary Sciences, CAS. The research of S. Zhang is supported partially by the National Natural Science Foundation of China with Grant Nos. 11471026 and 11871465 and National Centre for Mathematics and Interdisciplinary Sciences, CAS
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What’s modeling?
A model is an abstraction, a physical abstraction of an object or a conceptual abstraction of a situation. An architect might use a cardboard physical model to illustrate a proposed building. Conceptual models can represent complex systems like population dynamics, economics, or schooling fish. By “complex systems” I mean the things described in Blog 2 and Blog 3, situations with many independent agents governed by nonlinear rules of interaction among the agents and their surroundings. A conceptual model often takes the form of a set of equations with which the system can be simulated by computer, thereby becoming a “computer model.” Note I said simulated by computer, not solved by computer.
Let’s consider things that evolve in time, like the economy, the weather, or traffic. The term, “solved” implies a closed-form solution, which is an equation (or set of equations) into which you can put a value of time and calculate the future situation in a single step. For example, if you travel at 50 miles per hour, your distance is 50 multiplied by time.
X = 50*T
Put in any value for time; you get the value for distance. In 10 hours X is 500 miles. But a complex system has a whole set of equations, such as one equation for the movement of each ant around an anthill, where each ant moves randomly until it either finds food or it finds smells left on the ground by other ants that carry food. For such a system, there is no closed-form solution. You need a computer to adjust the direction and speed of each ant at a particular instant of time, then move each ant slightly and update the map of smells before performing the whole calculation all over again for another small step in time. This is a simulation, resulting in a movie that depicts the location of every ant, the food moved, and the smelly tracks of the food-carriers. You can see a short clip of the anthill simulation, or click for the entire 7-minute presentation offered by the Santa Fe Institute.
A model can reveal emergent properties of the system—for example, how the ants form tracks to food. Repeated runs of the model with different parameters can reveal the effects of a rapidly dissipating smell-track, or even the effects of garlic in the limburger if the equations include the dietary preferences of ants.
Modeling can disclose unexpected effects, but in each case you must run the model from time zero and watch as the situation evolves. You can’t jump to the situation at an arbitrary future time as you could with a closed-form solution. In many complex systems, a slight change in the starting condition can dramatically alter the situation much later. That’s called the “butterfly effect.” If your social system is susceptible to a butterfly effect, you want to know it.
So why does this matter?
As our society and the world become increasingly complex, decisions regarding regulation, subsidies, social security, taxes, and immigration need examination of both the intended consequences and the unintended consequences. Instead, most political decisions are based on simplistic statements made in ignorance of (or in denial of) the causes and effects. Because the world is a set of nested complex systems, modeling provides the best means for exploring and predicting the emergent consequences of our collective decisions.
I’ll invite other opinions by stating that our most crucial social decisions today deal with climate change. Climate-related political debates are often based on partisan interpretations of today’s temperature, or today’s cash flow, or today’s ideology, although all of our knowledge about future climate development comes from increasingly sophisticated modeling tested against historical data.
I suggest that the future of the earth depends on how well we do our modeling, and whether we pay attention to the results or proceed in ignorance. A good model wouldn’t have predicted the sinking of the Titanic with certainty, but it would have predicted the odds of collision with an iceberg. A good model would not have predicted current values for the Dow Jones, but it might have predicted the probablility of economic recession from the known indebtedness combined with the ideological deregulation of the banking industry. We should apply good models during decisions regarding our complex society, and pay attention to the predictions during our debates. | https://neeper.net/blog-13-models-and-modeling/ |
CROSS-REFERENCE TO RELATED APPLICATIONS
BACKGROUND OF THE INVENTION
DESCRIPTION OF PRIOR AND RELATED ART
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE INVENTION
This application is a continuation-in-part of U.S. application Ser. No. 14/294,115 filed on Jun. 2, 2014, currently pending, which is a continuation-in-part of U.S. application Ser. No. 14/180,611 filed on Feb. 14, 2014, currently pending, which is a continuation-in-part of U.S. application Ser. No. 13/854,988, filed on Apr. 2, 2013, currently abandoned, which is a continuation-in-part of U.S. application Ser. No. 13/787,889 filed on Mar. 7, 2013, currently abandoned. The patent applications identified above are incorporated herein by reference in their entirety to provide continuity of disclosure.
1. Field of the Invention
The invention generally relates to the field of telecommunications and in particular to systems and methods that are able to both authenticate correct and true caller ID information, and detect fraudulent and spoofed caller ID information. The invention also uses authenticated caller ID information in conjunction with lists such as whitelists and blacklists for the purpose of call allowance and blockage. The invention further relates to authenticating the called party to the caller party and verifying the call has been connected to the intended party or has been forwarded to an unintended party telephone number.
This invention further relates to the application of caller ID authentication methodology to the fields of electronic mail and postal mail communications. And in particular to systems and methods that are able to both authenticate correct and true email and postal mail source address information and detect spoofed source address information.
2. Background
Caller ID (caller identification, CID), also called calling line identification (CLID), calling number delivery (CND), calling number identification (CNID) or calling line identification presentation (CLIP), is a telephone service, available in analog and digital phone systems and most voice over Internet Protocol (VoIP) applications, that transmits a caller's number to the called party's telephone equipment during the ringing signal, or when the call is being set up but before the call is answered. Where available, caller ID can also provide a name associated with the calling telephone number. The information made available to the called party may be displayed on a telephone's display, on a separately attached device, or be processed by an attached computer with appropriate interface hardware.
Caller ID may be used to display a caller's telephone number (and, in association with a database, name) on a called user's telephone. This works in most countries; although systems are incompatible, each country will have appropriate equipment. A modem can pass CLID information to a computer for purposes of call logging or blocking, but this can be problematic as modems in different countries have different systems, causing hardware or software incompatibilities. However, many modems are designed and programmed to handle multiple signaling methods and can be configured to use the local standard.
202
In the United States, caller ID information is sent to the called party by the telephone switch as an analogue data stream (similar to data passed between two modems), using Bell modulation between the first and second rings, while the telephone unit is still on hook. If the telephone call is answered too quickly after the first ring, caller ID information will not be transmitted to the recipient. There are two types of caller ID, number only and name+number. Number-only caller ID is called Single Data Message Format (SDMF), which provides the caller's telephone number, the date and time of the call. Name+number caller ID is called Multiple Data Message Format (MDMF), which in addition to the information provided by SDMF format, can also provide the directory listed name for the particular number. Caller ID readers which are compatible with MDMF can also read the simpler SDMF format, but an SDMF caller ID reader will not recognize an MDMF data stream, and will act as if there is no caller ID information present, e.g. as if the line is not equipped for caller ID.
Caller ID forging, or spoofing, is the practice of causing the telephone network to display a number on the recipient's Caller ID display that is not that of the actual originating station. The term is commonly used to describe situations in which the motivation is considered malicious by the speaker or writer. Just as e-mail spoofing can make it appear that a message came from any e-mail address the sender chooses, Caller ID spoofing can make a call appear to have come from any phone number the caller wishes. Because of the high trust people tend to have in the Caller ID system; spoofing can call the system's value into question.
There are both legitimate and illegitimate reasons for caller ID spoofing. Some of the legitimate reasons for spoofing caller ID information are for example, calls from a large company, especially with multiple branches, where sending the main number makes sense. A hospital might have the primary number 555-1000, with perhaps 250 lines inside the main building, and another 100 at the clinic five miles away. While some of the phones will have 555-10XX numbers, many won't have any identifiable line. Having all calls come from 555-1000 lets the recipients know it's a hospital call. As another example, a company with a toll-free telephone number may prefer the caller ID to display this number. In another example, a call center making calls on behalf of many clients may prefer the caller ID to display a different number for each client's calls.
Some of the illegitimate reasons for caller ID spoofing can be where a telemarketer is making calls that it is not authorized to make, such as calling a number on the National Do Not Call Registry. The Do Not Call Registry is maintained by the FTC that allows consumers to register their phone number and is intended to give consumers an opportunity to limit the telemarketing calls they receive. Any telemarketer who violates the Do Not Call Registry can be fined. Another instance of illegitimate caller ID spoofing can be where an individual wishes to impersonate another entity, such as a bank, to acquire sensitive information from the called party through deception.
Invention Objectives
It is an objective of this invention to authenticate caller ID. An authenticated caller ID is where the information supplied by the caller ID is determined to be correct and true. Another objective of this invention is to detect a spoofed caller ID. A spoofed caller ID is where the information supplied by the caller ID is determined to be incorrect and fraudulent. Another objective of this invention is to determine a caller ID as irresolute if it cannot be determined that a caller ID is authenticated or spoofed. It is an objective of this invention to provide caller ID authentication utilizing systems and methods which are cost effective, which are not technically complex and which can be added to any telephone system. It is an objective of this invention to leverage authenticated caller ID's into a call handling system. The call handling system utilizes local, global and private lists for call handling. It is also an objective of this invention to use and adapt the described methods within electronic and postal mail systems for the purpose of authentication and spoof detection of source addresses.
FIG. 1
With respect to the disclosed invention, a relatable prior art is U.S. Pat. No. 7,113,577 by Cook et al, herein incorporated by reference in its entirety. The system architecture disclosed in FIG. 1 of the '577 patent is similar to the system architecture in this invention disclosure. describes a system where the caller and the called parties both have access to a PSTN and a data network. The parties initiate a telephone call over the PSTN and are able to access an “information server” over the data network. In this disclosure, the caller and called parties also have access to the PSTN and an authentication device connected to the party phones provides access to the data network. Also in this disclosure, the authentication devices access an authentication server, similar to the information server described in the '577 patent. In the patent, “ . . . the information server further receives a caller system identifier and a called system identifier from the caller system”. Similarly in this disclosure, the authentication device of the caller transmits the caller number and the called number to the authentication server. There are also other similarities that the '577 patent discusses, such as the information server pushing information to the called party instead of the called party pulling information from the information server. The patent also discusses where an information server is unavailable and the method of the patent is conducted in a peer-to-peer manner. Those concepts are similar to concepts that will be disclosed in this document.
The patent is mainly summarized as “ . . . An information server according to one embodiment of the invention comprises a communication interface configured to communicate over a communication network and a processing system connected to the communication interface. The processing system is configured to receive one or more digital content sets, receive a caller system identifier and a called system identifier from the caller system, select a digital content set from among the one or more digital content sets using the caller system identifier and the called system identifier, and provide the selected digital content set to a called system corresponding to the called system identifier prior to the caller system establishing a communication channel to the called system . . . . For example, the digital content set may include text/graphics identifying the caller as an officer or representative of an institution, or such as a doctor affiliated with a certain hospital, wherein both the doctor and the hospital are identified by the digital content set, etc. In addition, the digital content set may identify a family member or acquaintance who is using a public telephone . . . ”. Thus, the patent is mainly concerned with providing digital content to the called party based on the caller party.
Despite the similarities of the '577 patent to this invention disclosure, the '577 patent does not consider or disclose a way of authenticating a received caller ID as true and non-fraudulent. The patent never contemplates using the information server for the purpose of tracking calls placed by the caller and where the called party can inquire this tracking information. In the patent, the called party never contacts the information server for the purpose of cross checking the received caller ID against the information sent from the caller party to the information server. The patent does not disclose any method of detecting a spoofed caller ID. Thus, the '577 patent does not introduce a method of caller ID authentication, as will be described in this disclosure. However, the building blocks of the '577 patent are sufficiently similar, that any system utilizing the '577 patent may be easily be modified to make use of the methods proposed in this disclosure to authenticate caller ID.
Other known methods of caller ID authentication and spoof detection are inadequate and cumbersome. Such examples of caller ID authentication systems are described in U.S. Pat. No. 8,254,541 by Cai et al, U.S. Pat. No. 7,912,036 by Moore, U.S. Pat. No. 7,385,992 by Koch et al, U.S. Pat. No. 8,238,532 by Cox et al, and United States Patent Applications 2009/0046839 by Chow et al, 2005/0190904 by Anupam et al, herein incorporate by reference in their entirety. As will be elaborated, none of the known prior art systems utilize a combination of an authentication device coupled with an authentication server, where the calling authentication device transmits the source and destination of the call to the authentication server. And none of the prior art systems describe a system where caller ID is authenticated by the called authentication device; by transmitting the call source and destination to the authentication server for authentication. As will be explained in greater detail, the embodiments of the current invention do not rely on the underlying telephone system to feature the methods described in this invention disclosure to authenticate caller ID's. The caller ID authentication system described herein may be added on to any telephone system through a combination of hardware and software at various points throughout the telephone network; such as at the household NID level, integrated into household phones, integrated into smartphones operating systems, installed as smartphone software applications, and integrated into VoIP systems. Thus, the proposed methods of caller ID authentication may be rolled out over a period of time and in a staggered fashion.
The eventual benefit of caller ID authentication is call handling based on the authenticated caller ID information. Examples of call handling systems are described in U.S. Pat. No. 7,295,660 by Higginbotham et al, U.S. Pat. No. 7,222,158 by Wexelblat, and United States Patent Applications 2007/0143422 by Cai, 2011/0283349 by Wu, 2007/0071200 by Brouwer, 2011/0026699 by Amir et al, herein incorporated by reference in their entirety. Any call handling system must first authenticate caller ID, otherwise, the concept of a call handling system becomes moot. If a problematic telemarketer is banned from initiating further calls to a customer by being placed on a blacklist and all that is needed to circumvent the blacklist is for the telemarketer to spoof its' caller ID, then for all intensive purposes, the notion of a blacklist is pointless. None of the known prior art systems incorporate the disclosed caller ID authentication system within a call handling system.
In conclusion, none of the prior art demonstrates a method of authenticating and classifying caller ID as authenticated or spoofed. And if the caller ID cannot be determined to be authenticated or spoofed, it is classified as irresolute. None of the prior art demonstrates a method where the caller sends a message with the source and destination of the call to an authentication server. And where the called party can query the authentication server to crosscheck the received caller ID against the caller message received by the server. None of the prior art demonstrates other aspects of the invention, such as where the authentication server pushes the received message from caller to the called party, where the called party uses the pushed message to authenticate caller ID. None of the prior art demonstrates the aspect of the invention where the caller ID authentication is done in a peer-to-peer fashion, without the need for an authentication server. None of the prior art introduces the concept of certification within a caller ID authentication system. The concept of certification allows for the authentication server to certify that the authentication device transmitting a message containing a source telephone number in fact possesses ownership over the source telephone. None of the prior art combines such caller ID authentication within a call handling system, where the call handling system is separated into local, global and private call lists.
It is to be understood that both the following general description and the following example descriptions are explanatory only and are not restrictive of the invention as claimed.
In one embodiment of the proposed caller ID authentication and spoof detection method and system, a calling party and the called party have installed a caller ID authentication device that monitors all incoming calls and caller ID's and outgoing dialed numbers. Such an authentication device can be in the form of a software application, if the party is using a cellular smartphone. If the party is using a landline, the authentication device can be implemented at the telephone providers switching system. Or the landline party can install an authentication device at the phone line coming into the residence.
The authentication device is able to communicate with an authentication server. When a calling party, which is at the phone number 555-1000 for example, dials a number, such as 555-2000, the authentication device is able to intercept this number. As the call is being connected, the authentication device transmits the calling party's own number and dialed number to the authentication server. When the called party receives the call and the caller ID information of the calling party, the called party's authentication device immediately contacts the authentication server. The called party's authentication device inquires the authentication server if a call has been placed to the called party's number, which is 555-2000, from the calling party's phone number obtained from the caller ID, which is 555-1000. If the authentication server responds in the affirmative, then the authentication device determines that the received caller ID is authentic. If the authentication server responds in the negative, the authentication device determines that the received caller ID is not authentic. If the authentication server indicates the calling party does not currently have the caller ID authentication device installed, the authentication device at the called party determines the caller ID can't be authenticated. The authentication device can then inform the called party of the calling party's caller ID authentication status through various methods, one example of which is modifying the caller ID transmitted to the called phone.
In another embodiment, the authentication server certifies authentication devices in order to prevent an authentication device from fraudulently updating or inquiring the authentication server. Such certification is done by the authentication device contacting the authentication server, transmitting the phone number of authentication device and requesting a password. The authentication server then dials the received phone number of the authentication device and transmits the certification password. The authentication device now must use the received password each time it contacts the authentication server. Thus, the authentication server can be assured that the certified authentication device is at the appropriate phone number and is not fraudulently stating a different number from the one it is connected to.
In another embodiment of the invention, the authentication device is able to authenticate blocked caller ID's. In another embodiment, the authentication device is able to authenticate customized caller ID's that do not disclose the phone number of the caller. In yet another embodiment, the authentication device is able to authenticate customized VoIP caller ID's.
In another embodiment of the invention, the authentication device is able to handle and filter incoming calls. The authentication device maintains an internal whitelist and an internal blacklist. The internal whitelist is populated automatically by the authentication device. The internal whitelist is populated with telephone numbers that the user of the authentication device receives calls from and numbers that the user dials on a regular basis. The internal blacklist is populated by the user with a dialed star code after or during a call from an unwanted number. The authentication device thus uses these lists for call handling. Any incoming call with it's telephone number on the internal whitelist is allowed to ring and contact the user of the authentication device. Any incoming call with a telephone number on the internal blacklist is prevented from ringing the user's phone. Any number that is not on the internal whitelist or blacklist is treated as a gray number. The handling of incoming calls from gray number can be determined and customized by the user. The user is also able to customize how calls are handled based on the combination of the authentication status of the incoming calls caller ID and which internal list the incoming calls' telephone number is on.
In another embodiment, each authentication device uploads their internal lists to a global list server. The global list server can preferably be the same server used for caller ID authentication or a separate server. The uploaded internal lists are aggregated by the global list server. The global list server maintains a global whitelist and a global blacklist. If a phone number is on a certain number of authentication devices whitelists, the global list server adds that number to the global whitelist. Likewise, if a phone number is on a certain number of authentication devices blacklists, the global list server adds that number to the global blacklist. Thus, when a call is received, the authentication device can allow or drop the call if it is on the internal whitelist or internal blacklist. If the number is not contained on any internal lists, the device can inquire the global list server and allow or drop the call if it is on the global whitelist or global blacklist. If a number is not on any list, the call is handled as a gray number.
In yet another embodiment, the global list server contains a private whitelist and blacklist for each authentication device. If a called user who is using an authentication device receives a call from a caller with a blocked caller ID and who is also using an authentication device, the called party has the ability to add the blocked caller ID to the private whitelist or blacklist. Since the authentication device of the caller with a blocked caller ID is sending to the authentication server the phone number of the caller and if the global list server has access to the authentication server, that phone number may be added to the private whitelist or blacklist of the called party. Once on the private whitelist or blacklist, if the same caller calls again, when the called authentication device contacts the authentication server to authenticate the blocked caller ID, the authentication server can query the global list server and indicate that the called party that the calling number is on user's private whitelist or blacklist. Thus the authentication device of the called party may handle calls from blocked caller ID's.
The concept of certification is extended to passive certification. Passive certification allows for computing of the certification password through calls made and received by the authentication device. The certification password is updated to include the caller ID information sent to the authentication server. Through such a system, the certification password is changed on each call and the old certification password invalidated.
In another embodiment, the concepts of revealing caller ID's to certain or all parties are disclosed. The concept of caller ID revelation is then applied to VoIP calls. The concepts of caller ID authentication is discussed and elaborated as they relate other forms communications, such as SMS source address authentication, email source address authentication, and postal mail source address authentication.
The concept of caller ID authentication is further used to authenticate the destination of a call to the caller party. The authentication server uses the fact that an authentication request has been received by the intended called party to authenticate the called party to the caller party. And similarly, if an authentication request has not been received by the intended called party, the call is indicated to the caller party as having been forwarded.
1
This disclosure of the invention is submitted in furtherance of the constitutional purposes of the U.S. Patent Laws “to promote the progress of science and useful arts” (Article , Section 8). The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate embodiments of the invention and together with the general description, serve to explain the principles of the invention. All, patents, patent applications, publications and references cited in this specification are incorporated herein by reference in their entirety.
The invention disclosure is separated into Sections and Subsections. The Section and Subsection titles are followed by the figures those sections reference, if any. The Section may be referenced with the notation “Section (1)”. A Subsection would be referenced by the notation “Section (1-a)”, which would be a reference to the first Section (1) “Caller ID Authentication Device Implementations” and the Subsection (a) “Residential Landline Implementation”, which is the first Section and Subsection below.
FIG. 1A
FIG. 2B
1) Caller ID Authentication Device Implementations (-)
FIG. 1A
a) Residential Landline Implementation ()
FIG. 1A
The block diagram of shows one implementation of the invention embodiment.
FIG. 1A
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The block diagram of shows the caller ID authentication device connected to a residential household telephone system. The household telephones are serviced by a telephone service provider . Telephone service providers may consist of companies such as Verizon, AT&T, and the like. The telephone service provider provides telephone service through a communication medium , such as twisted pair, fiber optic, or a coaxial system. The communication medium is usually referred to as the local loop. The telephone service provider provides telephone service to the residence through public switched telephone network (PSTN) . The telephone service provider communication medium is connected to the residence through a Network Interface Device (NID) . The NID serves as a demarcation point between the telephone service provider's local loop and the residential premises wiring . The residential premises wiring usually consists of a twisted pair or a pair of copper wires which provide analog voice service to the household phones . The caller ID authentication device is connected between the NID and the household line. The caller ID authentication device may employ any of the described embodiments mentioned in this document. The authentication device is able to connect to an authentication server . The authentication server can consist of any server system known in the art and any required data storage and retrieval systems such as a database. The authentication server also has access to the PSTN . The connection to the authentication server from the authentication device can be cellular, DSL, coaxial and the like. The connection to the authentication server may be done through the same residential wiring , for example if the household is served by a DSL service, and the authentication device contains an integrated DSL modem. Alternatively, the caller ID authentication device can connect to the Internet through a Wi-Fi or an Ethernet connection, made available by the residence. Preferably, the authentication server is accessible from the Internet with connection to the authentication server being a broadband Internet connection which also serves the residence. However, it should be clear that the connection to the authentication server can be made on any medium which supports transmission of data.
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Alternatively, each household phone can have an authentication device integrated within. For example, a household corded or cordless phone can come equipped with an authentication device integrated into the phone by the manufacturer. The authentication device could have access to the authentication server through the residence wireless internet access, an Ethernet cable, a coaxial cable or through the residential telephone line itself. If there are more than one household phones, each authentication device of each phone may act independently, or they may form a master and slave relationship. That is, each authentication device may transmit and receive from the authentication server independently. Or one authentication device may declare itself to be a master device, and can route information to and from other slave authentication devices to the authentication server on behalf of those devices.
The installation of the authentication device to the residence can be performed by the telephone service provider by sending a technician to the residence and the technician installing the authentication device between the residence wiring and the NID. Alternatively, the telephone service provider can make available for purchase self install kits for the residential customers. In which the residential customers would perform a similar installation procedure as the technician.
The authentication device may consist of a hardware solution, such as an electronic device with one phone jack for receiving the telephone line from the NID. And another phone jack for connecting to the premises phone line. Thus, the authentication device would be able to relay calls to and from the telephone network to the household and provide all the functionality required by the embodiments presented. The authentication device would also contain an Ethernet port for connecting to the internet. Alternatively, the authentication device may contain a Wi-Fi card for connecting to the residence wireless internet connection. Or the authentication device may contain an integrated DSL modem. The DSL modem would access the DSL internet connection through the same phone jack connected to the NID line.
The authentication device may also be integrated into an Intelligent Network Interface Device (iNID). Such as setup would allow for the hardware and software required for implementing the following embodiments to be integrated into the NID itself.
FIG. 1B
b) Telephone Service Provider Implementation ()
FIG. 1B
FIG. 1A
FIG. 1B
FIG. 1A
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The block diagram of shows the same type of authentication device as was described in . However, the authentication device is implemented at the telephone service provider instead of at the household residence. The block diagram shown in is similar to the one shown in , except the authentication device is implemented at the telephone service provider. The telephone service provider may implement the caller ID authentication device through any combination hardware or software required to perform the functions of the authentication device as will be described. Thus, all the customers serviced by the telephone service provider may utilize the functions of the authentication device, without each individual customer having to install the device.
The authentication device may also be implemented as software within PBX systems. An example of such a software system is the Asterisk PBX system (www.asterisk.org). The authentication device would be implemented as software, since such an Asterisk system would already contain most or all the necessary hardware for connecting to an authentication server, handling incoming and outgoing calls in accordance within the embodiments described herein. Readers are encouraged to consult appropriate Asterisk documentation at asterisk.org website (www.asterisk.org/community/documentation) and consult the book “Asterisk: The Definitive Guide, 3rd Edition” by Leif Madsen (cdn.oreillystatic.com/books/9780596510480.pdf), herein incorporated by reference in its entirety.
c) Cellular and Smartphone Implementation
In the cases where the caller ID authentication is wished to be provided for cellular phones, such as Apple or Android smartphones and the like, the caller ID authentication device can consist of software routines integrated into the phone operating system (OS) or the authentication device can be integrated into the hardware of the cellular phone device. Alternatively, the caller ID authentication software may be downloaded and installed as an application, from an appropriate application store such as Apple App Store or Google Play App Store if present, by the cellular phone owner. The caller ID authentication software, either integrated into the OS or downloaded, would be able to carry out the embodiments of the invention described herein. The caller ID authentication software would be able to contact the authentication server through the cellular phone data connection. The cellular data connection could be obtained the by cellular customer through subscribing to the appropriate cellular data service, or the cellular telephone provider may offer the data connection to the authentication server free of charge for caller ID authentication purposes. The caller ID authentication software would also be able to change caller ID information, or instruct the caller ID information to be displayed in a certain color for authenticated caller ID's, for example.
FIG. 1B
Alternatively, the authentication device can be implemented at the telephone service provider of the cellular customer, similar to the system shown in .
FIG. 2A
FIG. 2B
d) VoIP Implementations (-)
FIG. 2A
FIG. 2A
FIG. 1A
FIG. 1B
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The block diagram of shows one example of an embodiment of the invention implemented on a VoIP system. The block diagram shows a VoIP provider implementing an authentication device . A VoIP provider may consist of companies such as Vonage, Skype, and the like. Such VoIP providers allow its' customers to use a computer phone software or a physical phone type device connected to an internet connection to call phones on the telephone network . The VoIP customers usually are not assigned a phone number, but may be in some instances. Thus, most VoIP customers are mainly concerned with ability to connect to customers on PSTN networks, and not mainly interested in receiving calls. Again, this not always the case. VoIP customers such as on the Skype provider may receive calls from other VoIP customers who do not have a phone number. One Skype customer may simply contact another Skype customer through their Skype username, and none of the parties may possess a phone number. In , the VoIP customer may or may not possess a phone number. The VoIP customers with a VoIP phone , which may be in the form of a hardware phone or a software phone program running on a computer, connects to the VoIP provider over a connection . The connection is usually an internet connection. The VoIP provider may implement the authentication device as a combination of hardware or software. The VoIP authentication device connects to an authentication server through a data connection . Data connection is preferably an internet connection. The authentication server also has access to the PSTN . The authentication device for the VoIP provider might have its' own phone number. As will be elaborated later, the concept of certification of an authentication device requires the authentication device to be connected to a phone number. In previous implementations shown in and , the authentication devices had a phone number which was the same phone number of the customer. However, in a VoIP system, the VoIP customer might not possess a phone number, thus the authentication device of a VoIP system will in most cases require its own phone number.
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If the VoIP customer has his own phone number, the authentication device may be implemented at the customer end, as is shown in . The customer VoIP phone may be in the form of phone software running on the customers PC. The authentication device may thus simply be in the form of a software add-on or plug-in running alongside the phone software. Alternatively, the authentication device functionality may be integrated within the phone software. If the VoIP customer is using hardware phone connected to the internet through a VoIP adapter (not shown), then the authentication device may be integrated into the VoIP adapter.
2) Caller ID Authentication Device Functional Overview
This section and the next section describe and elaborate some of the functionalities of the authentication device and the authentication server. These two sections are provided mainly as reference. Certain readers who are versed in the field of telecommunications may choose to skip to Section (4), and refer to these sections at a later time.
The caller ID authentication device is able to perform some or all of the following functions depending on the embodiment of the invention;
b) Present interactive voice response (IVR) to the user of the device for presenting the device settings to the user and allowing the changing of those settings,
c) Pass through voice communication
d) Access a data network
e) Access an authentication server over the data network
f) Intercept incoming calls
g) Selectively ring the user phone
h) Extract caller ID from an incoming call
i) Communicate with the authentication server for the purpose of caller ID authentication
j) Modify caller ID presented to the user to notify the user of the caller ID authentication status, and/or notify the user of the caller ID authentication status through other means
k) Maintain internal database of whitelisted and blacklisted telephone numbers
l) Implement logic to autonomously place telephone numbers on the internal whitelist
m) Implement logic to allow the user to manually place telephone numbers on the internal blacklist
n) Handle incoming calls according to the list the incoming telephone number is contained in
o) Access a global list server
p) Upload contents of the internal list to the global list server
q) Check the telephone number of an incoming call against the global list server for call handling purposes
r) Evaluate calls for certain criteria for the purpose of automated blacklisting of incoming calls
a) Intercept outgoing calls and extract outgoing dialed numbers,
However, not all functionalities stated may be required in all implementations and in all embodiments. The authentication device with the required functionalities may be realized in hardware or software, or combinations thereof, by a person skilled in the art. The authentication device may be required to perform other functionalities in certain embodiments, than are disclosed in this section. Some of the functionalities will be elaborated in this section, while other functionalities will be explored later in this document in more detail through disclosure of the invention embodiments.
a) Intercept Outgoing Calls and Extract Outgoing Dialed Numbers,
For outgoing calls from the user phone to the telephone network, the caller ID authentication device is able to extract dialed numbers by the user. The device may immediately pass through the dialed numbers by the user to the telephone network, or delay them. The purpose of delaying passing dialed numbers to the telephone network is twofold. If the user dials a star code that is specific to the authentication device functionality or settings, the star code would not be passed to the telephone network and the device can respond appropriately, through voice menu prompts for instance. Another benefit of delaying passing the dialed numbers is to give the caller ID authentication process a time buffer. If a user dials a valid telephone number, for example a ten digit number, the device may delay passing these numbers to the telephone network, for example by half a second to two seconds. The device can then use this delay to contact the authentication server and relay the dialed numbers. This delay would allow more time for the authentication server to be updated. The device can then put the call through. The delay would allow the authentication process to be perceived by the user as being more transparent and closer to real-time functionality.
The authentication device may drop outgoing calls and may play voice menu to the user For example, if a user on a residential phone dials *17, the authentication device may disconnect the call from the telephone network, pick up the call and play a voice menu to the user.
b) Present Interactive Voice Response (IVR) to the User of the Device for Presenting the Device Settings to the User and Allowing the Changing of Those Settings
Interactive voice response (IVR) is a technology that allows a computer to interact with humans through the use of voice and DTMF tones input via keypad. In telecommunications, IVR allows customers to interact with a company's host system via a telephone keypad or by speech recognition, after which they can service their own inquiries by following the IVR dialogue. IVR systems can respond with prerecorded or dynamically generated audio to further direct users on how to proceed. IVR applications can be used to control almost any function where the interface can be broken down into a series of simple interactions.
FIG. 1B
Similarly, the customer may access the authentication device settings through voice prompts or IVR or through an Internet interface. In the case of voice prompts, the residential customer can dial a star code, such as *09. The authentication device can intercept the dialed numbers and immediately answer the call with a voice menu. The voice menu can then allow the customer to modify or set the options of the authentication device. The authentication device settings can also be controlled over the internet. A connection to the authentication device can be made over the Internet and the settings can be modified. Alternatively, the authentication device can store its settings on an Internet server. The customer can then connect to the Internet server where the settings are stored and make changes, ideally through a graphical interface. Once the changes are made, the authentication device can download the changed settings from the server and apply the changes to itself. If the authentication device is installed at the telephone service provider as was shown in , the provider can allow changes to be made through the internet or through voice prompts, in a similar fashion. If the authentication device is installed on a smartphone, the settings can be modified through the smartphone application.
c) Pass Through Voice Communication
Once a call is connected, the authentication device passes through all voice communications.
d) Access a Data Network
The authentication device has an onboard communication module. The communication module can be any such module known in the art that allows access to any known wired or wireless data network. Such a communication module may be an Ethernet networking card, telephone modem, DSL modem, cellular modem, Wi-Fi card, and the like.
e) Access an Authentication Server Over the Data Network
For both outgoing and incoming calls, the authentication device is able to contact an authentication server.
Once the authentication device is installed, the authentication device may need to contact the authentication server to request certification password and send heartbeat messages to indicate that it is installed at a particular number. The number of the phone the authentication device is connected to can be directly inputted into the authentication device by the installer. For example, if the authentication device is installed at a landline, then the phone number can be inputted into the authentication device through the voice menu and prompts. Or the phone number can be inputted through an internet connection. If the authentication device is installed at a cellular phone, then the phone number can be inputted into the software application through the settings menu of the application.
Alternatively, the authentication device can discover the phone number the device is connected to autonomously. For example, the authentication device can dial an Automatic Number Announcement Circuit (ANAC) phone number and determine its' own phone number. If the authentication device is installed at a cellular phone, the device may discover the cellular phone number through the operating system.
f) Intercept Incoming Calls
Once the authentication device knows the phone number it is connected to, it may contact the authentication server and send a heartbeat message to indicate that it is installed at the particular number. The authentication server can then use this information to determine if a particular phone at a particular phone number has an authentication device installed or not installed. The installation status of the authentication device is used to determine and differentiate between cases; where the authentication device is installed and not currently placing calls, and where the authentication device is not installed. This determination becomes important where one caller is trying to spoof the caller ID of another caller. Such cases will be explained through examples later. The contents of the heartbeat message can consist simply of the phone number the authentication device is connected to. The authentication server can thus store the phone number in the heartbeat message. The heartbeat message can be sent at regular intervals, such as hourly, every 2 hours or daily. The heartbeat message received at the authentication server may be time stamped by the server. The time stamping would allow for the server to determine if the authentication device has transmitted the heartbeat message in the predetermined time interval, such as hourly, every 2 hours or daily. If not, the authentication server may determine that the authentication device has stopped functioning. As will be elaborated, the status of the authentication device will be utilized by the server for caller ID authentication and spoof detection.
In addition, the caller ID authentication device may intercept, mute, or drop incoming calls from the telephone network. As was explained previously, the authentication device is installed before the customer phone and all calls pass through the authentication device before reaching the customer phone. If a call is received by the authentication device, the authentication device may intercept the call and play a message to the caller, without ringing the household phones. Alternatively, the authentication device may ask the caller to verify they are human, as will be explained in later embodiments. In another embodiment, the authentication device may take a message from the caller, without ringing the household phones. In another embodiment, the authentication device may pick up and immediately hang up the incoming call, without ringing the customer phone. In another embodiment, the authentication device may play a tone to the incoming call, such as a tone to indicate that the line is disconnected, without ringing any of the household phones.
g) Selectively Ring the User Phone
In other embodiments of the invention, the caller ID authentication device is able to drop calls without ringing the customer phone or the cellular phone it is connected to, based on white and black lists maintained by the authentication device and the global list server. Besides the modifications described, the authentication device passes through all voice calls to and from the household phones transparently.
h) Extract Caller ID from an Incoming Call
The authentication device is able to extract caller ID information from an incoming call. In most embodiments, the device of the called phone will not pass through the incoming call ring until the caller ID is extracted and authenticated. In most embodiments, the device of the called phone rings the called phone after the caller ID is authenticated and the authentication status is shown to the user, such as through modified caller ID information.
i) Communicate with the Authentication Server for the Purpose of Caller ID Authentication
The authentication device implements logic to communicate with the authentication server and transmit and receive messages for the purpose of caller ID authentication.
j) Modify Caller ID Presented to the User to Notify the User of the Caller ID Authentication Status, and/or Notify the User of the Caller ID Authentication Status Through Other Means
For incoming calls from the telephone network to the user phone, the caller ID authentication device is able to intercept, extract, modify and transmit the caller ID information received from the telephone service provider. The caller ID authentication device for instance might receive from the telephone service provider a call with a caller ID in the form of “John Smith 555-1000”. The caller ID authentication device might then change this caller ID to “[A] John Smith 555-1000”. The added “[A]” prefix signifying that the caller ID has been authenticated, as will be elaborated and explained later in this document. Once the caller ID has been intercepted and modified if need be, the authentication device can then pass the caller ID information to the called phone. The customer phone can then display the caller ID information passed by the authentication device. The customer phone can also be manufactured to take advantage of such modified caller ID information. For example, if a caller ID has the prefix “[A]”, the caller ID display can light up in a green color. If the modified caller ID has the prefix “[S]” to indicate a caller ID that is spoofed, the caller ID display can light up in a red color for example. Likewise, the customer phones can ring in a different ring tone depending on the caller ID prefix received.
As was explained previously, the caller ID information is relayed to a phone between the first and second rings. This causes several problems with respect to caller ID authentication. If the called authentication device wishes to modify the received caller ID with a prefix to indicate the authentication status, it must do so within the first and second ring. Unfortunately, this might not be possible. The called authentication device receives the caller ID of the incoming call within the first and second ring. Thus, the authentication device must extract the caller ID information, contact the authentication server to authenticate the caller ID, receive a response from the server and send the modified caller ID to the called phone in the time between the first and second ring. The time between the first and second rings might not be sufficient to carry out the steps of caller ID authentication and caller ID modification by the authentication device. Therefore, the called authentication device in most cases might need to delay the ringing of the called phone for a certain period of time to provide a time buffer for the purpose of caller ID authentication.
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For instance, the authentication device might receive a first ring of an incoming call. The authentication device might delay the first ring and not transmit the ring to the called phone. Between when the first and second ring comes, the authentication device will have already received the caller ID information from the telephone service provider. The authentication device can then contact the authentication server, authenticate the caller ID and modify the received caller ID to indicate its' authentication status. Once the caller ID is authenticated the authentication device can then ring the household phone a first time, send the modified caller ID, and ring it a second time. Depending on the connection speed of the medium or to the authentication server and the overall time required for the authentication server to respond, the delay maybe one or two rings. If the delay required is a single ring, the authentication device begins ringing the household phone on the second ring received from the telephone service provider. If the delay required is two rings, then the authentication device begins ringing the household phones on the third ring received from the telephone service provider. And so on for additionally needed rings. Rings can also be delayed on a time basis. Such that the authentication device begins ringing the household phones, 5 seconds for example, after receiving the first ring from the telephone service provider. Solutions to overcome the delayed ringing will be explored later in this document.
k) Maintain Internal Database of Whitelisted and Blacklisted Telephone Numbers
The authentication device maintains an internal database of phone numbers. These phone numbers may then be used for the purpose of call handling.
l) Implement Logic to Autonomously Place Telephone Numbers on the Internal Whitelist
The authentication device implements logic to autonomously place telephone numbers of incoming calls on the internal whitelist.
m) Implement Logic to Allow the User to Manually Place Telephone Numbers on the Internal Blacklist
n) Handle incoming calls according to the list the incoming telephone number is contained in
The user may also place the telephone number of an incoming call to the internal blacklist through star codes.
The authentication device may then use the internal whitelist and blacklist to handle incoming calls. The handling of incoming calls may also take into account the authentication status of the incoming caller ID.
o) Access a global list server
The authentication device may contact a global list server. The global list server may be implemented within the authentication server.
p) Upload Contents of the Internal List to the Global List Server
The authentication device can upload the contents of the internal lists to the global list server.
q) Check the Telephone Number of an Incoming Call Against the Global List Server for Call Handling Purposes
The authentication device may contact the global list server and check the incoming caller ID against the global list server for the purpose of handling such calls.
r) Evaluate Calls for Certain Criteria for the Purpose of Automated Blacklisting of Incoming Calls
The authentication device may also evaluate calls according to certain metrics, such as call duration, which party talked, which party talked the most, voice recognition, etc. These metrics may then be used to autonomously place the telephone number of the incoming call on the internal blacklist or identify such calls as potential blacklisted calls.
3) Authentication Server Functional Overview
The authentication server may be implemented in hardware or software, or combinations thereof known in the art. The authentication server is able to receive communication from authentication devices installed at each user's phone. The authentication server contains logic to analyze the communications it receives. The authentication server has a database to store and process received information. The server is able to reply to inquiries made by the authentication devices. The server is able to dial phone numbers, connect to authentication devices over the telephone network, and send data on the telephone system through any available means, such as modem, DTMF tones, or through a computerized voice. The server may also maintain call handling lists on behalf of the authentication devices. The server may aggregate call handling lists of authentication devices. These and other functionalities will be elaborated in following embodiments.
The authentication server may have a static point of contact, such as a static IP address. Conversely, the authentication server may have a dynamic point of contact, such as a dynamic IP address. In cases of a dynamic point of contact, the dynamic contact point of the authentication server would be registered with a central service, such as a DNS type service. Through a DNS server, the authentication devices may look up the dynamic IP address through a previously known authentication server domain name. Similar schemes may be used on other types of telecommunication systems in which the authentication devices and authentication server resides.
FIG. 3A
FIG. 3B
4) Caller ID Authentication Device Operational Overview (-)
FIG. 3A
a) Called Authentication Device Contacts Server Variation ()
FIG. 3A
FIG. 3A
represents the flowchart and message diagram the caller ID authentication device, at both the caller and called party, uses to authenticate caller ID according to an embodiment of the invention. In , both the caller and the called parties have a caller ID authentication device installed. In most instances, in order to authenticate, both caller and called parties will be required to have implemented or installed the caller ID authentication devices at their ends. Situations where only one party has the caller ID authentication device implemented will be elaborated later and solutions explored. The heartbeat message sent by the authentication devices will not be explicitly shown or elaborated in this embodiment and will be explained in more detail in other embodiments, such as in Section (5-b).
FIG. 3A
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In , the sequence of events is shown where the caller phone is calling the called phone . The caller and the called phones might both be on a traditional landline phone system. In such a case, the authentication device of the caller and the authentication device of the called might be implemented as was described in Section (1-a) or (1-b), or any combination thereof. Such as, the caller might have installed the authentication device between his phone and the NID. And the authentication device of the called phone might be implemented at the telephone service provider. Alternatively, either or both parties could be using a cellular phone, and the authentication device might consist of a computer program integrated into the phone OS or a downloadable application. The authentication server can consist of any server and accompanying database and logic known in the art and which can be reached by the authentication devices.
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FIG. 3A
The caller in initiates a call by dialing a number (Step S). Before the dialed number is passed to and connected by the telephone service provider, the authentication device is able to intercept and capture the dialed numbers (Step S). The device may either collect the dialed numbers and only pass them to the telephone network when a valid telephone number is entered; or the device may pass through and capture the dialed numbers transparently. As soon as the number is dialed and the call is placed to the called phone through the PSTN (Step S), the authentication device of the caller contacts the authentication server (Step S). The authentication device of the caller transmits the caller's phone number and the phone number of the called phone, hereafter referred to as the calling status message. For example, the transmitted calling status message to the authentication server from the caller authentication device can be in the form of “CALLING_STATUS-[Caller: ###-###-####, Called: ###-###-####]”, where the # signs are numbers which together signify a telephone number. The calling status message contains the information that the “Caller” at the following phone number is currently calling the “Called” party at the following number. The transmitted information can be in the form of regular text data, or could be encrypted before transmission. The transmitted information could also be in any other format. Thus, the authentication server receives and stores the information that the caller's phone number is calling the called phone number (Step S). When the call from the caller reaches the called phone , the authentication device of the called phone intercepts the call (Step S). The call is not allowed to ring the called phone. The authentication device of the called phone receives and extracts the incoming caller ID information (Step S). The caller ID information contains the number of the caller phone, however, the information within the caller ID is assumed not to be reliable and may have been spoofed. Thus, the authentication device of the called phone contacts the authentication server (Step S). The authentication device of the called phone inquires the authentication server as to whether the caller phone as indicated by the caller ID is calling the called phone number. The called phone number is the phone number that is currently connected to the authentication device of the called phone. Such an inquiry, hereafter referred to as an authentication request message, may be in the form of “AUTHENTICATION_REQUEST-[Caller: ###-###-####, Called: ###-###-####]”, where the # signs are numbers which together signify a telephone number. The transmitted information contains an authentication request that the “Caller” at the following phone number is currently calling the “Called” party at the following number. As was previously discussed, the “Caller” number is the number obtained by the authentication device through the received caller ID. The “Called” number is the number that the authentication device is currently connected to and is currently authorizing caller ID's for. The transmitted information can be in the form of plaintext data, or could be encrypted before transmission. The transmitted information could also be in any other format. The authentication server looks up if there has been a calling status message received where the caller and the called parties are identical to the parties in the authentication request message (Step S). If so, the authentication server replies with indication that the caller phone is currently is indeed calling the called phone (Step S). The authentication server reply, hereafter referred to as the authenticated message, can be in the form of “AUTHENTICATED-[Caller: ###-###-####, Called: ###-###-####]”. Once the caller ID is checked against the authentication server, the authentication device allows the call to ring the called phone. The authentication device modifies caller ID information (Step S) and passes the modified caller ID to the called phone (Step S). If the caller ID is determined to be authenticated, the caller ID is prefixed with the text “[A]”. The user of the called phone can thus see the authentication status of the caller ID.
As will be elaborated in later examples, the authentication server may determine the caller ID as being spoofed. Such a spoof determination can be made if the authentication server did not receive a calling status message from the caller authentication device. Another method of spoof determination can be where the calling status message was sent from the caller authentication device, but the calling status message indicated a called number different from the called number which is contained in the authentication request. If the authentication server determines the caller ID to be spoofed, the server reply can be in the form of “SPOOF_VERIFED-[Caller: ###-###-####, Called: ###-###-####]”, hereafter referred to as spoof verified message. If the caller ID is determined be spoofed, the authentication device can modify the caller ID by prefixing it with the text “[S]”.
In this embodiment, the authentication status of the caller ID was passed to the called customer through a modified caller ID. The notification of the caller ID authentication status is not limited to modifying the caller ID of the called phone. Other means of notifying the customer of the caller ID authentication status are possible; such as a distinctive ring, changing the color of the caller ID display, changing the display color of the smartphone, etc.
The authentication server may time stamp received messages. Such time stamping would allow messages to be in effect for a certain period of time. For example, the calling status message may be time stamped by the server. The calling status message can then be set to expire after a certain period of time. After the calling status message has expired, the server can determine that the calling party has most likely ended the call initiation to the called party and the called party will not be sending an authentication request. Alternatively, the authentication device may send a message that the call has ended and to expire certain messages sent from the authentication device. The server may also expire matching calling status and authentication request messages once the caller ID has been determined to be authentic. Expiring a message may consist of marking the message as expired or deleting the message from the server records. The advantage marking a message as expired and not deleting the message will become apparent when the concept of auditing authentication devices is explained.
It should be noted that some terminology used in this document may vary or be used interchangeably with others; however, those skilled in the art should be able to comprehend the subject matter without any explicit definitions of such terminologies. For example, with respect to this section, Section (4-a), the terms caller and called phones were used to signify the two parties involved in the call. The terms such as source caller, source number, origin number, and source phone may refer to and identify aspects of the calling party. Similarly, destination phone, destination caller, and called phone may refer to and identify aspects of the called party. Thus, such terminologies may be used interchangeably throughout this document. As such, all interchangeable terminologies known in the art may not be explicitly defined.
FIG. 3B
b) Server Contacts Called Authentication Device Variation ()
The authentication device of the called phone contacting the authentication server requires the decoding of caller ID information after the first ring. As was discussed, this might induce a delay in ringing the called phone and as a result, the caller may hang up if the delay is too long. To alleviate this problem, once the authentication server receives information from the caller authentication device, the authentication server may contact the authentication device of the called party instead of waiting for the called phone authentication device contacting the server. Each authentication device may store its' own internet IP address corresponding to its telephone number, with the authentication server. The authentication server may thus contact the appropriate authentication device at this IP address. Such an IP address, or other contact address, can be transmitted to the server alongside a heartbeat message, as will be elaborated in later embodiments. Once the authentication server contacts the authentication device of the called party, the server can transmit that the caller at a certain number is calling. The authentication device can then store this information. When the call is received at the called phone, the authentication device can intercept the call, extract the caller ID, match the caller ID of the caller to the one transmitted from the server, and put the call through with the modified caller ID information to indicate authentication status. It should be clear that such a system would reduce delay of ringing the called phone and may in fact remove any delays. If the called authentication device receives the caller number from the server before the called phone starts ringing, the authentication device will not need to delay the ringing of the called phone. The authentication device may pass through the first ring in real time, extract the caller ID after the first ring, check that the caller ID matches caller's telephone number received from the server, modify the caller ID information, send the modified caller ID to the called phone, and pass through subsequent rings to the called phone.
The server may send a password when contacting the called party authentication device to prove that the server itself is authentic, and not being impersonated. The sent password may be the same certification password sent to the authentication device by the server, as will be elaborated in later embodiments. The sent password may alternatively be a hash of the certification password. The hashing function may consist of MD5, SHA 256, and the like. The authentication device may then verify the received password and authenticate the server as trustworthy.
This variation can also be interpreted as the push variation. In Section (4-a), the called authentication device contacted or pulled the authentication status of the caller from the server. In this Section and embodiment, the authentication server will push the authentication status of the caller to the called authentication device.
FIG. 3B
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illustrates the sequence where the authentication server initiates contact with the called phone authentication device instead of the authentication device initiating contact with the server. The caller phone initiates a call to the called phone (Step S). The authentication device of the caller phone extracts the dialed numbers (Step S), passes the call through to the telephone network (Step S) and sends a calling status message to the authentication server (Step S). The calling status message contains the information that the caller phone at a particular number is calling the called phone at a particular number. The authentication server receives the information and contacts the authentication device of the called phone (Step S). The authentication server contains the IP address or other appropriate communication contact address of the authentication device of the called party. The address of the device would be previously transmitted by the device to the server. The transmission from the server to the device, hereafter referred to as the calling status server message, may be in the form of “CALLING_STATUS_SERVER-[Caller: ###-###-####, Called: ###-###-####]”. The authentication device of the called phone receives and stores the message. The authentication device waits to receive a call. It is likely that the calling status message from the server will arrive before or at the same time as the call is received from the telephone network. When the call from the caller is received and intercepted (Step S), the authentication device does not need to delay the ringing of the phone if the calling status message has already been received. The first ring from the telephone line it passed through to the called phone. The called authentication device extracts the caller ID after the first ring (Step S). The caller ID number is authenticated and matched against the caller number contained in the calling status server message (Step S). If the caller ID number is identical to the caller number in the calling status server message, the caller ID is modified to indicate an authenticated caller ID (Step S) and passed to the called phone before the second ring (Step S). All subsequent rings are passed to the called phone.
If a call is received by the called authentication device, and the called authentication device was expecting a calling status message, the called authentication device may contact the authentication server. Such a scenario could be the result of a caller with a spoofed caller ID. In such a scenario, the called authentication device would contact the authentication server and request authentication. If the caller hadn't called the called party then the authentication server can reply with a spoof verified message. If the caller had called the called party, then it can reply with an authenticated message.
The following embodiments will mainly be demonstrated using the variation where the called authentication device contacts the server. It should be noted that, where technically possible, the server contacts called authentication device variation may be used instead.
FIG. 4
FIG. 5B
5) Caller ID Authentication Device Operational Examples (-)
FIG. 4
a) Caller Using a Valid Caller ID ()
FIG. 4
FIG. 3A
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represents, by way of example, the flowchart and messaging diagram the caller ID authentication device at both the caller and called party uses to authenticate caller ID, according to one embodiment of the current invention. The caller phone with the phone number 555-1000 initiates a call to the called phone with the phone number 555-2000 (Step S, S, and S). As certain steps are the same in this embodiment as was explained in , and Section (4-a), they will not elaborated in detail. The caller phone is using the correct caller ID and is transmitting the correct calling ID of 555-1000 to the called phone, that is the caller phone is not trying to spoof or fraudulently change its' caller ID in this example. The caller ID of the calling party sent to the called party is usually determined by the telephone provider of the calling party. Thus, the callers ID provided by the telephone provider is usually the correct caller ID. There may be instances where the caller ID can be set up, changed or modified by the calling party. As soon as the caller phone dials and initiates the call to the called phone, the authentication device of the caller contacts the authentication server (Step S). The authentication device of the caller transmits to the authentication server that the caller at 555-1000 is calling 555-2000. The calling status massage sent to the authentication server can be in the form of “CALLING_STATUS-[Caller:555-1000, Called: 555-2000]”. The authentication server stores the calling status message (Step S). The call is received at the called phone and is intercepted by the authentication device of the called phone before the called phone rings (Step S). The called phone is prevented from ringing by the authentication device of the called phone. The authentication device of the called phone receives the caller ID of the caller phone, which is 555-1000 (Step S). The authentication device then contacts the authentication server and inquires if the caller phone with the number 555-1000, which is obtained from the caller ID, is calling the called phone with the number 555-2000 (Step S). The authentication request message sent by the authentication device to the authentication server can be in the form of “AUTHENTICATION_REQUEST-[Caller:555-1000, Called:555-2000]. The authentication server compares the calling status message against the authentication request message (Step S). Since the caller and called fields match in the calling status message and the authentication request message, the authentication server replies back to the authentication device of the called phone in the affirmative with an authenticated reply message (Step S). The authenticated message from the authentication server can be in the form of “AUTHENTICATED-[Caller:555-1000, Called:555-2000]”. The authentication device modifies the caller ID (Step S) and rings of the called phone (Step S). The authentication device of the called phone transmits the updated caller ID of “[A] 555-1000” to indicate that the caller ID has indeed been authenticated. The user of the called phone can now be assured that he is indeed answering a call from 555-1000.
FIG. 5A
b) Caller Using a Spoofed Caller ID, where the Spoofed Number has an Authentication Device ()
FIG. 5A
represents the flowchart and messaging diagram the caller ID authentication device at both the spoofing victim and called party uses to authenticate caller ID, according to one embodiment of the current invention and by way of example.
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In the previous examples outlined in Sections (4-a, 4-b and 5-a), the heartbeat message the authentication devices sent to the authentication server where not explicitly shown. The heartbeat message in those cases was not critical in explaining those embodiments. In this example, the heartbeat message that the authentication devices send to the authentication server is explicitly shown. The heartbeat message is a message sent by the authentication device to the authentication server which indicates to the server that an authentication device is installed and functioning at a particular number. The heartbeat message of the good caller authentication device is sent to the authentication server (Step SA). The heartbeat message in this example and embodiment only contains the phone number that the authentication device is connected to, namely 555-1000. The contents of the heartbeat message sent to the authentication server can be in the form of “HEARTBEAT-[555-1000]”. The authentication server stores, or if already stored, updates its database with the heartbeat message (Step SC). The authentication server thus contains the information that the number at 555-1000 has an authentication device installed and is functioning. Likewise, the called phone authentication device sends a heartbeat message (Step SB). And the authentication server is now informed that the number at 555-2000 has an authentication device installed (Step SC). As will be explained shortly, the heartbeat message will be critical in determining if a certain caller ID information is being spoofed. The heartbeat message may also incorporate the certification password of the authentication device, as will be elaborated later. The heartbeat message can consist of a single message which informs the server that an authentication device is installed at a particular number. Such a single message can be in the form of a certification request, as will be elaborated in later sections. The heartbeat message may also contain any other desirable supplemental information. Supplemental information may include the status of the party the authentication status is connected to. For example, the party status information transmitted to the server may be that the party is not currently calling anyone, the party is taking an incoming call, the party is not home, or similar information. The server may then use this information if need be. The heartbeat message may be time stamped by the server. The server may then determine that an authentication device is not functioning if the next heartbeat message is not received in a certain period of time relative to the previous heartbeat message.
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The bad caller with the phone number 555-9999 wishes to call the called party at 555-2000 with a spoofed caller ID of 555-1000. The spoofed caller ID belongs to the good caller . The good caller has the caller ID authentication device installed. The bad caller initiates a call to the called party at 555-2000 with a spoofed caller ID of 555-1000 (Step S). The called party authentication device intercepts the call (Step S) and extracts the caller ID (Step S). The called party authentication device receives the caller ID of the bad caller as 555-1000. The authentication device contacts the authentication server and inquires if the caller phone at 555-1000 is calling the called phone at 555-2000 (Step S). The authentication request message sent by the authentication device to the authentication server can be in the form of “AUTHENTICATE-[Caller:555-1000, Called:555-2000]”. The authentication server determines through the previously received heartbeat message in Step SA that the caller at 555-1000 has an authentication device installed (Step S). The authentication server also determines that the authentication device has not informed the authentication server of calling the number 555-2000 through a calling status message. Thus, the authentication server determines that the call that the called phone at 555-2000 is receiving has not come from the caller 555-1000. The authentication server responds with a spoof verified message (Step S). The message sent by the authentication server may be in the form of “SPOOF_VERIFIED-[Caller:555-1000, Called:555-2000]”. The authentication device receives the message, and puts the call through by modifying the caller ID with the “[S]” tag to indicate a spoofed caller ID (Step S). The user of the called phone now sees the caller ID “[S] 555-1000” is indeed a spoofed caller ID, and can decide if he wishes to pick up the phone.
It can be seen that the heartbeat message allows for the authentication server to determine a caller ID is being spoofed. The heartbeat message is used to determine that an authentication device is installed at a particular location. And in this example, that the authentication device of the good caller did not indicate it dialed the called number, thus the caller ID received by the called phone was determined fraudulent.
There could be instances where the caller authentication device has indeed sent a calling status message; however, the authentication server has not received the transmission due to a bad connection, network loss, network delay, or the like. To mitigate such situations, the authentication server may contact the caller authentication in cases where an authentication request message is received and a corresponding calling status message has not been sent by the caller authentication device. In such a case, the authentication server may contact the caller authentication device and request it to transmit a calling status message for any calls in progress. The caller authentication device may then retransmit a lost calling status message, or transmit that there are no calls currently in progress. The authentication server may then determine the authentication status of the caller ID. As is apparent to the reader, the authentication server determines that the caller has an authentication device installed through received heartbeat messages or a previously received certification request, as will be elaborated in the next section. If a heartbeat message has not been transmitted within a certain timeframe, the authentication server may also contact the device and request a retransmission of the heartbeat message. Such safeguards would ensure that a call with a legitimate caller ID is not marked as spoofed due to network or equipment malfunctions. The authentication server may also acknowledge received messages to the authentication devices, such that if the authentication device does not receive an acknowledgement from the server, the device may retransmit any messages.
The heartbeat message can be in the form of a single message that indicates to the authentication server that an authentication device is installed at a particular phone number. The authentication device can thus register with the server of its presence and would not need to transmit a heartbeat message at regular intervals. As was described, the authentication server may simply contact the caller authentication device in cases where an authentication request message is received and a corresponding calling status message has not been sent by the caller authentication device. If the authentication server contacts an authentication device for a certain phone number and does not receive a reply for certain amount of time, such as a couple of hours or a day, the server may determine that the authentication device has been removed from the phone number or is simply malfunctioning.
FIG. 5B
c) Caller Using a Spoofed Caller ID, where the Spoofed Number does not have an Authentication Device ()
FIG. 5B
FIG. 5B
FIG. 5A
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represents the flowchart and message diagram that illustrates determining an irresolute caller ID, according to one embodiment of the current invention and by way of example. is similar to the example shown in , however, in this example the good caller does not have an authentication device installed.
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FIG. 5A
The authentication device of the called phone sends a heartbeat message to the authentication server (Step SA). Similar to the previous example of , the heartbeat message from the called phone will not be used to determined caller ID authentication status. It is merely shown in this example to signify that there are no other heartbeat messages, such as from a caller device. The authentication server stores the received heartbeat message (Step SB).
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The bad caller with the phone number 555-9999 wishes to call the called party at 555-2000 with a spoofed caller ID of 555-1000. The spoofed caller ID belongs to the good caller . The good caller does not have a caller ID authentication device installed. The bad caller initiates a call to the called party at 555-2000 with a spoofed caller ID of 555-1000 (Step S). The called party authentication device intercepts the call (Step S). The called party authentication device receives the caller ID of the bad caller as 555-1000 (Step S). The authentication device contacts the authentication server and inquires if the caller phone at 555-1000 is calling the called phone at 555-2000 (Step S).
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The authentication server determines that the caller at 555-1000 does not have a caller ID authentication device installed, since a heartbeat message was never received from the authentication device at that number and neither was a calling status message (Step S). Since the caller at 555-1000 does not have an authentication device installed, it is not clear if this number is indeed being spoofed or not. This call could indeed be coming from the caller at 555-1000, or it could be coming from another caller impersonating 555-1000. The authentication server thus does not have a way of knowing. As a result, the authentication server replies with an irresolute reply (Step S). The data format of the irresolute reply sent from the server, hereafter referred to as a irresolute message, can be in the form of “IRRESOLUTE-[Caller:555-1000, Called:555-2000]. The authentication device receives the irresolute reply, and puts the call through by modifying the caller ID with the “[I]” tag to indicate a irresolute caller ID (Step S). The user of the called phone now sees the caller ID as “[I] 555-1000” is indeed an irresolute caller ID, and can decide if he wishes to pick up the phone. The user can see that the caller ID in question is has not been authenticated as a true and valid caller ID, and nor has it been determined as a spoofed caller ID.
The significance of the heartbeat message can once again be seen. The lack of a heartbeat message from the caller authentication device allows the authentication server to determine that the caller does not have an authentication device installed. Thus, the authentication server determines that the caller may in fact be calling the called party or may not be calling the called party. Therefore, the authentication server replies with an irresolute reply. The authentication device of the called party can now indicate that the caller ID is irresolute.
Summarizing Sections (4) and (5), the authentication server is able to classify an authentication request as authenticated, spoofed or irresolute. The authenticated classification requires that; the caller phone number transmitted in the calling status message be identical to the caller ID extracted by the called authentication device and transmitted in the authentication request, and the caller destination phone number extracted by the caller authentication device and transmitted in the calling status message be identical to the called authentication device phone number transmitted in the authentication request.
The spoofed classification requires that; the caller phone number transmitted in the calling status message not be identical to the caller ID extracted by the called authentication device and transmitted in the authentication request, or the caller destination phone number extracted by the caller authentication device and transmitted in the calling status message be not identical to the called authentication device phone number transmitted in the authentication request.
The spoofed classification can also occur if the caller identified by the called party is calling another party which is not the called party making the authentication request. Such that; the caller phone number transmitted in the calling status message is identical to the caller ID extracted by the called authentication device and transmitted in the authentication request, and the caller destination phone number extracted by the caller authentication device and transmitted in the calling status message is not identical to the called authentication device phone number transmitted in the authentication request.
The spoofed classification can also result from the authentication server not receiving a calling status message from the caller authentication device indicating a call to the called party, but the authentication server receiving a heartbeat message from the caller authentication device which indicates that the caller authentication device is functioning.
The irresolute classification is the result authentication server determining that the caller party does not utilize an authentication device, such that the authentication device never transmitted a heartbeat message. The irresolute classification can also result from the caller authentication device malfunctioning, and stopping the transmission of heartbeat messages. And, where the caller authentication device does not respond to requests from the authentication server to retransmit heartbeat or calling status messages.
FIG. 6A
FIG. 6B
6) Securing Against Authentication Device Impersonation Through Authentication Device Certification (-)
FIG. 6A
a) Certification Process ()
One weakness in the previously shown embodiments is the fact that the authentication server relies on the messages it receives from the authentication devices as true. If an authentication device states that the phone number 555-1000 is calling 555-2000, the authentication server accepts this to be true. There could be instances where the authentication device is tampered with and the authentication device itself is modified to impersonate another authentication device installed at another phone number. Thus, there needs to be a way to certify that an authentication device is installed at a particular number.
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shows the flowchart and message diagram for one embodiment of the invention that certifies that an authentication device is indeed installed at a particular phone number and is acting on behalf of the phone number. When an authentication device is initially installed at a phone and it initiates a certification request (Step S). The certification request may be the result of the device being newly installed on the phone line, or the certification request may be because of the previous password being lost due to a power outage. Or the certification request could be the result of the previous password being compromised. Or the certification request could be initiated at regular intervals for added security. The device contacts the authentication server and transmits a request for a certification password (Step S). Along with the password request, the authentication device sends its' own phone number. As a practical example, the request for a password transmitted to the authentication server by an authentication device connected to the phone number 555-1000 may be in the form of “PASSWORD_REQUEST-[555-1000]”. Hereafter, such a transmission will be referred to as a password request message. The transmission of the message may be over any known medium, for example, over a telephone network, such as through a modem or through a data network, such as through a DSL connection. The authentication server receives the password request message and initiates a telephone call to the phone number contained in the request (Step S). The call would be connected through the PSTN, or a similar network could be utilized to connect to the phone number 555-1000 of the calling party. The authentication device intercepts the call without ringing the phone it is connected to and picks up. The device can blindly determine that the authentication server is calling simply by the fact that a password request message was sent recently. The device can also verify that the authentication server is calling by extracting the caller ID of the authentication server. The device can also authenticate the caller ID of the calling server, by sending an authentication request message to the server. Such safeguards would ensure that the authentication server cannot be impersonated by a third party. The authentication server generates a unique password for the authentication device (Step S) and transmits it (Step S) while the call is still in progress. The transmission of the password may be in the form of DTMF tones, assuming the password consists of digits 0-9. The transmission of the password may also be in the form of an authentication server generated computerized voice, where the authentication device is able to decode the spoken password. The transmission of the password can also be in the form of a modem or fax type transmission, where the authentication device possesses circuitry to decode such a transmission. As a practical example, the password transmission from the authentication server to the authentication device may be in the form of “PASSWORD-[Pass1000]”. Hereafter, such a message will be referred to as a password message. The authentication server stores the generated password along with the phone number for which it was generated for (Step S). The authentication device receives the generated password and also stores it (Step S).
If the certification password is transmitted to a device through DTMF tones generated by the server, the certification password would consist of just digits, such as “1234”. Those digits could then be transmitted back to the authentication server as the certification password over any communications medium, such as a cellular data connection, in the form of a data packet which contains those digits. The transmitted DTMF tones could represent upper and lower case characters through an encoding scheme, for instance a transmitted DTMF tone of “26” could represent “a”, “27” could represent “A”, “28” could represent “b”, and so on. So a DTMF certification password of “262728” would be decoded “aAb”. These characters can then constitute the certification password transmitted to the server. If the certification password is a fax, the fax can contain letters or characters and the authentication device can use optical character recognition to recognize the letters or characters. The certification password in the form of a fax can be the fax image itself, where the any transmissions to the server would consist of the fax image in any image format, such as GIF, TIFF, JPEG, etc, over any communication medium. The certification password transmitted through a modem interface can consist of characters, images, audio files, etc, which could be sent back to the authentication server over any communication medium and system. The certification password may be prefixed or postfixed with tags, and the authentication device may extract the certification password between such tags. The certification password may be any form of secret which is passed from the server to the device and able to be received at the device connected to the phone number. Another example of such a secret may be a unique encryption key corresponding to the device phone number. The device may then use the unique encryption key to encrypt messages sent to the server, and the server can then decrypt the communication by a corresponding unique decryption key corresponding to the device phone number.
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The password generated in this embodiment was “Pass1000”. That password was given only as an example, but it should be clear that such a password would likely be insufficient. The password generated by the authentication server must be unique and different, such that no two authentication devices posses the same password. Otherwise, one authentication device might impersonate the other authentication device, if it is discovered that they share the same password. Ideally, the authentication server transmits to the authentication device a password of at least a 10 character alphanumeric password, such as “UxYt6h&2!9”. A 10 character alphanumeric password, including symbols, is sufficient to generate at least 4.2×10possible password combinations. Thus, it is relatively easy to generate a unique password for each authentication device connected to 1×10possible phone numbers and that is also resilient to brute force attacks. The length and complexity of the password within the password message may be modified as needed.
The password received by the authentication device can now serve as a certification that the device is indeed connected to and has ownership of the phone number in question. Every time the authentication device contacts authentication server, the device can transmit its assigned password to certify that it is connected to and possesses ownership over the phone number. In the previously given example, the authentication device for the phone number 555-1000 was transmitted the password “Pass1000”. The authentication device can now send this password with each transmission to the authentication server to certify that the authentication device is indeed connected to the 555-1000 phone number. If the authentication device for the caller 555-1000 wishes to transmit to the authentication server that the caller is calling 555-2000, the calling status message transmitted to the authentication server can consist of “CALLING_STATUS-[Caller:555-1000, Password:Pass1000, Calling:555-2000]”. The authentication server can thus look up the password transmitted to the number 555-1000, verify that it matches the previously generated and stored password “Pass1000”, and certify that the transmission is originating from the authentication device connected to 555-1000. If the password “Pass1000” didn't match the password sent to the 555-1000 number, the authentication server can determine that this transmission is fraudulent and simply ignore it. If the authentication device at 555-1000 never requested to be certified, the authentication server could again simply ignore the transmission or reply back instructing the authentication device to re-certify. The reader should note that in the given calling status message example above, the certification password field is inserted after the phone number the authentication device installed at and wishes to certify. Thus, the convention of the phone number of the authentication device followed by the certification password, such as “CALLING_STATUS-[Caller:555-1000, Password:Pass1000 . . . ”, will be used throughout this document. Other conventions may be used instead where required and desirable. Instead of the authentication device transmitting the certification password to the authentication server, the transmission can consist of the hash of the password. Or the transmission can consist of a challenge from the server to the device, such as the server instructing the device to append specific characters to the certification password and transmit the hash of the resultant string. The server would then append the specific characters to stored password for the device, compute the hash and compare it to the received string from the device. Through utilizing such techniques known in the art, the authentication device may prove knowledge of the certification password to the server by transmitting the actual certification password to the server or by transmitting information derived from the password. It should also be evident that a sufficiently complex password would make the transmission of the phone number of the authentication device redundant. For instance, the calling status message can take the form of “CALLING_STATUS-[Password:3a834yh3o2y6xy3m7, Calling:555-2000]”. The authentication server can simply look up which authentication device the password corresponds to and determine telephone number of the authentication device.
It should be apparent to the reader that the certification process described in this embodiment of the invention is sufficient to overcome the possibility of some person or device fraudulently impersonating another authentication device. Only the authentication device connected to a particular phone number is sent a specific password by the authentication server. This password can only be used by the authentication device to transmit messages to the authentication server where the caller number matches its' own number. The authentication device cannot pretend to be a different number since it does not poses the authentication server assigned password for that number. In order to increase security, the authentication device can request a new password daily or weekly.
The certification process can also serve as the heartbeat message. The certification request, the sending of the certification password or the acknowledgement of the receipt of the password can serve as the initial heartbeat message. Thus, once an authentication device is certified, it would serve as registration with the authentication server that the authentication device is installed at a particular phone number. The authentication device may then send heartbeat messages at regular intervals, or omit doing so. In either case, the authentication server would have a record that an authentication device is installed at a particular phone number. If the authentication device is wished to be uninstalled, the user can initiate a uninstall routine on the authentication device, where the authentication device would notify the server that it is being uninstalled. The device could send a un-installation message to the server with the number of the device and the certification password of the device. The server would then delete the stored certification password and all heartbeat messages. Thus, the server would be able to determine that the authentication device is no longer installed at that particular number for future authentication tasks. The authentication device may indicate if the initial certification will be the only heartbeat or additional heartbeat messages will be sent and at which intervals. Alternatively, the server may request heartbeat messages at server specified intervals. Or the device and server can negotiate the intervals and also contents of the heartbeat messages.
Where applicable and unless otherwise stated for subsequent embodiments, the caller authentication device will be assumed to have received the certification password “Pass1000”, and the called authentication device will be assumed to have received the certification password “Pass2000” from the authentication server.
FIG. 6B
b) Caller ID Authentication and Certification without an Authentication Server ()
There may be instances where the calling and the called party wish to authenticate caller ID, but there isn't access to an authentication server or the authentication server is not present. In such cases, the caller ID authentication can be performed between the two parties in one embodiment of the invention. Such an embodiment can be considered a peer-to-peer method of certification and caller ID authentication. The caller authentication device would transmit a calling status message to the called party over a data network to indicate that the caller is calling the called party. The called party authentication device would receive the message and compare it to the received caller ID information of the incoming call. And thus the called party authentication device can notify to the called customer the caller ID authentication status. The certification process would also be performed between the two parties, without the need for an authentication server. The first party authentication device would contact the second party authentication device and transmit a certification password request. The second party authentication device would call the first party and transmit a certification password. The second party authentication device would then send a request for a certification password to first party authentication device. The first party authentication device would then call the second party and send a certification password. Thus, anytime the first party authentication device transmits to the second party authentication device, it transmits the received certification password from the second party. The second party can thus verify the first party is at the particular phone number. And anytime the second party authentication device contacts the first party device, it can send the received certification password. Clearly, the process of certification and caller ID authentication may take place between more than solely 2 parties. The caller authentication device may have received a certification password from 5 parties ahead of time, for example. Anytime the caller initiates a call to any of the 5 parties, the caller authentication device may transmit the certification password received from the respective party within the calling status message. Thus numerous users may authenticate caller ID's amongst themselves.
FIG. 6B
FIG. 6B
FIG. 6B
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Such an embodiment is shown in . represents the flowchart and messaging diagram that shows certification and caller ID authentication performed in a peer-to-peer fashion, where calling party forwards the calling status message to the called party. Steps S through S of illustrates the self certification process, and steps S through S illustrate the caller ID authentication process. Similar to the certification processes described previously, the self certification process would be performed once or at regular intervals. The certification process would take place before a call is placed. As was previously demonstrated, the caller ID authentication process requires that only the caller authentication device be certified. The certification of the called authentication device is not critical for caller ID authentication, but will be shown nonetheless. The caller phone authentication device initiates a certification request (Step S) and contacts the called phone authentication device and requests a certification password (Step S). The caller device can contact the called device through any transmission medium, such as an internet connection. The caller authentication device may look up the called authentication device contact address through a DNS type system set up for the purpose of authentication device IP address resolution. For example, the caller device can perform a DNS type lookup for the called phone number of 555-2000. The DNS type server would return the IP address for the authentication device of the called phone . Such a DNS type lookup system could also be implemented with regards to the embodiment described in Section (4-b). Once the caller device contacts the called device it sends a password request message. As was previously specified, the password request message contains the phone number of the device requesting a password; in this case the phone number is 555-1000. The called authentication device places a call to the caller phone (Step S). The caller authentication device intercepts the call and picks up. The called device generates a unique password (Step S) and transmits the password message (Step S). The called device stores the unique password generated for the caller phone number along with the phone number it was generated for. And the caller device stores the received password for the specific called phone number (not explicitly shown). Thus, when the caller authentication device wishes to send messages, such as a calling status message, to the called authentication device, the caller device can look up the certification password it received from the called device and transmit the certification password with the calling status message. And when the called device receives the calling status message with the accompanying password, it can compare the caller password to the one that it generated during the certification process. In this embodiment, the certification password that the called authentication device generated for the caller device is “Pass1000”. Thus, anytime the caller authentication device wishes to transmit a message to the called authentication device, it must send to password “Pass1000” to certify that the caller authentication device is indeed installed at the 555-1000 phone number.
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The called device then begins the process of certification with the caller device (Step S). It should be apparent that the certification process of each device may take place in any order, such that steps S-S could have taken place before steps S-S. The called authentication device initiates the certification process and contacts the caller authentication device for the purpose of certification and sends password request message (Step S). The password request message contains the phone number the authentication device is connected to. The caller device then calls the called phone (Step S), generates a unique password (Step S) and transmits the uniquely generated password (Step S). Thus, both the caller and the called device are certified against each other. If the caller device wishes to contact the called device, the caller device can send the certification password to prove that it is residing at the stated phone number.
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Once the caller device is certified, the caller phone may now place a call to the called phone . The caller phone initiates a call to the called phone (Step S). The authentication device captures the dialed numbers (Step S). However, in this embodiment, the authentication device of the caller phone is delaying the placing of the call. As was previously elaborated, such a delay allows more time for the caller ID authentication to take place. Thus, the authentication device of the caller phone extracts the called phone number and transmits a calling status message to the called phone authentication device (Step S). The calling status message sent to the called phone device includes the phone number of the caller, the called number, and the caller certification password as proof the device is connected to the caller phone. The caller status message may in the form of “CALLING_STATUS-[Caller:555-1000, Password:Pass1000, Called: 555-2000]”. After the caller device transmits the calling status message (Step S), the call is placed to the called phone (Step S). The called phone authentication device receives the calling status message and stores it (Step S). Since the calling status message was transmitted before the call was placed on the telephone network by the caller authentication device, the likelihood of the called party receiving the calling status message before receiving the call is very high. The called party receives the call, which is intercepted by the called party authentication device (Step S). Since the calling status message was received, the called authentication device is already expecting a call, and allows the call to ring. The called authentication device extracts the caller ID after the first ring (Step S). The device begins the process authenticating the caller ID (Step S). First, the certification password received within the calling status message is compared to the certification password transmitted to the caller phone during the certification process. If it is determined that the certification password belongs to the caller device as indicated in the calling status message, the message is determined to be originating from the caller authentication device. The device compares the caller ID of the caller to the caller field of the calling status message received in Step S. If they match the caller ID is authenticated. The caller ID is modified (Step S) and the caller ID is transmitted to the called phone before the second ring (Step S) to indicate the caller ID authentication status to the called customer.
If a call is received by the called authentication device, and the called authentication device was expecting a calling status message but did not receive one, the called authentication device may contact the caller authentication device. Such a scenario could be the result of a caller with a spoofed caller ID. In such a scenario, the called authentication device would contact the caller authentication device and request authentication. If the caller hadn't called the called party then it can reply with a spoof verified message. If the caller had called the called party, then it can reply with an authenticated message.
FIG. 6C
c) Caller ID Authentication and Certification without an Authentication Server—Pull Variation ()
FIG. 6C
shows a variation of the embodiment presented in Section (6-b). In Section (6-b), the calling status message was transmitted by the caller authentication device to the called authentication device. However, there is no requirement for the caller authentication device to transmit a calling status message. The caller device may simply pass the call to the telephone network. Once the called device receives the call, the called device may contact the caller authentication device to authenticate that the caller is calling the called party. In other words, the embodiment presented in Section (6-b) can be considered as peer-to-peer certification and authentication where the authentication status is pushed by the caller authentication device. Whereas, the following embodiment can be considered as peer-to-peer certification and authentication where the authentication status is pulled by the called authentication device.
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FIG. 6B
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represents the flowchart and messaging diagram that shows certification and caller ID authentication performed in a peer-to-peer fashion, where called party requests authentication status from the calling party. In , a caller with the phone number 555-1000 using an authentication device wishes to call the called party at with a phone number 555-2000 with an authentication device . Since in this embodiment, an authentication server is not available, the parties self-certify and self-authenticate. In , the Step S through Step S is executed identically as was explained in . In , Step S and Step S of is not executed. The caller authentication device does not transmit a calling status message. The caller device merely passes the call to the telephone network (Step S). The called authentication device intercepts the call (Step S). The called authentication device may delay ringing of the called phone to provide a buffer for the transmission and reception of messages required for caller ID authentication. The called authentication device extracts the caller ID of the caller (Step S). Since a calling status message was not received and an authentication server is not available, the called authentication device contacts the caller authentication device. The called device may look up the caller authentication device network address through a DNS type system. The called authentication device transmits an authentication request to the caller authentication device (Step S). The message may be in the format of “AUTHENTICATION_REQUEST-[Caller:555-1000, Called:555-2000, Password:Pass2000]”. The authentication request message requests authentication that the caller at the number 555-1000 has called the party at 555-2000. The caller number was obtained from the extracted caller ID, and the called number is the number of the called party. The password “Pass2000” was the password transmitted by the caller to the called party. It should be clear that in this case, transmission of the password is not critical for the purpose of caller ID authentication since the called device is merely inquiring if the caller device called it. The caller device does not necessarily need to certify the called device. The caller device may simply reply with an answer without certification. One motivation for including the certification password within the authentication request could be to reduce the chance of a rouge device sending random authentication requests and trying to obtain information from caller authentication devices. The transmission of the certification password may also protect against a compromise of DNS system, where the DNS resolves the address of the caller authentication device to an illegitimate party device. Once the caller authentication device receives the authentication request, the device compares the called number of the authentication request with the numbers captured from the caller phone in Step S. If they match, it replies with an authenticated message (Step S). If the caller authentication device was not calling the called party, then the transmitted message would have been a spoof verified message. The format of the authenticated message may be in the form of “AUTHENTICATED-[Caller:555-1000, Password: Pass1000, Called: 555-2000]”. The message authenticates that the caller with the number 555-1000 is calling the called party at 555-2000. As was previously discussed, the transmitted certification password is not critical in this embodiment, but might be desirable for numerous reasons. Once the called device receives the authenticated message, the device prepares the modified caller ID indicating the authentication status (Step S). The device rings the customer phone and transmits the modified caller ID (Step S).
With regards to the embodiments shown in explained in Sections (6-b) and (6-c), it should be noted that to some extent, role of the authentication server is being assumed by the caller authentication device. In the embodiment explained in Section (6-b), the caller authentication device is assuming the role of the authentication server by transmitting calling status messages to the called device. In the embodiment explained in Section (6-c), the caller authentication device is assuming the role of the authentication server by receiving an authentication request and transmitting an authentication reply.
i) Transmitting the Certification Password Through Caller ID
d) Alternate Transmission Methods of the Certification Password
In Sections (6-a to 6-c), the certification process consisted of a phone call from the certifying entity to the entity wishing to be certified. Such as the certifying device or server placing a phone call to the device wishing to be certified, and transmitting the certification password through the telephone network while the call is in progress by utilizing DTMF tones, computer generated voice, modem or fax transmission, and the like. The device wishing to be certified thus used the received password and transmitted it to the other entity in all communications, through any available transmission means, such as internet, cellular, Wi-Fi and the like, to prove to the other entity the ownership of the phone number in question.
One issue with such a system is the load that would be placed on the telephone network with a large number of users. For instance, if there are millions of authentication devices wishing to be certified, the authentication server would have to call each number, wait for the other entity to pick up the phone, transmit the certification password, hang up, and call the next number. Such a system might be impractical for several reasons, especially if the certification process is performed at regular intervals, such as every night or week, as was previously discussed. The issue becomes even more concerning if the authentication device is in the form of software running on a cell phone with a prepaid plan. Thus, each call from the authentication server to the customer's prepaid cell phone would result in using up a minute of customers air time or maybe more.
To mitigate such issues, instead of transmitting the certification password through the phone call, the certification password can be transmitted through the caller ID system itself. Thus, such a system would not require the authentication device to pick up the phone and wait as the authentication server transmits the password during the call. And since the call is not picked up, the customers would not be deducted minutes from their cell phone plans. Such a system would also be quicker in terms of transmitting passwords.
To transmit the certification password through the caller ID system, the certifying entity would call the entity wishing to be certified, such as the authentication server calling the authentication device, or one authentication device calling another in a peer-to-peer system. However, the certifying entity, such as the authentication server, would modify the caller ID of the call to include the certification password within the caller ID. For instance, if the authentication server wished to transmit the certification password “Password1000” to the authentication device, the password can be placed in the name field of the caller ID. Thus, the caller ID of the call from the authentication server to the authentication device can include the phone number of the server followed by the certification password, as an example the caller ID might be “1-800-555-1000 Password1000”. This caller ID would indicate that the calling number is 1-800-555-1000, which would be the number of the authentication server, and the certification password would be “Password1000”. Alternatively, the authentication server may omit sending the calling number within the caller ID, and simply transmit the certification password within the calling number field. As an example the caller ID of a call from the authentication server might be “2249850325231”. Thus, the certification password for the device would be the numbers in the calling number field of the caller ID, which is “2249850325231”. The password within the number field, or name field, of the caller ID can be prefixed or post-fixed with special characters, such as the ‘#’ characters, for example “#2249850325231”, “2249850325231#”, or “#2249850325231#”. Such characters would indicate that a password is being transmitted from the authentication server. Alternatively, the password can be prefixed or post-fixed with special digit sequences, such as the sequence ‘555’, for example “5552249850325231”, “2249850325231555”, or “5552249850325231555”. Since the certification password is contained within the caller ID, the authentication device can simply receive and store the certification password without picking up the phone.
Summarizing such a certification procedure, the authentication device requests a certification password from the authentication server by transmitting a password request message. The message may be transmitted over any medium, such as internet, cellular, or any other communications protocol. The password request message would include, at least, the phone number of the authentication device. The authentication server would receive the password request message. The authentication server would generate a unique password for the authentication device. The authentication server would then place a call to the authentication device over the telephone network. The authentication server would place the call with a modified caller ID which would include the certification password within the caller ID. The certification password could be in the name field of the caller ID, or the certification password can be placed in the number field of the caller ID. The authentication device then receives the call from the authentication server, but does not pick up the call. The authentication device can blindly determine that the authentication server is calling since the password request message was just sent. The authentication device can also determine the server is calling through the caller ID of the call, if the caller ID still contains the telephone number of the server, and is not replaced by the certification password. The authentication device may additionally determine that the server is calling with a certification password by checking for special character or digit prefixes or postfixes within the caller ID. The authentication device extracts the caller ID of the call and stores the password contained within the caller ID. The placement of the password within the caller ID, such as within the telephone number field or name field, can be predetermined or agreed upon. The predetermination can be preprogrammed into the system. Alternatively, the authentication server and authentication device can negotiate the placement when the authentication device sends the password request message. For instance, the password request message from the authentication device to the authentication server can indicate that the certification password should be placed in the telephone number field of the caller ID. Once the authentication device stores the certification password, it can merely wait for the authentication server to end the call. The authentication server can end the call after a predetermined number of rings. Alternatively, the authentication device can transmit to the server, over any available transmission means such as internet, cellular, Wi-Fi, or the like, that it has received the password and that it can end the call, at which point the server would end the call. The authentication device can transmit the received password in all future communications to prove to the server that the authentication device is actually installed and is connected to the telephone number in question. The server can then compare the telephone number of the authentication device transmitted within the message and the corresponding certification password to determine that the authentication device is indeed connected to and has ownership of such number.
ii) Other Certification Methods
The certification password may also be transmitted through multiple calls from the authentication server. Such that the first calls caller ID can constitute the first half of the certification password, and the second calls caller ID can constitute the second half of the certification password.
Another possible method of transmitting the certification password could be performed through a text message. If the authentication device is installed on a cellular phone, the authentication server can transmit the certification password as a text message to the authentication device phone number. Such text messaging protocols may include but not be limited to Short Message Service (SMS), Multimedia Messaging Service (MMS), and Enhanced Messaging Service (EMS). The authentication device can then receive and extract the password from the text message. If the authentication device cannot extract a certification password, such as from an SMS or a computerized voice, due to hardware or software limitations, the user may retrieve the password and manually enter it into the authentication device.
In addition to the certification methods discussed, the certification password may also consist of a password transmitted by postal mail to the customer. The customer may then enter the received password into the authentication device. Thus, the authentication device may use the postal password as proof of the mailing address of the customer. The postal password may also be used in conjunction with the certification password sent by the authentication server to the customer's phone, through the server calling or caller ID methods as previously discussed, Such combinations of postal and telephone certification passwords would serve as proof that the authentication device has ownership of both the customer's phone number and proof of a mailing address. Instances of where such combinations of certification passwords might be desirable would consist of sensitive establishments using an authentication device. For example, it might be desirable for a bank using an authentication device for their toll free customer service telephone number to be certified by both a postal certification password and a certification password transmitted by phone. Thus, an attacker would need both passwords to impersonate the bank. Certification attacks will be discussed in more depth in the next section. Similar to the phone or caller ID transmitted certification passwords, postal certification passwords may be sent at regular intervals. The authentication device may also contact the authentication server to request a postal certification password. The server may then send the request to the appropriate personnel or automated postal mailing system for processing.
Although the preferred embodiment of certification is where the authentication device contacts the server through a data connection to request certification and the authentication server contacts the device through a PSTN to transmit the password, other communications mediums are possible. The certification request made by the authentication device may take place over the telephone network. The device may dial a number which belongs to the authentication server. The server may then pick up the call and receive the request for certification by the device. The device may for instance transmit the appropriate DTMF tones to request a certification password and transmit the phone number of the device also through DTMF tones. The authentication server may then contact the device at the phone number. An advantage to such a system would be where the authentication server may employ more sophisticated forms of caller ID authentication with respect to the call received from the authentication device. Such forms of caller ID authentication may be those which are described in U.S. Pat. No. 8,238,532 and U.S. Patent Application US 2013/0109358. Such caller ID authentication methods employed by the authentication server would add another layer of security and safeguards to the certification process, since the caller ID of the authentication device would be authenticated by the server before sending a certification password back to the authentication device phone number. The authentication server may also require that the caller ID of the device, which is determined to be authentic through such sophisticated means, match the telephone number transmitted within the certification request.
The certification password transmitted by the authentication server to the authentication device can represent credentials required to retrieve a secondary certification password. The authentication server can transmit the initial certification password and create an appropriate web page to contain a secondary certification password to be retrieved. The authentication device can then query the domain address for the web page given by the certification password to retrieve a more complex secondary certification password, as can be referred to as the full certification password. The full certification password may then be used instead of the initial certification password in further communications with the server. For instance, if the authentication device received the initial certification password “1234”, through DTMF tones for example, the authentication device can navigate to the web page “1234.html” at a predetermined domain address of “authenticationserver.com”. Thus, the authentication device would navigate to “http://authenticationserver.com/1234.html” and retrieve the full certification password. The full certification password may be a string of characters, binary data, or any other form of information contained in the web page. The webpage “1234.html” could be made to expire after a certain time has passed, such as 30 seconds after transmitting the initial certification password, which would make it less likely for an attacker to discover the full certification password page. The full certification password web page can also be retrieved by transmitting the authentication device phone number along with the initially received certification password, thus the initial certification password would serve as credentials. For instance, the full certification password may be retrieved at the web address “http://authenticationserver.com/certificationpassword?phonenumber=555-2000&password=1234”. The “certificationpassword” field indicates that this is a certification password request at the server name “authenticationserver.com” address. The “phonenumber” field indicates the phone number of the requesting authentication device, which is given as 555-2000 in this example. The “&” is the separator between the two fields. The “password” field indicates the certification password received by the authentication device, which is given as “1234”. Again, password “1234” may be made to expire after a certain time to prevent instances of attackers trying to guess the initial certification password of the authentication device and retrieving the full certification password from the server. The full certification password retrieved by the authentication device may be the public key of the authentication server, which may be uniquely generated for each authentication device. The authentication device can then encrypt any communications, such as calling status messages and other messages, with the server public key. The server would then decrypt the messages with the server private key. The authentication device may also upload the authentication device public key to the server through the web page given by the initial certification password. Thus the initial certification password may be used to exchange any type of cryptographic information. Alternatively, the initial certification password can be used by the device to construct a more complex certification password, based on known cryptographic techniques and methods. The construction algorithm can be predetermined or negotiated between the device and server. The resultant full certification password would then be used as proof of ownership of the telephone number and used in transmissions with the server.
The authentication device can transmit within the certification request to the authentication server the preferred method of receiving the certification password, such as through DTMF, caller ID, and other methods as are disclosed. Or the authentication server can transmit available methods of sending the certification password and allow the authentication device to select which method to receive the password. The authentication device and server may negotiate which method to use based on the capabilities of the device and server, the cost of each method as incurred by the device or server, and the time constraints incurred by the device or server in transmitting the certification password.
iii) Passive Certification
The certification methods thus described were all examples of active certification. Active certification is defined as the authentication device wishing to be certified receiving the certification password from the certifying device or server. The concept of passive certification will now be introduced. In a passive certification system, the authentication device does not request or receive a certification password from any entity. Thus, the authentication device does not contact the server and request a certification password. Passive certification still utilizes the concept of transmitting the certification password as proof that the authentication device is connected to the phone number in question. However, the certification password is generated passively.
The passive certification password is generated as a result of the authentication device receiving phone calls which are determined as having authenticated caller ID's. If an authentication device receives a call with an authentic caller ID, it implies that the calling party is using an authentication device and the caller ID matches the calling party telephone number. An authentic caller ID also implies that an authentication server has received a calling status message from the calling authentication device. Thus, the authentication server knows all the calls placed from callers with authentication devices to other called parties, including parties with authentication devices. If the authentication server were to record and store all calls made by all authentication devices, then an authentication device wishing to be certified can send information about the calls it received from callers with authentication devices and use it in place of an active certification password. Instead of using an active certification password to prove that an authentication device is connected to a particular phone number, the authentication device can construct a passive certification password from calling parties caller ID's. For example, an authentication device can construct a passive certification password by taking last digit of the last three received calls with authentic caller ID's. For instance, if the last three authentic received caller ID's were “555-1000”, “555-1021”, and “555-4028”, the authentication device can construct a passive certification password by chaining the last digits together, such as “018”. Or, the passive certification password can be the chaining of the last two digits of the last three calls, such as “002128”, or it can be the sum of the last digits of the last three calls, such as “9”. Or, the passive certification can simply be the chained caller ID's of the last three calls, such as “555100055510215554028”. As can be seen, the construction of the passive certification password may be done in several ways, which can be dependent on several factors such as the last digits of a telephone number of received calls, or sums of digits, etc. However, the passive certification is constructed from the received caller ID's which have been determined to be authentic. The construction of the passive certification password can utilize any aspect of the received caller ID's, including, but not limited to; the digits of the telephone number of caller ID, the characters of the name field of the caller ID, and the time the caller ID was received. The authentication device of the called party can transmit additional aspects of the received caller ID, such as name of the caller, time of the call, or any information contained within the caller ID, for the purpose of having such information available to the authentication server for constructing a new password. Transmission of such information would preferably be done within the authentication request message. The construction of the passive certification password can be preprogrammed into the authentication devices. Or the authentication devices and the authentication server can negotiate a construction on their own. The authentication server may for example instruct the authentication devices to construct the passive certification password as the multiplication product of the last digit of the last 4 calls with authentic caller ID's. The authentication device can then use that construction until instructed otherwise. After the passive certification password is constructed, it can be transmitted to the server during each message.
The only restriction is for the passive certification password to be constructed from caller ID's of callers with authentication devices. Such calls would consist of authenticated caller ID's. Caller ID's which are determined to be spoofed or irresolute would not be used to construct the passive certification password. The reasoning behind this is that the authentication server does not receive a calling status message from spoofed or irresolute calls since those callers are not using an authentication device.
As can be seen, the passive certification password can serve as proof that the authentication device resides at a certain telephone number since only the authentication device at that telephone number would have knowledge of the caller ID's received by that number. The authentication server can then compare the passive certification password against the stored calling status messages, determine that the construction of the passive certification password matches the caller ID's received by the authentication device, and determine that the passive certification password is correct. The passive certification password also is more secure, since it changes every time the authentication device receives a call with an authentic caller ID. The passive certification password also does not need to be renewed at regular intervals because of this fact.
Passive certification can also be combined with active certification. For example, sensitive entities, such as businesses, banks, and so on, can be actively certified. And non-sensitive entities can be passively certified. The active certification of the certain entities would also make the system more robust. If active certified entities called individuals who are using passive certification, the passive certification can be more trusted, since the received caller ID's are of entities with active certification. For instance, the authentication server can require that the passive certification password incorporate a predetermined number of actively certified caller ID's before the passive certification password is to accepted by the server as proof of ownership of the telephone number. Such a requirement would prevent an attacker from constructing a fake passive certification password by making calls to himself from phones that he controls. Hence, the server would delay authenticating calls made the device not meeting such a requirement and the called parties may be replied with an irresolute determination.
Alternatively, an authentication device during initial installation can be actively certified and then switch over to passive certification after a predetermined time or after receiving a predetermined number of authentic caller ID's. Or the authentication device can be actively certified at initial installation and transmit the active certification password alongside the passive certification password. Or the authentication device can be actively certified at initial installation and at regular intervals and transmit the active certification password alongside the passive certification password.
Summarizing the process of passive certification by way of example, an authentication device at the phone number “555-7000” is initially active certified at installation. The active certification password sent by the authentication server is “Password1000”. The passive certification password construction is predetermined to be the last 2 digits of the last two received calls with authentic caller ID's. The full certification password is predetermined as the chained active certification password and the passive certification password. Thus, at initial installation, the full certification password for the authentication device is “Password1000”, since the authentication device has not received any calls. If the authentication device received a call where the caller was utilizing an authentication device, that caller ID would be authenticated. As an example, the authentic caller ID of the call is given as “555-1027”. The certification password would now be “Password100027”, which is the active certification password and the chained passive certification password of last two digits of the last two calls. Since the device only received 1 call, it only chains that caller ID's last two digits. If the authentication device transmitted this certification password along with its phone number “555-7000” to the authentication server, such as during a calling status message, the server would first check that the active certification password matches the password sent to the authentication device at the number “555-7000”. The authentication server would then check if the passive certification password matches the calling status messages on record. That is, did a caller with a caller ID ending in “27” call the number “555-7000”. If so, the certification password is determined to be correct. The server may thus accept any messages from the authentication device, and the server is assured that the authentication device is connected to the phone number “555-7000”. Since the certification password is the product of active certification, which was delivered by calling the “555-7000” number, and passive certification, which is the result of the “555-7000” received caller ID's. Thereafter, if the authentication device receives another call with an authentic caller ID of “555-1073”, then the full certification password becomes “Password10002773”. If the authentication device receives another call with an authentic caller ID of “555-1077”, then the full certification password becomes “Password10007377”. And so on.
Although not discussed in detail, the passive certification password may also be constructed from the dialed numbers. Since the authentication device is already capturing dialed numbers by the customer, these may be used in conjunction with received caller ID's to construct the passive certification password. However, since the certification password is used to certify that an authentication device is installed at a particular telephone number, the dialed numbers would not ideally be used for such a certification.
iv) Automatic Password Revocation
In another embodiment of the invention, the concept of passive certification is further extended to create a new password on each transmission of a message to the authentication server and revoke the old password.
The authentication device of the caller can construct a new password by incorporating the called number which is contained in the calling status message sent to the authentication server. The construction of the password would be performed according to a predetermined algorithm that incorporates the called number information within the current password. Such construction algorithms of the new password would be similar to the previous examples given, such as summing the last two digits of the called number to the current password. For example if the current password is “1234” and the caller authentication device transmitted to the server a calling status message with the called number “545-0055”, the number “55” can be summed with the current password “1234” and the new password would be “1289”. The password could only consist of just the last 4 digits of the summation to prevent the password becoming too large. Another example can be where the certification password has two fields, first field being the active certification password and the second field the automatic revocation field, such as “1234:56” and the last two digits of the called party are only added to the last field, such that the new password is “1234:11”, and the carry is disregarded. In case of an attack, the two fields would allow for the authentication server to verify that an authentication device is in possession of the active certification password before determining that the revocation field is incorrect and revoking the entire certification password. The authentication server can then use the same predetermined algorithm to construct the new password for the caller authentication device. Similarly, the called authentication device can use the “Caller” number transmitted within an authentication request to construct a new certification password each time it contacts the server. And the authentication server can construct a new certification password for the called authentication device. In automatic password revocation scheme, there would be no requirement that the caller ID of the caller be determined by the server as authentic, as would be performed through an authentication request and authenticated reply.
An advantage such password revocation is that if certification password of a victim is stolen, as soon as the attacker with an authentication device dials a number and sends a calling status message to authenticate a call with the victim's spoofed caller ID, the password of the victim is changed and the old password is revoked. Thus, any call made or received by the victim would result in the authentication server receiving the old password. The authentication server can immediately determine the password has been compromised and instruct the victim authentication device to recertify with a password request. Such an attack can further be mitigated by the authentication devices transmitting heartbeat messages at regular intervals which contain a certification password. The authentication server can then immediately discover if a password is stolen when a heartbeat contains an old password of a victim which is changed by an attacker making a call. Similarly, as soon as the attacker receives a call where the authentication device of the attacker sends an authentication request message, the password of the victim is changed and the old password is revoked.
v) Common-Secret Certification
Another method of certification can utilize a common secret between the user of the authentication device and the server. For instance, if the authentication server belongs to a bank and is certifying calls made from smartphones using the bank software which employs the authentication device, then a common secret certification may be used. An example of a common secret might consist of the customer's credit card number. The customer may enter the credit card number into the authentication device. When a call is made from the customer to the bank, the device may transmit a calling status message with the customer's phone number, and the credit card number as the certification password. The server would then look up the phone number, determine the customer name and verify the credit card number within the certification password field belongs to the customer. The server can then authenticate authentication request received from the bank call center, for example.
A common secret could consist of various secrets that are known between the authentication server and the user of the authentication device, where the server can match the common secret to a specific customer, but would be hard to obtain by an attacker. Such common secrets may include but is not limited to; social security number, birthday, credit card number, credit card PIN, postal mailing address, user biometric information, phone IMEI number, device MAC address, last purchase price made to a credit card, bank account balance, and last statement balance. A common secret would thus identify the customer, but would not necessarily verify the caller ID since the secret is not sent to the device phone number.
Therefore, a common secret password may be combined with active or passive passwords to construct the certification password, and both authenticate the customer and the caller ID of the call.
e) Attacks Against Active Certification
This section will explore attacks against the active certification process. Attacks against the passive certification process are very difficult since it would involve tapping into the telephone line of the authentication device and recording each caller ID received. And if an attacker can tap into a line, then the line is already compromised and the attacker can make any calls from that line and achieve impersonation.
The eventual objective of an attack would be to impersonate the caller ID of another customer. For instance, an attacker would want to be able to impersonate a banking institution and use the caller ID of the bank and call customers in the hopes of extracting sensitive banking information. If the bank and the customer both have authentication devices installed, then the attacker can never spoof its caller ID to match the banks caller ID. The reasoning behind this was explained in Section (5-b). And through the process of certification, attacks where the authentication device is hacked to transmit fraudulent messages to the server are eliminated.
Thus, the attacker might resort to certain avenues of attacking the authentication and certification system. Assuming the attacker wishes to impersonate a bank, the first option is for attacker can take an authentication device, and plug it into the banks telephone line. As was explained previously, the authentication device can be in the form of a hardware device which plugs into the phone line, or integrated into landline phones. The authentication device can then contact the authentication server and request a certification password, if the bank's authentication device is actively certified. The server can then call the bank and the attacker authentication device can answer the call and receive the password, or the device can retrieve the password through the caller ID, as was explained previously. The attacker can then remove the device, move to a new location, and call from his new location. This kind of attack would be equivalent to other forms of attack which obtains a certification password from the authentication server. Since the authentication device has the proper certification password, the server will assume that all messages sent from the device as valid and the device is still connected to the banks phone line. And if the attacker can spoof its caller ID to be the banks caller ID then his calls will be authenticated. Each time the attacker makes a call, his authentication device will contact the server and transmit a calling status message with the number of the bank, the certification password of the banks phone number, and the called number. The called party authentication device would then receive the call with the spoofed caller ID to match the bank's telephone number. The called authentication device would contact the server and send an authentication request with the spoofed caller ID of the bank and the called party telephone number. Since the server did receive a matching calling status message with the correct certification password, the server replies that the caller ID is authentic.
There are two safety mechanisms that mitigate this attack. The first is where the bank has its own authentication device installed. If the bank has its own authentication device installed, the process of certification will most likely fail. When the attacker's authentication device contacts the server for a certification password, the server calls the banks telephone number. If the certification password is to be transmitted during the call, then there is a chance that the banks authentication device might pick up the phone before the attackers device, since the caller ID of the call would be the telephone number of the authentication server. If the certification password is to be transmitted within the caller ID, then the bank's authentication device will notice that a certification password being transmitted within the caller ID. The bank's authentication device can notice that a certification password is being transmitted through either the authentication server's telephone number within the caller ID or through the special character or number sequences within the caller ID. If the bank's authentication device does notice a certification password within the caller ID, it can contact the authentication server and indicate that it did not request a certification password. Similarly, if the authentication device of the bank picked up the call from the authentication server and the server began to transmit a certification password, it can also indicate that it did not request a certification password. At which point, the authentication server would be alerted to an attack to the bank's telephone system and can alert the proper personnel. At the same time, the server may invalidate all certification passwords for the bank. In addition, the server may suspend all caller ID authentications for the banks telephone number. The appropriate personnel can then determine the nature of the attack, and once the attack determined to be eliminated, the authentication device of the bank can request a new certification password, and caller ID authentication may resume.
In the unlikely scenario that the attacker is able to obtain the certification password for the bank by plugging in an authentication device, the attack is mitigated in two ways. The first way is through the fact that the bank's device has the old certification password, while the attacker's device has the new certification password. Thus, as soon as the bank's device sends a calling status message, it's going to present the old certification password. At which point the authentication server is going to determine that even though a new certification password was requested and received, the bank is still using the old certification password. At which point, the authentication server would be alerted to an attack to the bank's telephone system and can alert the proper personnel. At the same time, the server may invalidate all certification passwords for the bank. In addition, the server may suspend all caller ID authentications for the banks telephone number. The appropriate personnel can then determine the nature of the attack, and once the attack determined to be eliminated, the authentication device of the bank can request a new certification password, and caller ID authentication may resume. The second way that the attack is mitigated is through the regular updates of the certification password. If certification passwords are updated on a daily basis, then the attacker would have to return to the bank every day, otherwise the certification password would expire and the attacker can't impersonate the bank with the correct certification password. The updating of certification passwords also mitigate attacks where the bank in question does not have an authentication device installed.
If it is discovered that an attacker was able to impersonate the bank, the authentication server would have all calls placed by the attacker on record, through the received calling status messages. Thus, those individuals the attacker called can be contacted by authorities and the attack can be mitigated. If the attacker is still making calls, those calls can be traced, tapped or recorded by authorities. Such factors would deter any attacker trying to compromise the authentication and certification process.
In addition to such safeguards, the authentication server may audit certain authentication devices randomly or when the server determines the authentication device is suspect. An authentication device may be determined suspect if it is making unusual amount of calls, or being placed on a large number of blacklists, blacklists will be described later in this document. The authentication device may be determined suspect by any other known means. The audit may consist of the server transmitting a request to the authentication device to send the last 10, 20, or any number of authenticated caller ID's it received. The audit may consist of the server invalidating the certification password of the authentication device and requesting that the authentication device recertify.
An attacker may also try to brute force the active certification password by transmitting messages to the server with a guessed password until the attacker gets a confirmation by the server that the password is correct. Such an attack is easy to mitigate since every bad guess is registered by the server. The server may block or disregard messages from certain IP addresses after 3 bad passwords attempts, for instance. Since the server is aware of attacks against certain numbers, those numbers could be transmitted stronger passwords by the server in subsequent certification request or the server may require those numbers to recertify in order to receive a stronger certification password.
Unless otherwise stated, for the remainder of this document any reference to certification will consist of active certification as were explained in Sections (6-a to 6-c). However, any form of certification may be substituted and used, where technically feasible.
f) Authentication and Spoof Detection Summary
FIG. 6D
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is a flowchart which summarizes the steps the authentication server uses in authenticating and detecting caller ID spoofing. When the authentication server receives an authentication request from the called authentication device (step ), it first checks if there is a matching calling status message (decision step ). If a matching calling status message exists, it replies to the called device with an authentic determination (step ). A matching calling status message could have been received but could have timed out if too much time had elapsed between the calling status message and the authentication request message, indicating the call referred to by the calling status message had ended before the authentication request message was received. In such a case, there would be no matching calling status message. Or a matching calling status message could have been expired due to the caller authentication device sending a call ended, or call busy message, which would indicate the call was terminated or was not connected, before the authentication request message was received. In such a case, there would be no matching calling status message. If there is no matching calling status message, the server checks if an authentication device is installed at the caller phone number (decision step ). The server determines an authentication device is installed at the caller phone number if it has received the expected heartbeat messages or, if heartbeat messages were not to be sent by the device, the device has performed the certification process. If neither of these events have taken place or if the caller phone has uninstalled the authentication device as determined by an uninstall message, the authentication server replies with an irresolute determination (step ). If the caller phone has an authentication device installed, the server checks if it has registered a communication contact address and is contactable (decision step ). The caller authentication device can register it's contact address with the server, such a network IP address, through heartbeat messages and allow for the server to contact it and query it for calling status messages. If the device either; has not registered its contact address, or has registered its contact address and is not responding to queries from the server, the server replies to the called device with a spoofed determination (step ). If the caller authentication device is able to be contacted, the server interrogates it to determine if it is currently calling the called party or which party it is currently calling (decision step ). If the caller device replies that it is not calling the called party, the server replies to the called device with a spoofed determination (step ). If the caller device replies that it is indeed calling the called party then the server replies with an authentic determination (step ). It is important to note that although the spoof determinations in steps and have been treated as being interchangeable with respect to the embodiments thus given and will be given in this document, they can be treated differently by the server in embodiments which require such differentiation. That is, the spoof determination in step can be treated by the server as a non-guaranteed spoof determination, and can indicated thusly to the called authentication device. The rationale of treating the spoof determination in step as non-guaranteed is that the caller authentication device is not contacted to confirm that it is not currently calling the called party. The caller party may indeed be calling the called party, but due to software, hardware or network errors, the server may not be made aware of it. Likewise, the spoof determination in step can be treated as a guaranteed spoof determination, and can be indicated by the server to the called authentication device when replying to the authentication request. The rationale for treating the spoof determination in step as guaranteed is that the caller authentication device confirms that it is not calling the called party. Therefore, if the called authentication device did receive such determinations, it can notify the user, such as prefixing the caller ID with a “[GS]” tag for a guaranteed spoof determination, and a “[NGS]” for a non-guaranteed spoof determination.
FIG. 6D
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The server or authentication device may modify the process of authentication and spoof detection given in . For instance, the server may skip steps , , and . That is the server can skip the steps of contacting the authentication device and querying the caller authentication device. The server would simply assume the device is not able to be contacted and proceed to step . Alternatively, the server may change the indication of non-guaranteed spoofed caller ID to irresolute, that is the determination given in step may be changed to irresolute caller ID. Such a change may be where the authentication device of the called party indicates to the server that it will only accept a guaranteed spoof determination, as shown in step , and if the spoof detection is non-guaranteed, then it should mark it as irresolute. Such a change may also be implemented by the server in implementations which require it. Therefore, the server or authentication device of the caller or called parties may modify and negotiate the handling of guaranteed and non-guaranteed spoof detection.
FIG. 7A
FIG. 7B
FIG. 7A
a) Authenticating Callers with a Blocked Caller ID ()
7) Authenticating Callers with a Blocked Caller ID's (-)
There could be cases where the caller has a blocked caller ID. A blocked caller ID is a caller ID where the number and name of the caller is not disclosed to the called party through the caller ID service. The caller ID of such calls usually displays “Private” as the caller ID or may display other text or no text at all. In one embodiment of the invention, the authentication device is able to authenticate a blocked caller ID. An authenticated blocked caller ID would be where the caller has a blocked caller ID and has an authentication device installed and is indeed calling the called number. The significance of a authenticating a blocked caller ID lies in the fact that the caller is using an authentication device. Most illegitimate telemarketers and individuals with bad intentions would be hesitant to install an authentication device on their line. As will be described in later embodiments, a caller with a blocked caller ID can be put on private and global whitelists and blacklists, which would prevent the caller from initiating further calls to certain or all parties.
FIG. 7A
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illustrates the flowchart and messaging diagram for authenticating a caller with a blocked caller ID. The caller and the called authentication device both send a heartbeat message to the authentication server (Step SA and Step SB). As was explained earlier, the heartbeat message may have been sent when the authentication device was first installed and turned on, and the heartbeat message could be sent at regular intervals thereafter. The heartbeat message along with all other messages sent to the authentication server may also contain the certification password. The authentication server stores the heartbeat messages, and if already stored, updates the storage of the heartbeat messages to signify that the authentication devices are installed at a particular phone number and still functioning. The caller with a blocked caller ID of “Private” initiates a call to the called party at the phone number 555-2000 (Step S). The authentication device of the caller captures the dialed numbers (Step S) and passes the call to the telephone network (Step S). As soon as the caller dials and initiates the call to the called phone, the caller authentication device contacts the authentication server (Step S). The authentication device of the caller transmits to the authentication server that the caller with the phone number 555-1000 is calling 555-2000 with a blocked caller ID. Revealing the phone number of the caller 555-1000 to the authentication server does not violate the privacy of the caller, since this number will not be revealed to the called party or the called authentication device. The transmitted calling status message sent to the authentication server from the authentication device may be in the form of “CALLING_STATUS-[Caller:555-1000, Caller ID: Private, Password:Pass1000, Calling:555-2000]”. The transmitted data indicates that the caller is calling from the phone number 555-1000 to 555-2000. The field of “Caller ID:Private” signifies the fact that the caller is calling with a caller ID which will show up at the called phone as “Private”. The password field signifies that this authentication device of the caller is certified and is sending the received certification password from the server as proof the device is residing at the particular phone number. The authentication server may thus use the phone number 555-1000 in the “Caller” field and the password Pass1000 in the “Password” field and compare it to the certification password sent to the phone number 555-1000 and if they match, the server may thus be assured that the authentication device is actually installed at the phone number 555-1000. The call is received at the called phone line, is intercepted by the authentication device of the called phone (Step S) and the called phone is prevented from ringing. The authentication device of the called phone extracts the blocked caller ID of “Private” (Step S). The authentication device then contacts the authentication server and inquires if a caller with a blocked caller ID of “Private” is calling the called phone at the number 555-2000 (Step S). It is important to note that the called phone does not know the number of the caller phone. The authentication request message sent by the called phone authentication device to the authentication server may be in the form of “AUTHENTICATION_REQUEST-[Caller:Blocked, Caller ID:Private, Called: 555-2000, Password:Pass2000]”. The transmitted data requests an authentication by the authentication server for a caller with a blocked number. The transmitted data also states that the caller ID of the caller is “Private”. The password field after the called phone number indicates that the authentication device of the called number has been certified. The authentication server receives the authentication request and looks up if a call has been made to the called number of 555-2000 by a caller with a blocked caller ID of “Private” (Step S). If such a call has been placed, the authentication server replies to the called authentication device in the affirmative (Step S). Otherwise, the authentication server can reply that the caller ID is irresolute. An irresolute caller with a blocked caller ID would signify that the caller is not using or does not have an authentication device. In the case of a blocked caller ID without an authentication device, the caller ID cannot be determined to be spoofed, since the caller ID is not displaying a phone number of the caller, only displaying that it is “Private”. The authentication device of the called phone receives the affirmative reply from Step S and puts the call through (Step S). The authentication device modifies the caller ID of the caller with the prefix “[A]”, thus the caller ID that the called phone receives would be “[A] Private”.
A blocked caller ID may display any type of text to signify a blocked caller ID or could simply be blank. It should be apparent to the reader that the embodiment thus presented did not rely on the text of the blocked caller ID. The caller ID authentication system would be able to function without any stated text for the caller ID field. The called phone authentication device does not need to supply any caller ID text to the authentication server. The caller authentication device can simply inform that a caller is calling 555-2000 with a blocked caller ID, without any indication of the transmitted caller ID text to the called phone. The called phone authentication device can inquire if any caller with a blocked caller ID is calling the called phone at 555-2000. The authentication server can simply look up if a caller with a blocked caller ID is calling the called party at 555-2000. And the authentication server can inform the called authentication device of the authentication status. A blank caller ID would be an unusual situation, but is presented to further the understanding of the embodiment.
A race condition could take place where there are two callers with blocked caller ID's are calling the same called party at the same time. Such a race condition has 3 possible variations. The first variation is where both callers are not using an authentication device. In such a variation, the authentication server would reply to the authentication device of the called party in the irresolute since the authentication server would never receive a calling status message. The second variation is where both callers are using an authentication device. In such a variation, the authentication server receives a calling status message from both callers and the server would reply with an authenticated reply to the called party authentication device. However, the server cannot determine which party was connected to the called party first. To mitigate such a scenario, the caller authentication device may send a call ended or called party busy message to the server if the caller party is unable to reach the called party. Such a message would ultimately allow the server to determine which party contacted the called party. Determination of the calling party phone number will be required for placing such a number on a private list, as will be elaborated later in this disclosure. The third variation is where one caller has an authentication device and the other does not. In such a scenario, the risk is where the caller without the authentication device connects with the called party, and the server authenticates the blocked caller ID because it received a calling status message from the caller with the authentication device. The risk can be mitigated by a call ended or called party busy message from the caller with the device which is unable to connect.
In addition to the caller party authentication device sending a call connected, call busy or called ended message to the server to signify the progress of the call for the purpose of mitigating race conditions, the called party authentication device may send additional authentication requests before the call is answered, while the call is being answered, and after the call has ended. Likewise, the server may send a message which updates the authentication status of the caller to the called party before, during and after the call. For instance, if one caller with a blocked caller ID without an authentication device calls the called party at the same time as another caller with a blocked caller ID and with an authentication device, the server would be updated with a calling status message. If the caller without an authentication device connected first, the called authentication device would send an authentication request with the blocked caller ID. The authentication server would mistakenly determine the caller ID as authentic and that of the caller party with the authentication device. The called party authentication device would then notify the called party that the blocked caller ID is authentic. If the caller with the authentication device sends a call busy to the authentication server, the authentication server would then be aware of the mistaken authentic determination that the caller with a blocked caller ID without an authentication device has actually connected before the caller with an authentication device. The authentication server may then send an update to the called party authentication device that the previous caller ID determination is invalid, and the new caller ID determination is irresolute. Alternatively, the called party authentication device may check back with the server for any updates to the caller ID authenticity determination before, during and after the call, and notify the user in such cases. The same methodology can be used to differentiate between two callers with blocked caller ID and with authentication devices calling the called party. Thus, if the caller is using a blocked caller ID, the authentication server might instruct the caller authentication device to submit call connected, call busy, or call ended messages. Or the authentication device might be configured to send such messages if the user is using a blocked caller ID. The authentication server might also instruct the called authentication device to re-authenticate after a certain time if a caller with a blocked caller ID is authenticated. At the least, a call busy or call ended message would likely be required for callers with blocked caller ID's to mitigate race issues.
FIG. 7B
b) Authenticating Callers with a Caller ID Nickname ()
In another embodiment of the current invention, the authentication system is able to authenticate callers with customized caller ID's or caller ID nicknames. A caller ID nickname would be an alternative to and a method of blocking a caller ID, where the caller is able to block his name and phone number from being displayed to the called party. A caller ID nickname is a nickname chosen by the caller who does not wish to display his name and phone number. The caller ID nickname is displayed on the called caller ID. The caller ID nickname could simply be the caller's real name, for example “John Smith”. Or the caller ID nickname could be a fictitious nickname chosen by the caller, for example “Smith_1337”. Such caller ID customization would be offered to customers by the telephone service provider for a fee, or for free. The caller ID nickname would either be unique within a telephone service provider or ideally the caller ID nickname would be unique within the entire telephone network employing caller ID nicknames. The caller ID nickname must be unique to prevent two callers from having the same nickname and being able to impersonate each other. Thus, a caller can choose a unique nickname within the telephone network, for example if a caller has chosen the nickname “John Smith”, that nickname would be unavailable to every other caller. Each telephone service provider may employ nicknames internally. For example, a caller being serviced by the telephone provider Verizon might chose the nickname “John Smith”, where that nickname is unique to the Verizon telephone provider. The Verizon telephone provider would then prefix the telephone provider's unique tag in front of the nickname. If the unique tag for Verizon is “Verizon:”, then the full caller ID nickname displayed to called parties would be “Verizon: John Smith”. Thus, the nickname “John Smith” would be unique to the Verizon telephone provider, and the full nickname “Verizon: John Smith” would be unique within the entire telephone network and all telephone service providers. For example, another caller may choose “John Smith” within the AT&T telephone service provider. And his full caller ID nickname would be “ATT: John Smith”. Thus, those two nicknames would be unique and be uniquely identifiable. As previously stated, ideally the nicknames would be chosen by the telephone customers that are unique to the entire telephone network, instead of being unique to the telephone providers and the telephone provider prefixing their unique tag in front of the nickname.
The concept of caller ID nicknames is analogous to an internet forum username. An internet forum username is a unique name that allows a person to post on an internet forum, instead of using private information such as an email address. Thus, the username allows for someone to post on a forum without revealing personal information and allows for other forum members to uniquely identify the individual through the unique username. The concept of telephone provider prefixes is analogous to the concept of sub-domains in DNS. For example, the blog host blogspot.com is able to host blogs through the use of sub-domains, such as googleblog.blogspot.com. Thus, blogspot.com is similar in concept to the telephone service provider prefix, and the googleblog is similar in concept to a caller ID nickname which is unique to blogspot.com. Thus, the full address googleblog.blogspot.com is unique to all address on the internet.
The nicknames for the entire telephone network would ideally be maintained by a global nickname server. The global nickname server would serve as a clearinghouse and prevent nickname conflicts between two users. Thus, the telephone service provider would allow its' customers to choose nicknames, and the nickname would be checked against the global nickname server to determine that it is unique to all the other nicknames. Once a caller calls a called party and wishes to use a caller ID nickname, the telephone provider looks up the caller nickname, retrieves the unique nickname and transmits the nickname to the called party as the caller ID. Thus the caller ID nickname is used instead of revealing the callers full name and phone number as would be the case of a non-blocked caller ID. In the case of a blocked caller ID, the nickname allows the called party to uniquely identify the call instead of receiving a caller ID of “Private” or a blank caller ID. The called user can use the caller ID nickname to determine if the caller has called before, or if the called party is someone who he talks to on a regular basis. A detailed technical explanation of methods and systems for implementing customized caller ID's or caller ID nicknames is beyond the scope of this invention disclosure. Readers wishing to learn more about technical details of such systems should consult US Patent Applications 2003/0007616, 2002/0080942 and U.S. Pat. Nos. 8,306,202, 8,077,850, 8,290,131, herein incorporated by reference in their entirety. This disclosure will only focus on authenticating such customized caller ID's. In the embodiments explained later, the caller ID nickname will be used for call handling. Caller ID nicknames can also be used for VoIP calls, as will also be explained shortly.
FIG. 7B
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illustrates the flowchart and signaling diagram for authenticating a caller with a caller ID nickname. The caller and the called authentication device both send a heartbeat message to the authentication server (Step SA and Step SB). As was explained earlier, the heartbeat message may have been sent when the authentication device was first installed and turned on, and the heartbeat message could be sent at regular intervals thereafter. The authentication server stores the heartbeat messages (Step SC). The caller with a caller ID nickname of “Smith_1337” initiates a call to the called party at the phone number 555-2000 (Step S). This caller ID nickname would be chosen by the caller through his telephone service provider. For every call, the telephone provider would transmit the caller ID of the caller as the appropriate caller ID nickname. In this example, the caller ID nickname of “Smith_1337” is unique to all caller ID nicknames within the telephone network, and no telephone provider prefix need to be affixed. The authentication device captures the dialed digits (Step S) and passes the call to the telephone network (Step S). As soon as the caller dials and initiates the call to the called phone, the authentication device of the caller contacts the authentication server (Step S). The authentication device of the caller transmits to the authentication server that the caller is calling 555-2000 with a caller ID nickname of “Smith_1337”. The phone number of the caller may or may not be revealed to the authentication server, in this example, the caller number is not revealed. Revealing the phone number of the caller 555-1000 to the authentication server would not violate the privacy of the caller, since this number will not be revealed to the called party or the called authentication device. The calling status message transmitted to the authentication server from the authentication device may be in the form of “CALLING_STATUS-[Caller:Blocked, Unique Caller ID: Smith 1337, Password:Pass1000, Calling:555-2000]”. The calling status message indicates that the caller is calling the phone number 555-2000. The field of “Unique Caller ID:Smith_1337” signifies the fact that the caller is calling with a unique caller ID which will show up at the called phone as “Smith_1337”. The password field signifies that this authentication device of the caller is certified. The authentication server verifies that the password of the caller authentication device is correct, and stores transmitted message (Not explicitly shown on figure). The call is received at the called phone line and is intercepted by the authentication device of the called phone and the called phone is prevented from ringing (Step S). The authentication device of the called phone extracts the caller ID nickname “Smith_1337” (Step S). The telephone company may transmit an identifier to indicate that the caller ID is a unique nickname, such as through a prefix, for example such as “Unique”, and the resultant caller ID would be “Unique:Smith_1337”. The receiving telephone may then display or remove the prefix. The authentication device may thus identify unique caller ID's and process them accordingly, including any removal of the prefix before being presented to the called party. In Step S, the authentication device contacts the authentication server and inquires if a caller with a blocked caller ID of “Smith_1337” is calling the called phone with the number 555-2000. It is important to note that the called phone does not know the number of the caller phone. The authentication request message transmitted by the called phone authentication device to the authentication server may be in the form of “AUTHENTICATION_REQUEST-[Caller: Blocked, Unique Caller ID:”Smith_1337″, Called: 555-2000, Password:Pass2000]”. The transmitted data requests an authentication by the authentication server for a caller with a caller ID nickname. The transmitted data also states that the unique caller ID of the caller is “Smith_1337”. The password field after the called phone number indicates that the authentication device of the called phone number has been certified. The authentication server receives the authentication request and verifies the password of the called authentication device. Then the authentication server looks up if a call has been made to the called number of 555-2000 by a caller with a unique caller ID nickname of “Smith_1337” (Step S). If such a call has been placed, the authentication server replies to the called authentication device in the affirmative (Step S). Otherwise, the authentication server can reply that the caller ID is either spoofed or irresolute. If the caller in question with the unique nickname of “Smith_1337” has an authentication device, which is determined to be functioning through a received heartbeat message, and is not calling anyone, then the calling party may be determined to be using a spoofed caller ID. If the caller in question with the unique nickname of “Smith_1337” doesn't have an authentication device, then the caller ID cannot be determined to be spoofed or authenticated, therefore, it would be irresolute. The authentication device of the called phone receives the affirmative reply from Step S and puts the call through (Step S). The authentication device modifies the caller ID of the caller with the prefix “[A]”, thus the caller ID that the called phone receives would be “[A] Smith_1337”.
c) Authenticating Callers within a VoIP Infrastructure
A VoIP infrastructure is where the caller is using an internet connection to connect to a VoIP provider, which relays the call to a traditional telephone network. Implementations of VoIP systems were discussed in Section (1-d). Such known VoIP providers are Skype, Vonage and the like. With Skype, the caller has software installed at a personal PC. The user uses the Skype software to initiate a call. The Skype VoIP software uses the internet to relay the call to Skype servers. The Skype server then relays the call to traditional telephone network and connects the call. The user then talks through a microphone connected to the PC. Thus, the Skype VoIP servers' handles the call from the user's PC to the traditional phone network over the internet. The Skype VoIP system is used generally where the user does not have a phone number. A Skype user usually has a Skype username, which allows others to connect to the user over the Skype system. The Skype system can assign users phone numbers. The Vonage VoIP system is more tailored at replacing residential phone service. With Vonage, the user installs a Vonage VoIP router. The router is connected to an internet service provided by the residence. The household phone line then is connected to the Vonage VoIP router. When a user picks up the phone, a dial tone is provided by the router. When the user dials a call, the Vonage router contacts the Vonage VoIP server. The Vonage VoIP servers then relay the call to the traditional phone network. Vonage also provides a phone number to the user. Thus, a caller from the traditional phone network can call a Vonage user. The traditional phone network then relays the call to the Vonage server, where it is relayed to the router and the called phone is rang by the Vonage VoIP router.
Since such VoIP systems sometimes do not assign phone numbers to their customers, the VoIP callers' caller ID usually shows up as “Private”, “Out of Area”, or other similar text. As was described, such caller ID can be thought of as a blocked caller ID since a phone number is not displayed. Thus, such VoIP systems would be ideal candidates for caller ID nicknames described previously. And through the use caller ID nicknames, caller ID authentication may be performed as was described in the previous embodiment.
The system and methods for caller ID nicknames assignment and authentication were previously explained, but will be recapped for a VoIP system. A VoIP provider such as Vonage may institute unique caller ID nicknames within it's′ network. A Vonage VoIP customer would be asked to pick a caller ID nickname which is unique to all Vonage customers, for example “CommanderData”. To the customer's unique nickname, the VoIP provider would prefix its unique tag, such as “Vonage:”. Thus, the complete caller ID nickname “Vonage:CommanderData” would be the customers caller ID nickname. The nickname “CommanderData” would be unique to the Vonage provider, and the nickname “Vonage:CommanderData” would be unique to the telephone network. Thus any outgoing calls by the customer would display the caller ID of “Vonage:CommanderData”. The called party can thus use nickname to determine if the calling party is someone familiar or not. The Vonage service provider may install an authentication device and for every outgoing call authenticate the caller ID nicknames with an authentication server. Alternatively, as was discussed, the caller ID nickname chosen by the customer may be unique to the entire telephone network, such that the VoIP provider need not prefix its tag to make it unique.
One issue with authenticating caller ID's within a VoIP system is if the VoIP customer does not have an assigned phone number, the process of certification becomes more complicated. There are several scenarios that can occur within such a VoIP system. The first scenario is if the VoIP customer does not have a phone number, but has an authentication device installed locally, such as in the form of authentication device software running on the customers PC or the authentication device integrated within the customers VoIP router. In such instances, the certification process will use the customer's uniquely assigned caller ID nickname. The authentication device can contact the authentication server and request a certification password. The device would submit along with the request the caller ID nickname. The authentication server would then resolve the caller ID nickname to the VoIP provider. For example, if the nickname was “Vonage:CommanderData”, the server would resolve this nickname as the customer “CommanderData” on the Vonage VoIP network. The server would contact the Vonage VoIP network and connect to the customer with the caller ID nickname “CommanderData”. The Vonage system would then connect the server to the user. The certification process can then proceed as explained in previous embodiments. The authentication device can intercept the call from the server and receive the certification password. And the device can use the certification password in all future communications with the authentication server.
FIG. 2A
The second scenario is where the VoIP customer has a caller ID nickname but does not have a phone number and does not have the authentication device installed locally. In such scenarios, the authentication device would be installed at the VoIP provider, as was shown in . In such an instance, the VoIP provider would make available the authentication device for each user. Such an authentication device would be added to the VoIP provider system through hardware or software or a combination thereof. Such that that the authentication device at the VoIP provider would be able to perform all the necessary functions for each customer, such as intercepting dialed number, contacting an authentication server, modifying caller ID, and so on. The authentication device at the VoIP provider would also possess its' own phone number for the purpose of certifying itself with the authentication server. The phone number of the VoIP authentication device would most likely be registered with the authentication server. The registration would indicate to the authentication server that the authentication device at that number belongs to the VoIP provider. And any message sent by the authentication device with the caller number as that number is acting on behalf of the VoIP provider. Thus, when a VoIP customer dials a number, the VoIP provider would send the calling status message, such as “CALLING_STATUS-[Caller:VoIP_provider_authentication_device_phone_number, CallerID:Vonage:CommanderData, Password:VoipPass1000, Called:555-2000]”. Thus, the authentication server can match the certification password in the calling status message against the password generated and sent to the VoIP provider authentication device and certify that the calling status message is being received from the VoIP provider. The authentication server can then receive authentication request from the called party and authenticate the caller ID nicknames.
It should be understood that if the VoIP customer has a phone number, the methods of caller ID authentication and process of certification would be accomplished in the same manner as were disclosed in the previous embodiments.
As an alternative to the requirement of phone numbers for certification and caller ID nicknames within a VoIP system may be email addresses. For example, the caller ID of the caller in a VoIP system can be the email address of the caller. The email of the caller would be the email address the VoIP provider has on record for the caller. Thus, the called parties may identify the caller through their unique email address, such as “[email protected]”. This email address can then be used in messages sent to the authentication server in place of the caller ID. The certification process can also use the email address. Such as the caller device can contact the authentication server and send a certification request containing the email address of the caller. The server can then send a certification password to the email address. The authentication device of the caller can then retrieve the certification password from the callers email address and use the password in all messages for the purpose of certification.
FIG. 7C
d) Spoof Determination Summary ()
FIG. 8
8) Authenticating Callers with Legitimate Caller ID Spoofing ( (A-C))
It should be apparent to the reader that the authentication system described in the preceding embodiments can be used within a PBX, business phone systems, call centers, IP PBX, Hybrid PBX, or similar phone systems system without any issues. In general, such systems have multiple user phones or stations connected to the PSTN through an exchange system, such as PBX and the like. Thus, when an outgoing call is placed from a user phone or a station of the exchange, the exchange would connect the call to an outside number. If the exchange system is not spoofing the caller ID information, then the caller ID of the exchange system is authenticated as discussed in previous embodiments. The exchange system would have an authentication device installed. The authentication device would certify with the authentication server upon installation and send heartbeat messages to the authentication server on a regular basis to indicate it is installed and functioning properly. The authentication device would then send to the server the telephone numbers of outgoing calls to be authenticated by the called authentication devices. And the authentication device of called parties would authenticate incoming calls.
As discussed previously, there might be instance where a caller will legitimately spoof their caller ID information. In the cases where there are two or more exchanges and the exchanges wish to spoof caller ID information to display a single number, caller ID authentication is possible. One example of such a system would be a business with two offices in separate locations. The business would like to display a single toll free number as the caller ID of all employees outbound calls to customers. Although this embodiment of the invention by way of example will show how one exchange may legitimately spoof another exchanges caller ID, it should be understood the embodiment may be implemented not just between exchanges but can be implemented between any two parties on a landline, on smartphones, or VoIP phones in the same manner.
FIGs. 8A-8C
FIG. 8
FIG. 8
FIG. 8
FIG. 8A
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In this embodiment of the invention legitimate caller ID spoofing and accompanying authentication methods are illustrated. , collectively referred as , shows the flowchart and signaling diagram for an embodiment of the invention of authenticating legitimately spoofed caller ID's. In , one exchange wishes to spoof the caller ID of another exchange. An exchange can be a PBX and the like. The steps performed by the exchanges in would be performed the same for any other system or phones which wish to perform legitimate caller ID spoofing. Referring to , the first exchange with a telephone number 555-1000 has an authentication device . The first exchange can be referred to as the spoofing party who wishes to spoof its caller ID to that of the spoofed party. The second exchange with a telephone number 555-2000 has an authentication device . The second exchange can be referred to as the spoofed party who gives permission to the spoofing party to spoof its' caller ID. The first exchange wishes to spoof the caller ID of the second exchange. The process begins where the first exchange authentication device certifies itself (Step S). The authentication device for the first exchange begins the certification process by contacting the authentication server and requesting a password (Step S). It should be noted that the certification process could have already taken place for the first exchange authentication device, in which case the process would skip to Step S. The authentication server then calls the first exchange (Step S). The call is intercepted by the authentication device and picked up. The authentication server generates (Step S) and sends a unique password (Step S). The authentication device may, during the call back from the authentication server, or anytime after a certification password is received may initiate a request to spoof its' caller ID to be the telephone number of the second exchange, namely 555-2000(Step S). This spoof request may take place at any time and there is no restriction that a spoof request be submitted after certification, but there is a restriction that the spoof request is only considered by the server which is received from certified devices. The spoof request may be initiated by the customer served by the authentication device. The spoof request may be indicated to the authentication device through IVR or through connecting to the device through an internet connection and adjust the device settings through a web page. If the authentication device is implemented as software, for example on a smartphone, the spoof request may be initiated by going to the software settings menu. The authentication device contacts the server and transmits a spoof request (Step S). The transmitted spoof request data maybe in the form of “SPOOF_REQUEST-[Caller555-1000, Password:Pass1000, Spoof:555-2000]” The “Caller” field signifies the requesting phone number. The “Password” field signifies that the authentication device is certified by transmitting the password received during certification. The “Spoof” field indicates to the server the phone number that is wished to be spoofed. The authentication server stores the spoof request (Step S) and replies that the spoof request is pending permission from the second exchange at 555-2000 (Step S). The replied message from the server may be in the form of “SPOOF_PENDING-[Caller:555-1000, Spoof:555-2000]”. It should be noted that the spoof permission by the second exchange could have already been sent to the authentication server, in which case the authentication server would have replied back as spoof request granted, as is shown in Step S.
FIG. 8B
FIG. 8A
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Referring to , which is a continuation of , the second authentication device initiates the certification process (Step S) by transmitting a certification password request to the server (Step S). It should be noted that the certification process could have already taken place for the second exchange authentication device, in which case the process would skip to Step S. The authentication server calls the second exchange phone number (Step S), which is intercepted by the authentication device and picked up. The authentication server then generates a unique password (Step S) and sends it to the authentication device (Step S). The authentication device then initiates a spoof permission (Step S). The authentication device of the second exchange may during the callback, or through an internet connection, grant permission for first exchange authentication device at 555-1000 to spoof its number. It should be understood that any permission to grant a spoof would be sent with the authentication device certification password received in Step S. The device sends a message to the server authorizing the number at 555-1000 to spoof its' number. (Step S). The transmitted message may be in the form of “SPOOF_PERMISSION-[Caller:555-1000, Spoof:555-2000, Password:Pass2000]” The “Spoof” fields signifies the phone number of the spoof permission granting customer. The “Password” field signifies the certification password of the device, transmitted by the server. The “Caller” field indicates that the phone number at “555-1000” may spoof the number of the spoof granting customer “555-2000”. The authentication server stores the spoof permission, and marks all matching spoof requests as granted (Step S). It should be apparent that matching requests would consist of requests in which the “Caller” and “Spoof” fields match, such as the messages in Steps S and S. The server replies with a message that the spoof permission is received and the spoof permission is now in effect (Step S). The in effect reply signifies that the spoof is granted to the party signified in the “Caller” field until it is revoked. The reply message may consist of “SPOOF_IN_EFFECT-[Caller:555-1000, Spoof:555-2000]”. The message confirms that the “Caller” with the phone number 555-1000 has been granted by the “Spoof” party at the phone number 555-2000 to spoof its' number.
FIG. 8C
FIG. 8B
868
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Referring to , which is a continuation of , the first authentication device checks the status of the spoof request by contacting the authentication server (Step S). The spoof request status check would only be performed if the server had replied that the previous request by the spoof requesting party was pending, as was indicated in Step S.
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Alternatively, the server may contact the spoof requesting party once the spoof permission is received by the server. In Step S, the transmitted message to the server may be in the form of “SPOOF_REQUEST_CHECK-[Caller:555-1000, Password:Pass1000, Spoof:555-2000]”. The authentication server replies that the first exchange authentication device may spoof its' caller ID as 555-2000 (Step S). The transmitted message to the device may be in the form of “SPOOF_GRANTED-[Caller:555-1000, Spoof:555-2000]”. Thereafter, the first exchange initiates a call to another party (called party not shown in figure) at the phone number 555-7000 with a spoofed caller ID of 555-2000 (Step S). The first exchange authentication device captures the dialed numbers (Step S) and passes the call to the telephone network (Step S). The authentication device contacts the authentication server and transmits the called number and spoofs the calling number as 555-2000 (Step S). The calling status message transmitted to the authentication server can consist of “CALLING_STATUS-[Caller:555-2000, Password: Pass1000, Called: 555-7000]”. The password “Pass1000” would be the password generated by the authentication server and transmitted to the authentication device in Step S. Thus the authentication device is using its' own password. The authentication server looks up the phone number of the password holder, and determines it to be 555-1000. However, since the authentication device of 555-2000 has given permission for the authentication device at 555-1000 to spoof its number, the authentication server accepts the “Caller:555-2000” field in the transmitted data. Then, the call is received at the called party with the phone number 555-7000. If the called party has an authentication device installed, the authentication device contacts the authentication server and authenticates that the caller ID of the call. The authentication server may either indicate that the caller ID is authentic or it can indicate to the called party that the caller ID is legitimately spoofed. Such indication would be performed by the authentication server transmitting the appropriate messages to the authentication device of the called party.
The calling status message can also be in the form of “CALLING_STATUS-[Caller:555-1000, Spoofing:555-2000, Password: Pass1000, Called: 555-7000]”. The “Caller” field would indicate the caller number, the “Password” field would indicate the password of the caller. The “Spoofing” field would indicate the number that the caller is spoofing its caller ID as. The server would first verify the password matches the password of the caller, verify that the caller has permission to spoof the number contained in the “Spoofing” field, and then store the calling status message. In either calling status message formats, the authentication server would have received a message that the caller is calling with a spoofed caller ID and the server would then check the permissions on file if the caller is allowed to spoof the number. The server can then authenticate an authentication request sent by the called party which contains spoofed the caller ID of the caller.
A spoof permission may also be rescinded by the party who granted the spoof permission. Such a rescind message can be in the form of “SPOOF_PERMISSION_RESCIND-[Caller:555-1000, Spoof:555-2000, Password:Pass2000]”. Such a rescind message is self explanatory and will not be elaborated. According to such a rescind message, the server would no longer authenticate spoofed caller ID's by the respective caller party. The server may also contact the spoofing party authentication device to notify it of the spoof permission rescind.
FIG. 9
FIG. 14
9) Authentication Device Call Handling (-)
The next embodiments will introduce the call handling features of the authentication device. If the authentication device is able to authenticate caller ID's, then it becomes possible to handle calls based on such authenticated caller ID's. Calls from numbers that are deemed trustworthy and which the user wishes to be contacted by; can be connected to the user by the authentication device. Such calls can be which the user speaks to on a daily basis, such as spouse, relatives, etc. Calls from numbers that are deemed untrustworthy and which the user does not wish to be contacted by; can be intercepted by the authentication device. Such calls can be calls from telemarketers, robocallers, etc. The authentication device can thus maintain a list of such trustworthy and untrustworthy phone numbers in lists. And since the authentication device is able to both authenticate caller ID and detect caller ID spoofing, such lists can form a very effective method for call handling not presently known in the art.
The call handling concepts introduced in the following section will utilize the caller ID authentication systems and methods discussed in the previous embodiments. However, other methods of caller ID authentication, either known in the prior art or discovered after the date of this disclosure, may be used instead.
a) Authentication Device Internal Lists: Whitelist
In this embodiment of the current invention, the authentication device maintains a local or internal whitelist. The internal whitelist is a list of numbers that the authentication device determines to be trustworthy. Trustworthy numbers are defined as numbers that the customer of the phone for which the authentication device is connected to wishes to receive calls from. Such trustworthy numbers may be the customer's spouse, children, relatives and the like. The internal whitelist of the authentication device would consist of a database contained in the device for storing phone numbers. The phone numbers are added to the internal whitelist by the authentication device autonomously. The authentication device adds numbers that the customer dials a certain amount of times to the internal whitelist automatically. The authentication device adds numbers that call the customer a certain amount of times to the internal whitelist automatically. The phone numbers of received calls would be identified by caller ID and would only be added to the whitelist if the caller ID can be authenticated according to the previously described embodiments of the invention. Thus, the authentication device autonomously complies and populates an internal whitelist of phone numbers. A number may also be manually added to the whitelist through the authentication device IVR prompts, or through the device settings accessed through the internet.
If the incoming call has a blocked caller ID, that is it displays “Private” or no text as the caller ID, then this call cannot be placed in the internal whitelist. Since the caller ID is blocked, there isn't a way to differentiate one blocked caller ID from another. However, if the incoming call is utilizing a customized caller ID, such as a caller ID nickname, then it is possible to add such a call to the internal whitelist. Since a caller ID nickname by definition is unique, it is possible to differentiate one nickname from another, and handle such calls appropriately, as will be explained. Adding a caller ID nickname to the whitelist would be done through a similar mechanism to adding incoming calls with non-blocked caller ID's to the internal whitelist. If the customer receives a certain number of calls from a caller with a caller ID nickname, that nickname can be added to the internal whitelist autonomously.
All numbers and caller ID nicknames which are added to the internal whitelist, or any other list as will be explained, must be authenticated first. Such authentication may be performed by any embodiment previously described. Authentication ensures that a caller is actually calling from the number which is stated on the caller ID. Numbers added to the internal whitelist must also not be on the internal blacklist.
b) Authentication Device Internal Lists: Blacklist
The authentication device also maintains a local or internal blacklist. The internal blacklist is a list of numbers that the authentication device stores that are determined to be banned from contacting the customer. A number is added to the internal blacklist in one of two ways, autonomously or manually. The autonomous method of adding a number to the internal blacklist will be discussed later. The user of the authentication device can manually add a number to the blacklist by entering a star code, during a call or after a call has ended. For example, if the user of the authentication device receives a call from a telemarketer, and the user wishes to ban the telemarketer from further contacting him, the user can press a star code such as *10. If the star code is pressed during a call, the caller ID number of the call is added to the internal blacklist. If the star code is pressed after a call has ended, the preceding calls telephone number is added to the blacklist. The process of adding a telephone number to a blacklist after a call has ended can be where the user hangs up the phone, picks up the phone, hears a dial tone, and dials *10. Since the authentication device is able to monitors outgoing dialed numbers, the device can intercept the star code, pick up the phone and announce through a voice prompt that the last called number has been added to the internal blacklist. A number may also be manually added to the blacklist through the authentication device IVR prompts, or through the device settings accessed through the internet. Only authenticated caller ID's, such as determined according to the previously described embodiments, can be added to the blacklist.
There might be instances where a number calls the customer several times, and the device mistakenly determines this to be a whitelisted number and autonomously adds it to the whitelist, as described previously. However, such a number may be a telemarketer, and it might be the case that the customer forgets to add the number to the blacklist after such a call. In such a case, if the customer eventually adds the number to the blacklist after receiving a subsequent call, the number would be removed from the whitelist and added to the blacklist.
If the incoming call has a blocked caller ID, that is it displays “Private” or no text as the caller ID, then this call cannot be placed in the internal blacklist. Since the caller ID is blocked, there isn't a way to differentiate one blocked caller ID from another. However, if the incoming call is utilizing a customized caller ID, such as a caller ID nickname, then it is possible to add such a call to the internal blacklist. Since a caller ID nickname by definition is unique, it is possible to differentiate one nickname from another, and handle such calls appropriately, as will be explained. Adding a caller ID nickname to the blacklist would be done through a similar mechanism to adding incoming calls without blocked caller ID's to the internal blacklist. If the customer receives a call with a caller ID nickname and wishes to add it to the internal blacklist, then the customer presses the appropriate star code during or after the call has ended, as was explained, and the appropriate nickname can be added to the blacklist.
If the authentication device is implemented as a software program running on a smartphone for instance, then the user may add a number to the blacklist by pressing a button on the call screen, before, during or after the call has ended. Such a button may be in the form of a thumbs down button, to indicate that the user does not wish to receive calls from the caller again. A thumbs up button may be provided to allow for the user to place the caller ID on the whitelist manually. If the user does not press any button, then the call is handled as a graylisted number until it is automatically put on the internal whitelist or; automatically or manually placed on the blacklist at a later time.
a
FIG. 9
c) Authentication Device Call Control and Internal Graylisting for a Non-Blocked Caller Id's ()
4
First, call handling with respect to non-blocked caller ID's will be presented, then call handling with respect to blocked caller ID's will be elaborated. The authentication device has 2 compiled lists, an internal whitelist and an internal blacklist. Any incoming call number that is on the internal whitelist and the caller's caller ID is authenticated, the call is allowed to ring the customers' phone. Any incoming call that is on the internal blacklist and the calls caller ID is authenticated it not allowed to ring the customers' phone. If an incoming call is not on either the internal whitelist or blacklist, but the caller ID of the caller can be authenticated (authenticated caller ID prefix “[A]”), the customer can choose to have those calls handled as a graylisted number. A graylisted number is any number not on either a whitelist or a blacklist. A graylisted number, at the choosing of the customer, may be connected to the customer directly, after which it can be put on the internal blacklist or internal whitelist by the authentication device. A graylisted number may also be forced to verify that the caller is a human or be forwarded to voicemail. The human verification can consist of the authentication device picking up the call, without ringing the customers' phone. The called authentication device can play a message that verifies that the caller is human by requiring a correct answer. The answer by the caller can be input through a touch-tone phone. Human verification can consist of asking a question such as a math question. A sample math question can consist of “what is two times two?”. The caller would then have to press on a touch-tone phone as the correct answer and to pass the human verification test. Another type of question could be a math question with a knowledge component, such as what is two times the number of wheels on a motorcycle?”. This question requires both math and knowledge. Such questions could be uploaded from a central server, preferably such as the authentication server or the global list server (as will be disclosed shortly) or any other server. The uploading of the questions and answers from a central server on a regular basis would ensure that new questions can be introduced to calling systems and calling system would not be able to circumvent and recognize such questions autonomously. The ultimate purpose of such questions is to verify that a human is calling rather than an automated caller, such as robocallers or automated telemarketers. Once the human verification test is passed by the caller, the call is allowed to ring the customers' phone. If the human verification test is failed, the call could be dropped or forwarded to voice mail. If an incoming call is not on either the internal whitelist or blacklist, and the caller ID of the caller cannot be authenticated (irresolute caller ID prefix “[I]”), the customer can customize how those calls are handled. If an incoming call is not on either the internal whitelist or blacklist, and the caller ID of the caller is determined to be spoofed (spoofed caller ID prefix “[S]”), the customer can customize how those calls are handled, but mostly likely will choose it to be handled as a blacklisted number.
In the previous embodiments, the authentication device modified the caller ID of the caller to indicate the authentication status, such as “[A]”, “[S]” or “[I]”. The authentication device can further modify the caller ID of an incoming call to indicate which list the number is currently belongs to. For example, if a calling number is both authenticated and is on the internal whitelist, then the caller ID can be modified with the prefix “[W-A]”, to indicate the caller is on the internal whitelist and is authenticated. Likewise, the caller can indicate a caller's number is on the whitelist, blacklist, or graylist with the prefixes “[W]”, “[B]”, “[G]”. Since blacklisted calls will almost always be dropped, the called user will likely never see a caller ID with the prefix “[B]”. Therefore, there are a total of nine combinations of caller ID prefixes that can be applied to a caller's caller ID, which are “[W-A]”, “[W-S]”, “[W-I]”, “[B-A]”, “[B-S]”, “[B-S]”, “[G-A]”, “[G-S]”, “[G-I]”.
The customer can specify how calls are handled based on the combination of caller ID authentication status and which list the call number is currently stored in. There are three possible caller ID authentication statuses, namely; authenticated, irresolute, and spoofed. And there are three possible lists that a caller number can be on, whitelist, blacklist and graylist. Thus there are a total of 9 combinations of call handling options. These 9 call handling options can be set by the authentication device manufacturer. And the customer of the authentication device may choose to modify these call handling options.
FIG. 9A
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shows, by way of example, the flow chart that is used to handle calls with non-blocked caller ID's by the authentication device. Once a call is received in step by the authentication device, the call is intercepted. The caller ID of the intercepted call is determined and authenticated against the authentication server, according to the previously described embodiments. If the caller ID is determined to be authentic in decision step the process moves to the next decision step . The decision step checks if the authenticated caller ID number is contained in the internal whitelist. If so, the process moves to step and the call is connected to the user's phone. The connection to the user's phone is made by the authentication device ringing the user's phone and sending the modified caller ID which indicates that the caller is on the internal whitelist and has an authentic caller ID. The prefix added to the caller ID would be “[W-A]”. If the caller ID number of the caller was not on the internal whitelist, the decision step would proceed to decision step . The decision step would determine if the caller ID number is contained in the internal blacklist. If the number is determined to be on the internal blacklist, then the decision step would proceed to step . In step , the call would be dropped. Dropping the call would consist of the authentication device not ringing and not connecting the call to the user. Dropping the call could also consist of the authentication device picking up the phone and hanging it up. Dropping the call could also consist of a message played to the caller that the called party does not wish to receive calls from the caller. Dropping the call could also consist of a message played to the caller that the line has been disconnected, in other words misrepresenting the telephone line status in order to persuade the caller not to call the number. Dropping the call could also consist of certain tones being played to the caller, such as a disconnected tone.
FIG. 9A
FIG. 9A
923
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The rest of the steps in will not be explained in detail. The remaining steps presented in are self evident and can be followed by a reader trained in the art. Steps with the letter “C”, such as step signify connected calls. Steps with the letter “H”, such as step , signify calls that will be subject to human verification by the authentication device. Steps with the letter “V”, such as step , signify calls that will be forwarded to voice mail, by the authentication device. Steps with the letter “D”, such as step , signify calls that will be dropped by the authentication device.
FIG. 9A
FIG. 9A
FIG. 9A
FIG. 9A
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The flowchart shown in is given as an example only, and the manufacturer or the end user of the authentication device may wish to change how some calls will be handled depending on the caller ID authentication status and the list the calling number is contained in. However, the central premise of this embodiment of the invention is allowing authenticated whitelisted numbers to ring the customer phone number, as shown in step of . And authenticated blacklisted number to be dropped and prevented from ringing the customer phone and bothering the customer, as shown in step of . And having an authenticated call that is either not on the whitelist or blacklist be treated as a gray number, where the treatment of the gray number is definable by the customer. In the embodiment shown in , such a gray number is chosen to be subject to human verification. Thus, the authentication device is able to handle incoming calls based the combination of caller ID authentication and the list the calling number is contained in.
b
FIG. 9
d) Authentication Device Call Control and Internal Graylisting for a Blocked Caller ID ()
FIG. 9B
FIG. 9B
shows the flowchart an exemplary embodiment of the invention uses in handling incoming calls with blocked caller ID's. is shown as an example, the specific options for handling incoming calls with blocked caller ID's may be modified by the customer or the telephone service provider as seen fit.
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981
982
When a call is received by the called authentication device, the call is checked to see if the caller ID of the call is blocked. If so, the process proceeds to step . After step , the process moves to decision step . Decision step determines if the authenticated caller ID is in the form of a caller ID nickname. If so, the process moves to step . The decision step checks to see if the caller ID nickname of the incoming call is contained in the internal whitelist. If so, the process moves to step and the call is allowed to ring the customer phone. If the caller ID nickname is not on the internal whitelist, the process moves to step . The decision step checks to see if the caller ID nickname is on the internal blacklist, if so the process moves to step and the call is dropped. If the nickname is not on the blacklist, the process moves to step and the call is sent to voicemail by the authentication device.
951
952
952
963
952
964
If the incoming call is did not have a caller ID nickname, the decision step would have branched into step . The decision step checks to see if the caller ID is blocked but is authenticated. If so, the process moves to step and the call is sent to voicemail by the device. If the caller ID had not been authenticated, the decision step would have branched into step where the call would be dropped by the device.
FIG. 10A
FIG. 10B
i) Constructing the Global List Through Internal Lists
e) Authentication Device Global Lists (-)
In another embodiment of the current invention, the authentication devices of each user help compile a global whitelist and a global blacklist. Since the authentication device would be installed on each phone line of many users, it becomes possible to aggregate each user's internal whitelist and blacklist into a global whitelist and blacklist. If a phone number is on more than a certain number of authentication devices internal whitelists, that number can be deemed to be trustworthy to all customers. Therefore, it would be safe to add such a phone number to the global whitelist. An example of such a phone number could belong to a bank. Likewise, if a phone number is on more than a certain number of authentication devices internal blacklists, that number can be deed to be untrustworthy to all customers. Therefore, it would be correct to add such a phone number to the global blacklist. An example of such a phone number could belong to a problematic telemarketer. If the telemarketer's phone numbers is on a certain number of authentication devices blacklists, then that number can be put on a global blacklist.
For the purpose of aggregating and maintaining global lists, a global list server is required. The global list server is accessible by the authentication devices, preferably over a data connection such as the internet. The authentication devices may be able to access the global list server through the same connection as the authentication server. The functionality of the global list server may be integrated into the authentication server. The global list server maintains a database of numbers, a global whitelist, and a global blacklist. The global list server also has logic to aggregate received numbers from the authentication devices and place them on the appropriate lists based on a predetermined algorithm. Alternatively, the global list server can also maintain the internal lists of authentication devices.
The global list server can also use intermediary lists for the purpose of aiding which numbers are added to the global lists. For example, the global list server can place all uploaded internal whitelist numbers from the authentication devices in an intermediate whitelist. The global list server may place additional uploaded internal whitelists in the intermediate whitelist as they are uploaded throughout the day. At the end of each day, month or year, the global list server may then choose which number from the intermediary whitelist to nominate to the global whitelist.
Thus, for each incoming call, the authentication device can check if the calling number is on an internal whitelist or internal blacklist. If so, the authentication device can handle the call as described in the previous embodiment. If the calling number is not on an internal whitelist or internal blacklist, the authentication device can query the global list server. If the calling number is on the global list server's global whitelist, the call may be accepted. If the calling number is on the global list server's global blacklist, the call may be rejected. If the calling number is not on the global list server's global whitelist or blacklist, the number can be treated as a gray number. The gray number can thus be handled according to preset rules.
FIG. 10A
FIG. 10A
FIG. 10A
FIG. 10A
FIG. 10B
FIG. 10B
FIG. 10B
FIG. 10B
1010
1011
1010
1011
1012
1020
1020
shows, by way of example, an embodiment of the current invention. shows the internal whitelists and blacklists for three user authentication devices. The authentication device for user A has an internal whitelist with three numbers. These three numbers would have been automatically added to the internal whitelist by the authentication device. The first number 555-1000 for example, could have been added by the authentication device to the internal whitelist because user A dialed that number more that a certain number of times, such as dialed 555-1000 more than three times. The second number 555-1001 for example, could have been added by the authentication device to the internal whitelist because user A received calls by that number more than a certain number of times, such as 555-1001 called the user more than three times. Telephone numbers are only added to the internal whitelist if the user does not indicate such a number to be blacklisted. shows the telephone number 555-9003 to be on the authentication device for user B , internal blacklist. The telephone number 555-9003 would have been added to the internal blacklist manually by the user through entering in the appropriate star code during or after receiving a call from such a number. At regular intervals, such as every hour, six hours, or day, each authentication device for each user uploads its' internal lists to the global list server. Instead of uploading at regular intervals, the authentication devices may upload its' lists every time a new telephone number is added to a list, or every time a certain numbers of telephone numbers are added to its' list. The global list server receives the internal lists of each authentication device and aggregates and populates them into the appropriate global whitelist and global blacklists. A telephone number is placed into the global whitelist if it is contained and is received from a certain number of authentication devices internal whitelists. Likewise, a telephone number is placed into the global blacklist if it is contained and is received from a certain number of authentication devices internal blacklists. shows the authentications devices of user A , user B and user C transmitting the internal list to the global list server . As can be seen, the global list server maintains a global whitelist and a global blacklist. In the shown example, the global lists are empty. In , the global list server aggregates the received internal lists and populates the global lists. In example of , a telephone number on more than 2 users authentication device lists are placed into global list. However, as is apparent to the reader, the threshold for placing telephone numbers into the global list can be any number, as seen fit, such as 2, 4, 100, etc. Instead of choosing a threshold number, the threshold can be a percentage. Such a percentage can be in the form of; if a telephone number is in the internal whitelist of 10% or more of all the users' internal whitelist, the telephone number would be placed into the global whitelist by the global list server. Returning to , the example used the number 2 as the threshold for placing a telephone number into the global list. From , it can be seen that the telephone number 555-9003 has been placed into the global blacklist. The reason 555-9003 was placed into the global blacklist is because that number is contained in user B's internal blacklist, and user C's internal blacklist. Since the number of internal lists the telephone number is contained in is equal to the predetermined threshold, the telephone number is placed into the appropriate global list. The predetermined threshold for placing telephone numbers into the list can be the same threshold for both whitelists and blacklist or different. Such as telephone numbers on 10% or more internal whitelists are added to the global whitelist, but telephone numbers on 20% or more internal blacklists are added to the global blacklist.
One issue with blacklists is to when and how to remove a certain number from the blacklist. For instance, if a telemarketer's phone number is placed on a blacklist, the telemarketer can close the number. The blacklisted number can then be assigned to a residence by the telephone company. In such cases, the residence can have a hard time contacting individuals or companies since the number maybe on internal, private or global blacklists. With regards to globally blacklisted numbers, the numbers could be put on probation after a certain amount of time has passed since they were put on the global blacklist. When the number is put on probation, the number would be taken off the global blacklist, private blacklists, and the global list server would instruct the number be removed from all internal blacklists. The probation status of the number would mean that if a certain number or percentage of user's placed the number on their internal or private blacklists, it would be put back on the global blacklist. And the threshold for putting a number on probation back on the global blacklist would be much lower than a number not on probation. For instance, if the requirement for being put on a global blacklist was being on more than 20% of internal or private blacklists, then the threshold for being put back on the blacklist for a number on probation could be 2%. If the threshold for the number on probation to be put on the global blacklist is exceeded, such as more than 2% of users put the number back on their internal or private blacklist, then the number would be put back on the global blacklist. If the threshold has not been passed while the number is on probation and after a certain period on probation, the number is taken off the probation status and treated like any other number with respect to threshold for being put on global white or black list. A similar methodology could be used for internal or private blacklisted numbers not on the global blacklist. After a certain period of time, a blacklisted number can be automatically taken off the internal or private blacklist. The user can be allowed to determine the time period, or asked to verify that the blacklisted number be taken off the list. Such methodologies are not necessary with respect to internal or private or global whitelists since as soon as a whitelisted number becomes problematic, those numbers would be placed on internal or private and possibly global blacklists through already described methods.
ii) Compiling the Global Whitelist Through Call Tally
In conjunction to compiling the global whitelist through aggregating internal whitelists, the authentication server can keep a tally of calls being made to a telephone number, and if the number of calls exceeds a certain threshold, that number can be placed on the global whitelist. The reason being that callers would not regularly place calls to a blacklisted number. The authentication server can determine the threshold through a blind tally of calls having been made to that number. Or the server can determine the threshold as number of unique callers who have made a call to that number. The threshold can be a total number, such as 1000 calls made to that number. Or the threshold can be time based, such as 100 calls per day, or 100 calls average per day during the past month. A separate threshold can be used to remove a number from the whitelist, such as if less than 10 call per day are being made to that number. The threshold can also take into account the number of calls made to a telephone number from callers with authentication devices versus those without. Likewise, the authentication server can keep a tally of calls that have been received by a telephone number and place the telephone number in the global whitelist if the amount of calls received passes a threshold. The number of calls being made to a telephone number and being received by that number would not necessarily be equal since there might be callers who do not have authentication devices installed. Alternatively, the tally might consist tallying caller telephone numbers which have been authenticated, and placing such numbers within the global whitelist if the tally exceeds a certain number. Similar to the previously given tally examples, the tally may only take into account authentic caller telephone numbers which have been placed to unique called numbers, such that two calls from an authentic caller number to the same called number would only be counted once.
FIG. 11A
FIG. 11B
j) Call Handling with Global Lists (-)
After a global whitelist and global blacklist has been compiled, call handling by the authentication device is done through both the internal lists and global lists. As was explained in Section (9c), the device handles calls through internal lists. If a device receives an incoming call which is not on any internal list, the authentication device may query the global list server to check if the number is on any global lists. The query may consist of simply sending the caller ID of the caller and the server replying with the corresponding global list identifier the number is on. After the authentication device queries the server to check if a number is on the global lists, the call can then be handled based on which global list the number is on.
Alternatively, if the authentication server and the global list server are integrated within the same server or are able to communicate with each other, the global list identifier, such as global whitelist or blacklist, may be sent by the authentication server when replying to caller ID authentication requests. For example, if the authentication server is replying with an authenticated message, the global list of the caller number may be indicated thus so; “AUTHENTICATED-[Caller:555-1000, Caller List: Global Whitelist, Called:555-2000]”. The “Caller List: Global Whitelist” would indicate that the caller with the 555-1000 number is currently on the global whitelist. The authentication device can then handle the call appropriately knowing which global list the calling number is on.
FIG. 11A
FIG. 11B
FIGS. 11A and 11B
FIGS. 9A and 9B
FIGS. 9A and 9B
FIG. 9A
FIG. 11A
FIG. 9A
FIG. 11A
FIGS. 11A and 11B
920
920
1120
1120
shows, by way of example, the flowchart that is used to handle calls with non-blocked caller ID's by the authentication device. shows, by way of example, the flow chart that is used to handle calls with blocked caller ID's by the authentication device. The are similar to explained in Sections 9-C and Section 9-D. The only differences are that the decision steps in which checked if a certain number was on a certain internal list have been updated to check if they are also on a certain global list. For instance, in the decision step checked if an authenticated caller ID was on the internal whitelist and the call was handled appropriately. In , the decision step from is replace by decision step . Decision step is updated to take into account that a number can either be on the internal whitelist or on the global whitelist. In the example given in , numbers that are on either the internal and global whitelist are handled the same. However, it should be understood that they could be treated differently. Such that a number on the internal whitelist is allowed to connect to the customer phone, while a number not on the internal whitelist, but on the global whitelist is put through to voice mail. Thus a call may be handled based on the combination of internal lists, global lists, and authentication status of the calling party. The rest of is self explanatory and will not be elaborated in detail.
One issue that might arise when handling calls based on both internal and global lists is which lists take precedence. For instance, if a number is on the internal blacklist but is also on the global whitelist. In such a case, the internal blacklist would preferably take precedence over the global whitelist, and the call would be handled as an internal blacklisted number. Call handling with regards to list precedence could be configured by the telephone provider, authentication device maker, or the customer as seen fit.
3
In Section (9-c), there were a total of 9 call handling options. With the introduction of global lists, three more lists are introduced. Thus there are internal white, black and gray lists, and in addition global white, black and gray lists. Therefore, there are 3 internal lists, 3 global lists, and 3 authentication combinations. The total number of call handling options is 3 to the power , which is 27. For example, an incoming telephone number could be authenticated, on the internal graylist, and on the global whitelist. Or an incoming telephone number could be authenticated, on the internal whitelist, and on the global graylist. The telephone customer or the authentication device manufacturer may adjust how calls are handled in each of the 27 combinations. There are however some redundant combinations, such as an incoming telephone number could be spoofed, on the internal blacklist, and on the global graylist. In the example of a spoofed number, the combinations of internal and global lists would likely not affect the handling of the call since if the number is spoofed it would most likely be handled regardless of the internal or global lists.
FIG. 12
k) Autonomous Determination of Blacklisted Numbers ()
Incoming calls can also be added to the internal blacklist by the authentication devices autonomously. An autonomous blacklist determination of an incoming call can consist of the duration of the call. For example, if a number calls, and the customer picks up and hangs up within a short period of time, such calls may be inferred as a nuisance call and a call which the customer does not wish to receive. Another autonomous determination metric of a blacklisted call can be which party talked the most during the call. For example, if a call is received by the customer, and the caller talked most of the time and the customer only said “Hello”, then such a call may be inferred as robocall or an automated message call. Another autonomous determination metric of a blacklisted call can be which party was talking when the call was hung up. For example, if a call is received and the customer hangs up the call while the caller is still talking, then it may be inferred that this is a call the customer does not wish to receive in the future. Another autonomous determination of a blacklisted call can consist of voice recognition. For example, if the called party says “Not interested” before hanging up, this call can be inferred as a telemarketing call by the device. Likewise, recognizing, through voice recognition, certain keywords spoken by the caller and the called parties, a call may be determined to be a blacklisted number. The keywords may be those which are used within normal conversation, such as “No thank you”, or the keywords may be directed at the authentication device for the purpose of placing the call on the blacklist, such as “spam caller”, “place on blacklist”, “fraud caller”, or “report caller”. Therefore, autonomous determination metrics can include call duration, called party total talk time, called party total talk time as a percentage of call duration, caller party total talk time, called party total talk time as a percentage of call duration, which party was talking when the call was hung up, keywords spoken during the call, and which party spoke certain keywords. The autonomous determination metrics can also include which parties were talking in which portion of the call, such as if the caller was talking during the first half of the call, or the called party talking during the last half of the call, and so on.
The authentication device would have the necessary combination of hardware and software to measure the metrics described herein for the purpose of autonomously blacklisting a certain call. For example, the call duration metric can simply be measured by the authentication device starting a timer when a call is picked and stopping the timer when the call is hung up. Similarly, other metric can be measure through systems and methods known in the art.
Using such metrics, the authentication device may autonomously add calls to the internal blacklist. The determination that a call is to be placed on the internal blacklist can consist of several metrics used in combination. Such metrics can be individually combined through a score system. Each metric can be assigned a score from 0 to 100, where 0 is that the call is certain not to be a blacklisted call and where 100 is where the call is certain to be a blacklisted call. If one or more metrics are used, the average score of these metrics computed and if the average score is past a predetermined threshold, the number can be added to the internal blacklist.
FIG. 12
FIG. 12
is a flowchart illustrating one example of adding the caller number to the internal blacklist autonomously. The flowchart illustrates the use of two metrics for determining if the caller number should be added to the blacklist. The flowchart is shown as an example only. Adding a called number to the internal blacklist may entail the use of one or more metrics, where each metric and metric criterion is weighted differently, and the resultant formula for weighing each metric may be different that what is presented in the figure. The metrics and the resultant formula would be determined most likely through trial and error, and would likely need to be modified on an ongoing basis to adapt to undesired call behavior. In , the first metric used is call duration, with the metric criterion being the length of the call, namely; 20 seconds, 10 seconds, 5 seconds, and 1 second. The second metric used is which party talked and the metric criteria being; called party talked, caller party talked, or both parties talked. Each of the metric criterions is assigned a score. The determination formula is the average of the two metric criterion scores. The determination formula adds an incoming caller ID to the internal blacklist if the average of the 2 metric scores is greater than the determination threshold of 50. If not, the phone number is not added to the internal blacklist.
FIG. 12
1201
4
1210
1215
1220
1230
1240
1250
1255
1260
1270
1295
1285
The process of starts when a call with an authenticated caller ID has ended . The authenticated caller ID may be a call in the form authenticated non-blocked caller ID or an authenticated caller ID nickname. The caller number must preferably be graylisted, in that it is neither on any whitelist or blacklist, whether internal or global. After the call has ended, the process moves through decision steps in measuring the call duration metric. The first decision step determines if the call has had a duration of more than 20 seconds. If so, the decision step moves to step where the first score is assigned 20 points. Thus, the call duration metric criterion of a call lasting more than 20 seconds assigns 20 points to the first score. The decision steps , , and are self explanatory. Each decision step corresponds to a specific call duration criteria metric. If a metric criterion is satisfied, such as a call duration being a certain time, the corresponding score is assigned. One should note that the shorter the duration of the call, the more points the first score is assigned. The reasoning behind this is that the short duration calls have a higher likelihood of being unwanted calls and should be placed on the internal blacklist. The process then moves to the next three decision steps which correspond to the, which side talked metric. The which side talked metric determines which side talked during the call, and each metric criteria assigns a number to the second store. If only the called party talked, as is determined by the decision step , then the process moves to step and the second score is assigned 70 points. The rest of which side called decision steps and are self explanatory. Once all the metric criteria are determined the process moves to the determination formula 1280. In this embodiment the determination formula determines the average of the first score and the second score. If the average is greater than a predetermined threshold, 50 in this instance, the call number is added to the internal blacklist as shown in step . If not, the call number is not added to the internal blacklist, as is shown in step .
FIG. 13A
FIG. 13B
I) Internal Potential Blacklist for Use with Autonomous Determination of Blacklisted Numbers (-)
FIG. 12
In the previous section, the authentication device autonomously determined blacklisted numbers and added them to the internal blacklist. However, as one might realize, there might be instances where the autonomous determination of blacklisted numbers would be subject to errors. For example, a call may be received from an automated robocall system from a dentist's office to remind customers of upcoming appointments. The customer might pick up the phone, answer “hello”, realize that it is the dentist's office reminding an upcoming appointment, and hang up within 5 seconds. According to the previous embodiment show in , since the call lasted less than 5, the first score is assigned a value of 80. And since both parties were talking, the second score is assigned a value of 40. The average of the two values is 60, which would surpass the predetermined threshold of 50 for placing this call on the blacklist. As the phone number of the dentist's appointment reminder system is placed on the internal blacklist, the customer will not receive further appointment reminders from his dentist. The prevention of such situations will be explored in this section.
One solution to the problem of mistakenly adding numbers to the internal blacklist is to introduce a new internal list. The new internal list would be called an internal potential blacklist. Thus, instead of autonomously placing a number in the internal blacklist, autonomously determined telephone numbers would be placed in the internal potential blacklist. The potential blacklist would contain numbers which are suspected of being blacklisted numbers. The numbers on the potential blacklist would not be subject to call handling procedures. The potential blacklist would only function as an internal storage of numbers. The internal potential blacklisted numbers would then be uploaded to the global list server. The global list server would also have a new list, called the global potential blacklist. Similar to the internal potential blacklist, the global potential blacklist number would not be subject to call handling procedures, and only serves as storage of numbers. The global list server would aggregate all the internal potential blacklists of the authentication devices to the global potential blacklist. If a phone number is on more than a certain number of internal potential blacklists then that phone number can be added to the global blacklist. Any number added to the global blacklist would be treated according to the global blacklist rules. Thus, potential blacklisted numbers compiled by the authentication device may be checked against each other and errors in the system can be reduced.
FIG. 13A
FIG. 13B
FIG. 13A
1310
1311
1312
1320
and illustrate, according to one embodiment of the invention, the concept of potential blacklists. shows three user authentication devices , , for users A, B, and C, respectively. It can be seen that the three authentication devices have an internal whitelist and an internal blacklist. The concepts of internal whitelists and internal blacklists and how they are populated were discussed previously. The authentication devices also have an internal potential blacklist. The internal potential blacklist contains numbers which have been autonomously identified to be potentially blacklisted numbers by the authentication device. The internal potential blacklist is not used for call handling. The numbers on the internal potential blacklists are uploaded to the global list server . The global list server maintains a global potential blacklist. The global list server receives the numbers from internal potential blacklists from all the users' authentication devices.
FIG. 13B
FIG. 13A
FIG. 13B
illustrates the result of each authentication device uploading the internal lists to the global list server. The population of the global whitelist and global blacklist were discussed previously. In example of and , a telephone number on more than 2 users authentication device lists are placed into global list. The blacklisted number 555-9000, 555-9001, and 555-9002 are added to the global blacklist since they are on more than 2 internal blacklists. In addition, the internal potential blacklists are aggregated by the global list server. The global potential blacklist contains all the aggregated numbers transmitted from the authentication devices internal potential blacklist. In this example, a telephone number on more than 2 internal potential blacklists are added to the global blacklist. Thus, the number 555-7000 and 555-7001 are added to the global blacklist from the global potential blacklist, since two number are on the internal potential blacklists of user A and user C. As was discussed previously with regards to global whitelists and blacklists, the determination of how a number on the global potential blacklist gets added to the global blacklist may be modified as seen fit. For example, a telephone number on global potential blacklisted number may be added to the global blacklist if it is on more than a certain number of internal blacklists, or it is on more than a certain percentage of internal potential blacklists, and so on.
The second solution to the problem of mistakenly adding numbers to the internal blacklist is for the global list server to crosscheck the numbers on the global potential blacklist against the internal blacklists received from the authentication devices. Such a solution would be in addition to the global list server moving numbers from the global potential blacklist to the global blacklist. The global list server may check if a number on the global potential blacklist is on a certain number of internal blacklists uploaded by the authentication devices. If so, the number on the global potential blacklist may be moved to the global blacklist. Thus, by checking the autonomously determined blacklisted numbers against the manually determined blacklisted numbers, the process can be made more robust.
The concept of a potential internal and global blacklist can be similarly applied to whitelists. If a user calls a number once, the number can be added to the internal potential whitelist. The global list server can then aggregate the internal potential whitelists of users, and promote numbers that are on a certain number or percentage of users.
m) Server Determination of Received Autonomous Blacklist Determination Metrics
The authentication device can transmit the autonomous blacklist determination metrics, such as call duration, talk time, etc, directly to the authentication server. The server can then aggregate such information and determine if a number should be placed on the global blacklist. Alternatively, the authentication device may upload a recording of the telephone call to the authentication server and the server would measure certain metrics for blacklist determination. The authentication device may only upload a recording of the telephone call if certain metrics criterions have been met and consented to by the user, or the user may initiate an upload of the recording manually.
FIG. 14A-14C
n) Adding a Blocked Caller ID to a Private Whitelist or Blacklist ()
As was described, a number may be put on an internal list for the purpose of handling such calls in a different manner, such as allowing or dropping those calls. However, one restriction in the previous embodiments were that blocked caller ID's, with the exception of caller ID nicknames, could not be placed on the internal lists. As was described, a blocked caller ID has a caller ID text of “Private” or a blank caller ID text, and does not display a phone number. Thus, a blocked caller ID cannot be placed on any internal list, since one blocked caller ID cannot be distinguished from another blocked caller ID. However, if a blocked caller ID is authenticated, the caller phone number of the blocked caller ID is transmitted to the authentication server. Authenticating a blocked caller ID was explained in Section (7-a). If the global list server has access to the authentication server, or if the global list server and authentication server are implemented as a single system, it is possible to add a blocked phone number to a private list stored on the global list server. Since, if the global list server has access to the phone number of a blocked caller ID, this fact can be used to place such a phone number on the private list maintained for the customer.
In this embodiment, the global list server will maintain a private whitelist and blacklist for each individual customer. The private lists on the global list server will contain phone numbers of callers with blocked caller ID's. Thus, if a called party receives a call with a blocked caller ID, the authentication device of the called party proceeds to authenticate it. The authentication server can authenticate the caller with the blocked caller ID, assuming the caller has an authentication device installed. Once authenticated, the authentication device of the called party can put the call through. Such a process was described in Section (7-a). The called party may then answer the call and may wish to add the caller to the blacklist through dialing a star code. Since the caller ID is blocked, the authentication device of the called party cannot add the phone number to the internal blacklist. The authentication device of the called party contacts the authentication server and asks to put the number of the caller on the private blacklist. The authentication server contacts the global list server with the caller number and asks for it to be put on the called customer's private blacklist. The global list server thus adds the number to the customer's private blacklist. The next time the same caller with the blocked caller ID calls, the authentication device proceeds to authenticate the call. The authentication server passes the caller number to the global list server to check if it is contained on any lists. The global list server replies that it is contained on the customer's private blacklist. The authentication server replies with an authenticated message along with the list the blocked number is contained in. The authentication device of the called party handles the call accordingly.
FIG. 14
FIG. 14A-14C
FIG. 14
1410
1411
1420
1421
1430
illustrates the flowchart and messaging diagram for adding a blocked caller ID to the user's private blacklist, by way of example. , collectively referred to as , shows a caller with a blocked caller ID using an authentication device which is calling the called party with an authentication device . The authentication server has also an integrated global list server. The authentication server implements all the functionalities of a global list server, as was described in the previous embodiments. The integration of the authentication and global list server allows for a much more practical and robust system. On each authenticated message transmitted to the authentication device, the authentication server can also look up which list the caller number is on and include the list identifier within the authenticated message. Thus, the authentication device does not need to contact a separate global list server to inquire the calling numbers global or private list status. Alternatively, the global list server may also maintain the internal list of the authentication device.
FIG. 14A
FIG. 7A
1461
1468
1461
1468
761
768
Referring to , steps SA through S have been explained and elaborated in previous embodiments. For instance steps SA through S correspond identically to steps S through S as illustrated in and as elaborated in Section (7-a). Therefore, explanation of those steps will be skipped.
1469
1470
1471
In Step S the authentication server receives the authentication request and looks up if a call has been made to the called number of 555-2000 by a caller with a blocked caller ID of “Private”. If so, then the caller ID is determined to be authentic. Since the authentication server also has an integrated global list server, the authentication server looks up the caller's number of 555-1000 against all the lists in the global list server, namely; global whitelist, global blacklist, private whitelist, and private blacklist. In this example, the number is not on any lists of the global list server. In Step S, the authentication server replies with an authenticated message. The contents of the message in this example are given as “AUTHENTICATED-[Caller:Blocked, Caller ID:Private, CallerList:None, Called: 555-2000]”. The field “CallerList:None” of the message indicates that the caller's telephone number is not on any list of the global list server. In Step S, the authentication device of the called party passes the modified caller ID of “[A] Private” to the called party.
FIG. 14B
FIG. 14A
1472
10
1473
1474
1475
1476
1465
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Referring to , which is a continuation of , after the called party receives the call, the called party decides to block the number from initiating further calls (Step S) by adding the number to the blacklist. The called party adds the number to the blacklist by dialing star code, such as * (Step S). The called authentication device captures the star code and does not pass it to the telephone network (Step S). The authentication device determines that the last caller had a blocked caller ID, therefore the last caller may not be added to the internal blacklist. The authentication device determines that the caller must be placed on the private blacklist. The authentication device sends a message to the authentication server to indicate that the previous caller is to be placed on the private blacklist (Step S). The message sent to the authentication server in this example is given as “ADD_TO_PRIVATE_BLACKLIST-[Caller:Blocked, Caller ID:Private, Called: 555-2000, Password:Pass2000]”. The message indicates that the last caller with the blocked caller ID and with a caller ID of “Private” is to be added to the private blacklist. The field “Caller ID” would not be critical to the adding the blocked number to the private blacklist. The message can simply be interpreted as adding the last caller with a blocked caller ID to the called party's private blacklist. The authentication server receives the message and looks up the telephone number of the last caller with the blocked caller ID (Step S). The caller's number was transmitted with the calling status message in Step S. The authentication server places the caller's number, 555-1000, on the called party's private blacklist. The authentication server sends a confirmation message to the authentication device (Step S). The confirmation message in this example is given as “ADDED_TO_PRIVATE_BLACKLIST-[Caller:Blocked, Caller ID:Private, CallerList: PrivateBlacklist, Called: 555-2000, Password:Pass2000]”. The field “CallerList: PrivateBlacklist” indicates and confirms that the caller was added to the private blacklist. The authentication device sends a confirmation to the called party that the caller was added to the blacklist (Step S). The authentication device confirmation may be in the form of playing a confirmation voice message to the customer after the customer presses the star code.
FIG. 14C
FIG. 14B
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Referring to , which is a continuation of , the caller with the blocked caller ID calls the called party a second time. Steps S through S are identical to steps S through S and are processed in the same manner. Therefore, they will not be elaborated. In Step S the authentication server receives the authentication request and looks up if a call has been made to the called number of 555-2000 by a caller with a blocked caller ID of “Private”. If so, then the caller ID is determined to be authentic. Since the authentication server also has an integrated global list server, the authentication server looks up the caller's number of 555-1000 against all the lists in the global list server, namely; global whitelist, global blacklist, private whitelist, and private blacklist. Since the called party has previously added the caller to his private blacklist, the authentication server determines the caller to be a privately blacklisted number. In Step S, the authentication server replies with an authenticated message. The contents of the message in this example are given as “AUTHENTICATED-[Caller:Blocked, Caller ID:Private, CallerList:PrivateBlacklist, Called: 555-2000]”. The field “CallerList:PrivateBlacklist” of the message indicates that the caller is on the private blacklist of the called party. The authentication device of the called party receives the authenticated message which indicates the caller is on the private blacklist of the called party. The authentication device then drops the call (Step S). The decision to drop the call would be dependent on the call handling settings of the authentication device, but ideally, privately blacklisted numbers would be dropped. Similarly, privately whitelisted numbers would be allowed to connect.
With regards to the private whitelist, the authentication server may be set to place a blocked caller ID number which is not on either the private whitelist or the private blacklist, on the private whitelist if the number calls the called party a certain number of times. Such placement would be similar to the placement of numbers on the internal whitelist by the authentication device, as was explained in previous sections.
The called party may set call handling options such that the authentication device handles calls with blocked caller ID's based on the private whitelist or the private blacklist determination received from the authentication server. As was explained in previous sections, call handling options may include allowing the call to connect to the called party, dropping the call, sending the call to voicemail, etc. The call handling option can also indicate handling calls with blocked caller ID's which are not on either private whitelist or blacklist.
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The concepts introduced in this section of the combined authentication server and global list server maintaining the private lists of the authentication devices may be used to maintain the internal lists of the authentication devices. There isn't a requirement that the internal list of the authentication device be maintained at the authentication device. Instead the global list server, or preferably a combined authentication and global list server, may maintain the internal lists of each authentication device. The authentication/global list server can thus add number that the customer receives or dials a certain number of times to the customer's internal list, since the authentication server is already receiving the called number and received numbers for each authentication device through calling status and authentication request messages. The authentication/global list server can then pass the list status of the calling number within the authentication request reply messages, as was illustrated in Step S.
The concepts introduced in Section (9-e) with regards to aggregating internal lists to global lists may be used to aggregate private lists also. Since the global list server maintains a private whitelist and blacklist for each customer, the global list server may aggregate the private lists and move telephone numbers which are on more than a certain number of private lists to the global lists.
FIG. 15A-15E
10) Caller ID Revealing Techniques ()
FIG. 15A-15C
a) Revealing a Blocked Caller ID to Certain or all Called Parties ()
Since a number of a blocked caller ID is sent to the authentication server by the caller party authentication device, the server may be given permission to reveal the blocked caller ID to certain or all called authentication devices.
FIG. 15A
FIG. 15C
FIG. 15C
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illustrates an example where a caller gives permission for her blocked caller ID to be revealed to all called parties. All called parties would be parties which haven an authentication device installed. The callers' caller ID would still be blocked for any calls to called parties who do not have an authentication device installed. In this example, the authentication server stores all permissions regarding which called numbers to reveal caller ID's to. In the next example, the server will not store such numbers. The caller phone with blocked caller ID initiates a request, Step S, to the caller authentication device to reveal its phone number to all called phones, an example of such a phone is the called phone . Such a request can be in the form of the user going to the authentication device settings in a web interface, or smartphone interface, or IVR, or any other means already discussed. The authentication device receives the request, Step S, and sends a “CALLER_ID_REVEAL” message to the authentication server. The caller ID reveal message contains the caller phone number “555-7000” and the certification password for that number “Pass7000”. The authentication server receives the request, Step S, and first verifies the certification password. The authentication server then stores the caller ID reveal field for the specific called number. In the message the caller ID reveal field is “555-7000”, which indicates that the caller wants to reveal its caller ID as that number. A caller with a spoof permission may reveal its number to be that of a spoofed party which gave the permission. Since the caller ID reveal field is the same as the caller party number, it does not need a spoof permission in this case. The server then stores this reveal number for the party to be called. The called filed in the reveal request determines the number for which to reveal caller ID's for, in this example that field is given as “All Parties”. “All Parties” would mean that the caller ID will be revealed for all parties. The called field could contain a phone number such as “555-2000”, in which case the caller ID will only be revealed for that number. The called field could be a listing of numbers for which the caller wishes to reveal caller ID's for. The “Time-In-Effect” field in the message is given as “1 h”. The time in effect is the time this reveal is valid for, which is given as 1 hour in this example. The server will thus reveal the callers' caller ID to all parties for a duration of 1 hour, after which time the reveal request will be deleted and caller ID will no longer be revealed by the server to called authentication devices. The server will then pass along the caller ID reveal number in an authenticated message transmitted to any number or number contained in the called field for the duration of the time in effect field, which will be shown by example in . The server may acknowledge the results of the caller ID reveal message as being successfully received and in effect. The server may also respond with any error messages should the message have the incorrect password, or would need spoof permission if it wished to spoof another party phone number. The request in Step S, by the user, would ideally be performed before a user makes a call. However, there could be instances where the user has called a number and the called party wishes the caller to reveal her caller ID during the call. In such cases the user can initiate the caller ID reveal request for the call and the steps outlined in this embodiment can be performed similarly. If the server determines that the caller is talking to the called party, the server can transmit a second “AUTHENTICATED” message with the caller ID revealed. Authenticated messages which reveal the caller ID will be explained later with respect to . The called authentication device can receive the second authenticated message with the revealed caller ID and notify the user during the call. If the caller wished to rescind the caller ID reveal request, the caller can initiate a rescind request to the authentication device. The rescind request may be in the form of “CALLER_ID_REVEAL_RESCIND-[Caller:555-7000, Caller ID Reveal:555-7000, Time-In-Effect:1 h, Password:Pass7000, Called:All Parties]”. The “Time-In-Effect” field of the rescind request allows for the rescind to be in effect for only a certain amount of time or indefinitely. The remainder of the rescind request fields are self explanatory and will not be elaborated.
The caller ID reveal message could also contain additional information besides the caller party telephone number. For instance, the message could also reveal the caller name, such as “CALLER_ID_REVEAL-[Caller:555-7000,Caller ID Reveal Name: John Smith, Caller ID Reveal:555-7000, Time-In-Effect:1 h, Password:Pass7000, Called:All Parties]”. The name within the “Caller ID Reveal Name” field could then be transmitted with the revealed caller number to the called party. The user can set what type of information is revealed and which called numbers are to be revealed which type of information. The server could either allow any information to placed within the caller ID reveal name field or the server could verify such information. The verification could consist of the server performing a lookup of the caller ID reveal number in a phonebook and only allow for the caller ID reveal name to match the name in the phonebook.
FIG. 15B
FIG. 15A
FIG. 15B
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shows an alternative embodiment than that was shown in . In , a caller with a blocked caller ID is calling another party. In this example, the authentication device of the caller keeps a list of all parties which to reveal caller ID's for. The listing of called numbers for which to reveal caller ID's for can be compiled manually by the user of the authentication device. Or the listing can be extracted from the user's internal whitelist. Or the listing can be extracted from the users contact list on a smartphone, for instances where the authentication device is a software program running on such a phone. Such listing compilation methods can be selected and the listing itself can be modified by the user. This listing is not transmitted to the authentication server in the form of caller ID reveal messages. Instead, the caller ID reveal indication is transmitted within the calling status messages. Steps S through S are executed the same as in previously described embodiments and figures. Step S is the calling status message transmitted to the authentication server. The calling status message contains the caller number and certification password. The calling status message also contains the “Caller ID Reveal” number. This number is then stored by the authentication server for the purpose of revealing it to the called party number, as contained in the “Called” field of the calling status message. The caller ID reveal number can be the number of the caller or another number that the authentication device has spoofing permission for, as was described in the previous embodiment. Once the server verifies the certification password and checks if the caller ID reveal number is the caller number or another number which spoof permission was given, the calling status message is stored. The calling status message does not contain a “Time-In-Effect” field as in the previous embodiment since this reveal request is applies to the call being placed only, and not other called numbers.
FIG. 15C
FIG. 15C
FIG. 15A
FIG. 15C
FIG. 15B
FIG. 15C
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shows, by example, the flowchart and messaging diagram of the authentication server revealing the caller ID of the caller with a blocked caller ID to the called party authentication device. The example given in could have been the result of the reveal request shown in , since that request was for a request to reveal caller ID's for all parties. Or the example in could have been the result of the calling status message given in and where the call is received by the called party for which caller ID was reveal given permission for. In , the called party authentication device for the called party at the phone number 555-2000 intercepts the call, Step S, and extracts the caller ID, Step S. The authentication device then sends an authentication request, Step S with the blocked caller ID indication. Such steps were already elaborated in previous embodiments. The authentication server receives the request, Step S, verifies the called party certification password, determines that the caller ID is authentic, and also determines that the calling party requested the caller ID to be revealed to the called party. The authentication server then transmits an authenticated message, Step S, with the caller ID of the called party revealed, as shown in the “Caller ID:555-7000”. The authentication device receives the authenticated message, appends the [A] in front of the received caller ID of the calling party, and displays the resulting caller ID “[A] 555-7000” to the user, Step S. Thus, a caller with a blocked caller is both authenticated and the caller ID revealed.
FIG. 15D-15E
b) Caller ID Revealing within a VoIP Infrastructure ()
In Section (7-C), authenticating caller ID's within a VoIP infrastructure was disclosed. The authentication system used the concept of caller ID nicknames and disclosing the nickname within the caller ID presented to the called party in order to uniquely identify and differentiate each VoIP caller. However, VoIP calls can be treated as call with a blocked caller ID. That is, a called party on a landline would not receive a caller ID telephone number from a VoIP user who only has a VoIP username at Skype for instance. Even if the Skype caller has a telephone number assigned to her by Skype where she can receive calls, Skype might not pass this telephone number in her caller ID to called parties. In such instances, we can treat a VoIP call that does not transmit either the VoIP unique username or VoIP user telephone number, as a blocked caller ID. And such a VoIP call can be authenticated in the same manner as a blocked caller ID, as was elaborated in Section (7-a). And such a VoIP call can have its' caller ID revealed to the called party.
FIGs. 15D and 15E
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illustrate the messaging diagram of an embodiment of the invention for VoIP caller ID authentication and revelation. The process is similar to revealing a blocked caller ID. The caller is using a Skype software client on his PC. His Skype username, also known as the “Skype Name” or “Skype Contact Name”, is “JohnSmith” and does not have an accompanying telephone number connected to the VoIP account. Thus, when he calls a telephone such as a landline, his caller ID may be displayed as blank, “Out of Area”, or as “Skype Caller 1-XXX-XXX . . . ”, where the name “Skype Caller” and number “1-XXX-XXX . . . ” is shared between all Skype callers and does not serve the function of a uniquely identifying caller ID or caller ID nickname. The caller authentication device may be an addon to the Skype software client, or software implemented by Skype within the client, or any number of other implementations already discussed. The device initiates certification by contacting the authentication server, Step S, and sends a password request, Step S. The contact to the server would preferably be through the internet as the user is assumed to be connecting to Skype over the internet on his PC. The password request consists of the VoIP provider the user is at, “Skype”, and the Skype username the authentication device is connected to “JohnSmith”. The server receives the request and initiates a VoIP call to the username over the VoIP provider contained in the request, Step S. The contact to the Skype username would also occur through the internet. This is somewhat different with respect to previous examples where the contact to the server by the device was preferably performed through the internet, but the certification password was transmitted by the server to the device over telephone or SMS networks. The authentication server thus contacts “JohnSmith” at Skype. This contact can be done by the server using the Skype API, or through any other means. Alternatively, the caller may have a Session Initiation Protocol (SIP) address and the server can contact the caller authentication device at the SIP address. The server generates a password, Step S, and transmits it Step S. The authentication device receives the call and the transmitted the password. The password can be transmitted by the server through DTMF tones during the call and the device can then decode the DTMF tones and determine the password. The certification password can be transmitted during the call through computerized voice, fax, or modem, as previously disclosed. Also previously disclosed, the password can be transmitted through the caller ID of the call from the server to the device. Alternatively, Skype users can receive instant messages at their username, thus the password can be transmitted within an instant message to “JohnSmith” at Skype without the need to place a call. And the authentication device can intercept the instant message and extract the password.
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The authentication device replies with the received certification password, Step S. The acknowledgement would preferably contain the password to verify that it has received the password correctly. The server would then store the username and the accompanying password, Step S. This would also serve as the first heartbeat and prove that the authentication device is installed at the username. In this example, no other heartbeats are sent by the device. The device can now present the username “Skype:JohnSmith” and password “7890” to the server and prove that it is the actual authentication device at that username. The certification of the called party is not explicitly shown, but is assumed to have taken place with the called party receiving the password “Pass2000”.
After certification, the caller can initiate calls and have its username authenticated and if desired, revealed to the called party. This process is similar to authenticating a blocked caller ID and revealing such a blocked caller ID to certain or all parties.
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In Step S, the Skype caller initiates a call to the called party . The authentication device captures the dialed numbers, Step S. The call is passed to the called party, Step S. The device contacts the server with a calling status message, Step S, containing the Skype username as the caller field, the certification password, the caller ID to be revealed, and the called party telephone number. The called party authentication device intercepts the call S, and extracts the caller ID, S. The called authentication device contacts the server, Step S, with an authentication request with the Caller field indicated to be blocked and the caller ID field blank to signify no caller ID number or name has been received. The authentication server verifies the called party authentication password, Step S, and checks if a caller is currently calling the called party. The server determines that the Skype caller is currently initiating a call to the called party. The authentication server sends an authenticated message, Step S, with the caller ID field revealing the caller username and the VoIP provider of the user, which is “Skype:JohnSmith”. The server also indicates the caller is currently not on the called party internal lists or global lists. The authentication device of the called party then modifies the caller ID presented to the called party, Step S, to indicate the revealed VoIP username and the authentication status. Thus, the VoIP username is revealed and the VoIP username is authenticated.
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After the caller authentication device determines the call has ended, Step S, the device sends a call end status message to the server to signify that the caller has ended the call with called party, Step S. Such messages mitigate race conditions and allow the server to determine which party connected to the called party when caller ID information is unavailable to the called party. As elaborated, such messages can be transmitted by the caller and called parties to signify stages of each call being made or received, such as call ringing, call busy, and call ended messages. However, such messages use communication bandwidth and other resources, especially if the authentication device is implemented on a smartphone as software and is contacting the server through a wireless data plan with a bandwidth cap. Thus, it would only be desirable to have callers with blocked caller ID's to transmit such messages, especially a call busy or call end message. If the caller's caller ID is not blocked, then a race condition is mitigated by the called party transmitting the caller ID of the connected caller to the authentication server. In cases where the authentication server receives a calling status message and does not receive an accompanying authentication request from the called party within a certain amount of time, such as 10 seconds, the server may infer that the caller either hung up the call before being connected or the called party was busy taking another call or the called party was not out of coverage area, if on a smartphone. The server can then expire the calling status message and would not need a call ended status message or a call busy message from the caller party. The tradeoff in such a system is the possibility that an attacker might call the called party with the spoofed caller ID at the same time the victim is calling the in.
called party and connect to the called party before the victim and be authenticated. The risks of an attacker guessing the caller number which made a call and did not connect within the same timeframe are low but not impossible, and would have to be weighed against the available network and server resources to send and process additional messages.
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In another embodiment of the invention, the server may measure the average time between receiving a calling status message and an authentication request message, and use the time measurement in determining when to expire such messages. The time between the two messages reflecting when the call is initiated by the caller party and when the call is received by the called party, which is the time period it takes to connect the call. If the server determines that the average time between the two messages is 5 seconds, then the server may expire such messages after 10 seconds, allowing a margin of safety of 2. The server may measure the time between the two messages with respect to area codes, geographical locations, telephone networks, and other aspects of the caller and called parties. Thus, the server may infer that the caller being in a 555 area code takes at least 3 seconds to connect to a called party. Or the server may infer that a caller being in a 555 area code calling a 444 area code takes on average 4 seconds to connect. Such time measurements may be used to differentiate between race conditions as previously explained. If a calling status message was received from a 555 area code caller, and at the same time a calling status was received from a 444 area code caller, and if a authentication request was received by a area code called party with 3 seconds, the server may use the previously measured connection times and infer that it is most likely that the 555 area code caller connected within that time frame, and not the 444 area code caller. These time measurements may further be utilized to determine a caller with an illegitimate spoofed caller ID. For instance, if a spoofer with a spoofed caller ID area code of 555 sends a calling status message and the called party at the 444 area code sends an authentication request within 3 seconds, but average time for a 555 caller to connect to a 444 party was previously determined to be 4 seconds, as measured by the authentication server, then the server may determine the caller ID is spoofed. Or the server may scrutinize the caller party. The authentication server may request that the caller authentication device recertify. Or the server may audit the caller authentication device and request the last 3 calls received or made by that device, and compare it to the records for that device on the server. The same concept would equally be utilized for VoIP callers and called parties. To aid in more accurate measurement of the connection time, the calling status message may be transmitted to the server at an exact time or instance within the call initiation, such as after the last digit of a valid telephone number is dialed by the caller party. Likewise, the authentication request may be transmitted to the server at an exact time or instance within the call reception, such as when the first or second ring is received by the called party. The authentication devices may send messages separate from the calling status and authentication request, to aid in measuring the connection time of the call. Such steps would allow for very accurate measurements of the call connection times by the server.
11) Privacy Protections
Since the authentication server is notified of each call that is being placed and each call that is being received, the users of such a service might have privacy concerns. To resolve such issues, the information that is sent to the authentication server could be anonymized. For instance in a calling status message, the number of the caller and called parties are transmitted to the authentication server, where the authentication server is made aware of who is calling whom. Instead of transmitting the phone numbers, an encoding of the phone number may be transmitted. The encoding would be performed through a predetermined algorithm. Such encoding algorithms would allow for one-way encoding of information and not allow for the encoded information to be decoded, such examples of encoding are hashes (e.g. MD5, SHA256).
An overly simplified example of such an encoding might be the sum of adding up all the numbers contained in the phone number. Thus, instead of a calling status message in the form of “CALLING_STATUS-[Caller:555-2000, Password:Pass1000, Called: 555-7000]”, the calling status message with the phone numbers encoded through summing all the digits would be “CALLING_STATUS-[Caller:17, Password: Pass1000, Called: 22]”. The caller number “555-2000” is encoded as 17 by adding the digits 5,5,5, and 2. The same for the called phone number. And the authentication request “AUTHENTICATION_REQUEST-[Caller:555-2000, Called: 555-7000, Password:Pass7000]” would be encoded as “AUTHENTICATION_REQUEST-[Caller:17, Called: 22, Password:Pass7000]”. Thus the authentication server can match the encoded caller and called numbers, and authenticate the caller ID. The password would also be paired with the encoded phone number. Thus, the server would store the certification password for the caller with the encoded phone number of “17” as “Pass1000”. During certification the server would have to receive the actual telephone number in order to contact the number and transmit the certification password. However, after the password is transmitted, the server may then encode the phone number, store it alongside the password and destroy the non-encoded phone number. Where the caller would only transmit the encoded phone number and the server can verify that the password matches the password on file for that encoded phone number.
Such privacy protections would allow for the authentication server to authenticate caller ID's without knowing the actual phone numbers of users and whom they are calling or receiving calls from. Such encoding would break the processes of caller ID revealing. The encoded caller ID may be revealed, and the called party could then associate the encoded caller ID with a contact name on her phone. However, such encoding still makes it possible to add a number, even blocked numbers, to internal, private and public lists. The requirement is that the encoding results in a unique string for a unique phone number and that no collisions occur where two different phone numbers have the same encoded string.
Safeguarding the communication between authentication devices and the authentication server through the use of cryptographic techniques, such as utilizing public/private keys, signed certificates, and other techniques are beyond the scope of this invention disclosure and will not be elaborated upon. Similarly, safeguarding the transmission of the certification password from the authentication server to the authentication device phone number will also not be discussed. A practitioner in the art may utilize any and all currently known network and telecommunications cryptology and security methods to safeguard such communications.
12) Authentication Server Load Balancing and Scaling
Since caller ID authentication and spoof detection is being performed by a single centralized entity, the authentication server, a potential issue is the computational and network load the server will have to endure. If it is assumed that the authentication server is performing caller ID authentication for the entire telephone system, the server will likely have to process thousands of calls per second. In order to mitigate such issues, sub-servers can be set up to segment the number of customers served and each sub-server can contact other sub-servers in cases where a caller or called party is served by another sub-server. For instance, one sub-server can be set up to serve the 555 area code. Thus, all 555 area code callers and called parties would contact the 555 sub-server for the purpose of caller ID authentication. Furthering the example, the 555 sub-server can be located at the domain name “555.authenticationserver.com”. Anytime a caller or a called party with a 555 area code phone numbers, such as 555-444-3333, makes or receives a call, it will contact the “555.authenticationserver.com” sub-server. The primary motivation with such segmentation is to segment the caller and called parties based on area codes, with the assumption being parties are most likely to call other parties within close geographical proximity and hence same area codes. Thus all calls within the 555 area code are served by a single sub-server without the need to contact another sub-server. It is also preferable that the sub-server which serves a certain area code be physically located within the location served by the area code and thus reduces any latency between the devices and the sub-server. Once all the parties within a geographical proximity are served by a single sub-server, if a need arises to contact a party not served by the sub-server, the call information can be passed onto the proper authentication sub-server. As a result, the caller party with a 555-444-1000 phone number can contact the “555.authenticationserver.com” sub-server. If he is calling a number which is not served by this sub-server, such as 560-444-2000, the 555 sub-server can contact the 560 sub-server and forward the calling status message. Alternatively, if the 560 area code server receives an authentication request, it can contact the 555 sub-server to authenticate the caller ID of the 555 caller. The methodologies of one sub-server authenticating itself with respect to another authentication sub-server are beyond the scope of this invention disclosure and will not be elaborated upon, but would ideally be performed via public-key infrastructure (PKI). Further segmentation can be achieved based on prefixes, also known as exchange codes. For instance all parties with a phone number area code 555 and 444 prefix, such as 555-444-3333, can contact the “444.555.authenticationserver.com” sub-server. Or certain area codes can be combined, such as all parties with a 555, 556, or 557 area code phone numbers can contact the “555-557.authenticationserver.com” sub-server. Again, the primary motivation is to segment users based on geographical proximity. And if 555, 556, and 557 area codes are in close geographical proximity to each other, it would be assumed that majority calls to and from the 555 area code originate to and from the 555, 556, or 557 area codes. Or the segmentation can take into account the caller party area code and the called party area code, such as if a caller with a 555 area code is calling a party with a 557 area code it can contact the “555.calling.557.authenticationserver.com” sub-server. Again, the motivating assumption being that a majority of calls from the 555 area code connect to the 557 area code, and vice versa.
The same concept can be applied to VoIP callers without telephone numbers. The IP address of the VoIP caller can be mapped to a geographic location. The methodology of deriving an approximate geographic location from an IP address is well known in the art and will not be elaborated upon. Once the geographic location of the VoIP caller or called party is known, the location can be mapped to a corresponding telephone area code. The authentication device of the VoIP caller or called party can then contact appropriate authentication sub-server. As a practical example, let us assume that a VoIP caller had an IP address “173.77.1.1”. The IP address of the user would be mapped to a geographical location, let us assume the IP address was mapped to the city of New York. And let us assume that specific geographical location corresponded to the “212” telephone area code. Thus, the VoIP caller with IP address “173.77.1.1” would contact the authentication sub-server at “212.authenticationserver.com” for the purpose of transmitting a calling status message. Therefore, the benefit of segmenting users based on geographical proximity can be adapted to VoIP users.
Alternative to segmentation by area code, segmentation can be performed by obtaining the physical location of the telephone, such as through GPS data, and the authentication device contacting the nearest geographical authentication sub-server. The GPS data can be mapped to an area code instead, and the appropriate sub-server can be selected.
There could be present a central top-level authentication server for the purpose of authentication devices inquiring which sub-server to contact based on its telephone area code, GPS location, or network/IP address. The top-level server can also be utilized for authentication of one sub-server with respect to another sub-server.
13) Caller ID Authentication without an Authentication Device
The previous sections described caller ID authentication utilizing authentication devices in communication with an authentication server. Also peer-to-peer authentication has been described where there isn't an authentication server available. In both cases, an authentication device was necessary for caller ID authentication. In this section, authenticating caller ID without an authentication device will be discussed.
If a caller did not have an authentication device, and the called party had an authentication device, the caller ID was determined to be irresolute in the previous examples and sections. However, if the authentication server had access to the caller party information, such as calls the caller party was making in real-time, then the authentication server would have the same information as would be received within a calling status message. The authentication server then would be able to determine a caller ID as authentic, spoofed or irresolute in reply to an authentication request. For instance, if a caller is on the Verizon telephone provider and does not have an authentication device installed, the caller may opt-in for the Verizon provider to make his calling information available to the authentication server. The authentication server would thus receive all calls being made and received by that individual. If the caller then made a call to a party with an authentication device, the authentication server would receive the caller telephone number and the destination telephone number. The called party authentication device would send an authentication request. The server could then reply with the authenticated message. Even though the Verizon telephone provider is not implementing the full functionality of an authentication device for each customer, the called party can authenticate caller ID's from any Verizon customer who opted-in. Also, any opted-in Verizon customer would be protected from spoofing with respect to called customers using authentication devices.
The previous example can be further developed where both the caller and the called parties do not use an authentication device. If the authentication server had access to a called party's caller ID information in real-time, then the authentication server would have the same information as contained in the authentication request message. The authentication server can then authenticate and detect spoofing if it had access to both the caller's and the called party's call information in real-time. However, since the telephone provider is not implementing the full functionality of an authentication device for each customer but only providing call information, the customer would have to be made aware of caller ID authentication determination in another way. If the authentication server detects that a called party is receiving a spoofed caller ID, then the authentication server can call the called party during the spoofed call. If the called party has a call waiting feature, she can put the caller on hold and receive the call from the authentication server. The authentication server can then inform the called party of the spoofed caller ID though a computerized voice. If the called party does not have call waiting, the authentication server can wait until the called party line is not busy, and then contact the called party. The called party can then be made aware of the previous caller's caller ID status. Or the authentication server can send a text to a phone number, send an email to a designated address or simply log spoofed callers where the user can view such information about her received calls at a website. A user may set how she wants to be warned by calling the authentication server service telephone number and inputting her preferences through IVR, or visiting the authentication server website and logging in.
If a landline called party is not utilizing an authentication device, but the server is able to capture the caller ID's the called party receives, the authentication server may forward the caller ID authentication status to another device, such as a smartphone. Authentication device software may be installed at a smartphone. The authentication device software may authenticate caller ID's for the calls made or received by the smartphone telephone number. However, in this example the smartphone authentication device is primarily used to display caller ID status of the calls the landline phone receives. The authentication device would contact the authentication server and request a password for the landline phone by transmitting the landline telephone number. The authentication server would then call the landline phone and transmit the password, preferably through a computerized voice. The landline customer would pick up the phone, hear and determine the password being said by the voice, and input the received password into the authentication device on the smartphone. The authentication device would then be able to authenticate itself as the landline authentication device to the server. When a call is received by the called party on the landline, the authentication server can then contact the authentication device on the smartphone and transmit the caller ID status of the call being received by the landline phone. Such a system could also allow for a user to manually input, by hand, into the smartphone authentication device telephone numbers of calls made or caller ID's received by the landline phone and receive authentication status of such calls.
Such a system could be used in cases where the caller has an authentication device and the called party does not have an authentication device, such as a called party using a landline, and where the authentication server is not capturing caller or called party call information in real-time. The called party can designate the authentication device software on her smartphone as the authentication device of the landline. Thus, the server would forward or push the authentication status of calls being made to the landline phone, as described Section (4-b), to the designated smartphone authentication device. However, in such a case the server would only know authentic caller ID's being received by the landline and would not be able to inform the landline designated software of spoofed or irresolute caller ID's. Such a system would also allow for the smartphone to legitimately spoof its caller ID to that of the landline phone, where entering in the password of the landline phone into the smartphone authentication device would serve as the permission.
A telephone provider, such as Verizon, can determine the caller ID status of a call received by one of its customers without the need for implementing authentication devices or an authentication server. For instance, if Chase Bank was a Verizon customer and called a bank customer who is also a Verizon customer, Verizon can determine with certainty that the caller ID displayed on the bank customer is authentic. Since Verizon is aware of the calls and caller ID's being received by the bank customer, who is a Verizon customer, Verizon can look up if the caller in the caller ID field is also a Verizon customer. If the caller is a Verizon customer, Verizon can simply look up who the caller is currently calling, and if the caller is indeed calling the called customer, it can determine the caller ID to be authentic. If a Verizon customer received a call claiming to be Chase Bank, Verizon can look up calls currently being made by Chase Bank, and if Chase Bank is currently not making a call to the Verizon customer in question, Verizon can determine this to be a spoofed caller ID. Thus, for any Verizon caller who calls another Verizon customer, Verizon can determine the caller ID is authentic. And anytime an attacker calls a Verizon customer and spoofs the caller ID of another Verizon customer, Verizon can determine the caller ID spoofed. Likewise, if a non-Verizon customer called a Verizon customer with a true caller ID, the caller ID could not be determined by Verizon to be authentic or spoofed and would thus be irresolute. For irresolute caller ID's, Verizon could contact an authentication server in the hope that the caller is utilizing an authentication device, and can then receive caller ID authenticity status. The authentication server can be implemented by Verizon for the purpose of authenticating out of Verizon network caller and their caller ID's or the server can be implemented by a third party and Verizon can be given access to it. Indication of caller ID authenticity can be performed by Verizon through caller ID modification, as was previously described. Verizon could prefix an “[A]” tag to authentic caller ID″s and an “[S]” tag spoofed caller ID's. Alternatively, the caller ID status could be pushed to an authentication device software running on the customer's smartphone, as previously discussed. The authentication device as it applies to this example can be, but need not be certified, neither actively or passively. Since the authentication device only serves to display the caller ID status, it does not need to prove its ownership of a telephone number. Therefore, the telephone customer can download the software from the telephone provider and provide the software the same customer credentials that were used to sign into his online telephone account, such as email address and password. The software can then use these credentials to look up the caller ID status of calls that he receives and display them to the customer. One issue with such an embodiment is legitimately spoofed caller ID's. Continuing the given example, if Chase bank had a call center in a foreign country and called the bank customer with a spoofed caller ID to that of Chase Bank, then Verizon would check calls being made by Chase Bank, determine that Chase is not currently calling the called party, and mark the caller ID as spoofed. To solve such issues, Verizon could set up an authentication server where Chase Bank can give spoofing permission to the foreign country call center and the call center can send calling status messages to the authentication server and have the legitimately spoofed caller ID authenticated. The caller center at the foreign country would preferably be actively certified, or can be passively certified.
14) Advantages, Ramifications and Authenticating Other Forms of Communications
The caller ID authentication system can be used selectively, instead of applied to all calls made and received by the customer. For instance, a customer may install a software app provided by his bank onto his smartphone. The bank app may track calls made by the customer and only contact the authentication server when the customer dials the bank customer support number or the bank branch. The authentication server may also be set up by the bank and only service the bank app and the bank. Thus, when the customer calls the bank, the bank app can send a calling status message to the banks authentication server. The bank customer support operator or the bank branch operator can verify the authenticity of the caller ID of the customer through software running on their systems contacting the authentication server. And if the bank called the customers cell phone, the bank app can indicate the authenticity of the banks caller ID to the customer. In such a fashion, each institution may release a smartphone app that caters to their customers.
It should be understood that the system and methods introduced in this invention disclosure may be used in any telecommunication system and any aspect of telecommunications. For example, the method of caller ID authentication of this invention may be utilized in a voice mail system. In such a voice mail system, calls from certain authenticated caller ID's, such as from the owners of the voice mail box, can be put through to the voicemail inbox and voicemail settings without the requirement of a password challenge. In another example, a fax machine may utilize the disclosed invention to authenticating caller ID's through an integrated authentication device. Thus, the fax machine can indicate the authentication status of the fax sender in the page margin of the fax document. In another example, a financial institution may utilize the disclosed invention to authenticating caller ID's to verify that the account holder of record is calling the financial institution.
a) Applications of Caller ID Authentication Methodology in Other Communications Systems
One should also realize that the described methods of caller ID authentication and list based call handling may be applicable to other fields of communication, such as email, phone texting, radio communication and the like.
For example, in an email system, the described invention of caller ID authentication may be used to authenticate the email sender address. The email sender and the email receiver can implement an authentication device, such as software running on the user's PC, or the email provider may implement the authentication device within their email servers as software, hardware or combination thereof. Thus, as soon as an email is sent from the email sender to the recipient, the sender authentication device can transmit an email sent status message to an authentication server. The email sent message can include the sender email address and the recipient email address. As soon as the email recipient receives the email, the recipient authentication device can contact the authentication server, and send an authentication request. The authentication request can include the sender email address, as determined by the “From:” field of the email (similar to the concept of caller ID), and recipient email address. The authentication server can then compare the email sent message sent by the sender and the authentication request sent by the recipient to determine that the email sender address is authenticated, spoofed or irresolute. The authentication device of the recipient can then modify the email sender address to indicate the authentication status of the sender's email address. Or the authentication device may indicate the authentication status of the sender email address in other forms. The concept of certification may also be directly used in email address authentication. The authentication device may transmit a certification request to the authentication server containing the email address that the authentication device is installed at, and the authentication server may then email a certification password the email address. The authentication device may then use the received password for all transmissions sent to the authentication server. Thus, the authentication server can verify that the authentication device possesses ownership of the email address. Thus, one skilled in the art can utilize any and all concepts introduced in this disclosure, such as caller ID authentication, caller ID spoof detection, certification, peer-to-peer authentication and certification, local lists, global lists, private lists, in email authentication and email handling system. The concepts of the invention may thus be used in other field of communications where it is advantageous to authenticate the sender address, and where the field of communication allows for a sender address to be authenticated according to this invention disclosure.
The concepts of active and passive may also be applied to other fields of communications, such as the email sender authentication system thus described. An example of passive authentication in an email system would be constructing a passive certification password from the email “From:” or “To:” fields.
As should be apparent to those skilled in the art, the same concept of caller ID and email authentication just discussed can be used similarly in SMS, MMS or any other form of text messaging in order to authenticate the source address of such a message.
Another field of communications that would benefit from the described methodology of sender authentication and list based message handling is postal mail. Although the concepts described thus far can be easily be utilized to authenticate the sender of a piece of postal mail by those skilled in the art, a non-limiting example of how such a system might be implemented will be discussed. The postal mail authentication system would have an authentication server which would receive from a communication network, preferably over the internet, the mailing status messages and mail authentication request messages from authentication devices. The authentication server would then transmit the authentication status of the source address of the piece of postal mail to the recipient authentication device. The authentication device may be implemented by the sender of the postal mail within a digital postage system, such as those offered by Stamps.com or Pitney-Bowes. In such systems, the sender of a postal mail is allowed to print postage directly from their PC. Such systems also offer a digital scale connected to a computer for adjusting the printed postal rate based on the mail weight, given a specific destination. Thus, when the sender of the mail item prints postage for a specific destination, the integrated authentication device within the digital postage system would contact the authentication server and send a mailing status message containing the sender postal address and destination postal addresses. The authentication server would store the mailing status message. As is apparent, the sent postal mail would have written on it the source postal address and the destination postal address. The source postal address on the mail article might or might not be the sender address of a mail item. For instance, an attacker might send a postal mail which shows the source address to be a bank and try to trick a postal recipient by instructing them to call a certain telephone number, which might be controlled by the attacker and used to record sensitive financial information disclosed by the victim.
The authentication device at the recipient might consist of a software program running a PC. Once the recipient receives the postal mail, she can merely enter the source postal address that is indicated on the piece of mail into the software program. The authentication device would then contact the authentication server with an authentication request containing the recipient postal address and the source postal address of the mail item. The authentication server would then determine the source address is authentic if the source address matches the sender address, and the recipient address matches the destination address contained in the mailing status message and the authentication request message. Such matching has already been described with regards to caller ID authentication.
The authentication device at the recipient might be in the form of an imaging scanner connected to a PC and accompanying software running on the PC. In such a system, the imaging scanner would perform optical character recognition (OCR) and extract the source address indicated on the postal mail, and the user can be saved from the inconvenience of entering in the source address of every mail item she receives. If needed, the address the machine is residing at, that is the address the recipient is at, can be entered manually by the user at the installation of the authentication device. The authentication device may be in the form of a smartphone with an imaging device, such as a camera, and accompanying software. The smartphone of the recipient may be used to take a picture of the postal mail, the software may then perform OCR and extract the source address. Instead of performing the OCR locally on the device, the authentication device may transmit the taken picture directly to the server, where the server would perform the OCR. The sender mailing system might encode the sender address and destination address in a machine readable form on the mailing, such as a barcode (i.e. 1 dimensional bar code), Quick Response Code (i.e. QR code or 2 dimensional bar code), or the like. Next to the QR code on a mailing envelope, a message such as “Scan this code to authenticate the source address of this mail item” might be placed. The recipient can then scan the QR code, and the authentication device can extract the address of the sender and contact the authentication server with an authentication request. Such encoding would also make possible for other kinds imaging systems to extract the source and destination addresses, such as barcode scanners. The described smartphone system may also be utilized by the sender of the postal mail. The sender could enter the mail destination address manually into the software for an article of mail or scan the mail article and the software application would then process the destination address through OCR. As should be apparent, the types of authentication devices given in the examples for the sender and the recipient can be interchanged such that one type of device can also be used by the other party.
The certification process would be similar to the certification processes described thus far as it relates to other communication systems. The authentication device would send a certification request to the authentication server. The authentication server would then initiate the mailing of the certification password through the postal mail system. The recipient of the certification password postal mail would either enter in the certification password manually into the authentication device or scan it with an imaging device of the authentication device. Such a certification process would prove that the authentication device is residing at the postal address it claims to be at. The authentication device could contact the authentication server and verify certification password. This initial contact would present the first heartbeat and prove to the server that an authentication device is installed for that postal mail address. The authentication device may send heartbeats at regular intervals to indicate that it is functioning at that address or may omit doing so. The concepts of the previously elaborated passive certification password would also apply to postal mail authentication. The passive certification password would be constructed from destination and source addresses of mail sent or received.
Thus, the mailing status message would contain the sender address, destination address and the certification password received at the sender address by postal mail from the authentication server. The authentication server would thus compare the sender address and the certification password in the mailing status message to the certification password sent to that sender address and only accept the mailing status message if they match. Likewise, the authentication request message would contain the source address, destination address, and the certification password of the destination address. The server would thus compare the destination address and the certification password in the request to the certification password sent to the recipient address, which is the destination address, and only reply to the request if they match.
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Such a system could be vulnerable to a race condition such that a bank could send a legitimate postal mail at the same time an attacker sends an illegitimate postal mail claiming to be from the same bank. Such a situation could lead to the illegitimate mail being authenticated since it arrived first, and the legitimate mail being determined to be spoofed since it arrived later. To mitigate such issues, the piece postal mail may have associated with it a race string, such a string consisting of letters or numbers that would identify the postal mail and help differentiate it from other sent pieces of mail. The race string would be constructed by the authentication device of the sender, and sent to the authentication server within the mailing status message. The race string would be indicated on the postal mail or be able to be derived from the postal mail. Such a race string can simply be in the form of a date sent and a couple of random digits, such as “1/1/2014:12” or the date and time, such as “1/1/2014-1:01:01 pm”. Thus, the attacker would have to send the postal mail on the same date “1/1/2014”, and guess the later two digits or exact time. A more complicated system could be 8 or more random digits or characters, which would make it very difficult for an attacker to guess. Another system could consist of the sender keeping a running tally of the mail items sent, and indicating the tally on the mail item. Such that the race string can be “10” if the sender has sent 9 pieces of mail and this is the 10mail item. The tally could be the number of all mail sent by the sender or just mail sent by the sender to that specific recipient. Such a system would require the attacker to know the total number of mail items the sender sent in total or the number of total mail items the recipient received from the sender. Such a system would also allow for the recipient to become aware of lost or delayed mail. If the recipient is authenticating a mail item with a “10” race string, but never received or authenticated a mail item with a “9” race string, then she could be made aware of the missing mail item with the “9” race string by the authentication system. The race string could be the weight of the postal mail, where the sender weights the mail, sends the weight to the authentication server within the mailing status message, and the recipient weights the mail and sends the weight to the authentication server in the authentication request. Thus, the mail can be indentified according to its weight from other mail that might have the same source and destination addresses. Similarly, the race sting could be the shape, dimensions, color, or an actual photographic image of the postal mail. The image can consist of the mail envelope or the mail contents. The image transmitted by the recipient authentication device can then be accepted by the server if it is sufficiently similar to the image transmitted by the sender authentication device. Another example of a race string that can be derived from the mail could be the hash of the contents of the mail. The determination of the race string from the mail item can be through a predefined algorithm, such as simply using the weight, or for instance, weight multiplied by its circumference. The race string may be a combination of thus described examples and the race string can be indicated explicitly on the mail, such as the date and time when it was mailed, or not explicitly indicated on the mail and able to be derived from the mail, such as its weight. Thus, the sender authentication can construct the race string, send it with the mailing status message to the server. The recipient authentication device would then send the race string of the received mail. The server would then be able to differentiate between two postal mailings with the same source and destination addresses and authenticate them individually. Such authentication would consist of the server verifying that a mailing status message and the authentication message both containing the same race string. If the authentication server can't match an authentication message with a mailing status message that has the same race string, then the source address would be determined to be spoofed. The authentication device of the recipient may then indicate the authentication status of the source address. Summarizing, the race string can be explicitly indicated on the mail, such as random string of characters, date and time the mail item was sent, and so on. Or the race string can be implicit and not indicated on the mail item, but can be derived from the mail item itself, such as its weight, shape, dimensions and hash of the contents of the mail. There is no requirement that a race string is used and may be omitted if desired. If the race string is omitted, it would be desirable for the server to only authenticate mail items in which are received by the recipient in a certain time. That certain time would depend on the postage class used, which would be indicated by the sender to the server, and the geographical distance between the sender and recipient addresses. The server would hence construct and maintain an expected arrival time for mail based on such factors, and if the recipient tried to authenticate a mail item past the calculated time, the server may warn the recipient.
If the race string is constructed by the sender authentication device to be a sufficiently unique and hard to guess string of characters, such as 10 random alpha-numeric characters, an attacker would not be able guess the race string easily. In such an example, a QR code can placed on the mail by the sender which would only encode the race string, and nothing else. Again, the mailing item might indicate “Scan this QR code to authenticate”. The sender authentication device would still send a mailing status message with the sender postal address, recipient postal address, certification password of the sender, and race string. The recipient authentication device however, need only scan the QR code, extract the race string, and send the authentication request containing only the race string, the recipient postal address and certification password of the recipient. The certification password of the recipient is not vital to the authentication process, but is desirable for other security issues. Such issues could be where an attacker brute forces the race string and tries to determine the senders that a recipient is receiving mail from. The QR code could be a URL that contains the race string, where the URL points to the authentication server and race string is appended to the URL as a query string (i.e. HTTP GET method). The authentication device may then append the certification password to the URL as another query string. For security purposes, the certification password sent to the server may instead be a hash computed by appending the race string to the password and taking the hash of the resulting string. The recipient postal address may or may not be sent in the authentication request. Since the server would receive the race string in the authentication request, the race string allows the server to match a corresponding mailing status message with the same race string and determine the destination address. And if the certification password transmitted in the authentication request matches certification request for the destination address, then the server can authenticate the postal mail. Thus, after receiving the authentication request and verifying the certification password, the server can look up the race string, confirm that the sender sent a message with the race string to the recipient address, and send the address of the sender to the recipient authentication device. The recipient can then determine the source address of the mail. The source address of the mail can be indicated on the mail piece, which the recipient can cross check with the sender address supplied by the authentication server. The source address of the mail can be omitted from the mail piece, and the recipient can discover the address of the sender through the message sent by the server.
Such scanning of a QR code by the recipient would also serve as a confirmation that the recipient has received the item, known in the art as delivery confirmation or return receipt. Since the recipient device is transmitting the certification password of the device in the authentication request, the request proves that the intended recipient has received the mail item. Such a system could be utilized in credit card mailings for example, where the customer who receives the credit card mailing could simply scan the QR code to verify they have received the credit card, and the bank can automatically activate the credit card. The authentication request transmitted to the authentication server can include both the recipient postal mail certification password, and the recipient telephone certification password, thus verifying the postal address and phone number of the recipient. Alternatively, a QR code sticker could be placed on the credit card. The QR code would instruct the authentication device to place a call to a telephone number indicated on the QR code. The authentication device would then call the QR code telephone number and transmit a calling status message to the server. The calling status message could also contain a race string contained in the QR code. The called party (i.e. Bank) would then authenticate the caller ID of caller, verify the caller has transmitted the appropriate race string to the server, and activate the credit card. The race string could be the credit card number that is wished to be activated. The race string could be an encoded form of the credit card number, or the last 4 digits of the credit card number.
Such a race condition could also apply to electronic mail previously described. In an electronic mail, the race string can be constructed from the subject line, if the subject line is sufficiently unique. Or the race string can consist of a hash of the email message body, computed using hashing algorithms such as MD5, SHA256, and the like. Or a constructed race string, such as 8 random characters, can be inserted into the email by the authentication device of the sender, such as within the email body, subject line or From field. The race string can also be when the email was transmitted, where the transmission time and date are usually indicted on the email header. If the race string is inserted into the mail by the sender, then the race string could have a tag prefixed or postfixed which allow the recipient to find and extract the race string from the email. For example, if the race string is “1234”, then the tag can be prefixed such as, “<Race String:1234>”. The recipient can thus find the text “Race String” between the two enclosures “<”, “>”, and extract the string. Summarizing, the race string is used to uniquely identify a message in a communication system such that a delay in the transmission does not result in a message with a spoofed source address from being mistakenly determined as having an authentic source address.
Another issue that relates to all forms of communications is matching the source address of a message with a corresponding institution. For instance, if a caller ID is given as “XYZ Bank, 555-1000”, even if a caller ID of 555-100 is authenticated according to previously described methods, the name “XYZ Bank” would have to be authenticated through other means. The same issue applies to electronic and postal mail. Given an address of “XYZ Bank, 123 Any St, Any State, 12345”, the address “123 Any St, Any State 12345” can be authenticated according to the previously described methods, but there needs to be a mechanism to match “XYZ Bank” to that address. In a telephone system, such matching can occur via a phone book lookup. The phone number “555-1000” would be looked up in a phone book database and the corresponding name can be compared that of one displayed in the caller ID. The phone book database could be maintained by each telephone provider and allowed to be searched. Such databases are already present on the internet and available for search, such as whitepages and yellowpages maintained by Verizon, AT&T, and others. The lookup can be performed by the authentication server or the authentication device of the called party. If looked up by the authentication server, the authentication status of the name could be transmitted to the authentication device within the caller ID number authentication message. The authentication server may also transmit the name corresponding to the phone number and allow the called authentication device to perform a match between the caller ID and the name transmitted by the server. The lookup can instead be performed by the authentication device of the called party and the verification of the name matching a given telephone number can be indicated to the user. Such a lookup can also be applied to electronic mail authentication system. If an email address is given as “XYZ Bank, [email protected]”, then the lookup can be performed against a database of email addresses and corresponding entities. Or the lookup can be performed by navigating to domain address of the email address, “https://xyzbank.com”, and determining if the site domain in the SSL certificate corresponds to name given in the email address, “XYZ Bank”. Or the lookup can be performed by navigating to the domain address and querying the domain as to the name associated with a given email address.
As was the case with caller ID authentication, the eventual benefit of authenticating the source address is to handle incoming communications based on such authenticated addresses. Thus, the authenticated postal addresses of the sender may be added to local and global white and black lists. Such lists and the process of adding addresses to such lists were described with respect to caller ID and can be similarly applied to postal addresses. The handling of mail dependent on such lists may be done manually. For instance, if a mail room in a bank branch is utilizing an embodiment of the invention through a handheld scanner, then the mail room clerk can place whitelisted incoming emails in one pile and blacklisted emails in another pile. The mail room clerk can also place the source addresses from advertisers (i.e. Junk Mail) in the blacklist of the bank branch manually. The authentication device of the bank branch may add authentic source addresses to the bank whitelist autonomously if the mail has been received a number of times. Likewise, the authentication device of the bank branch may add destination addresses to the bank whitelist autonomously if mail has been sent to a destination address a certain number of times. Such concepts were elaborated as they related to caller ID. The mail room clerk may also forward spoofed mail to appropriate personnel or law enforcement. The bank branch office clerk may then choose to only receive whitelisted postal mail addressed to her. The office clerk may then place the source addresses of a received mail in the blacklist of the clerk, similar to the concept of a local blacklist in caller ID. The office clerk may perform adding an address to her local blacklist through a software interface on her computer, or through scanning the mail and indicating it be placed on her blacklist. The local lists of all the bank clerks may then be aggregated and white and black lists can be compiled for the bank branch, similar to the concept of a global lists in caller ID. The global lists for each bank branch can be compiled into an institutional global list for all bank branches. All the lists for the institutions can be compiled into multi-institutional global list for many institutions, such as banks, hospitals and so on.
If the postal mail source address authentication is done autonomously, such as through a mail sorting machine with an integrated OCR, the sorting machine may mark whitelisted mail with a green marker, blacklisted mail with a black marker, and spoofed mail with a red marker. Such marking would allow for the customer to determine how to treat the mail. The sorting machine could also sort mail for an intended customer into separate white or black listed piles. The postal worker can then deliver two piles to the customer and the customer can prioritize the opening of incoming mail to whitelisted mail first. The delivery of the postal mail can be prioritized by the postal service whereby whitelisted mail is delivered on a first come basis and blacklisted mail is delivered only on certain days of the week. Such machines could be implemented by postal companies such as UPS or FedEx, or they could part of a country postal system, such as USPS.
The concepts of caller ID authentication and postal mail authentication can be combined in a fax transmission. For instance, a fax can contain the sender and recipient postal addresses, in either typed or hand written form ready for OCR, or in a barcoded form. The fax can then be sent to the recipient fax machine through a telecommunication system. The postal mail authentication server would be updated by the fax sender as having sent a mail from the sender to the destination mail address. The fax machine can determine the authenticity of the caller ID of the fax sender through the described embodiments of the invention. The fax machine can also determine the authenticity of the sender postal mail address by reading the sender and recipient postal addresses and performing authentication of the sender postal address, as described previously.
The concepts of caller ID authentication, email authentication and postal mail authentication can be combined in an electronic fax system. Electronic fax providers maintain a fax telephone number for a customer and then sends the received faxes as emails to the customers, an example of such a system is the service provided by eFax.com, myfax.com and others. In such a system, the electronic fax provider can authenticate caller ID of incoming faxes and indicate the result on the fax. The electronic fax provider can then send the fax to the customers email. The email provider can then authenticate the source address of the email from the electronic fax provider and indicate it to the customer. The customer can then authenticate the source postal address contained in the fax.
FIG. 16A
c
FIG. 16
b) Authenticating Email Source Address and Adding it to a Whitelist. (-)
Although the process of adapting the described invention has been discussed with respect to email source address verification, an example embodiment will now be presented. The concept as it relates to email source address verification would apply similarly to other forms of electronic messaging such as text messaging, SMS and the like.
FIG. 16A through 16C
FIG. 16
FIG. 16
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The flowchart and messaging diagram , collectively referred to as , illustrates, by way of example, an embodiment of the invention that authenticates email source address. shows an email sender with an email address of “[email protected]”. Such a sender could be the customer service center of a bank. The sender has an authentication device associated with the sender email address. The authentication device can be implemented at the sender email client, email server, ISP, or through any other means. Preferably, the authentication device is a software program running at the email provider of the sender. The authentication device has access to all emails sent and received by the sender email account , which has the address “[email protected]”. The authentication device is able to extract the “From:” and “To:” fields of such emails. The recipient has an email address of “[email protected]”, such a recipient could be the customer of the bank. The recipient also has an authentication device . The authentication device can be implemented by the recipient email client, email server, ISP, or through any other means. Alternatively, the authentication device may be a browser plug-in which works in conjunction with an Internet based email provider, such as Gmail. With such a browser plug-in, the plug-in would be able to determine the user visiting the Gmail site, determine if the user is opening an incoming mail or sending a composed mail, and perform the steps of contacting the authentication server and receive source address authentication. The results of such determination would be presented to the user as a pop up, through a toolbar, or modifying the Gmail web page presented to the user. In this example, it is a plug-in running on the Microsoft Outlook email client. The authentication server is a server which receives messages from the authentication devices, replies back authentication status, and performs certification of the authentication devices. The authentication server also has integrated private and global lists. The server holds private white and blacklists for each user, and uses these lists in the construction of global white and black lists, which were discussed previously with respect to caller ID. The authentication server can indicate the lists status of the source address to the recipient, which will not be explicitly shown in this example, but the process would be similar to examples shown previously with respect to caller ID. The authentication server may authenticate more than just emails, for instance, the same authentication server may also be authenticating caller ID's, postal mail, and other forms of communications.
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The process first begins by each authentication device certifying itself. Such certification would take place when the user first uses the authentication device, such as when first installing the device or when the user is first allowed to use the email account. In Step S, the sender device initiates certification and sends a certification request to the server, as shown in Step S. The request can be sent over any communication medium and system, and not necessarily over the medium and system which is being used to send and receive emails. The certification request contains the email address the device is connected to and authenticating sender addresses for. The server receives the request and generates a unique password, Step S. The password is sent to the device by the server, Step S. The unique password is sent by the authentication server to the sender authentication device email address at “[email protected]” through any communication medium and system that is able to transmit an email to the specified email address. The sender authentication device receives the email at the “[email protected]” email account, extracts the password and stores the password and presents it to the server in further communications as proof the device resides at and is in possession of the email address “[email protected]”. The password may be a string of characters, an image file, a sound file, an attachment, or any other form of data that is available to be transmitted within an email and which can be sent back to the server by the device as the certification password. The password maybe encoded in an image, such as through a CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart), which would require the user of the authentication device to decode the password from the CAPTCHA image and enter it into the authentication device. Such password techniques maybe used to verify a human is the owner of the email account. The authentication device may intercept the certification email, extract the password, and delete the email and prevent it from going to the sender account. Such a step would make for a more seamless and less confusing process for the user. The authentication device may contact the server with the email address and the just received certification password to indicate that it received the certification password and is now installed and functioning at the email address. In Steps S through S the steps are repeated for the recipient device. In this example, the process of certification serves as the only heartbeat messages and no other heartbeat messages are sent. The certification is recorded by the server as the only heartbeat and verifies that the authentication device is installed for that email address.
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In Step S, the sender initiates an email to the recipient. In Step S, the authentication device of the sender intercepts the mail, and extracts the recipient address. In Step S, the email is sent to its destination. In Step S, the authentication device of the sender notifies the server of the sent email through an sent status message, The “SENT_STATUS” message contains the email address of the sender and the certification password of the sender.
The delay techniques described with respect to caller ID authentication may also be used similarly in email communications. The authentication device of the sender may intercept and delay the transmission of the email and send the sent status message to the authentication server first. After some delay, the email may then be transmitted to the recipient by the sender authentication device. Such a delay would allow the authentication server to be updated with the sent status message and be ready to reply to the authentication request message. Also, the authentication server may wait a certain amount of time for the sent status message to arrive before replying to the authentication request message. Similar to the technique of server push discussed with respect to caller ID may also be used, such that the server may push the sent status message to the recipient device. Thus, the recipient would not have to contact the server with an authentication request. As was discussed with respect to caller ID, such push methodology would require the authentication server to have knowledge of the authentication device communication address, such as an IP address, which the authentication device can inform to the server by including the device IP address within heartbeat messages.
When the “SENT_STATUS” message is received by the server, the certification password is compared against the certification password sent to the sender email address, the step not explicitly shown in the figure. If they match, the message is processed, if not, the server notifies the device that the password is incorrect. The message also contains a “Race String” constructed by the sender device from the email. In this example, the “Race String” is constructed by computing the hash of the email and taking the last 4 characters of the hash, which is given as an example as “e717”. A race string that is derived from the email and not explicitly contained within the email will be referred to as an implicit race string. As was explained, the race string construction can be done through other means, such as generating a random string of characters or digits, and including the string in a predetermined location in the email subject, email body, the email sender address, or any other location in the email. The recipient authentication device would then be able extract the random string from the email. If the race string is contained in the email, such as a string “Authentication race string: e717”, such a race string will be defined as an explicit race string. In this example, the race string is derived from the email message and not explicitly stated within the message, thus it is an implicit race string. The sent status message also contains the destination of the email. The server then stores the received sent status message.
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In Step S, the authentication device of the recipient , intercepts the email and determines the email source address by extracting the “From:” field of the email, Step S. The email source address may be the address of the sender or it may not. An attacker might have spoofed the email “From” field to be that of another party. Or the email source address might indeed be the sender email address. In Step S, the device sends an authentication request with the recipient email address and recipient certification password. The request also contains the “Race String” constructed from the received email. The “Race String” would be constructed identically to how the sender constructed the string, which was taking the hash of the email message and using the last 4 characters of the hash. The rules or algorithms for constructing the “Race String” would be predetermined. The predetermination could be hard coded within the authentication device software or hardware. Alternatively, the predetermination could be the authentication device being informed by the authentication server which rules or algorithms to use for constructing the string. The authentication request also contains the sender address extracted through the “From:” field of the received email. The server receives the authentication request, verifies that the certification password matches the one sent to the recipient email address. Requiring a certification password by the recipient would prevent cases where an attacker might try to determine who has sent an email to the recipient by sending many authentication requests to the server with guessed sender email addresses. Then, in Step S, the server compares the authentication request against any matching sent status messages. If there is a matching sent status message with the same sender and same destination, and same race string, the authentication server sends an authenticated message to the recipient authentication device, Step S. An authenticated message indicates that the source address of the email is indeed the sender address, and the source address is authentic. In Step S, the authentication of the sender address is indicated by the authentication device of the recipient modifying the email by prefixing a “[A]” before the sender address, such that the recipient sees the sender as “[A] [email protected]” and would know that the email has been verified as coming from that address. Other indications of the authentication status are possible, such as through an icon, pop up, changing the color of the email source address, subject text, etc. If the authentication server did not find a matching sent status message but received a heartbeat from the sender, the server would reply with a “SPOOF_VERIFIED” message. If the sender does not have an authentication device installed, as determined by the missing heartbeat message, then the server would reply with a “IRRESOLUTE” reply. These scenarios were discussed in detail with respect to caller ID authentication.
If the recipient receives an email from a sender that does not have an authentication device installed, as determined through an irresolute message by the authentication server, the recipient may manually or the recipient authentication device may automatically reply to the email sender and request that she install an authentication device. The email reply may contain a link to the appropriate site that instructs how to install an authentication device, such as a site to download a plug-in for an email client or for a browser.
Once the server has determined an email address authentication status, the server may mark the sent status message and authentication request message as old or delete the messages. Such steps would prevent race string collisions where an old sent status message might be used to verify a current authentication request message because they have the identical race string due to random chance. The received messages by the server may be time stamped and be deleted after a certain amount of time residing on the server.
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After the user receives and reviews the email, the user initiates adding the sender to her private whitelist, Step S. The authentication device of the user might provide for adding the sender address to the whitelist through a button. The button could be part of a toolbar, if the device is implemented as an email client plugin. The button could be a part of the user interface implemented by the email provider. The user presses the button, Step S, the authentication device detects the user action, Step S, and sends ADD_TO_PRIVATE_WHITELIST message to the authentication server, Step . The message contains the sender address, recipient address and recipient password. The authentication server verifies the recipient address and corresponding password, Step S. Once the password is verified, the server adds the sender address to the recipients' private whitelist. The server acknowledges the action, Step S, and the device indicates the acknowledgement that the sender address has been added to the user's whitelist to the user, Step S. The authentication server would thus indicate the particular email sender is on the recipients whitelist on all authenticated messages sent to the recipient authentication device, which is not shown in this example. The authentication device may then handle incoming mail based on list status, similarly to how incoming calls where handled based on caller ID list status. An email from a whitelisted sender may be shown with a green font or put into a special email folder. An email from a blacklisted sender may be placed in the trash folder or presented with black font.
In addition to the recipient manually placing a sender address to the private whitelist, the authentication server may automatically add a sender address to the private whitelist. The authentication server can keep a tally of the number of times a sender has sent an email to a recipient, and after a predetermined number of sent emails, the server can add the sender address to the recipients private whitelist. And list status of the automatically whitelisted sender can be sent within the authenticated messages.
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Similar to the embodiments described with respect to caller ID authentication, the authentication server may push the sent status message to the recipient authentication device. The authentication server may thus send a message to the recipient authentication device in the form of “SENT_STATUS_SERVER-[[email protected], Race String: e717, Destination: [email protected]]”. The recipient authentication device may then store the sent status message and authenticate the email from the sender once it is received, without the need to send an authentication request to the server. Similar to embodiments described with respect to caller ID authentication, the authentication server may receive an authentication request message and not a corresponding sent status message. Such situations might arise due to network transmission errors, packet loss, or combination of software or hardware issues which prevent the timely delivery of the sent status message. The authentication server may first determine if an authentication device is installed at the sender email address as indicated within the authentication request. The server may use received heartbeat messages or the certification process to determine that an authentication devices is indeed installed at the sender address. The server may then contact the sender authentication device and request a sent status message. The authentication server may thus send a message to the sender authentication device in the form of “AUTHENTICATION_REQUEST_SERVER-[[email protected], Race String: e, Destination: [email protected]]”. The sender authentication device may then verify that an email was sent to the destination address with the specific race string.
FIG. 8
FIG. 16
There may be instances where legitimate spoofing of email sender addresses may be desirable, such as a mailing service placing the clients return address on an email to a customer of the client. Legitimate spoofing was covered with respect to caller ID in Section (8) and illustrated in . The process of allowing a spoofing party to legitimately spoof the email address of another party would be performed similarly for email systems. Summarizing such a process, the authentication device of the spoofed party that wishes to allow for another party to spoof it's email address transmits a “SPOOF_PERMISSION” message with its own email address and certification password, and the spoofing party address which it is allowing to spoof its address. The spoofing party is then granted permission by the server to spoof its email address to that of the spoofed party giving permission. Such a permission is given by the server transmitting a “SPOOF_GRANTED” message to the spoofing party. The spoofing party may then spoof its email address to be that of the spoofed party. When sending a spoofed email, the spoofing party sends a “SENT_STATUS” message as before, but with the spoofed address as the “Sender” address, and with its own certification password. The authentication device of the sender may automatically realize the mail is being spoofed if the sender address and the stated “From” address do not match. Or the authentication device may be manually set up by the user to indicate that certain sent emails are being spoofed. The server then looks up the email address of the password holder and determines that the owner has permission to spoof the email address given as the “Sender” address. Alternatively, the “SENT STATUS” message can contain both the sender true address and the email address that it is spoofing. Such a message would make the process simpler for the server to verify the message by comparing the sender address and certification password, and then determine if the sender was given permission to spoof its address to the one stated in the message. Using the example in , such a “SENT_STATUS” message with legitimate spoofing might be in the form of “SENT_STATUS-[Sendennoreply@mailservercom, Password: 4321, Spoofed [email protected], Race String: e717, Destination: [email protected]]”. The server would verify the password “4321” is the one which was sent to “[email protected]”. The server would then determine if “[email protected]” gave permission to “[email protected]” to spoof its email address. If so, the server would store the message. If not, the server may instruct the spoofing party to request permission or check with the spoofed party if it wishes to grant permission. The recipient authentication device would receive the email with the spoofed address and send an authentication request, such as “AUTHENTICATION_REQUEST-[[email protected], Race String: e717, Destination: [email protected], Password: 5678]”. The authentication server would then send an “AUTHENTICATED” message back to the recipient, since the spoofed party had given permission for the spoofing party. Thus, it should be clear to those skilled in the art that all aspects of the invention as they have been described with respect telephone communications can be used in any communications system for authenticating sources addresses of such communications. As such, aspects of the invention such as compiling whitelists, blacklists, list based communications handling, active and passive certification methods, legitimate spoofing, blocked source addresses, revealing blocked source addresses can all be applied to and utilized in all forms of communications, such as electronic mail.
The recipient in the given previous embodiment had an authentication device installed and was actively certified. However, the recipient need not have an authentication device installed and need not have active certification to authenticate the email message. The email message can include an URL inserted within the email by the sender authentication device for the purpose of authenticating the email source address or discovering if the source address is spoofed. For instance, the email sent by the sender in the previous example could have an URL inserted within the body of the email, such as “Click this link to authenticate the source address of this email: “https://authenticationservercom/verify?racestring=e717&[email protected]”. The domain name “authenticationserver.com” is the name where the authentication server resides. The “verify” is where the email race strings are sent within the server. The “racestring” and the following “e717” indicate to the server that the recipient wishes to authenticate an email with a race string “e717”. The “destination=” field and the following address “[email protected]” indicate to the server the destination of the email. Such a URL would be provided where the race string is considered to be sufficiently unique as to uniquely identify this email against all other emails sent and being verified by the authentication server. In such a case, the sender address is redundant and may be omitted, since the server may determine the sender address by looking up the sent status message with the identical race string and destination. Preferably, the race string in such a case would be constructed hashing the full email text, including the email sender address, email recipient address, and email body. The race string can also be with a unique serial number or a unique serial number can be appended to the race string. Alternatively, the URL may include the sender and recipient addresses, such as “https://authenticationservercom/verify?racestring=e717&[email protected]&desti [email protected]”. When the recipient, who does not possess an authentication device, receives the email, the recipient may then click on the URL. The click action would open a browser and navigate to the URL on the recipient's browser. The authentication server would then process the URL. Navigating to the URL would be similar in functionality as an authentication request message sent by a recipient authentication device. However, such a URL would not contain the certification password of the recipient and the recipient would not prove ownership of her email address to the server. The fact that the recipient provided the server with the URL proves that the recipient has received the email in question, and most likely is in ownership of the destination email address. The server would determine if it has received an accompanying sent status message with the same race string and destination address. If the server has received the appropriate sent status message, the server may return a web page that displays the source address of the email and that the source address is authentic. If the server has not received the appropriate sent status message from the sender, the server may contact the sender authentication device and request the missing sent status message, assuming the sender has an authentication device installed indicated by an appropriate heartbeat message or successful certification process. If the server determines that the sender has an authentication device and has not sent the email by not having received a sent status message and not received a satisfactory response when the server contacted the sender authentication device, the server may then return a web page to the recipient that the source address has not sent such an email and the source address is likely fraudulent. The server may then ask if the recipient would like to upload the email for the purpose of reporting it to the appropriate authorities. If the server determines that the sender does not have an authentication device installed, the source address is not determined to be irresolute. Since the inclusion of the URL into the email is performed by the authentication device, the fact that the sender does not have an authentication device installed makes the source address fraudulent, which would then be indicated by the web page returned to the recipient. An issue with including an URL for retrieving authentication information is that an attacker can include a URL with a misspelled domain name similar to the legitimate domain and serve the user with a fake web page stating source email address authentication. Or even more troublesome, the URL could point a webpage which serves exploit software for hijacking the visitor's browser or operating system through vulnerabilities in their systems. Thus, an authentication device installed at the recipient, such as an email client plug-in or a browser plug-in is more desirable rather than URL type authentication systems.
FIG. 17A
FIG. 17E
c) Authenticating Postal Source Address (-)
Although the process of adapting the described invention has been discussed with respect to postal source address verification, an example embodiment will now be presented. The concept as it relates to postal source address verification would apply similarly to other forms of messaging, such as fax transmission, as was disclosed earlier.
FIG. 17A-17E
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The messaging diagrams, illustrations, and flowchart as shown in illustrate, by way of example, an embodiment of the invention that authenticates postal source address. FIG. A shows a postal sender with a postal address of “100 Any St., Any Town, AA 11111”. The postal address convention used is that of U.S. postal address, given with imaginary street, town, state and zip code. Other conventions may be used as they apply to other countries and other addressing conventions. For instance, the zip code may be given as a “5+4” zip code, such as “11111-1234”. The recipient authentication device may disregard the last 4 digits of the zip code when sending an authentication request message and server would still be able to authenticate the source postal address. Abbreviations may be written in full, such as “Street” rather than “St.”, and the authentication server can determine they are both identical as they relate to addresses and authenticate the source address. The authentication server can determine that two addresses are identical if they are sufficiently equivalent through the use of such means. The sender postal address has an authentication device associated with it and utilized by the sender. As was elaborated, the authentication device can be implemented at the sender printer used to print postal labels, postal stamps, or can be implemented as software running on a smartphone. The authentication device can be implemented through multiple means, but in this example it is a part of an electronic postage and metering system, such as provided through “stamps.com”. Ideally, the authentication device determines the destination address of mailings through automated means, by virtue of being integrated within an electronic postage system. However, the destination address can be manually inputted into the software by the sender. The authentication device would also have knowledge of the sender address, either entered manually, or obtained through automated means, such as determining the “from” address of the mailing, determining the GPS location of the authentication device and mapping the location to a street address, or through any other known means in the art. However, there is no requirement that the sender authentication device physically reside at the sender postal address. The sender may be utilizing the authentication device for postal mailings for a post office box rented by the sender that is located at the local post office. Thus, the authentication device is merely associated with a certain postal address and the device proves and asserts this association by presenting its certification password to the server. The recipient has a postal address of “200 Any St., Any Town, BB 22222”. The recipient authentication device can be implemented through multiple means as previously elaborated, but in this example, it is implemented as software running on a smartphone with a camera. The authentication server is a server which receives messages from authentication devices, and replies back authentication status, and perform certification of authentication devices. The messages transmitted from authentication devices to the server can be through any communication means, but it is preferably over the Internet through a broadband or cellular medium. The server may also have integrated private and global lists, as have been discussed with respect to call and email handling and would function similarly for postal mail. The authentication server may authenticate more than just postal mail, for instance, the same authentication server may also be authenticating caller ID's, electronic mail, and other forms of communications.
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FIG. 17B
FIG. 17B
FIG. 17B
The process first begins by each authentication device certifying itself. Such certification would commonly take place when the user first uses the authentication device, such as when first installing the device. As will be discussed later, the recipient need not necessarily be certified. In Step S, the sender device initiates certification and sends a certification request to the server, as shown in Step S. The request can be sent over any communication medium and system, such as internet, cellular, and also over the medium and system which is being used to send and receive postal mail, such as a mailed postcard. The certification request contains the postal address the device is associated with and authenticating source addresses for. The server receives the request and generates a unique password, Step S. The password is sent to the device by the server, Step S. The unique password is sent by the authentication server to the sender authentication device postal address through the postal system. However, the password can be sent through any communication medium and system that is able to transmit a message to the specified postal address. The sender authentication device receives the postal mail with the password at the postal address. An example of such a postal mail envelope with the certification password is given in . The certification password postal mailing in contains the “From” or source address of the mail, which is the authentication servers address . It also contains the “To” or destination address of the mail , which is the address the authentication device is associated with and is being certified. The postal mailing also contains the certification password . The certification password transmitted to the sender in this example is “1234”. The password can consist of any length of letters, numbers, or special characters. The password maybe encoded in an image, such as through a CAPTCHA, which would require the user of the authentication device to determine the password from the CAPTCHA image and enter it into the authentication device. Such password techniques maybe used to verify a human is the owner of the postal address. In this example, the certification password is also given as a QR code and instructions to scan the QR code to initialize and certify the authentication device of the user. The QR code contains the string “Certification Password:1234” and is given as an example, and can be replaced by any other machine readable coding scheme, such as barcodes, and so on. The password can also consist of other forms of data on other mediums that can be sent to a postal address, such as data on RFID tags, data or encryption certificates or keys on CD, DVD, magnetic disk, and so on. The password itself may be used by the authentication device as credentials to retrieve a more complex password, such as a digital certificate from a website, the website being maintained by the authentication server. Retrieval of passwords was elaborated with respect to caller ID certification passwords in Section (6-D-ii) and applies equally to the application of postal mail. Once the certification password mailing is received, the user can read the certification password and manually input the password into the authentication device. Alternatively, if the authentication device is able to read barcodes or QR codes, the mailing can be scanned and the device can read the code and automatically determine and store the certification password. Another alternative is the authentication device visually scanning the mailing and performing OCR to read and extract the certification password . The postage stamp that would be affixed to the envelope is not shown in . Once the password is stored, the device presents it to the server in further communications as proof the device is associated with and is in ownership of the postal address “100 Any St., Any Town, AA 11111”. If for instance, authentication device resides within an automated mail sorting system, the device may intercept the postal mail, extract the password, and destroy the mail and prevent it from going to the sender address. The authentication device may contact the server, preferably through an internet connection, and the just received certification password to verify the password and indicate that it received the certification password and is now installed and functioning for the postal address. The steps S through S are repeated for the recipient device. In this example, the process of certification serves as the only heartbeat messages and no other heartbeat messages are sent. The certification is recorded by the server as the only heartbeat and verifies that the authentication device is installed for that postal address.
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FIG. 17C
In Step S, the sender initiates sending a postal mail to the recipient. The initiation can consist of the sender entering in the destination address and the postal article information, such as size and weight, into an electronic postage system. In Step S, the authentication device of the sender determines the destination mail address. In Step S, the mail item is sent to its destination, such as being placed in a mail box. In Step S, the authentication device of the sender notifies the server of the sent mail through a sent status message. The “SENT_STATUS” message contains the mail address of the sender, the certification password of the sender, and the mail destination address. Instead of the authentication device transmitting the certification password to the authentication server, the transmission can consist of the hash of the password. Or the transmission can consist of a challenge from the server to the device, such as the server instructing the device to append specific characters to the certification password and transmit the hash of the resultant string. The server would then append the specific characters to stored password for the device, compute the hash and compare it to the received string from the device. Through utilizing such techniques known in the art, the authentication device may prove knowledge of the certification password to the server by transmitting the actual certification password to the server or by transmitting information derived from the password. It should also be evident that a sufficiently complex password would make the transmission of the postal address of the authentication device redundant. For instance, the sent status message can take the form of “SENT_STATUS-[Password: 5i38x75s2y6xy3s2t, Race String: 1/1/2014-10:10 AM, Destination: 200 Any St., Any Town, BB 22222]”. The authentication server can simply look up which authentication device the password corresponds to and determine postal address of the authentication device. The sent status message also contains the race string. In this example, the race string represents the date and time the mailing was initiated by the user. The race string can consist of random characters, mail weight, mail size, or any other information which uniquely identifies this item against other items sent by the sender to the recipient. As was explained, the race string ensures the correct item is authenticated regardless of spoofed items arriving before the legitimate item. In this example, the race string is printed onto the mail item. The printing can be done by the electronic postage system onto the envelope. Or the race string can be printed onto the postage label. The race string can be printed within the contents of the envelope. The mailing envelope that is sent by the sender to the recipient in this example is shown in . The mail has a source address and a destination address . The explicit race string is printed on the envelope. The source, destination and race string are also encoded on the envelope as a QR code . The data encoded in QR code in this example is “From:Any Person, 100 Any St., Any Town, AA 11111 To: Any Person, 200 Any St. Any Town, BB 22222 Authentication Race String: 1/1/2014-10:10 AM”. The QR code allows for the recipient authentication device to read the mail information autonomously. The “To” field in the QR code might be redundant in some embodiments, for instance where the recipient authentication device is authenticating postal mail for a single address and the device already knows the address it is associated with. However, if the authentication device is authenticating multiple recipient postal addresses, for example if the recipient has multiple post office mailboxes, then the “To” field would be desirable. Below the QR code is text which instructs the recipient to scan the code to authenticate the mail. The text also includes a website URL. If the recipient did not encounter such a QR code previously, the website would instruct the user on how to download the appropriate software to authenticate the mail item. The software may be a smartphone application, or a laptop application which utilized the laptop camera to scan the mail QR code. Alternatively, the website may prompt the user to enter in the source address, destination address, and race string manually into a web page to authenticate the source mail address.
When the “SENT_STATUS” message is received by the server, the certification password is compared against the certification password sent to the sender mail address, the step not explicitly shown in the figure. If they match, the message is processed, if not, the server notifies the device that the password is incorrect. The server then stores the received sent status message.
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FIG. 17E
FIG. 6D
FIG. 16D
After the recipient receives the mail, she scans the mail QR code on her smartphone, Step S. In Step S the authentication device software on the smartphone extracts the information contained within the QR code , including the mail source address and race string. The mail source address may be the address of the sender or it may not. An attacker might have spoofed the mail “From” field to be that of another party and generated a fake QR code. Or the mail source address might indeed be the sender email address. If the recipient did not have an authentication device with a camera, steps S and S would be replaced by the user reading the envelope manually and entering the appropriate pieces of information into the device manually. In Step S, the recipient authentication device sends an authentication request with the recipient mail address and recipient certification password. The request also contains the “Race String” extracted from the mail. The authentication request also contains the source address extracted through the “From:” field of the received email. If there wasn't a race string indicated, the recipient authentication device would derive the race string, such as from the mail weight, dimensions and so on. The mail weight can be determined through a scale connected to the recipient authentication device or can be entered manually by the user. The mail dimensions could be determined by recipient authentication device through a camera or can be entered manually by the user. Which aspects of the mail item constitute the implicit race string can be predetermined or can be indicated by the authentication server to the authentication devices. The authentication device may also send an authentication request without a race string, and the server would then look up if a matching sent status message exists, and if so, instruct the authentication device to provide the implicit race string, and which aspects of the mail item constitute the implicit race string. In such a case the sender authentication device may indicate to the server how the implicit race string was constructed, and the server can relay the construction instructions to the recipient device. In Step S, the server receives the authentication request, verifies that the certification password matches the one sent to the recipient mail address. Requiring a certification password by the recipient would prevent cases where an attacker might try to determine who has sent a mail to the recipient by sending many authentication requests to the server with guessed sender mail addresses. The server then performs the authentication and spoof detection as shown in the flowchart . The flowchart is similar to the procedure used to determine caller ID authentication and spoof detection as was given in , with the only exception being the addition of decision step , which will be explained shortly. The flowchart shows the server receiving the authentication request from the recipient authentication device (step ). As was explained, the server checks for the correct recipient certification password. Once the correct password is verified, the server checks if there is a matching sent status message with the same sender and same destination, and same race string (decision step ). In this example, such a message was received and the server replies with an authentic source address determination (step ) and sends an authenticated message to the recipient authentication device, as shown Step S in . An authenticated message indicates that the source address of the mail is indeed the sender address, and the source address is authentic. The authenticated message in this example contains the information received within the authentication request, however, much of the information is redundant, such as including the recipient address, and the message may not contain all the information shown in the figure. The authenticated message may also indicate if the same QR code, or the same piece of mail as identified by the source address and the race string was previously authenticated by the server and if so, the date and time of the previous authentication. Such safeguards would used to prevent an attacker obtaining a mailing from the recipient's garbage, copying the QR code or authenticating information, and sending a piece of mail with the same QR code to the recipient. The recipient can thus see that the same mailing had been authenticated at a previous date and time, and act accordingly. The QR code could be protected and covered by a tamper evident tape or tamper evident scratch-able paint, which would need to be removed before being able to be scanned. Such safeguards would alert the recipient that the QR code had been tampered or viewed before arriving at the recipient. Once the recipient authentication device receives the authenticated message, it notifies the user of the authentic source address status, step S. The notification can consist of the software stating that the source address is authentic. Or for instance, the image of the envelope taken by the recipient authentication software that is presented to the user can overlay a green check icon next to the sender address or QR code. Once the server has determined a mail address authentication status, the server may mark the sent status message and authentication request message as old or delete the messages. The received messages by the server may be time stamped and be deleted after a certain amount of time residing on the server. Such steps would prevent race string collisions where an old sent status message might be used to verify a current authentication request message because they have the identical race string due to random chance.
FIG. 17E
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Continuing from , a matching sent status message could have been received but could have been timed out if too much time had elapsed between the sent status message and the authentication request message, indicating the mail referred to by the sent status message had been lost before the authentication request message was received and the authentication request message refers to a spoofed mail item. In such a case, there would be no matching sent status message. Or a matching sent status message could have been expired due to the sender authentication device sending a message to the server indicating the mail was undeliverable, indicating the mail was returned to the sender by the post office. In such a case, there would be no matching sent status message. If the authentication server had not received a matching sent status message, it would check if the sender had an authentication device installed (decision step ). The server can determine if the sender had an authentication device installed if it had performed the certification process or had sent the appropriate heartbeat messages. If the server determines that the sender did not have an authentication device installed, the process moves to decision step . The server then checks if the mail item contains an explicit race string or if the race string is implied. In the example given, the race string was explicit, as was shown in element , and was chosen by the sender to be the time the mail item was generated. As have been disclosed earlier, the race string can be implicit and can be derived from the mail item, such as its weight. If the race string is explicit, and the sender does not have an authentication device installed, it implies an attacker has included the race string on the mail item in the hope that the recipient might not actually verify the source address by contacting the server and assume the source address is authentic. Thus, the source address is indicated by the server as spoofed (step ). The server can determine if a race string is explicit from the race string type indicated by the sender and recipient, for instance if it is a date/time combination, random characters, and so on. The server can also determine if the race string is implicit if the race string type is the mail weight, color, dimensions, and so on. Alternatively, the sent status message can include an indication stating if the race string is explicit or implicit. If the mail item did not contain an explicit race string, then the server determines the source address to be irresolute (step ). The reasoning for the irresolute determination was elaborated with respect to caller ID. The server cannot determine a source address as being authentic or spoofed if the sender does not have an authentication device installed.
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If the sender did have an authentication device installed, the server checks if the authentication device of the sender is able to be contacted, decision step . As was explained, authentication devices can register their communication addresses, such as an IP address, for the purpose of receiving queries from the authentication server. If the server determines the authentication device did not register a contact address with the server, the server determines the source address to be spoofed, step . If the device is able to be contacted, the server queries the device to determine if the mail item as indicated within the authentication request was sent by the sender, decision step . Such a query can be in the form of a message sent to the authentication device of the sender from the server in the form of “SERVER_AUTHENTICATION_REQUEST-[Sender: 100 Any St., Any Town, AA 11111, Race String: 1/1/2014-10:10 AM, Destination: 200 Any St., Any Town, BB 22222, Password: 5678]”. The sender device can then examine the query, determine if it has a record of it indeed mailing an item to the stated destination with the stated race string. The sender authentication device may also query the user if such a mailing was made and not registered with the authentication device. If so, the device can reply to the server with an affirmative message, and the server can determine the source mail address to be authentic, step . If the authentication device of the sender replies that it did not send the mail item in question, the server can determine the source mail address to be spoofed, step . As was explained with respect to the flowchart in , the spoofed determinations , , and , can be treated as being equal or can be differentiated. Spoof determinations and can be indicated by the server in reply to the recipient authentication device to be a guaranteed spoof determination, and spoof determination can be indicated to be a non-guaranteed spoof determination. The rationale for the differentiation is the same as was given with respect to . Spoof determination is non-guaranteed since the server is assuming that the mail source address is spoofed without actually interrogating the sender authentication device if it indeed mailed the item and the sent status message was lost or never sent due to a failure of some kid. Spoof determination is a guaranteed spoof determination by the server knowing the sender given in the source address does not have an authentication device installed and thus the mailing should never have an explicit race string indicated, and any race string implies an attacker crafted it. Spoof determination is guaranteed spoof determination since the server contacted the sender authentication device as identified by the source address and the sender device indicated it did not send the mail in question.
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FIG. 17D
The QR code given in could have instead consisted of a URL which pointed to the authentication server with the proper information to authenticate the source address. If the QR code was a URL, then there is no requirement for the recipient to have an authentication device installed and certified, as was similarly explained with respect to email source address authentication. For instance, the QR code in could have consisted of the URL “https://authenticationserver.com/authenticate?sender=”100 Any St., Any Town, AA 11111″&recipient=“200 Any St., Any Town, BB 22222”&racestring=“1/1/2014-10:10 AM””. The scanning of such QR code would launch a browser on the scanning device and navigate to the authenticationserver.com domain and pass the sender, recipient and race string information to the server. The server would then determine if a matching sent status message exists, and indicate the authentication status of the source address. Thus, the information that is sent to the server is the same as an authentication request in , except for the certification password. In this instance, it is assumed that since the recipient is able to provide the sender, recipient and race string information, that the recipient has received the mail in question. And that an attacker cannot easily guess such information for the purpose of obtaining information about the mail the sender or recipient has sent or received. In the case of such URL implementations, it would be preferable for security purposes to generate a more secure race string, such as random characters of sufficient length. If the recipient did have an authentication device installed, then the device could have scanned the code, and contacted the authentication server with the fields identified in the URL as sender, recipient and race string. The contacted domain by the authentication device could be different from the one given in the URL “authenticationserver.com”.
As was elaborated with respect to caller ID authentication in Section (6-D-iii), the concept of constructing a passive certification password applies equally to postal mail. The reader is encouraged to refer to the section since all the concepts equally apply to postal mail. The central concept of a passive certification password was that the authentication device could construct a password based on the authentic caller ID's received. This password would then prove to the authentication server that the authentication device was in fact in ownership of the telephone number it claimed to be associated and linked with. Similarly, the authentication device of the postal recipient can construct a password which is based on and incorporates authentic source addresses of received mail items. The construction would be performed according to a predefined algorithm either determined manually by the user or manufacturer of the device, or the algorithm can be negotiated between the authentication server and authentication device. For instance, the passive certification password can be constructed from the received last 3 authentic source address zip codes. The recipient authentication device can append each source address zip code and construct the passive certification password. If the recipient received source addresses with zip codes “11111”, “22222”, and “33333”, then the passive certification password for the device would be “111112222233333”. The recipient device could then use this password in conjunction with an active certification password or the device may never be actively certified and use the passive certification password instead. Since the authentication server would have received the appropriate sent status messages of mail items sent to the recipient, the server would have the necessary information to determine the passive certification password and construct the passive certification password for the device using the same predetermined algorithm. The passive certification password can be constructed from and incorporate any aspect of the source address or the implicit or explicit race strings. As was elaborated with respect to caller ID authentication in Section (6-D-iv), the concept of automatic password revocation applies similarly to postal mail. That is, the sender authentication device can incorporate the destination address of the mail item within the password to construct a new password. For instance, if the certification password that was sent by the authentication server to the sender postal address was “12345” and the sender initiated a mail to a destination address with the zip code “11111”, then a new password can be constructed by adding the old password and the zip code of the mail destination address. Thus the new password would be “23456”. The authentication server would also perform the addition of the old password with the zip code received within the sent status message and be able to verify the new password. As was explained with respect to caller ID passive authentication, the construction of the new password would be performed according to a predetermined algorithm, such as adding the zip code of the mail item destination address, or appending the last 3 zip codes of mail item destination addresses, or adding the street numbers of the last 2 mail item destination addresses. The construction of the new password can incorporate any aspect of the destination address.
FIG. 17E
Additional concepts that were disclosed with respect to caller ID authentication and email source address authentication would apply similarly to postal mail source address authentication, where applicable. Such concepts are autonomous white list construction, manual blacklist construction, global blacklist construction, legitimate spoofing, authenticating legitimately spoofed source addresses, and so on. Any new concepts that were introduced with respect to postal mail source address authentication could thus be applied to caller ID and email authentication, where applicable. For instance, with respect to , such a flowchart was not explicitly shown for email authentication but would apply similarly since the concepts of explicit race strings, implicit race strings, contactable authentication devices and other concepts apply equally.
15) Call Destination Authentication and Call Forwarding Detection
a) Call Forwarding Background and Forwarding Based Attacks
Call forwarding or call diversion is a process where a telephone user may forward incoming calls to another number. For instance if a user is on vacation, the user may forward all incoming calls to his house instead to the phone number of his vacation rental. Such forwarding is usually performed through dialing a digit sequence, such as ‘*72’ followed by the phone number to forward all calls to. All calls that are forwarded are charged to the forwarding number (user's house in the above example) and not to the forwarded number (vacation rental in the above example). Call forwarding can be disabled by also dialing a digit sequence, such as ‘*73’.
As convenient as call forwarding may be, it introduces several security issues. Such security issues start where an attacker tricks the victim into dialing the digit sequences and the forwarded phone number into her phone to enable forwarding, the victim being unaware she is imitating forwarding on her line. The attacker can then take advantage of this fact by the forwarded number belonging to his friend. The attacker can thus place collect calls to the victim's phone number. The victim's number forwards to the attacker's friend. The friend accepts the collect call, and the call is charged to the victims account. This attack is known in the art as the “Prison Scam”. Another way an attacker can take advantage of this type of attack is to have the forwarded number be a phone the attacker controls. The attacker can thus answer as the victim, for instance if the victims bank is calling her to verify a suspicious charge on her credit card. The attacker, most likely being the perpetrator of the suspicious charge to her credit card, can verify the fraudulent charge and the bank will not be aware that the call has been forwarded and process the credit card charge. There are other variants of such attacks and scams which rely on an attacker forwarding the victims calls to another number.
Such call forwarding scams are hard to detect by both the caller and victim parties. The caller party cannot detect forwarding unless he knows the called party personally and recognizes her voice, or the caller party verifies the called party through asking for personally identifiable information, assuming such information has not been compromised by the attacker who has forwarded the call. The victim of the call forwarding attack usually discovers her number has been forwarded by callers personally informing that they can't reach her, after some time has passed and after the damage has been done. The victim might be informed her number is currently forwarded by the telephone company through a special dial tone, but she would have to have knowledge that such a dial done indicates her number is forwarded. The telephone company may also indicate an incoming call is being forwarded to another number by the ringing the victim's phone once and then forwarding the call. Again, the victim would have to have knowledge that such a single ring indicates forwarding, otherwise she might mistake such a single ring to a caller hanging up.
b) Call Forwarding Detection
The methodology of caller ID authentication will now be adapted to detect call forwarding and notify the calling party of this fact. The previous sections discussed identifying the calling party to the called party; this section and following embodiments will discuss identifying the called party to the calling party.
Call destination authentication and call forwarding detection will be performed analogously to the concept of delivery confirmation or return receipt as was described with respect to postal mail authentication. That is, the fact that the called party device contacts the server to authenticate the caller's caller ID will be used as verification that the intended called party has received the call. And the information of the called party device not contacting the server will be used as verification that the call has been forwarded and the intended called party has not received the call. There are several variations of forwarding detection where the called party and the attacker has or has not installed an authentication device. The variations where the attacker installs an authentication device on the forwarded phone will be discussed in the next section.
FIG. 18A
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illustrates one embodiment of the invention that detects call forwarding and notifies the caller party of this fact. In this embodiment it is assumed that the called party has not had her number forwarded by her or an attacker unknown to her. The Steps S to S, with the exception of Steps S and S, are performed in the same manner as have been previously described with respect to caller ID authentication and will thus be summarized without going into detail. The caller and called parties have authentication devices, and respectively, installed and are certified. In this example, the certification process serves as the only heartbeat. In this example certification passwords sent by the server are “123456” for the caller phone number and “987654” for the called phone number. The passwords may be sent through DTMF, caller ID, etc. The password may also be sent through a combination of means, such as DTMF and caller ID, where the password is constructed from both the received DTMF password and the caller ID password. As described previously, the authentication device may then send the received password to prove ownership of the phone number it is authenticating. Each device may transmit the actual password, derivatives of the password, such as its hash, or any other method which proves knowledge of the password, such as the device receiving a challenge from the server which is then responded to by computing an outcome based on the password and the received challenge. The caller party calls the called party and the authentication devices of each party contacts the server with the appropriate messages for the purpose of caller ID authentication. In Step S, in addition to the authentication server determining the caller ID of the caller is authentic, it determines that the appropriate called party has been reached. The server uses the fact that a calling status message and a matching authentication request has been received to determine the call destination is authentic. In Step S, the authentication server transmits to the caller authentication device a “DESTINATION-AUTHENTICATED” message. The destination authenticated message may contain additional information, such as the caller and called phones as transmitted within the calling status message. Such a message indicates to the caller party that the call has not been forwarded and has reached the intended called party. The caller authentication device may then indicate to the caller party the called number and destination has been authenticated. Such an indication might consist of a special tone that is generated by the caller authentication device and played during the phone call. Or the indication might consist of a visual message or icon, for instance if the authentication device is installed on a smartphone. The user may adjust such indications, or choose only to be notified if call forwarding is detected.
FIG. 18B
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illustrates an embodiment of the invention where the called party has her phone number illegitimately forwarded by an attacker to the attackers phone number or a phone number the attacker controls. In this embodiment, it is assumed that the attacker does not have an authentication device installed. The attacker may choose not to install and authentication device for reasons previously discussed. The attacker might not have the means to install an authentication device, such as lacking the proper hardware or software. Another reason was outlined in Section 6-e, namely the authentication device of the attacker notifies the calls he is making and receiving to the authentication server, which would be undesirable to an attacker. Steps S and S are processed as in the previous example. In Step S, instead of the call being connected to the called party, the call is forwarded to the attacker phone number. Since the attacker does not have an authentication device installed, the authentication server never receives an authentication request message from the called authentication device. In Step S the authentication server determines that a matching authentication request has not been received from the called party authentication device, the call has been forwarded to another number and that the called party has not received the call. In Step S, the authentication server transmits a “DESTINATION FORWARDED” message to the caller party authentication device. Such a determination indicates to the caller party that the call has not reached its intended destination and has been forwarded to another number. The caller party device may then notify the user of the forwarded determination.
FIG. 18C
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is a flowchart which shows the steps the authentication server uses in authenticating the called party and detecting call forwarding. When the authentication server receives a calling status message from the caller authentication device (step ), it first checks if an authentication device is installed at the called phone number (decision step ). The server determines an authentication device is installed at the called phone number if it has received the expected heartbeat messages or, if heartbeat messages were not to be sent by the device and the device has performed the certification process. If neither of these events have taken place or if the called phone has uninstalled the authentication device as determined by an uninstall message, the authentication server replies with a call destination irresolute determination (step ). This is similar to an irresolute caller ID determination discussed earlier, that is the server cannot determine if the called destination has been forwarded or not. If the called phone has an authentication device installed, the server checks if a matching authentication request message has been received (decision step ). If a matching authentication request status message exists, it replies to the caller device with a called destination authenticated determination (step ). A matching authentication request message would be where such a message is received within a certain time frame after the receipt of the calling status message, such as 0-10 seconds. Or if the caller authentication device is expected to send additional messages such as call ringing, call connected, and call ended, a matching authentication request message would be where such a message is received from the receipt of a calling status message until the caller authentication device sends a call connected message, or a certain time period after a call connected message, or until a call ended message is received. The requisite time frame the server waits for the authentication request message can be preset or can be based on measured times of previous reception of such messages. If there is no matching authentication request message, the server checks if the called authentication device has registered a communication contact address and is contactable (decision step ). The called authentication device can register it's contact address with the server, such a network IP address, through heartbeat messages and allow for the server to contact it and query it for authentication request messages. If the device either; has not registered its contact address, or has registered its contact address and is not responding to queries from the server, the server replies to the caller device with a call destination forwarded determination (step ). If the called authentication device is able to be contacted, the server queries it to determine if it is currently receiving a call from the caller party or which party it is currently receiving a call from (decision step ). If the called device replies that it has not received a call from the caller party, the server replies to the caller device with a called destination forwarded determination (step ). If the called device replies that it is indeed receiving a call from the caller party then the server replies with a call destination authentic determination (step ). Similar to spoof detection as described with respect to , the call forwarded determinations and can be differentiated by the server and indicated to the caller party. For instance, forwarded determination can be indicated as a guaranteed forwarding indication, since the server contacts the called authentication device and verifies the call from the caller has not been received. And forwarding determination can be indicated as a non-guaranteed forwarding indication since the server blindly determines the call has been forwarded without verifying the fact with the called authentication device. Preferably but not necessarily, the call forwarding detection is used in combination with caller ID authentication and spoof detection embodiments previously described.
c) Detecting Forwarding where the Attacker Installs an Authentication Device
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If the attacker who is forwarding the calls of the victim has an authentication device installed, the detection and mitigation of such instances become more involved. In the first instance of such a case, we will assume the victim does not have an authentication device installed. The attacker tricks the victim to forward all her calls to the attacker's phone. The attacker then installs an authentication device and proceeds with certification. The authentication server receives the certification request with the victim's phone number. The server then contacts the victims phone number, which is forwarded to the attackers phone, and the authentication server then transmits the certification password, either through DTMF tones, caller ID, modem transmission, computerized voice, etc. The attacker now has a certified authentication device installed at the forwarded phone number. Thus if a called party with an authentication device receives a call from the attacker, such calls will be determined by the authentication server as having an authentic caller ID. And any calls a caller places to the attacker will be determined by the authentication server as an authentic call destination. As a result, the concept of authentication device reputation score will be introduced to mitigate this type of attack. It is anticipated that an attacker who has forwarded a victims calls only has a certain time frame to carry out his attacks before being discovered. And that the calls he places will be put on blacklist of certain individuals he calls. Thus, the authentication server will use these facts to compute a reputation score of authentication devices, the score reflecting the reliability and trustworthiness of the authentication determination, both caller ID authentication and call destination authentication. Such reputation can be based on how long the authentication device has been installed or have been certified, for instance a device which has been installed for 5 days would have a higher reputation score than a device which has been installed 1 hour ago. Or a device which received a certification password 5 days ago would have a higher reputation score than a device which was certified in the last hour. Noting that there may be instances where the device may be installed and not have been certified or passively certified. There could also be instances where the device loses the password some time after being installed, due to hardware or software failures, and requesting a new password instead. Another aspect of the score can be based on the number of calls the authentication device made or received since being installed or certified, with the reputation score increasing as more calls are made and received. Another aspect of the score can be based on the authentication device being actively and passively certified, indicating that the device has received a password from the server and has knowledge of the calls received and made by the phone number. A device which is both actively and passively certified would have a higher value added to its reputation score versus a device which is just actively certified, and the least value added to its reputation score would be a device which is just passively certified. Another criterion of the score can be based on the number of white and black lists the authentication device phone number is on, for instance an authentication device phone number which is on 10 internal or private white lists would have a higher reputation score than an authentication device phone number which is on 2 internal or white lists. Another aspect of the reputation score can consist of the IP address of the authentication device. If the geographical location of the authentication device IP address, corresponds to or is in close proximity to the geographical location of the authentication device phone number, as determined by its area code, then such proximity can be used towards the reputation score. If the IP address of the authentication device is determined to be not in close geographical proximity to the phone number it is authenticating, such as an U.S. telephone number and the authentication device has a European IP address, then such a determination can be used against the reputation score. Another aspect of the reputation score can be based on whether the device has registered a communication address, such as an IP address, with the authentication server. A device which registers a communication address would have a higher reputation score since the server is able to contact the device and verify caller ID authentication and call destination authentication as previously described. Another avenue of mitigation of such a forwarding attack is to allow for the caller to indicate that the called number has been forwarded. If a caller calls the victim and the call is forwarded to the attacker and the caller does not recognize the attacker, the caller can indicate that the call has been illicitly forwarded. Such indication can consist of the caller pressing a special digit sequence which is detected by the caller authentication device. Or the user can indicate such forwarding using the authentication device user interface or device web interface. The authentication device of the caller can then inform the server of such indication. The server can then integrate any received illicit forwarding indication during the computation of the reputation score of the called authentication device. The reputation score would be lowered based on the percentage of users indicating illicit forwarding in a given time frame. If sufficient percentage of callers indicates such illicit forwarding, the server may suspend all authentications for the called party in question. The server may require recertification of the called party authentication device, or contact the telephone service provider to discover forwarding state of the telephone number. The authentication server may notify the called party authentication device of received forwarding notification from caller parties. If the called party indicates that forwarding is unauthorized, the server may instruct the called party authentication device to dial digit sequences to disable forwarding on the called line. Similarly, a called party may indicate that a caller has a spoofed caller ID, and the called authentication device may forward such an indication to the authentication server and can be used by the server within the reputation score calculation. Another aspect of the score can account for the fact when the server is able to contact the authentication device for the purpose of verifying spoofing and call forwarding. Such contact based methods were given as step in and step in . The server can thus assign higher confidence scores to spoofing and forwarding determinations that have been verified by respective authentication devices. As discussed in section 6-d-ii, the server may employ additional methods of caller ID or forwarding detection, for instance as described in U.S. Patent Application 2013/0109358 by Vijay Balasubramaniyan, et al. If the server calls the authentication device for the purpose of transmitting the certification password through DTMF or caller ID, the authentication device may pick up the call and play an audio sample. The audio sample transmitted over the telephone network and received at the server can then be analyzed for purpose of discovering call forwarding, as outlined in the '9358 patent application. The reputation score of the particular authentication device can be take into account such additional call forwarding detection methods employed by the server when it contacted the authentication device.
Once a reputation score is computed dependent on such factors, the reputation score can then be transmitted alongside the caller ID and call destination determinations to caller and called authentication devices. The reputation score of the caller authentication device can be transmitted along with caller ID determination to the called authentication device as indication of the trustworthiness of the caller ID determination. And likewise, the call destination determination would also include the reputation score of the called party authentication device. In both cases, the reputation score is used to allocate a certain time frame for call forwarding attacks to be discovered, either by the victim or parties which are called by or place a call to the victim and are forwarded to the attacker. Such as a score can be a percentage out of 100, with 100 being the highest score. Or such a score can be on a scale of 1 to 10. The reputation score can then be used to by users to determine the accuracy of caller ID and call destination determinations. Or the users can indicate to the authentication devices that they only trust reputation scores greater than 50%. As a result, if a called party receives a call from a caller party with an authentic caller ID but a reputation score of 48%, the authentication device of the called party would disregard the authentic caller ID determination and indicate the caller ID as irresolute. Similarly if a caller receives a call destination authentication with a reputation score of 37%, she might be less inclined to trust it and take additional measures. The authentication server can also disregard authentication devices with a reputation score below certain threshold, and process those calls as irresolute until a higher reputation score is gained by the authentication device. The reputation score can thus be used internally by the server or revealed to the party authentication devices.
Another variation of the call forwarding attack can be where the victim has an authentication device installed and the attacker forwards her number and proceeds to install an authentication device of his own. These types of attacks are easier to detect by the authentication server, some of which were outlined in Section 6-e. Methods similar to detecting attacks against active certification will be used to detect call forwarding attacks where the attacker installs an authentication device of his own. If the attacker installs an authentication device and certifies it, there will now be two authentication devices authenticating outgoing and incoming calls for the same number, but with two different certification passwords. As soon as the victim makes a call, the victim's authentication device will contact the server with a calling status message, along with the old password. The authentication server will immediately determine that this is the original authentication device and that a new authentication device has been installed instead. The server can perform several procedures at this junction. The server can transmit a query to the original authentication device to verify that the user of the device has knowledge of the new authentication device. If the user replies in the affirmative, for instance in the case of legitimate forwarding, the server can send a code to the old authentication device. The old authentication device may then instruct the user to enter in the code on the new authentication device, thus proving she has control of and owns both devices. The user can also initiate installing an authentication device on the legitimately forwarded telephone number using the same process. If the user replies that the new authentication device does not belong to her, the server may instruct the old and new authentication devices to dial a special sequence, such as *73, to disable call forwarding on their lines. Disabling call forwarding may involve differing digit sequences depending on the telephone server provider and geographical region. Disabling call forwarding may also take the form of the user of the device storing account credentials on the authentication device or server, such as username and password that is used to log in to the users telephone account on the telephone service providers website. The authentication device or the server may then use such credentials to log into the telephone provider web site on behalf of the user and disable call forwarding. If the account management of the telephone provider reveals the forwarded number, the device or the server may transmit the forwarded number to appropriate law enforcement. The server may also invalidate the certification passwords of both the old and new devices and reset their reputation scores. It is also important to note that the new authentication device installed by the attacker will have a low reputation score, by the fact of being newly installed and certified. This low reputation score would help mitigate attacks in the meantime of being installed and being discovered by the server. Another method the server can use when presented with the attacker installing a new authentication device is to contact the old authentication device contact address, such as IP address, if there is one provided during heartbeat messages. Thus, when the attacker wishes to install and certify his authentication device, the server would contact the IP address of the old authentication device and verify that either the old device is no longer installed, or if it is installed, the user of the old device gives permission to the new device to perform authentication, such as per legitimate call forwarding. If the authentication devices have a unique serial number or a digital certificate assigned to them for the purpose of uniquely identifying each device, then such information can be used by the server to detect a new device being installed for the same phone number which has been forwarded. During a certification request or heartbeat transmission such unique device identity information may be transmitted by the authentication device to the server for such a purpose.
A user who has an authentication device installed at his house may wish to forward his calls to another phone (his vacation house) and have those calls authenticated. One method of performing legitimate forwarding was discussed previously. The user installs an authentication device at the forwarded number. When a call is made from the new device, the authentication server can then contact the old device for the appropriate permission. The user may instead indicate to the old authentication device that a new device at a forwarded number is to be installed. Such indication can be performed through IVR or the device user interface or web interface. The user can type in the serial number of the new device to give permission. Or the user can be given a code by the server to type into the new device, and the new device can then contact the server with the code to confirm the permission. Thus, any messages transmitted by the new device would be treated by the server as acting on behalf of the old device and be authenticated as such. And any queries by the server, such as requesting a calling status message, would be sent to the new device. The user may then rescind permission for the new device when forwarding is no longer desired.
d) Various Call Destination Authentication Topologies
As were described with caller ID authentication peer-to-peer topologies and topologies without an authentication device or server, as described in Section 13, equally apply to call destination authentication. For instance, in the example given in Section 13, if Chase Bank calls a Verizon customer, Verizon can check the calls being received by the called customer and notify Chase Bank of the call destination authenticity. Or Verizon can check if there has been any forwarding enabled on the call destination account, and notify the caller party, assuming both the caller and called parties are Verizon customers and share the same telephone service provider. If the called party is not a Verizon customer, Verizon can contact an authentication server in anticipation that the called party has an authentication device installed and the authentication server can thus determine and reply with the call destination authenticity determination. Verizon can also transmit the caller call information to the authentication server for the purpose of the server transmitting the caller ID authentication determination to the called party authentication device. The process of Verizon or a telephone service provider authenticating itself with respect to the authentication server is beyond the scope of this document, but would preferably be done through utilizing public key infrastructure or comparable methodologies known in the art.
16) Summary
Summarizing an embodiment of the invention as it relates to any and all applicable communication systems, an authentication device is installed at both sender and recipient. The sender is transmitting a message to the recipient. The recipient authentication device authenticates the source address of the message. When the sender initiates the transmission of the message, the sender authentication device contacts the authentication server. The contact from the device to the server may be through any communication medium, regardless of the medium used to transmit the message. The sender authentication device notifies the server of the sender address and the message destination. When the recipient receives the message, the recipient authentication device contacts the authentication server, through any communication medium. The recipient authentication device requests that the transmission with a source address contained in the message and received at the recipient address be authenticated. The authentication server compares the received notification by the sender device and compares the sender destination against the recipient address, and sender address against the source address. The authentication server then notifies the recipient authentication device that the source address of the transmission is authentic, spoofed or irresolute.
The authentication devices are certified that they are indeed installed or have access to a particular communication address for a particular communication medium through certification passwords. A certification password is sent by the authentication server to an authentication device by transmitting to the authentication device at its communication address using any communication medium that the address is connected to. The authentication device then receives the certification password transmitted by the server. Thus the certification password can be presented to the server by the authentication device as proof that the authentication device is residing at a particular communication address. Hence, the authentication devices are sent a certification password in-band and the all other communications of messages that relate to authentication may be done in-band or out-of-band.
There are three variations of the authentication messaging protocol. The first variation is the sender contacts the server with a sent status message when a transmission is initiated. And the recipient contacts the server with an authentication request when a transmission is received. The second variation is the server notifies the recipient of the transmission a sender has made to the recipient. The third variation where a sent status message is not received by the server and the server upon receiving an authentication request from the recipient contacts the sender and verifies the transmission.
In the variation where the sender sends a sent status message, the recipient need not include the source address in the authentication request, such as where the source address is blocked. Since the authentication server is already made aware of the transmission from the sender to the recipient, the recipient can inquire who the current transmission is from or which senders are initiating a transmission to the recipient. Applications of this concept were explained as they related to authenticating blocked caller ID's and VoIP calls.
Applications of such a system of message source address authentication have been presented in detail as they relate to telephone, email, and postal mail. Also, variations of the system such as of peer-to-peer, server push, and others have been presented. Furthermore, the use of such a system within a whitelist and blacklist handling system has been disclosed. Those skilled in the art can apply the described system of source address authentication and accompanying variations and list handling to any currently known communication system, or communication systems yet to be discovered.
Some parts of the invention has been has been described in language specific to structural features and/or methodological steps, however it is to be understood that those parts of the invention defined in the appended claims is not necessarily limited to specific features or steps described. Rather, the specific features and steps are disclosed as forms of implementing the claimed invention. Those skilled in the art will recognize that the invention can be practiced with modifications and in the spirit and scope of the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
The same reference number represents the same element or the same type of element on all drawings.
FIG. 1A
represents a simplified block diagram of a landline communications network utilizing the caller ID authentication device within a residential setting in an exemplary embodiment of the invention.
FIG. 1B
represents a simplified block diagram of a landline communications network utilizing the caller ID authentication device within the telephone service provider in an exemplary embodiment of the invention.
FIG. 2A
represents a simplified block diagram of an embodiment of the invention implemented on a VoIP system where the authentication device is implemented at the VoIP provider.
FIG. 2B
represents a simplified block diagram of an embodiment of the invention implemented on a VoIP system where the authentication device is implemented at the customer.
FIG. 3A
represents the flowchart and messaging diagram the caller ID authentication device at both the caller and called party uses to authenticate caller ID.
FIG. 3B
illustrates the sequence where the authentication server initiates contact with the called phone authentication device instead of the authentication device initiating contact with the server.
FIG. 4
represents, by way of example, the flowchart and messaging diagram the caller ID authentication device at both the caller and called party uses to authenticate caller ID.
FIG. 5A
represents the flowchart and messaging diagram the caller ID authentication device at both the spoofing victim and called party uses to authenticate caller ID.
FIG. 5B
represents the flowchart and message diagram the caller ID authentication device at the called party uses to determine an irresolute caller ID.
FIG. 6A
shows the flowchart and message diagram for one embodiment of the invention that certifies that an authentication device is indeed installed at a particular phone number and is acting on behalf of the phone number.
FIG. 6B
represents the flowchart and messaging diagram that shows certification and caller ID authentication performed in a peer-to-peer fashion, where calling party forwards the calling status message to the called party.
FIG. 6C
represents the flowchart and messaging diagram that shows certification and caller ID authentication performed in a peer-to-peer fashion, where called party requests authentication status from the calling party.
FIG. 6D
illustrates the flowchart diagram showing one embodiment of the invention that is used by the server to determine caller ID authenticity.
FIG. 7A
illustrates the flowchart and messaging diagram for authenticating a caller with a blocked caller ID.
FIG. 7B
illustrates the signaling diagram for authenticating a caller with a caller ID nickname.
FIGs. 8A-8C
FIG. 8
, collectively referred as , shows the flowchart and signaling diagram for an embodiment of the invention of authenticating legitimately spoofed caller ID's.
FIG. 9A
shows, by way of example, the flow chart that is used to handle calls with non-blocked caller ID's by the authentication device.
FIG. 9B
shows, by way of example, the flow chart that is used in handling incoming calls with blocked caller ID's by the authentication device.
FIG. 10A
shows, by way of example, the internal lists maintained by the authentication device and the global lists maintained by the global list server.
FIG. 10B
shows, by way of example, the state of the global list server after the internal lists of the authentication devices are uploaded to the global list server.
FIG. 11A
shows, by way of example, the flowchart that is used to handle calls with non-blocked caller ID's by the authentication device utilizing both global and internal lists.
FIG. 11B
shows, by way of example, the flowchart that is used to handle calls with blocked caller ID's by the authentication device utilizing both global and internal lists.
FIG. 12
illustrates, by way of example, the flowchart used in adding an incoming telephone number to the internal blacklist of the called authentication device autonomously.
FIG. 13A
shows, by way of example, the internal lists and internal potential blacklists maintained by the authentication device, and the global lists and global potential blacklists maintained by the global list server.
FIG. 13B
shows, by way of example, the internal lists and internal potential blacklists maintained by the authentication device and the global lists and global potential blacklists maintained by the global list server after the authentication devices have uploaded their lists to the global list server.
FIGs. 14A-14C
FIG. 14
, collectively referred to as , illustrates the flowchart and messaging diagram for adding a blocked caller ID to the user's private blacklist, by way of example.
FIG. 15A-15C
illustrates, by way of example, the flowchart and messaging diagram for revealing a blocked caller ID certain called parties.
FIG. 15D-15E
illustrates, by way of example, the flowchart and messaging diagram for authenticating caller ID within a VoIP infrastructure.
FIGs. 16A-16C
FIG. 16
, collectively referred to as , illustrates the flowchart and messaging diagram for an embodiment of the invention which authenticates email sender address and adding it to a local whitelist of the recipient.
FIG. 17A
illustrates, by way of example, the flowchart and messaging diagram for certifying authentication devices in a postal system.
FIG. 17B
illustrates an example certification mailing envelope that is sent to the postal authentication device by the authentication server.
FIG. 17C
illustrates an example mailing envelope sent from a sender to a recipient.
FIG. 17D
is a flowchart and messaging diagram that illustrates an embodiment of the invention for authenticating postal mail source address.
FIG. 17E
illustrates the flowchart diagram showing one embodiment of the invention used by the server to determine postal mail source address authenticity.
FIG. 18A
illustrates, by way of example, the messaging diagram for authenticating the destination of a call.
FIG. 18B
illustrates, by way of example, the messaging diagram for determining a call has been forwarded.
FIG. 18C
illustrates the flowchart diagram showing one embodiment of the invention that is used by the authentication server to determine call destination authentication and call forwarding detection. | |
BACKGROUND OF THE INVENTION
The formation of slimes by microorganisms is a problem that is encountered in many aqueous systems. For example, the problem is not only found in natural waters such as lagoons, lakes, ponds, etc., and confined waters as in pools, but also in such industrial systems as cooling water systems, air washer systems and pulp and paper mill systems. All possess conditions which are conducive to the growth and reproduction of slime- forming microorganisms. In both once-through and recirculating cooling systems, for example, which employ large quantities of water as a cooling medium, the formation of slime by microorganisms is an extensive and constant problem.
Airborne organisms are readily entrained in the water from cooling towers and find this warm medium an ideal environment for growth and multiplication. Aerobic and heliotropic organisms flourish on the tower proper while other organisms colonize and grow in such areas as the tower sump and the piping and passages of the cooling system. The slime formation not only aids in the deterioration of the tower structure in the case of wooden towers, but also promotes corrosion when it deposits on metal surfaces. Slime carried through the cooling system plugs and fouls lines, valves, strainers, etc., and deposits on heat exchange surfaces. In the latter case, the impedance of heat transfer can greatly reduce the efficiency of the cooling system.
In pulp and paper mill systems, slime formed by microorganisms is commonly encountered and causes fouling, plugging, or corrosion of the system. The slime also becomes entrained in the paper produced to cause breakouts on the paper machines, which results in work stoppages and the loss of production time. The slime is also responsible for unsightly blemishes in the final product, which result in rejects and wasted output.
The previously discussed problems have resulted in the extensive utilization of biocides in cooling water and pulp and paper mill systems. Materials which have enjoyed widespread use in such applications include chlorine, chlorinated phenols, organo-bromines, and various organo-sulfur compounds. All of these compounds are generally useful for this purpose but each is attended by a variety of impediments. For example, chlorination is limited both by its specific toxicity for slime- forming organisms at economic levels and by the tendency of chlorine to react, which results in the expenditure of the chlorine before its full biocidal function is achieved. Other biocides are attended by odor problems and hazards with respect to storage, use or handling which limit their utility. To date, no one compound or type of compound has achieved a clearly established predominance with respect to the applications discussed. Likewise, lagoons, ponds, lakes, and even pools, either used for pleasure purposes or used for industrial purposes for the disposal and storage of industrial wastes, become, during the warm weather, besieged by slime due to microorganism growth and reproduction. In the case of industrial storage or disposal of industrial materials, the microorganisms cause additional problems which must be eliminated prior to the materials' use or disposal of the waste.
Naturally, economy is a major consideration with respect to all of these biocides. Such economic considerations attach to both the cost of the biocide and the expense of its application. The cost performance index of any biocide is derived from the basic cost of the material, its effectiveness per unit of weight, the duration of its biocidal or biostatic effect in the system treated, and the ease and frequency of its addition to the system treated. To date, none of the commercially available biocides has exhibited a prolonged biocidal effect. Instead, their effectiveness is rapidly reduced as a result of exposure to physical conditions such as temperature, association with ingredients contained by the system toward which they exhibit an affinity or substantivity, etc., with a resultant restriction or elimination of their biocidal effectiveness, or by dilution.
As a consequence, the use of such biocides involves their continuous or frequent addition to systems to be treated and their addition to multiple points or zones in the systems to be treated. Accordingly, the cost of the biocide and the labor cost of applying it are considerable. In other instances, the difficulty of access to the zone in which slime formation is experienced precludes the effective use of a biocide. For example, if in a particular system there is no access to an area at which slime formation occurs the biocide can only be applied at a point which is upstream in the flow system. However, the physical or chemical conditions, e.g., chemical reactivity, thermal degradation, etc., which exist between the point at which the biocide may be added to the system and the point at which its biocidal effect is desired render the effective use of a biocide impossible.
Similarly, in a system experiencing relatively slow flow, such as a paper mill, if a biocide is added at the beginning of the system, its biocidal effect may be completely dissipated before it has reached all of the points at which this effect is desired or required. As a consequence, the biocide must be added at multiple points, and even then a diminishing biocidal effect will be experienced between one point of addition to the system and the next point downstream at which the biocides may be added. In addition to the increased cost of utilizing and maintaining multiple feed points, gross ineconomies with respect to the cost of the biocide are experienced. Specifically, at each point of addition, an excess of the biocide is added to the system in order to compensate for that portion of the biocide which will be expended in reacting with other constituents present in the system or experience physical changes which impair its biocidal activity.
SUMMARY OF THE INVENTION
The biocidal compositions of the present invention comprise, as active ingredients, 1) diiodomethyl-p-tolylsulfone (DIMPS) and 2) 2- bromo-2- nitropropane-1,3-diol (BNPD). These constituents are commercially available. The synergistic effect obtained by combining DIMPS and BNPD has not been previously disclosed.
DETAILED DESCRIPTION OF THE INVENTION
Surprisingly, the present inventors have found that mixtures of DIMPS and BNPD are especially efficacious in controlling the growth of bacterial microbes, specifically the Klebsiella pneumoniae species. This particular species is a member of the capsulated, facultative class of bacteria and is generally present in air, water and soil. These bacteria continually contaminate open cooling systems and pulping and papermaking systems and are among the most common slime formers. The slime may be viewed as being a mass of agglomerated cells stuck together by the cementing action of the gelatinous polysaccharide or proteinaceious secretions around each cell. The slimy mass entraps other debris, restricts water flow and heat transfer, and may serve as a site for corrosion.
The fact that the Klebsiella species used in the tests is a facultative species is important as, by definition, such bacteria may thrive under either aerobic or anaerobic conditions. Accordingly, by reason of demonstrated efficacy in the growth inhibition of this particular species, one can expect similar growth inhibition attributes when other aerobic or anaerobic bacterial species are encountered. It is also expected that these compositions will exhibit similar growth inhibition attributes when fungi and algae species are encountered.
In accordance with the present invention, the combined DIMPS and BNPD treatment may be added to the desired aqueous system in need of biocidal treatment, in an amount of from about 0.1 to about 200 parts of the combined treatment to one million parts (by weight) of the aqueous medium. Preferably, about 5 to about 50 parts of the combined treatment per one million parts (by weight) of the aqueous medium is added.
The combined treatment is added, for example, to cooling water systems, paper and pulp mill systems, pools, ponds, lagoons, lakes, etc., to control the formation of bacterial microorganisms, which may be contained by, or which may become entrained in, the system to be treated. It has been found that the compositions and methods of utilization of the treatment are efficacious in controlling the facultative bacterium, Klebsiella pneumoniae, which may populate these systems. It is thought that the combined treatment composition and method of the present invention will also be efficacious in inhibiting and controlling all types of aerobic and anaerobic bacteria.
Surprisingly, it has been found that when the ingredients are mixed, in certain instances, the resulting mixtures possess a higher degree of bactericidal activity than that of the individual ingredients comprising the mixture. Accordingly, it is possible to produce a highly efficacious bactericide. Because of the enhanced activity of the mixture, the total quantity of the bacterial treatment may be reduced. In addition, the high degree of bactericidal effectiveness which is provided by each of the ingredients may be exploited without use of higher concentrations of each.
The following experimental data were developed. It is to be remembered that the following examples are to be regarded solely as being illustrative and not as restricting the scope of the invention.
DESCRIPTION OF PREFERRED EMBODIMENT
DIMPS and BNPD were added in varying ratios and over a wide range of concentrations to a liquid nutrient medium which was subsequently inoculated with a standard volume of a suspension of the facultative bacterium Klebsiella pneumoniae. Growth was measured by determining the amount of radioactivity accumulated by the cells when 14C- glucose was added as the sole source of carbon in the nutrient medium. The effect of the biocide chemicals, alone and in combination, is to reduce the rate and amount of 14C incorporation into the cells during incubation, as compared to controls not treated with the chemicals. Additions of the biocides, alone and in varying combinations and concentrations, were made according to the accepted "checkerboard" technique described by M. T. Kelley and J. M. Matsen, Antimicrobial Agents and Chemotherapy. 9:440 (1976). Following a two hour incubation, the amount of radioactivity incorporated in the cells was determined by counting (14C liquid scintillation procedures) for all treated and untreated samples. The percent reduction of each treated sample was calculated from the relationship: ##EQU1##
Plotting the % reduction of 14C level against the concentration of each biocide acting alone results in a dose-response curve, from which the biocide dose necessary to achieve any given % reduction can be interpolated.
Synergism was determined by the method of calculation described by F. C. Kull, P. C. Eisman, H. D. Sylwestrowicz and R. L. Mayer, Applied Microbiology 9,538 (1961) using the relationship: ##EQU2## where: Q.sub.a =quantity of compound A, acting alone, producing an end point
Q.sub.b =quantity of compound B, acting alone, producing an end point
Q.sub.A =quantity of compound A in mixture, producing an end point
Q.sub.B =quantity of compound B in mixture, producing an end point
The end point used in the calculations is the % reduction caused by each mixture of A and B. Q.sub.A and Q.sub.B are the individual concentrations in the A/B mixture causing a given % reduction. Q.sub.a and Q.sub.b are determined by interpolation from the respective dose- response curves of A and B as those concentrations of A and B acting alone which produce the same % reduction as each specific mixture produced.
Dose-response curves for each active acting alone were determined by linear regression analysis of the dose-response data. Data were fitted to a curve represented by the equation shown with each data set. After linearizing the data, the contributions of each biocide component in the biocide mixtures to the inhibition of radioisotope uptake were determined by interpolation with the dose-response curve of the respective biocide. If, for example, quantities of Q.sub.A plus Q.sub. B are sufficient to give a 50% reduction in 14C content, Q.sub.a and Q. sub. b are those quantities of A or B acting alone, respectively, found to give 50% reduction in 14C content. A synergism index (SI) is calculated for each combination of A and B.
Where the SI is less than 1, synergism exists. Where the SI=1, additivity exists. Where SI is greater than 1, antagonism exists.
The data in the following tables come from treating Klebsiella pneumoniae, a common nuisance bacterial type found in industrial cooling waters and in pulping and paper making systems, with varying ratios and concentrations of DIMPS and BNPD. Shown for each combination is the % reduction of 14C content (% I), the calculated SI, and the weight ratio of DIMPS and BNPD.
TABLE I
______________________________________
DIMPS vs. BNPD
ppm ppm Ratio
DIMPS.sup.1
BNPD.sup.2 DIMPS:BNPD % I SI
______________________________________
6 0 100:0 88
3 0 100:0 76
1.5 0 100:0 48
0.75 0 100:0 28
0.38 0 100:0 20
0.19 0 100:0 12
0 1200 0:100 80
0 600 0:100 73
0 300 0:100 60
0 150 0:100 42
0 75 0:100 29
0 37.5 0:100 19
6 1200 1:200 92 1.53
3 1200 1:400 89 1.27
1.5 1200 1:800 76 1.96
0.75 1200 1:1600 82 1.26
0.38 1200 1:3158 81 1.25
0.19 1200 1:6316 49 6.10
6 600 1:100 91 1.22
3 600 1:200 87 0.95
1.5 600 1:400 88 0.94*
0.75 600 1:800 77 0.90*
0.38 600 1:1600 75 0.92*
0.19 600 1:3158 35 6.62
6 300 1:50 91 1.06
3 300 1:100 85 0.79*
1.5 300 1:200 76 0.80*
0.75 300 1:400 70 0.78*
0.38 300 1:800 66 0.81*
0.19 300 1:1600 28 5.05
6 150 1:25 91 0.97
3 150 1:50 86 0.67*
1.5 150 1:100 74 0.66*
0.75 150 1:200 64 0.68*
0.38 150 1:400 82 0.22*
0.19 150 1:800 56 0.63*
6 75 1:12.5 92 0.91*
3 75 1:25 85 0.64*
1.5 75 1:50 69 0.67*
0.75 75 1:100 52 0.85*
0.38 75 1:200 73 0.22*
0.19 75 1:400 41 0.78*
6 37.5 1:6.25 91 0.92*
3 37.5 1:12.5 83 0.65*
1.5 37.5 1:25 63 0.80*
0.75 37.5 1:50 40 1.17
0.38 37.5 1:100 62 0.28*
0.19 37.5 1:200 33 0.72*
______________________________________
.sup.1 product containing 40% actives DIMPS
.sup.2 product containing 95% actives BNPD
TABLE II
______________________________________
DIMPS vs. BNPD
ppm ppm Ratio
DIMPS.sup.1
BNPD.sup.2 DIMPS:BNPD % I SI
______________________________________
6 0 100:0 91
3 0 100:0 81
1.5 0 100:0 59
0.75 0 100:0 26
0.38 0 100:0 0
0.19 0 100:0 0
0 1200 0:100 81
0 600 0:100 71
0 300 0:100 56
0 150 0:100 38
0 75 0:100 22
0 37.5 0:100 6
6 1200 1:200 95 1.72
3 1200 1:400 91 1.38
1.5 1200 1:800 87 1.24
0.75 1200 1:1600 84 1.23
0.38 1200 1:3158 82 1.22
0.19 1200 1:6316 82 1.18
6 600 1:100 94 1.43
3 600 1:200 90 1.04
1.5 600 1:400 83 0.94*
0.75 600 1:800 78 0.89*
0.38 600 1:1600 74 0.94*
0.19 600 1:3158 72 0.96
6 300 1:50 94 1.30
3 300 1:100 88 0.89*
1.5 300 1:200 79 0.76*
0.75 300 1:400 73 0.69*
0.38 300 1:800 65 0.76*
0.19 300 1:1600 60 0.87*
6 150 1:25 93 1.24
3 150 1:50 87 0.83*
1.5 150 1:100 77 0.66*
0.75 150 1:200 67 0.60*
0.38 150 1:400 56 0.69*
0.19 150 1:800 44 0.93*
6 75 1:12.5 94 1.18
3 75 1:25 88 0.77*
1.5 75 1:50 76 0.59*
0.75 75 1:100 59 0.62*
0.38 75 1:200 38 0.89*
0.19 75 1:400 21 1.39
6 37.5 1:6.25 93 1.19
3 37.5 1:12.5 86 0.76*
1.5 37.5 1:25 73 0.60*
0.75 37.5 1:50 48 0.76*
0.38 37.5 1:100 22 1.20
0.19 37.5 1:200 11 1.32
______________________________________
.sup.1 product containing 40% actives DIMPS
.sup.2 product containing 95% actives BNPD
Asterisks in the SI column indicate synergistic combinations in accordance with the Kull method supra.
In Tables I and II, differences seen between the replicates are due to normal experimental variance.
In accordance with Tables I-II supra., unexpected results occurred more frequently within the product ratios of DIMPS to BNPD of from about 1:1579 to 1:6.25. Since the DIMPS product contains about 40% active biocidal component and the BNPD product contains about 95% active biocidal component, when based on the active biocidal component, unexpected results appear more frequently within the range of active component of DIMPS:BNPD of about 1:3750 to 1:14.8. At present, it is most preferred that any commercial product embodying the invention comprises a weight ratio of active component of about 1:15 DIMPS:BNPD.
While this invention has been described with respect to particular embodiments thereof, it is apparent that numerous other forms and modifications of this invention will be obvious to those skilled in the art. The appended claims and this invention generally should be construed to cover all such obvious forms and modifications which are within the true spirit and scope of the present invention. | |
Math is not fun for many people. It is very important when it comes to precious metal. Often you will find yourself in a situation where you need to convert your pennyweight into the ounces to figure out how much money it is worth.
Below is a simple chart that converts pennyweight, grams, ounces, pounds, troy ounces and troy pounds Simply use the number in the right hand column to multiply by in order to convert one weight unit to another weight unit.
How Heavy Are Gold Bricks? | http://www.sellgoldhq.com/convert-weight/ |
The utility model relates to an electroslag remelting furnace's scratch start device, include scratch start backing plate, scratch start bottom plate and the striking piece that contacts with the electrode base metal, scratch start bottom plate is cut in the disc of self -electrode mother metal and is welded on scratch start backing plate for radial, the striking piece for take from the electrode base metal, length is lighter than the radial strip piece of scratch start bottom plate, this striking block welding in scratch start bottom plate, and at least for six and the centre of a circle of using scratch start bottom plate as center along the radial equipartition in scratch start bottom plate border, form the space that is used for placing the fritting sediment between each adjacent striking piece, each striking piece top is used the scratch start bottom plate centre of a circle and is placed the plane as center formation electrode bar. The device can guarantee that scratch start stage electric current, voltage are steady, avoids the scratch start stage easily to beat arc, easy adhesion, the easy scheduling problem of opening a way, and the electrode base metal is all taken from to scratch start bottom plate and striking piece in addition, and is consequently pollution -free, is the electroslag remelting furnace's scratch start device that is used for of a high efficiency, safety. | |
Q:
how to solve a cubic congruence
I have to solve this congruence: $x^3 \equiv 1\,(\mod\;101)$ , but I I can't understand some steps,
$$
x^3 \equiv 1\,(\mod\;101) \\
\gcd(3,100)=1\\
3a+100b=1\\
x^1\equiv x^{3a+100b} \equiv1^a\cdot1^b\equiv 1\;(\mod\;101)
$$
Solution:$x \equiv 1 \;(\mod\;101)$
In particular, I don't understand what the $3$ of the greatest common divisor represents(is it the exponent?) and why $x^{3a+100b}$ is equivalent to $1^a\cdot1^b$.
A:
Yes, the $3$ in $\gcd(3,100)=1$ is the exponent in the equation (which is a cubic equation b.t.w.).
As to $x^{3a+100b}$, it is equal to $x^{3a}x^{100b}=(x^3)^a(x^{100})^b=1^a(x^{100})^b$, and $x^{100}\equiv 1\bmod 101$ by lil' Fermat, since $101$ is prime.
| |
The derivative of x is 1. A derivative of a function in terms of x can be thought of as the rate of change of the function at a value of x. In the case of f(x) = x, the rate of change is 1 at all values of x.
The derivative of a function f(x) is defined as the limit as h tends towards zero of the expression (f(x+h) - f(x))/h.
If f(x) = x, then (f(x+h) - f(x))/h = ((x+h) - x)/h = h/h. For every value of h not equal to zero, h/h is equal to one. Therefore, as h approaches zero, (f(x+h) - f(x))/h = 1. | https://www.reference.com/world-view/derivative-x-80435ef3f2e41c7 |
Where did the name “Stephen’s Table” come from?
The name comes from Acts 6:1-7 in the Bible where Stephen was chosen among the first deacons of the early Christian community to “wait on tables” while the Twelve attended to “prayer and ministry of the word.” In the same way, our volunteers have been called into service on behalf of the broader Chattanooga community — so that the needs of those who have been marginalized are not forgotten.
What is the purpose?
The purpose is to preserve the social bonds between prisoners and their hometown — which is known to improve the transition from prison to society.
Who do you serve?
Anyone from Hamilton County who is currently serving time in a state or federal prison. We send them all postcards as soon as we are informed of their sentence and invite them to participate in Stephen’s Table. To be eligible they must be:
- Incarcerated in a state or federal prison on a felony conviction
- From Hamilton County and planning to return to Hamilton County if released
- At least 24 months before he or she is released
- And willing to actively participate in Stephen’s Table
Why do you need volunteers?
For two reason: (1) Because they are uninfluenced by a paycheck, community volunteers are often viewed by prisoners as more legitimate representatives of their hometown than an organization’s paid personnel, and (2) given the number of people from Hamilton County who are presently incarcerated (c. 1,000), it would be impracticable for a few hired hands to develop individual relationships with them all.
What do you need me to do?
Mostly write letters.
How many people will I be expected to write?
No more than two — unless you have the time and really want to. We would rather you spend more time on fewer people than less time on more.
How often?
The research we are following demonstrates that one visit a month during the last 12 months in prison improves the rate people stay out of prison by 30% — and that two letters have the same effect as one visit. Therefore, the goal is get two pieces of mail to each person you are writing every month.
For how long?
Twelve months. However, you can volunteer as long as you want. We’ll invite you to renew your commitment every year in order to manage expectation.
Can I visit the person I’m writing?
Not as a Stephen’s Table volunteer. If you would like to visit the person you’re writing, you must complete an additional training and certification program for visitation.
Does it cost anything?
Not necessarily. We discourage spending money on the person you’re writing — except to contribute to holiday programs sponsored by the institution, purchase magazine or newspaper subscriptions if possible, or donate to Chattanooga Endeavors to offset the costs of managing your letter exchange ($10 a moth per match). The main cost associated with volunteering for Stephen’s Table is your time. Whether you spend any money is up to you.
What are my responsibilities after they get out?
None. You’re responsibilities are transferred to Chattanooga Endeavors during the three months prior to their release. If you would like to reengage with them you must complete an additional training and certification program for mentors.
Will the person I’m writing know where I live?
No. All letters from prisoners are sent to a PO Box which Chattanooga Endeavors manages. We actually require this and prohibit volunteers from using any other address.
How about my name?
Just your first name. Volunteers write on behalf of Stephen’s Table and attempt to form a bond between the person they’re writing and the group not themselves.
What about personal information?
All they will know about you is what you tell them. We encourage minimal self-disclosure — only when it is pertinent to an exchange and never in a way that reveals personal identity.
Doesn’t this extra caution make it hard to build a relationship?
Not at all. Whereas self-disclosure can help build rapport, you will get further by being dependable, truthful, and non-judgmental.
Do you screen people before matching them with volunteers?
Yes. We screen for risk and match those with higher risk to an internal committee of seasoned volunteers with staff support.
Do you accept sex offenders?
Yes. However, they are matched with the same high risk committee.
What about murderers?
We do. In fact, many people with murder convictions are actually very low risk. This is because they tend to be older, have served much more time in prison, and have fewer prior convictions. Nevertheless, we encourage you to decline writing someone who you are uncomfortable with. As a Stephen’s Table volunteer, you have the right to discontinue writing anyone at any time for any reason.
How about someone with a life or death sentence?
Absolutely. We believe that we have a special calling to reach out to people who may never come home. If you would like to write someone with a life or death sentence, please let us know.
Can I email the person I’m writing?
Sort of. We send letters to you by email and you reply to the letters by email. However, the email comes to us. We print your letter and drop it in the mail to the institution with a postage-paid return envelop. This creates an additional layer of anonymity for you and makes the exchange extremely efficient — even when you’re out of town.
Can I try it out without making a year-long commitment?
Yes. You can attend a monthly letter writing group where you can write someone on our waiting list without being matched to him or her. Come just once — or every month. It’s up to you. And there is no commitment. You’ll learn about our mission, get a taste for how we do our work, and meet some people who share your concern for prisoners.
Where can I find out more?
You can look through our material at www.stephenstable.org or give us a call at (423) 266-1888 ext 103
How do I get started?
If you would like to get started, please complete our online volunteer registration. | https://chattanoogaendeavors.org/service/stephens-table/faq/ |
Suppose is a prime number.
where is the elementary abelian group of prime-square order and is the group of prime order .
The group is isomorphic to elementary abelian group of prime-cube order . Equivalently, it is a three-dimensional vector space over the field of elements.
The behavior is somewhat different for and odd primes. For the anomalous case , see second cohomology group for trivial group action of V4 on Z2.
The cohomology group can be computed as an abstract group using the group cohomology of elementary abelian group of prime-square order, which in turn can be computed using the Kunneth formula for group cohomology combined with the group cohomology of finite cyclic groups.
Here, is the quotient of by and .
which is the elementary abelian group of order .
As mentioned earlier, the information here does not apply to the case . For the case , see second cohomology group for trivial group action of V4 on Z2.
Consider an extension group with central subgroup isomorphic to (group of prime order) and quotient group isomorphic to elementary abelian group of prime-square order. Denote by a basis for (i.e., two non-identity elements of that do not generate the same cyclic subgroup) and by elements of that map to respectively. Denote by a non-identity element of the central subgroup.
Then, is generated by the elements . Further, we can get a power-commutator presentation for using these generators. Specifically, we know that . We also know that the elements are each equal to some power of .
is the power of that equals. It is 0 if (i.e., is the identity element) and nonzero otherwise.
The total number of possibilities is . Further, the mapping from that sends a cohomology class to the tuple is an isomorphism of additive groups. This means that to add two cohomology classes, we can add the corresponding tuples.
If, however, we consider the action of the direct product of these groups, with one acting by pre-composition and the other by post-composition (since the actions are at opposite ends, they commute by associativity, so this is an action of the direct product), then the orbits are precisely the four cohomology class types described here, i.e., each cohomology class type gives one orbit.
The discussion here relies on the explicit description of cohomology classes in terms of the invariants .
The automorphism group of the acting group, as noted above, is the general linear group . It turns out that its action on is given precisely by the action as matrices, and it also induces an action on the by a group automorphism of .
The automorphism group of the base group, which is concretely , acts on all coordinates via multiplication by the corresponding element mod . Thus, this is a scalar multiplication action. The orbits of non-identity elements are thus the lines in , or equivalently, the set of orbits can be described as two-dimensional projective space over . There is also the orbit of the identity or zero element, so the total number of orbits is .
where the image of is , i.e., the group of cohomology classes represented by symmetric 2-cocycles and corresponding to the abelian group extensions. We also know, again from the general theory, that the short exact sequence above splits, i.e., has a complement inside . However, there need not in general be a natural or even an automorphism-invariant choice of splitting.
where is the subgroup of comprising those cohomology classes that have as a representative an alternating bihomomorphism from to . In this case, is an elementary abelian group of prime-square order and is a group of prime order.
The map on that sends a cohomology class to the skew of any representative 2-cocycle, has as its kernel and its effect on is to output the alternating bihomomorphism that is twice of the alternating bihomomorphism representing the cohomology class.
This can also be interpreted in terms of the Baer correspondence. See the #Baer Lie rings section.
In this case, we do not get a direct sum decomposition of the above sort, because there is no notion of division by two or halving that would allow us to get an alternating bihomomorphism by halving the commutator map (which equals the skew of the 2-cocycle). For more, see second cohomology group for trivial group action of V4 on Z2#Direct sum decomposition.
where is the subgroup of the second cohomology group comprising those classes that can be represented by alternating bihomomorphisms.
This can be used to obtain the Baer correspondence (the class two version of the Lazard correspondence) as follows: an element of corresponds to an extension group that is a group of nilpotency class two (which includes the abelian and non-abelian cases). It has a unique direct sum decomposition as the sum of a symmetric 2-cocycle (whose cohomology class is an element of ) and an alternating bihomomorphism (whose cohomology class is an element of ). The alternating bihomomorphism is given by halving the commutator map (which in turn is the skew of the 2-cocycle).
The Baer Lie ring for the group is defined as follows: the additive group is the extension corresponding to the symmetric 2-cocycle, and the Lie bracket is given by the alternating bihomomorphism.
Explicitly, for in the extension group, we define the alternating bihomomorphism as and the addition as .
unitriangular matrix group:UT(3,p) (there are such rows) semidirect product of cyclic group of prime-square order and cyclic group of prime order (this occurs in columns and rows).
group of all 1-cochains for trivial group action 4 elementary abelian group of prime-fourth order all set maps from to with pointwise addition, so the dimension is the cardinality of .
group of all 2-coboundaries for trivial group action 2 elementary abelian group of prime-square order By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all 2-cocycles for trivial group action 5 elementary abelian group of prime-fifth order ?
second cohomology group for trivial group action 3 elementary abelian group of prime-cube order This is , so dimensions subtract and orders divide.
In particular, what this means is that for every cohomology class, there are different choices of 2-cocycles that represent that cohomology class.
group of normalized 1-cochains for trivial group action 3 elementary abelian group of prime-cube order all set maps from to with pointwise addition, that send the identity to the identity. There are thus 3 elements that can be mapped arbitrarily.
group of all normalized 2-coboundaries for trivial group action 1 group of prime order By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all normalized 2-cocycles for trivial group action 4 elementary abelian group of prime-fourth order ?
second cohomology group for trivial group action 3 elementary abelian group:E8 This is , so dimensions subtract and orders divide.
In particular, what this means is that for every cohomology class, there are different choices of normalized 2-cocycles that represent that cohomology class.
Since these are short exact sequences of vector spaces, they must split. Further, a splitting of the latter also gives a splitting of the former.
The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF. Begin by setting equal to a specific prime number value.
The precise output depends on the value of .
The extensions can be constructed using the additional command Extensions. Begin by setting equal to a specific prime number value.
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives. Begin by setting equal to a specific prime number value.
This page was last edited on 4 September 2012, at 20:19. | https://groupprops.subwiki.org/wiki/Second_cohomology_group_for_trivial_group_action_of_elementary_abelian_group_of_prime-square_order_on_group_of_prime_order |
TECHNICAL FIELD
BACKGROUND ART
DISCLOSURE OF THE INVENTION
BRIEF DESCRIPTION OF THE DRAWINGS
BEST MODE FOR CARRYING OUT THE INVENTION
The present invention relates to an adaptive antenna transmitting/receiving apparatus which is a transmitting/receiving apparatus using an adaptive antenna, and more particularly, to an adaptive antenna transmitting/receiving apparatus which generates transmission antenna weights based on reception antenna weights.
Generally, in mobile transmission systems, interference from other users, i.e., multi-user interference is a significant factor to limit the system capacity. Thus, for increasing the system capacity, an adaptive antenna technique is considered effective since it suppresses a received signal in a particular direction (interfering wave direction) during reception, and avoids interference in unnecessary directions during transmission.
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Fig. 1 illustrates a prior art example of a transmitting/receiving apparatus using such an adaptive antenna (adaptive antenna transmitting/receiving apparatus). As illustrated in Fig. 1, this conventional adaptive antenna transmitting/receiving apparatus comprises N antenna elements 101 - 101, N multipliers 102 - 102, adder 103, reception antenna weight generating circuit 104, antenna weight converting circuit 105, and N multipliers 106 - 106.
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Multipliers 102 - 102 perform weighted multiplications by multiplying received signals from antenna elements 101 - 101 by reception antenna weights W=(w, w, ..., w) generated by reception antenna weight generating circuit 104, respectively. Adder 103 adds together the received signals of the respective antennas weighted by multipliers 102 - 102 to deliver as single received data.
Reception antenna weight generating circuit 104 is applied with the received data delivered from adder 103, and generates weighting coefficients which permit the received data to have an optimal value as reception antenna weights W.
The operation of reception antenna weight generating circuit 104 is generally implemented by MMSE (minimum Mean Squared Error) control which adaptively updates weighting coefficients to minimize a mean squared error of weighted and added received signals and a reference signal. Known algorithms for realizing MMSE include LMS (Least Mean Square) and RLS (Recursive Least Square) algorithms, and the like. In the present invention, no limitations are particularly imposed to the algorithm used in reception antenna weight generating circuit 104.
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Reception antenna weights W=(w, w, ..., w) generated by reception antenna weight generating circuit 104 are used for weighting received signals in multipliers 102 - 102, and are also applied to antenna weight converting circuit 105.
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Antenna weight converting circuit 105 converts reception antenna weights W generated by reception antenna weight generating circuit 104 to transmission antenna weights W'=(w', w', ..., w'). This conversion is performed for correcting an amplitude/phase deviation among branches of an RF transmitting circuit, or for correcting a frequency difference of transmission/reception when transmission/reception are different in frequency as an FDD (Frequency Division Duplex) system, and is basically performed with the intention to generate transmission weights W' for forming a similar pattern to a directivity pattern formed in the reception.
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Multipliers 106 - 106 multiply data to be transmitted by transmission antenna weights W' delivered from antenna weight converting circuit 105, respectively.
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In the conventional adaptive antenna transmitting/receiving apparatus illustrated in Fig. 1, a plurality of N antenna elements 101 - 101 are arranged in an array, and weighting coefficients of respective antenna elements 101 - 101 are adaptively controlled through signal processing to conduct a control for increasing an antenna gain (directivity) in a particular direction. On the other hand, in transmission, the reception antenna weights are calibrated to generate transmission antenna weights which are multiplied by signals to be transmitted of respective users, such that the resulting signal are transmitted to have the directivity in a desired wave user direction and to reduce interference given to other user directions. An adaptive antenna transmitting/receiving apparatus in such a configuration is described in literature [1] listed below.
Literature [2] listed below shows a system for controlling reception antenna weights so as to minimize a mean squared error after RAKE combination generated using a despread pilot symbol and a determination information data symbol as reference signals, for example, as a method of calculating optimal reception weights.
[1] NTT DoCoMo Technical Journal, Vol. 8, No. 1 (Apr. 2000);
[2] S. Tanaka, M. Sawahashi and F. Adachi, "Pilot symbol-assisted decision-directed coherent adaptive array diversity for DS-CDMA mobile radio reverse link", IEICE Trans. Fundamentals, Vol. E80-A, pp.2445-2454, (Dec, 1997); and
[3] Tanaka, Harada, Ihara, Sawahashi and Adachi, "Outdoor experiment characteristics of adaptive antenna array diversity reception in W-CDMA, Technical Report RCS99-127, pp.45-50 (Oct. 1999).
Also, literature [3] listed below describes a method of using transmission antenna weights which are generated based on antenna weights generated as described above, as a method of generating transmission antenna weights.
(1) when the generation of reception antenna weights is in course of convergence, so that optimal weights have not been formed;
(2) when the reception line quality is so bad that the generation of optimal reception antenna weights encounters difficulties;
(3) when the reception line is instantaneously interrupted due to shadowing or the like;
(4) when a failure in the apparatus or the like results in defective reception, and the like,
In the conventional adaptive antenna transmitting/receiving apparatus illustrated in Fig. 1, the transmission antenna weights are generated based on the reception antenna weights generated for reception. This is intended to apply the directivity pattern formed during reception, as it is, to transmission. However, if the apparatus fails to generate optimal antenna weights, for example:
the transmission antenna weights are not either optimized. For this reason, it is contemplated that not only the directivity is not formed for a desired station, but also larger interference is given to other users by contraries, resulting in a significant degradation of the transmission characteristic as well as the reception characteristic.
Conventional techniques for controlling transmission antenna weights to have optimal values are described in JP-2000-22611-A, JP-2001-217759-A, and the like.
JP-2000-22611-A describes an adaptive antenna transmitting/receiving apparatus which comprises an incoming direction estimating circuit, and estimates an incoming direction by the incoming direction estimating circuit to correct transmission antenna weights. However, this conventional adaptive antenna transmitting/receiving apparatus cannot estimate an incoming direction if the reception line quality is degraded, resulting in the inability to control optimal transmission weights, and possible erroneous control conducted in some cases to significantly degrade the transmission characteristic.
JP-2001-217759-A in turn describes an adaptive antenna transmitting/receiving apparatus which controls transmission antenna weights based on an interference degree calculated by an interference degree calculating unit. However, this conventional adaptive antenna transmitting/receiving apparatus is similar in that it cannot calculate the interference degree itself if the reception line quality is degraded, resulting in the inability to control optimal transmission weights, and possible erroneous control conducted in some cases to significantly degrade the transmission characteristic.
It is an object of the present invention to provide an adaptive antenna transmitting/receiving apparatus which is capable of preventing the transmission characteristic from significantly degrading when optimal reception antenna weights cannot be generated.
a plurality of antenna elements;
a plurality of first multipliers for performing a weighted multiplication by multiplying received signals from the plurality of antenna elements by reception antenna weights, respectively;
an adder for adding together the received signals of the respective antennas weighted by the plurality of first multipliers to deliver as single received data;
a reception quality measuring circuit for measuring a reception quality of the received data delivered from the adder;
reception antenna weight generating circuit applied with the received data delivered from the adder for generating weighting coefficients which permit the received data to have an optimal value, as the reception antenna weights;
an antenna weight converting circuit for converting the reception antenna weights generated by the reception antenna weight generating circuit into transmission antenna weights;
a transmission antenna weight control circuit for delivering the transmission antenna weights delivered from the antenna weight converting circuit and stored therein the delivered transmission antenna weights when the reception quality measured by the reception quality measuring circuit exceeds a previously set threshold value, and for delivering the stored transmission antenna weights instead of the transmission antenna weights delivered from the antenna weight converting circuit when the reception quality is less than the threshold value; and
a plurality of second multipliers for multiplying data to be transmitted by the transmission antenna weights delivered from the transmission antenna weight control circuit, respectively.
To achieve the above object, an adaptive antenna transmitting/receiving apparatus of the present invention comprises:
According to the present invention, the reception quality circuit measures the reception quality of received data, and the transmission antenna weight control circuit does not use the transmission antenna weights generated by the antenna weight converting circuit as they are, but uses stored transmission antenna weights, i.e., the transmission antenna weights when the reception quality exceeded the threshold value at the last time, when the reception quality is less than the previously set threshold value, thus making it possible to prevent the antenna elements from being controlled with erroneous transmission antenna weights to significantly degrade the transmission characteristic.
Also, another adaptive antenna transmitting/receiving apparatus of the present invention further comprises a reception quality measuring circuit for measuring the reception quality of received data delivered from the adder, and a transmission antenna weight control circuit, wherein the transmission antenna weight control circuit delivers the transmission antenna weights delivered from the antenna weight converting circuit, as they are, when the reception quality measured by the reception quality measuring circuit exceeds the previously set threshold value, and delivers previously set values instead of the transmission antenna weights delivered from the antenna weight converting circuit, as the transmission antenna weights, when the reception quality is less than the threshold value.
According to the present invention, the reception quality circuit measures the reception quality of received data, and the transmission antenna weight control circuit does not use the transmission antenna weights generated by the antenna weight converting circuit as they are, but uses the previously set values as the transmission antennas, when the reception quality is less than the previously set threshold value, thus making it possible to prevent the antenna elements from being controlled with erroneous transmission antenna weights to significantly degrade the transmission characteristic.
Also, the transmission antenna weight control circuit may deliver the previously set values which have a value only for a weight corresponding to a particular antenna element and zero for all the remaining weights corresponding to the other antenna elements, as the transmission antenna weights.
Further, the reception quality measuring circuit may measure the reception quality based on any of a signal-to-noise power ratio, a bit error rate, and a block error rate of the received data which is the output of the adder circuit.
It is an object of the present invention to provide an adaptive antenna transmitting/receiving apparatus which is capable of preventing the transmission characteristic from significantly degrading when optimal reception antenna weights cannot be generated.
a plurality of antenna elements;
a plurality of first multipliers for performing a weighted multiplication by multiplying received signals from the plurality of antenna elements by reception antenna weights, respectively;
an adder for adding together the received signals of the respective antennas weighted by the plurality of first multipliers to deliver as single received data;
a reception quality measuring circuit for measuring a reception quality of the received data delivered from the adder;
reception antenna weight generating circuit applied with the received data delivered from the adder for generating weighting coefficients which permit the received data to have an optimal value, as the reception antenna weights;
an antenna weight converting circuit for converting the reception antenna weights generated by the reception antenna weight generating circuit into transmission antenna weights;
a transmission antenna weight control circuit for delivering the transmission antenna weights delivered from the antenna weight converting circuit and stored therein the delivered transmission antenna weights when the reception quality measured by the reception quality measuring circuit exceeds a previously set threshold value, and for delivering the stored transmission antenna weights instead of the transmission antenna weights delivered from the antenna weight converting circuit when the reception quality is less than the threshold value; and
a plurality of second multipliers for multiplying data to be transmitted by the transmission antenna weights delivered from the transmission antenna weight control circuit, respectively.
To achieve the above object, an adaptive antenna transmitting/receiving apparatus of the present invention comprises:
According to the present invention, the reception quality circuit measures the reception quality of received data, and the transmission antenna weight control circuit does not use the transmission antenna weights generated by the antenna weight converting circuit as they are, but uses stored transmission antenna weights, i.e., the transmission antenna weights when the reception quality exceeded the threshold value at the last time, when the reception quality is less than the previously set threshold value, thus making it possible to prevent the antenna elements from being controlled with erroneous transmission antenna weights to significantly degrade the transmission characteristic.
Also, another adaptive antenna transmitting/receiving apparatus of the present invention further comprises a reception quality measuring circuit for measuring the reception quality of received data delivered from the adder, and a transmission antenna weight control circuit, wherein the transmission antenna weight control circuit delivers the transmission antenna weights delivered from the antenna weight converting circuit, as they are, when the reception quality measured by the reception quality measuring circuit exceeds the previously set threshold value, and delivers previously set values instead of the transmission antenna weights delivered from the antenna weight converting circuit, as the transmission antenna weights, when the reception quality is less than the threshold value.
According to the present invention, the reception quality circuit measures the reception quality of received data, and the transmission antenna weight control circuit does not use the transmission antenna weights generated by the antenna weight converting circuit as they are, but uses the previously set values as the transmission antennas, when the reception quality is less than the previously set threshold value, thus making it possible to prevent the antenna elements from being controlled with erroneous transmission antenna weights to significantly degrade the transmission characteristic.
Also, the transmission antenna weight control circuit may deliver the previously set values which has a value only for a weight corresponding to a particular antenna element and zero for all the remaining weights corresponding to the other antenna elements, as the transmission antenna weights.
Further, the reception quality measuring circuit may measures the reception quality based on any of signal-to-noise power ratio, a bit error rate, and a block error rate of the received data which is the output of the adder circuit.
Fig. 1 is a block diagram illustrating the configuration of a conventional adaptive antenna transmitting/receiving apparatus; and
Fig. 2 is a block diagram illustrating the configuration of an adaptive antenna transmitting/receiving apparatus according to one embodiment of the present invention.
Next, one embodiment of the present invention will be described in detail with reference to the drawings.
Fig. 2 is a block diagram illustrating the configuration of an adaptive antenna transmitting/receiving apparatus according to one embodiment of the present invention. In Fig. 2, components identical to the components in Fig. 1 are designated the same reference numerals, and description thereon is omitted.
The adaptive antenna transmitting/receiving apparatus of this embodiment additionally comprises reception quality measuring circuit 107, and transmission antenna weight control circuit 108, added to the conventional adaptive antenna transmitting/receiving apparatus illustrated in Fig. 1.
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In the adaptive antenna transmitting/receiving apparatus of this embodiment, since transmission antenna weight control circuit 108 is provided between antenna weight converting circuit 105 and multipliers 106 - 106, multipliers 106 - 106 in this embodiment do not multiply data to be transmitted by transmission antenna weights W' delivered from antenna weight converting circuit 105, but multiply the data to be transmitted by transmission antenna weight W" delivered from transmission antenna weight control circuit 108.
This embodiment is characterized in that reception quality measuring circuit 107 and transmission antenna weight control circuit 108 are provided in an adaptive antenna transmitting/receiving apparatus for generating transmission antenna weights based on reception antenna weights.
Reception quality measuring circuit 107 measures the reception quality of received data delivered from adder 103 by calculating SIR (Signal-to-Noise power Ratio), BER (Bit Error Rate), BLER (BLock Error Rate), or the like for notification to transmission antenna weight control circuit 108.
Transmission antenna weight control circuit 108 is applied with transmission antenna weights W' which are the outputs of antenna weight converting circuit 105, and the reception quality which is the output of reception quality measuring circuit 107 for determining transmission antenna weights W". Specifically, transmission antenna weight control circuit 108 determines whether or not transmission antenna weights W' converted by antenna weight converting circuit 105 are actually used. The criterion for the determination used herein is the reception quality measured by reception quality measuring circuit 107.
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Transmission antenna weight control circuit 108 delivers transmission antenna weights W' from antenna weight converting circuit 105, as they are, to multipliers 106 - 106, when the reception quality measured by reception quality measuring circuit 107 exceeds a previously set threshold value.
Then, transmission antenna weight control circuit 108 conducts a control such that the transmission antenna weights, which are the outputs of antenna weight converting circuit 105, are not used as they are, when the reception quality is less than the threshold value.
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As an example of this control, previously calculated antenna weights may be continuously used as they are, for example, when the reception quality is less than a certain threshold value, such that the transmission antenna weights are not updated. In this event, transmission antenna weight control circuit 108 compares the reception quality, which is the output of reception quality measuring circuit 108, with a previously set threshold value, and delivers the transmission antenna weights, which are the outputs of antenna weight converting circuit 105, to multipliers 106 - 106 as they are, when the reception quality exceeds the threshold value, and stores therein the delivered transmission antenna weights. Then, when the reception quality is less than the threshold value, transmission antenna weight control circuit 108 delivers the stored transmission antenna weights (the transmission antenna weight when the reception quality exceeded the threshold value at the last time) instead of the transmission antenna weights which are the outputs of antenna weight converting circuit 105.
In this way, an erroneous detection in an incoming direction due to a degradation in reception characteristic can be prevented from affecting the transmission antenna weights.
1
N
Alternatively, transmission antenna weight control circuit 108 may deliver previously set values to multipliers 106 - 106 as transmission antenna weights W" instead of transmission antenna weights W' delivered from antenna weight converting circuit 105 when the reception quality measured by reception quality measuring circuit 107 is less than a previously defined threshold value.
1
2
N
The previously set values are, for example, W"=(1,0,...,0), i.e., weighting coefficients having a value only for a weight corresponding to a particular antenna element, and zero for the others, such as w=(1.0,0.0), w - w=(0.0,0.0).
Such a control, if conducted, will lose a directivity gain during transmission which has been provided when reception antenna weight generating circuit 104 is correctly estimating an incoming direction, but can avoid reception antenna weight generating circuit 104 from making an error in estimating an incoming direction to cause a danger of erroneously forming the directivity to other than a desired station upon transmission, and giving interference to other users. Also, when an instantaneous interruption extends for a long time to cause a large change in incoming direction, it is possible to prevent a degradation in characteristic, on the contrary, because the transmission directivity cannot follow. | |
---
abstract: 'In this note we study solitary wave solutions of a class of Whitham–Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnstr[ö]{}m, Groves and Wahl[é]{}n [@EGW]. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni [@Buffoni2004], Groves and Wahl[é]{}n [@MR2847283], together with the concentration-compactness principle.'
address:
- 'Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway.'
- 'Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway.'
author:
- Dag Nilsson
- Yuexun Wang
title: 'Solitary wave solutions to a class of Whitham–Boussinesq systems'
---
[^1]
Introduction {#Introduction}
============
This work is devoted to the study of solitary wave solutions of the Whitham–Boussinesq system $$\label{eq:bdw}
\begin{aligned}
\partial_t\eta &=-K\partial_xu-\partial_x(\eta u)\\
\partial_tu &=-\partial_x\eta-u\partial_xu.
\end{aligned}$$ A solitary wave is a solution of the form $$\begin{aligned}
\label{eq:travelling wave ansatz}
\eta(x,t)=\eta(x-ct),\ u(x,t)=u(x-ct),\end{aligned}$$ such that $\eta(x-ct),u(x-ct)\longrightarrow 0$ as $|x-ct|\longrightarrow \infty$. Here, $\eta$ denotes the surface elevation, $u$ is the rightward velocity at the surface, and $K$ is a Fourier multiplier operator defined by $$\label{eq:K}
\mathcal{F}(Kf)(k)=m(k)\hat{f}(k),$$ for all $f$ in the Schwartz space $\mathcal{S}(\mathbb{R})$. More specifically, we require that
- The symbol $m\in S_\infty^{m_0}({\mathbb{R}})$ for some $m_0<0$, that is $$\begin{aligned}
|m^{(\alpha)}(k)|\leq C_\alpha(1+|k|)^{m_0-\alpha},\ \alpha\in \mathbb{N}_0. \end{aligned}$$
- The symbol $m:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is even and satisfies $m(0)>0$, $m(k)<m(0),\ \text{for}\ k\neq 0$ and $$\begin{aligned}
m(k)=m(0)+\frac{m^{(2j_*)}(0)}{(2j_*)!}k^{2j_*}+r(k),\end{aligned}$$ for some $j_*\in \mathbb{N}_+$, where $m^{(2j_*)}(0)<0$ and $r(k)=\mathcal{O}(k^{2j_*+2})$ as $k\rightarrow 0$.\
As an example we have $m(k)=\tanh(k)k^{-1}$, which yields the bi-directional Whitham (BDW) system, and this choice of symbol is the main motivation for studying . The BDW system was formally derived in [@MR2991247; @MR3390078] from the incompressible Euler equations to model fully dispersive shallow water waves whose propagation is allowed to be both left- and rightward, and appeared in [@MR3060183; @MR3668593] as a full dispersion system in the Boussinesq regime with the dispersion of the water waves system. There have been several investigations on the BDW system: local well-posedness [@2017arXiv170804551E; @MR3763731] (in homogeneous Sobolev spaces at a positive background), a logarithmically cusped wave of greatest height [@Ehrnström2018]. There are also numerical results, investigating the validity of the BDW system as a model of waves on shallow water [@MR3844340], numerical bifurcation and spectral stability [@doi:10.1111/sapm.12221] and the observation of dispersive shock waves [@MR3523508]. However there are no results on the existence of solitary wave solutions.
We also mention that one can include the effects of surface tension in the BDW system by choosing $m(k)=\tanh(k)k^{-1}(1+\beta k^2), \beta>0$. It was recently shown in [@2018arXiv180504372K] that is locally well-posed for this choice of symbol. However, the above symbol with $\beta>0$ is not included in the class of symbols considered in the present work. Moreover, in [@MR3749383; @DDK; @ED], other types of fully dispersive Whitham-Boussinesq systems are considered. We also mention the generalized class of Green–Nagdhi equations introduced in [@MR3564304], which was shown to posses solitary wave solutions in [@MR3841973].
Solitary wave solutions to the Whitham equation
===============================================
In order to prove existence of solitary wave solutions of our strategy will be to reduce this to a problem that is similar to one studied in [@EGW]. For this reason we first discuss the results and methods of that paper. In [@EGW] the authors prove the existence of solitary wave solutions of the pseudodifferential equation $$\label{EGW-eq}
u_t+\big(Ku+\tilde{n}(u)\big)_x=0,$$ where $K$ have properties $(A1)$, $(A2)$ and the nonlinearity $\tilde{n}$ satisfies
- The nonlinearity $\tilde{n}$ is a twice continuously differentiable function $\mathbb{R}\rightarrow \mathbb{R}$ with $$\label{egw}
\tilde{n}(x)=\tilde{n}_p(x)+\tilde{n}_r(x),$$ in which the leading order part of the nonlinearity takes the form $\tilde{n}_p(x)=c_p{\lvertx\rvert}^p$ for some $c_p\neq 0$ and $p\in [2,4j_*+1)$ or $\tilde{n}_p(x)=c_px^p$ for some $c_p>0$ and odd integer $p$ in the range $p\in [2,4j_*+1)$, while $$\tilde{n}_r(x)=\mathcal{O}({\lvertx\rvert}^{p+\delta}),\quad \tilde{n}_r'(x)=\mathcal{O}({\lvertx\rvert}^{p+\delta-1})$$ for some $\delta>0$ as $x\rightarrow 0$.
In particular, the uni-directional Whitham equation, introduced in [@MR0671107], belongs to this class of equations , with $m(k)=\sqrt{\tanh(k)k^{-1}}$. The Whitham equation possesses periodic travelling waves [@MR2555644] and solitary waves [@EGW], moreover the solitary waves decay exponentially [@MR3603270]. It was recently confirmed that the Whitham equation possesses a highest cusped wave [@2016arXiv160205384E], as conjectured by Whitham.
Under the travelling wave ansatz: $u(t,x)=u(x-c t)$, the equation becomes $$\label{travellingwave-EGW}
Ku-c u+\tilde{n}(u)=0.$$ The existence of solutions of is established via a related minimization problem. Let $$\tilde{\mathcal{E}}(u)=-\frac{1}{2}\int_\mathbb{R}uKu\ \mathrm{d}x-\int_\mathbb{R}\tilde{N}(u)\ \mathrm{d}x,\quad \mathcal{I}(u)=\frac{1}{2}\int_\mathbb{R}u^2\ \mathrm{d}x$$ with $$\begin{aligned}
\tilde{N}(x)&=\tilde{N}_{p+1}(x)+\tilde{N}_r(x),\\
\tilde{N}_{p+1}(x)&=\int_0^x\tilde{n}_p(s)\ \mathrm{d}s=\frac{ c_px^{p+1}}{p+1}, \text{ or }\frac{ c_px{\lvertx\rvert}^{p}}{p+1}, \\
\tilde{N}_r(x)=&\int_0^x\tilde{n}_r(s)\ \mathrm{d}s=\mathcal{O}({\lvertx\rvert}^{p+1+\delta}).\end{aligned}$$ Let $q,R>0$ and $$V_{q,R}:=\{u\in H^1(\mathbb{R}):\ \mathcal{I}(u)=q,\ {\left\lVertu\right\rVert}_{H^1}<R\}.$$ Minimizers of $\tilde{\mathcal{E}}$ over $V_{q,R}$ (that are not on the boundary) satisfy the Euler-Lagrange equation $$\label{EL-egw}
\mathrm{d}\tilde{\mathcal{E}}(u)+\nu\mathrm{d}\mathcal{I}(u)=0,$$ for a Lagrange multiplier $\nu$, and is precisely , with $c=\nu$. In [@EGW] the authors show that there exist solutions of the minimization problem $$\arg \underset{V_{q,R}} \inf \tilde{\mathcal{E}}(u),$$ which by the above argument yields travelling wave solutions of . The existence of minimizers is established using methods developed in [@Buffoni2004; @MR2847283] and we give here a brief outline of the proof. The functional $\tilde{\mathcal{E}}$ is not coercive and since the domain is unbounded one cannot use the Rellich–Kondrachov theorem. In particular, direct methods cannot be used to obtain a minimizer. Because of this one needs to study a related penalized functional acting on periodic functions. Let $P>0$ and $L_P^2$ be the space of $P$- periodic, locally square-integrable functions with Fourier-series representation $$w(x)=\frac{1}{\sqrt{P}}\sum_{k\in{\mathbb{Z}}}\widehat{w}(k)\exp(2\pi ikx/P),$$ with $$\widehat{w}(k):=\frac{1}{\sqrt{P}}\int_{-\frac{P}{2}}^{\frac{P}{2}}w(x)\exp(-2\pi ikx/P)\,{\mathrm{d}}x.$$ For $s\geq0$, we define $$H^s_P:=\{w\in L^2_P:\ \|w\|_{H^s_P}<\infty\},$$ where the norm is given by $$\|w\|_{H^s_P}:=\left(\sum_{k\in {\mathbb{Z}}}\bigg(1+\frac{4\pi^2k^2}{P^2}\bigg)^s|\widehat{w}(k)|^2 \right)^{\frac{1}{2}}.$$ The authors [@EGW] studied the following penalized functional $$\tilde{\mathcal{E}}_{P,\varrho}(u):=\varrho({\left\lVertu\right\rVert}_{H_P^1}^2)+\tilde{\mathcal{E}}_p(u),$$ over the set $$V_{P,q,R}:=\{u\in H_P^1:\ \mathcal{I}_P(u)=q, \ {\left\lVertu\right\rVert}_{H_P^1}<2R\},$$ where $\tilde{\mathcal{E}}_P, \ \tilde{\mathcal{I}}_P$ are the same functionals as $\tilde{\mathcal{E}},\ \tilde{\mathcal{I}}$ but where the integration is over $[-P/2,P/2]$, and $\varrho:[0,(2R)^2]\mapsto [0,\infty)$ is a penalization function such that $\varrho(t)=0$ whenever $t\in[0,R^2]$ and $\varrho(t)\rightarrow \infty$ as $t\rightarrow (2R)^2$. The penalization function makes $\tilde{\mathcal{E}}_{P,\varrho}$ coercive, and the fact that we are now working in $H_P^1$ allows the use of the Rellich-Kondrachov theorem. It is then an easy task to show that there exists a minimizer $u_P\in V_{P,q,2R}$, of $\tilde{\mathcal{E}}_{P,\varrho}$. The next step is to show that $u_P$ in fact minimizes $\tilde{\mathcal{E}}_P$ over $V_{q,R}$. This is immediate after showing that $$\label{main-ineq}
{\left\lVertu_P\right\rVert}_{H_P^1}^2\leq Cq,$$ and choosing $q$ sufficiently small. The other key ingredient of the proof is the concentration compactness theorem [@MR778974]. In the application of this theorem, the main task is to show that ‘dichotomy’ does not occur. This is done using proof by contradiction, where the contradiction is arrived at using the strict subadditivity of $$I_q:=\arg\underset{V_{q,R}}\inf\tilde{\mathcal{E}}(u),$$ as a function of $q$. The strict subadditivity of $I_q$ is established by using a special minimizing sequence for $\tilde{\mathcal{E}}$, constructed from the minimizers $u_P$. In addition it is necessary to decompose $u$ into high and low frequencies in order to get satisfactory estimates on ${\left\lVertu\right\rVert}_{L^\infty}$, see [@EGW Corollary 4.5]. It is an easy task to show that ‘vanishing’ cannot occur either. Therefore, from the concentration compactness theorem, ‘concentration’ is the only possibility and the existence of minimizers then follows from a standard argument.
Under the additional assumption that
- $\tilde{n}\in C^{2j_*}(\mathbb{R})$ with $$\tilde{n}_r^{(j)}(x)=\mathcal{O}({\lvertx\rvert}^{p+\delta-j}),\ j=0,\ldots, ,2j_*,$$
it is possible to relate the minimizers of $\tilde{\mathcal{E}}$ to those of $\tilde{\mathcal{E}}_{lw}$, where $$\tilde{\mathcal{E}}_{lw}(u)=-\int_\mathbb{R}\left(\frac{m^{(2j_*)}(0)}{2(2j_*)!}(u^{(j_*)})^2+ \tilde{N}_{p+1}(u)\right)\, \mathrm{d}x.$$ More specifically, $$\sup_{u\in \tilde{D}_q}\text{dist}_{H^{j_*}(\mathbb{R})}(S_{lw}^{-1}u,\tilde{D}_{lw})\rightarrow 0,\quad \text{ as }q\rightarrow 0,$$ where $\tilde{D}_{lw}$ is the set of minimizers of $\tilde{\mathcal{E}}_{lw}$ over the set $$\{u\in H^{j_*}(\mathbb{R}):\ \mathcal{I}(u)=1\},$$ and $\tilde{D}_q$ is the set of minimizers of $\tilde{\mathcal{E}}$ over $V_{q,R}$ and $$(S_{lw}u)(x):=q^\alpha u(q^\beta x)$$ is the ’long-wave test function’ with $$\label{alpha-beta}
\alpha=\frac{2 j_*}{4j_*+1-p},\quad \beta=\frac{p-1}{4j_*+1-p}.$$ The numbers $\alpha$ and $\beta$ are chosen in such a way that $$2\alpha-\beta=1,\quad (p-1)\alpha=2j_*\beta.$$ This choice of $\alpha$, $\beta$ appear naturally when deriving the long-wave approximation of . The functional $\tilde{\mathcal{E}}_{lw}$ is related to $\tilde{\mathcal{E}}$ via (see [@EGW Lemma 3.2]) $$\tilde{\mathcal{E}}(S_{lw}u)=-qm(0)+q^{1+(p-1)\alpha}\tilde{\mathcal{E}}_{lw}(u)+o(q^{1+(p-1)\alpha}),$$ for any $u\in W:=\{u\in H^{2j_*}({\mathbb{R}}):\ {\left\lVertu\right\rVert}_{H^{2j_*}}<S\}$ with $S$ being a positive constant.
We mention here a recent work [@arXiv:1802.10040] where they use an entirely different approach to prove the existence of small amplitude solitary wave solutions of the Whitham equation.
Solitary wave solutions to the Whitham–Boussinesq system {#sec-bi-directional-whitham}
========================================================
Formulation as a constrained minimization problem
-------------------------------------------------
In the present work we seek solitary wave solutions of , and the idea is to reformulate in such a way that the method of [@EGW] can be applied. Under the travelling wave ansatz , the system then becomes $$\begin{aligned}
c\eta &=Ku+\eta u, \label{2}\\
cu &=\eta+\frac{u^2}{2} \label{3}.\end{aligned}$$ It follows from that $\eta=u(c-\frac{u}{2})$, and if we insert this into then we find that $$\label{4}
Ku-u(u-c)(\frac{u}{2}-c)=0.$$
We first formally assume that $\|u\|_{L^\infty}\ll c $ to formulate into a variational problem. This is no restriction since the constructed solutions will automatically satisfy this smallness condition (see Theorem ). Let $w=\frac{u}{c}(\frac{u}{c}-2)$, so that $u=c-c\sqrt{1+w}$. The map $w \mapsto u$ is well-defined, since $$\begin{aligned}
{\left\lVertw\right\rVert}_{L^\infty}\leq {\left\lVert\frac{u}{c}\right\rVert}_{L^\infty}{\left\lVert\frac{u}{c}-2\right\rVert}_{L^\infty}\lesssim {\left\lVert\frac{u}{c}\right\rVert}_{L^\infty}\ll1,\\\end{aligned}$$ We then may rewrite the equation using the new unknown $w$ as $$\begin{aligned}
\label{vareq}
\frac{2}{\sqrt{1+w}}K(\sqrt{1+w}-1)-\lambda w=0,\end{aligned}$$ with $\lambda=c^2$. We now define $$\mathcal{E}(w)=\underbrace{-\frac{1}{2}\int_\mathbb{R}wKw\ \mathrm{d}x}_{:= \mathcal{K}(w)}\underbrace{-\int_\mathbb{R}N(w)\ \mathrm{d}x}_{:=\mathcal{N}(w)},$$ where $$\begin{aligned}
N(w)=2\Psi(w)Kw+2\Psi(w)K(\Psi(w)), \end{aligned}$$ $$\begin{aligned}
\Psi(w)=\sqrt{1+w}-1-\frac{w}{2}=-\frac{w^2}{8}+\Psi_r(w),\end{aligned}$$ $$\begin{aligned}
\ \Psi_r(x)=\mathcal{O}(x^3).\end{aligned}$$ To extract the lower-order parts we also write $$N(w)=N_h(w)+N_l(w),$$ with $$N_h(w)=-\frac{w^2}{4}Kw,\quad N_l(w)=2\Psi(w)Kw+2\Psi(w)K(\Psi(w)).$$ We then note that $${\mathrm{d}}\mathcal{E}(w)+\lambda {\mathrm{d}}\mathcal{I}(w)=0$$ is precisely . Hence, $w$ is a critical point of $\mathcal{E}$ under the constraint $\mathcal{I}(w)=q$, if and only if $u=c-c\sqrt{1+w}$ is a solution of , with $\lambda=c^2$. We will find critical points of $\mathcal{E}(w)+\lambda \mathcal{I}(w)$ by considering the minimization problem $$\arg \underset{V_{q,R}} \inf \mathcal{E}(w).$$ Here we are minimizing a functional $\mathcal{E}$ of almost the same type as in [@EGW], with $p=2$, but with a slightly different nonlinearity. In our case, the nonlocal operator $K$ appears in the nonlinear term $N$. However, since $K$ is a bounded smoothing operator, it is not hard to show that the methods used in [@EGW] can be applied to the functional $\mathcal{E}$. However, the results [@EGW Lemma 2.3, Lemma 3.2, Lemma 3.3] require a bit more care, in particular it is important to know how $\mathcal{N}$ acts under the long-wave scaling, and we therefore include the proofs of these results in the next subsection. We finally have the following existence result:
\[th:main\] There exists $q_*>0$ such that the following statements hold for each $q\in(0,q_*)$.
($\romannumeral1$) The set $D_q$ of minimizers of $\mathcal{E}$ over the set $V_{q,R}$ is nonempty and the estimate $\|w\|_{H^1({\mathbb{R}})}^2={\mathcal{O}}(q)$ holds uniformly over $w\in D_q$. Each element of $D_q$ is a solution of the travelling wave equation ; the squared wave speed $c^2$ is the Lagrange multiplier in this constrained variational principle.
($\romannumeral2$) Let $s<1$ and suppose that $\{w_n\}_{n\in\mathbb{N}_0}$ is a minimizing sequence for $\mathcal{E}$ over $V_{q,R}$. There exists a sequence $\{x_n\}_{n\in\mathbb{N}_0}$ of real numbers such that a subsequence of $\{w_n(\cdot+x_n)\}_{n\in\mathbb{N}_0}$ converges in $H^s({\mathbb{R}})$ to a function in $D_q$.
Technical results
-----------------
In our case the long-wave functional $\mathcal{E}_{lw}$ is given by $$\begin{aligned}
\mathcal{E}_{lw}(w):=-\int_{\mathbb{R}}\left(\frac{m^{2j_*}(0)}{2(2j_*)!}(w^{(j_*)})^2-\frac{m(0)}{4}w^3\right){\mathrm{d}}x,\end{aligned}$$ and we also recall the long-wave scaling: $$S_{lw}w(x)=\mu^\alpha w(\mu^\beta x),$$ with $$\label{alphabeta_specialcase}
\alpha=\frac{2j_*}{4j_*-1} \quad \text{and} \quad \beta=\frac{1}{4j_*-1}.$$ Note that is a special case of , with $p=2$.
We first present a result corresponding to [@EGW Lemma 3.2], which relates $\mathcal{E}$ with $\mathcal{E}_{lw}$.
\[le:long wave\] Let $w\in W$ with ${\left\lVertw\right\rVert}_{L^\infty}\ll 1$ and $\mathcal{I}(w)=1$. Then $$\begin{aligned}
\label{eq:long wave test}
\mathcal{E}(S_{lw}w)=-q m(0)+q^{1+\alpha}\mathcal{E}_{lw}(w)+o(q^{1+\alpha}).
\end{aligned}$$
Recall the definition $$\mathcal{E}(S_{lw}w)=\mathcal{K}(S_{lw}w)+\mathcal{N}(S_{lw}w).$$ We first calculate that $$\begin{aligned}
&\mathcal{K}(S_{lw}w)\\
&=-\frac{1}{2}\int_{\mathbb{R}}q^{2\alpha}w(q^\beta x)Kw(q^\beta \cdot)(x)\, \mathrm{d}x\\
&=-\frac{1}{2}\int_\mathbb{R}q^{2\alpha}m(k){\lvert\mathcal{F}(w(q^\beta\cdot))(k)\rvert}^2\, \mathrm{d}k\\
&=-\frac{1}{2}\int_\mathbb{R}q^{2\alpha-\beta}\left(m(0)+q^{2j_*\beta}\frac{m^{(2j_*)}(0)}{(2j_*)!}k^{2j_*}+r(q^\beta k)\right){\lvert\hat{w}(k)\rvert}^2\ \mathrm{d}k\\
&=-q m(0)-q^{2\alpha+(2j_*-1)\beta}\int_\mathbb{R}\frac{m^{(2j_*)}(0)}{2(2j_*)!}(w^{j_*})^2\ \mathrm{d}x-\frac{q^{2\alpha-\beta}}{2}\int_\mathbb{R}r(q^\beta k){\lvert\hat{w}(k)\rvert}^2\, \mathrm{d}k,\end{aligned}$$ and one may continuously estimate the last term as $${\lvert\frac{q^{2\alpha-\beta}}{2}\int_\mathbb{R}r(q^\beta k){\lvert\hat{w}(k)\rvert}^2\, \mathrm{d}k\rvert}\lesssim q^{2\alpha+(2j_*+1)\beta}\int_\mathbb{R}k^{2j_*+2}{\lvert\hat{w}(k)\rvert}^2\, \mathrm{d}k,$$ and $\int_\mathbb{R}k^{2j_*+2}{\lvert\hat{w}(k)\rvert}^2\ \mathrm{d}k$ is uniformly bounded, since $w\in W$. We next consider $$\mathcal{N}(S_{lw}w)=-\int_\mathbb{R}N_h(S_{lw}w)+N_l(S_{lw}w)\, \mathrm{d}x.$$ A direct calculation shows that $$\begin{aligned}
&-\int_\mathbb{R}N_h(S_{lw}w)\ \mathrm{d}x=\int_{\mathbb{R}}\frac{q^{3\alpha}}{4}w^2(q^\beta x)Kw(q^\beta\cdot)(x)\ \mathrm{d}x\\
&=\int_\mathbb{R}\frac{q^{3\alpha-\beta}}{4}\overline{\mathcal{F}(w^2)(k)}\hat{w}(k)\left(m(0)+q^{2j_*\beta}\frac{m^{(2j_*)}(0)}{(2j_*)!}k^{2j_*}+r(q^\beta k)\right)\, \mathrm{d}k\\
&=q^{3\alpha-\beta}\int_{\mathbb{R}}\frac{m(0)}{4}w^3\, \mathrm{d}x+o(q^{3\alpha-\beta}),\end{aligned}$$ where we again used that $w\in W$ in order to estimate the remaining terms. The term $\int_\mathbb{R}N_l(S_{lw}w)\ \mathrm{d}x$ is of lower order and can be estimated in the same way.
Combining all the above estimates yields the identity .
We next move to the corresponding result of [@EGW Lemma 3.2].
\[lemma:approximate\] Let $$\begin{aligned}
\mathcal{K}_P(w)=-\frac{1}{2}\int_{-\frac{P}{2}}^{\frac{P}{2}}wKw\ \mathrm{d}x,\quad \mathcal{N}_P(w)=-\int_{-\frac{P}{2}}^{\frac{P}{2}}N(w)\ \mathrm{d}x,\end{aligned}$$ $$\begin{aligned}
\ \mathcal{E}_P(w)=\mathcal{K}_P(w)+\mathcal{N}_P(w),\end{aligned}$$ and let $\{\tilde{w}_P\}$ be a bounded family of functions in $H^1(\mathbb{R})$ with ${\left\lVert\tilde{w}_P\right\rVert}_{L^\infty(\mathbb{R})}\ll1$ such that $$\mathrm{supp}(\tilde{w}_P)\subset (-\frac{P}{2},\frac{P}{2}) \quad \mathrm{and}\quad \mathrm{dist}\big(\pm \frac{P}{2},\mathrm{supp}(\tilde{w}_P)\big)\geq\frac{1}{2}P^\frac{1}{4},$$ and define $w_P\in H_P^1$ by the formula $$w_P=\sum_{j\in\mathbb{Z}}\tilde{w}_P(\cdot +jP).$$
- The function $w_P$ satisfies $$\lim_{P\rightarrow \infty}{\left\lVertK\tilde{w}_P-Kw_P\right\rVert}_{H^1(-\frac{P}{2},\frac{P}{2})}=0,\quad \lim_{P\rightarrow \infty}{\left\lVertK\tilde{w}_P\right\rVert}_{H^1({\lvertx\rvert}>\frac{P}{2})}=0.$$
- The functionals $\mathcal{E}$, $\mathcal{I}$ and $\mathcal{E}_P$, $\mathcal{I}_P$ have the properties that $$\lim_{P\rightarrow \infty}\big(\mathcal{E}(\tilde{w}_P)-\mathcal{E}_P(w_P)\big)=0,\quad \mathcal{I}(\tilde{w}_P)=\mathcal{I}_P(w_P),$$ and $$\begin{aligned}
&\lim_{P\rightarrow \infty}{\left\lVert\mathcal{E}'(\tilde{w}_P)-\mathcal{E}'_P(w_P)\right\rVert}_{H^1(-\frac{P}{2},\frac{P}{2})}=0, &&\lim_{P\rightarrow \infty}{\left\lVert\mathcal{E}'(\tilde{w}_P)\right\rVert}_{H^1(-\frac{P}{2},\frac{P}{2})}=0\\
&{\left\lVert\mathcal{I}'(\tilde{w}_P)-\mathcal{I}'_P(w_P)\right\rVert}_{H^1(-\frac{P}{2},\frac{P}{2})}=0,&& {\left\lVert\mathcal{I}'(\tilde{w}_P)\right\rVert}_{H^1({\lvertx\rvert}>\frac{P}{2})}=0.\end{aligned}$$
To prove Lemma \[lemma:approximate\], we need the following technical result of [@EGW Proposition 2.1].
\[proposition:technical\] The linear operator $K$ satisfies\
(a) $K$ belongs to $C^\infty(H^s({\mathbb{R}}),H^{s+|m_0|}({\mathbb{R}}))$ $\cap$ $C^\infty(\mathcal{S}({\mathbb{R}}),\mathcal{S}({\mathbb{R}}))$ for each $s\geq0$.\
(b) For each $j\in {\mathbb{N}}$ there exists a constant $C_l=C(\|m^{(l)}\|_{L^2({\mathbb{R}})})>0$ such that $$|Kf(x)|\leq \frac{C_l\|f\|_{L^2}}{ \mathrm{dist}\big(x,\operatorname{supp}(f)\big)^l},\quad x\in {\mathbb{R}}\setminus \operatorname{supp}(f),$$ for all $f\in L^2_c({\mathbb{R}})$.
The limits in ($\romannumeral1$) are proved in [@EGW Proposition 2.1], so we turn to ($\romannumeral2$). Using ($\romannumeral1$) we get that $\mathcal{K}(\tilde{w}_P)-\mathcal{K}(w_P)\rightarrow 0$, as $P\rightarrow \infty$. Note that $$\label{estimate:N, step 1}
\begin{aligned}
\mathcal{N}(\tilde{w}_P)&=-2\int_\mathbb{R}\Psi(\tilde{w}_P)K\tilde{w}_P+\Psi(\tilde{w}_P)K(p(\tilde{w}_P))\ \mathrm{d}x\\
&=-2\int_{-\frac{P}{2}}^{\frac{P}{2}}\Psi(w_P)K\tilde{w}_P+\Psi(w_P)K(\Psi(\tilde{w}_P))\ \mathrm{d}x\\
&=-2\int_{-\frac{P}{2}}^\frac{P}{2}\Psi(w_P)K(\tilde{w}_P-w_P)+\Psi(w_P)K\big(\Psi(\tilde{w}_P)-\Psi(w_P)\big)\ \mathrm{d}x\\
&\quad+\mathcal{N}_P(w_P).
\end{aligned}$$ In light of ($\romannumeral1$) we have $$\label{estimate:N, step 2}
\begin{aligned}
&\bigg|\int_{-\frac{P}{2}}^{\frac{P}{2}}\Psi(w_P)K(\tilde{w}_P-w_P)\ \mathrm{d}x\bigg|\\
&\leq {\left\lVert\Psi(w_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}{\left\lVertK(\tilde{w}_P-w_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\rightarrow 0, \quad \text{as}\ P\longrightarrow \infty.
\end{aligned}$$
Since ${\left\lVert\tilde{w}_P\right\rVert}_{L^\infty}\ll1$, we have ${\left\lVertw_P\right\rVert}_{L^\infty}\ll1$. To estimate the second term on the right hand side of , one first calculates $$\begin{aligned}
\Psi(\tilde{w}_P)-\Psi(w_P)&=\sqrt{1+\tilde{w}_P}-\sqrt{1+\sum_{j\in\mathbb{Z}}\tilde{w}_P(\cdot+jP)}+\frac{1}{2}\sum_{{\lvertj\rvert}\geq 1}\tilde{w}_P(\cdot jP)\\
&=-\frac{\sum_{{\lvertj\rvert}\geq 1}\tilde{w}_P(\cdot +jP)}{\sqrt{1+\tilde{w}_P}+\sqrt{1+w_P}}+\frac{1}{2}\sum_{{\lvertj\rvert}\geq 1}\tilde{w}_P(\cdot +jP)\\
&=\left(\frac{1}{2}-\frac{1}{\sqrt{1+\tilde{w}_P}+\sqrt{1+w_P}}\right)\sum_{{\lvertj\rvert}\geq 1}\tilde{w}_P(\cdot +jP),\end{aligned}$$ and then applies Proposition \[proposition:technical\] to get $$\label{estimate:N, step 3}
\begin{aligned}
&\int_{-\frac{P}{2}}^\frac{P}{2}\big|K\big(\Psi(\tilde{w}_P)-\Psi(w_P)\big)\big|^2\, \mathrm{d}x\\
&\leq \int_{-\frac{P}{2}}^\frac{P}{2}\bigg|\sum_{{\lvertj\rvert}\geq 1}K\left[\tilde{w}_P(\cdot +jP)\big(\frac{1}{2}-\frac{1}{\sqrt{1+\tilde{w}_P}+\sqrt{1+w_P}}\big)\right]\bigg|^2\, \mathrm{d}x\\
&\lesssim \int_{-\frac{P}{2}}^\frac{P}{2}\left(\sum_{{\lvertj\rvert}\geq 1}\frac{{\left\lVert\tilde{w}_P(\cdot +jP)\left(\frac{1}{2}-\frac{1}{\sqrt{1+\tilde{w}_P}+\sqrt{1+w_P}}\right)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}}{\text{dist}\big(x+jP,\text{supp}(\tilde{w}_P)\big)^3}\right)^2\, \mathrm{d}x\\
&\lesssim \|\tilde{w}_P\|_{L^2}\int_{-\frac{P}{2}}^\frac{P}{2}\big(\sum_{{\lvertj\rvert}\geq 1}\frac{1}{(jP+\frac{1}{2}P^{\frac{1}{4}})^3}\big)^2\, \mathrm{d}x\\
&\rightarrow 0, \quad \text{as}\ P\longrightarrow \infty.
\end{aligned}$$ Hence we obtain $$\label{estimate:N, step 4}
\begin{aligned}
&\bigg|\int_{-\frac{P}{2}}^\frac{P}{2}\Psi(w_P)K\big(\Psi(\tilde{w}_P)-\Psi(w_P)\big)\ \mathrm{d}x\bigg|\\
&\leq{\left\lVert\Psi(w_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}{\left\lVertK\big(\Psi(\tilde{w}_P)-\Psi(w_P)\big)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\rightarrow 0, \quad \text{as}\ P\longrightarrow \infty.
\end{aligned}$$ From , and , it follows that $\mathcal{N}(\tilde{w}_P)-\mathcal{N}_P(w_P)\rightarrow 0$, which in turn implies that $$\mathcal{E}(\tilde{w}_P)-\mathcal{E}_P(w_P)\rightarrow 0, \quad \text{as}\ P\longrightarrow \infty.$$
The equality $\mathcal{I}(\tilde{w}_P)=\mathcal{I}_P(w_P)$ is immediate.
A direct calculation yields $$\mathcal{N}'(w)=-\left(\frac{1}{\sqrt{1+w}}-1\right)Kw-\frac{2}{\sqrt{1+w}}K(\Psi(w)),$$ so we may estimate $$\begin{aligned}
&{\left\lVert\mathcal{N}'(\tilde{w}_P)-\mathcal{N}'_P(w_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\\
&\leq {\left\lVert\left(\frac{1}{\sqrt{1+w_P}}-1\right)(K\tilde{w}_P-Kw_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\\
&\quad +{\left\lVert\frac{2}{\sqrt{1+w_P}}K\big(\Psi(\tilde{w}_P)-\Psi(w_P)\big)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\\
&\rightarrow 0, \quad \text{as}\ P\longrightarrow \infty, \end{aligned}$$ where we have used ($\romannumeral1$) and . One can similarly show that $${\left\lVert\frac{{\mathrm{d}}}{{\mathrm{d}}x}\mathcal{N}'(\tilde{w}_P)-\frac{{\mathrm{d}}}{{\mathrm{d}}x}\mathcal{N}'_P(w_P)\right\rVert}_{L^2(-\frac{P}{2},\frac{P}{2})}\rightarrow 0,\quad \text{as}\ P\rightarrow \infty.$$ Hence $${\left\lVert\mathcal{E}'(\tilde{w}_P)-\mathcal{E}'_P(w_P)\right\rVert}_{H^1(-\frac{P}{2},\frac{P}{2})}\rightarrow 0,\quad \text{as}\ P\rightarrow \infty.$$
Note that $\frac{1}{\sqrt{1+\tilde{w}_P}}-1=0$ for ${\lvertx\rvert}>\frac{P}{2}$, we calculate $$\begin{aligned}
&{\left\lVert\mathcal{N}'(\tilde{w}_P)\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}\\
&={\left\lVert\left(\frac{1}{\sqrt{1+\tilde{w}_P}}-1\right)K\tilde{w}_P+\frac{2}{\sqrt{1+\tilde{w}_P}}K(\Psi(\tilde{w}_P))\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}\\
&={\left\lVert\frac{2}{\sqrt{1+\tilde{w}_P}}K(\Psi(\tilde{w}_P))\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}.\end{aligned}$$ Since $\operatorname{supp}(\Psi(\tilde{w}_P))=\operatorname{supp}(\tilde{w}_P)$, we have ${\left\lVertK(\Psi(\tilde{w}_P))\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}\rightarrow 0$. It follows that $${\left\lVert\mathcal{N}'(\tilde{w}_P)\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}\rightarrow 0,\quad \text{as}\ P\rightarrow \infty.$$ A similar calculation shows that $${\left\lVert\frac{{\mathrm{d}}}{{\mathrm{d}}x}\mathcal{N}'(\tilde{w}_P)\right\rVert}_{L^2({\lvertx\rvert}>\frac{P}{2})}\rightarrow 0.$$ Consequently, we have $${\left\lVert\mathcal{N}'(\tilde{w}_P)\right\rVert}_{H^1({\lvertx\rvert}>\frac{P}{2})}\rightarrow 0, \quad \text{as}\ P\rightarrow \infty.$$
Just as in [@EGW Theorem 6.3] we can relate the minimizers of $\mathcal{E}$ with those of $\mathcal{E}_{lw}$: $$\sup_{w\in D_q}\text{dist}_{H^{j_*}(\mathbb{R})}(S_{lw}^{-1}w,D_{lw})\rightarrow 0,\quad \text{ as }q\rightarrow 0,$$ where $D_{lw}$ is the set of minimizers of $\mathcal{E}_{lw}$ over the set $$\{w\in H^{j_*}(\mathbb{R}):\ \mathcal{I}(w)=1\},$$ and $D_q$ is the set of minimizers of $\mathcal{E}$ over $V_{q,R}$. We finally include a regularity result for the travelling wave solutions of which corresponds to [@EGW Lemma 2.3].
\[le:regularity\] Let $w$ be a solution of in with ${\left\lVertw\right\rVert}_{L^\infty}\ll1$. Then for any $k\in \mathbb{N}_+$, $w\in H^{k}$ and satisfies $$\|w\|_{H^k}\leq C(k,\|w\|_{H^1}).$$
Let $$f=\sqrt{1+w}-1,$$ then one has $\|f\|_{L^\infty}\ll1$ due to ${\left\lVertw\right\rVert}_{L^\infty}\ll1$. In view of , $f$ solves $$\label{eqf}
f=\frac{2}{\lambda (1+f)(2+f)}Kf.$$ Differentiating in yields $$\begin{aligned}
\label{eq: direvative}
\partial_xf=\frac{2}{\lambda [(1+f)(2+f)+f(2+f)+f(1+f)]}K\partial_xf.
\end{aligned}$$ The denominator is positive due to $\|f\|_{L^\infty}\ll1$.
Let $l\in \{1,2,\cdots,k\}$. For each fixed $f\in H^l$ we define a formula $\phi_f$ by $$\phi_f(g)=\frac{2}{\lambda [(1+f)(2+f)+f(2+f)+f(1+f)]}g.$$ Then one now may follow the argument in \[EGW, Lemma 2.3\] by using the properties of $\phi_f$ and $K$ to show $$\begin{aligned}
\|\partial_xf\|_{H^l}\leq C(\|f\|_{H^1})\|\partial_xf\|_{L^2}.
\end{aligned}$$
For completeness, we give its proof here. For any $s\in [0,l]$, it is easy to see that $\phi_f$ and $K$ define an operator in $B(H^s,H^s)$ and $B(H^s,H^{s+|m_0|})$, respectively. Thus the composition $$\psi_f=\phi_f\circ K\in B(H^s,H^{s_*}),\quad s_*=\min\{l,s+|m_0|\},$$ and the norm of $\psi_f$ depends upon $\|f\|_{H^l}$. Consequently, any solution $g$ of $g=\psi_f(g)$ belongs to $H^{s_*}$ and satisfies $$\|g\|_{H^{s_*}}\leq C_{l,\|f\|_{H^l}}\|g\|_{H^s}.$$ Applying this argument recursively, one finds that any solution $g\in L^2$ belongs to $H^l$ and satisfies $$\|g\|_{H^l}\leq C(l,\|f\|_{H^l})\|g\|_{L^2}.$$
Since is equivalent to $\partial_xf=\psi_f(\partial_xf)$, a bootstrap argument shows that $f^{\prime}\in H^l$ with $$\|\partial_xf\|_{H^l}\leq C(l,\|f\|_{H^1})\|\partial_xf\|_{L^2},\ l=1,2,\cdots,k.$$ So far we have shown that $$\|f\|_{H^k}\leq C(k,\|f\|_{H^1}).$$ Finally, recalling that $w=f^2+2f$ and $H^l$ is an algebra, we therefore obtain $$\|w\|_{H^k}\leq C(k,\|f\|_{H^1})\leq C(k,\|w\|_{H^1}),$$ where we have used $\|w\|_{L^\infty}\ll1$ in the last inequality.
The results of the present work may be extended to a more general class of nonlinearities $N$. On the one hand, we have that the leading order part of $N$ is cubic, but this could be extended to higher power nonlinearities. On the other hand, the multiplier operator $K$ appearing in $N$ can be replaced by an operator $K'$ belonging to a wider class of Fourier multipliers. For instance, it is not necessary for the symbol of this $K'$ to be of negative order. An example is $K'=\text{Id}$, which yields the nonlinearities studied in [@EGW].
Acknowledgment
==============
Both authors would like to thank M. Ehrnstr[ö]{}m and E. Wahl[é]{}n for suggesting this topic.
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[^1]: D.N.was supported by an ERCIM ‘Alain Bensoussan’ Fellowship. Y.W. acknowledges the support by grants nos. 231668 and 250070 from the Research Council of Norway.
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Q:
Constructing a monotonic function
(I'm not sure how to tag this; I'm tagging it math.CO because that's where it arose, but math.FA or something might be appropriate as well. Feel free to edit or comment on what this should be.)
I have a polynomial in a bunch of variables $P(x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_m)$. I know that if we evaluate the polynomial at
$x_1 = x_2 = \ldots = x_n = 1, y_1 = y_2 = \ldots = y_m = 0$
then the result is greater than the evaluation at
$x_1 = x_2 = \ldots = x_n = 0, y_1 = y_2 = \ldots = y_m = 1$.
Then are there necessarily functions
$f_1, \ldots, f_n$, with $f_i \in C^1[0,1], f_i(0) = 0, f_i(1) = 1$
and
$g_1, \ldots, g_m$, with $g_j \in C^1[0,1], g_j(0) = 1, g_j(1) = 0$
such that
$Q(t) = P(f_1(t), f_2(t), \ldots, f_n(t), g_1(t), \ldots, g_m(t))$
is monotonically decreasing on $[0,1]$?
A:
The answer is no. Is suffices to construct a polynomial such that these points are local minima. An example when $n=1$ is
$$x^2(y-1)^2(\epsilon+y^2(x-1)^2)$$
with $0<\epsilon$ small.
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NEW DELHI : Starting July 1, private hospitals in the country will no longer be allowed to directly procure COVID-19 vaccines from manufacturers and have to place orders on the Centre’s CoWIN portal.
All private hospitals have to register on CoWIN as a Private COVID Vaccination Centre (PCVC) to participate in the aggregation mechanism. Most hospitals have already registered on the portal, sources stated.
“States and UTs will be informed by MoHFW, about the total quantum of doses available for private CVCs in a month for them. They will aggregate the demand from private CVCs, keeping these quantities in mind,” a statement from the Union health ministry said.
There will be no need for approval from the government. Successful submission of the procurement orders on the government portal will be sufficient.
The Union health ministry will facilitate the supply of these vaccines to the private hospitals after payment is made electronically through the NHA portal.
“The likely monthly consumption shall be estimated by multiplying the daily average consumption during the week of the choice of the Private CVC (PCVC) in the previous month by 30. The maximum limit will be twice this quantity,” the instructions read.
For example, if for a PCVC, while submitting the order for the month of July 2021, the 7 day period selected by the PCVC is 10 June to 16 June and in that period, if 630 doses are reflected on CoWIN as administered, the daily average number of doses will be 630/7 i.e 90. Hence, the Maximum Order Quantity (MOQ) for the month of July 2021 = 90 x 30 x 2 = 5,400.
This process has already started. The following refinements in the process to be followed by the hospitals for procurement and supply of vaccines will come into effect from July 1.
PCVCs can place orders for Covishield or Covaxinin up to four instalments and payment should be made within three days of placement of orders, accepted only through electronic or digital mode, the SOP specified. | https://kashmirindepth.com/2021/06/30/private-hospitals-cant-directly-procure-covid-vaccines-need-to-place-orders-on-cowin-from-july-1/ |
The most ludicrous parent-child age gaps in movies
The silver sheen of the movie screen can forgive a lot, but it can’t always turn back the clock. These 10 parent/child movie pairings might have looked good on celluloid, but on paper, the actor’s age gaps make for a different story…
‘Indiana Jones And The Last Crusade’ (1989)
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Indiana Jones and The Last Crusade (Paramount)
The father: Sean Connery The child: Harrison Ford The age gap: 12 years
To be fair, Sean Connery and Harrison Ford are two ageless men: movie stars who seem forever preserved in cinematic amber. However, ever since Connery hung up Bond’s tux he’s operated in Kindly Grandpa Mode (even when romancing Catherine Zeta-Jones in ‘Entrapment’), and his ice white beard and scolding tone made him the ideal candidate to knock Harrison Ford’s Indy – just 12 years his junior – into shape. Ford, 74, still dating Calista Flockhart, 52, has never let an age gap worry him.
‘Forrest Gump’ (1994)
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Forrest Gump (Paramount)
The mother: Sally Field The child: Tom Hanks The age gap: 9 years
To this day, Tom Hanks still calls Sally Field ‘Mama’. The actress played Hanks’ mother from when Forrest was a boy right through to adulthood, which does help explain the rather paltry nine year age gap between the pair.
Six years earlier, however, Field played Tom Hanks’ love interest in the movie ‘Punchline’. This is a pertinent example for those who like to illustrate how quickly middle-aged women in Hollywood go from being a ‘viable love interest’ to getting cast as ‘wrinkly old spinster’.
‘The Graduate’ (1967)
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The Graduate (MGM)
The mother: Anne Bancroft The child: Katherine Ross The age gap: 9 years
Okay, so ‘Mother’ needed to be relatively youthful and attractive in order to believably seduce Dustin Hoffman in this ’60s classic (that iconic, poster-worthy shot of Anne Bancroft’s legs wouldn’t be quite so effective if she was rocking varicose veins) but she was just nine years older than her screen daughter. That just doesn’t add up; maybe that’s what Hoffman and Katherine Ross are each pondering on the back of that bus in the film’s iconic closing shot.
‘Prometheus’ (2012)
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Prometheus (20th Century Fox)
The father: Guy Pearce The child: Charlize Theron The age gap: 9 years
For all of ‘Prometheus”s flaws, the one that’s overlooked the most was the inexplicable decision to cast Guy Pearce as 103-year-old billionaire industrialist and Rupert Murdoch lookalike Peter Weyland, slathering the poor Aussie in layers of prosthetics. That deeply weird casting choice gave us the gift of Pearce, 44, wearing old man makeup, talking to his screen daughter Vickers, played by 36-year-old actress Charlize Theron. Is this weird or not? We can’t wrap our heads around it.
‘Star Trek’ (2009)
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Star Trek (Paramount)
The mother: Winona Ryder The child: Zachary Quinto The age gap: 5 years
Winona Ryder seemed like she’d stay young forever thanks to iconic roles in teen movies like ‘Heathers’ and ‘Beetlejuice’, and to be fair, she has aged extremely well, looking half her 45 years. However, the choice to cast her as Spock’s mother in JJ Abrams’ ‘Star Trek’ reboot was a puzzler, given that Zachary Quinto and Ryder were both teenagers at the same time. Vulcan women just have good genes, we guess.
‘Alexander’ (2004)
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Alexander (Warner Bros.)
The mother: Angelina Jolie The child: Colin Farrell The age gap: 1 year
Even if you discount the super-weird incestuous thing going on between them, the screen relationship between mother Angelina Jolie and son Colin Farrell in Oliver Stone’s historical flub is deeply peculiar. “I think seeing us together, he fits Alexander and I fit Olympias, and somehow together we became mother and son,” commented Jolie, “somehow” being the operative word. Still, they did things differently in the olden days. Maybe there were lots of Macedonian women giving birth at the age of one.
‘Riding In Cars With Boys’ (2001)
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Riding In Cars With Boys (Columbia)
The mother: Drew Barrymore The child: Adam Garcia The age gap: Minus 2 years
Now we’re getting into bizarre territory: films where the children are actually older than their movie parents in real life. Drew Barrymore plays a girl who gets pregnant at age 15; it’s testament to her youthful looks that she could play the same role 20 years later, but less believable was the casting of her 20-year-old son. Adam Garcia was 28 when he played Drew’s son, making him two years older than his screen mother. We bet that made for some interesting off-screen chemistry…
‘Back To The Future’ (1985)
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Back To The Future (Universal)
The father: Crispin Glover The child: Michael J Fox The age gap: Minus 3 years
Does ‘Back To The Future’ get a free pass for casting Crispin Glover as Marty McFly’s father, despite the fact that he was three years younger than even the eternally youthful Michael J Fox? The twisted temporal timelines, and the need for Glover to play George McFly as both a young and older man suggests that lenience is in order, but it’s worth noting just how well Glover sells being Marty’s old man, even though he’s not. Leah Thompson, for what it’s worth, is three months younger than her screen son.
‘Blow’ (2001)
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Blow (New Line)
The mother: Rachel Griffiths The child: Johnny Depp The age gap: Minus 5 years
Sure, Johnny Depp has cheekbones that could slice cheese and granted, they take years off his age (he’s currently 53), but only someone who has been snorting copious amounts of Colombian marching powder could believe that Australian access Rachel Griffiths could pass for his mother. She’s five years younger than him for a start; it’s another classic case of casting a mum against a male character seen as both young and old and hoping nobody notices. Well, we noticed.
‘Hamlet’ (1948)
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Hamlet (Rank Organisation)
The mother: Eileen Herlie The child: Laurence Olivier The age gap: Minus 11 years
Laurence Olivier was almost old enough to be co-star Eileen Herlie’s father, but the roles were reversed in the Oscar-winning 1948 version of ‘Hamlet, with 30-year-old Herlie playing Queen Gertrude, mother of 41-year-old Larry’s crown prince. The actress played Queen Gertrude once more on Broadway in 1964, opposite Richard Burton’s Hamlet, where she was seven whole years older than her onstage son. | |
I’m going to do a couple of postings about causality, because it seems to me that how humans experience causality is of paramount importance to a lot of ideas such as “free will”, responsibility, and knowledge.
Presenting a philosophical framework in which to defend these ideas, however, is beyond me – so I’m going to approach the discussion casually, and I’ll be less rigorous about terminology than I’d have to be if I were going to try to defend my thoughts against a full-on skeptical enquiry.
I’m going to try to avoid making asssertions, since this discussion is sort of an oblique critique of Aristotelian ideas of causality. Let’s start by rejecting those as arbitrary.
Aristotle asserts that there are a variety of different kinds of causes: proximal causes, necessary causes, sufficient causes, final cause. Other than their potential usefulness as labels, we aren’t given a way to tell which kinds of causality are which, except for after the fact – which is how we humans appear to experience causality, anyway. In other words, after a tree falls in the forest, I can say “it was rotten at the roots, so it fell” or “there was strong wind so it fell” but it doesn’t make sense for me to say that before it falls (since it hasn’t fallen) and once it does fall, I can no longer really be sure which of the causes I assign to the event was really the cause. I start to have problems with cause and effect because the way we humans seem to assign them is 1:1 – there is A cause and AN effect, but reality appears to be much more complicated than that. Especially because reality doesn’t courteously come to a pause while we ponder cause and effect, it keeps right on rolling: what other things fell when the tree fell?
It appears to me that when something happens, it has multiple causes: the tree’s roots have to be weak or it wouldn’t blow over. But humans’ way of talking about cause and effect is to try to assign a 1:1 cause/effect relationship – which makes sense because otherwise we’re going to be unable to talk about cause and effect at all: were the roots 40% of the cause and the wind 60%? When Aristotle starts breaking causality into various types of causes, that’s pretty much what he’s saying. There are some causes without which an effect would never happen – the final cause – but there are other causes which lead up to the final cause, without which the final cause cannot happen. Aristotle’s example is teeth growing: they appear to grow because there are certain material causes (they must be physically assembled by the body via some process) but their final cause is teleological – we need teeth to chew with. Aristotle slides right by the problem that “having teeth” is a vague concept. At what point can we say someone has teeth? If we have trouble saying at what point someone has teeth at all, how can we say that something caused us to have teeth? If we’re going to say “the tree fell because the roots were rotten” we slide past the problem of saying what “rotten roots” means, and a pyrhhonian skeptic would probably appear to point out that our definition of “rotten roots” may be circular: we have rotten roots if they are so weak the tree falls down, therefore we can say that the tree fell down because it fell down. Ooops.
As I said in the beginning: I am not trying to present a philosophical world-view – I don’t have a framework in which to answer to these problems. In truth, I tend to be genuinely suspicious of people who do feel they have an answer, because I suspect they are engaged in motivated reasoning. Most of us will have encountered cause/effect based reasoning for the existence of god: there must be some un-caused cause, etc. I first started worrying about cause and effect because of that argument – the closer I looked at it the less I understood about cause and effect. Rather than believing in god, I just wound up confused.
“Why did the chicken cross the road?”
One of the many problems with that question is that it implies there’s a single cause for the actions of the chicken. But it seems to me that there are nearly an infinity of causes, if you want to start assigning causes:
- Because humans bred chickens
- Because there was a coyote chasing it
- Because of evolution
- Because chickens like to play in roads
- Because of the chicxulub event, which wiped out the dinosaurs and opened up new niches for life to evolve into, therefore: birds
- Because god said “let there be poultry, and roads”
- Because of The Big Bang
- It was trying to escape from Twitter
If we were to enumerate Aristotle’s causes, several of the ones on my list above are material causes, i.e.: “that out of which” the effect is made. After all, something made the chicken. But “evolution” or “humans breeding birds” or “eggs” or “grandma chicken” are equally good responses. We might say “let’s favor the ‘largest’ response on the list – i.e.: evolution had more to do with the creation of chickens than the chicxulub event, therefore it had more effect. But that’s a human conceit: the efficient cause – “the primary source of change” is simply whatever this particular human says it is; we have no means for determining primary source of change so our instinct appears to be to pick the earliest or most recent: the big bang, or the coyote.
Every effect has more than one cause. Simple cause and effect appears to be an illusion our brains create so we can cope with reality.
This is where I get off the trail and wander out into the weeds of nihilism. It appears to me that “cause and effect” is nothing more than a word we use – a shorthand, if you will – for our personal understanding of something that we observe. And our understanding is always limited by our senses and perspective, of course. Let’s reconsider the chicken, and let’s suppose for the sake of argument that there is a coyote stalking it. I observe the coyote and assign the coyote’s threatening behavior as one of the causes of the chicken’s crossing the road. But if the person standing next to me is positioned such that they don’t see the coyote (due to a philosophically convenient intervening rock) they may interpret the chicken’s road-crossing as due to an evolved-in fondness for roads, if they are an evolutionary psychologist. Then we’re right back to extreme skepticism: because of our inability to know anything due to the fallability of sense input (#include <skepticism/pyrrhonism.h>) we can’t make claims to knowledge about why the chicken did anything, but worse still: the tool we would use to talk about it between eachother is mere language, which is not reliable enough. We are left wondering if there’s a thing we would agree is a “chicken” at all, let alone a road-crossing event such as we would collectively agree that there had been a road, and it had been crossed by a chicken, and there was a thing we call a “coyote” involved, ad infinitum.
If I did have a framework for talking about causality, it would probably sound almost like woo-woo: “cause and effect” are the words we use to talk about our limited understanding of the events we are discussing, as a way of packaging up those events that we perceive as directly affecting them. Meanwhile, it seems to me now that cause and effect are not a simple linear chain of one event leading to another – the causes are the totality of prior events and the effects are all subsequent events. Causality is not a line, it is a meshwork that branches infinitely, because the closer we wish to examine any event, the more causes and prior causes we can identify. “Make me one with everything,” the buddhist said to the hot dog vendor. The hot dog vendor replied, “The big bang already did. That’ll be $5.”
So now I am back to my starting-point: “we experience causality.” It is difficult to convince me that we know anything about causality, aside from its being a handy organizing principle we can apply verbally when we are trying to communicate about something. That could go some way toward explaining why, when humans talk about cause and effect, we are all over the place – some of us think the chicken’s actions were purely internally-driven (“free will”) while others think the chicken was acting in a context where its options were constrained by the coyote (“it’s a meat robot programmed to avoid coyotes”) and – in the same breath, we can ascribe the entire thing to god or the big bang or both or neither.
From a practical standpoint, in order to make science work, we need to think about cause and effect as if it’s a simple linear model. In order to eat, our distant ancestors had to model game animals’ motivations in a simple linear model of cause and effect: if I wait for the chickens’ ancestors to cross the road and come to the watering-hole, I can eat one of them. It’s tempting, at this point, to adopt an evolutionary psychology-style “just so story” – our brains’ ability to interpret cause and effect is optimized to assign proximal cause to the point where we are most likely to be able to take an action that will confer a survival advantage. | https://freethoughtblogs.com/stderr/2016/10/15/causingproblems/ |
Like most worlds, investing has its own terms. I can’t remember when I first heard the term alpha in stocks and investing, but it must have been over 25 years ago. Since then, it has become a very common part of my vocabulary as a financial advisor and part of the management team at a broker-dealer.
So what is Alpha in stocks? The term alpha in stocks denotes an investment strategy’s ability to generate excess return minus costs above a market index. The term “Alpha” started originally when a portfolio manager that achieves a positive alpha will outperform a market index by enough to cover not only his/her own costs but also to earn additional profit from the risk he/she is taking.
Alpha generally means excess return (greater than a risk-free market return, such as the one earned by a simple bond) on an investment portfolio. In other words, the ratio of the excess return to the risk-free rate. In this article, alpha refers to excess return.
In finance, an alternative interpretation of alpha differentiates it from beta rather than the risk/reward profile, as is done in academia. This alternative interpretation is also sometimes used to describe any investment strategy seeking above-market returns, including Dynamic asset allocation and efficient frontier strategies.
Nowadays, alpha is sometimes used as a general term to describe any investment strategy seeking above-market returns. However, the alternative interpretation could be considered misleading, as alpha generally means the excess return on an investment portfolio. It is comparable to the use of beta in academia and research.
Table of Contents
- 1 How is Alpha measured?
- 2 Risk and return
- 3 What is a risk-free rate?
- 4 Capital Asset Pricing Model (CAPM)
- 5 Is positive Alpha good?
- 6 What is a benchmark index?
- 7 Individual Investor Alpha
- 8 Modern Portfolio Theory
- 9 Using alpha in benchmarks
- 10 Tell me the difference between alpha and beta?
- 11 What is beta in stocks?
How is Alpha measured?
Alpha is measured on a simple decimal fraction (e.g., 0.7 or 1.2). A value that is higher than 1.0 indicates that the investment has more potential for gains than losses and vice versa. An alpha of zero represents an absolute performance track record against that of the market index. Whereas an Alpha with a positive number means the returns are above or in excess of the benchmark. However, an Alpha with a negative number means the investment performed below the benchmark index.
There are many ways to measure an investment strategy’s ability to generate an excess return, most are based on the standard deviation of returns or volatility of returns.
Alpha is usually measured as a percentage to express the magnitude of this expected excess return, and you can look up Alpha before your potential investments in various financial instruments. But if you want to see how much an investment could gain rather than lose, you should use Sharpe Ratio or Sortino Ratio.
Risk and return
When buying bonds or stocks, investors look for the potential income opportunities that could be created from such a transaction. In this situation, if an individual doesn’t secure an investment, she can settle on only one low-income level. If the company faces a higher risk of losing a large portion of its investment, more income will be to pay for it. Alpha finance calculation helps reduce risk by giving investors a way to predict a stock’s performance. You can invest in equity stocks you may not like, but you may risk doing something for the long term based on the risk. Probably losing. You’re investing in something if they fail.
What is a risk-free rate?
This is an investment that has theoretically zero risks. Most investments do have some risk, even if it is very small, like US Treasuries.
Capital Asset Pricing Model (CAPM)
The capital asset pricing model is the whole or systematic risk and expected return (ROI) for investments. It helps investors understand the true risk vs reward of potential investments such as stocks.
Is positive Alpha good?
Yes, as mentioned above, a positive Alpha is generally good. This means that the investment or portfolio performed better than the benchmark index.
Most portfolio managers and investors try to generate excess returns or positive Alpha. This is because if the Alpha is not positive, then the investor may have been better off just investing in the benchmark index.
What is a benchmark index?
A benchmark index is an index used by the investment community as a point of reference for the performance of different stock segments.
Numerous financial indexes track the prices or returns on categories of stocks, bonds, mutual funds, or other types of investment vehicles.
Dow Jones Industrial Average (DJIA), S&P 500, and Nasdaq Composite Indexes are the most well-known and widely used benchmarks.
An index may also refer to a published list showing comparative values for shares in that particular market at a given time. The most highly regarded indices were those compiled by Dow Jones & Company and Standard & Poor’s Corporation (S&P).
Individual Investor Alpha
As an individual investor, the concept of Alpha can be used to evaluate current and new stock investments. Select stocks of lower risk with high potential gains for an increased alpha. For example, you may have to compare competitor companies and choose one that shows the most potential. For instance, the FedEx and UPS competitors may compare profitability and growth metrics like the ratio of prices to earnings.
Modern Portfolio Theory
It is a statistical measure from investment disciplines called Modern Portfolio Theory. According to MPT, estimated investment returns take account of quantifiable risk volatility and are risk-free. Standard Deviation, the squared, smooth ratio, beta, and alpha are statistics to determine the standard deviation and the R-squared. The alpha indicates the effect of investments on the returns of the larger asset. The formula is dependent on the standard deviation of the rsquared.
Using alpha in benchmarks
Your benchmark can be a stock index or exchange-traded fund. According to the NASDAQ 100 index, some U.S. technology companies are currently included in such indices. Compare the profits of your investment portfolio with those of some of these indexes, including the S&P Dividend Aristocrats.
Tell me the difference between alpha and beta?
Defining Beta indicates volatile bourse relative to market movements. Beta is indicative of movement in the same way as the market as a whole. Alpha is more based on the addition of return provided in the investment in the market. When the beta value is negative, the investing has an inverse correlation to the value of the market. So it tends to go up versus the downside for the value of. A beta of 1 indicates that a company is as volatile as a market and that its price pairs closely with that of the market. When Beta is zero, investment and market do not correlate. A beta of more than zero and less than 1 means that investments are less affected than the market.
What is beta in stocks?
In finance, beta () is a measure of the instantaneous relative volatility of a stock. A beta value of less than 1 indicates that the stock is cheap, and a beta greater than 1 indicates that the stock is expensive. While other measures of volatility, such as standard deviation, can be used to measure overall volatility, they are not particularly useful for measuring relative volatility because standard deviation estimates vary with time and with how much information about past price movements has been known.
Beta is expressed as a decimal fraction (e.g., 0.9 or 2), but it can also be measured in percentage terms (e.g., 99% or 6%).or in decibels (e.g., 15 or 10). | https://alphabetastock.com/what-is-the-meaning-of-alpha-in-the-stock-market/286383/ |
The Balance of Payments is a complicated international economic formula used to understand all of the transactions that a country conducts with those in another country. The transactions include everything that is undertaken by that country’s people, companies and government bodies and consists of all imports and exports. Goods, services and capital are included in these transactions, in addition to foreign aid or remittances.
TL;DR (Too Long; Didn't Read)
The balance of payments, also known as the balance of international payments, represents the difference in value between payments into and out of a country over a specific period.
Accounting for Transactions Between Countries
If all transactions between the two countries are properly included, then the payments and receipts between the two countries will be equal. For example, if a country exports an item, then it technically imports foreign capital as payment for the item exported. However, sometimes a country cannot fund its purchases and ends up dipping into its reserves to make payments. When this happens, the country has a Balance of Payment deficit. Statistical discrepancies often occur as it’s difficult to account for every transaction between two countries accurately.
Tallying up Accounts
To calculate the BOP of a country, you need to review three main accounts: the current account, the capital account and the financial account. Each of these accounts contains inflows and outflows. The current account includes merchandise trade goods, services, income receipts and one-way foreign transfers. Transfers of financial assets, including tax payments and transfers of titles to assets, are included in the capital account. The financial account includes stocks, bonds, commodities and real estate. Sometimes, the capital account and the financial account are looked at together as one entity because they both include financial transactions.
Calculating the BOP
To calculate the BOP, you need to calculate the sum of the country’s exports and imports. Exports are written as a credit entry while imports are written as a debit entry. For example, if a country has exports of $400 million and imports of minus $500 million, then they have a trade deficit of $100 million, or a BOP of minus $100 million. If the numbers were reversed and the number of exports exceeded the number of imports, then the country would have a trade surplus.
How to Interpret the Results
The BOP helps economists evaluate the strength of a country’s economy in comparison to the economies of other countries. When a country has a deficit, they are technically borrowing money to purchase goods and services from the rest of the world. However, if they have a surplus, they are in a better financial position and can afford to import additional goods and services. Imbalances in the BOP can create political tensions between countries and disrupt the world’s political climate.
References
- Investopedia: Balance of Payments (BOP)
- The Concise Encyclopedia of Economics: Balance of Payments
- Internal Economics: Balance of Payments - Calculation
- Investing Answers: Balance of Payments (BOP)
- Federal Reserve Bank of New York. "Balance of Payments." Accessed Jan. 16, 2020.
- The Library of Economics and Liberty. "Balance of Payments." Accessed Jan. 16, 2020.
- Khan Academy. "Lesson Summary: The Balance of Payments." Accessed Jan. 16, 2020.
- The Wharton Business School of the University of Pennsylvania. "Chapter 3: Balance of Payments Suggested Answers and Solutions to End-of-Chapter Questions and Problems." Accessed Jan. 16, 2020.
- The World Bank. "Current Account Balance (BoP, Current US$) - United States," Accessed Jan 2, 2020.
- CBO Testimony. “Foreign Holdings of U.S. Government Securities and the U.S. Current Account, June 26, 2007." Accessed Jan. 16, 2020.
- Bureau of Economic Analysis. “National Income and Product Accounts Tables," Table 1.1.5. Go to "Modify" and set to "Annual." Accessed Mar. 6, 2020.
- United States Census Bureau. "Top Trading Partners - January 2020," Accessed Jan. 22, 2020.
- Statista. “The 20 Countries with the Highest Trade Balance Deficit in 2017 (in Billion U.S. Dollars)." Accessed Jan. 15, 2020.
Writer Bio
Anam Ahmed is a Toronto-based writer and editor with over a decade of experience helping small businesses and entrepreneurs reach new heights. She has experience ghostwriting and editing business books, especially those in the "For Dummies" series, in addition to writing and editing web content for the brand. Anam works as a marketing strategist and copywriter, collaborating with everyone from Fortune 500 companies to start-ups, lifestyle bloggers to professional athletes. As a small business owner herself, she is well-versed in what it takes to run and market a small business. Anam earned an M.A. from the University of Toronto and a B.A.H. from Queen's University. Learn more at www.anamahmed.ca. | https://bizfluent.com/how-6631597-calculate-balance-payments.html |
We are going to learn about a Cartesian Plane.
Success Criteria: you will know you have been successful when you can:
# Give a definition for a Cartesian Plane.
# Identify and name parts of a Cartesian Plane.
# Plot points on a Cartesian Plane.
Intro to the coordinate plane:
Points on the coordinate plane
Interactive Cartesian Coordinates
https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview-hist-alg/v/descartes-and-cartesian-coordinates
17 thoughts on “Cartesian Planes”
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I do not know a single thing about cartesian plane but i am excited on learning abount them.
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For Cartesian planes I haven’t heard, seen or worked with a Cartesian plane at all.
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What I need to learn:
I have never heard of a cartesian plane before and I will need some help learning about them.
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I do not know what Cartesian Planes are.
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I have never done Cartesian planes before, so this will be a new learning activity :).
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I kind of know how to find a cartesian plane.
I need to work on learning the mathmatical terms for them and how to write them.
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I am good at finding things on a map with coordinates but I have never ever heard of cartesian planes.
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Abs
Cartesian Planes are basically a grid with coordinates with ten squares on the Y axis and ten on the X axis.
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I don’t really know what Cartesian Planes are.
I need to learn all the Success Criteria.
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I have not done these before.
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I do not know what a cortisone plane is or what to do with it. I have never heard of it before.
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I can identify and name a cartesian plane and locate points on a plane.
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I do not have an understanding of a Cartesian Plane. I will be able to successfully identify and name a Cartesian plane
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Cartesian Planes
Yes, I do understand what cartesian planes are. Cartesian planes are a type of grid, with a Y-axis, & an X-axix, as in all types of grids. Their are typically 10 lines on each axis, but their can be as many or as little as you want. However many you use to draw on one axis, their must be the same amount in the other. For example, if there are 10 lines in the Y axis, you must draw ten on the X axis. The number that you have chosen on each axis can be squared, and that is the amount of squares you will have throughout the entire grid/plane.
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I think that the websites helped me understand what Cartesian Planes are, but I think that the pirate game was easier than the rest it was more obvious. I could not access the Billy Bug website but that doesn’t matter. | https://ehssjadej.edublogs.org/2018/11/28/coordinates/ |
The Queer Healing Art Center buzzed with excitement this past Saturday as artists prepared their bright white canvases, paintbrushes, and acrylic paint for an Art Battle.
This was the Queer Healing Art Center’s one-year anniversary of hosting Art Battles—live competitions where artists paint blank canvases while surrounded by an audience. About 100 people attended Saturday’s event.
“As soon as the paintbrush hits the canvas, everyone is electrified,” said Kin Folkz, né Monica Anderson, co-founder and executive director of the Queer Healing Art Center.
There were three 20-minute rounds. Six artists competed in the first round, followed by another six in the second. Guests used their phones after each round to vote, and five artists advanced to the final round.
Artist Juliette Leong immediately stood out, in part because of her big yellow bow and light-up shoes that sparkled as she approached her canvas. Leong is Art Battle’s youngest-ever competitor at age 6. But she was not nervous, she said.
When asked what she was going to paint, Leong replied, “You just have to wait and see.”
Juliette painted a cityscape with fireworks in a black sky and people wearing glittery dresses.
The Art Center, located in the heart of Lakeshore Avenue’s commercial corridor, opened as a brick-and-mortar in 2020 just weeks before shelter-in-place.
“This pandemic was, for many of us, one of the latest in a long line of various pandemics,” Folkz said. “Everyone was experiencing heartache.”
The Art Center’s members wanted to find ways to support their community. They started a trans and non-binary caravan to circle Lake Merritt with their art-adorned vehicles and bring awareness to their COVID experiences. They served close to 20,000 meals to unhoused and housed people. And in 2021, they started Oakland Art Battles.
Shortly after 8:00 p.m., the artists dropped their brushes and the audience inspected the art and cast their final votes.
Diego Gomez, 40, of San Francisco captivated the audience with two monster-themed masterpieces that night. Gomez walked away with gold, qualifying for the Art Battle regional competition.
The evening ended with a silent auction of the freshly painted artwork.
Diane Chan, 72, reflected on the paintings. “None are alike. They’re all beautiful,” she said. “This is my very first event and it will not be my last.”
For Folkz, these art competitions are about community, art as healing, but also, survival.
This past year the Queer Healing Art Center struggled to keep its doors open. The revenue from Art Battles helps to pay the rent. Saturday’s Art Battle raised about $2,000 from ticket sales.
More than anything, Folkz says, the events are like a family gathering.
“It’s an environment of collective self-care. We’re all in it together. It’s less about the creation itself and more about the process of [making art] together,” they said.
The Queer Healing Art Center has two upcoming exhibits: Marsha Marsha Marsha and Bejeweled. The next Art Battle is scheduled for Dec. 3, 2022.
This story was produced by and co-published with Oakland North. | https://oaklandside.org/2022/10/04/queer-healing-art-center-lakeshore-oakland-art-battles%EF%BF%BC/ |
Under the guidance of an instructor your team not only gets to climb but also learns some technical skills; each climber needs a belayer (person who controls the rope), so as part of the introduction everyone learns how to belay. The team are therefore responsible for each others safety, as well as providing encouragement. There are many climbing routes to choose from and each individual can choose where and how high they wish to climb. | http://www.distant-horizons.co.uk/outdoor-activities/climbing-wall-team-building |
Decision making models fall into two general categories: rational and intuitive. These two broad categories provide variations to arrive at a decision in any situation. The rational decision-making model includes the Vroom-Jago system and a seven-step process. Recognition-primed decision-making models are considered intuitive methods. Managers and leaders often combine rational and intuitive models when faced with a problem or opportunity.
Rational decision making models employ a structured approach that is orderly and logical. A sequence of steps starts with identifying the problem or situation at hand, followed by compiling all the facts and information necessary to create a solution. Next, the data is analyzed for various options to determine which action might achieve the desired result. The final step in rational decision making involves acting on the preferred option and setting aside adequate resources to make it work.
Decision making models are tools that help individuals and organizations make better choices. There are a variety of different models available, but all share a common goal: to improve the quality of decisions. Some common Decision making models include the rational decision making model, the bounded rationality model (the satisficing model), Vroom-Yetton decision model, recognition-primed decision making model, and the intuitive decision making model. Each of these models has its own strengths and weaknesses, but all can be useful in the right situation. The key is to understand when to use each model and how to apply it in a way that will produce the best results. With the right decision making model, organizations and individuals can make choices that are more informed, efficient, and effective.
A decision-making model is a structured process used to guide teams to make decisions. Each decision-maker model uses different methods to help you analyze and overcome a particular challenge. Because decision-maker models take different approaches, they’re useful for people with different learning styles or time constraints.
Types:
Creative decision model
The creative decision model uses original ideas to create innovative solutions that achieve goals or overcome obstacles. This involves thinking through a situation and inventing a solution without referencing similar situations.
Often, you can use this model for situations you haven’t experienced before, like new projects or production issues. Using the creative decision model typically requires flexible thinking to create successful, unique solutions. You may follow these steps when using the creative decision-making model:
- Define your goal or obstacle: You may not have experience with your goal or obstacle, so it’s useful to define it as clearly as possible to help you understand what you need to do. This may involve meetings with your team or other colleagues, like business partners or managers.
- Consider relevant information: Do research on the challenge you need to solve to learn everything you can about it. This includes trying to find any similar projects, reports or companies that may inspire your ideas.
- Consider the information over time: You can choose how long to consider the information, but it’s helpful to take at least a day to think about your challenge passively. To do this, you may brainstorm ideas, talk with colleagues or make a word-association list.
- Create a usable solution: With the creative decision model, your idea may come naturally after a period of thinking about your goal or obstacle and the information relevant to it. Think through your solution logically to make sure it’s usable for your situation.
- Finalize your decision and take action: After considering the details of your solution, you may finalize your decision with your team and take action to solve your challenge. It’s helpful to have a draft or presentation of your creative solution to explain it to your team more easily.
Recognition-primed decision model
The recognition-primed decision model, created by Gary A. Klein in his book “Sources of Power,” uses quick thinking and prior experience to make decisions, often in fast-paced environments.
Team leaders may use this model to assess the basics of a situation and create a potential solution and then think through the solution to determine if it’s usable. This may require you to have a lot of experience with the goal or obstacle for you to create a suitable solution. The following steps can help you use a recognition-primed decision model:
- Define your goal or obstacle: Clearly define the goal or obstacle your team wants to achieve or overcome to make it easier for you to create a solution quickly. While the idea can be broad, try to identify the most important thing you need to decide.
- Consider relevant information and similar situations: Using your prior experience, quickly assess the situation and determine what information or prior situations can help you make a usable solution. If you have time, do more research on how to solve your goal or obstacle.
- Create a potential solution: Create at least one potential solution using your prior experience or additional knowledge about the situation. To quicken your decision process, try to create a generic solution so you can change or add details as you think through it.
- Consider if the solution works: Think through your solution to determine if it can really solve your challenge. Start by considering the most obvious issues and then consider the smaller details of the solution.
- If needed, change the solution: Your first solution may not produce the best outcomes, so change details about it if you need to. This may involve adding new actions to your solution, making it more specific or changing it altogether.
- Finalize your decision and take action: Once you’re confident in your solution, finalize the decision with your team and take action. In a fast-paced situation, you may have to change your solution again if you learn new information while taking action.
Rational/Classical decision model
The rational decision-making model focuses on using logical steps to come to the best solution possible. This often involves analyzing multiple solutions at once to choose the one that offers the best quality outcome.
Teams typically use the rational decision model when they have time for meetings and research, which allows them to create a list of potential solutions and discuss the pros and cons of each.
Features of Classical Model:
- Problems are clear.
- Objectives are clear.
- People agree on criteria and weights.
- All alternatives are known.
- All consequences can be anticipated.
- Decision makes are rational.
- They are not biased in recognizing problems.
- They are capable of processing ail relevant information
- They anticipate present and future consequences of decisions.
- They search for all alternatives that maximizes the desired results.
Steps:
- Define your goal or obstacle: First, you must define the goal or obstacle you wish to achieve or overcome. Defining this helps you understand exactly what outcome your solution should produce.
- Determine the relevant information: For this step, consider delegating research tasks to your team or brainstorming during a team meeting. Determine what information about your goal or obstacle is relevant to finding a solution.
- Create a list of options: Using the relevant information, your team can create a list of potential options for solutions. Try to support your options with evidence for why they would solve achieve your goal or overcome your obstacle.
- Arrange options by their value: After creating a list of options, arrange them by their likelihood of success. Options that have a higher chance of success also have a higher value, while options with little evidence may have a lower value.
- Choose the best option: Consider the value of each option and how it can help your company succeed. With your team, come to a consensus about the best option for a solution using the information you’ve gathered.
- Finalize your decision and take action: Once your team decides on the best solution, clearly state your commitment to the solution and ask if any team members have concerns. After this, you can implement your solution in your company.
Intuitive decision model
Rather than logical reasoning, the intuitive decision model uses feelings and instinct to make decisions. Often, team leaders or managers use this model to make quick decisions when they don’t have a lot of time for research or planning.
The process of an intuitive decision is less structured and may use previous knowledge of similar goals or obstacles to determine a useful solution.
Steps:
- Define your goal or obstacle: Even with little time, it’s important to define your goal or obstacle clearly, especially if you’re making a decision without your team. This can help you explain the decision and its effects later.
- Identify similar goals or obstacles: Brainstorm similar goals or obstacles you’ve encountered before and how you solved them. Use this information as a basis for creating your own solution.
- Recognize possible biases: Recognizing your biases is especially important when you don’t have input from your team. Consider how your decision may affect yourself, your team and your company as you think of potential solutions.
- Determine a usable solution: Determine the best solution using your prior experience and the values of your company. An ideal solution helps your company achieve its goals or overcome an obstacle while also benefitting your team and other employees.
- Finalize your decision and take action: After choosing a usable solution, you can alert your company and team of your decision. If you have to make the decision quickly, you may have to put it into action without discussing with your team.
Bounded Rationality Model or Administrative Man Model:
Decision-making involves the achievement of a goal. Rationality demands that the decision-maker should properly understand the alternative courses of action for reaching the goals.
He should also have full information and the ability to analyse properly various alternative courses of action in the light of goals sought. There should also be a desire to select the best solutions by selecting the alternative which will satisfy the goal achievement.
Herbert A. Simon defines rationality in terms of objective and intelligent action. It is characterised by behavioural nexus between ends and means. If appropriate means are chosen to reach desired ends the decision is rational.
Bounded Rationality model is based on the concept developed by Herbert Simon. This model does not assume individual rationality in the decision process.
Instead, it assumes that people, while they may seek the best solution, normally settle for much less, because the decisions they confront typically demand greater information, time, processing capabilities than they possess. They settle for “bounded rationality or limited rationality in decisions. This model is based on certain basic concepts.
- Sequential Attention to alternative solution:
Normally it is the tendency for people to examine possible solution one at a time instead of identifying all possible solutions and stop searching once an acceptable (though not necessarily the best) solution is found.
- Heuristic:
These are the assumptions that guide the search for alternatives into areas that have a high probability for yielding success.
- Satisficing:
Herbert Simon called this “satisficing” that is picking a course of action that is satisfactory or “good enough” under the circumstances. It is the tendency for decision makers to accept the first alternative that meets their minimally acceptable requirements rather than pushing them further for an alternative that produces the best results.
Satisficing is preferred for decisions of small significance when time is the major constraint or where most of the alternatives are essentially similar.
Thus, while the rational or classic model indicates how decisions should be made (i.e. it works as a prescriptive model), it falls somewhat short concerning how decisions are actually made (i.e. as a descriptive model). | https://theintactone.com/2022/11/25/decisions-models-concept-and-types/ |
Érd is located in Hungary country, in Europe continent (or region). DMS latitude longitude coordinates for Érd are: 47°23'41.6"N, 18°54'49"E.
• Latitude position: Equator ⇐ 5270km (3275mi) ⇐ Érd ⇒ 4737km (2944mi) ⇒ North pole.
• Longitude position: Prime meridian ⇒ 1420km (882mi) ⇒ Érd. GMT: +2h.
• Local time in Érd: Sunday 2:55 am, May 31, 2020. [*time info]
Step for grid lines is 15°. On second map 45° latitude and 15° longitude line is numbered. Map pointer shows the 47.39, 18.91 lat-long coordinates.
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Q:
Last digit of $235!^{69}$
Problem
What is the last digit of $235!^{69}$?
It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back.
My initial thought goes to the last digit $5$ in $235$. If $235!$ can be shown to have a $5$ as last digit, raising it to any natural number shouldn't change that, right?
A:
It is zero. Actually $n!$ is a multiple of $10$ for $n\ge 5$, and raising a multiple of $10$ to any natural power results in a multiple of $10$ again.
To elaborate, $235!$ ends in $\lfloor\frac{235}5\rfloor+\lfloor\frac{235}{25}\rfloor+\lfloor\frac{235}{125}\rfloor+\ldots = 57$ zeroes, hence $235!^{69}$ ends in $57\cdot 69=3933$ zeroes.
The last non-zero digit of $235!$ turns out to be a $6$ (however, showing that is not that trivial), and for that your argument applies that this does not change by raising to natural powers (because already $6\cdot 6\equiv 6\pmod{10}$. We conclude that the last non-zero digit of $235!^{69}$ is a $6$.
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Innovative B2B Marketing
# PRAISE FOR *INNOVATIVE B2B MARKETING*
‘A long-overdue resource for anyone working in the B2B environment or wishing to enter into it. Simon Hall’s development of practical and relevant B2B marketing models make this essential reading for marketers and other professionals.’ **Dawn Southgate, Head of Knowledge, The Chartered Institute of Marketing**
‘If you read any B2B marketing publication this year, this should be it! Simon Hall provides us with a refreshing view, relevant content and truly practical advice, touching on many areas pertinent in the B2B marketing world right now and for the future. Absolutely fantastic insight.’ **Catherine Howard, UK and Ireland Marketing Director, Atos**
‘B2B marketers today need to be more dynamic than ever before, but this can be difficult in an environment where the complexity of the tools, technologies, techniques and channels has exploded, leaving many of us floundering. *Innovative B2B Marketing* provides the guidance that all B2B marketers need to help them navigate for success. I will be keeping this book very close to hand.’ **Richard Robinson, Chair, B2B Marketing Council, The IDM**
‘How do you build best-in-class B2B strategy in today’s digital economy? How are great B2B brands built? How do you select the right lead nurturing agency? *Innovative B2B Marketing* is a rich treasure trove of B2B strategic wisdom.’ **Shenda Loughnane, Global Chief Strategy Officer, iProspect**
‘*Innovative B2B Marketing* is a comprehensive and compelling book that every B2B marketer should read.’ **Ashraf Kamel, Head of Technology Vertical EMEA, LinkedIn Marketing Solutions**
**Innovative B2B Marketing**
New models, processes and theory
Simon Hall
*This book is dedicated to Laura for her unwavering support and inspiration*
# contents
1. Cover
2. Title Page
3. Copyright
4. Contents
5. *List of figures and tables*
6. *About the author*
7. *Preface*
8. *About this book*
9. **01 Introduction to business marketing**
1. The changing business landscape
2. What is B2B marketing?
3. Business market segmentation and size
4. The digital economy
5. Core trends and this book
6. References and further reading
10. **PART ONE** Developing your marketing strategy
1. **02 The new marketing mix**
1. The marketing mix defined
2. Is the marketing mix still relevant?
3. The shift away from the four Ps
4. The new B2B marketing mix
5. Marketing mix and trade-offs
6. Reference
2. **03 B2B marketing strategy and planning**
1. Introduction
2. The traditional and new B2B marketing planning process
3. B2B marketing audits
4. Marketing strategy formulation
5. The marketing funnel and beyond
6. B2B marketing strategies
11. **PART TWO** Improve B2B customer-centric marketing
1. **04 Business customers and buying behaviours**
1. The importance of business buying behaviours
2. Identifying business buying behaviours
3. Business buying influencers
4. Buyer types
5. Types of buying situation
6. Changes to business buying behaviour
7. Buyer triggers
8. Further reading
2. **05 Acquisition marketing**
1. Customer acquisition and customer lifecycle
2. The REAP model
3. Acquisition marketing and types of acquisition customer
4. The acquisition marketing process
5. Re-acquisition marketing process
6. Acquisition marketing strategies and tactics
7. Measuring acquisition
8. Further reading
3. **06 Retention and loyalty marketing**
1. The REAP model and retention/loyalty marketing
2. Customer loyalty
3. The communication factor
4. How to increase customer loyalty
5. Customer satisfaction and marketing
6. Customer lifecycle communication
7. Customer loyalty measurement
8. References and further reading
4. **07 C-suite marketing**
1. C-suite and C-suite marketing
2. The importance of C-suite
3. C-suite marketing challenges
4. C-suite persona and persona creation
5. Key success factors in C-suite marketing
6. C-suite contact strategies and tactics
7. C-suite loyalty
8. Measuring C-suite marketing
9. Reference and further reading
5. **08 From product to solutions marketing**
1. Product and portfolio marketing
2. Product marketing: hardware, software and services
3. The shift away from ‘products’
4. Building effective value propositions
5. Towards solutions marketing
6. **09 B2B influencer marketing**
1. What is influencer marketing?
2. Types of influencer marketing
3. Why use influencer marketing?
4. Influencer types
5. The influencer marketing process
6. Influencer marketing for the long term
7. References
12. **PART THREE** Transform through content and digital marketing
1. **10 Digital and content marketing**
1. The digital marketing evolution
2. Changing B2B digital marketing channels
3. Driving forces
4. The changing face of B2B
5. Digital and sales
6. References and further reading
2. **11 Digital marketing strategy and planning**
1. Digital marketing strategy objectives
2. Digital challenges and opportunities
3. Digital marketing value chain
4. Paid, earned and owned
5. Digital marketing measurement
6. Attribution models
7. Digital tools and technologies
8. Reference
3. **12 B2B digital marketing channels**
1. What is a digital channel?
2. The business website: the top priority
3. SEO and search
4. B2B e-mail
5. Mobile marketing
6. Online PR
7. Digital advertising
8. Webinars and webcasts
9. Digital video
10. Communication channel integration
11. References and further reading
4. **13 Content marketing**
1. Content and content marketing
2. The benefits of B2B content marketing
3. Types of content
4. What is great content?
5. Content creation
6. References
5. **14 B2B social media marketing**
1. Social media evolution
2. Benefits of social media for business
3. Social media channels
4. Social sales and marketing
5. The social media marketing process
6. Social media marketing enablers
7. Social and the customer cycle
8. Implementing a B2B social media advocacy programme
9. Social listening and monitoring
10. Social media channel selection
11. Reference and further reading
6. **15 B2B brand building**
1. Introduction
2. B2B brand touch points
3. Brand building obstacles
4. B2B brand building goals and programmes
5. Brand loyalty
6. Importance of brand consistency
7. Brand building stages
8. Reference
13. **PART FOUR** Collaborating with channel partners
1. **16 The modern B2B channel landscape**
1. B2B channel partner marketing
2. Benefits of channel partners and channel partner marketing
3. Channel partner types
4. The changing channel landscape
5. Channel selection
6. Channel partner and marketing alignment
2. **17 Marketing to channel partners and enabling them**
1. Marketing to channel partners
2. Digital communication
3. Offline communication
4. Selecting channel marketing communications
5. Channel partner enablement
6. New approaches: social for marketing to channel partners
7. Marketing to channel partners: challenges and solutions
8. Digital applications
9. Creating one marketing team: vendor and channel partner
3. **18 Marketing through channel partners**
1. Definition
2. Types of marketing through channel partners
3. Selecting and sharing marketing
4. Marketing through channel partners: lead generation
5. Gaining channel partner mindshare
6. Social media and channel partner marketing
7. Through-partner marketing applications
8. Reference
4. **19 Channel marketing strategy and control**
1. Channel marketing strategy
2. B2B channel marketing goals and objectives
3. Channel partner lifecycle marketing
4. Acquisition channel partner marketing
5. Channel partner for customer acquisition
6. Channel contact strategy
7. Channel planning and control
8. Channel marketing budgets
14. **PART FIVE** Optimizing marketing execution
1. **20 Sales and marketing alignment**
1. The importance of alignment
2. Benefits of sales and marketing alignment
3. Organizational obstacles to alignment
4. Establishing sales and marketing alignment
5. Alignment areas
6. Alignment initiative: sales integration
7. References and further reading
2. **21 Account-based marketing**
1. Definition
2. Benefits of ABM
3. ABM types
4. The ABM process
5. ABM methods
6. Is ABM right for you?
7. Measuring ABM
8. References and further reading
3. **22 Lead generation**
1. Introduction
2. Challenges impacting lead generation
3. Types of leads
4. Lead generation and the customer buying cycle
5. The lead generation process
6. Lead generation tactics and strategies
7. Lead generation agencies
8. Lead nurturing
9. Further reading
4. **23 Modern B2B events marketing**
1. B2B events marketing
2. B2B event types and segmentation
3. Events marketing goals
4. Events marketing and busines buying stages
5. Events marketing problems and success factors
6. New events technologies
7. Social media and events marketing
8. Maximizing event lead capture
9. Reference and further reading
5. **24 Modern marketing operations**
1. Marketing operations
2. Marketing data management
3. Marketing budget management
4. Reporting and analytics
5. References and further reading
15. *Index*
16. *Backcover*
# List of Pages
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# Guide
1. Cover
2. Table of Contents
3. Start Reading
# List of figures and tables
FIGURE 1.1 Innovative B2B marketing
FIGURE 2.1 The new marketing mix (updated)
FIGURE 2.2 Marketing mix process
FIGURE 3.1 Traditional vs new B2B marketing planning
FIGURE 3.2 Marketing strategy formulation
FIGURE 3.3 From funnels to cycles
FIGURE 3.4 B2B marketing strategies
FIGURE 4.1 The B2B buyer process
FIGURE 5.1 Customer lifecycle
FIGURE 5.2 Acquisition process
FIGURE 5.3 How to segment for customer acquisition
FIGURE 6.1 Customer loyalty process
FIGURE 6.2 Customer loyalty
FIGURE 6.3 Measuring customer loyalty
FIGURE 7.1 C-suite messaging
FIGURE 7.2 C-suite and their gatekeepers
FIGURE 8.1 Value proposition creation
FIGURE 8.2 Integrated solutions marketing
FIGURE 9.1 Influencer marketing process
FIGURE 9.2 Influencer identifier
FIGURE 10.1 Content marketing tactic usage
FIGURE 11.1 Digital marketing strategy framework
FIGURE 11.2 The digital marketing value chain
FIGURE 11.3 Attribution models
FIGURE 12.1 Hub and spoke
FIGURE 12.2 E-mail and the B2B buyer journey
FIGURE 12.3 Webinar marketing goals
FIGURE 13.1 Content creation process
FIGURE 13.2 Stakeholder content mapping
FIGURE 14.1 B2B social media navigator
FIGURE 14.2 Social media marketing process
FIGURE 14.3 CLC social media
FIGURE 15.1 B2B brand touch points
FIGURE 15.2 B2B brand building process
FIGURE 15.3 B2B brand positioning map
FIGURE 16.1 Main channel routes to market: IT
FIGURE 17.1 Marketing to channel partners (key areas)
FIGURE 18.1 Marketing through channel partners
FIGURE 19.1 Channel partner lifecycle
FIGURE 19.2 Channel partner acquisition process
FIGURE 19.3 Customer acquisition through channel
FIGURE 19.4 Channel partner contact strategy
FIGURE 20.1 Alignment radar
FIGURE 20.2 Key areas of alignment
FIGURE 21.1 ABM process
FIGURE 21.2 The ABM process and key applications
FIGURE 21.3 ABM and the buying cycle
FIGURE 22.1 Leads and the buying cycle
FIGURE 22.2 Lead scoring
FIGURE 22.3 Lead nurture mechanics
FIGURE 23.1 Events and the buyer journey
FIGURE 23.2 Events pre- and post-engaging the customer
FIGURE 24.1 MPM areas
FIGURE 24.2 Budgeting process
TABLE 1.1 The changing shape of industry
TABLE 1.2 B2B marketing types
TABLE 1.3 Estimated number of businesses, UK private sector (thousands)
TABLE 2.1 Marketing mix: mix/trade-off review
TABLE 3.1 Relevant differentiation source
TABLE 5.1 Differences between retention and acquisition
TABLE 6.1 Marketing and the customer lifecycle
TABLE 9.1 Identifying the right influencers
TABLE 10.1 Transactional digital vs relationship digital
TABLE 13.1 Content map
TABLE 16.1 Distributors vs resellers
TABLE 16.2 Marketing and channel partners
TABLE 17.1 Digital and offline marketing vehicles
TABLE 17.2 Audience and communication/content type
TABLE 18.1 Digital and offline marketing vehicles
TABLE 18.2 Leads tracking via channel partners
TABLE 19.1 Goal and channel partners
TABLE 23.1 Social media and events
TABLE 24.1 KPI alignment and goals
# ABOUT THE AUTHOR
Simon Hall is a marketing innovator with over 20 years’ experience in technology and services marketing. In his former career, he served as UK Chief Marketing Officer for Dell, and held senior roles at Acer, Microsoft and Toshiba. In 2016 Simon founded NextGen Marketing Solutions, with the aim to help companies of all sizes capitalize on the latest marketing techniques. He is passionate about pioneering new and exciting initiatives and sharing his knowledge and experience with the current and next generation of marketers, and you can find him regularly speaking at conferences and events, as well as on the board of several associations.
Simon is a course director with the Chartered Institute of Marketing (CIM), where he runs public courses as well as in-company training, a B2B council member with the Institute of Direct and Digital Marketing (IDM), the Internal Consulting Group’s (ICG) Global Thought Leader for B2B Marketing and a member of both the CMO Council and European CMO Council.
# PREFACE
Undeniably, the B2B environment has undergone huge change in the last decade, probably more so than in the 30 years preceding. Business customers are demonstrating new buying habits, the digital era has brought about new ways of marketing, the industry landscape has produced an enormous range of digital applications which impact, support and enable the full spectrum of B2B marketing disciplines, eg content, marketing performance marketing, brand building and channel partner marketing.
This change has meant that a number of the marketing models and approaches developed 20 to 30 years ago for B2B marketers have become less relevant and applicable; these models needed to be adapted, updated or even replaced.
For B2B business and marketing professionals, it may seem overwhelming to try to understand and grasp what is going on across all the different areas of B2B marketing, assuming they have the time to read and interpret literally hundreds of articles. *Innovative B2B Marketing* is a clear, practical guide that demystifies modern aspects of B2B marketing, including new marketing models, processes and thought leadership pieces. This book divides these multiple themes and areas into five core sections: marketing strategy creation, customer-centric marketing, content and digital marketing, channel partner marketing, and marketing related to marketing execution.
You will also find several case studies throughout the book. A good percentage of these come from the information and technology sector, because IT has been the home of digital applications and digital technology, and is therefore one of the first sectors to pilot and implement digital marketing practices.
*Innovative B2B Marketing* is perfect for any marketer working in, or with an interest in, the B2B space. It is also suitable for non-marketers wanting to get a grasp of the core elements of modern B2B marketing. In short, this book is great for marketers at any stage in their career.
# ABOUT THIS BOOK
### Part One: Developing your marketing strategy
The newly evolved business landscape has impacted the traditional marketing mix over the past decades driving a shift away from product-centric approaches to customer-centric and more recently towards a more dynamic marketing mix including social and offline communities. Associated with the changing business, marketing strategy and planning activities have needed to be adapted; this part looks at how to deal with this new landscape and provides a framework for a new marketing mix as well as guidance on how to create and plan strategies.
### Part Two: Improve B2B customer-centric marketing
Customer-centric marketing is about putting the customer at the forefront of any marketing initiative; it’s about viewing customers as valued entities. A customer-centric marketing approach can help a business add value and identify ways to differentiate through leveraging customer relationships and insights.
In this part we look at different stages of the customer lifecycle from customer acquisition through to retention and loyalty. Also included is a section on C-suite marketing; the ability to market to senior business executives is also becoming more important for vendors’ future success and profitability. Customer solutions marketing is about how vendors can market solutions and offerings as opposed to product marketing where the focus is on providing solutions according to customers’ needs. Finally, you can learn about how influencers can be capitalized on for B2B.
### Part Three: Transform through content and digital marketing
With shifts in buyers’ journeys, in media consumption, offline and online channels, and in technologies, it’s more important than ever before for marketers to think strategically and embrace different tactics that support business goals. In this part we’ll cover the main aspects of B2B digital and content marketing, from digital channels to B2B brand building.
### Part Four: Collaborating with channel partners
The channel plays a critical role in supporting businesses in providing solutions, in growth across geographies, as well as many other business goals. Marketing to or through channel partners is an area all B2B marketers still wrestle with. What are the techniques, are there technologies making our lives easier? This and many other questions are addressed in this part.
### Part Five: Optimizing marketing execution
With new digital applications such as marketing automation, CRM, etc, B2B marketers can and need to engage sales better and maximize marketing performance. This part spans topics such as how to engage and market to accounts, how to engage customers better through events, and how to improve lead generation and lead nurturing. Across all these areas B2B marketers need to understand the new role of sales and of marketing and work more effectively as one sales and marketing unit.
Supporting online resources for *Innovative B2B Marketing* can be downloaded from koganpage.com/Innovative-B2B.
# 01
# Introduction to business marketing
# The changing business landscape
In less than 100 years we’ve moved through multiple eras of business and industry, from the production age through to the marketing age and recently from the globalization era to the era of digital and applications.
Various factors have affected a shift away from internationalization and globalization in the 21st century; businesses during this period moved from a model of focusing on globalizing to a more balanced model where businesses benefited from globalization but reverted to a more localized business approach; sometimes called the ‘Glocal’ model.
Business today looks and feels very different even compared to 10 years ago; they need to deal with increased data and information; to understand and capitalize on digital technologies; to operate with rapidly shifting business models and new working behaviours where location becomes less relevant.
As a result of this dynamically changing business environment, business-to-business marketing has needed to change, new models have needed to be developed and old models adapted. This book looks at the core areas of marketing where change has occurred and provides up-to-date tools for B2B marketers to deal with this challenging new business environment.
# What is B2B marketing?
B2B marketing, sometimes referred to as ‘business marketing’ or ‘industrial marketing’, is the practice of individuals or organizations marketing products or services to other companies or organizations. The customer in this case is an organization rather than an individual customer or consumer, though the products, services and solutions they purchase may sometimes be the same or similar.
Table 1.1 The changing shape of industry
| **Business era (20th and 21st centuries)** | **Description** | **Period** |
| Production era | Emphasis on goods production, innovation in production techniques enabling faster manufacturing | 1900–1930 |
| Marketing era | Focus on consumer and satisfying their needs and preferences | Since 1950s |
| Relationship era | Deepening engagement with customers, employees and suppliers | Since 1990s |
| Globalization | Greater focus on internationalisation of business, going global | Since 1990s |
| Internet age | Greater use of the internet for information, for commerce | Since mid-1990s |
| Digital (or Application) age | Characterized by digital applications | Since late 2000s |
## B2B vs B2C marketing
B2B marketing differs from consumer (or B2C) marketing in several ways, such as time to purchase, number of stakeholders involved, complexity of offerings, and the buying process. For example, the buying period for a consumer to purchase a new mobile phone could involve a few days: the buyer is influenced by a positive review of a mobile phone and the next day decides to purchase it. In business this can be very different: a business buying new mobile phones for its salesforce may want to understand more about the technology, capability and business-related services before deciding to purchase. The stakeholder involved in a consumer purchase is typically one person; in a large business there may be multiple departments involved, such as operations, finance, sales and procurement before deciding to purchase.
A B2B marketing audience expects more detailed and richer information as well as different forms of content to help guide them in their decision process. A B2C marketing audience may expect content to be simpler, more engaging and stimulating rather than informative.
### Channel
Despite the many differences there are points of cross-over where B2B is like B2C marketing. For example, businesses targeting small businesses or home offices may use similar distribution channels and retailers as those for consumers; small business owners may use retailers or other consumer channels to purchase goods in small quantities.
### The emotional factor
Consumer purchases are often regarded as being emotionally based, as opposed to being purely rational. In business this may also be the case as those buying for themselves may be small business owners. Even in large organizations an individual who has the ultimate responsibility for procuring for the business may be emotionally influenced as the purchase decision, if it goes wrong, could negatively impact the business and its performance.
### Large audiences to target
Consumer marketing tends to be aimed at large groups of consumers through mass media and retailers; likewise, small businesses and small office users may also fall into large audience groups. For example, a business selling to a small business community needs to send messages to thousands or millions of businesses, if not segmented.
### Segmentation
Another difference between B2C and B2B marketing can occur with segmentation. In B2C segmentation can occur across life stage and life style, eg families, teens, parents, to make marketing and budgets work better. In business the segmentation can be based on organizational factors such as industrial sector, size of business or buying behaviour.
## B2B marketing types
Within B2B marketing there are clear distinctions between marketing to small businesses or small office customers and medium or large enterprises. This distinction is transactional marketing for small business and relationship or enterprise marketing for medium and large businesses. The main differences are summarized in Table 1.2. The core factors are: the type of product or solution involved, which is typically more elaborate and requires more scrutiny; the number of buyers or departments involved; and the drivers of the buying process.
Table 1.2 B2B marketing types
| **Transactional**
**B2B marketing – SOHO/small business** | **Relationship**
**B2B marketing – scaling business/medium/large** |
| Product driven | Trust-based relationships |
| Large target audience = small businesses (>4m in UK) | Small, focused audience (<100k in UK) |
| Simpler buying process | More complex process |
| Small size of purchase; more quantity | Large size of purchase and bids |
| Single stakeholder in buying process | Multi-stakeholder buying process |
| Emotional buying decisions | Emotions invested + rational buying decisions |
| Increasing and generating revenue | Increasing and generating leads
Increasing pipeline and revenue |
B2B insights
According to the latest B2B buyer research published by DemandGen Report (Demandbase, 2015), the world of B2B marketing is continuing to get more complex and buyers more sophisticated. Some of the findings of the survey are:
* 53 per cent of buyers say their time to purchase has increased with 80 per cent of those saying they are taking more time to research and 82 per cent saying they are viewing more sources.
* 43 per cent stated there is an increase in the number of people who are part of the buying committee with 48 per cent saying there are one to three people involved and 35 per cent stating there are four to six.
* Buyers are increasingly using the internet as a source with 53 per cent saying that social media played a role in their purchase decision.
* 52 per cent of B2B buyers stated they viewed two to four pieces of content before making a decision and 28 per cent stated they viewed five to seven.
* 85 per cent stated they wanted a vendor with knowledge of their industry.
All these statistics point to the fact that marketers need to embrace different marketing vehicles and better tailor their messages and content to the customer. Marketing has the challenging role of influencing buyers more than before so that when customers are about to purchase they’re already considering the vendors’ products or services.
# Business market segmentation and size
There are many ways a vendor can segment its business audience; the most typical being by number of employees a company employs. Table 1.3 shows how the UK Department of Business, Innovation and Skills splits the UK business market into four categories: micro, small, medium and large.
Table 1.3 Estimated number of businesses, UK private sector (thousands)
| **Size** | **2012** | **2013** | **2014** | **2015** |
| All private sector business | 4,818 | 4,914 | 5,243 | 5,389 |
| All SME (1–249) | 4,811 | 4,907 | 5,236 | 5,382 |
| Micro (1–9) | 1,023 | 987 | 1,044 | 1,069 |
| Small (10–49) | 178 | 187 | 195 | 204 |
| Medium (50–249) | 30 | 31 | 31 | 33 |
| Large (250+) | 6 | 7 | 7 | 7 |
In 2015, within the UK, the medium and large enterprise segment accounted for less than 1 per cent of the total business population; organizations with fewer than 50 employees accounted for over 99 per cent. According to these figures the medium and large business segment has grown about 20 per cent over the past 10 years (Department for Business Innovation and Skills, 2015).
## Business market segmentation
Segmenting a business market means dividing the market into different homogeneous groups of companies. Some ways to segment business markets are by demographics, buyer behaviour, situational behaviours, industry classifications, by region and operations variables.
The aim of segmentation in B2B is to gather businesses that have similarities into groupings. These groupings help a business’ effectiveness in targeting, engaging and selling to them. Businesses classify and segment audiences not only for marketing reasons but for business reasons such as how they go to market, for structuring sales or entire divisions and for better tracking business performance.
Marketing departments can help with buyer behaviour segmentation by understanding the customer’s likelihood of purchasing a product or service based on sector, previous purchase history, size of business, etc. This is sometimes called ‘propensity modelling’ – the customer’s propensity to purchase certain products is determined by a set of common identified factors or criteria. Arming the business with key information about a customer’s propensity to purchase goods helps sales and marketing be more effective in delivering customer leads.
Behavioural segmentation looks at knowledge, attitudes, usage rate, response rates or readiness stage for a product or service. Marketers can also segment according to multiple geographic sub-segmentations, eg regions, countries, cities, neighbourhoods, or postal codes. A small business providing services may target only local businesses rather than national or international organizations.
## SIC classification system
Businesses can be segmented according to the sector they operate in, using Standard Industrial Classification or SIC codes. Established in the United States in 1937, this is a system for classifying industries by a four-digit code, and is used by government agencies to classify industry areas. The SIC system is used for classifying business activities in the United Kingdom, and it correlates the European Union’s industrial classification system, NACE (Nomenclature Générale des Activités Économiques dans les Communautés Européennes).
Business clusters are important for business marketers and can influence the use of locations and communication
* *Business clusters*: geographical concentrations of interconnected businesses or suppliers in a field that can exist on a local, national or even global scale.
* *High-tech clusters*: technology-oriented groupings, often built around universities or research centres. Examples are Silicon Valley and the South West and Thames Valley areas in the United Kingdom.
* *Know-how-based clusters*: more traditional in their nature and have built up over many years. In London one can find a concentration of financial companies in the City.
* *Government-incentivized clusters*: some countries use economic incentives to attract companies or industries. An example is the software industries and call-centres in India.
* *Low-cost manufacturing clusters*: these clusters have typically emerged in developing countries within particular industries, such as automotive production, electronics, or textiles. Examples include electronics clusters in Mexico (eg Guadalajara) and Argentina (eg Córdoba).
# The digital economy
The digital application economy has been one of the strongest forces of change in the business world. It has brought about multiple new business models, new business growth perspectives and helped shift economies and economic status for some countries. Some examples of these new business models are e-commerce, the cloud and online advertising.
E-commerce in the form of online transactions has impacted almost all traditional business forms; some businesses have made a move to a full e-commerce approach while others have opted for a hybrid approach. One example is buying flight tickets, where purchasing and checking-in have almost fully moved online.
Cloud business, where services are provided via the internet, has the key benefits of lower set-up costs, fewer operational resources requirements, flexibility, ability to grow and scale, and ease of collaboration. Examples of cloud date back to the 1990s where free e-mail services in the cloud were provided by Hotmail, Yahoo, Google and so on.
Online advertising has also arrived and grown as a result of the digital economy. New advertising-based models include free or subsidized services and new payment methods such as cost-per-mille (CPM), cost-per-action (CPA) and cost-per-click (CPC).
# Core trends and this book
The past 10 years have been characterized by several specific trends in B2B marketing (see Figure 1.1); this book takes into consideration those core trends as well as essential topics to support B2B marketers in developing more innovative marketing plans and strategies and optimizing their execution:
**Core trend 1: Increased customer-centricity**
With changing business behaviours and changing organization dynamics B2B marketers need to have a better grasp of organizational influences, and to better engage and communicate with businesses and stakeholders through more targeted and tailored communication and content. C-suite marketing and influencer marketing have become hot topics in B2B as they are about leveraging different relationships and influencers all to be found in Chapters 4 to 9.
**Core trend 2: Rise of digital and content marketing**
There has been a shift in marketing communications to include a greater emphasis on content; in turn content formats have diversified, and generally become richer. There has also been a move away from interruptive marketing towards inbound marketing. Interruptive marketing is promoting a product or service through continued advertising, promotions, public relations and sales; sometimes it is regarded as an annoyance rather than a help for customers. With inbound marketing customers are receptive to your marketing messages and it is marketing that attracts and pulls customers, rather than marketing pushing messages. Ten years ago marketers didn’t have the capability to track customer buying behaviour or media consumption across multiple marketing vehicles; also customers didn’t use such a variety of marketing vehicles in one purchase. Digital and associated technologies have changed all that: businesses today are able to track customer media consumption and journey in a more connected manner. Digital applications allow companies to understand how customers engage across digital channels such as mobile, blogging and social, how many pieces of content they use, and how they engage with such content. Chapters 10 to 15 shed light on how to leverage new B2B digital and content marketing to support the business.
**Core trend 3: From single routes to market to multi-channel marketing**
Businesses and marketers were once constrained to think about singular physical routes to market, directly or indirectly. With the growth of digital technologies and applications companies today need to embrace digital and physical distribution channels and integrate them in their plans, to support and accelerate their growth. Chapters 16 to 19 outline key B2B channels as well as different forms of marketing through and with channel partners.
**Core trend 4: Increased pressure to optimize marketing execution**
With new digital applications such as marketing automation, CRM, etc, marketers today can and need to engage sales better and maximize marketing performance. With the changing nature of buying decisions, the tactics marketers use to generate leads have become richer, and new technologies and digital applications have increased the ability to track customers’ behaviours across different communication channels. You’ll find more about this in Chapters 20 to 24.
Figure 1.1 Innovative B2B marketing
Innovative B2B marketing
# References and further reading
AT&T, Cisco, Citi, PwC and SAP (2011) The new digital economy: how it will transform business, Oxford Economics, Oxford, available at: http://www.pwc.com/mt/en/publications/assets/the-new-digital-economy.pdf (accessed 8 February 2017)
Demandbase (2015) B2B buyers survey report, available at: http://www.demandgenreport.com/resources/reports/the-2015-b2b-buyer-s-survey (accessed 8 February 2017)
Department for Business Innovation and Skills (2015) *Business Population Estimated for the UK and Regions 2015*, DBIS, London
Rayna, T (2008) *Understanding the Challenges of the Digital Economy: The nature of digital goods*, Imperial College, London
# PART ONE
# Developing your marketing strategy
# 02
# The new marketing mix
This chapter will give you an understanding of:
* the benefits of the marketing mix
* the new marketing mix
* the marketing mix process
* how to trade-off on marketing mix elements
# The marketing mix defined
The marketing mix is a framework for marketers to strategize, plan and execute their marketing activities. The marketing mix was originally defined as putting the right product (or combination thereof) in the right place at the right time and at the right price, sometimes called the 4Ps. Some of the original work on the 4Ps has evolved to be more customer-centric.
The purpose of the marketing mix is to capture all the key variables that need to be considered in marketing strategy, planning and implementation, and to reach an optimum balance between the marketing variables. One focus of the marketing mix has remained core to all recent marketing mix approaches: that the needs of the target customers are met. The challenges in defining the marketing mix are many, such as market forces and changes in business environments and customers’ needs.
# Is the marketing mix still relevant?
The marketing mix provides multiple benefits for marketers; it is a framework for assessing resource allocation within the team. For example, where the marketing department is required to put greater emphasis on social media or digital marketing, the marketing head of department may need to consider allocating more resources.
The marketing mix also serves as a checklist for marketers, whether in creating strategy, planning, implementing or reviewing activities. It’s a steer for responsibilities: each mix element can have accountable owners or one owner, for example some can be accountable for channel marketing, others for pricing, others for solutions marketing. It also facilitates better internal communication due to assigned SPOCs (sales point of contacts), so sales understand who to go to for what; in turn clearer communication reduces confusion or conflict between sales and marketing departments.
# The shift away from the four Ps
The original marketing mix was associated with the four Ps: price, product, promotion, and place. It assumed little difference existed between B2B marketing and B2C marketing, marketing of products in volume or marketing of complex solutions, and marketing supporting a simple sales process compared to marketing to customers requiring higher sales touches.
After the 1980s it was felt the original four Ps were centred on the business rather than the customers and an adapted set of mix variables or descriptors was required to become more customer-centric. The four Ps were adapted as follows:
* *Product* was originally about tangible goods or intangible services a customer requires. Most products under the original 4Ps were considered subject to a product lifecycle, from launch to end of life. The changed focus was centred on customer solutions.
* *Place* was originally about where the product was found and its associated distribution channel; this has evolved into channel and accessibility to the product or service and how convenient it is for customers to receive or access products and services.
* *Price*: the focus on the amount the customer pays for the product shifted to cost; cost became the new determining factor behind pricing where marketers help businesses correctly price products or service according to needs or buying capabilities.
* *Promotion* represented different communication vehicles; ‘communication’ or channels of communication became the new elements, where they allow for customers wanting two-way communication with the companies that make the products or services.
Since 2000 there has been a greater move to the internet as a buying channel, which has given customers a greater ease in purchasing. The ability to access data through more powerful mobile devices means customers and business customers could use different types of content. There has been a greater richness in the content customers use, which has placed more demands on business to align through content marketing. Additionally, with more sophistication in social platforms and more real-time usage, business customers can engage quicker, better and with more people than even five years ago.
# The new B2B marketing mix
These influences have brought about a new B2B marketing mix (see Figure 2.1) and the five Cs to help marketers deal with the new B2B situation:
**Channels of communication**
Since the 1990s communication channels have changed to reflect a higher mix of digital communication. Existing digital channels have also evolved and new channels have emerged such as social, digital PR, and webinars.
**Content**
Tied to communication channels, content has increased in importance. B2B marketers are now dealing with more diversified and richer content formats and more content overall. Businesses also have more possibilities to create content.
**Customer solutions marketing**
This has evolved due to commoditization of products in areas such as telecommunications and IT, as well as customers who expect more than simply products. This trend has influenced companies to think of true solutions as well as end-to-end solutions. Social listening and digital monitoring tools have allowed companies to track and listen to customers and come up with solutions that are more relevant and targeted.
**Channels to market**
This area has changed to include a digital channel mix. Some companies will offer multiple options for customers to purchase products, potentially with different customer experiences. Companies can face the challenge of managing an omnichannel approach due to the transparency brought about by digital and customers demanding greater consistency between channel offerings.
**Communities**
These are groups or grouping of people, mainly centred on common interests or themes. Social has played a big part in bringing these communities together and in reinforcing offline communities. Due to the impact communities have on businesses it’s important for marketers to adopt communities as part of their mix, whether centred on associations, offline groups, memberships or social forums.
Figure 2.1 The new marketing mix (updated)
The new marketing mix (updated)
© Simon Hall, 2017
With the need for businesses and marketers to meet the needs of business customers through combined partner approaches, whether alliances, channel partners or marketing partners, collaborative marketing has become even more important.
## How to determine the marketing mix
Marketers need to determine the optimum mix of the different marketing Cs to satisfy the needs and wants of the customer and create maximum impact from their activities. With customers’ needs and environmental factors in constant flux this is not always an easy task. In the words of Philip Kotler, ‘Marketing mix represents the setting of the firm’s marketing decision variables at a particular point of time’ (Chand, 2016).
**TIP** The marketing mix process
1. Target customer identification: this first step is about defining the target customer; the more detailed the description the better the plan and quality of marketing mix elements.
2. Needs and requirement analysis: once the target customer is identified a needs/requirement analysis should be carried out; this can use in-house resources and knowledge such as customer service feedback, sales account plans, customer input and forums where customer insights can be collected directly. Marketers should use the analysis to define buying power, motives of customers, and key current and future needs associated with the business offerings.
3. Once customer research is complete the next step is to look at product offerings and solutions aligned to the customer and their needs.
4. The next stage is to determine the channel to the customer; for some customers the organization may need to offer alternative routes to purchase, eg online and by phone.
5. This step is about selecting the communication channel and content formats according to the previous steps.
6. Next is aligning communities, and cost as well as price structure; for example, depending on channel the price and cost structure will be different.
7. Before finalizing the mix the elements are reviewed in light of constraints or parameters defined by the business or marketing departments. For example, the route to market may require adjustment through business strategy plans; the price, pricing policy and cost may need operations and sales buy-in or may need market testing; and channels of communication and content may be determined by the marketing budget.
8. Implementation – in this phase the marketing mix is implemented and reviewed on a regular basis to assess the need for adjustment.
In general, parameters can come in the form of budget limitations, sales-force limitations, business growth stage and channel development.
# Marketing mix and trade-offs
In defining the marketing mix the most difficult task for marketers is to incorporate trade-offs. For example, if marketing decides on a low price then it is likely there will be less budget to advertise; if marketing decides on a heavily tiered route to market the budget will need to be able to support this approach.
## Deciding on the trade-offs
Where to trade-off and how needs to be decided according to the target customer segment. The first step in the marketing mix could be to look at customer solutions to fulfil customers’ current or future needs (see Figure 2.2). Following that, the optimum route to deliver the solution to the customer can be decided upon, depending on the cost and the price the customer is prepared to pay, or according to market forces. With this knowledge, the content and its forms can be built according to the communication mix; eg where it’s a more complex solution a webinar, slideshow and face-to-face events may form part of the mix. If the customer is a large enterprise, the trade-offs may be between richer and tailored content to reach specific target stakeholders, or multiple formats of less rich content that support a corporate reseller and engage the same customer on the vendors’ behalf.
Figure 2.2 Marketing mix process
Marketing mix process
The channels of communication will depend on how much marketing budget there is, how many resellers the business is considering and the bandwidth to provide and promote directly or indirectly. If the strategy is to reach and influence an executive set of decision makers via social media at the customer side, the trade-off on content and channels may be to over-index content for social platforms rather than advertise through other digital channels or even offline channels. Such trade-off decisions can be mapped against parameters and interdependencies can be called out or highlighted on a grid like the one shown in Table 2.1.
Table 2.1 Marketing mix: mix/trade-off review
| **Trade-off review** | **Communication and content** | **Channels** | **Customer solution** | **Cost** | **Community** | **Collaborative partnership** |
| **Communication and content** | X | | | | | |
| **Channels** | | X | | | | |
| **Customer solution** | | | X | | | |
| **Cost** | | | | X | | |
| **Community** | | | | | X | |
| **Collaborative partnerships** | | | | | | X |
| **FOCUS**
**Communications and content** | **Channels** | **Customer solution** | **Cost** | **Community** | **Collaborative partnership** |
| VARIABLE A: Number of content formats | | | | | |
| VARIABLE B: Content mix | | | | | |
| VARIABLE C: Digital channel mix | | | | | |
Other external factors affecting marketing mix decisions can include the following:
* Shift in customers’ buying behaviour – where customers make a shift to buy through a channel rather than directly. Customers’ change to using more social or less TV will influence media/vehicle mix.
* Technologies changing the game, for example where customers move from booking a flight via travel operators to buying directly from the airline online.
* Trade behaviour, where customers find they can benefit from easier purchasing across borders.
* Competitor consolidation – where a consolidation of providers leads to the remaining vendors having to market harder to highlight their benefits.
* One organization leaving a market provides an opportunity for others, through aggressive acquisition, to sweep up customers left adrift.
* Government legislation such as placing a ceiling on prices or restricting price dumping will affect a company’s approach to price setting.
ACTIVITIES
Review your recent marketing plan for the new marketing mix and the five Cs. Where do you see opportunities to adapt in terms of collaborative partnerships or in taking advantage of customer communities?
Choose one marketing mix element and look at decision making and dependencies between that one and the other marketing mix elements.
# Reference
Chand, S (2016) Marketing-mix: definition, nature and determining the marketing-mix, available at: http://www.yourarticlelibrary.com/marketing/marketing-management/marketing-mix-definition-nature-and-determining-the-marketing-mix/27951 (accessed 10 February 2017)
# 03
# B2B marketing strategy and planning
This chapter will give you an understanding of:
* B2B marketing strategy and planning
* the marketing planning process
* market audits and key aspects
* how to formulate marketing strategy
* how to develop effective B2B strategies
# Introduction
A marketing plan outlines a company’s marketing activities for a coming period whether three months, a year or longer. It describes the business activities involved in achieving specific marketing objectives within a given period. Typically, those using marketing plans are marketing departments to ensure key elements are implemented and reviewed. Other departments such as sales and finance will need to view marketing plans to understand what marketing delivers to the business.
The planning process entails two core activity areas: marketing research to understand the market, competition, environment and customers; and planning the marketing mix. In this chapter we look at marketing planning in a B2B context, which has undergone many new influences, meaning that some of the more traditional marketing planning models need adapting to address such changes.
# The traditional and new B2B marketing planning process
The process of marketing planning usually follows a number of stages from the development of objectives, through to strategy setting, the implementation of activities to achieve the strategic objectives, and the control and review of how the marketing implementation occurred. Figure 3.1 outlines the main steps and some adaptations based on changes in B2B marketing that have occurred in the past 10 to 15 years.
Figure 3.1 Traditional vs new B2B marketing planning
Traditional vs new B2B marketing planning
## Goal setting
Within this stage, goals are clarified that align to the business overall mission and business purpose. Marketing alongside other departments needs to work to support the business and corporate goals and marketing need to understand and appreciate corporate objectives. Goal setting in the 21st century needs to allow for more flexibility due to the rapid shifts in technologies, customer behaviour, B2B social marketing potential and digital technologies.
## The audit – situation analysis
The next main stage is situation analysis and during this phase a marketing audit is carried out. The marketing audit helps marketers analyse and evaluate the marketing strategies, current activities, goals and how marketing is performing against those marketing and business metrics.
New aspects that B2B marketers need to invest in are a situation analysis of business customers in terms of needs, behaviours, buying process and other related aspects; with the rapidly evolving changes in customers and their context, existing assumptions from marketing departments can quickly become out of date. Also new to the audit should be technology landscape and audits associated with the business type and marketing type.
## Marketing strategy
The next main phase is the development of the marketing strategy, which involves developing objectives and addressing the outcomes of a SWOT analysis (strengths, weaknesses, opportunities and threats). During this phase weaknesses and threats are defined which the business should address, and opportunities are incorporated into the plan, leveraging existing strengths. The marketing objectives reflect what the business is wanting to achieve in the coming period. Marketing goals usually follow a SMART approach (specific, measurable, achievable, results based and timed). After identification of opportunities and challenges, the next step is to develop marketing objectives that indicate the end state to achieve.
## Marketing planning
Following the strategy formulation, the next stage is to define the marketing plan for the new marketing mix; based on mix definition the plan is created, which includes overall goals, timings and customers to target.
## Implementation and evaluation
During this phase the marketing activities are allocated resources, and activities implemented and evaluated. Marketing budget is determined and reviewed based on defined outcomes and therefore return on investment (ROI). The marketing team implements plans by launching programmes and campaigns, and spends budgets on various activities. During this phase evaluation takes place; evaluation can be an informal review.
New digital technologies as well as changes in business customers using more digital applications means marketers can more easily track and monitor customers and behaviours. Digital applications such as marketing automation and CRM support marketing departments in implementing and monitoring activities as well as tracking spend.
# B2B marketing audits
Comprehensive marketing audits include the following.
## SWOT analysis
The SWOT analysis (strengths, weaknesses, opportunities and threats) should answer key questions such as what strengths customers believe your business has, what it could do better, what weaknesses it needs to address, and what opportunities it needs to exploit. Conducting this analysis helps the business understand how it performs in the eyes of the customer.
**TIP** Use the SWOT to build customer relationships
The questions related to the SWOT can be a way to build better relationships with customers as the business is seen to be listening to its customers; of course, it is important for the business to subsequently address those concerns from a customer’s perspective. Opportunities and threats need to be understood so companies can deal with threats and go after opportunities.
## Competitor analysis
During this part of the audit competitors are also reviewed using SWOT. The outcome from competitor analysis can be defensive as well as offensive. This analysis should include a strategic as well as tactical component as this helps your business incorporate short-term tactics as well as longer-term turn-around activities. The objective here is to identify strengths to match, weaknesses to exploit and to predict potential opportunities that the competitor may capitalize on, and to prevent this through offensive and defensive tactics.
Defensive tactics may be short term in dealing with price, promotion or other competitive tactics that impact current business performance. Offensive approaches are usually longer term and strategic in nature.
## External market factor review
Of course, competitor analyses are part of an external audit; there are several other factors that need to be included, distilled into the PESTEL model, ie political, economic, social, technological, environmental and legal aspects. For example, political factors such as elections can impact business confidence; new technologies coming to market may lead to a downturn in purchases of current technology or, where business stability is key and staying with older technology deemed to be stable and reliable, current technology could be seen as a more attractive purchase. Economic factors such as a recession may impact the business overall; new legislation such as laws about the use of cookies or storage of contact details and privacy can impact how businesses market to customers.
## Customer research
During this part of the audit a business directly or indirectly researches into customers’ needs; their decision criteria; why, when and how frequently they buy; and why they buy certain products and not others. In defining the customers’ needs it’s prudent to understand those of both current and future potential customers. Often the tendency is to focus on today’s customers and their needs rather than define future potential customers and future needs. Incorporating this into customer research will help businesses position themselves for the future.
## Internal marketing review
The current marketing plan, activities and performance are reviewed. In a B2B marketing context this can be carried out via a digital review of online clicks and conversion through to deals, or through a relationship marketing approach reviewing how leads or outbound activities convert to opportunities, pipeline and revenue. Sub-segmenting activities according to the marketing source can help marketing assess the performance at a tactical level as well as support marketing in addressing those performance gaps.
# Marketing strategy formulation
The formulating of the marketing strategy starts with competitive advantage and differentiation review (see Figure 3.2).
Figure 3.2 Marketing strategy formulation
Marketing strategy formulation
Competitive advantage can be identified by conducting a SWOT and a competitor analysis. Differentiators can be identified by conducting a differentiator review, ie a gap analysis of major competitors. Traditionally B2B organizations differentiated themselves through new technologies or new product offerings, or better cost or price and/or geographical presence. With increased globalization and commoditization of products and technologies, differentiation now tends to arise from digital applications, organizational reputation, and omnichannel purchase possibilities.
Much has been written about competitive advantage or unique value propositions; B2B marketers need to think about ‘relevant’ differentiation. This is not only about having an advantage over competitors that is unique, it is vital that it is aligned to customers’ needs.
Table 3.1 Relevant differentiation source
| **Channel mix** | **Customer targeting** | **Perception and CSR** | **Customer engagement** | **New digital applications** |
| Mix of online/offline
Change routes to market
Offer online possibilities | 1) Target niche customers
2) Leverage account-based marketing | Invest in perception building and corporate social responsibility | 1) Offer possibilities to engage before purchase
2) New educational material and formats
Social engagement
CLC marketing | New service and application offering |
# The marketing funnel and beyond
Marketing funnels have been used extensively by marketers in the consumer and B2B environments and have been effective tools in planning, implementing and reviewing. Using funnels can be misleading as it assumes a linear view of customers’ behaviour in marketing activities; it also assumes that a business starts with a large target customer set and steadily shrinks down to smaller customers to engage.
While funnels can be useful as ways to track media, activities and leads, marketers should consider that customers’ behaviour is less linear, their decision process is more organic and receptive to multiple stimuli from the market and other customers. Also, customers tend to follow a cyclical path in purchasing and deciding on purchases; see Figure 3.3.
Figure 3.3 From funnels to cycles
From funnels to cycles
## Applying the customer cycle
Essentially the marketing funnel is there to help marketers plan from one end of the customer journey to the next. For planning purposes marketers still need to consider that those who are aware of or are becoming aware of needs will be a larger group than those exploring options or purchases. Where things differ is the ability of companies to better target customers and prospects and in being able to re-engage customers through new approaches in marketing and new technologies. Businesses are potentially more capable of identifying and engaging customers and maintaining a greater engagement and connection with prospects across the customer purchase journey.
Also new in 21st century approaches is the ability of companies to influence customers indirectly through influencers, partners and other third parties. Collaborating with third parties to identify and engage customers is required rather than recommended.
# B2B marketing strategies
Marketing strategy is fundamentally about gaining and sustaining competitive edge; it should facilitate and support the business’s growth. It needs to includes short- and long-term activities. Most of the current successful B2B marketing strategies can be developed along the areas covered in this book (see Figure 3.4):
1. *Customer-centric strategies* that focus on areas of the customer lifecycle, or focus on a niche target segment such as C-suite, or crafting a strategy based on true solutions for customer needs. Influencer marketing is now evolving in B2B as an effective way to gain mind-share and reach new audiences. Other possibilities are to refocus business and marketing investments across the customer cycle: vertical sector orientation means tailoring marketing content, messaging and budget to specific vertical markets.
2. *Transform through digital and content.* Other ways are to define a new digital marketing mix or create compelling content. Some successful strategies have included capitalizing on social and leveraging platforms to support customer relationship management and amplification of brand-building programmes. More frequent are B2B brands that invest to reposition themselves or to change perceptions of customers. Where the perception of the business differs from the reality or the desired view of the brand, a business will decide to invest in communication and branding that changes the customers’ perceptions.
3. *Collaborate with channel partners.* Some B2B brands find success in redefining relationships with existing channel partners or creating new channel partners. An example would be to work with an independent software vendor that has experience of working in a vertical segment and its customers.
4. *Improving marketing execution.* The fourth area relates to improving areas of marketing execution, either by improving the quality of lead generation and nurturing, becoming more targeted with budgets and resources in account-based marketing (ABM), through greater sales and marketing alignment to be more effective as a business in achieving goals, or through improving events marketing.
5. *Collaboration with partners.* Across all the above; marketing partnerships can serve to further strengthen the strategy as well as improve the probability of achieving strategic goals. An example would be collaborating with an industrial association and its marketing resources to reach an industrial business segment.
Figure 3.4 B2B marketing strategies
B2B marketing strategies
**EXAMPLE** CSR and business marketing strategy
It has become more and more important for companies to show social responsibility, and having a Corporate Social Responsibility policy in place has been a growing trend. CSR is a way for businesses to connect on an emotional level, to help serve the communities and to demonstrate responsibility to the environment. As a knock-on effect this approach helps businesses distinguish themselves from other companies. B2B businesses may support CSR initiatives through environmental issues and employee programmes. For some, CSR has been an investment issue: public organizations may view purchasing from companies with a CSR initiative as a plus as they can indirectly support those companies in their CSR initiatives.
Grainger is a B2B distributor of products used to maintain, repair or operate facilities. Many businesses and institutions across the world rely on Grainger for pumps, motors, hand tools, janitorial supplies, fasteners and much more. Grainger has put particular emphasis on CSR across four core areas of people, operating responsibility, sustaining environment and serving communities.
# PART TWO
# Improve B2B customer-centric marketing
# Business customers and buying behaviours
This chapter will give you an understanding of:
* the importance of business buying behaviours
* the new business buying process
* buyer triggers and how to identify them
* identifying business buyer behaviours
* influences on purchase decisions
# The importance of business buying behaviours
Critical to your success as a B2B marketer in creating marketing strategies, plans and activities is understanding how businesses make purchase decisions, how they move from one stage to the next and what influences them within and between the stages.
## The buying stages
Six main stages, shown in Figure 4.1, can be identified as part of the business buying process:
1. *Need recognition.* The business recognizes a potential need; the recognition can come from an individual or a group of people within the business and may be the output from a study or other outside stimuli such as vendors or partners. Typically, the need is solved by purchasing a product, a service, or a portfolio of products.
2. *Need quantification and research*. Once a need is identified the next step is to gain commitment to fulfilling the need; in larger businesses this can be a department convincing stakeholders to release capital to pay for a product or service.
3. *Vendor review.* During this stage, the people involved in the buying process seek out information and search for vendors who could supply potential solutions to their needs. Most buyers start with an online search, which can then be followed up with attendance at seminars, trade shows or further searches online. Smaller businesses as well as bigger ones may use contacts as sources of information.
The potential suppliers are then evaluated and suppliers compared. Typically, buyers will weigh vendor alternatives based on a set of purchasing criteria. Different organizations will weigh parts of a proposal differently, depending on their goals and the products they purchase. For example, price may be an important factor for some whereas others may place an emphasis on service and service level agreements. Larger business customers have a more structured process to tender for vendors through RFPs (request for proposals).
4. *Purchase.* Based on findings during the evaluation phase a customer selects a vendor or vendors(s) and proceeds to ordering. The ordering of the products or services can be structured within a longer-term agreement or may be a simple transaction.
5. *Post-purchase.* Following the initial purchase the customer could repurchase from the vendors when necessary, move to a new vendor, revert to a previous vendor or increase purchasing activity.
Figure 4.1 The B2B buyer process
The B2B buyer process
# Identifying business buying behaviours
Identifying customer buying behaviour can help in business planning across logistics and finance. It can also remove the element of surprise; for example if a business establishes that two-thirds of its customers conduct all their repeat purchases in one quarter, it can plan its resources accordingly. The education sector tends to make most of its purchases of computing equipment in the second quarter, ie between April and June. Other benefits come from improved marketing and improved selling as sales and marketing can engage customers better, in the right way at the right time.
**TIP** How to identify buying behaviours
Identifying business buying behaviour includes answering key questions such as how customers buy, when and where they buy, how they inform themselves about purchases and who they engage. Identifying buying behaviour is also about understanding business perspectives and outlooks.
Organizations can identify buying behaviours through different approaches; they could set up customer advisory councils or conduct a survey. Other approaches include learning from sales account managers who have in-depth understanding of their customer accounts; or leveraging third-party associations that can engage customers on the organization’s behalf.
## What to do with the knowledge
Businesses should use their findings to look at commonality in behaviours between business types. This can be based on a similar industry sector or similar business size. Capturing information based on sector or size can help businesses structure their organization; marketing can look at channels to engage, communications routes to use, ways to engage, and using more relevant language.
# Business buying influencers
## Internal
Purchase process and complexity of purchase
Purchasing can be affected by the complexity of the product or solution; where it is a single product or one bought in smaller amounts the purchase process may be a simple one. For example, a quick review of specifications is all that is required prior to purchasing online; this process resembles that of consumer buying. Where the product or solution is more complex, such as IT infrastructure for a larger business, the purchase process can be longer due to the number of stakeholders involved; there are also typically more steps involved and more time required to move between the buying stages.
**The size of the order or number of products ordered**
Where the number of products ordered is large the time to purchase typically takes longer, as more products need to be evaluated.
**Capital outlay**
Where the capital outlay is sizeable, the customer will need more time to review and select a product or service as there are potentially more stakeholders involved in understanding the justification of the spend. In larger organizations it may also require more time, decisions and stakeholders to ‘free’ up capital.
**Company objectives**
A company’s goal to grow the business at different speeds at different stages may affect how and when it buys.
**Purchasing policies**
Purchase policies can influence the buying process, and the size of the procurement department plays a role.
**Fiscal timelines**
A company’s fiscal structures and calendar, such as its financial year, can impact when products are purchased. For example, Dell’s fiscal year runs from February to January whereas Microsoft’s is from July to June.
**Organizational structures and interpersonal factors**
How people engage as a result of stakeholder personalities, organizational structures or rigid processes could influence the buying process negatively or positively. For example, where the steps in purchasing are cumbersome this may protect against invalid purchases but also dissuade from pursuing opportunities to purchase something that eventually helps the business.
**Size of business**
Smaller companies will have fewer processes, steps, policies and stakeholders, which could mean that decisions can be made faster. The flip side is that due diligence in reviewing and evaluating may be skipped, leading to less than optimal purchases.
## External
**Competitive influences** may be changes in competitive activity that impact short- or long-term purchase decisions. New entrants to the market that employ disruptive pricing or terms and conditions can affect purchasing quantities or timing of decisions.
**Legal factors** affecting purchase decisions may be related to the financial and stock markets and black-out periods.
**Technological factors** may be new technology that renders existing technologies obsolete; customers may decide to shift to the newer technology.
**Governmental aspects** include businesses delaying or accelerating purchases ahead of governmental election results.
# Buyer types
In organizations, buying is usually undertaken by two or more individuals and so it is often referred to as ‘group buying’. Sometimes the buying process will involve technical issues or financial decisions whereby finance work with procurement on the best purchase for the business. As organizations grow and evolve so do the number of stakeholders and roles within the organization that influence the buying process. In general, the main types of buyers or roles are:
* *Users.* These are people who will directly use the product or service; in business, this can be a group of individuals who would benefit from the purchase.
* *Influencers*. These are individuals or groups who steer or impact the purchase decision; they may be experts in an area, they may even come from outside the organization where specialist knowledge not found within the company is required.
* *Buyers.* In smaller companies buyers and users may be the same; in larger companies, they can be separate, even from different departments. They typically have formal authority to select vendors but may not have the power to make the final decision.
* *Deciders.* These can be the same as buyers in some organizations or for some purchases. Deciders have the ultimate power to make decisions on purchases; they can be CEOs or heads of department.
* *Gatekeepers.* These formally control information and access to other groups in the buying process.
# Types of buying situation
Business buying can take different forms depending on a number of factors, which can include the internal and external influences mentioned above. The main forms of buying situations are repeat purchase, modified re-purchase and new purchase.
## Repeat and modified re-purchase
Repeat purchases of the same product or set of products may, if straightforward, occur without any engagement. *Modified re-purchase* is where the buyer amends the previous purchase; the modifications can be:
* *To include new products or services.* An example could be a business buying notebooks and modifying the purchase to include additional or different notebooks due to new processes or technologies.
* *To account for new prices or used to renegotiate new prices*. This could be a business renting office space from a provider and, based on changes in the office rental market, may mean the business wants to renegotiate a better (lower) price.
* *New packaging or other customization.* The buyer may want different packaging or delivery, or the product customized slightly differently.
## New purchase
New purchase can come in the form of spot purchase – products are bought as a one-off. This type of purchase may occur online if the product in question is easily understood and the purchase process is straightforward. Alternatively, new purchases can be a range or portfolio of products; the buyer may expect phone contact or a face-to-face meeting to better understand the product or portfolio of products under consideration.
**EXAMPLE** Domain website service vendors and purchasers
A new purchase could be the customer making the first purchase with the website provider. Repeat purchase could be a repeat service purchase on a monthly or annual basis. A modified re-purchase could be buying extended services such as access to richer content and images or expert online support services that the website domain provider offers.
# Changes to business buying behaviour
Business buying behaviour has altered greatly in the past decade. Business buyers used to rely on inputs from vendors for the majority of information for the purchasing process. Because of the wide availability of information online, customers are now able to research on their own and evaluate potential solutions without needing to engage vendors directly for a large part of the buying process.
Nowadays, business customers will typically engage vendors much later in the buying process and the process has shifted from vendor-led to customer-led. This aspect of customer activity means that customers could either be better informed or more misinformed, where they form unrealistic opinions of a possible vendor. Potential vendors may lose out on opportunities to pitch and explain their business offerings.
## Selling and the customer process
With customers changing how they engage sales, vendors also need to adapt. Customers decide to make their first engagement with potential vendors further along the buying process, so sales departments need to leverage marketing’s support, to influence customers through content and information, supporting them in their purchase process.
The process can shorten as sales staff take on the role of facilitator by providing information via their communication channels. Some businesses have seen real benefit in accelerating buying processes through employee advocacy, as their networks help customers access information more readily, directly or indirectly.
# Buyer triggers
A buyer trigger is an event that causes a customer to recognize a clear need. It usually moves a buyer from a state of curiosity or consideration to a more urgent state of need.
**EXAMPLE** Microsoft XP end-of-life support
In April 2014 Microsoft withdrew its support for the Windows XP operating system; this was announced some time in advance to allow customers and businesses to take the change on board. This in turn led to different buying behaviours: some businesses made short-term decisions to purchase new computer products with the more up-to-date operating system; other bigger organizations invested large sums of money in purchasing and extending support. For example, the Dutch government signed a ‘multi-million euro’ deal with Microsoft for continued support for its Windows XP systems.
Other examples of business triggers are:
* a product fails and is out of warranty;
* a warranty period is about to expire;
* a company intends to move to the next stage of growth and buy new office space;
* a new technology and related products are launched.
## Identifying and exploiting buyer triggers
Organizations can be effective in sales and marketing by identifying and exploiting customer triggers. Identification can come from monitoring and listening activities that help to not only identify triggers but anticipate them. Some possible routes to buyer trigger identification are:
* Taking part in a social forum where customers participate.
* Undertaking needs assessment research with existing/new customers.
* Monitoring market trends through the web or research firms.
* Monitoring technology trends.
* Engaging industry associations that represent the target customer segment.
Organizations can capitalize on the triggers by some of the following:
* Creating customer persona(s) for the trigger event identified.
* Looking at potential reactions the trigger event may initiate.
* Including messaging and content that tap into trigger event reactions.
* Ensuring messaging includes a solution your business can offer for the trigger event.
ACTIVITIES
Look at your own business and industry and try to identify trigger events.
As a customer yourself can you notice any trigger events that have caused you to buy something?
Think about purchase influences for customers you’re marketing to and identify at least three internal and external influences.
# Further reading
Forbes Insights (2009) The rise of the digital C-suite: how executives locate and filter business information, *Forbes Insights*, June
# 05
# Acquisition marketing
This chapter will give you an understanding of:
* the REAP model
* differences between retention and acquisition customers
* types of acquisition marketing
* the acquisition marketing process
* re-acquisition marketing
* re-acquisition marketing tactics
* acquisition marketing strategies
* measuring acquisition marketing
# Customer acquisition and customer lifecycle
Customer acquisition refers to the activity of identifying and engaging potential customers for the purpose of persuading them to purchase a company’s products and/or services. Potential customers are sometimes called ‘prospects’.
## Customer lifecycle
A customer lifecycle outlines the progression of steps a customer makes from point of awareness of an organization’s products to the point of becoming a repeat customer of an organization. The main phases are reach, acquisition, onboarding, retention, upsell and cross-sell, and loyalty. The main phases, shown in Figure 5.1, can be described as:
* Reach. Where an organization targets a customer and tries to get the attention of the businesses they want to reach.
* *Acquisition*. Where an organization attracts and engages the business for purchasing its products and/or services.
* *Onboarding*. This involves the prospect becoming familiar with the organization’s products and/or services, understanding and making use of the offerings.
* *Repeat – retention*. During this phase the organization sells its products to the customer. The customer is a repeat customer for similar products and/or services.
* *Expansion – upsell and cross-sell*. During this phase the organization looks to expand its offering to the customer through upselling and/or cross-selling.
* *Loyalty*. Loyal customers can be defined by the number of purchases made over a period of time.
Figure 5.1 Customer lifecycle
Customer lifecycle
# The REAP model
One model to categorize customers is the REAP model; customers according to this model can be defined by level and scope of purchase by retention, expansion, acquisition and preferred customers, where:
* *Retention* accounts are existing customers who may be purchasing more than one of the vendor’s products and may be repeat purchasing. They show a degree of loyalty.
* *Expansion* accounts are existing customers identified for purposes of developing business further either by increasing the volume of products/services they purchase or for expanding the portfolio they purchase from a vendor.
* *Acquisition* accounts are customers not purchasing today; they may also be customers recently acquired but needing to be managed via acquisition sales account managers for account relationship reasons.
* *Preferred* accounts are retention accounts that buy in large quantities and show a degree of loyalty over time; without these accounts the business would be struggling to survive. These accounts are also advocates of the company’s brand and generally require extra handling and care beyond the key account management activities.
## Retention vs acquisition
Acquisition customers differ in several ways from retention customers. As they are not yet customers of the organization they are not yet account managed and, generally, no previous data exists about the customer and there is no pre-existing relationship. Thus the responsibility falls generally on marketing to understand and share insights about such customers. More differences are summarized in Table 5.1.
Table 5.1 Differences between retention and acquisition
| **Retention** | **Acquisition** |
| Existing relationship | Not account managed |
| Data about the customer exists and is accessible | No relationship pre-exists |
| Future project needs are documented | No current or up-to-date data exists |
| Behavioural data | No buying behaviour information exists |
| Easier to engage C-suite | It is typically challenging to engage higher management |
| Their details are logged in a central database or CRM system | |
| Account management is possible | |
# Acquisition marketing and types of acquisition customer
Acquisition marketing is any marketing activity with the specific goal of acquiring new customers. In this case new customers are customers who have never purchased from the organization or who are no longer actively doing so. Acquisition customers can be sub-divided as follows:
* *Prospects*. A prospect is a potential customer or sales lead that has been qualified as fitting a set of criteria.
* *Greenfield customers are completely new customers for the business*. These customers have not previously bought from the company.
* *Re-acquisition customers*. These are customers who used to buy from the business but no longer do so and have been identified as customers to re-acquire.
* *Lapsed customers*. Customers who recently lapsed and no longer purchase beyond an expected period of time. Lapsed customers could be categorized as lapsing if they are purchasing once within three or six months but have not purchased for six months.
* *Inactive customers*. Inactive customers are accounts that have not purchased for an extensive period, eg not purchased for more than 12 months.
# The acquisition marketing process
The acquisition process, shown in Figure 5.2, can be split into six main stages:
1. *Analysis*. A company needs to understand the background to customers purchasing competitor products, and what competitors have in terms of value propositions. Other factors to consider are whether customers are aware of the organization’s products and offers and their perception of them.
2. *Preparation*. In this phase the organization prepares itself before reaching out to the customer; preparation activities include aligning with sales on the acquisition process and approach; defining the customers and their pain points (ie their main needs or challenges); adapting the business value proposition and agreeing on the hook or how to entice the customer.
3. *Education*. During this phase customers are typically approached indirectly with information to help them in their research phase; this corresponds to the buyer journey where customers quantify needs and research alternative solutions for their needs.
4. *Customer engagement initiated*. During this phase the customer is contacted by the business or a business representative.
5. *Follow-up*. During the follow-up stage sales become more actively engaged with the customer and help in answering any final questions the customer may have.
6. *Acquisition*. The final stage of the process is where the customer decides to purchase and completes a transaction.
Figure 5.2 Acquisition process
Acquisition process
## Analysis – insights and data
One gap a company has in acquisition marketing compared to marketing to existing customers is a lack of data and insights about new customers. This type of information is critical if a company intends to effectively target and engage customers in the right way at the right time via the right channels.
**Tip Key** questions and areas businesses need to investigate
* Competition. Why is the customer buying competitor products? What are their weaknesses and strengths? What is their business model? What are their tactics and strategies at a business and marketing level?
* Resistance. Why is there resistance to purchasing our business products and/or services?
* Customer engagement. Who is the customer? Who are the influencers?
* Customer buying behaviours. What do they buy? Why do they purchase today? When do they purchase? How do they purchase?
* Customer behaviours. What are their pain points? What are their top needs?
The ultimate objective of the analysis is to identify a unique angle or opportunity and allow the business to uniquely position itself in engaging the customer.
## Preparation
As part of the preparation stage the role of B2B marketing is to support the business with the right go-to-market (GTM) models, define and align on the sales approach and prepare marketing materials.
### Right GTM model
The right GTM model approach is dictated by the prospect and the most suitable engagement and channel model for the vendor. For example, in targeting small businesses a company can choose different direct channel approaches to engaging and acquiring, eg via call-centres, online investment or an outbound salesforce. Alternatively, it could look at indirect channel routes, eg online mail order or retailers and their physical or online stores. The selection of the channel can be based on financial capital, in-house competencies, in-house resources, operating locations, and efficiencies in reaching and engaging customers.
### Sales
Compared to marketing to existing customers, the efforts in acquisition marketing are typically greater and the cost is usually higher for the business, so ensuring sales alignment and the correct structure is key. Below are some of the main considerations in preparing with sales:
**Align to the correct sales group** for the acquisition campaign; if the marketing activity is about bringing customers in front of specialists and thought leaders in the organization then be careful not to only include sales generalists who may not be able to answer specialist questions from key customers.
**Align on customer type and need**. According to the business portfolio gaps and business goals, the business should align on the right customer target, their profile and pain points. Inputs into this phase can come from existing customers who mirror the target customer profile.
**Align on process for engaging customer**. Marketing and sales need to define when and how to engage within the marketing/sales process. For example, will marketing leverage sales to make outbound inquiries or phone calls? Will sales lead by using hooks or door openers?
### Marketing preparation
Marketing in this phase should define messaging, content and materials about the business proposition. The agreed types of hooks needed, eg incentives, compelling articles, services or other types, can be prepared.
## Education
The education phase is typically managed by an organization’s marketing department. This phase is the major pivotal change point compared to10 years ago, when organizations had the luxury of being contacted by customers. Marketing needs to position and place information at customer touch points or across media that customers use. The challenge for marketers is identifying which media or information sources are being used by which customers and when.
## Engaging the prospects
During this stage marketing typically takes the lead on the first steps by engaging through third parties, through forms, an online website, offline associations, events or by engaging through marketing directly. Depending on the customers and their decision journey the number of touch points before sales engage can vary: some industries highlight an average of 12 touch points before sales are engaged; others talk about three or five. The mix of marketing touch points leading to the final sale can also vary depending on customer type and marketing type.
## Follow-up
During this final stage the prospect is not yet a customer but sales need to follow up based on engagement or contact activities. It is generally pre-agreed when and what stage the handover to sales occurs. Acquisition marketers and acquisition sales need to agree in advance and correctly assess the likelihood and timing of conversion to a sale. For example, if a customer purchasing a complex solution is in the early stages of the decision process it is probable they will require more information, more engagement and time compared to existing customers who buy simpler products.
## Acquisition
The customer is acquired by the organization as a customer. A customer may remain an ‘acquisition’ customer for some time in the eyes of the organization until the customer’s new business is developed into something more established.
# Re-acquisition marketing process
The process for re-acquiring customers can follow a similar process to that of acquisition, although there are differences. During the analysis stage, a company should understand why customers ceased purchasing products or ceased business. These insights can shed important light on how to proceed in the next stages or even whether to proceed. Businesses can find this out through approaches such as ad hoc surveys directly or indirectly with lapsed customers, surveys via customer service, or from reports such as Net Promoter Score reports.
During the preparation phase, channel selection and agreeing on a different sales engagement (eg via more senior sales) may be prepared. Content may be incorporated that addresses solutions to the specific customer challenge, which may have been the reason behind the customer ceasing business.
The engagement phase can include engaging via a third-party partner, channel or even marketing agencies. More than is the case with acquisition customers, the follow-up in a timely and relevant fashion while addressing any input or response from the customer is critical.
**EXAMPLE** Net Promoter Score
General Electric, Honeywell, Dell, HP and many more companies use Net Promoter Score to identify customer satisfaction. As part of the report there are three categories of customers: promoters, detractors and passives:
**Promoters** are loyal, enthusiastic fans. They are more likely than other customers to increase or maintain their purchase activity over time.
**Detractors** are unhappy customers. They are more likely to decrease or cease purchase over time.
**Passives** are customers who repurchase but are only passively satisfied.
The Net Promoter Score can help organizations in their acquisition marketing or sales by investigating the detractors and understanding why customers lapse or don’t purchase; they can help an organization identify areas to fix before acquiring customers.
## Why customers lapse
Customers lapse for various reasons; here are just some of them:
* *Poor ‘service’ from the business.* The business failed to meet the customer’s expectations in terms of ‘how’ it delivered the product or service; this may be the way it handled a complaint.
* *Neglect*. In the initial stages of acquisition the customer was engaged regularly but over time this engagement changed or stopped, essentially making the customer feel neglected.
* *Poor quality of product/service offering*. The quality of the product or service did not meet expectations or was of inferior quality.
* *Competitor acquisition programmes*. A competitor offering a similar product or service increased its price aggression or engagement with the customer, which will lead to customers switching supplier.
* *Change of price position from the vendor*. Some companies over time decide to increase margins on their products or service through reducing costs or repositioning their products; this can affect the purchasing behaviour of customers in that they either buy less or stop altogether.
* *Irrelevant communication*. If the communication form isn’t appropriate to the customer needs or not aligned to customer wishes, it may put off customers and their business. Some companies have been criticized for excessive e-mailing to their customers and eventually annoying them.
**TIP** How to segment for re-acquisition
To better ensure the achievement of marketing efforts, markets need to be more targeted to re-acquire customers. Figure 5.3 is an example of how to segment lapsed customers based on four criteria: 1) how recent their latest purchase was, 2) type of purchase, 3) size of business, and 4) propensity for purchasing high-margin products.
Figure 5.3 How to segment for customer acquisition
How to segment for customer acquisition
Using these criteria, classify customers against a scale, A, B and C, where segment A is most recent purchase, most frequent purchase, largest purchase and highest propensity for purchasing high margin products. Segment A is the most desirable segment, Segment C the least. The customers that tick all the boxes with an ‘A’ will be the priority segment.
**TIP** Tactics in re-acquisition marketing
There are a number of different approaches to re-acquiring customers:
* *Lapse red flags*. The business can define red flags or alert systems to understand changes in customer touch rate or engagement rate. Sales or marketing operations can sometimes help with reporting to identify accounts to target in advance.
* *Reason to re-engage*. Trigger activities or events can be included in messaging, highlighting why the business is re-engaging. Such tactics can be effective if they show that the vendor is aware the customer no longer buys and shows the importance of the customer to the vendor.
* *Change communication channel*. Where traditional e-mail hasn’t worked; the vendor can employ different approaches such as social InMail, direct mail or other.
* *Use fresh new content*. Use new creatives to engage lapsed customers; use new ways to engage if budgets permit, for example video slabs.
* *Listen and respond*. Organizations can conduct random surveys that highlight key customer needs and therefore provide opportunities to organizations in their marketing and messaging.
# Acquisition marketing strategies and tactics
## Door openers and hooks
A science has been built up around ‘door openers’ and ‘hooks’. Door openers are usually designed to be effective in capturing attention and in giving the business an opportunity to engage with the customer. Types of door openers include low-cost introductory letters, vouchers and coupons as well as more sophisticated approaches that involve a sales person delivering something of value at a meeting with the customer. Door openers should appeal to the stakeholder targeted for engaging. Door openers can also be service related; for example, some IT hardware companies offer low-cost or free services to demonstrate their capabilities.
## Acquisition marketing strategies
There are different schools of thought on how to acquire customers. Less efficient is the ‘persistence’ approach where a company or its salespeople continuously knock at the door or pester potential clients. Others that embrace marketing techniques are more effective by being more strategic, sophisticated in planning and targeting. Some examples of the more effective acquisition strategies are:
**Segment the market; segment the customers**
It is important to create a detailed segmentation that can actually be used. For example, a business may have the objective to target all small businesses in the UK, but with over 4 million small businesses, marketing to all of them and subsequently engaging them is not realistic or cost-effective. Segmentation of this large audience could be by vertical type, by region, or by further segmenting by employee numbers. Ideally segmentation should follow a set of criteria, eg all small businesses with more than five employees in each region to be offered consulting services.
**Create compelling stories and clear call-to-actions for follow-up**
Companies of all sizes have stories to tell; they have something to say that will capture the attention of potential customers. Few businesses think about creating and sharing their proposition or message in an interesting way that then leads to something tangible. As an example, they could offer an article or a thought leadership piece, linking it to an invitation to a webinar or to call and discuss the unique challenge in person.
**Advocacy**
Creating advocacy programmes is also important for acquisition. Customer advocates are probably the strongest influence on potential customers as customers listen to other customers; this is why Amazon ratings and similar review-type sites are so popular. Advocacy can come in different forms. For example, Glassdoor is an online site giving employees and non-employees opportunities to talk about companies and rate them. Negative comments will dissuade other people from seeking employment, or engaging with the company; positive comments and high ratings can help the organization attract talent and business.
**New GTM approach; new resellers**
Sometimes the challenge is that a customer wants to buy an end-to-end solution comprised of multiple products, services and applications, whereas the organization usually offers a single solution. Here it’s important to understand whether a business should adapt its go-to-market approach if it wants to sell fully fledged end-to-end solutions.
# Measuring acquisition
The most obvious way of measuring acquisition marketing activities is by tracking the number of new customers and the revenue attributed to new customer business. Typically, in any given period, while businesses acquire new customers some customers will lapse or stop their purchases, so an important measure of the true impact is ‘net new’ customers and ‘net new’ revenue calculated as:
Net new customers = new customers – lapsed customers in any given period
Net new revenue = new revenue attributed to new customers – revenue attributed to lost or lapsed customers in the same given period
It’s important to have a long-term approach to measurement as acquiring completely new customers usually takes longer than expected. Typical quarterly approaches to reviewing for ROI (return on investment) will not work.
In general, the more complex the product or solutions the longer the sales cycle, so for ROI to be determined one needs to take a longer-term view. The reason behind this is not just the large size of business involved but the greater number of decision makers in the purchase of more complex products or solutions.
## Cost and benefit of acquisition
Cost
The cost of customer acquisition (CAC) means the price you pay to acquire a new customer. From a marketing perspective this is calculated by tracking marketing activities and campaigns and associated business wins. In its simplest form, it can be worked out by:
Dividing the total costs associated with acquisition by total new customers, within a specific time period
For example, if a campaign is built on a new proposition and targeting 100 customers, marketing could include e-mail, door opener, event or webinar invite. The costs would be as follows:
Door opener (× 200): £1,000 (for 200 door openers)
Webinar: £5,000 (infrastructure)
Event: £15,000 (for venue, logistics, etc)
E-mail: £1,000 (for content, creation)
Total: £22,000
### Benefit
The benefit is the number of sales resulting from a marketing campaign. For example, where a campaign resulted in acquiring 20 new customers in each period, each delivering £4,000 in revenue and £1,500 in profit, the total benefit in revenue and margin would be £80,000 and £30,000. Using the cost calculation above, the net profit (sales minus costs) would be £8,000; £30,000 minus £22,000.
ACTIVITIES
Look at the key questions to customers box and check whether you’re able to answer them. Where there are gaps carry out further research.
Where you see resistance or blocks to customer engagement and acquisition, conduct a mini survey to understand what the resisting forces are and why they exist.
Look at the three core areas of sales alignment and check with senior management on whether this is the situation today.
# Further reading
Richardson, N, James, J and Kelley, N (2015)*Customer-centric Marketing: Supporting sustainability in the digital age*, Kogan Page, London
# 06
# Retention and loyalty marketing
This chapter will give you an understanding of:
* customer retention marketing
* customer loyalty
* communication and improving and maintaining loyalty
* top marketing strategies to increase customer loyalty
* customer satisfaction and marketing
* leveraging customer advocacy for marketing
* customer lifecycle communication
* mapping marketing and customer lifecycle stage
* how to increase customer share-of-wallet
# The REAP model and retention/loyalty marketing
As we learnt in Chapter 5, there are different stages in the customer lifecycle. Existing customers can be split further into the following categories: retention customers, expansion customers and preferred customers.
## Before acquisition
From the current portfolio of customers an organization needs to identify which customers are core to its revenue, which are important as part of its future growth and which accounts to focus to collaborate or promote more. Sometimes this part of the business is ignored: organizations continue to pursue new customers, leaving behind and slightly neglecting their existing accounts.
## After acquiring customers
Marketing’s role doesn’t finish with the acquisition stage. The role of marketing is about supporting the organization in developing business further with customers, in improving relationships, and in identifying which customers are more likely to grow and buy products and/or services either in increased volumes or branch out further by purchasing a wider range of products. This can be tracked based on customers’ buying power.
Customers’ buying power is their ability or extent to which they ‘could’ purchase a vendor’s products. This is one way to segment accounts. For example, if we define expansion accounts as those with between 10 and 40 per cent buying power this means that they are only spending up to 40 per cent of the potential they could be spending on a vendor’s products; there is a further 60 per cent opportunity within the account for the vendor to sell its products. This percentage is sometimes referred to as share-of-wallet (SoW).
One way a business can segment its customers by buying power or SoW is as follows:
* preferred accounts – those that spend over 70 per cent SoW;
* acquisition accounts – spend below 10 per cent SoW;
* retention – spend over 40 per cent but less than 70 per cent;
* expansion accounts – spend between 10 and 40 per cent.
Preferred accounts are those customers that carry a strategic importance for the business; they will typically receive greater attention from their supplier. The role of the supplier is to nurture and maintain close relationships through offering additional benefits compared to the expansion and retention accounts. Expansion customers typically are at the business development stage for the vendor; they may be relatively new customers.
## Why focus on customer retention marketing?
According to Bain and Company (Stillwagon, 1990), a 5 per cent increase in customer retention can generate increases of 25 to 95 per cent in profits. Ignoring existing customers can lead to higher customer churn and loss in revenue and profit; and we see indications from Gartner Group that neglect can mean consequences as they tell us that 80 per cent of a company’s future revenue will come from just 20 per cent of existing customers (Marsh, 2015).
Some of the main initiatives on developing existing customers are how to increase loyalty, increasing business with the customer and share-of-wallet and building strategic relationships.
# Customer loyalty
Customer loyalty is defined as the likelihood of existing or previous customers to continue buying from a specific organization. Great attention is given to marketing and customer service to retain current customers by increasing customer loyalty. Some organizations have structures within the business to support customer loyalty; others go as far as to create loyalty programmes to reward customers for repeat business.
## Why invest in customer loyalty?
As we learnt from the previous chapter the cost of marketing in re-acquiring or acquiring customers is significantly more than the cost of marketing to existing customers, so it pays to market to existing customers.
There are some exceptions to the rule. It is sometimes beneficial for businesses to focus away from certain customers that have become unprofitable and disproportionally use up the organization’s resources in managing them or serving them.
## Customer loyalty process
The main steps organizations take to ensure they maintain or increase customer loyalty are shown in Figure 6.1 and as follows:
1. *Customer research.* During this phase an organization researches customers and their view of the business so as to understand what would make them stay with the supplier or, more importantly, what would make them switch to another supplier. Sometimes the least obvious responses can come back, eg price in B2B is not always the most important factor. Other aspects to identify are how the customer would like the customer-supplier relationship to change or improve.
2. *Summarize and prioritize.* During this phase the supplier summarizes back to the customer what it will do to improve or maintain the relationship in the short- and mid-term, outlining priorities.
3. *Short-term fixes.* During this phase short-term fixes are carried out; this phase is important as it is about delivering according to the supplier’s promises.
4. *Implementation.* Longer-term fixes are implemented and communicated to customers; examples of this could be new services or products, improved value propositions, assigning a sales person with more specialized experience or a dedicated service manager.
5. *Review.* This is about reviewing more formally the progress with the customer, and ensuring all facets noted within phase one have been captured and addressed successfully.
Figure 6.1 Customer loyalty process
Customer loyalty process
## Customer loyalty spectrum
Customer loyalty is the result of consistent positive experiences, positive emotional experiences and perceived positive value of an experience. Customer loyalty can be viewed as a spectrum from a minimum or a low degree of loyalty to the customer being not only an advocate but a champion for the company.
Five different types of customer loyalty can be identified (see Figure 6.2):
1. *Spot customer.* A customer with low degree of loyalty. Looks at provider as a short-term and tactical means to support its business; may decide to purchase again but not to be relied upon.
2. *Repeat customers* are customers who come back to purchase more from the supplier; the loyalty is confined to re-ordering or purchasing a second time from the organization.
3. *Customer advocate* is a customer who advocates your products or services; if contacted by another customer or potential customer it would rate your organization positively overall; can be relied on.
4. *Customer champion* is a customer who champions the organization’s products and rates them very positively; if impacted by a negative experience it may quickly shift to a lower degree of loyalty.
5. *Strategic champion* is a customer that demonstrates a strong customer loyalty, champions your products and is heavily ingrained with the supplier; this could be due to collaborative ties, even family ties. Champions are typically not put off by one-off negative experiences.
Figure 6.2 Customer loyalty
Customer loyalty
# The communication factor
How businesses retain or lose business customers can be influenced by how they communicate with them. The main success factors in retaining and building relationships with customers are:
* *Communicate consistently and regularly.* It is often the case in the acquisition phase that businesses over-communicate, but once the customer is acquired there can be a drop-off in communication. It’s important that the frequency of communication is consistent and, with digital technologies and applications, it is easier than ever before to achieve this.
* *Appropriate communication.* Appropriate communication is also important. Where customers are experts in their field and knowledgeable about their business needs, the communication needs to account for this. Organizations’ marketing departments need to arm their sales people with relevant information to support these more technical or specialized conversations.
* *Listen and respond.* Improving stickiness and loyalty with customers is down to better listening and responding. Digital marketing and new applications support organizations both in listening to customers’ conversations and in responding with content that meets their needs.
# How to increase customer loyalty
The ways organizations can increase customer loyalty include the following:
**Create customer-tailored content**
Create content that is tailored or specific to the customer, the customer’s business or industry. Creating and including content relevant to the target group can be effective in improving engagement rates and response rates.
**Redefine customer value across the company**
Increasingly customers place more emphasis on value compared to price. In the technology sector, commoditization means technologies can be offered to more people at lower prices, forcing organizations to provide more value through combined offerings and services and products.
**Reward customer loyalty**
Many organizations offer rewards to channel partners through programmes providing incentives to sell more of their products. With more incentive points, channel salespeople win more prizes.
**Reward large volume purchases**
Most businesses offer discounts based on sales volume over a period. For example, large framework agreements in the public sector set pricing over a period of two or three years as well as discounts over that period.
**Reward repeat purchase**
Discounts are based on quantities over time; this can be through framework agreements that can run for extended periods of one or two years or even longer.
**Update with valuable information and news**
Customer satisfaction can be maintained or improved through timely updating of information. Some organizations inform customers about trends or activities that could impact them both positively and negatively. In updating customers it’s important to understand how customers wish to be informed, and ideally offer solutions to capitalize on opportunities as well as dealing with potential challenges.
**Personalized treatment**
Personalized treatment can be through dealing directly with a customer contact or set of contacts. Marketing may decide to turn the spotlight on customers by telling their story through its communication channels, which in turn helps promote customers and their business. Another way to offer a more personal touch is to invite the customer to speak at an event the supplier has set up.
**Facilitate peer-to-peer networking**
One way for businesses to support their customers is through helping them network with other customers, either from the same industry or facing similar challenges due to their position. Facilitating customer-to-customer networking through online information, social forums or webinars and events is a powerful way of offering value to customers without a business overtly pushing its products or services.
## Leveraging customer advocacy for marketing
Building a referral programme can come in different forms. These can range from simply developing and communicating customer stories and case studies to more sophisticated approaches involving organizations in branding campaigns and leveraging customer advocates to promote the business across multiple media: online video, adverts, social, event promotion, etc.
# Customer satisfaction and marketing
Although often an initiative led by the wider business, customer satisfaction and the process of surveying it touches on aspects for which marketing is accountable. Typical surveys include questions about product or service satisfaction, business operations such as logistics, customer services levels, the quality of sales engagement, pricing, and the company overall. Such surveys can come in the form of Net Promoter Scores, a reporting system used by many organizations. Marketing can influence these aspects directly and indirectly across three core areas:
1. *Marketing mix elements.* Product quality may not meet customers’ requirements or may not satisfy their need as expected. Pricing may too high for the product they’re purchasing. Communication, whether excessive or not aligned or appropriate, can impact on customer satisfaction.
2. *Sales.* With sales, the customer relationship or satisfaction can be impacted by a change of sales account manager or reducing the frequency of the manager’s contact with a customer. Time of engagement can also have an impact: sales engagement can be improved through marketing identifying customer needs, and providing sales with information and content that address those needs, in a timely fashion. Marketing can help by providing material through third parties so customers still stay informed but aren’t bombarded by e-mails or too-frequent sales calls that may be regarded as a nuisance over time.
3. *Corporate and wider business.* Other factors impacting customer satisfaction are company values and reputation; in modern day business, an organization’s business affairs are transparent and very public so where organizations don’t conduct themselves well per customer values this can also impact customer satisfaction. Marketing can help improve perception of the company through PR and branding activities.
# Customer lifecycle communication
Customer expansion and retention marketing is about following customers’ journeys and their business growth. Customer communication can also revolve around the lifecycle of engagement where there may be dips and peaks in customer/vendor relationships for different reasons. The dips can be due to customer business troughs and therefore lack of need or ability to purchase, or vendor-specific due to rotation of sales people as well as customer satisfaction factors mentioned before.
Customer lifecycle dips can be mitigated through marketing initiatives as marketing can craft news and messaging as well as provide unique opportunities to improve customer engagement.
## Mapping marketing and customer lifecycle stage (post-acquisition)
In managing customer lifecycle post-acquisition marketers can prepare for the different stages such as retention, customer satisfaction dips, expansion and premium-based marketing. Table 6.1 outlines some marketing activities for the different lifecycle stages.
Table 6.1 Marketing and the customer lifecycle
| **Stage post-acquisition** | **Customer behaviour example** | **Marketing tactics** |
| Retention | Repeat purchasing | Update via newsletter
Invitation to retention account events
Support sales in regular communication
Marketing supporting sales phone calls with company news
Share latest on market trends, information |
| Dips in vendor/customer relationship | Customer fails to repeat purchase in a given period | Reminder e-mails
Invite to webinar, events
Offer incentives to purchase
Invite to customer advisory council
Use detail from NPS report to engage customers |
| Expansion | Customer increases purchase level through increased amounts or range of vendor’s products | Marketing expand content, engagement
Support increased sales engagement through broader set of marketing activities
Offers-based marketing to upsell
Highlight benefits of other products in portfolio |
| Promotion- preferred | Customer becomes champion or strategic partner | Collaborative marketing
Inclusion at vendor-led events
Include in brand marketing
Hospitality marketing
1x1 marketing at senior level
Include in executive visits |
# Customer loyalty measurement
Customer loyalty can be measured across the core areas of engagement, purchasing behaviours and advocacy (see Figure 6.3 for more detail).
**Engagement metrics** differ between business type and their marketing channel usage; for example, a digital channel savvy small businesses could be measured on time spent online and returning visits. Other metrics for customers who are account managed may be their frequency of use of a customer portal or their usage of customer solution tools; or downloading of information.
**Purchasing behaviours** reflecting loyalty can cover the number of repeat purchases in each period, renewal rate if the product is software based, share-of-wallet and churn rate. Low churn rate for a given customer group or segment would indicate a good degree of loyalty.
**Advocacy** could be based on likelihood to recommend or leave; these are typically qualitative and measured through surveys such as Net Promoter Score.
Figure 6.3 Measuring customer loyalty
Measuring customer loyalty
The degree of usage of a loyalty programme and membership numbers are a measure of loyalty, although as an absolute measurement this may be misleading as customers may be members but be completely disengaged. It’s important to understand how engaged the customer is with the loyalty programme.
Activities
Identify marketing activities today that are oriented towards customer retention; do you see any gaps?
Look at how you measure customer retention or loyalty marketing. What do those metrics tell you? Are there any different metrics that can be used?
# References and further reading
Ahmad, R and Buttle, F (2001) Customer retention: a potentially potent marketing management strategy, *Journal of Strategic Marketing*, **9**, pp 29–45
Ahmad, R and Buttle, F (2002) Customer retention management: a reflection on theory and practice, *Marketing Intelligence and Planning*, **20** (3), pp 149–61
Marsh, B (2015) 4 musts for serious customer success in 2015, Salesforce Blog, 16 January, available at: https://www.salesforce.com/blog/2015/01/4-musts-serious-customer-success-2015-gp.html (accessed 10 February 2017)
Payne, C M and Ballantyne, D (1991) *Relationship Marketing*, Butterworth-Heinemann, Oxford
Stillwagon, A (1990) Did you know: a 5% increase in retention increases profits by up to 95%, available at: https://smallbiztrends.com/2014/09/increase-in-customer-retention-increases-profits.html (accessed 10 February 2017)
# 07
# C-suite marketing
This chapter will give you an understanding of:
* C-suite – who they are
* importance of C-suite
* C-suite marketing challenges
* C-suite and personas
* C-suite marketing success factors
* C-suite contact strategies
* measuring C-suite marketing
# C-suite and C-suite marketing
‘C-suite’ is a term commonly used to refer to the senior executive level of companies and includes all executives described as CEOs, CIOs, CMOs or other ‘Chief’ or ‘C-level’ positions. ‘C-suite’ for the purposes of this chapter also includes those who may not have a C-ba sed acronym to their title but who operate at the senior executive level within an organization. In a similar fashion ‘C-suite marketing’ refers to all aspects of marketing oriented towards C-level or senior executive decision makers in a business.
## C-suite roles
The main C-suite roles are as follows:
* *CEO*: concerned with the success of the company and is involved in creating its vision. CEOs hold their management team accountable; they think about revenues, margin and performance of the business on a regular basis. In engaging a CEO, marketing needs to communicate at a business level rather than on a technical or product level.
* *CFO*: manages and reports on the financials of the business. CFOs report financials and highlight any red flags; they’re interested in marketing from a point of view of what it can deliver and the return on the overall marketing resources – budgets, people, etc.
* *CIO*: handles all IT requirements for the company. CIOs handle the internal IT support and ensure the business and employees are supported through equipment and infrastructure.
* *CSO*: sometimes known as the general manager, head of sales. CSOs lead and direct the salespeople in the organization; they are responsible for sales targets and all aspects of sales.
* *COO*: concerned about the operations of the business from a logistics and supply perspective.
* *CMO*: is concerned about the brand, marketing communications and supporting the business growth through delivering new business opportunities through leads, clicks or calls, or supporting sales through sales enablement. CMOs are also typically responsible for market research, market insights and providing greater customer insights through data.
## C-suite trends
Within the executive level of companies there have been some recent shifts and changes. CMOs are gaining more influence due to the advent of digital technologies and applications in marketing that can help businesses become more efficient and effective. From a recent study conducted by the CMO Council (2013), according to both marketers (85 per cent) and IT executives (85 per cent) who participated in this study, the relationship between the two is critical to the execution of customer-centric programmes. In the future it is expected new positions such as chief data officers will be in place to support CEOs and COOs in managing streamlined companies by using the mass of data available to them and to draw insights.
**EXAMPLE** The emerging CDO role
CDOs or Chief Digital Officers were unheard of 10 years ago. They act as the main driver behind digital transformation, leading companies into the next stage of their development. They are also looked to for coordinating digital initiatives across the enterprise, not just marketing. Various sectors have seen CDOs appear, such as higher education, healthcare and manufacturing. They work closely with the CMO and are concerned with the digital ecosystem and company touch points as well as heavily invested in data use and applications.
# The importance of C-suite
## Why C-suite?
C-suite marketing is for organizations that want to think of longer-term, strategic and essentially more profitable business relationships with customers, through building more senior executive relationships and marketing to them. Where the sale is thought of as a transaction and to non-senior executives, the chances are that the supplier is missing out on potential opportunities. Typically, engaging the senior executive level of companies means suppliers can enter into more strategic conversations, propose different value propositions and solidify business relationships, which opens up wider business opportunities.
## Broader view of needs
Senior level executives or C-suite will have a broader view of the company direction and will be interested in unlocking potential through capital, resource or other means. They could also be looking for partners in the forms of suppliers who can offer a different relationship. It may be that the vendor organization can offer something the company hasn’t thought about or that the proposition being offered needs to be considered with a long-term perspective, which at a lower level in the company may not be appreciated. For example, a proposition which in the short term costs a lot more but over a year reduces energy bills or other operating costs may be more positively considered by a financial head who is looking at long-term benefits rather than a more junior person in the purchasing department who is taking a short-term view of the business.
## Early in decision cycle
We discovered from earlier chapters that the main challenge today for B2B marketers is to influence or engage customers or decision makers earlier in the buyer journey so they are more informed and influenced to purchase an organization’s products or services. One way to resolve this challenge comes from deepening engagement with the C-suite of a customer who then can influence purchase decisions within the organization earlier on in the purchase decision process.
# C-suite marketing challenges
C-suite are probably particularly challenging to engage; they’re generally extremely busy people, with packed calendars. They tune out of traditional marketing messages more than non-C-suite, skip TV ads and don’t click on digital advertising. In the office they receive masses of e-mails and brochures and, more than others, need to be very discerning in what they read and respond to.
C-suite typically have a set of gatekeepers that protect them, such as PAs, heads of department reporting into them, assistants who filter e-mails, phone calls, etc. C-suite executives will also typically have very specific interests and motivations depending on their area of responsibility and topical business challenges, so unless potential vendors are on top of the latest trends and challenges and unless they tailor those to the sector or language used by such C-suite executives they may find very little or no response to marketing.
Once a C-suite executive is engaged and interested in a vendor’s value proposition and what it can deliver, the time between the first engagement and concluding a deal is likely to be much longer than engaging purchasing departments directly. However, the potential benefits are much greater in terms of broader business arrangements and more profitable business. It does mean that success from marketing or business activities aimed at the C-suite can be difficult to demonstrate due to the length of the purchase process. Other challenges are that in some organizations the C-suite is large, consisting of many stakeholders, so detailed targeting and sub-segmentation are required.
With the above taken into considerations it’s clear that approaches such as database or e-mail marketing will probably not work to engage a C-suite executive.
# C-suite persona and persona creation
Before marketing to a C-suite it’s important to gather insights about the different personas within the executive suite of companies being targeted. In building personas, it’s important to define exactly which C-suite to target, as in larger organizations C-suite can be large teams.
The main process for creating C-suite personas is as follows:
1. *C-suite buyer/potential buyer research.* The first step is to conduct research of the C-suite or executives. Ideally this is conducted by C-suite type to be able to structure and group feedback later. The research could be groups of CEOs, of CMOs, of CIOs or of CFOs depending on the objectives of the marketing department. Typically interviews and insights can cover such as things as latest challenges, future objectives, what information sources are used to keep up to date, and how they want to be engaged by companies such as yours.
2. *Capture insights.* In this stage common behaviours are captured across different groupings. Assuming it’s CIOs who are the intended target, groupings could be by size of company where, for example, CIOs in smaller companies may be concerned about building IT infrastructure in line with business growth.
3. *Build personas.* Next is building personas per type, age, interests, role, size of company, main concerns, main challenges and main requirements from potential suppliers.
4. *Content mapping.* Finally, content and messaging are created in line with the gathered insights and mapped per industry or C-suite type; ideally the content is per C-suite buyer behaviour and decision path.
# Key success factors in C-suite marketing
Unlike marketing to heads of departments, marketing to a senior executive group of people means adjusting elements of the marketing mix and strategy such as communication vehicle, content, niche target group, contact strategy, go-to-models involving vendor sales or other groups. For example, a newly appointed account manager with few years’ experience is probably not the best person to engage a C-level executive in the prospect company. In general, marketing to C-suite will differ from the usual sales and marketing channels, and typically requires extra knowledge, resource and budget. Few companies conduct C-suite marketing successfully.
**TIP** Core areas of success factors in carrying out C-suite marketing
**Consistency in segmentation**
Contrary to perceptions, tribalism exists between and within the C-suite; it’s important when engaging C-suite in forums, etc, to be sensitive to this. A CEO of a small business will have different challenges from a CEO of a midsize or large company.
**Taxonomy**
C-suite tend to respond to different content compared to others in an organization; their concerns are generally more strategic so if marketing is to be successful it’s worth creating tailored content.
**Business vs product benefits**
Executives tend to care about how to solve business problems; they are less interested in engaging potential vendors about technologies and products. Therefore, when thinking about products or services businesses should position solutions in terms of the bottom line and what can help grow the business. Peer stories of how executives of other companies managed can lend credibility to business propositions. Marketing should employ multi-step or business benefit thinking in communicating with senior executives, ie rather than the immediate benefits of the product, highlight the indirect benefits to the business derived from rolling out the offerings (see Figure 7.1).
Figure 7.1 C-suite messaging
C-suite messaging
**Niche segmentation – C-level**
Where the focus of a C-suite marketing programme is to engage only specific representatives, separating out these roles and contact details from the rest of the C-suite is critical.
# C-suite contact strategies and tactics
One of the main challenges organizations have with C-suite marketing is contacting and engaging C-suite people. Even vendors that have been doing business with a customer may find there is a complete lack of engagement with any senior executive at the customer site.
## Why the challenge in contacting?
As mentioned earlier, C-suite members will have gatekeepers in the form of department personnel or personal assistants who manage post and e-mail on their behalf. As you can see from Figure 7.2 the gatekeeper is not just one person and if the marketing department needs to target more than one C-level member the challenge multiplies.
Figure 7.2 C-suite and their gatekeepers
C-suite and their gatekeepers
Other stumbling blocks can be behavioural. Where an organization has started to engage a customer through a sales person, that person may be reluctant to refer a vendor to someone more senior in their organization. This could be for a number of behavioural rationales, such as protecting territory.
The contact strategy and tactics will differ depending on whether they are new or existing customers; here are some of the possibilities of engaging C-suite. With new customers:
**Third-party research**
Use third-party research companies to improve contact data; marketing can help to identify organizations that have contact data profiles.
**Via industry associations**
Executives are often part of multiple associations depending on function, background or business interests. For example, a CFO may be part of a financial organization or an accounting organization as well as leadership organizations. Reaching CFOs may be more effective through financial bodies or associations that have a long-standing relationship with the company’s executives.
**New content, reports or research**
One way to engage executives may be through data or research that is particularly relevant to their interests.
**New services**
Provide information on a new service or product and how this benefits the top- and bottom-line of customers. For example, a new service may improve efficiencies such that financial or headcount resources are freed up, or it may help the customer achieve the same thing faster and so support them in gaining competitive edge in their field.
**C-suite forums and memberships**
Vendors can ask their executive level to engage and participate via forums on social platforms; sometimes this type of activity is supported and managed by PR to ensure the message is right and the right social platforms are leveraged.
With existing customers:
**Vendor executive engagement**
Marketing can enlist the company CEO or other C-suite representative to engage an existing or potential customer at a more senior level. Through having the C-suite of the vendor engaged, the communication would be directed to the right level in the customer.
**Create a C-level event**
Creating a specific event exclusive to C-suite will encourage customers to ensure the right level of people attend.
**Build PA peer-to-peer relationships**
A vendor could request their executive PA to build a relationship with the client’s own PA by offering opportunities for lunch or other forms of engagement opportunities. In turn this relationship-building serves as a way to remove the barrier of the main gatekeeper to the C-suite contact person.
**Two-pronged communication**
E-mails directly to executives will either go to PAs or into a spam folder, so having a two-pronged communication approach involving e-mail plus some other form of communication will ensure a higher probability of the message reaching the intended target person.
# C-suite loyalty
How do companies gain C-suite loyalty? How do they sustain such loyalty with executives, and what is the role of marketing? Essentially C-suite loyalty is a two-way process where both supplier executives and customer executives see a benefit in the executive relationship. There are a number of ways such executive level loyalty can be sustained; some are:
* *Offer information that only executives are privy to.* This can be through simply creating content only meant for executives, their pain points, or through more sophisticated approaches by providing access to specific online sites only for C-suite executives.
* *Three-way networking.* Facilitate a relationship between the customer’s executives and between executives at the vendor and customer site. Marketing’s role here is to develop executive engagement opportunities for executives to come together.
* *Provide insights.* A vendor could share interesting insights or tips with the customers’ C-suite to keep them abreast of the latest news and market trends.
# Measuring C-suite marketing
One challenge marketers face is measuring the impact of C-suite programmes. Some stumbling blocks for marketing are the inability to articulate the benefit of C-suite programmes, which can often can lead to such programmes being dismantled or kept for vanity purposes. Poor information-capture mechanisms or untimely capture can mean that the data is not captured or the wrong type of data is captured.
Short-term measuring can impact C-suite marketing success. A C-suite marketing activity is typically strategic in form and a final purchase is tactical, so the time to move from C-suite engagement to purchase can be lengthy. Where a vendor fails to incorporate a long-term measurement approach the wrong assumptions about ROI can be made.
**TIP** How to measure
**Define metrics**
Define the right metrics and articulate how they tie to the business and marketing goals; eg if a C-suite programme is defined by the number of C-level executives engaged, where the number has been reached this is one KPI achieved. Another possible metric is the number of touch points a target C-suite makes over different marketing vehicles, eg digital banners, e-mail, social, advertising or events.
**Interpret metrics**
A company can do this itself or engage an external company to do so. Marketing can look at CRM and data analysis by doing control group vs random sampling to understand the impact of C-suite programmes.
**Blend the metrics**
Ideally marketing needs to use metrics that include the beginning, middle and end of the customer journey for the intended C-suite marketing activity. By highlighting metrics in this way marketing can show how metrics evolve over time; for example, early C-suite marketing metrics could be awareness and engagement reach, and later on metrics could be opportunities and pipeline based.
**Time the metrics**
Marketing should define times according to measurements, eg the early phase metrics may be reviewed in the first three to six months and include engagement and participation level metrics, while metrics towards the end of the activity (eg after 12 months) may include revenue.
ACTIVITIES
Look at your top 30 accounts and ask your account manager what level of relationship the company has with their C-suite. Is it just with the CEO or with others?
Where you have an existing C-suite relationship, look to build a forum or an event to discuss with them the latest trends regarding C-suite, sharing any observations from your company and its executive leadership.
# Reference and further reading
Accenture (2014a) Cutting across the CMO-CIO divide, Accenture Interactive, available at: https://www.accenture.com/au-en/~/media/Accenture/Conversion-Assets/DotCom/Documents/Global/PDF/Dualpub_16/Accenture-Interactive-Cutting-Across-the-CMO-CIO-Divide-Pdf.pdf (accessed10 February 2017)
Accenture (2014b) CMOs: time for digital transformation – or risk being left on the side lines, Accenture Interactive, available at: https://www.accenture.com/th-en/~/media/Accenture/Conversion-Assets/DotCom/Documents/Global/PDF/Industries_14/Accenture-CMO-Insights-Web.pdf (accessed 10 February 2017)
CMO Council (2013) Big data’s biggest role: aligning the CMO and CIO, CMO Council, available at: http://www.sas.com/resources/asset/big-data-biggest-role-106349.pdf (accessed 10 February 2017)
# 08
# From product to solutions marketing
This chapter will give you an understanding of:
* product and portfolio marketing
* product marketing: hardware, software and services
* building effective value propositions
* how to create a solution
* challenges in marketing a solution
* enabling the solutions marketing integration
* solutions marketing strategies
# Product and portfolio marketing
Product marketing is the science behind marketing products, where products take the centre stage. Product marketers are sometimes called ‘brand managers’ as they manage the product as a brand.
Twenty years ago hardware product marketers were common; increasingly one finds more software and service marketers as businesses look to differentiate themselves and their offering, reflecting also the economy moving more to solutions, software and services marketing.
## Portfolio marketing
Few marketers today market just one product; most will market multiple products across hardware, software, services or a combination so the correct description is marketing of ‘offerings’ or ‘portfolio’ marketing. Portfolio marketing is about managing multiple products, each as discrete products or product categories.
## Portfolio management
Marketing departments typically have the responsibility of managing product and service offerings and to manage the product mix to achieve business revenue and margin targets. Often there are products that serve to achieve unit/revenue targets while other products or services serve to achieve margin targets; it is rare that a business looks at the same product to achieve and maximize both margin and unit volumes.
Portfolio marketing for many marketers brings the challenge of managing a marketing mix per product, and with multiple products or product types it means that marketers or marketing departments need to be able to manage different marketing mixes across each product type or category. Marketers and businesses tend to create portfolios according to customer type, size and segment or tailored to a different mix of customer need, eg where the vendor’s portfolio is broad.
# Product marketing: hardware, softwareand services
Product marketing can differ depending on whether the product is hardware, software or a service. Some of the similarities and differences are:
* Product hardware marketing refers to something tangible, whereas services and software tends to be less tangible.
* Sometimes product vendors will use services to differentiate or to help with selling products, eg free services to capture the customers’ attention.
* Tangible products are more susceptible to lifecycle impacts than services, which tend to be more easily adapted.
* Both hardware and software can come in packaged forms; for example, Microsoft offers customers delivery via the internet or offline channels such as retailers or the post.
* Hardware (and sometimes software) products need storing so the logistics in delivery and warehousing become more time-consuming.
* Services are based on people delivering them. They are dependent on their skillsets and the quality of their output; they are less controllable than hardware or software products. One could view the person delivering the service as part of the product.
* The quality of services is based more on subjective opinion of the customer; hence frameworks such as checklists or quality controllers are important.
* Measurement of service is subjective so evaluation approaches are needed such as questionnaires and surveys.
# The shift away from ‘products’
## The core product and the extended product and solutions
In product marketing there is typically one or multiple core products which can be hardware, software or service based. Beyond the core product there are additional aspects such as warranty, packaging, brand and customer care known as the ‘augmented product’. In solutions marketing the core product can be multiple products tied together.
Within B2B the augmented product is of great importance. For example, for software and hardware core products, service is a major way that businesses can differentiate themselves and demonstrate value add. Even for software, vendors such as Microsoft and Apple make the product tangible by using cards for buying iTunes or boxes for office applications. Services augmentation is through customer care, brand or other aspects. A business needs to leverage augmented aspects of the solution in order to connect with customers, engage them and appeal to them.
## Why evolve to become more solutions focused?
Solutions marketing is all about being customer-centric; businesses have seen real benefits from moving to a customer-centric business model. By identifying customer needs, organizations can create better solutions; often customers have a need but are not aware of how to articulate the solutions or features they require.
Take the example of dictation software. The customer needs to reduce the burden of typing at a keyboard throughout the day but may be unable to state that he or she needs a software application that can perform this task.
## Product focused vs customer-centric
So what is the difference between product focused and customer-centric product marketing? Product focused marketing typically has a more specific value proposition or a value proposition that doesn’t align with customer needs; the products developed are off-the-shelf rather than oriented towards customers. Customer-centric product marketing is demonstrated through needs-focused messaging and products, configuring and tailoring offerings to the customer, and the design of products is bottoms-up or built on expertise from customer input, whether directly or indirectly.
## From product to solutions
In B2B one hears more frequently the term ‘solutions marketing’. The background to this focus will depend on sector and segment: in the IT sector, personal computers, laptops and notebooks that were once regarded as luxury items or business purchases are now affordable for all. Businesses are under greater pressure to demonstrate the value they offer and how they can support business customers with solving current challenges.
# Building effective value propositions
One of the main ways to effectively market to customers is in creating and messaging value propositions. Value propositions explain the value and experience a customer will receive from a product, service or solution. It states specific benefits and highlights why it is a superior offering compared to alternatives.
## Creating the value proposition
Creation of the value proposition is the foundation to effectively engaging customers in solutions marketing. The main process, outlined in Figure 8.1, is as follows:
1. *Identify customer needs, desires, challenges.* What are customer pain-points today? What keeps them awake and what problems do they have that are not getting solved? Identification of needs can come through talking to customers, engaging them in exploratory discussions or through listening forums via social, etc.
2. *Identify benefits.* Benefits from the product or offering are defined and adapted to customer needs, desires and challenges. Product value is then mapped according to features, benefits and experiences for the customer. In this phase competition is reviewed.
3. *Create value proposition.* Insights are gathered and the value proposition created.
4. *Create and test messaging.* The value proposition is translated into salesforce and market messaging, and tested.
5. *Implementation.* The final stage is implementing the value proposition internally by ensuring enterprise-wide adoption, and integrated into external marketing materials and messaging.
Figure 8.1 Value proposition creation
Value proposition creation
**TIP** Key questions to answer
In creating a value proposition, organizations need to answer the following key questions regarding product and service offering:
What value does it deliver for the customer?
What needs does it fulfil?
What problems does it solve?
Why should a customer purchase it over the competition?
# Towards solutions marketing
The difference between solutions marketing and portfolio marketing is that the marketing of a portfolio tends not to be messaged or designed around a solution; it tends to be presented in the form of ranges of business offers.
Portfolio marketing or marketing of offerings can sometimes be marketed as a solution or in piecemeal fashion as some customers may not require or want to purchase a complete ‘solution’. They may have already purchased some of the solution, for example a customer who is creating a home office may already have a computer and only need a screen and a printer.
## Types of solution
Solutions can come in different forms to fill different needs and gaps; the main ones are:
* *Technology*: where the customer needs to draw on the benefits of different technologies to fulfil a need.
* *Business*: where the customer has internal or other organization needs, eg office space, and looks to a vendor to fulfil them.
* *Specialized*: where the solution may need adapting to the customer’s specific needs, or involve purchasing a specialized service or software. For example, an organization may provide specialized consultants to help with a customer’s challenge; in the IT sector this may be designing a datacentre specific to the customer’s back-office space requirements.
* *Verticalized*: like the specialized solution but required for a vertical customer; this could be for the construction industry where the customer requires more robust and rugged outdoor computing equipment.
**TIP** Mechanisms for creating solutions
Marketers are increasingly challenged with constructing a solution; this can be done from the main options below:
* *Adaptation.* The organization creates a solution from its offering or through adapting its portfolio to be able to offer a solution. For example, IT vendors such as HP and Dell have acquired many companies in the past decade to be able to offer not only the hardware for a solution but also the software and service aspects.
* *Alliances.* Some organizations will create an alliance with another company; these typically are organizations that complement each other’s portfolio and are generally not competitors.
* *Channel partners.* Organizations decide to sell their products through a channel partner with the objective of offering a full range of products or a solution to a customer. This can be a loose arrangement or a specific arrangement when using value-added resellers.
* *Specialized channel resellers.* Independent software vendors or specialized channel resellers that offer solutions for a specific sector can also be used by vendors to offer end-to-end solutions.
## Challenges in marketing a solution
One of the main challenges facing marketers is in marketing a solution, even where all the components of the solution are owned by one vendor. Marketing a solution is about integrating marketing, whether within one company or between companies. Integration means integrating marketing processes, marketing information and people.
Five areas can be identified that support solutions marketing:
1. *Customer-oriented value proposition development.* This can be achieved through better PR, improved customer segmentation and account-based marketing.
2. *Customer intimacy.* Customer intimacy is about maximizing exposure to customers through customer events, where companies listen to existing and potential customers. Customer engagement is facilitated via social forums or through loyalty programmes.
3. *Internal integration processes and approaches.* Supporting the external marketing, internally there needs to be cooperation between departments and product units in formulating solutions. This can be done using processes that orient the business towards customer segments rather than inward approaches focusing on the product.
4. *Strategic alignment.* This is about aligning the organization strategically towards relevant market segments for the current and future business and building capabilities to support the business in offering solutions.
5. *Sales integration.* Sales integration within and across the organization can help with future implementation; where the solutions require handover between departments any niggles can be resolved in advance. Sales integration can ensure a level of buy-in is already in place when it’s time to roll out.
**TIP** How to integrate for solutions marketing
1. *Integration starting point.* The integration starting point is to create a forum for different companies or departments to engage, share and agree on the objective, scope and process. This can be through regular calls and meetings.
2. *Prioritize customer segment.* The different companies or departments need to agree on the customer or priority customer segment(s), eg where there are multiple segments in play. This step is critical to ensure the next steps in integration take place smoothly.
3. *Define customer challenges, needs.* Next the combined departments/organizations need to define the core needs and challenges that customers face.
4. *Ensure the solution is complete according to detailed customer challenges.* From the previous step it may become clear that an aspect of a solution is missing, eg for an office documentation solution up-to-date software that helps archive documents may be missing even if all the hardware is present.
5. *Integrate messaging.* With solutions marketing the biggest challenge marketers face is in integrating components of marketing into one cohesive message. It is important to highlight the overall benefit as well as individual benefits of each of the elements of the solution.
6. *Test the message.* Before proceeding with marketing materials and content it’s prudent to test the message and solution with customers to understand if the solution and its message resonate; this should then be incorporated and tweaks or changes made if necessary.
7. *Create combined marketing materials and content.* Once the message is defined, marketing should create marketing materials or content, such as a digital brochure, a video or advert.
8. *Create combined marketing activities*. These can be combined webinars, events or other marketing vehicles.
## Enabling the solutions integration
For solutions integration of messaging and marketing to occur a few things need to be established. The first is executive sponsorship. A business needs to visibly support this through formal communication and executive sponsorship; this will allow for any process or other stumbling blocks to be overcome and ensure buy-in across business units. The second requirement is overall marketing leadership; it should be agreed that one of the partners needs to lead the overall process. Where both regard themselves as leading this can result in duplication and potential friction in the process.
Figure 8.2 Integrated solutions marketing
Integrated solutions marketing
**EXAMPLE** Dell solutions marketing programme
In 2013 Dell UK Public Marketing decided to embark on a new approach to marketing its public information technology solutions. The impetus for this initiative was sales feedback which highlighted the need for a more end-to-end view of the Dell information technology offerings as well as the need to make customers aware of the full extent of Dell’s portfolio.
The solutions marketing programme involved a multi-step process that kicked-off creation of the outline to support this new solutions marketing approach, buy-in from the public sector followed by alignment sessions with the senior sales executives to map out content requirements. Following this first draft of requirements, whiteboard sessions were set up to document information technology needs and blueprint end-to-end solutions. The external marketing agency then compiled the solutions content in the form of a guide which would be reviewed by sales people from external and executive sales. In the background, marketing communications (marcom) would ensure compliance with brand guidelines. The final stage involved the document being reviewed again by the general manager for sign-up prior to the solutions guide being launched.
The success of this programme hinged on various aspects; sales integration and full support were key in ensuring the content and final brochure were relevant for the customer. The external marketing agency played a facilitator role for sales and marketing as well as an integrator role between different departments. The general manager played an executive sponsor role, which ensured all departments were bought into the process.
The initial feedback from sales and customers was strongly positive as sales were able to enter into new conversations with customers, new and existing, and sell a broader range of information technology products and solutions. Customers were vocal about how easy the guide was to understand and how clear they found the information and diagrams. Now the solutions guide is a successfully established format for all public verticals and some verticals in the non-public space within Dell.
## Dealing with competition
The strength of a vendor’s solutions will be directly linked to its resilience in resisting competition; the strength can be the comprehensiveness, the level of customization, the level of integration with customer’s business or quality of its components. Where a solution is missing some of these it will be more exposed and vulnerable to competitive displacement.
Marketing has the ability to articulate clearly the benefits of a solution that customers find important. Although a solution may be the sum of different products, vendors still need to ensure that organizations are managing product transitions and lifecycles to keep them as up to date as possible. Competition may offer more up-to-date aspects of the solution, which can persuade customers to move away from their existing supplier.
## Updating solutions
Probably contrary to perception, solutions have a shelf-life; even those more complex and tailored to customers’ needs will reach maturity or end-of-life. The complexity and sum of the parts of a solution tend to mask some of the needs for updating or replacing. Situations where solutions need updating or changing include:
* *Where customers’ needs change.* If this is a dramatic change of business needs, business purpose or growth, a vendor needs to ensure it capitalizes in time.
* *Products within the solution reach end-of-life.* For example, application software as part of a home office solution is replaced by updated office software and is no longer supported.
* *Products within a high-end technology solution* need to be updated with new technologies.
* *Competition challenges by offering new benefits.* For example, a telecommunication provider offering new tariffs for multiple phone contracts will influence alternative vendors in updating and improving their solution or offering.
# 09
# B2B influencer marketing
This chapter will give you an understanding of:
* types of influencer marketing
* what an influencer is
* influencer marketing process
* preparing the influencer strategy and programme
* finding influencers
* identifying the right influencers
* engaging influencers
* measuring influencer marketing
# What is influencer marketing?
Influencer marketing is a form of marketing where the focus is placed on specific individuals (or groups of individuals) as a means of influencing the target customer. It identifies individuals who have influence over potential buyers and orients marketing activities around such influencers with the goal of leveraging their influence.
The idea of using influencers to market a product or service is not new. Most consumer commercials include influencers (professional athletes, movie stars, etc) though most B2B marketing budgets are not sufficient to cover the costs of such influencers and fortunately these influencers may not be the most suitable to influence business customers. In B2B there are influencers with a smaller reach but influence over a very specific set of target customers.
# Types of influencer marketing
Influencer marketing can come in different forms: testimonial advertising, key events where speakers and influencers are advertised, bloggers who have a reach.
**EXAMPLE** Dell
In 2010 and 2011 Dell employed a brand campaign ‘Take Your Own Path’ which included some of their main customers at that time. The campaign meant the owner of the company became the main advocate and influencer for highlighting the benefits of buying solutions from Dell. As part of the campaign multiple advertising vehicles were employed as well as multiple content formats across adverts, banners, videos, PR and TV interviews. This campaign was designed to build credibility for Dell as a B2B solutions provider.
# Why use influencer marketing?
The application of influencer marketing in B2B is increasing. This is due to three factors:
1. Social platforms have matured and become more sophisticated.
2. Customers in general are using social more and in more versatile ways in the business world where Twitter, LinkedIn and YouTube have grown in usage and sophistication.
3. Content is growing at an overwhelming rate and influencers provide a different angle with which to engage customers.
At the heart of influencer marketing is the fact that people trust people, especially people who have credentials of some sort. This part of B2B marketing is sometimes called B2P as it’s about the human component of B2B marketing.
Influencer marketing is great where one is trying to reach an audience or sway an audience in a direction. Influencer marketing can overcome the following issues:
* *Audience reach.* The current perception of the brand or marketing hinders its ability to reach a given audience.
* *Credibility.* The influencer’s advocacy can help the company improve its credibility.
* *Resonance.* Current marketing messages and activities aren’t resonating with the target customer; they’re not compelling enough. Influencer marketing helps support and substantiate messages with a customer.
Recent research shows:
* 72 per cent of B2B buyers use social media to research a purchase (Demand Gen, 2014).
* More business buyers (53 per cent) are relying on peer recommendations when they make a purchase. This number has increased from 19 per cent in 2012 (Demand Gen, 2014).
* B2B customers want to know what subject matter experts think. In fact, vendor subject matter experts were rated the most trustworthy or credible source of information, beating peers, analysts and digital influencers (ITSMA, 2015).
# Influencer types
An influencer is a person who is well-connected and regarded as influential and in the know; someone who is looked to for advice, direction, knowledge and opinions. Business buyers can be influenced by these types of people:
* peers who are regarded as leading in their field;
* analysts for a subject area or market – through white papers or papers they are deemed experts;
* magazine writers or journalists;
* specialists in each field who are regarded as experts by customers;
* bloggers: typically specialists or experts in their subject who communicate opinions and views via blogs.
# The influencer marketing process
Influencer marketing can be split into five steps, shown in Figure 9.1:
1. *Prepare.* The customer or audience intended to be reached is defined. Activities to carry out in this phase are creation of stakeholder maps, identification of how stakeholders are influenced, purchasing influencers within the buying journey.
2. *Identify key influencers.* The next step is to discover or identify the right influencers. During this phase digital applications can be used to identify the best placed influencer(s).
3. *Select influencers.* In this stage influencers are selected based on criteria such as followers, reach, strength of recommendation, level and so on.
4. *Engage and activate.* During this phase influencers are engaged directly or indirectly.
5. *Measure results.* Results from the influencer campaign or activity are measured.
Figure 9.1 Influencer marketing process
Influencer marketing process
## Step 1. Preparation
Before launching into the influencer search, engagement, etc, one needs to prepare the ground. It’s important to define the customer target and then map the influencer ecosystem for the customer target set.
### Influencer marketing goals
Influencer marketing goals are defined formally or in brainstorming forums. Defined goals can be about reaching a customer set, or increasing credibility or opening new business.
### Customer target set
The customer target set can be a customer segment, sub-segment or another grouping. Typically, influencers are engaged to reach a specific niche target segment; where there are multiple segments it’s likely that more than one influencer is required. This specificity will help with the impact and effectiveness of the influencer marketing campaign later in the process.
### Building the influencer stakeholder map
The influencer stakeholder map is created using two approaches that may substitute or complement each other. The first is auditing the existing customer base using in-house resources. Customers can be engaged through sales or directly to understand their influencers during their purchasing journey. Customers may not refer to such people as ‘influencers’ but as key contacts, information sources or experts. This method is ad hoc and low in cost, which means most businesses can carry this out; it assumes a good relationship between vendor and supplier. The second method is using a research company where organizations can use existing and/or external customers; the benefits of this approach are that in-house resources are not burdened with additional tasks and external expertise can be leveraged.
## Steps 2 and 3. Identify and select influencers
In B2B the process of finding influencers can be different to that of the consumer space, where influencers have great exposure and are often very public. If the business is actively engaged in business conversations, themes and members of associations there’s a likelihood that there are already several influencers in its network, whether digitally, socially or offline. Some ways of identifying influencers are:
* *Social listening tools.* Using social listening applications with influencer search plug-ins and key words or terms one can find several influencers.
* *Twitter activity.* Influencers can be found by viewing Twitter activity and associated number of connections and number of followers. It’s likely that influential speakers or speaker associations are already being engaged.
* *Industry associations.* Industry associations will typically be connected to key experts in the industry and know of influencers. Within the industry association itself, some of the senior executives may be influencers.
* *Events and speakers.* Event management organizers will probably know of relevant keynote speakers who are influencers.
* *Peers.* One can ask peers, in the company or in other companies, who may know of influencers or be in touch with them.
* *Subject matter experts.* In-house subject matter experts who are bloggers may be influencers themselves or be in a social network that includes influencers.
Figure 9.2 Influencer identifier
Influencer identifier
**TIP** Finding influencers in the B2B enterprise marketing space
It is possible that the influencer being searched for is not yet using digital or social in the same way as in the consumer space. If the B2B customer being targeted is a niche one, the influencer and their way of engaging customers may be selective. So how does one find such influencers? For example, how would you find an influencer talking about technology usage in healthcare? Here are some thoughts:
* Ask contacts working in the healthcare or IT industry.
* Look at trade fair events and key speakers.
* Look at trade or industry associations and senior people who are responsible for public relations.
* Ask a PR agency.
* Look at editors of healthcare IT journals.
* Ask customers who they think of as credible spokespeople.
Where research finds multiple influencers, marketers will need to select the optimum influencer type and the best one per category. Even if the marketing budget supports the use of multiple influencers at the same time, this may lead to different messages, which could be confusing for the customer.
One can narrow down the choice of influencers as follows (see Table 9.1): 1) measure using digital metrics such as Klout score, reach or posts. Reach can include aspects such as Twitter reach, and activity levels in terms of number of blogs, posts and how many people like, share or comment on such posts. 2) Offline, which may require qualitative research, eg a survey with customers asking them to highlight their sources of information or influencers. Overall the digital and non-digital weighting can be across influencer type and based on type and level of influencer strength.
Table 9.1 Identifying the right influencers
| **Stage 1: Influencer type selection** |
| **Criteria** | **Journalist** | **Analyst** | **Peer** | **Industry** | **Subject matter expert** |
| **Reach** | | | | | |
| **Posts** | | | | | |
| **Activity –Klout** | | | | | |
| **Followers** | | | | | |
| **Customer mention** | | | | | |
| **Qualitative** | | | | | |
| **Stage 2: Influencer weighting** |
| **By type** | **Journalist 1** | **Journalist 2** | **Journalist 3** |
| **Reach** | | | |
| **Posts** | | | |
| **Activity – Klout** | | | |
| **Followers** | | | |
| **Customer mention** | | | |
| **Qualitative** | | | |
## Step 4. Engage and activate
Once the influencers have been identified and the choice narrowed down to the right influencer(s) the next stage is to engage them. Ways of doing so include:
* Be their fan and follow them online.
* Participate in their forums.
* Engage through customer referral and recommendation, on- and offline.
* Engage through PR agencies or industrial association referral.
### Engagement model
Engaging influencers is about building relationships. Every influencer will probably require a different engagement approach so it’s important to think about how to engage.
### Incentivizing them
Engaging influencers, like most business relationships, is a two-way approach. It involves a balance between offering advantages for the influencer partnership and receiving benefits; the partnerships tend to be long term so creating the right engagement environment is important.
Advantages offered to influencers may be providing access to market information or specialized information about the business’s products and solutions. Like partnerships the influencer relationship can be a loose one or a heavily integrated one where the influencer is treated as an extended member of the company; however the risk of over-integration of influencers is that they are no longer perceived as impartial by the customers they’re trying to influence. Other advantages may be participating in company or customer events, offering them platforms to speak to customers and access to information. Benefits from the influencer are reach, credibility, resonance and ultimately in converting prospects to customers or in increasing customer stickiness.
## Step 5. Measure results
Before an influencer marketing campaign is implemented KPIs (key performance indicators) linked to objectives need to be determined. Measurement of influencer marketing will vary and depend on goals stipulated within the preparation phase. Types of measurement can be vehicle level, campaign level and according to the customer journey.
The influencer KPIs should align with the influencer goals. For example, if the objective of the influencer campaign is customer reach then possible metrics could be audience reach by social platform; or circulation of an article or advert if the activity is an insert within a magazine. Where the objective is to improve awareness for a particular product or service, the KPI can be a perception survey linked to the influencer activity. If the objective is in generating leads, the measurement can be about leads or opportunities linked to the influencer campaign.
# Influencer marketing for the long term
Once influencer campaigns have been completed the typical behaviour for businesses is to move to the next thing; unless this is a deliberate choice, a business should try to avoid this as influencer relationships take time to build and to leverage and maximize. Stopping an activity before ROI stage means wasting budgets. Influencers hold the key to customers, so switching on and off a campaign may mean customers follow the same behaviour. An influencer campaign may take time to move the customers along the buyer journey; where the business objective is in improving consideration or lead generation, the vendor needs to ensure enough time is taken for the campaign to take effect.
**EXAMPLE** IBM and influencer marketing – Skyword
IBM’s main challenge was to build presence and awareness of its digital applications; for this it turned to influencer marketing. IBM set up an influencer blogging programme known as the ‘Skyword programme’ focused on midsize business marketing. It brought together different influencers from IT analysts and independent bloggers with the focused theme for the activity being competitive advantage and how midsize business can achieve it through information technology.
As part of the campaign influencers created content which in turn educated the target market. The objective was to create a knock-on effect of reaching and engaging intelligent individuals within midsize businesses to be able to impact change within that company. It challenged these individuals to use information differently, to empower their teams to make better decisions that would ultimately drive business growth.
The campaign’s success was built on identifying influencers correctly aligned to business goals. Where the business goals were aligned to product and service portfolios, IBM chose particular influencers as they had credibility and were authentic; the independent nature of followers and networks was a key component of the campaign.
ACTIVITIES
Based on your industry, start to identify potential influencers through digital applications such as Klout and Twitter.
Use partners, associations, PR or anything else to build influencer identification through non-digital means.
Compare the results from the digital and non-digital approaches in terms of influencer names. What are your observations?
# References
Demand Gen (2014) B2B buyer behaviour survey, available at: https://www.demandgenreport.com/industry-resources/research/2508-the-2014-b2b-buyer-behaviour-survey (accessed 10 February 2017)
ITSMA (2015) *How Buyers Consume Information Study*, abbreviated summary, ITSMA, London
# PART THREE
# Transform through content and digital marketing
# 10
# Digital and content marketing
This chapter will give you an understanding of:
* digital marketing evolution
* B2B digital marketing channels
* how media has changed
* digital and sales
* digital and marketing
# The digital marketing evolution
Even in the past five years B2B marketing has undergone some dramatic shifts. Recent years have seen the rise of content marketing, an increase in content formats, the use of digital in marketing, the increase in marketing technologies and the changing behaviours of customers in how they use digital and online information.
Within B2B marketing there are two distinct forms of digital marketing: transactional and relationship (or enterprise as it is sometimes called); see Table 10.1. Transactional digital marketing is digital marketing centred on business purchases of product; this is very like consumer marketing and is typically the form used to market to small business customers. The customer buying process and behaviours across the journey are easier to measure as almost all of it occurs online. Relationship digital marketing is built on a business requiring sales relationships to market and sell; this requires integration between offline and online or at least an appreciation that some aspects of the decision journey are not all digital.
Table 10.1 Transactional digital vs relationship digital
| | **Transactional digital marketing** | **Relationship digital marketing** |
| **Products** | Point products | Multiple products |
| **Customer** | One decision maker | Multiple decision makers |
| **Targeting** | Large sets of consumers based on segment/persona | Fewer groupings of customers based on segment/account/persona |
| **Price** | Important | Less important |
| **Time to purchase** | Short | Long |
| **Main digital marketing objective** | Website traffic and conversion | Lead generation |
| **Communication channel mix: online vs offline** | High mix (to full use) of digital channels | Balanced mix of digital vs offline channels used |
B2B customers are becoming more selective, more demanding and vocal about how they want to be engaged by potential suppliers. E-mail marketing in some areas has become overused or applied inappropriately, leading to customers perceiving it as a nuisance and subsequently requesting e-mails from vendors to be stopped. Because of digital marketing and technologies, business customers expect more from an organization; they expect to be engaged in a timelier fashion per their needs.
# Changing B2B digital marketing channels
Digital channels are the channels used to reach customers via online; over recent years their breadth and versatility have increased. For companies still making the shift, this will feel like a maze and somewhat impossible to navigate.
Digital marketing channels are sometimes called vehicles; the main ones are SEO (search engine optimization), online PR, social media, e-mail, video, webinars, digital advertising, website marketing and mobile. Their application and mix will depend on the customers, their buying phase, a vendor’s goals and marketing budget.
Marketers’ growing digital focus is being driven by technology and an increase in user demand via mobile and social. As a consequence, marketers need to adapt their marketing content and communication channel mix to include more digital channels.
# Driving forces
So why the change in digital marketing and in digital marketing applications? We know from previous chapters that customers’ behaviours have had an impact. The three main driving forces that have influenced the usage of digital marketing and digital applications are: accessibility, availability and versatility of information.
## Accessibility
In the past 10 years mobile devices have changed. As recently as 2003 smartphones were regarded as a luxury item used by senior level executives, determined by pricing. Today virtually everyone can own a smartphone. Smartphones have become more powerful for using, consuming and sharing information. Ultra-mobile devices, tablets and such have become more prevalent. According to *Forbes*, the top 25 per cent of SMEs (small and medium-sized enterprises) are seeing big gains from mobile. In some parts of the world, over 90 per cent of SMEs use their smartphone to help manage their business.
Because of this shift businesses provide more information and content that can be accessed through mobile device. Terms such as ‘mobile optimized’ or ‘mobile responsive’ have become common. B2B marketers in turn are allocating more of their budgets to mobile marketing to ensure they reach potential buyers.
Some recent trends show total mobile advertising spend was expected to reach $40 billion in 2016 (Stengard, 2016); it is surpassing desktop spend. In-app mobile advertising spend will account for almost 75 per cent of total mobile ad spend.
Customers can obtain information in real time; they are able to be engaged almost any time of the day. However, being able to access a customer all the time doesn’t mean vendors should do it.
### Cross-device marketing
A new feature of mobile advertising is that of cross-device marketing; this provides advertisers with information on the user’s use of devices and their interplay by looking at mobile exchange, desktop activity and geo-specific locations. Cross-device is a hot topic in B2C marketing and now a growing trend in B2B. While B2C is about targeting an individual across multiple devices, in B2B it’s about both the individual and the company targeting across multiple devices. If multiple employees from the same company have visited an organization’s website with some regularity, this can mean either that they are more likely to convert to buyers or that they are somewhere along the buyer journey.
Understanding multiple device usage and behaviours gives marketers insight into building unique user profiles. In B2B it is important for creating content that is mobile friendly and helps understand when, how and where to engage prospects.
## Availability
Another major driving force is the availability of information; information and content is growing at a rate faster than users can consume it. Today the challenge is not about getting information but understanding which pieces of information to access, read and keep.
Management of information is becoming more the trend. In the business world the challenge is how to digest it, store it and then use it. ‘Big data’ as a term has become popular as some businesses require specialized applications on powerful servers with massive storage to crunch data and draw insights from it to help them improve their operations and marketing.
Online one can find how data is being created in real time. For example, in 60 seconds over 100 LinkedIn accounts are created, over 70 domains are registered, 700,000 search queries are carried out and 168 million e-mails are sent.
## Versatility
The final driving force is versatility of information. Although the types of content have mostly been available for a long time, the range of content formats used by B2B marketers has grown in the past five years; for example, B2B marketers are employing 12–14 formats of content on average (CMI, 2016).
This is probably due to customers spending more time online and business customers looking for more information and possibly to access customers. Users expect information to be easy and quick to read, which has led to easily digested pieces of content being created. As a knock-on effect, infographics have become popular, SlideShare has become more widespread and the technology has become more accessible for B2B marketers, from both an ease of use and cost perspective.
Figure 10.1 Content marketing tactic usage
Content marketing tactic usage
**SOURCE** CMI (2016) B2B content marketing: 2016 benchmarks, budgets and trends – North America, Content Marketing Institute, http://contentmarketinginstitute.com/wp-content/uploads/2015/09/2016_B2B_Report_Final.pdf
# The changing face of B2B
Associated to the evolving and changing digital landscape, B2B marketers have experienced a change in media and media consumption. Video has become more widely used; more and more B2B customers turn to video to learn about products they’re interested in, and the increase in YouTube views, year on year, is in double digits.
Social media has matured and evolved to become more than just a conversational platform. LinkedIn, which was regarded by businesses as a great networking or HR recruitment platform, has now evolved to become a marketing and sales platform. Twitter is used by businesses to connect and engage potential customers. According to recent research (Bullas, 2012), articles with images get almost double the views of articles without.
**Example** YouTube
YouTube, founded in 2005, is today the most used video site in the world. In 2016 it was used by around 1.3 billion people, 72 hours of video was uploaded every minute and four hours of video were watched monthly.
YouTube originated as a site for amateur videos but has become a site to effectively distribute original content. B2B marketers use YouTube at different stages in the buyer journey not only to promote awareness of products and solutions but to nurture customer relationships and even to help in closing sales.
More than half of YouTube views come from mobile devices and 9 per cent of US small businesses use YouTube. According to the CMI, in 2016 76 per cent of B2B marketers use video marketing.
## Marketing automation
Businesses that use marketing automation to nurture prospects experience as much as a 451 per cent increase in qualified leads (Moore, 2016). Marketing automation, which was perceived initially as an e-mail management tool, now can manage and track all marketing channels, measure them and in doing so help marketing better understand customers’ behaviours online and offline.
Marketing automation has evolved to support easier and quicker integration of offline tactics. For example, at events badges can be more easily scanned and sent to CRM systems, allowing leads to be captured faster and more easily nurtured and tracked.
# Digital and sales
In the past the emphasis was on salespeople discovering customers’ problems or challenges and then presenting them with the right solution; this allowed salespeople to steer customers towards vendors’ own solutions or offerings. The salespeople could provide the solution and articulate how their particular solution was better.
Digital has inverted the relationship between sales and customer, giving customers more power to manage their purchase decision journey, to educate themselves and to decide when and how to engage a potential vendor. By inverting the relationship, digital has also meant that engagement with a vendor can and does take place much later in the journey.
This shift means sales need to appreciate the new digital channels and for some to adopt and use them. For example, some sales specialists have become bloggers in their field, and others have adopted social media platforms such as Twitter and LinkedIn to engage a wider audience. Others work more closely with marketing to help with the creation of content that they can use later to send to a potential prospect. Advocacy programmes have become more important where sales as well as other employees embrace and use social content.
Sales are becoming marketers and the most successful salespeople are the ones embracing different forms of digital, adopting social platforms to engage customers in different ways and understanding how to use different forms of content to manage customer relationships across all purchase phases.
## Changing models and approaches
Digital marketing has also impacted the linearity of customer buying phases. Customers used to move from one step in the process to the next, using a piece of content or marketing tactics as they moved through their journey; today customers can, in an instant, digest a piece of content, share on social; then be messaged through their mobile device via e-mail or SMS. At each stage customers can digest different content and use different marketing vehicles. This means that the traditional way of tracking and measuring media and ROI by each channel in isolation is no longer possible or if it is done it does not reflect reality. New measurement and tracking models are being developed to account for this lack of linearity, such as attribution models across the journey in a blended or allocated fashion.
Funnels are also becoming more complex. Customers don’t always move clearly from awareness, through consideration to purchase. The reality is probably more stages for a business customer, extending from early in the buyer journey to the end. Instead of using marketing funnels in their pure form, marketers need to complement the funnel approach with buyer cycle stages, breaking it up into early, mid and late in the cycle.
# References and further reading
Bullas, J (2012) 6 powerful reason why you should include images in your marketing – infographic, availabe at: http://www.jeffbullas.com/2012/05/28/6-powerful-reasons-why-you-should-include-images-in-your-marketing-infographic/ (accessed 10 February 2017)
CMI (2016) B2B content marketing: 2016 benchmarks, budgets and trends – North America, Content Marketing Institute, available at: http://contentmarketinginstitute.com/wp-content/uploads/2015/09/2016_B2B_Report_Final.pdf (accessed 10 February 2017)
Columbus, L (2015) Mobile technologies becoming a growth engine for small and medium businesses, *Forbes*, availabe at: https://www.forbes.com/sites/louiscolumbus/2015/02/08/mobile-technologies-becoming-a-growth-engine-for-small-and-medium-businesses/#45be4f574edb (accessed 10 February 2017)
Moore, K (2016) 25 marketing automation stats for 2016, SessionCam, available at: https://sessioncam.com/25-marketing-automation-stats-for-2016/ (accessed 10 February 2017)
Stengard, J (2016) 15 must know marketing automation stats, emarketeer, avail-able at: http://www.emarketeer.com/blog/mustknow-marketing-automation-stats/ (accessed 10 February 2017)
# Digital marketing strategy and planning
This chapter will give you an understanding of:
* digital strategy objectives
* digital marketing framework
* digital planning/strategy enablers
* digital challenges and opportunities
* digital marketing value chain
* paid, earned and owned digital
* digital marketing measurement
* digital marketing technologies
In this chapter, we’ll look at digital marketing objectives, a framework for digital marketing planning, and how to embrace new digital technologies that enable digital marketing success.
# Digital marketing strategy objectives
As digital marketing has become part of every stage of a buyer journey and digital technologies flourish, the objectives behind a digital marketing strategy can vary greatly as can the implementation and use of digital marketing techniques. Some examples of digital strategy objectives include increasing leads and customer engagement, reaching new customer segments, raising brand awareness, reducing marketing and customer service costs. For each objective the application of digital marketing can be very different.
## Strategy and planning framework
To develop digital marketing plans and strategies we need a framework to work against; Figure 11.1 outlines how this is broken into six stages:
1. Goal definition.
2. Strategic analysis, forecasting (market assessment).
3. Strategic options development/marketing mix development.
4. Evaluating strategic options.
5. Implementing and activating, optimizing the marketing of the digital content and channels.
6. Analysing and reviewing.
Figure 11.1 Digital marketing strategy framework
Digital marketing strategy framework
© Simon Hall, 2017
### Enablers
Supporting the strategy and planning framework are several enablers for carrying out and implementing a digital marketing strategy:
**Capabilities** are marketers’ know-how; they are important in designing the right digital marketing strategy and in implementing it effectively.
**Technology selection**. Technologies and applications may be needed to collect and crunch data related to the strategy as well as track customers across the journey. If the focus of the company is to further embrace social media then social monitoring and listening applications may need to be invested in.
**Content optimization**. Content may need to be optimized according to gaps in the buyer journey, or in creating new formats not yet available. For example, where a customer is expecting SlideShare content to help later in the evaluation phase, the organization should make sure to have this available in time.
**Buyer journey mapping**. Although it seems like an obvious task, it may not be so for some companies. In this case it’s not just about mapping out stages in the buyer journey but what customers do with which information, leveraging which channels, whether digital and non-digital.
**Vehicle mix testing**. Where there is little existing information it’s prudent for marketers to test digital mix assumptions on a target group either directly or via an agency.
## Step 1. Goal definition
So how can digital marketing support different marketing strategies? To understand more about this, below are some examples of B2B marketing strategies and how digital marketing and applications can be aligned.
### Targeting and reaching customers
Traditionally, if a marketing department wanted to reach masses of business customers it would need to think about hefty marketing budgets to cover direct mail, e-mail, public relations, advertising and maybe TV advertising. Businesses today can suddenly reach thousands of customers at very low or even no cost.
Digital marketing plays an important role in targeting and reaching customers through the organic multiplier effect digital channels have: through shared, liked and referred content businesses today can quickly target and reach customers and gain exposure not possible in the same way with offline marketing. Marketers putting together a digital marketing strategy designed to reach customers have to think about how to do so effectively, ie how to reach the right customers at the right time in the right way.
### Engage
Marketing automation technologies allow businesses to engage huge volumes of customers through nurtured communications and targeted communications. Social platforms and online communities mean that businesses now can engage potential customers much easier through questions, dedicated forums or other aspects of social media platforms.
Ad-serving technologies make it possible for advertising to be placed on website pages whose content is deemed relevant. Native advertising means customers get to see additional content similar to what they are already viewing in a similar format; this means organizations can share content without interrupting the customer. Other forms of engagement are through online chat functions; vendors employ chat possibilities for customers to ask short questions in real time rather than being burdened with the time and cost of making phone calls.
Online webcasts or webinars allow businesses to present directly or in partnerships with others key messages or information. Customers, regardless of location, can attend lectures, seminars or virtual events.
### Customer nurture
Prior to the increase in digital and digital technologies businesses were limited in ways to nurture customers along their buyer journey; digital has made it possible to nurture customers through their purchase process. A nurture strategy could involve a mix of the following digital marketing applications: software to retarget customers with additional content, e-mail management software or marketing automation software that manages e-mail send and response rates and sends subsequent e-mails and associated social banners to customer profiles and industries. Digital channels and formats for nurturing customers along their buyer journey can be social forums, own social downloads, webcasts to update on latest product information and new videos.
### Customer conversion
Digital can help with converting customers to purchase through webinar platforms geared to answering pre-order questions. As part of events, digital plays a role in helping customers engage better with content or to view content before, during and after the event.
## Step 2. Strategic analysis, forecasting
The second stage is about assessing the market and forecasting digital market dynamics. During this step, marketers need to identify what is changing, why it’s changing and how the market is changing.
## Step 3. Strategic options development
During this phase different marketing mix options are developed according to goals. Options for reaching, engaging, nurturing and converting customers are explored. New elements of the marketing mix such as social are also defined within this phase, including which social channels to use and how.
## Step 4. Evaluate strategic options
Options are evaluated against budget, resources available, internal skills and capabilities as well as reviewing ROI projections.
## Step 5. Implement and activate
Digital plans are implemented according to schedules. Activities are readied where needed to support implementation and technologies leveraged so that marketing can optimize the use of digital content and channels.
## Step 6. Analyse
During this stage the activities and plan related to goals, objectives and KPIs are reviewed and analysed.
# Digital challenges and opportunities
With digital marketing technology there are several challenges facing marketers; on the flip side, digital technologies have brought with them some solutions and answers to alleviate these challenges. It’s important marketers embrace and capitalize on these digital opportunities.
## Digital challenges
The main challenges digital marketing and technologies have brought have been in content diversification, the evolving social media marketing space and the move away from traditional marketing funnels. Regarding the increased diversification of content formats, marketers need to create not only more content but content in a richer variety of forms. Some of these content formats would, only 10 years ago, be challenging or even impossible due to limited resources, bandwidths and capabilities.
The social landscape is evolving as business customers are using social more and in different ways due to the increased sophistication of social media platforms. LinkedIn, for example, now offers new and different capabilities beyond helping people network; it supports advertising, lead generation and different content formats. Social has become part of business customers’ DNA and their business processes. Marketing funnels have been erased and in their place is a maze of interconnected channels reflecting the organic way in which customers use channels and consume information.
## Digital opportunities
Digital marketing and digital technologies have brought about new opportunities from which B2B marketers can benefit:
**Lower cost of digital marketing**
Those who used TV or offline media in the past will find digital more cost-effective in reaching the same customer segments.
**Speed of connecting**
With digital, the pace of marketing to customers and connecting with them has become much faster; at a touch of a button businesses can engage and share information with prospects. Digital and social platforms facilitate ease of engaging and reaching customers and even in building relationships with them.
**More detailed analysis**
Digital applications enable businesses and marketers to analyse behaviours at a micro-level; where customers use digital throughout their buying process marketers can track them and their behaviours more easily.
**Richer content**
Richer and more varied content has become easier and possible even for less experienced marketers.
**Content distribution**
New technologies mean marketers can distribute content easier in a targeted manner and through retargeting can tap into buyer journeys and areas of interest by retargeting customers with additional pieces of content.
**Media buying**
Digital applications have made it possible for marketers to purchase media more easily, either directly or through agencies.
**Competitive levelling**
Digital also supports smaller businesses in marketing in the way larger businesses do; they can compete regardless of their size. Traditionally a smaller retailer would struggle to match the finesse of the approach of its larger competitors.
**Native advertising**
With digital it is possible to use native advertising to catch content and use it to build trust and engagement with potential customers. Native advertising doesn’t disrupt the user experience and offers helpful information in a format similar to the other content on the site, so users engage with it more than they would with, say, a banner ad. According to the CMI, in 2015 34 per cent of B2B marketers used native advertising to promote content; although not a large number this trend is expected to increase in the coming years (CMI, 2015).
# Digital marketing value chain
The digital marketing value chain should be used as an auditing tool to review the value add of all the digital marketing-related elements according to business and customer objectives. By working through the value chain back-end to customer-facing front-end, businesses are able to understand how each digital marketing element impacts the next stage in the value chain and thus understand if there is a value-added or redundant (non- value-added) component. A typical digital marketing value chain is split into four core areas:
1. *Background infrastructure* can be split into owned online resource, in-house content resource and general cost of infrastructure. At this stage the challenge for marketers is to understand the potential need and value-add prior to incurring costs in using or purchasing such applications and technologies.
2. *Digital preparation – awareness based.* Such digital applications can be through using apps to buy media and serve ads. In the cost of creating content (as opposed to cost of resource) this can be the cost of paid search rather than organic search.
3. *Digital preparation – lead generation based* can include social paid targeting activities, content syndication associated with lead generation, webinars or other digital channels directly attributed to generating leads.
4. *Digital performance tracking and review* – such technologies can be CRM with marketing plug-ins, marketing automation to analyse marketing activities more broadly, or specific monitoring programs to monitor SEO quality, or social listening to monitor conversations on social.
Figure 11.2 The digital marketing value chain
The digital marketing value chain
## Why conduct a value chain audit?
The costs of digital marketing can mount up alarmingly unless controlled. Technology has grown at a phenomenal rate and marketers could spend all its budget on new technologies and none on ‘marketing’. Additionally, business leaders will need to understand how marketing can justify new investment and what such investments can deliver for the business; sometimes an audit can surface new opportunities for improving the business.
# Paid, earned and owned
Another way to develop strategy is to look at the mix of owned, paid and earned media channels:
**Owned media** is any online property that a vendor owns or controls that is unique to the brand. Examples are own websites, own blog sites and own social pages. The social media blogs are extensions of the own website.
**Paid media** is media the business pays for, and can include everything from social, through to online video to promote to customers, reach them, and increase traffic coming to owned media. Businesses do this to help achieve objectives not possible with pure owned media. Even with social you can pay for advertising (pay per click ads and display ads) that boosts content and gains exposure. Paid search also helps improve rankings, and one can pay influencers to promote content.
**Earned media** is media is neither owned nor paid; it can be described as online word-of-mouth. Examples are mentions, shares, reposts, reviews, recommendations or content picked up by third parties. One of the most effective driving forces of earned media is usually a combined result of strong organic rankings on the search engines, and content distributed by the brand. You can also have earned influencers who you haven’t paid for but support the brand’s messaging. One big driver behind earned success is content: compelling, timely, informative content such as blogs, infographics, videos, etc, can help improve the impact of earned media.
# Digital marketing measurement
Before building tables of metrics it’s important to answer a number of questions. Digital marketing and technologies produce more metrics than other areas simply because they have the ability to track more easily and support more granular views than traditional offline marketing. Marketers need to be extremely focused on what they measure and how, and in overall management of digital measurement to ensure success:
1. *Set measurable goals.* During this step goals that can be measured are defined, along with how they are to be measured.
2. *Define business-associated KPIs.* The KPIs should be linked to business performance in some way; even KPIs early in the buying cycle should be linked to engagement or reach. For example, the number of repeat visitors to a website or number of pieces of content being viewed can be linked to customer conversion statistics.
3. *Track and report.* During this phase the activity is tracked and reported back to key stakeholders.
4. *Review.* The activity is reviewed in terms of performance against set KPIs, how the activity is supporting business goals.
5. *Optimize.* Optimization may be needed when results are below plan. Where performance is above plan, optimization may still be possible by looking at implementation or how to take it to the next level, for instance where the activity was a pilot before the full implementation.
Digital marketing measurement can be split into some core categories:
* general performance – traffic, leads, reach;
* digital channel based – website, blog, social networks, search engines;
* customer lifecycle – acquisition of new customers; existing customers;
* campaign-based performance – lead generation, click-through, conversions.
Some other considerations and questions are:
* How will we use the metric? This question should help marketers think about which KPIs are going to be used rather than collecting sheets and sheets of metrics.
* How frequently will we need to use it? Some metrics may not need collecting or reviewing so frequently; limiting the number of metrics to review on a regular basis to a manageable number helps save time.
* Who is the audience of the metric? This question will help marketers think about how metrics are understood and whether they need interpreting or explaining before presenting. Rarely is it the case that marketers can understand and articulate all metrics they receive; this line of questioning will help marketers challenge the metric before sharing it further.
# Attribution models
One of the areas B2B marketers need to grasp is how and where to attribute marketing spend across the buyer journey; these engagements are referred to as touches. For some sectors these can stretch beyond 10 or 15 touches before a customer engages a vendor salesperson. Figure 11.3 shows the main attribution models:
**First touch** – this model means value is given to the first marketing touch in a sequence of touches. An example could be a business buyer receiving a message via a mobile phone and many steps later purchasing online from his or her desktop computer.
**Last touch** – this models means value is given the last touch in a sequence of customer touches. The example above means the weighting is given to the online website purchase.
**Non-weighted –** value is attributed equally across all touch points regardless of their relevance or importance.
**Time based** – value is attributed increasingly to touch points closest to conversion. This is often used because it is where it is easiest to measure and assign value; the risk is that investments earlier on in the buying cycle that are more awareness based are neglected and undervalued.
**Measurement based** – value is attributed to the touch points most easily measured; this can be where digital activities are measured even if offline activities contribute greatly to the purchase. This is the case in sectors where integration between offline and online marketing hasn’t been put in place; the risk is that offline marketing is undervalued.
**Position based** – value is attributed to touch points at the start and end; the middle touch points are usually not attributed.
Figure 11.3 Attribution models
Attribution models
© Simon Hall, 2017
# Digital tools and technologies
As mentioned before, marketing technology is vast with literally thousands of applications to choose from. To simplify things, one can place these in four core areas: back-office infrastructure, customer-facing, monitoring/review and intermediary marketing. Examples are:
* *Marketing back-office infrastructure*: includes CRM, marketing automation, content management and e-commerce engines.
* *Customer-facing*: includes webinars, search, e-mail marketing, blog, display.
* *Monitoring/review*: includes social analytics, SEO optimizing.
* *Intermediary marketing*: includes data management platforms (DMPs).
## Technology selection
* Various criteria will determine which technology to select, such as:
* business size: number of employees within the company;
* business goal: short, mid or long term, the goals to be achieved within one, two or more years;
* objectives and priorities: aligned to short- to mid-term goals;
* technology needs case: per above, highlighting benefit of the technology/ies;
* budget and budget ceiling: the budget available and the limit to spending;
* in-house vs external: where in-house is required this may depend on in-house competencies.
Activities
Look at your current marketing approach and identify which media are owned, paid and earned.
Consider your main target audience today. What do you know about their time spent online? Where do they go? What do they use online for?
# Reference
CMI (2015) B2B content marketing: 2015 benchmarks, budgets and trends – North America, Content Marketing Institute, availabe at: http://contentmarketinginstitute.com/wp-content/uploads/2014/10/2015_B2B_Research.pdf (accessed 10 February 2017)
# 12
# B2B digital marketing channels
This chapter will give you an understanding of:
* key digital marketing channels
* importance of website within B2B digital marketing
* how to use search and SEO
* B2B e-mail
* B2B mobile marketing
* online PR
* digital advertising
* webinar marketing
* video marketing
# What is a digital channel?
A digital channel can be thought of as a transport route, where the traffic is the content the digital channel delivers. Different content formats can be thought of as different types of transport vehicles. The ‘digital’ part of the term is because it’s via the internet or online. Non-digital is any form of communication channel not online or on the internet, eg hard copy magazines, newspapers and brochures, physical events, printed advertising and word-of-mouth.
In the digital world, some of these channels will merge and cross-over as they are used together (see Figure 12.1). Marketers need to view digital channels as an interconnected universe, in the same way as customers using digital do, not only using a mobile or a website for purchases. The main digital channels are:
**SEO:** SEO is search engine optimization and search engine marketing; it’s about being ranked high in search results when buyers search for topics related to your products.
**Online PR:** online PR marketing is linked to reputational marketing; it’s about telling stories through effective PR activities.
**Social media:** social media sites are born out of community participation, user-generated content, ratings and interactions.
**E-mail:** this continues to be integral in B2B marketing strategies and a great way to engage and nurture some business customers.
**Digital advertising:** this is sometimes referred to as ‘online advertising’.
**Mobile:** mobile marketing involves communicating with the customer via a cellular (or mobile) device, to send a simple marketing message, to introduce them to a new audience participation-based campaign or to invite them to visit a mobile website.
**Webinar:** a webinar is a presentation, lecture, workshop or seminar that is transmitted over the web using video conferencing software.
**Video:** video channels are used to provide digital video, eg YouTube.
**Website:** website marketing is using a company’s own website to support the communication and marketing of the business, value proposition and other information.
Figure 12.1 Hub and spoke
Hub and spoke
# The business website: the top priority
A company’s website is its window to customers; it can help drive perceptions either negatively or positively depending on how customers access the website, navigate it and how they perceive its content. The website should be the place customers come to from elsewhere in the digital world so having traffic directed to a business’s website is key. It should be thought of as the central hub for a business and for its channels, digital and otherwise.
**TIP** Traffic direction and business buying process
In Chapter 3 we learnt about business buying stages and behaviours. For a vendor it’s important to direct customers to its website in a timely manner. Too early may mean customers feel pushed into the sales process if the page they are directed to talks about pricing, commercial terms and products. Too late may mean that businesses lose out on opportunities where customers have already purchased something elsewhere.
## Business customer type and website development
Most businesses need to engage different stakeholders within their customer target set. Typical distinctions are between more specialized or technical targets and those more interested in business impact, such as business decision makers or senior executives. Therefore, it’s important that businesses create separate pages for each type of stakeholder.
## Measuring effectiveness
Effectiveness of websites can be measured by repeat visitors, time on site and/or bounce rate. A high bounce rate would indicate that the customers are leaving the website quickly and are not compelled to stay. A short amount of time on site indicates they are not staying long enough to read and digest the content. Another way to measure is by using heat maps, which show how much time is spent on different areas of the website pages.
# SEO and search
SEO and search marketing remain one of the top priorities for any B2B marketers. According to some recent surveys (eg Snyder and Hilal, 2015), 71 per cent of B2B researchers start their research with a generic search and it is the number one vehicle for customers.
In B2B ‘searching’ can be the customer researching more about the product/service solution they’re interested in. Vendors trying to engage prospects can facilitate search through better optimization of their web and setting up links and key words that will direct people to their website. This is search engine optimization, where the optimization refers to a business’s own website and related websites. The other main area is paid search; it refers to businesses spending money on key words. In line with the amount of spend against key words, the website is ranked based on search queries.
## Optimizing SEO
There are several digital applications businesses use to understand how or whether their website is SEO optimized; typical criteria are performance of the website, navigability, security and mobile-applicability. Optimization can be conducted in-house if the competence and resource are available. Essentially SEO is about being found without paying for keywords, through an intuitive website, by building links from content on other digital sites to the company’s own website.
## Optimizing paid search
One way B2B marketers can leverage paid search is by optimizing and selecting key words related specifically to a vendor’s business and its USP (unique selling proposition). The advice is to stay clear of peak seasons as bidding on key words may prove expensive: the more businesses bidding the higher the price of key words. Sometimes it may be more beneficial to bid on the key word directly after the peak season.
# B2B e-mail
B2B marketers use e-mail more than ever: 83 per cent of marketers use e-newsletters as a content marketing tactic according to CMI’s 2016 report. Within B2B there are different types of e-mail marketing ranging from a very basic e-mail through transactional e-mails to rich content e-mails.
Transactional e-mails are basic e-mails without any rich content; they are designed to influence a call-to-action or transaction. Direct e-mails are based on traditional marketing and don’t always take account of a customer’s journey or needs; they are there to communicate a promotional message. Personalized e-mails have some form of personalization whether through simply using the recipient’s name or individualizing the content throughout. Fully personalized e-mails can be very effective and are sometimes used to improve open and response rates. Rich content e-mails include an image, video link or other; the inclusion of richer content can help with improving open rates and engagement with the customer. Retargeted e-mails are used to retarget a customer who may have clicked or opened another form of content.
**TIP** Using e-mail in enterprise marketing
In enterprise or relationship marketing, e-mails are still widely used but in different forms; aside the above different forms of e-mails, the way e-mails are managed can differ where the relationship is being built or where sales want to lead the way information is communicated. The e-mail can be a lead into a phone call or be used to communicate something during a phone call or a way to follow up a phone call. This type of e-mail mechanic is sometimes called assisted e-mailing, as the e-mail is not sent on its own but assisted by another activity.
In some companies automated e-mail sends are restricted, as blanket e-mails are regarded as ineffective or even damaging to customer engagement. E-mails are therefore provided to sales to use according to their judgement.
**TIP** Nuisance e-mails – how to avoid them
If customers want to receive information or be engaged by a vendor at the right time in the buying process, e-mails can be welcome. They are only perceived as a nuisance if they’re not timely or if they are over-used. Where companies send too many e-mail to customers they can feel bombarded and ask the vendor to stop through suppression requests.
One way around the risk of over e-mailing is to create preference systems allowing customers to opt-in to e-mail only for selected topics. A common example of preference systems can be found in magazine or event companies that offer topic selection options for information or e-mail themes that customers are interested in.
Figure 12.2 E-mail and the B2B buyer journey
E-mail and the B2B buyer journey
© Simon Hall, 2017
## Social media
Social media channels cover a wide range of different types of social channels including social listening, social platforms – eg Facebook, LinkedIn, Twitter, social e-mail and social commerce. You’ll find more about how this digital channel can be used by B2B marketers in Chapter 15.
# Mobile marketing
Mobile marketing offers multiple benefits to B2B organizations, as well as being one of the main influencers of digital marketing. According to a survey by Google, 57 per cent of customers say they won’t recommend a business with a poorly designed mobile site and 40 per cent have turned to a competitor’s site after a poor experience. Another survey from emarketer (Sukhraj, 2016) highlighted that among the group of SMEs (small and medium-sized enterprises) that had or planned to create a website, just 33 per cent had a mobile-optimized site in September 2015.
Mobile provides customers with anytime, anywhere access; vendors looking to target customers can access customers in many more ways than with desktop-based activity. Other benefits of mobile marketing include the ability to personalize or target customers, the greater response rate and effectiveness of mobile channels, immediacy and distribution.
## B2B mobile marketing tactics
The main mobile marketing tactics include the following:
* *SMS* is simple and involves sending a text message. SMS can be effective in delivering very simple messages and in response rates as it boasts extremely high open rates; on average around 98 per cent open rate. Average response rates for SMS are also high.
* *QR codes*. Quick Response codes are like a barcode that can be read using smartphones; they can be used to provide access to additional content, eg as part of a campaign.
* *Mobile display ads*. Through different options such as click-to-call and click-to-video, mobile display is great for engaging prospects with the brand. Mobile adverts can be used in conjunction with geo-targeting to advertise to customers within a given area.
* *Mobile optimized and responsive websites.* Considering the amount of time prospects or customers are away from their desk using smartphones, having a mobile site allows users more possibilities to see a brand’s content; responsive content means adapting the content according to the device whether a mobile, a desktop or tablet.
* *Mobile applications* include software downloaded and installed on mobile devices. B2B event companies use mobile applications to better engage with audiences before, during and after events to share information and capture feedback and comments.
# Online PR
Online PR is very similar to traditional PR in the sense that it’s about influencing people rather than buying placement for brand content. The influence could result in a story in a magazine, newspaper or blog. It could also result in other online pick-up, including social media.
As people are influenced through more digital stimuli and usage, PR has made a shift in the past decade over to digital; we’ll explore this digital channel more in Chapter 15.
# Digital advertising
Similar in many ways to advertising in print media, with online advertising one pays for the space online, with the additional benefits that organizations can determine where the advert is placed and who sees it. With digital advertising one can target customers by behaviour, which can be a good way to reach professionals. This is done through advertising networks, which basically track the behaviour of users who visit their site network, then catalogue a user’s behaviour based on site content. Types of digital advertising in B2B covers mobile advertising, social media advertising and e-mail advertising.
## Retargeting
One of the main benefits of digital advertising over offline advertising is the ability to retarget customers. Retargeting in digital covers audience retargeting, key word retargeting and CRM retargeting.
## Ad serving
Ad serving is sometimes known as ‘advertising display’ and is about displaying text or rich media content online on a company or third-party website that has auctioned off digital real estate. Adverts are typically displayed as banners or pop-ups. Ad serving applications can offer the option to automatically optimize ads, thus reducing workload such as A/B testing. Many B2B ad serving and retargeting products, eg Demandbase and Everstring, are supporting more account-based ad targeting which is part of the Account Based Marketing movement (more on this in Chapter 21).
**TIP** Using Google AdWords
Google AdWords is an online advertising service enabling advertisers to displays ads in a timely manner to specific audiences. AdWords’ success for B2B marketing depends on three main factors: 1) carefully selecting relevant key words; 2) speaking to customers’ pain points; and 3) making sure ads are written for B2B buyers and have key word qualifiers that can help filter out B2C traffic.
# Webinars and webcasts
A webinar’s main feature is its interactive elements. Webinars come in the form of lectures and tutorials, presentations or mini events online. Webinars are useful in some of the following circumstances:
* When budgets don’t allow for physical events.
* When the target customers are in different locations or across different sectors.
* Where multiple topics at different time periods need to be planned.
* Where customers’ preference is for a short engagement approach.
* Smaller businesses may consider webinars as an alternative way to show-case their value proposition or present on a topic that is current and topical.
## Webinars as alternatives to events
Webinars can be considered as event alternatives as they allow different speakers to present from different locations. Where businesses have operations spread across different locations or even countries, a webinar allows them to include the highest quality internal speakers to be put in front of customers.
## Types of webinar
Webinars can be one of three types. They can one-way educational, where the focus is to update and educate the audience and where the communication is only to the audience; two-way interactive where audience response is facilitated; or awareness focused where the objective is to create awareness of a product or service. Webinars are also great tools for improving retention by keeping customers up to date on the latest products or solutions.
**TIP** Webinar optimization
The main areas where webinars can be optimized are:
* *Speaker quality* : involving the best speakers means webinars can be executed to the highest level possible.
* *Interactive where appropriate*: facilitating opportunities for interactions will allow participants to engage; this can be managed or unmanaged depending on the business preference.
* *Sales engagement/inclusion*: with sales involved in the process, opportunities arising from the webinar sales can be followed up in a more timely fashion.
Figure 12.3 Webinar marketing goals
Webinar marketing goals
**SOURCE** Rose, R (2015) *Webinars: they’re not just leads anymore*, CMI, www.slideshare.net/CMI/webinars-theyre-not-just-for-leads-anymore
# Digital video
Digital video is becoming more and more effective in B2B marketing and YouTube is increasing in popularity and is used by most businesses today. Typically, video channels are used for demonstrating or launching new products and technologies, in providing expertise, sharing events or demonstrating leadership in a subject area.
YouTube is one example of a video channel supporting SEO of its videos, which in turn helps boost rankings. Video can come in different forms: as pure advertising, as something to solve business challenges, or to demonstrate thought leadership.
## Video and the customer journey
Video can be used across the customer journey and is becoming increasingly effective not only in the awareness stage but during the consideration and final purchasing stages. This is through case study videos and videos demonstrating a solution or product.
# Communication channel integration
In B2B marketing, digital has become more important and marketers need to be aware of the customer’s information needs and consumption behaviours. Offering up a pure digital approach may mean that either some customers or stakeholders are left out, or customers are not included at certain buying stages, such as where they require more direct engagement via phone, or face-to-face in the consideration or purchase phase.
With that in mind, B2B marketers need to understand when, where and how to integrate digital and non-digital channels. There are different approaches to integration:
* *By stage of purchase*: one way is to look at early buying cycle mix, where the early part may be more digitally led and the late part may be mostly offline where customers may require some face-to-face interaction.
* *By customer type*: small businesses may be chosen for a pure digital approach while larger enterprises will require a mixed approach.
* *By customer decision maker*: even where targeting an organization may require a mix of digital and offline channels, certain decision makers could be regarded as being engaged purely through digital; for example, a CMO or chief digital officer in a company may be singled out as using mainly digital channels.
# References and further reading
CMI (2016) B2B content marketing: 2016 benchmarks, budgets and trend – North America, Content Marketing Institute, available at: http://contentmarketinginstitute.com/wp-content/uploads/2015/09/2016_B2B_Report_Final.pdf (accessed 10 February 2017)
Sukhraj, R (2016) 31 mobile marketing statistics to help you plan for 2017, emarketer, available at: https://www.impactbnd.com/blog/mobile-marketing-statistics-for-2016 (accessed 10 February 2017)
Snyder, K and Hilal, P (2015) The changing face of B2B marketing, available at: https://www.thinkwithgoogle.com/articles/the-changing-face-b2b-marketing.html (accessed 10 February 2017)
# 13
# Content marketing
This chapter will give you an understanding of:
* what content marketing is
* benefits of B2B content marketing
* types of B2B content
* the content marketing process
* content marketing planning
* key tools to support content marketing
* content curation and how to take advantage of it
# Content and content marketing
Content marketing can be defined as the process of creating and distributing relevant, timely, compelling content with the purpose of attracting, engaging and acquiring target customers. Content marketing in the past five years has become a hot topic; in Chapter 10 we reviewed the three main driving forces of digital, which has given marketers greater access to more information and content and the ability to create more types of content. Where customers are exposed to more information and content, the need to make that information more compelling, relevant, digestible, readable and timely becomes paramount, hence the increasing importance of content marketing.
# The benefits of B2B content marketing
Content marketing allows B2B marketers and businesses to reap different rewards; it builds a community of loyal customers; good quality free information can seal connections between a community of customers loyal to the vendor’s brand. Regular sharing of content will incentivize customers to stay connected to the brand. Content marketing also helps instil trust; with good quality content the brand can become more of an authority and trusted supplier or provider in the eyes of the customer; this trust will influence purchase decisions.
Compelling content will improve SEO; good quality content tends to get cited and shared, which will mean a greater likelihood of people going to a brand’s website. Content helps drive inbound traffic: with great content, businesses can benefit from more inbound traffic and more interest without having to go out and seek customers. Easy to read and understand content can help customers make better decisions; it can mean customers educate or self-educate better, which in turn allows them to make better decisions for their business.
According to other recent research from Forrester and Sirius Decisions (2013), about two-thirds of B2B buying decisions are being investigated, evaluated and compared before the brand is contacted. Most B2B purchasers say that content has a moderate or major impact on vendor selection.
# Types of content
Content can come in many different forms: videos, banners, mobile content, SMS messages, white papers, tutorials, advertorials, adverts, testimonials, SlideShare, demos, infographics, blogs, webinars and forum content. It encompasses digital and non-digital forms, although many marketing articles refer to digital as this is the area that is evolving faster and becoming more innovative.
B2B content marketing differs in several ways from B2C marketing; ultimately the user or recipient of the content is very different. There are similarities but there are also differences such as thought leadership, used more in B2B marketing; thought leadership content is content demonstrating leadership on a specific topic. The digital channels used can also vary between B2B and B2C; in B2B there are social media digital channels that are more effective in reaching B2B audiences and webinars are used a lot more. In B2C more traditional advertising such as TV or online TV is used; this is rarely the case with B2B. The message being delivered can also vary: in B2B this can be more rational, fact based and appealing to business concerns.
# What is great content?
Great content is:
* *Compelling.* It resonates with the audience and compels them to view more closely.
* *Timely.* It reaches customers at the right time for their needs; retargeting or redirected links make this possible.
* *Digestible/readable* for the target customers and their buying stage.
* *Relevant.* Information is relevant to the customer, their business, industry, etc.
* *Original.* It is something different, new, fresh rather than a repetition of what’s already out there.
* *Emotive.* It inspires an emotion such as an urgent need or a desire.
**EXAMPLE** American Express Open Forum
Amex’s Open Forum was launched in 2007 and has been regarded as the gold standard content marketing programme. The site was originally designed for small business: a major feature is facilitating small businesses discussing between themselves and sharing advice, asking questions and rating the advice given. The site features content by key business theme, eg raising finance, and multiple types of content, from articles through to videos. The forum is also a source of leads for new card members.
# Content creation
So how does one create content? Below is a six-step outline for content creation, shown in Figure 13.1:
1. *Customer needs assessment.* Customer needs and associated digital channels and content are identified. The needs of different organizational stakeholders are reviewed. Buyer triggers are identified, account profiles and personas are created. Core themes are then agreed upon based on the research.
2. *Planning.* In line with the budget content requirements are documented, timelines are drawn, processes are mapped out and editorial calendars built. The editorial calendar is the main tool to document content, timing of content, availability of distribution and different messages. There may be decisions to change content or reduce the amount of content based on the output of planning or to adjust the budget if it doesn’t allow for both creation and implementation.
3. *Content creation.* The main content is then created; this could be a large piece of research, or a piece involving a case study. The main content includes all messages and is the basis for the content creation that surrounds a marketing or brand campaign.
4. *The content portfolio is then created.* Different content formats are developed based on the main content. This part of the process is sometimes called ‘content fragmentation’.
5. *Content activation.* Content is distributed or activated. During this step content is made ready and accessible and distributed according to content calendars. Part of this step is content amplification where techniques are employed to maximize the audience reach for the content.
6. *Measure and optimize.* Content is measured in terms of effectiveness and against a set of criteria and metrics such as downloads, clicks, click-throughs, time spent or other if the content is offline.
Figure 13.1 Content creation process
Content creation process
## 1. Customer needs assessment
The first step in creating content is to understand the target customers, their needs, their pain points. One way to capture this information is in the form of a buyer persona. Buyer personas are fundamental to understanding the customer; despite this only 44 per cent of B2B marketers use them, according to ITSMA (Schwartz and Weaver, 2014). They require resources, effort and competency to build.
**TIP** How to create buyer personas
Are you finding it difficult to define the right content for your target audience? Is the challenge in mapping content to the step in the buyer journey? If yes then the answer lies in buyer personas.
Buyer personas can be created in three main ways: 1) interview with customers directly or via a research company; 2) using marketing automation or CRM database which captures information source insights regarding what buyers use; and 3) website download forms which include insights such as information sources and priorities.
A typical buyer persona includes the following:
* *Needs.* What are the pain points of the customers?
* *Drivers and motivators.* What motivates them workwise?
* *Role and level.* What role do they hold, and what level in the organization?
* *Key media used.* What media do they use?
* *Communication channels used.* How do they like to be communicated to?
* *Organization interdependencies.* Who do they engage within or outside the organization?
**TIP** Identify buyer triggers
Another step in creating compelling content is identifying buyer triggers. A buyer trigger is an event or occurrence that spurs buyers to make purchases. Buyer triggers come from different influences such as technology, legal, competitive, business growth, governmental or economic.
## 2. Planning
Within this stage content mapping, content audit and content plans are carried out. Timings for content creation, availability and distribution are defined as well as corresponding budgets.
### Content mapping
Content mapping is the mapping out of content formats and requirements according to buyers and their stages in the purchasing process. For example, in the awareness stage content formats could be videos, demonstrations and banners; as buyers move further along the buyer journey they require SlideShare, different types of video, infographics or even white papers.
Typically, content is mapped against two core areas (see Table 13.1): 1) buyer journey: content is created to cover each part of the journey and gaps in types of content or by buyer stage are identified; and 2) by format per buyer journey: the most used and required formats per buyer stage are created. Following the content mapping by buyer stage, the next step is to review content according to buyer type (see Figure 13.2). The buyer type can alter the content needs, formats and the buyer stage to focus on.
Figure 13.2 Stakeholder content mapping
Stakeholder content mapping
Table 13.1 Content map
| **Buying stage** | **Awareness** | **Evaluate** | **Compare** | **Buy and post- purchase** |
| Content stage | Educational content | Research support content | Comparison support content | Usage and optimization of content |
| Content types | SEO
Videos
Infographics
Research reports
Industry article | Webinars
Case studies
Testimonial
Content syndication | Webinars
Case studies
Testimonial
Content syndication
Comparison tools
SlideShare | User forum content
Tutorial
Forums
Blogs |
### Content audit
The content audit follows the content mapping stage. It is the analysis of the market communication material the organization developed for internal and external purposes. The objective is to determine content availability compared to content requirements. It’s about uncovering opportunities and gaps within the content portfolio: are there enough formats to capture buyers and help them move from a consideration stage to purchase stage?
A content audit can be conducted in its simplest form as an inventory list documenting all that is available in terms of content – white papers, infographics, etc, the date or age of the content, ownership of content and details regarding target customers. The next step is to understand the content’s usability, share-ability and tone: can customers easily find it, is it shareable, relevant, does it have a call-to-action?
### Other content marketing planning tools
* *Provisional editorial calendar* mapping out themes or requirements by time period – month, week, etc. It is quite possible that all content is already created before the start of the campaign and content is released and activated according to timings. The benefit of this is that the messaging is controlled and fewer messages are in the market; sub-themes can be emphasized by month. Also, the customer can digest and follow rather than receiving everything all at once.
* *Creation vs activation review.* This essentially is reviewing budget allocation between the content creation and content marketing; some marketers can get so excited about creating content that they leave little budget to share the content, promote and distribute it.
* *Budget planning.* Based on scope, themes, and personas, next is determining budget requirements to create different formats. According to a CMI report (2016) the average marketers in 2014 created 14 different formats for every theme compared to nearer seven in 2010.
## 3. Core content creation
Core content is the main, original core piece (or pieces) of content upon which the content portfolio is built. Examples of core content are:
* a piece of research;
* a story the company wants to tell;
* an event or trigger event in the market that the company has an answer to;
* a new message or theme the company promotes;
* a customer story;
* a market trend around which the business focuses key messages.
What the core content looks like depends on the customer target, the business goals and other influences.
## 4. Build the content portfolio
Once the core content is created the next stage is to fragment or atomize according to content format requirements, eg taking a research piece and the results and transforming them into infographics, banners, SlideShare or other formats according to the content map.
Business marketers can encounter numerous stumbling blocks in content creation; the main ones are:
* *Knowing how much content to create.* Putting a cap or ceiling on content is difficult, particularly when there is a trend of increasing content and content consumption. Defining the volume comes down to understanding the buyers and their use of content across the buyer stages.
* *Lack of buy-in to the value-add of content.* Having an organization not fully bought-in to the value of content can mean content is not created to the right scale or quality.
* *Knowing when to refresh.* The timing of refreshing may be misunderstood; updating content too late may mean the business loses out on opportunities to engage.
* *Too many formats.* With too many different content formats managing and sharing content can prove challenging; good content can get lost in the maze of poor content.
* *Content creation know-how.* In-house technical knowledge for creating content could be lacking.
**TIP** Content creation considerations and solutions
In creating content marketers should be aware of the following:
* *Co-creation.* Businesses tend to find greater response rates to content co-creation than purely building content in-house; co-creation can come in multiple forms such as through analysts, case studies with a customer, a collaborative provider, by teaming up with a channel, etc. The content collaborator can substantiate a brand’s credibility.
* *Ageing of content.* Content freshness can be very subjective and the B2B marketers behind the creation and managing of campaigns will typically have a different relationship to the content than its the recipients. Vendors are quick to view content as old after just three months as they are exposed to the content so frequently. B2B marketers should think about how much exposure customers have over a period of time and use that as an indicator, rather than internal fiscal quarters or planning cycles.
* *Curate it.* Where resource is limited or timelines are tight companies can look to partners for licensing content.
## 5. Content activation and distribution
During this phase content is promoted and distributed across different channels according to media plans and editorial calendars. Below are some of the mechanisms for distributing content:
* *Content syndication* through third parties who place company content on their website.
* *Media placement and distribution,* media or advertising slots are booked across online and other sites, directly or via media agencies.
* *Via content licensing,* through content licensing and posting companies who act on behalf of the vendors.
Content can be amplified beyond these distribution mechanisms through self-sharing of the content, eg through social; or by leveraging influencers or encouraging advocates to post and share content relevant to the topic.
**Example** Self-amplification
Business self-amplification has become popular in B2B, partly because marketers’ budgets have become more constrained but also because businesses have been conscious of how powerful people networks have become. An employee advocacy approach has helped companies such as Dell, HP and Cisco in promoting their brand and key messages.
### Content syndication – how to promote and distribute content
Content syndication is the process by which third-party sites cite or use your content. It drives more engagement with an organization’s content, and more traffic to a company’s website; it generally increases exposure of the brand. Content syndication normally forms part of a larger marketing campaign or initiative. It can be used when the content syndicator is regarded as a trusted provider, which can help instil credibility in the content, or when the content syndicator provides customer accessibility benefits, eg a network of different customers. It helps build awareness by offering additional exposure.
### Content curation
Content curation is content provided by other people, eg influencers. It involves organizing content from a selection of sources and using it to support messaging; it’s not about creating ‘new’ content. Content is curated when the content plan has gaps in terms of amount or type of content and when budgets and resources don’t allow for in-house creation; it is also a quick way to secure content.
The process of curation involves four core steps: 1) discovering – sorting the content, 2) compiling the content, 3) placing per themes, and 4) publishing the content. Using curated content means an organization can share other parties’ content, as well as including commentary on the content piece. It is important to credit the author of the content by adding an attribution.
**TIP** Checklist for content curation
1. Careful selection. Topics are carefully selecting according to initial customer research, understanding of needs, pain points, etc.
2. Content check for trustworthiness. Content is reviewed in terms of sources, stories.
3. Content should support marketing and/or brand efforts.
4. Relevancy. Content is checked for relevance to the target group and for the goal of the activity.
### Content amplification
This is about bringing the content to life. It can be thought of as the process of helping content reach wider audiences than through standard media planning and placement. Some ways to amplify the content are through influencer marketing (see Chapter 9) and the following:
* via influencers – by employing influencers to share content;
* encouraging referrals – customers post and share their stories;
* where partners or channel partners share and promote content further;
* native advertising using new technology applications.
## 6. Content measurement
The effectiveness of content marketing is something marketers need to demonstrate if they’re to protect their budgets. Measuring content marketing can be split into four core areas using the TELS framework:
**Traffic metrics**. Metrics relating to number of visits, unique visitors.
**Engagement metrics** include bounce rates, repeat visits and time spent. A bounce rate of 80 per cent means that 80 per cent of customer landing on a page and content leave immediately. A measure of how much time an audience spends on content can say a lot about the content quality. Finally, having a high repeat visit level is a positive sign as it shows customers are willing to come back.
**Lead generation metrics**. These include number of leads, leads converted to sales opportunities and ultimately leads closed as deals.
**Sentiment metrics**. These include metrics such as comments, likes and shares.
**TIP** Website heat maps
There are many great applications out there that illustrate how a vendor’s audience is engaging with a page and its content. Heat maps show where on the website the customer is clicking and how much of the website is used.
### Content marketing examples
Xerox developed the ‘Get optimistic’ campaign in early 2012; it was about positioning itself better through integrating content marketing with a sales outbound programme. Bi-weekly e-mails were sent to prospects that directed them to a personalized website. The campaign helped Xerox add 20,000 new contacts.
Sage wanted to engage with ‘micro-businesses’ (those with fewer than 10 employees) so it decided to focus on searches to meet objectives by targeting 100,000 business searches across seven defined user themes. Unique content was developed according to the themes used in an SEO and PPC (pay per click) campaign. Overall this resulted in 85 per cent of the UK micro-business sector being reached and 130 million impressions.
ACTIVITIES
Identify by campaign which pieces of content could be used for different stages of the buyer journey: beginning, middle and close.
Identify your key target personas of a recent campaign and check whether your content is really tailored to them through formal or ad hoc surveys.
What are your KPIs according to the TELS framework? Where do you focus today? How would you adjust the KPI discussion depending on a marketing or sales audience?
# References
CMI (2016) B2B content marketing: 2016 benchmarks, budgets and trends – North America, Content Marketing Institute, available at: http://contentmarketinginstitute.com/wp-content/uploads/2015/09/2016_B2B_Report_Final.pdf (accessed 10 February 2017)
Schwartz, J and Weaver, A (2014) The marketing strategist: what’s behind the customer mask? ITSMA, available at: http://www.itsma.com/whats-behind-the-customer-mask/ (accessed 10 February 2017)
Sirius Decisions (2013) How to get started on marketing and sales alignment, available at: https://www.salesforce.com/blog/2013/10/align-marketing-sales.html (accessed 10 February 2017)
# 14
# B2B social media marketing
This chapter will give you an understanding of:
* social media business benefits
* social media channels and how to select them
* social and the customer cycle
* how to implement an advocacy programme
* social media marketing strategies
# Social media evolution
Social has evolved in the past decade from a conversational forum to a full end-to-end marketing and business tool. Businesses are now embracing social in different ways and those that evangelize and embrace social are set to reap huge benefits across the business and business functions.
Social media is a collection of online platforms and tools that people use to share content, profiles, opinions, insights, experiences, perspectives and media itself, facilitating conversations and interactions online between groups of people. Social media is an ever-growing and evolving collection of online tools and toys, platforms and applications that enable all of us to interact with and share information.
# Benefits of social media for business
Social has shifted in the way it impacts and benefits business; about 10 years ago social media platforms were forums oriented towards consumers. Today businesses benefit from influencing customers without engaging them, in messaging and marketing at a low cost (or even no cost). Social has become a new element of the marketing mix that helps position and promote the brand as well as generating leads. For small businesses, it’s one of the easiest way to get started: beyond networking it can help them promote their offerings.
Social media can drive traffic to a website; can boost a site’s SEO. Great content on social platforms means search engines are more likely to pick up a business’s posts. Social can help businesses understand their audience; eg Facebook insights can help them learn about customers.
# Social media channels
B2B social media channels have evolved to support business needs and customer needs; they have become more fragmented into the following areas (see the ‘new’ B2B social media navigator in Figure 14.1):
* *Social networks.* Network platforms designed to allow people to connect with each other; the main social network platforms used by businesses in most countries are LinkedIn, Twitter, Google Plus and Facebook.
* *Social video.* Rich and streaming media sites including YouTube and Vimeo help to engage customers early in their buying journey through richer content.
* *Social display.* Banners, typically provided through social media platforms, are a form of paid advertising.
* *Social knowledge.* These are social networks used also for business reference purposes, like Yahoo! Answers, Quora, Ask and Wikipedia. Businesses use them for reference or to get quick answers on questions and topics.
* *Social business forums.* Business communities that may also be part of social networks or separate as paid or unpaid memberships. Such business forums include members from a similar industry, function or to discuss similar challenges. For example, IT decision makers (who are considering any IT purchase) use Spiceworks to engage other IT decision makers.
* *Social e-mail.* Social networks’ own e-mail services, for example LinkedIn ‘InMail’ can be an effective way for targeting customers and potential customers.
* *Social blogs.* Company blogs, industry specialists’ blogs or specialists blogging on platforms or forums.
Figure 14.1 B2B social media navigator
B2B social media navigator
**EXAMPLE** Spiceworks
Over the past 10 years, Spiceworks has been effective in creating a rich IT decision maker community; it is regarded as the world’s largest social business network for IT professionals. In 2011 alone, IT professionals spent almost 3 billion minutes in Spiceworks, making it one of the most active sites in the IT industry.
# Social sales and marketing
Social media is a channel used by business customers to self-educate. Here they can learn about new products and vendors, engage peers and potential vendors to ask questions. Now that customers are increasingly using social, both sales and marketing are using social more to educate customers, to target, identify, acquire, engage and retain customers; they can learn more about customers’ interests, behaviours, challenges and business needs.
So, who is responsible for social media within an organization? Sales need to become more social savvy to use social channels, to educate, engage, directly nurture prospects and customers. Marketers need to use social to learn about customers, to identify and attract them, to nurture them indirectly through their buying process. Clearly social media and the customers’ use of this channel has forced a stronger cross-over between sales and marketing.
# The social media marketing process
The six-step process in developing and rolling-out social media marketing activities, shown in Figure 14.2, is:
**Step 1: Social media objectives and goal setting**
The first step is to align with the business and marketing goals as well as the audience to target.
**Step 2: Social media audit**
The next step is to conduct a social media audit across four core areas: customer and social media channel usage; vendor employee business usage of social platforms; vendor-owned social platforms; and required social media platforms. The audit should start with the target customers’ social media usage: what, how and when. The next level of the audit is to what extent owned platforms are used internally within the company compared to the required extent; eg where social is being used simply to network and add contacts and the business needs to use the platform to target and engage through additional licences.
**Step 3: Social media channel selection**
According to the social media audit and budget, social media channels are selected and prioritized.
**Step 4: Social activation**
Once the channels are selected, content should be prepared and readied as part of the overall media and content marketing plan (see Chapter 13) where content timings, paid, owned and earned plans come together to activate social across different channels.
**Step 5: Social review**
Once implemented, social media activities should be reviewed against pre-agreed KPIs.
**Step 6: Social optimization**
Based on what has been learnt during the review stage, social media activities are optimized further, whether this is content, implementation, type of social advertising or marketing.
Figure 14.2 Social media marketing process
Social media marketing process
# Social media marketing enablers
The success of social media marketing not only hinges on good planning and innovative usage of social media but also on some key foundations or enablers. Success typically relies on sales engagement and support; ie their buy-in. Business buy-in in this context is the respective business stakeholders being fully aware of the social media benefits for the wider business as well as being supportive of social media activities.
Sales adoption, ie sales readily using social media platforms, is key. Some sales specialists may already be bloggers within the business. Where senior business stakeholders have adopted social media, whether for private or business purposes, this can also benefit the business.
Having the business actively using social media is fundamental to any business social media activities. Where a business has little or no social presence, a social media marketing strategy will struggle. Business customers today expect all businesses to have a presence and know how to use business social platforms.
**TIP** How to gain buy-in
So you need to get your business and sales department participating in your social media initiatives but there isn’t the buy-in? Here are some key steps to achieving this across four approaches:
1. *Leverage third-party advocates.* The typical scenario within businesses is that the marketing department takes the lead on social media and in using it for marketing and even broader business purposes. One way to support the marketing voice within a company is introducing outside advocates or experts who can explain the potential of social media for the business.
2. *Competition activities/monitoring.* Sometimes highlighting how competitors are using social and how they benefit from it can be a great way of showing how an organization needs to catch up.
3. *Customer insights and viewpoints.* Leveraging customer views on how they use social, when and to what extent can play to marketers or others trying to get buy-in for social media marketing. One simple way could be to get sales people to ask their key accounts about their use of social media.
4. *Identify internal non-marketing advocates.* Marketing departments are usually expected to be social media advocates so having non-marketers emphasizing and supporting the benefits of social can be a great way to influence other non-marketers.
# Social and the customer cycle
As social media platforms have become more sophisticated, so business can leverage social at each stage in the customer lifecycle, from finding them, to engaging, acquiring and retaining them. Following are examples of business goals aligned to the customer lifecycle and how social could be used for each of the stages; see Figure 14.3.
Figure 14.3 CLC social media
CLC social media
## Finding and targeting customers
Social media can be used in various ways for finding and targeting customers: through targeted display advertising on social media platforms or through monitoring third-party social forums. Vendors should check if the targeting mechanism is a help for the prospect or a disruption for the customer, eg banners that pop up like chat windows when a user is doing something completely unrelated may be off-putting.
## Engaging customers
Customers can be engaged via a social forum, whether the company’s own or a third party. Another possibility is sharing banners and retargeting customers when they click on a banner. Influencers or referrals can be used to engage and offer some credibility and trust to an unknown organization reaching out to potential customers.
**TIP** Nurture the engagement – don’t rush it
During this phase vendors can easily lose potential customers. With social marketing it is so easy to connect, contact and communicate with customers, sales may not be able to resist the temptation to ask for a phone call or to ask questions about potentially selling something. Businesses need to remember the word ‘social’; in B2B this is about building some form of relationship prior to any pitch or sale. Some social sites offer blocking mechanisms to restrict a vendor connecting or engaging unfamiliar contacts, so watch out!
Other possibilities for vendors to engage customers:
* *Invite to a forum based on a relevant topic.* The forum could be a tutorial via SlideShare.
* *Via InMail, personally engage a customer.* Based on defined personas or business types, companies can engage customers through social e-mails or texts.
* *Respond on social questions.* Where customers ask for help with a question or challenge, vendors can demonstrate capability and even thought leadership on a topic.
**EXAMPLE** Dell
Dell set up a social forum ‘idea storm’, which is a social crowdsourcing forum to capture input and ideas related to new products and services. Users post their own ideas and vote and comment on other ideas. One of the key inputs was the need to integrate cameras in notebooks to support online chatting and phone calls, leading to Dell’s notebook range being designed with integrated cameras. This was an effective way to engage on a topic that the customer was interested in as well as the vendor.
## Social for acquisition
Social can be used for acquiring customers by using social e-mails or messaging (eg LinkedIn, InMail) offering the latest solutions to customers’ needs or interest; or downloads from a social website with the opportunity to hear about latest offerings; or gated content in the form of SlideShare, infographics, e-books, or white papers.
## Social for customer retention and loyalty
Post-acquisition, social media is a great tool to increase loyalty and stickiness with existing customers. One of the ways is sales updates. These may include news about technologies, about the business, or latest offerings. Rather than using e-mails all the time, this form of updating can be more powerful and less invasive as information can be shared through links or likes, and it reduces traffic to the customers’ own e-mail box.
**Conversations**
Businesses can listen and respond on their own social site to customer mentions, or questions; this may be about sales or other business stakeholders actively engaging customers on key topics through the company’s social blogs and posts.
**Social mention**
Organizations can support customers through mentioning them in their news updates via social or posting about an achievement the customer has had; this can help the customer’s profile in the industry through ‘free’ marketing.
**Collaborative social marketing**
The vendor and customer participate together on social blogs, where they co-advertise through videos or banners.
**Post-event engagement**
Companies can include a follow-up message to an event through social in the form of additional content or news.
**Customer service**
Dealing with complaints in a timely manner can help put a customer at ease and address a problem. Integrating sales operations can also be beneficial where social is used to provide updates on delivery or other aspects.
# Implementing a B2B social media advocacy programme
More and more organizations are looking to actively encourage internal employees in using social for business purposes, known as employee advocacy programmes. Here are some of the main steps to implementing a targeted social media advocacy programme, known as the TIPSS model:
**Target audience and groupings**
The first step is defining the target customer audience, eg specialists within an industry sector or general customers in all businesses.
**Internal audience grouping**
Based on the target audience, the next stage is to build internal groups, which could be general sales, specialist sales, or different business functions.
**Prepare**
Internal employees are trained to use social, content is prepared, and mechanisms for internally sharing social content are created.
**Share content that is easily shareable**
Accessible content is sent on.
**Structured review against core KPIs**
KPIs could be reach, influence, activity level, etc.
**EXAMPLE** Dell SMAC programme
Dell, like many larger companies in the IT sector, has set up its own social media programme and policy known as SMAC (social media and communities). Dell was one of the first to create a social policy; the background is to ensure all its employees leverage social media channels responsibly and according to guidelines. This helps employees understand how to use social effectively as well as avoid misuse.
# Social listening and monitoring
Social listening and monitoring is another great way to understand social usage by different segments. Social listening or monitoring is about searching the web to see what’s being said about your company, your competitors and other topics of interest.
Social listening can be used in different ways to help businesses, for example to find out platforms and websites customers use online, to listen and respond and eventually generate leads, to identify influencers or to improve marketing effectiveness. For example, a business can monitor key words related to business’s pain points and by responding on those can generate new leads. A survey by Oracle (2013) found 43 per cent of users interact with brands on social media for a direct response to a problem or question; additionally, 31 per cent interact with brands to gain direct access to customer service representatives or product experts.
# Social media channel selection
Selecting the right social media channel depends on the target customer audience. For example, a healthcare professional may have a profile on LinkedIn, be present on Facebook and be in an online community with a separate platform but may not use Twitter. Some of the main criteria in social media channel selection are:
* *Audience usage and propensity to use.* What do they currently use? This information can be found in the persona profile (see Chapter 13).
* *Where is your competition on social?* Through understanding competitors’ social activity and channel use, one can avoid falling behind.
* *Buying stage focus.* What stage of the buying journey are you targeting? The answer may determine the type of social media channel.
* *Budget.* What budget do you have for social, or is the idea only to use owned and earned material?
* *Accessibility/reach.* Which social platform allows you to reach and/or engage your target customer?
**CASE STUDY** Schneider Electric/Invensys Better Together
In September 2014 Schneider Electric launched a communication campaign using social media channels to promote the merger/acquisition of Invensys with the aim of reaching global customers. The objective was to address two major concerns: 1) existing customers were concerned about the company’s future investment in products they used and thus were holding back on new projects; 2) new customers did not yet recognize the value of the combined Schneider Electric and Invensys portfolio and demonstrated higher resistance to moving to Schneider Electric/Invensys solutions.
The focus of the campaign was to demonstrate the combined value and benefits and to reassure customers in their business with either or both companies; as part of the campaign an internal competition was launched to encourage employees to use their personal accounts to tweet about the power of being ‘Better Together’. For every tweet posted using #BetterTogether the firm committed to donate a corresponding amount to ‘Habitat for Humanity’. After five months about €20,000 was raised and donated. Confidence in existing customers was re-established. Aside from the business numbers, internal productivity improved as both companies were working better together and both could see the benefits of a combined business.
ACTIVITIES
Identify which social media channels your current customers are using. Does this match the social media channels included in your marketing activities or campaigns?
How often are your customers using social and for what purpose?
Identify social forums, groups related to your function, industry or business challenge and join one of them.
# Reference and further reading
Khan, U (2016) ‘Social listening’ can help you play a marketing detective and unlocking customer success, available at: https://medium.com/kodenext/social-listening-can-help-you-play-a-marketing-detective-unlocking-customer-success-a2c3f28bd217#.no9e4zfhu (accessed 10 February 2017)
Oracle (2013) eMarketer mobile commerce roundup, available at: http://www.oracle.com/us/products/applications/emarketer-mobile-commerce-roundup-2188367.pdf (accessed 10 February 2017)
# 15
# B2B brand building
This chapter will give you an understanding of:
* brand building and impact on business growth
* key brand elements
* what prevents a business from building a brand
* the brand building process
* how to engage employees to communicate the brand
* how to build a B2B brand tracker
* B2B brand ROI and measurement
* how to ensure brand consistency
# Introduction
There are several differences in building a B2B brand compared to a consumer brand. Business brands don’t have the same day-to-day exposure as consumer brands; they also tend to use different communication channels. Business brands need to appeal to different needs and requirements and the messages they use aren’t as sexy. So how do B2B brands establish presence and how do brands that have B2B and B2C manage the different messages?
## Brand and brand building defined
A brand starts with the identity of a company through the way it presents itself: name, description and logo. For a company the consistency of these different elements forms a brand. A brand is much more than just a logo: it is the consistent value systems a company communicates to the outside world. Such brand consistency needs to cut across all its collateral and communication elements – this is the corporate identity. If there is inconsistency in look and feel, this can undo all the efforts of a company to position itself.
## Why invest in brand building?
Several indices point to links between business performance/profitability and brand strength. For example, Brandz creates a report tracking the top 500 US stock exchange companies, and according to its recent report (Brandz, 2015) the top 100 brands grew by 81 per cent in market value while the top 500 grew by 22 per cent. Additionally, research has shown that strong brands secure loyalty, drive choice and command a premium.
## Brand building in B2B is less emotive – wrong!
B2C customers may purchase the wrong brand of marmalade but that is easily rectified. What if the IT storage infrastructure for a business doesn’t function or worse, loses data; such things if known to a business’s customer can hurt a company’s performance, earnings or its credibility. This higher level of emotionality in B2B becomes more obvious on closer inspection: B2B purchases can entail personal risks, more so than for B2C. For example, the person responsible for having purchased the datacentre or choosing the supplier will have invested emotions as their credibility and time are impacted.
# B2B brand touch points
The B2B brand touch points are the points at which customers could engage with a brand directly or indirectly; the sum of the touch points usually helps the customer form a view of the brand. Figure 15.1 shows the main brand touch points across the customer journey: before, during and after purchase. Some of these elements are typically more specific to B2B, such as product training, channel partner reseller or distributors and the touch points where they represent the brand: unveiling events, user conferences, business cards and collateral that businesses receive.
Figure 15.1 B2B brand touch points
B2B brand touch points
Within B2B relationship marketing, customers are a lot more exposed to sales or physical opportunities to engage, such as webinars and events, so managing messages and associated branding elements is important if an organization intends to build its brand effectively. Even for small businesses, points shown in Figure 15.1. Early in their business life, getting these right and managing this mix could impact the future success and growth of the business.
# Brand building obstacles
So, if brand building is proven to be a success factor for a business and its profitability, why don’t more businesses invest in it? The reality is that businesses encounter some of these challenges:
**Pressure to invest elsewhere**
Pressures can come from different areas that influence other investment areas for marketing or even the business; pressures can come from shareholders, from internal stakeholders or even external stakeholders (non-shareholders).
**Short-term view of marketing investment**
Building a brand is typically a long-term investment and needs to be carried out consistently over time if it is to be impactful. Some companies that measure themselves every six or three months look for quicker results, which is not consistent with brand building or repositioning.
**Failure to articulate benefits**
If marketing fails to articulate the benefits of brand building or where sales fail to understand them, this can lead to brand building initiatives being negatively impacted.
**Competitive pressures**
When dealing with competitors that employ aggressive sales tactics such as dramatic price reductions, a vendors’ brand activities can be impacted as longer-term investments aren’t seen as a priority.
**Complexity in sustaining, and rolling out**
Building brands and maintaining brand building activities over time usually involve different skillsets, investment in resources and budget. Brand investment needs to be carefully managed to ensure the correct media mix and media placement. There is also a need to review return on advertising spend (ROAS).
# B2B brand building goals and programmes
The main goals for investing in brand building initiatives are repositioning of the brand, increasing awareness for the brand, acquiring new business, building loyalty or demonstrating value-add. The main types of B2B brand building initiatives linked to these goals are:
* *Brand positioning.* This relates to the positioning of the company. For example, in the last 20 years a lot of companies in the IT sector have repositioned their brands either into more premium price brands or as solution providers.
* *Sponsorship.* B2B sponsorships usually help brands in reaching different or extended audiences. Examples are sponsoring a piece of research intended for a specific customer, sponsoring an event or a campaign for an industrial association, or larger-scale sponsorships such as major sporting events. Ideally the sponsorship speaks to the audience and highlights to the intended customer the background of the B2B brand sponsoring the activity. Sponsorships can also help brands acquire new business and reinforce loyalty in existing customers.
* *Employee advocacy programmes* for brand building can be powerful in B2B; they can potentially serve all the above goals.
* *Social amplification programmes* are used more and more by B2B brands; they can serve to deliver on all goals. Social amplification through sales as well as partners can help disseminate messages and content much faster.
* *Partnerships.* Partnerships can include member partners that cover a target segment or sub-segment, alliance partners that offer complementary products or services to the B2B brand portfolio, or channel partners who work closely with the B2B brand to develop a market.
**Example** B2B sponsorship – Deloitte and the London 2012 Olympics
Deloitte provides an example of how a professional services organization can leverage a sports sponsorship to highlight its business expertise. Deloitte sponsored the London 2012 Olympics and was the official provider of professional services for the London Organizing Committee of the Olympic Games and Paralympic Games (LOCOG). This role involved responsibility for services to support delivery of the games through more than 750,000 hours of support. Deloitte helped the organizing committee through various complex challenges such as sourcing 1.8 million items of sporting equipment in six weeks. Deloitte’s professional services spanned a number of operational areas for LOCOG including catering, procurement, tax and technology; through this wide array of services Deloitte could showcase the breadth of its services and depth of expertise, while demonstrating its contribution to society.
# Brand loyalty
Brand loyalty is the level to which a customer stays with a brand; it can be viewed in terms of how tied a customer is to the brand, despite temporary issues or challenges it may have with the brand vendor.
Brand loyalty can be viewed as a two-way relationship: organizations that engage with their customers or potential customers; and customers that demonstrate how they engage with the vendor through purchases, positive feedback or even how they defend a brand where it is receiving unfavourable criticism.
Creating brand loyalty can be based on different brand elements or touch points. Loyalty benefits cover several areas: with increased loyalty, organizations don’t need to devote as much marketing to persuading customers to purchase or consider purchases: with greater brand loyalty the marketing costs reduce. Through brand loyalty more customers talk about the brand or advocate it. This can be thought of as ‘earned’ media. Earned media (or free media) is publicity gained through marketing efforts other than paid media marketing. Through loyal customers, other potential customers are persuaded to try the brand. With greater brand loyalty an organization is more likely to have stable repeat business.
# Importance of brand consistency
B2B brand consistency is very important: it’s about being consistent across customer touch points, ie where customers engage directly or indirectly with the brand. Ensuring consistency is about having a touch-point management programme spanning customers by buying stage. In establishing such a programme there are a number of characteristics which could be managed to support brand consistency. Creative, look and feel should be consistent. Messages across a campaign should also be consistent, as should overarching corporate messages. Where business unit silos exist this can lead to messaging silos where eventually the company appears fragmented and confusing for the customer.
Websites and micro-sites should be consistent in look and feel, and means of navigation. Having a policy governing how to use social media, how to post, how to engage customers and what to message, send and share should form part of the B2B brand social media policy (see Chapter 14). Employee communication includes how employees apply e-mail signatures, type of font used, usage and placement of any logos as well as how employees conduct themselves in terms of making themselves available and accessible.
# Brand building stages
In building a brand B2B marketers should use the five-point framework shown in Figure 15.2:
1. *Brand gap audit.* This stage is about understanding the current view of the brand, the perception of the company held by customers and business partners. Sometimes this outside view may not always align to the reality of what the business does and its goals.
2. *Positioning.* This includes a business’s messaging and overall brand elements; potentially the root cause of gaps found in the brand audit may be one or more brand elements. During this phase the positioning is clearly defined.
3. *Plan for building and activation.* The plan for brand building and activation is defined; marketing vehicles and channels are selected, brand building approaches are agreed to by core stakeholders (typically the CEO or business head as well as the marketing head).
4. *Activation. Internal*: ensure all internal employees are clear about brand values and tenets, business stakeholders are re-educated with messages, the brand campaign and how branding activities support the business. *External*: expose the brand. The brand initiatives, messages and campaigns are then rolled out externally; this can be direct marketing, through third parties, agency media placement as well as through employees who articulate the new messaging and share brand campaign assets.
5. *Review and assess the brand.* During this phase various aspects are measured such as brand equity, brand loyalty and brand awareness.
Figure 15.2 B2B brand building process
B2B brand building process
## 1. Brand gap audit
As mentioned above the purpose of the brand audit is to understand internal and external views of the brand compared to business goals. Audits should result in understanding the performance of the brand, areas or gaps to performance, as well as the competitive brand position. Failure to track brand efforts is one of the reasons why there could be lack of buy-in: there is either a belief that it is not necessary or, if it is, there is no evidence to show where to spend the brand investment.
**TIP** Building a B2B brand tracker
1. *Brief creation.* Create the brief, which can be a simple document outlining the intended audience, the objective of the brand survey, timings, and expected respondent numbers.
2. *Survey creation.* The survey questions are created based on the brief.
3. *Implementation.* The survey is carried out, directly or externally by an agency or marketing services company.
4. *Collect answers.* Responses to survey are collected.
5. *Report creation.* Report against key brand metrics, showing the response rates.
In implementing B2B trackers marketers should consider several aspects. For example, with more questions the cost of the tracker will increase as it requires more time to conduct the survey; it is therefore important to carefully select questions and their number. Larger numbers of respondents will also increase costs: companies should consider what is needed or representative in terms of an audience sample. Can the questions stand the test of time and run for multiple quarters or even years? If so, this will allow companies to compare the brand tracker movements over time.
## 2. Positioning
This is an important step for a vendor: it is defined as creating a brand offer or presence and supports a brand in positioning value in the mind of the target customer. Brand position goals should encompass the following core elements:
* *Differentiation.* Ensure uniqueness in brand compared to alternative vendors; this can form part of the value proposition.
* *Customer focused.* Includes what customers the brand is targeting.
* *Sustainable.* Positioning statements need to be sustainable over a long period. This allows customers to relate to the brand and understand it over time.
* *Geo-agnostic.* If a brand operates across different territories, positioning activities should not be focused on one area.
* *Support organizations’ business goals.* Brand positioning should serve to achieve business goals.
Taking the above into consideration, companies can use the brand positioning map shown in Figure 15.3 to understand current brand positioning against market criteria: competition and customers, customer needs, uniqueness, customer benefit and reason to buy.
Figure 15.3 B2B brand positioning map
B2B brand positioning map
## 3. Plan for building and activation
The brand plan typically includes key goals and focus uncovered in step 1, the brand audit, positioning statements and values from step 2, and focused target audiences that may not necessarily be the current target market of the business. Also included are key messages the business wishes to deliver, the media and communication outline defining which media will be used, and across which communication vehicles.
The brand plan should include key partners that will deliver it. In B2B it’s important to consider which partners will deliver the message and which marketing or business partners will be involved; partners in this sense include alliance partners, channel partners, influencer partners, associations or other organizations that the business is connected to. The partner selection and plan may indicate some gaps.
## 4. Activation
This stage should serve to prepare for external brand activation as well as accompany it. Similar to employee advocacy programmes for social marketing, for brand building organizations should follow the framework:
**Target audience and groupings**
The first step in creating a brand employee advocacy programme is defining the target audience: maybe specialists within an industry sector or a general customer-reach programme for all businesses.
**Internal audience grouping**
Based on the target audience, the next stage is to build internal groups; they could be general sales, specialist sales, or different business functions.
**Prepare**
Internal employees are trained in using social. Social-ready content is prepared, and mechanisms for internally sharing it are created. Content should be ready to share and send on.
**Structured review against core KPIs**
KPIs could be reach, influence, activity level or other measures.
The difference to the social employee advocacy programme is that social can be used to promote brand building, whereas the inverse is not true.
**TIP** Key considerations in internally activating B2B brand building
* *Ease of understanding:* sometimes the brand build initiative is not understood, so making the messaging clear is key to the success of brand advocacy initiatives.
* *Ease of access, easy to share*: organizations should make brand assets and content easy to access as well as easy to share.
* *Who messages and articulates*: internal activation success sometimes rests on internal leadership support and advocacy. Rather than leaving this to marketing, involve the CEO with public relations articles if possible; this way brand initiatives are seen as ‘business’ initiatives that all in the business own.
* *Who owns this*: there is a common misunderstanding that the brand is owned by marketing; in fact all employees and departments play a part in brand building. Marketing tends to own external activation and PR, and supports internal departments by providing brand messages.
* *Not only digital*: brand building, particularly in B2B, is about supporting face-to-face and non-digital engagement at events, in phone calls and meetings, and in printed material left for channel partners.
* *Communicate value of brand*: in getting campaign buy-in it’s important that organizations highlight the benefit and value of the brand for their customers.
Implementing brand activities or campaigns can occur in different ways. They can be a pure media and advertising exercise or can be an all-encompassing approach including media, messaging, influencers, internal champions, digital and offline media.
Content creation and activation form part of the brand plans as well as social media activities that help amplify the brand assets and messages (see Chapters 13 and 14). In activating the brand externally B2B marketers should consider the following:
* During this phase consistency in message is key, to give strength and impact to the brand messages.
* Social in B2B has evolved rapidly in the past 10 years both in capability and usage in business. Organizations are increasingly using social platforms to rapidly reach audiences and extend the reach of brand assets.
* In B2B, business partners are important in delivering brand messages; they allow them to reach the right audience and lend messages credibility.
* Depending on history or the extent of brand activations organizations should allow time for brand activities to take effect.
* Touch rate is the rate at which marketing content (or in this case branded content) is touched and seen by customers in whatever form. The rate differs between customer type, industry sector and solution type. With digital applications as well as digital marketing, touch rate can be tracked both on- and offline.
## 5. Review and assess the brand
How do you know how customers see your brand? Is it positioned the way you want it to be? One part of the brand process is to review and assess the brand. Organizations can measure brand loyalty, awareness, health and/or perception, where brand loyalty refers to likelihood to repurchase, recommend or refer; brand awareness is awareness of the brand overall (do potential customers know about your company and what it offers); brand health highlights how values, mission, culture are communicated; and brand perception relates to how the brand is perceived against value criteria or positioning (do customers see the company as a solutions provider rather than a seller of products).
It is important to look at different types of metrics to get a full picture of the brand. Some of the key metrics are behaviour metrics, which provide insight into brand performance relating to customer expectations, competitor actions; traffic metrics which include search engine rankings, website traffic; and customer engagement metrics which include time spent viewing content, time on a website. Qualitative assessment may be of a specific area such as brand campaign or initiative reviews; see also the above section on brand tracker creation.
ACTIVITIES
Identify potential areas where customers interact with your brand directly or indirectly. How many of them can be controlled or influenced by marketing; how many of them fall outside of the marketing scope?
What metrics do you use today to measure your brand activity and which part of the brand do they measure? Brand awareness, brand health, brand consistency or brand perception?
Review current marketing materials for a given period. Are the look and feel, message and creative all consistent with one another?
# Reference
Brandz (2015) Brandz top 100 most valuable global brands 2015, available at: https://www.millwardbrown.com/BrandZ/2015/Global/2015_BrandZ_Top100_Report.pdf (accessed 10 February 2017)
# PART FOUR
# Collaborating with channel partners
# 16
# The modern B2B channel landscape
This chapter will give you an understanding of:
* the main B2B channel partner types
* channel partners and alignment with B2B segments
* B2B channel partner marketing
* channel marketing trends
* how to select channel partners
* differences between distribution and channel reseller marketing
# B2B channel partner marketing
As most medium and large enterprises sell indirectly via channel partners and a large portion of small businesses need to engage a reseller or distributor, B2B marketers need to be equipped with skills and knowledge as to how to market indirectly. This chapter looks at B2B channel marketing and the main dynamics and trends within B2B channels. For clarity, channel partner marketing, sometimes known as ‘channel marketing’, is the practice of marketing indirectly via any channel partner or channel intermediary.
## What is a channel partner?
Channel partners are a person or organization providing services or selling products on behalf of a vendor. They come in different forms; eg value-added resellers (VARs), distributors, online resellers and, in technology sectors, there are also managed service providers (MSPs), consultants and systems integrators (SIs).
## What is B2B channel partner marketing?
B2B channel partner marketing involves channel resellers or channel partner systems oriented towards businesses; one could think of the approach as B2B2B where the channel systems serve business and where the channel partners are focused on selling to businesses. B2C channel marketing is the marketing via channels to consumers, where the channel can be retailers or online resellers.
# Benefits of channel partners and channel partner marketing
B2B organizations engage channel partners to achieve different goals; some of the benefits organizations derive from using channel partners are:
* *Gaining access to a customer segment.* Where channel partners already have a legacy in doing business in a market and have built credibility with a customer segment, a vendor engaging them can immediately avail itself of this established relationship.
* *Geographical reach.* Channel partners can help businesses by providing geographical reach; this can be by working with a channel partner in the region or one that has multiple sites across different areas.
* *Entering new market sectors.* Channel partners that have established credibility within an industry sector can provide value to vendors by leveraging their contacts and relationships.
* *Solution selling and marketing.* The channel partner acts as a value-added reseller and resells combined solutions for segments, or can help assemble one vendor’s products with another to create solutions.
For B2B marketers, channel partner marketing brings various benefits. Partners can raise the awareness of a product or product offering to a broader customer base or to a niche market. They can help to position a vendor’s products through channel partners or influence the perception of a product where channel partners are seen to support the product and service; the vendor’s products become more credible in the eyes of the customer.
# Channel partner types
‘Channel resellers’ is the umbrella term for those channel partners reselling a vendor’s products. The main channel partner types include VARs, large corporate resellers, micro-channel resellers, distributors, and online resellers:
* *Micro-resellers* are typically smaller in size; their range of products may be narrower. Often they support a set of customers based on geographic location.
* *Value-added resellers (VARs)* offer additional services to help businesses such as logistics, warehousing, stocking, credit checks, integration, advisory services and extended offerings. With changing dynamics and more services offered via the cloud, VARs need to justify their value-add even more. Some business customers still prefer to maintain their relationship with the VAR due to legacy relationships and trust in the capability of the VAR. Often a reseller will not need to be involved beyond the point of sale, enabling them to focus on generating sales and maximizing profits; the VAR takes over these efforts. Typically, a VAR helps provide exposure to new customers and the customer acquisition benefits tend to be mutual or on the side of the vendor.
* *Online and mail order resellers* are like retail stores, except they usually do not have traditional brick-and-mortar locations and so no additional expenditure on maintaining a physical location. With lower operating costs they can offer lower prices. Some resellers focus primarily on conducting business via mail order or online. Examples of online resellers selling to businesses in the UK are DABs, Insight and Ebuyer.
* *Distributors* primarily manage relationships with resellers; by managing multiple relationships with resellers they reduce time and manpower expenditure by the vendor. They typically manage relationships with smaller resellers. Where a good relationship exists, distributors may allow vendors access to their partner database for the purpose of acquiring new channel partners.
## IT channel partners
More typical to the IT sector are the following channel partner types, shown in Figure 16.1:
* *Independent software vendors (ISVs)* provide software-added services and make and sell software products that run on one or more computer hardware or operating platforms; they often focus on specific vertical or niche markets.
* *System integrators* are resellers that integrate technologies into systems; they tend to offer large-scale technology solutions including consulting. These types of resellers are fewer in number, larger in size, can be very demanding, and so tend to determine elements of the relationship dynamics between the vendor and the channel partner. Within the relationship, the vendor is the party bringing customers to the table. These channels tend not to be suitable for new businesses, start-ups or companies early in their stages of growth due to the resource-intensive nature of these companies.
* *Managed service providers (MSPs)* are service providers for business and IT departments offering several offsite or cloud-based services.
Figure 16.1 Main channel routes to market: IT
Main channel routes to market: IT
# The changing channel landscape
There are a number of trends directly or indirectly impacting channel partners. Increasingly, channel resellers are building their own software and service revenue streams; vendors need to be aware of this when discussing portfolios as part of the channel partnership. For some channel partners there is an increased focus on vertical integration of their offerings, allowing them to target potential new clients and increase account penetration with a richer set of products and services, leading to increased revenue and a more successful business. In the past decade there has been a move for channel resellers to increase their presence internationally; some have achieved this through acquiring channel resellers in a different region. Messaging for channel partners will become increasingly challenging, with technologies in some areas eroding the competitive advantage of product USPs. Channel partners will come under pressure to change their messaging to businesses to meet these challenges and needs so as to sustain or gain business.
## Key trends in channel partner marketing
Channel partner marketers need to consider some of the recent trends impacting marketing to and with channel partners. The shift to ‘solutions marketing’ thinking, as mentioned in Chapter 8, is becoming more marked for B2B marketers. One of the main areas to support this is working with channel partners to develop and market combined solutions. Combining solutions via a channel partner may be a result of some vendors being unable to offer the full suite of products or services that make up the solution, or because the channel reseller offers elements of the solution itself. This means B2B channel partner marketers need to create marketing content that can be assembled in a modular fashion so that channel resellers can incorporate assets, text, or visuals into their own creative.
B2B marketers need to think of different ways to collaborate with channel partners on marketing activities. Collaborative marketing can mean including channel partners in marketing activities and messaging such as events, webinars, online, website mentions and so on. An effective way to market is to sub-segment and create marketing that speaks to a niche or vertical market. This requires marketers to provide content tailored specifically to niche markets in collaboration with the relevant channel partners.
# Channel selection
The main B2B routes to market can be subdivided into three main types: directly to business customers, indirectly via channel resellers or VARs, and indirectly via distribution and channel resellers to business customers. Below are some thoughts on when to choose each route.
## When to engage channel resellers directly
* The vendor has the resources (sales, marketing, operations, etc) to engage the reseller or resellers directly.
* Channel reseller numbers are small enough to be managed by in-house resources such as sales and marketing.
* A direct engagement with the resellers is regarded as more beneficial, for example to build collaborative business, marketing or sales activities.
## When to engage a distributor directly
This engagement may be in addition to or instead of engaging channel resellers directly:
* The distributor network is strong and offers the ability to access new markets.
* The distributor provides reach to smaller businesses, and the number of companies to engage or reach is so large the vendor resources are unable to manage the business directly.
* The distributor warehouse is leveraged to house, store and deliver goods; this can be the case where a business has products in another location and needs to have more timely delivering of goods to supply immediate demand.
* The distributor provides manpower and/or competence to manage reseller relationships, for example for smaller or relatively new businesses unable to commit.
## When to engage a VAR directly
* The VAR offers specialized services to support a vendor.
* It offers specialized knowledge that the vendor doesn’t have in-house.
* It can provide access to specialized customer target sets.
* It has its own customer relationships within a specific business segment or sector that some vendors would like to benefit from.
Marketing to or with distributors can differ greatly from marketing via channel resellers; the requirements for marketing differ depending on the channel type. Some of the main differences are shown in Table 16.1. Although distributors can offer services like marketing, branding, etc, they are normally focused on buying and selling on ‘products’ rather than complete offerings or solutions. Resellers tend to avoid committing to sales or revenue targets and therefore aged inventory is a risk.
Table 16.1 Distributors vs resellers
| | **Distributor** | **Reseller** |
| **Buying and selling** | Buys products and sells to resellers | Buys products and sells to end customers |
| **Managing sales to channel** | Can manage sales to various resellers | NA |
| **Location** | Sells to physical resellers in their territory
Can have a wider area coverage than resellers | Can cover a wide area, depending on organizational size |
| **Inventory** | Large warehousing facilities to support big logistical operations
Normally buys inventory | Never takes inventory of products |
| **Margin and pricing** | Often requires higher margins than a reseller | Can work with lower margins than a distributor |
| **Merchandising promotions** | Promotions widely used | Manages promotions and all other forms of marketing |
| **Pipeline-based lead generation** | Rarely follows up leads | Often will take leads, manage leads and follow up |
# Channel partner and marketing alignment
The marketing methods of resellers and distributors can greatly vary depending on the size or type of channel partner. Some smaller channel partners that don’t have strong brand recognition may tend towards search or SEO activities while larger corporate resellers can leverage brand strength and brand budgets to generate or sustain awareness of their value proposition.
Table 16.2 shows major partners aligned to business and B2B marketing type. Before engaging channel partners, marketing departments need to consider marketing capability, competence and structure. The types of marketing are:
* *B2B transactional marketing* includes merchandising, promotional marketing, e-mail marketing, pricing and product management; typically there are strong ties between marketing and operations so as to manage product inventory.
* *B2B relationship marketing* typically includes provision of content and assets, lead generation, event marketing, webinar support, other inbound lead generation activities and account-based marketing.
* *B2B marketing enablement* is required by all channel vendors, though enablement materials will be different; see the next chapter for more information on this. Channel partner enablement is enabling channel partners to carry out any marketing on behalf of the vendor.
* *Vertical marketing* is very much like relationship marketing but oriented towards vertical market customers.
Table 16.2 Marketing and channel partners
| **Channel partner** | **Typical business vendor** | **B2B marketing type** | **Example** |
| Distributor | All sizes depending on goals | Transactional marketing | Search, SEO, website demand generation, advertising, promotional e-mails |
| Large corporate reseller | Large corporates | Relationship marketing | Inbound and outbound lead generation including webinars, events, content syndication, account-based marketing |
| Micro reseller | Small and medium business | Transactional marketing | Search, display, SEO, e-mail |
| VAR | All sizes | Vertical and/or relationship marketing | Inbound and outbound lead generation including webinars, events, content syndication, account-based marketing specific to vertical customers |
| SI | Medium and large business | Relationship marketing | Inbound and outbound lead generation including webinars, events, content syndication, account-based marketing |
| Online resellers | All | Transactional marketing | Search, SEO, website demand generation, advertising, promotional e-mails |
ACTIVITIES
Identify the different channel routes to customers for your industry.
Are you marketing to different channel partners today and is your marketing differentiated enough? Review marketing needs of the different channel partners by conducting brief surveys with key stakeholders within the different channel partners.
# 17
# Marketing to channel partners and enabling them
This chapter will give you an understanding of:
* what channel partner marketing is
* channel partner marketing tactics and vehicles
* key challenges and solutions
* channel partner strategies
* digital applications
# Marketing to channel partners
Vendors and suppliers can use a variety of approaches to engage and market to channel partners. The marketing mix and adoption of marketing vehicles will depend on channel partners and their ability to receive and use such marketing. The type of channel partner stakeholders can vary greatly, which means vendor marketing needs to be able to adapt accordingly.
Channel partner marketing is defined as anything intended for the channel partner and/or those working within it. Marketing to channel partners has similarities as well as differences to standard B2B marketing directly to customers. It is similar in that the channel partner in one respect should be considered a customer and recipient of marketing activities and messages. The differences lie in the scope or context of the working relationship where the channel partner plays the role of partner as well as customer. Marketing needs to take these two approaches into account when working with channel partners.
Marketing to channel partners covers five main elements (see Figure 17.1):
1. Clear and effective communication.
2. Provision of content and information in the right format across the communication channels.
3. Tools to use the information and content, which could be marketing campaign creation tools or content creation tools such as templates.
4. Training to support channel partners in effectively using the tools.
5. Marketing communities: some vendors have built forums or communities for marketers.
Figure 17.1 Marketing to channel partners (key areas)
Marketing to channel partners (key areas)
Channel partners often have a selection of providers and suppliers to work with. Almost like marketing to customers, vendors need to capture channel partners’ attention and win them over through channel marketing. Channel marketing helps to arm the channel partner with all the right materials and information so that they can in turn market a vendor’s products and services.
# Digital communication
The main marketing vehicles used to communicate to the channel partners are increasingly digital and online. Table 17.1 is a summary of the main communication vehicles split into digital and offline.
Table 17.1 Digital and offline marketing vehicles
| **Digital** | **Offline** |
| E-mail | F2F |
| E-shots | Events |
| E-newsletters | Magazines, brochures |
| Own website | Onsite demonstrations |
| Webinars | Onsite trainings |
| Webcasts | Direct mail |
| Portals | |
| Social | |
## E-mail marketing
In terms of e-mail marketing to channel partners, the principles are like e-mailing directly to customers. The main difference is that the receivers of e-mails at channel partners tend to be sales rather than purchasing departments.
E-mail and information management needs to be structured in terms of contact people, number of e-mails in a given period and overall number of e-mails from the brand if it has different parts of the organization communicating with the same channel partner. For example, as some channel resellers will have multiple stakeholders or sales people to engage, getting the names of the right contact people may be necessary to ensure messages reach their intended audiences or to drive accountability for forwarding and re-distributing e-mails within a channel reseller.
As with e-mailing customers it’s important to make e-mails easy to read, short, and with a call-to-action. With a channel partner the call-to-action may include follow-up accountabilities.
E-shots can be a form of e-mail and can be used to announce new products and services or promote offers or events. E-shots are designed to deliver clear and concise messages and are a good way of supporting other marketing activities or reaching a broad audience at low cost.
E-newsletters are used to include a selection of pieces of news and serve to provide the intended stakeholders with several pieces of information. It’s important to ensure the e-newsletter is easy to read rather than coming across like a lengthy e-mail. Images are important to engage the audience.
The choice between e-shots, e-mails and e-newsletters depends on different audiences, bandwidth and resource of the vendors.
## Website-vendor website
The website is important for housing information as well as providing access for channel partners to separate portals. The main website is also there for the public and business to view and serves as a great opportunity for vendors to redirect customers to channel resellers’ own websites or other useful points of information supporting the business through channel partners.
## Vendor portals for channel partners
Portals in this case are specific for the channel partner rather than a general website intended for anyone. Portals should be there as the hub for channel communication and all communication should drive traffic to them. They need to be easy to access and contain information in a structured manner.
Portals provide information only for the eyes of channel partners; some vendors segment content according to channel partner type. Types of information to provide are product information sheets, marketing campaign information, assets to support channel partners’ marketing in creating their own campaigns, training, information on rebates and bids budget, the ability to purchase directly for more sophisticated websites, and customer opportunity assessment tools.
## Webinars
Webinars can be a great way to update more complex or important information for channel partners. Some channel resellers or partners may have their operations some distance from the vendor, making physical contact impractical; webinars allow vendors to educate, update or even provide virtual events with interactive possibilities.
# Offline communication
Offline ways to market or communicate to channel partners include events, face-to-face communication, phone calls and printed brochures or leaflets.
## Events
Events can be used for updates on the latest product offerings, and allow a vendor to promote itself. Depending on the size of channel partner and relationships, events can be exclusive to one channel partner or to multiple channel partners. The benefits of events marketing include having a forum where it is possible to engage and answer questions and be more interactive. Events are also a great way to build better relationships between vendors and channel partners. For more complex solutions or technical products events are an ideal way to get across their value and benefits to a lot of people.
## Printed brochures and catalogues
Brands can use educational or promotional brochures to communicate product offerings to channel partners. Printed materials are also a great way for vendors to be top of mind if they’re placed on site at channel partner locations and can also be used by sales people as reference material.
# Selecting channel marketing communications
The purpose behind marketing to channel partners will determine the communication vehicle used; see Table 17.2. Is it to educate, to inform, to provide any necessary tools, or to help sell?
Table 17.2 Audience and communication/content type
| **Channel partner audience** | **Communication channel** | **Content type** |
| All | E-mail, E-newsletter | E-newsletter |
| Sales | Webinar, social media | SlideShare, digital brochure, digital product, demonstration |
| Sales specialist | Webinar, online portals, website | Technical, solutions-oriented, SlideShare, digital brochure, digital product demonstration |
| Marketing | E-mail, online portal access | All types |
| Senior management | E-newsletter, event, meeting | Informative meeting or event |
Having established the purpose, the next main criterion is the channel partner audience. For example, if the audience is sales specialists it is likely that more specialized information is required with more time to explain complex products. E-mails may not be the best means to communicate in this case; webinars, printed brochures and follow-up e-newsletters may be more appropriate.
Beyond the purpose and audience, the budget available would determine the communication possibilities. Where the vendor has a limited budget or resource, ‘pure’ digital approaches are probably more appropriate: e-mails, e-shots, website promotions and possibly webinars. For dispersed offices, digital marketing using webinars or webcasts is best suited for educating and engaging channel marketing audiences.
# Channel partner enablement
Channel partner enablement has become an increasing trend in recent years due to the changing role of channel partners and greater demands on channel partners to sell solutions and value rather than simply products. This shift requires help from vendors in providing different sets of content, tools and training. Channel partner enablement is essentially enabling the channel partner to perform effectively on behalf of the vendor; it is normally mentioned in conjunction with marketing where the channel partner is equipped by the vendor to carry out marketing activities.
## Scope
The vendors’ role can be challenging as multiple stakeholders need to be enabled to perform marketing. Examples of stakeholders may include marketers, sales and some senior executives. The four main areas of channel partner marketing enablement are:
1. *Tools*: access to tools and instructions on how to use them, eg tools to help identify and size customer opportunities.
2. *Content*: access to content, information, resendable or forwarding-friendly communication.
3. *Training*: product/portfolio benefits; information to help argue and position against the competition.
4. *Insights*: tips on customers and behavioural information that supports the sales process.
**TIP** How to enable channel partners
**Content**
Content for channel partner enablement is usually not intended for end customers. Types of content can be comparison sheets, product positioning data, demonstration software, and printed brochures highlighting the portfolio. Other assets can be product details or case material for channel partners to assemble into a campaign themselves.
**Tools**
These can include product positioning tools, marketing campaign creation tools, customer assessment tools that marketing roll out, planning tools, and a ess to online applications that allow both channel partner and vendor marketing to track marketing performance. Tools can also extend to budget, market development fund (MDF) and planning.
**Training**
Training can be very structured leading to certification. It can be carried out on- or offline, though with more sophisticated products or solutions it is better to conduct face-to-face training. Training can include how to use content and marketing tools, how to position products, and demonstration training of products.
**Insights**
Finally, insights used in partner enablement can be based on market or customer trends that the vendor has researched. These insights show how to better market or engage customers through marketing and convert them to sales.
## Measuring enablement
In enabling channel partner marketing, the channel partner needs to ensure the right sales stakeholders are using the right tools and information and attending any enablement event or activity. Channel partner enablement can be measured in different ways:
**Attendance**
Where the enablement activity is onsite training, a webinar or a face-to-face event, attendance numbers can be used as a measure.
**Channel partner portal sign-in and frequency**
Where enablement is content and information in an online partner portal, the total number of people using the tool can be one indicator. The frequency of use in each period per stakeholder as well as time on site can be used to understand how well the online portal is used. It can also provide insights back to the vendor on whether the content or tool is effective. Where frequency of usage is low this may point to the online tool not being easy to access or use and adaption or further training is required.
**Completion of training**
Where training covers more than one activity or module, completion of the training set can be another way to assess sales readiness.
# New approaches: social for marketing to channel partners
Social is increasingly used in B2B and it’s not different for channel partner marketing. The main platforms used are Twitter, LinkedIn, Facebook and YouTube.
For marketing to channel partners social can be used in different ways. The main objectives are to keep the channel partner network updated with the latest information, in educating channel partners about new products and services, in reinforcing communities between vendor and channel partners, and to improve the relationships between the channel partner and vendor.
For vendors and channel partners to use social effectively it’s important that both understand how to use it. The rules of engagement should be established in advance and social media sub-groups may need to be set up, based on subject matter expertise or a focused theme, to improve sharing.
EXAMPLES
Users of social forums with channel partners include Dell and Channel. Dell Channel Europe communicates the latest pieces of news to its channel partners as well as to the channel marketing community. HP Partner Ready is on Twitter to provide the latest news.
# Marketing to channel partners: challenges and solutions
## Challenges
Whether the vendor is a small or large business, marketing to channel partners can be challenging for the following reasons:
* *Too many stakeholders – no accountability*: for vendors one challenge can be reaching multiple stakeholders and gaining mindshare. Where the vendor’s marketing department communicates with many stakeholders within one channel partner this may lead to stakeholders’ apathy in receiving or reading such e-mails.
* *Resource*: smaller businesses typically have fewer resources and therefore may not have the capacity to communicate to multiple channel partners.
* *Channel partner organizational structure*: the structure may mean that there are too few marketing resources to deal with the many channel vendors that need to be engaged.
* *Over-communicating*: vendors can use too many market communication forms; this may lead to channel go-to people becoming numbed by the communication and not reading or digesting it.
* *Over-reliance on one vehicle*: e-mails or e-shots can become over-used leading to channel partner recipients not leveraging or responding to them.
* *Stakeholder relevance*: where a message is intended for a marketer but written for sales this may mean it is not followed up or acted upon.
## Solutions
Marketers need to see how they optimize communication with the channel partner sales and marketing. Some of the ways to address the challenges include clearly defining marketing go-to people and agreeing with the channel partner on who they are and who is accountable for any follow-up. Vendors could check with channel partners to capture any changes in personnel to ensure contact data is up to date.
Sending e-mails to fewer stakeholders can reduce the perception of over-communicating as well as supporting accountability measures for those go-to people at the channel partners. In addition, by selecting communication and better frequency management the number of e-mails can be controlled. E-mail should be easy to re-use or send on, allowing for sharing of information by channel partner stakeholders.
Having the vendor’s CMO involved can sometimes help increase the level of reading and responding to e-mails by the partner. Where the channel partner is a premium or very strategic one, and the budget allows, some companies opt for having someone on site to support messaging, sales, marketing and training.
**TIP** Solving communication challenges through one point of access
One way around trying to message multiple channel stakeholders while reducing traffic sent to their inboxes is using a single hub of information. Many larger companies are doing this but it can also work for smaller ones. Making the portal easily accessible and having a home page with all the information allows users to easily access it without any website navigation problems. Rather than worry about sending e-mails to multiple stakeholders, businesses can create forums on Twitter or LinkedIn to message updates or additions about the information hub.
# Digital applications
One solution to managing communication to channel partners can be using channel partner content syndication. One of the top challenges in campaign management is coordinating communications across multiple channel partners.
Channel content syndication replicates and customizes digital content across external web properties, ie to channel partner websites. Different types of content syndication can include social media, e-mail or asset syndication. The benefit is that channel partners, including all forms of resellers, have up-to-date information and content.
# Creating one marketing team: vendor and channel partner
One additional approach is to facilitate marketers in engaging and sharing information between vendors and channel partners. Such marketing forums or communities can be created either in a structured way by setting up forums or sub-groups on social media, by having an online space to share information not housed on a social platform, or by an offline engagement through meetings. The benefits of such forums are that the vendor can facilitate sharing of ideas and information to stimulate a team atmosphere and to motivate individuals through being part of a community.
ACTIVITIES
Review with a channel partner the e-mails sent to their stakeholders in a given period for quantity, relevance and whether they’re used. Identify improvements in communicating.
Review channel partner marketing needs and whether current vendor marketing efforts support those needs. For any challenges in channel partner marketing, review current digital applications.
# 18
# Marketing through channel partners
This chapter will give you an understanding of:
* goals of marketing through channel partners
* types of marketing through channel partners
* criteria for selecting activities
* main solutions for marketing through channel partners
* lead generation
* gaining mindshare
# Definition
Marketing through or with channel partners aims to engage existing or potential business customers; the marketing flows can happen in three ways (see Figure 18.1):
1. *Directly by the vendor*, ie vendor directly markets to the customer and directs the customers to the channel partner, for example through links on their website, or in redirecting to channel partner sales (B).
2. *Indirectly*, ie via the channel partner to the business customer where the channel partner markets on behalf of the vendor (A).
3. *With the channel partner* to the business customer, where the vendor collaborates with the channel partner to engage the customer as a joint marketing activity (C).
Figure 18.1 Marketing through channel partners
Marketing through channel partners
## Why market to a customer if channel partner marketing is used?
If a vendor has engaged a channel partner to reach and engage customers, why would the vendor use its own resources to market to customers? The reason could be the initial agreement between the vendor/supplier and channel partner regarding the purpose of the relationship, for example where the partnership was set up for logistical reasons or for customer reach. Other reasons could be that the vendor believes the core messages that the channel partner needs to deliver are not strong enough and that more direct messages to the target market are required, or that the channel partner’s use of marketing communication doesn’t cover certain aspects, for example if the channel partner relies on newsletter and e-shots and it’s felt that other forms of marketing should be used.
# Types of marketing through channel partners
The type of marketing will depend on customer audience, channel partner type as well as the marketing flow. Where the marketing is directly to the customer, the marketing can come in three forms, shown in Table 18.1. The first is awareness-based marketing such as PR, e-shots, social and co-branding; advertising aims to create awareness for the vendor’s brand and to direct customers or traffic to its channel partners. The second is lead generation, where vendor marketing invests in generating leads through content syndication, outbound telesales or other to capture leads and pass these on to channel partners. The third, demand generation, is about directing traffic to the channel partner’s website or sales through incentives, promotions or direct response marketing.
Table 18.1 Digital and offline marketing vehicles
| **Awareness tactics** | **Lead generation tactics** | **Demand generation tactics** |
| PR | E-shots | Newsletters |
| E-shots | Social lead generation | Content syndication |
| Promotions | Incentives | Seed units |
| Social | | |
| Branding | | |
| Advertising | | |
Where a channel partner markets on behalf of the vendor the channel partner typically does less marketing than the vendor would. It would do little awareness-based marketing as this is the vendor’s role. Lead generation may occur where the partner has such an approach in place and where the vendor has a market development fund (MDF) or other budget. Channel partner and vendor marketing collaboration examples can span all the types of marketing shown in Table 18.1; examples are collaborative PR, advertorials, testimonials, and lead generation marketing such as content syndication where the channel partner participates.
## Types according to channel partner type
Types of B2B marketing can vary depending on the type of channel partner; a supplier/distributor relationship may be based more on demand generation, lead generation and associated marketing vehicles.
With large value-added resellers the forms of marketing may be more relationship based; eg launch events or webinars with channel partners, combined online advertising, combined lead generation/content syndication, and outbound marketing forms. This type of marketing is looking at generating larger sales volumes from fewer sales transactions.
# Selecting and sharing marketing
So how are marketing activities selected between a channel and vendor? What are the selection criteria? How are the responsibilities divided up?
One approach is to split responsibilities based on budget level, ie the amount of budget made available from the vendor that the channel partner can commit to a set of marketing activities on its behalf. The channel’s type of business dictates the effectiveness and forms of marketing, eg a distributor will focus marketing on promotions or driving demand rather than driving leads which have a longer conversion time. Cost will dictate that phone calls may be too expensive and time-consuming to engage a huge volume of small business customers; on the flip side the cost to serve a large corporation may justify more elaborate means of marketing. Finally, the stage of relationship between channel partner and vendor will have a bearing on what role they play in supporting the business for each other: in the early phase there may be less marketing investment as the channel partner is testing and piloting the new vendor’s products as well as marketing them.
**TIP** Process for selecting and sharing activities
This should follow the below TMAR framework:
1. *Targets defined and co-defined*: vendor channel marketing and channel partners should agree on business targets for marketing activities; this can be unit volumes, website traffic or phone traffic to generate leads, opportunities and pipeline, and time for delivery of marketing activities.
2. *Marketing mix defined*: which customer solutions to include in portfolio, which themes or campaigns to focus on, which communication vehicles, which content, etc.
3. *Activity split and focus defined*: may already be set out based on existing marketing but may need redefining for incremental or larger-scale marketing activity. Key questions concern who invests and implements which activities; eg does the vendor set up the event or the channel partner or both?
4. *Roll-out and review*: marketing enablement assets are provided for activities the channel partner owns. Activities are implemented and then reviewed together in terms of achievement of KPIs.
# Marketing through channel partners: lead generation
Enterprise or relationship marketing (see Chapter 1) will tend to rely a lot on lead generation activities. Lead generation without a channel partner in the mix can be challenging in terms of tracking leads, ensuring follow-up, conversion, etc, and using multiple channel partners and their sales can bring additional challenges.
The first challenge is about accountability and ownership: who owns the lead and which stages of the lead generation process they are accountable for. Aside from ownership, how is the follow-up handled between vendor and channel partner or between a third party that generated the initial lead? How does the vendor ensure the leads are kept as warm as possible and don’t go cold? With different entities covering different stages of the lead journey, how are leads tracked? Finally, how are leads assigned if there are multiple channel partners working with the vendor?
## Managing lead generation with channel partners
Here are some ways to manage lead generation with channel partners.
### Stakeholder management
Clearly define steps in lead generation from marketing-ready lead through sales-ready lead to follow-up, opportunity identification, pipeline detection and closure. For each step define who owns the step: who is responsible for generating the lead, for nurturing the lead, for qualifying it.
### Assigning leads to channel partners
1. Create assignment rules for types of leads so they are automatically directed; rules may be based on customer location, specialist based, type of lead, product based.
2. Agree on business rules for leads: volume of leads, type of lead, timing, etc, for channel partner.
3. Review the effect of business rules: check timing and volume of leads meet expectations.
4. Maintain consistency of rules through lead assignment.
5. Optimize lead allocation by removing or adding channel partners where needed, eg where the channel partner doesn’t execute or decides not to take further part.
### Ensuring lead execution
1. Agree on follow-up timelines. Some companies apply zero tolerance beyond 24 hours for lead follow-up; ie no lead is left unfollowed beyond 24 hours. Others will agree to longer follow-up times.
2. Follow-up and execution. Define champions or accountable people within the channel partner who enforce a discipline on lead follow-up and conversion.
3. Quality of execution. Ensure that diligence is applied to qualifying the lead if this is the channel partner’s responsibility, and applied to nurturing the lead and following up; this will all mean better ROI for the channel partner and vendor.
4. Visible and transparent executive support. Get senior level support from the channel partner and make it visible; this in turn will mean sales are more bought-in to the process.
## How to track leads via channel partners
Tracking of leads can be cumbersome but there are ways to support this across different channel partners, outlined in Table 18.2.
Table 18.2 Leads tracking via channel partners
| **Lead tracking option** | **Detail** |
| Channel partner CRM | The channel partner uses its own CRM system to track and manage leads and provides the vendor with access to the CRM system for purposes of reviewing performance |
| Vendor CRM | The channel partner uses the vendor’s cloud-based CRM systems and inputs and tracks leads performance |
| Marketing automation platform (MAP) | The vendor uses a marketing automation platform that can integrate multiple CRM systems, providing access across channel partners |
| Manual | Reporting of leads is manually captured by channel partners and vendors and tracked through reporting devices such as Excel |
## Lead handover
Lead handover refers to leads being passed from one party to the next; usually handover aims to improve conversion. An example is where a telemarketing agency passes on leads to channel partners; another is where leads are handed over by the vendor sales. Handover typically occurs with help from a marketing automation or CRM system.
# Gaining channel partner mindshare
One of the challenges of marketing through channel partners is gaining mindshare. A partner needs to maximize its share of the marketing space that channel partners devote to their vendors, whether directly or indirectly competing or just taking up marketing communications space.
Even if agreements and structures are in place a business and its marketing department need to go a few steps further in motivating channel partners and gaining mindshare. The competition are doing the same so doing the bare minimum means the vendor will lose out. Having maximum mindshare and support from channel partners will in turn mean greater probability of achieving targets. There are a number of ways vendors can increase mindshare and support from channel partners:
* *Sales incentives* help to motivate sales to promote products; some vendors believe that having a contract and agreement in place is enough but if other vendors use channel incentives (and they often do) then attention will be drawn away from a vendor’s products.
* *Great content*: if the vendor can provide some compelling content, sales people are more likely to read and forward the content to colleagues and partners.
* *Lead generation*: providing high quality leads to channel partners.
* *MDF*: a larger contribution of marketing development fund from one vendor over the other will mean the channel partner invests more in communicating and marketing and therefore customers and sales have greater exposure.
* *Onsite sales activities*: onsite activity from vendors can be focus days or burst focus sessions.
* *Partner programmes*: partners participate in a vendor programme and receive rewards for achieving specified levels of sales.
* *Awards schemes*: partners can take part in a competition with other channel partners and thus will have increased focus and push for a vendor’s products; this can be effective in driving sales where the vendor has multiple channel partners.
According to a 2014 Canalys press release about its survey of 130 partners, on average partners rated the importance of programmes to their vendor relationships as 8 on a 10-point scale (Canalys, 2014). The results of the survey also highlighted the benefits of a partner programme in order of importance: rebates, deal registration, lead generation, account management and front-end discounts. Twenty-six per cent of partners said back-end rebates were the most important benefit, while 66 per cent of partners selected lead generation as a top-five benefit.
**EXAMPLE** The Oracle channel marketing handbook
Oracle has developed a handbook specifically for channel partners covering a broad range of marketing services. Channel partners’ marketing stakeholders can use different marketing services such as ‘campaign as a service’, events, marketing automation support, sales velocity days and other resources. Through this comprehensive handbook channel partner marketers are able to access everything in one document.
## Partner programmes
Partner programmes are a great way to gain mindshare and lock in channel partners. Most IT vendors operating via channels use them. They are typically segmented across three or four types of partner as follows:
**Level 1**. The highest level, where the channel partners are regarded as premium accounts that generate high revenues; they are typically account managed, sometimes exclusively by one account manager. These channel partners will usually have their own in-house marketing resources.
**Level 2**. Mid-level channel partners may or may not share an account manager; they may be managed via phone instead of a face-to-face arrangement. These channel partners may be reselling some products but not a wide or full portfolio.
**Level 3**. These may not be account managed but still have access to some vendor marketing resources.
**Level 4**. May be an additional tier of newly acquired channel partners.
Marketing support to these channel partner levels would differ:
1. Assignment of leads from vendor would typically only go to level 1 channel partners.
2. MDF would be provided to those channel partners actively marketing the vendor’s products and investing so level 3 and 4 probably would not receive it.
3. Type of content may differ: premium partners would receive richer content.
4. Marketing campaigns may only be provided to level 1 and 2 accounts, where channel partners are specifically included.
# Social media and channel partner marketing
The channel partner’s role in using social media as part of collaborative marketing may take different forms:
* *Co-marketing awareness activities*: channel partner and vendor support awareness of a new initiative through social presence. Social media can help in supporting thought leadership activities.
* *Lead generation*: the vendor generates leads via a social platform that are then handed over to the channel partner.
* *Vehicle-assisted*: a channel partner integrates use of social to promote an event and engage customers at or after the event.
* *Co-marketing employee-assisted*: employees of the channel partner use content and information and promote this through their own channels.
* *Combined social forums*: the channel partner and vendor agree to host a combined social group or discussion via a social platform.
# Through-partner marketing applications
Through-partner marketing applications are starting to grow, which is great for large businesses wanting to streamline efforts and drive efficiencies as well as for smaller or medium-sized businesses that need help with activities that are too resource intensive. Through-partner marketing applications include social syndication through channel partners, channel content syndication, and marketing automation platforms designed to work across multiple partners and automate efforts to track and nurture leads.
**EXAMPLE** EIMS Channel Energizer programme
EIMS is a global sales and marketing agency founded in 1996 by IT industry professionals who still run the business; it has consistently grown 25 per cent year on year. One of its specialisms is to build and enable a channel, feeding the partners with high quality leads to optimize channel value. One of the services in EIMS’s channel portfolio is called ‘Channel Energizer’.
The purpose of the EIMS Channel Energizer is to increase channel productivity and deliver high growth in a way that is cost-effective; through insights and data a tailored programme is created to target specific opportunities. In many cases, the programme enables vendors to reach more effectively into their channel, boosting the revenues of partners that have not enjoyed the benefit of high-touch account management and transforming the relationships with some partners that had previously not delivered significant revenue.
The EIMS method starts with analysis and planning, a study to identify the characteristics displayed by successful partners to really understand how to achieve agreed business goals. To assess potential, EIMS understands the partner’s business mix including other solutions and relationships that may be competitive or complimentary to that of the vendor.
Driving growth through increasing the productivity of existing partners is often the shortest route to success, but the initial analysis and assessment may conclude that to deliver aggressive revenue goals, especially in new markets, new partners may need to be recruited. The identified characteristics of successful partners informs targeting of potential new partners. The resulting shortlist of new and existing partners is engaged, typically by tele conference, to assess their levels of motivation for long-term success and alignment to the vendor’s goals. The EIMS approach is to then engage with the principal decision maker to evaluate the business ambition to grow with the vendor and understand alignment of strategic direction. As a result, the shortlist is further refined to create a final ‘hot’ target group of high propensity partners ready to enrol in the Channel Energizer programme.
EIMS has a unique technology platform to ensure optimal process and consistent partner experience; it includes a storefront allowing vendors to offer services to their channel from their own marketing teams and a multitude of preselected and approved external providers. Partners can self-select, guided by a concierge service where required, to ensure an optimal use of marketing development funds. This enables delivery of approved partner plays across a large partner base in a hassle free and a highly scalable way. Partner engagement and activity outcome is completely transparent and reportable and the partners can stay one step ahead of their competition.
This proven approach has helped many organizations drive accelerated growth in the channel while increasing the number of transacting partners. Vendor and partner engagement is enriched as each delivers more value to the other, supporting sustainable and predictable long-term success. For some customers this programme has allowed them to break into new international markets; for other customers it delivered incremental pipeline of around $19 million.
# Reference
Canalys (2014) Partner programs prove key to channel relationships, Canalys press release, 2014/060, 8 September, available at: https://www.canalys.com/static/press_release/2014/canalys-press-release-20140908-partner-programs-prove-key-channel-relationships.pdf (accessed 10 February 2017)
# 19
# Channel marketing strategy and control
This chapter will give you an understanding of:
* types of channel partner marketing strategies
* channel partner lifecycle marketing
* channel partner acquisition marketing
* channel planning and control
* channel partner marketing – measuring and planning
# Channel marketing strategy
Channel partner marketing strategy, called ‘channel marketing strategy’ for the purposes of this chapter, refers to the different approaches to marketing via channel partners or intermediaries to customers. Channel selection, management and strategy are important business decisions typically encountered by organizations’ business leaders and senior executives in sales, marketing, operations and financial departments.
In most organizations the marketing departments have insights into the market and market structure including the channel structure; the role of marketing regarding the channel partner marketing strategy is to support and guide business leaders in assessing optimal routes to customers, and in defining the optimal mix of channel types. Marketing can also help the business by providing an overview of the full channel partner universe, highlighting all the channel partners and intermediaries in a given region.
Based on the goals and agreed route(s) to market, the marketing department needs a structure that best supports the business and channel type. For example, some channel resellers require a high-touch sales process and marketing departments will focus resources on supporting distribution channel partners.
# B2B channel marketing goals and objectives
In defining channel marketing strategies and channel strategies it’s important to first determine the goals; below are some of the main ones influencing channel partner marketing strategy selection (see also Table 19.1):
**Market coverage**. Where a company’s goal is to cover a particular geographic area, channel selection would be based on location.
**Customer reach**. Where a company is unable to reach customers with its own resources, a channel partner that helps in reaching customers would be selected.
**Provision of end-to-end solutions**. Channel resellers can complement the brand’s offering to provide a complete solution.
**Accelerate growth**. Collaborating with a channel partner could help the firm grow its business faster.
**Vertical growth**. A vertically aligned channel partner strategy may be well suited to achieve such growth.
**Credibility**. Where a new brand or a brand in a new market needs to gain credibility, channel partners with established relationships in that given market can help organizations promote their products to the new audience.
Table 19.1 Goal and channel partners
| **Goal** | **Channel marketingstrategy type** |
| New market coverage | Channel partner acquisition |
| New customer reach | Channel partner acquisition |
| Provide end-to-end solutions | Channel partner integration marketing
Partner enablement |
| Accelerate growth | 1. Expansion marketing via channel partner through demand generation and lead generation
2. Channel partner acquisition |
| Brand and market presence building | Collaborative channel marketing
Channel partner enablement |
| Improve execution | Channel marketing operations |
Here are some of the main types of channel marketing strategies aligned to the above goals:
**Expansion channel partner marketing**. Where a vendor works with an existing channel partner to increase business with existing customers by marketing and selling more of the portfolio.
**Channel partner acquisition**. This involves adding new channel partners to support business goals such as market coverage, customer reach, or vertical market access.
**Customer acquisition via channel partners**. This is about working with channel partners to focus on specific customers to acquire.
**Integration marketing**. This hinges on collaborative marketing and integrating marketing messages across the channel partner and vendor’s portfolio to highlight end-to-end solutions.
**Brand and market presence building**. Marketing together with the channel partner intensifies brand building efforts; this may be a combined new solution both parties will benefit from, or where investment in co-branding is expected to reap returns through additional business.
**Alliance channel marketing**. Vendor and channel partner work together to promote the vendor’s product portfolio, for example where a technology vendor can contribute a lot of value to the relationship through thought leadership, budget or other aspects.
**Marketing operations**. This can be where digital applications are adopted or processes are optimized to support smoother marketing operations and execution. An example can be a new marketing automation system.
**EXAMPLE** Alliance channel marketing
In 2000 Datapipe and Equinix established a customer and vendor relationship which by 2016 developed into a real channel alliance partnership. Their two business complemented each other well: Datapipe acted as a managed hosting and cloud services provider needing to expand its business across multiple geographies while Equinix, an industry leading data centre company with global footprint, was also growing at a rapid rate. In 2016 they used a joint marketing approach with Datapipe and Equinix co-sponsoring marketing and industry events, as well as mutually investing in co-marketing material, success stories and lead generation.
# Channel partner lifecycle marketing
One approach to channel marketing is to manage and support the lifecycle of the partner through business- and marketing-aligned activities. Figure 19.1 shows the main stages of the partner lifecycle with the appropriately aligned marketing activities:
* *Acquisition.* A channel partner is identified, targeted, engaged and then added to the list of a vendor’s channel resellers.
* *Onboarding.* A channel intermediary becomes a channel partner for the vendor. The role of vendor marketing is to support the new channel partner in becoming familiar with the vendor’s products through introductory training, value propositions per product/solution and product positioning information.
* *Enabling.* Following onboarding the partner is enabled through provision of tools, systems, training, roadmaps, demo units and access to people who support channel partner marketing personnel.
* *Demand creation.* Once enabled the partner should be able to create demand on behalf of the vendor through marketing resources, advertising, marketing development funds and, in some cases, additional marketing budget.
* *Grow and expand.* During this phase the partner and vendor tend to work more closely together; the channel partner acts as an extension to the vendor business and vendor marketing can make available budget and additional resources to support lead generation and marketing development activities.
* *Retain and integrate.* The final stage is in retaining the channel partner and integrating some existing vendor practices; marketing activities could include marketing forums and co-partner communications.
Figure 19.1 Channel partner lifecycle
Channel partner lifecycle
# Acquisition channel partner marketing
Channel partner acquisition marketing can occur via two routes: directly from the vendor targeting, selecting, engaging and onboarding the channel partner, or the vendor and distributor work together to acquire a reseller. To understand marketing’s role in channel partner acquisition we need to first understand the typical steps in the overall process. Here is a simple six-step process (see also Figure 19.2):
1. Define target market, target customers and number of customers per territory or region. This information will help you later in selecting the most appropriate channel partner type and to understand if a region is under- or over-served.
2. Define offerings to meet target market needs. The needs of a certain target market may not be met by the existing portfolio and additional products or services are needed for a more complete offering. In this case the selection criterion is whether the channel partner can assemble its own products with those of the vendor to meet the customer’s needs. Sometimes a customer’s needs may be so specific that a specialized channel partner needs to be acquired.
3. Review current GTM approach, coverage for market. During this step, you look at existing channel partner coverage and structure to understand whether the established business and channel partners are sufficient. It may be that you don’t need to acquire more channel partners but rather to work better with existing partners.
4. Identify gaps between intended target market and current GTM model to create a channel partner checklist by, for example, geography, channel partner type and customer segment.
5. Evaluate and select channel partner based on a set of criteria that could include location, specialism, appetite to engage new vendor or ease of engagement.
6. Initiate acquisition process directly or indirectly. The vendor may decide to use a distributor to start acquiring channel partners. In some cases, the distributor may already have the intended channel partner in its established list of partners; in other cases, the distributor’s reputation may allow it to engage and acquire the channel partner more easily.
Figure 19.2 Channel partner acquisition process
Channel partner acquisition process
# Channel partner for customer acquisition
Acquiring customers via channel partners can follow a similar process to the one outlined in Chapter 5, although there are several differences that make it worthwhile for vendors to consider the channel partner marketing route. Figure 19.3 shows the different routes to acquiring customers via channel partners.
Figure 19.3 Customer acquisition through channel
Customer acquisition through channel
In scenario 1 the prospect is already a customer of the channel partner but only buys certain products from another vendor. This could be because the vendor only offers those products, or the customer prefers to buy only those products from the other vendor. The prospect is still interesting for both you and the channel partner to sell additional products and services not currently bought by the other vendor, if there is no conflict. The channel partner has an existing relationship or reputation that carries credibility with the target customers, and the buying process can occur faster than a vendor trying to acquire the customer directly. As the channel partner already has an established business relationship with the customers, social media can be a powerful tool to engage the customers through forums, YouTube and social networks to share information about additional products and services. The channel partner could follow up by inviting the customer to webinars or face-to-face events to explain and engage further.
In scenario 2 the prospect is a customer purchasing products/services through another channel partner (channel partner 2); in this respect the intended customer is a prospect for both you and channel partner 1. As there is no potential conflict of interest, ie the vendor is not a supplier of channel partner 1, the full products/services can be offered through channel partner 1 to the new customer. The benefits of this approach can be that channel partner 1 has a strong reputation in the market, sector or region, which improves the possibility of acquiring new customers rather than the vendor going directly. Vendor B and channel partner 1 can agree on how to create an initial market plan for educating and informing customers through a mix of advertising, social media, online and other media.
## Integration marketing
Integration marketing can occur in different ways, for example in integrating products to form solutions, integrating messages, or integrating channel partners further into scope of vendor’s business. Marketing’s role in these can be either to ensure messaging is delivered clearly to channel partners in the form of assets as well as integrated into the channel partner marketing tactics. In terms of channel reseller integrating within the business, marketing can support through extending lead generation activities to include the channel partner and also pass on opportunities to the partner. Other aspects are onsite enablement and sales activities supported by marketing to train and motivate the channel partner sales.
# Channel contact strategy
Effectiveness in marketing to, with and through channel partners can be heavily influenced by having a contact strategy. A contact strategy for channel partners is about creating different sub-segments of channel partners for the different business and marketing activities and aligning the marketing contact or engagement approaches accordingly. These channel sub-segments can be split into retention marketing, expansion/growth marketing, marketing based on acquiring channel partners and marketing based on promoting and further integrating the channel partner. Other aspects include improving efficiencies in selecting and engaging channel partners, in messaging to and through them, and in intensifying relationships through tailored channel marketing.
Possible activities supporting a promotion strategy could be to leverage social media channels to promote types of content, to include the channel partner in new online forums, or in reworking a communication strategy to add or remove contact people to distribution lists (see Figure 19.4).
Figure 19.4 Channel partner contact strategy
Channel partner contact strategy
**EXAMPLE** SAP – Our Business Runs Better
In 2011 SAP launched a marketing programme to improve its presence and visibility within the SME segment, called ‘Our Business Runs Better’ (OBRB). It approached this by demonstrating to an experienced network of channel partners who knew and understood the business challenges pertaining to SMEs, as well as demonstrating solutions to those pain points. Through co-marketing planning and provision of assets such as images, e-mail, landing pages and online banners it could develop new engagement opportunities with its network of channel partners across the EMEA territories. This resulted in over 1,000 leads and sales revenue of over €15 million and €30 million for 2011 and 2012 respectively.
# Channel planning and control
The vendor and the channel partner need to establish a discipline for sharing plans, activities, resources and marketing schedules. There should also be budget and marketing development funds provision. The difference between standard and channel marketing planning is that the channel partner will normally need to plan with multiple vendors, so it’s important the vendor maximizes its use of the partner’s resources and market communications vehicles.
In planning with partners, channel marketers have additional alignment challenges: they not only need to align internally with their stakeholders on priorities and budget spend but also with channel partner stakeholders. There need to be measurement systems in place to track activities, review them and optimize.
## Measuring and planning
The channel marketing measurements or KPIs are dependent on the type of marketing activity. In dealing with channel partners and potentially offsite remote measurement systems it’s important to keep KPIs simple and relevant. Here are some ways to measure different types of channel marketing activities:
* overall marketing: pipeline, revenue;
* lead generation: number of leads, leads converted to opportunities;
* demand generation: channel partner website traffic and conversion to leads;
* marketing enablement: quotes on the day, attendee level.
# Channel marketing budgets
Managing channel marketing budgets includes several aspects such as budget allocation methods, partner development fund management, budget tracking and budget review. If these elements are badly managed there can be budget wastage, poor execution and friction between the channel partner and vendor.
## Building and creating marketing budgets
Marketing budgets can be sourced from different areas. The main ones are from the vendor’s own resources, from suppliers that have a vested interest in the vendor’s sales, and from investment companies.
**TIP** How do you source additional budget?
Vendors may have hidden or undiscovered opportunities to find new sources of marketing budget. For example, Intel and Microsoft, which are both component and software vendors, provide additional marketing budget to IT companies through partner development programmes. Outside of the IT industry sources of additional funds can be suppliers, additional channel affiliates, or services companies that could benefit from collaboration. If you’re struggling to find budget, you may discover hidden opportunities with your suppliers or partners who have a vested interest in your business’s growth.
## Channel budgeting process
Here is a typical process for marketing budgeting for channel partners:
1. The channel marketing budget is defined for a given time.
2. Marketing budget is allocated according to business goals.
3. Marketing budget is then allocated for specific channel partner marketing activities.
4. Channel marketing budget allocation can be based on these criteria: premium channel partner status, legacy marketing activities where vendor channel marketing ring-fences budget in advance; potential growth in given period; seasonal business peaks.
5. Budgets are reviewed based on ROI.
6. Budget are optimized based on return, through changing the marketing mix or allocating more budget to those channel partners performing the strongest.
## Justifying your channel marketing budget
Where channel partners only require budget to train and enable sales, there are more challenges for marketing to justify its budget: proving the link between final sale and marketing budget may be difficult as the activity is further removed from the sale. In such situations it’s important that the value and measurement of training to sales is better articulated, for example the metric for enabling or training sales could be attendance at a training event, or how often sales accessed an online marketing training tool.
# PART FIVE
# Optimizing marketing execution
# 20
# Sales and marketing alignment
This chapter will give you an understanding of:
* importance of alignment
* benefits of alignment
* obstacles to sales and marketing alignment
* how to identify misalignment
* steps to establishing alignment
* integration best practice
# The importance of alignment
Sales cultures within organizations can in themselves be obstacles for marketing to engage or to build relationships. Some organizations have very complex sales structures, further complicating the marketing task in achieving alignment with sales.
Alignment is important as it highlights the value each party delivers for a goal. In an aligned situation, sales will get what they need to support the business and marketing achieve their goals through delivering to the right customers. The right messages are sent via the right sales structure and physical and communication channels. In B2B marketing the success of marketing often hinges on sales alignment, for example in lead handover, or in participating in an activity. Alignment is less of an issue in B2C marketing.
## Changing sales role
The modern buyer is digitally driven, socially connected, mobile and empowered, with virtually unlimited access to information and people. The role of sales has shifted according to changes in organizational buying behaviour and this means that sales need different support from marketing in the form of sales enablement, information and content tailored to buyer stages. On the flip-side marketing needs sales to adapt their role to understand and embrace how to educate customers indirectly and how to embrace new communications channels such as social media.
# Benefits of sales and marketing alignment
Research suggests that business productivity is greater and targets more easily achieved when sales and marketing are aligned and engaged. Recent reports have produced the following insights:
* Companies with dynamic, adaptable sales and marketing processes achieved an average of 32 per cent annual revenue growth while less aligned companies reported an average 7 per cent decline in revenue (Aberdeen Group Study, 2011).
* Organizations with tightly aligned sales and marketing operations achieved 24 per cent faster growth and 27 per cent faster profit growth over a three-year period (Setty, 2013).
* Organizations with tightly-aligned sales and marketing had 36 per cent higher customer retention rates and achieved 38 per cent higher sales win rates (MarketingProfs, 2015).
Overall the main benefits can be summarized as:
* *Increased sales conversion rates.* Sales convert more leads as marketing deliver leads that sales need and desire.
* *Faster sales conversion.* With tighter engagement between sales and marketing, leads are handed over to sales more efficiently and sales follow up in a more timely fashion.
* *Transparency on pipeline improves.* As marketing understands how pipeline builds and sales understands what is being delivered from marketing, there is a greater visibility to the pipeline progression.
* *Less leakage of leads.* Where leads are not followed up this can be rectified earlier and they can be reassigned to a different sales group or further nurtured by marketing.
* *Optimization of marketing budgets.* As there is greater input and direction from sales, marketing budgets are optimized for lead quality and conversion.
# Organizational obstacles to alignment
There are often several reasons why sales and marketing departments aren’t well aligned:
**Culture**
Sales cultures and marketing cultures tend to be very different. They use different terminology, value different things, work differently and have different behaviours. Some business cultures can be heavily sales focused, have silo departments, or emphasize the importance of sales over marketing. Cultures within sales and marketing can be based on legacy leaders, teams, etc. These cultural aspects can maintain divisions between sales and marketing departments.
**Targets**
Sales teams are set targets and incentivized differently to marketers. Sales targets can be based on pure pipeline and revenue whereas marketing looks at website traffic and quality of content. Where targets are set separately for marketing teams and sales teams this will lead to a lack of alignment.
**Lack of focus**
Organizations may simply not focus on marketing efforts, which can frustrate marketing teams who feel undervalued. This lack of focus can be due to seasonal business peaks where sales need to focus on opportunities to close rather than working on leads earlier on in the customer buying cycle.
**Multiple sales teams and priorities**
Where the organization has different sales teams, this can lead to different or conflicting requests; eg an internal sales team may prioritize leads whereas an external sales team may prioritize enablement tools and materials to support more customer-facing activities.
**Leadership**
Leadership is required to correct misalignment, not only from marketing but also from the overall business lead. Leaders can help unite different departments around common goals.
# Establishing sales and marketing alignment
For alignment to be established organizations should follow this four-step process:
1. *Recognize misalignment.* During this step the organization agrees and recognizes that there is a misalignment between sales and marketing, and acknowledges the need to address it. Stakeholders are identified, ideally representatives from sales and marketing, to take the lead in the process for alignment.
2. *Identify and audit.*Obstacles and contributors to sales and marketing alignment are identified. This is carried out by reviewing structures, cultures, KPIs, etc. This will typically be a qualitative exercise involving outsiders as well as key stakeholders from sales and marketing. Marketing or sales investigate and listen to feedback to understand the factors behind the misalignment. The results can then be plotted in a radar diagram like the one shown in Figure 20.1 to help plot the impact of each misalignment factor.
3. *Agree on fixes, priorities and timelines.* The organization and sales and marketing leaders agree on where to start and the priorities to address to solve the misalignment. As part of this process marketing may relook at business goals, challenges and opportunities as well as more granular behaviour and process aspects. These can be through service level agreements (SLAs) between sales and marketing teams holding both parties accountable for what they deliver; they can be extremely formal or less formal allowing for change where needed.
4. *Implement and review.*The steps and activities to resolve the misalignment are implemented and progress checked through regular reviews.
Figure 20.1 Alignment radar
Alignment radar
# Alignment areas
## Define marketing’s role in the organization
The role of marketing in midsize or larger organizations may stretch across multiple types of marketing. Where marketing resources are limited this means the department may not be able to deliver across all the marketing areas that the business and sales are asking for. The main types of marketing are:
* *Sales enablement.* The focus is on preparing and supporting sales through training and provision of marketing materials and assets that support the sales cycle.
* *Marketing communications and content.* Marketing’s focus is on managing the marketing communication channels and the content across those channels.
* *Brand building and awareness.* The focus is on building the brand and investing in activities that promote the company and its value proposition.
* *Demand and lead generation.* Marketing focuses on generating demand and leads and investing in activities to maximize traffic for the business.
To improve marketing effectiveness there needs to be an agreement within the executive team on the core priorities of marketing; this doesn’t preclude marketing from doing something in all areas but helps provide a core focus for marketing to deliver on.
## Target customer
It’s important to define the type of customer to target: whether existing or new and, if new, which sector. Other aspects to stipulate are size of business, geographical focus, key target stakeholders and department. Having customers defined in a granular way will help when planning the right marketing mix and materials and in supporting sales in the right and relevant way.
## Activity alignment
Once customers have been defined the next stage is for sales and marketing to agree on activities. Some activities will involve both or either sales and marketing; for example what type of content will be used to engage customers directly from marketing or via external parties, and which content is sales expected to use to engage customers themselves? What messages will be used to generate leads and who is responsible for nurturing and to what stage in the lead development process?
## Communication structure/governance
A communication policy should cover the following:
* *Forms of communication to use.* It should be agreed how sales want to receive updates from marketing and how marketing expect to be updated by sales. Is this through e-mail, phone conferences or face-to-face meetings?
* *Frequency.* Both parties need to agree on how frequently the updates will occur.
* *Timing.* Timing of reporting is key to allow for sales to address or correct their performance if it is not on plan.
## Shared KPIs
Both parties should agree on KPIs relevant to each other and this should form a two-way SLA. For example marketing’s commitment to sales could be on number and quality of leads in a given period, pipeline or revenue. Sales commitment to marketing could be time to follow up leads. Marketing and sales goals should both be aligned to the same business targets to ensure both departments speak the same KPI language.
# Alignment initiative: sales integration
One effective way for marketing and sales to align is to work together on common projects and for marketing to include or integrate sales within marketing initiatives. This could be at an activity level such as 1) co-creating content, 2) co-designing campaign focus, 3) in execution process or design, or 4) in promoting and marketing. In co-creating content or designing marketing campaigns, the focus can be in providing customer input indirectly from sales to support more relevant messaging, as well as using customer language.
Figure 20.2 Key areas of alignment
Key areas of alignment
Regarding 4) promotion and marketing, some organizations include sales people who are also specialists and/or influencers in the market. Other ways are to integrate sales in advocacy programmes for social marketing and brand building or, at a marketing vehicle level, by inviting sales to participate in an event, either as speakers or to support customer networking.
Having sales acting as part of the marketing team helps marketing through additional support as well as gaining further buy-in to other departments. Sales gains more understanding and appreciation of marketing efforts and can influence marketing to optimize their activities to improve impact and ROI.
**EXAMPLE** Atos
In February 2014, Catherine Howard stepped into the role as Atos UK Marketing Director. It was quickly apparent that there was a lack of alignment between marketing and business stakeholders and a lack of buy-in to marketing and its deliverables.
The process for turning around the marketing department kicked off with reviewing attitudes towards marketing and past delivery of marketing; this was captured through various research and surveys: one-to-one meetings with key business stakeholders including United Kingdom and Ireland board members, client facing team meetings, one-to-one meetings with the marketing team, and an online survey across the wider organization.
The investigation surfaced underlying issues which needed to be addressed; the marketing department led by Catherine developed a business-led marketing strategy focused on three core areas all supporting the objective of externally delivering business value. The plan was presented to the board and marketing secured an increased budget for implementing it.
To enable the implementation, the new marketing team was restructured and people and activities were better aligned to business and objectives. The new structure provided each key business stakeholder with marketing contacts; and behind the scenes stakeholder plans were developed to ensure each marketing team member was engaged with his or her respective business stakeholders in a regular and appropriate fashion. This new approach improved accountability and response rates within marketing and helped establish a new team collaborative culture between marketing and the rest of the business. Accompanying the team reinvigoration were daily guidance from the marketing leader, improved communications in the form of weekly calls, regular one-to-one meetings, pay realignment, as well as celebration of marketing successes both at an individual and team level.
Following the implementation, the marketing culture was transformed and team morale issues addressed and improved; additionally, new process and business leading programmes were introduced to support the business. The marketing team could deliver a much-improved return on investment and demonstrate a contribution to Atos’s business growth; the marketing team also received consistent praise from key business stakeholders.
ACTIVITIES
Based on the different types of marketing – sales enablement, brand building and lead generation – what percentage of time do you or your marketing department spend on each type of marketing? Is the allocation of time aligned to business requirements?
Conduct a mini survey with sales to understand their perception of marketing. Where do they see value-add and where do they see a need for improvement?
# References and further reading
Aberdeen Study Group (2011) Introduction to integrated marketing, Aberdeen Study Group, available at: http://docplayer.net/7387863-Sales-and-marketing-alignment.html (accessed 10 February 2017)
Alterra Group (2011) Account-based marketing: an approach on the rise in professional services – initial survey results, available at: http://alterra-aroup.com/wp-content/uploads/2011/04/AlterraGroupABMTopLevelResults.pdf (accessed 10 February 2017)
MarketingProfs (2015) In search of sales and marketing alignment, The Marketing Advisory Network, available at: http://marketingadvisorynetwork.com/2016/03/31/4173/ (accessed 10 February 2017)
Setty, R (2013) How to get started on marketing and sales alignment, Sirius Decisions, available at: https://www.salesforce.com/blog/2013/10/align-marketing-sales.html (accessed 10 February 2017)
# 21
# Account-based marketing
This chapter will give you an understanding of:
* the importance of account-based marketing (ABM)
* types of ABM programmes
* ABM process
* preparing for ABM
* challenges and pitfalls in ABM
* ABM technologies
* measuring ABM
# Definition
Account-based marketing (ABM), originally known as ‘key account marketing’, is a very strategic approach to marketing to companies. The philosophy of ABM is to treat an organization as a set of key individuals and decision makers to market to in a targeted fashion, rather than marketing to organizations as a whole hoping the right message will reach the right stakeholders. ABM is about targeting specific accounts with a company’s value proposition and relies on the vendor aligning its processes, resources and departments with those specific accounts.
ABM has been around since the mid-1990s and was mainly used by larger companies. It was originally part of a high-touch process by organizations’ sales teams to focus on accounts rather than generic marketing. HP and Xerox were notable early adopters. Today, thanks to new marketing technologies, ABM can be used by almost any size of business; smaller companies with very limited business and technical resources can target, assign and manage hundreds, even thousands, of prospective accounts and customers. They can keep track of their customers, segment lists and target accounts with personalized content for e-mail nurture campaigns or for generating new interest.
# Benefits of ABM
ABM is popular because it drives higher response rates; it is also a great way to bring together sales and marketing and for creating alignment between them. Fundamentally ABM focuses on the customer and means a business customizes its approach and messages to customers as it improves relevancy of sales.
Some insights from recent research have shown that 97 per cent achieved higher or much higher ROI with ABM than with other initiatives (Alterra Group, 2011), and 84 per cent found ABM provided significant benefits to retaining and expanding existing client relationships (Newman, 2016).
# ABM types
There are five distinct types of ABM:
1. *Strategic ABM.* A form of ABM that is usually reserved for strategic accounts and/or large accounts, and typically includes one-to-one marketing. It can involve a dedicated marketer and a dedicated customer account manager. Performance metrics and ROI are typically focused over the long term due to the nature of the account and type of marketing activity associated with it, for example hospitality, collaboration events and co-marketing are typically associated with large, strategic accounts.
2. *Segment-based ABM.* This is aligned to customer segments or customer sub-segments, eg targeting mid-market accounts per criteria of purchasing power, behaviour, or potential.
3. *Vertical account ABM.* Oriented towards public or private verticals; for example, where a company is looking to target senior clinicians for healthcare-related products.
4. *Campaign-based ABM.* Sometimes called programmatic ABM. Campaigns including messaging and themes oriented towards a set of accounts established through research; may involve some in-depth research that uncovers new insights into customer concerns. Campaign-based ABM usually has a set of content and vehicles behind it.
5. *Product/solution ABM.* Related to buying decisions; can sometimes be referred to as lifecycle as it’s based on the purchase lifecycle of customers. Examples could be previous product purchase and associated purchase behaviours centred on a group of accounts.
# The ABM process
The following framework, also shown in Figure 21.1, will allow organizations to effectively set up and implement ABM. Figure 21.2 shows the process and key applications.
1. *Strategic alignment.* During this step it’s important that business objectives are understood in terms of growth areas and customer focus, as well as marketing’s role in supporting such goals.
2. *Account analysis and identification.* The next stage is to build customer intelligence and profiles and use insights from sales about existing customers related to target customer segments. Target accounts are identified, working closely with sales based on a combination of sales input and third-party databases; predictive analytics may be used to determine the specific accounts most likely to buy.
3. *Define scope.* Scope of activities and size of accounts are defined and account lists are created. The scope will depend on sales force size, sales objectives, and ability of marketing to support sales with, for example, content.
4. *Asset and activity preparation.* Marketing builds assets and information for account-based initiatives; examples are qualification scripts, white papers, associated infographics and associated promotions.
5. *Launch ABM.* Account data is loaded, typically into a CRM application. The ABM activity is launched according to scheduled timings. Coaching or training accompanies this step to support sales in using marketing assets and tools.
6. *Execution.* During this phase sales follow up on leads or make outbound calls for the pre-defined set of accounts.
7. *Measure and review.* KPIs are reviewed for each of the activities according to the accounts targeted. Corrective actions are taken where KPIs are not achieved; eg where follow-up is behind plan sales executives may act to ensure greater focus during a given period.
Figure 21.1 ABM process
ABM process
Figure 21.2 The ABM process and key applications
The ABM process and key applications
# ABM methods
* Outbound ABM is most commonly used by larger businesses in relationship marketing; it is about leveraging marketing tactics such as e-mail, phone, direct mail and events to engage a target set of accounts for specific messages and content.
* Display-based ABM involves lining up ads so they only appear to people at specific companies. Some account-based advertising networks go one step further in targeting by role or title.
* Social is a great way to optimize ABM activities. By using social listening programmes, companies can gain insights into target account needs and pain points. Leveraging such insights, companies can post relevant content on social media sites, answering directly those customer needs. Using LinkedIn and Facebook, organizations can run very targeted adverts, down to named individuals of companies. LinkedIn allows for targeting adverts to people in companies according to role or job title.
**TIP** Where do you find account information for ABM?
One core success factor for ABM is correctly targeted accounts based on analysis and insights and can be drawn from different areas:
* *Past purchase and portfolio gaps.* These can be drawn from previous purchase history and can highlight purchase gaps, eg where only a few of the products/services are being purchased from a larger product portfolio.
* *Account propensity modelling.* A more sophisticated approach is to take multiple criteria such as sector, size of company and past purchase history, and use them to create a more accurate set of accounts based on richer detail. The idea is that a vendor can identify accounts with a higher propensity for purchasing its products or services.
* *Account plans.* Account manager input can help with customer insights and highlight likely prospects for targeting with additional products or services.
* Other sources of information include external telemarketing and data research companies.
Here are some key success factors drawn from companies that have demonstrated best practice:
* *Sales inclusion.* Co-leading or involving sales in the ABM process up-front can help with improved selection of customer accounts and improved focus in follow-up.
* *Insight-led.* Having some level of insights clearly helps in the identification process; although sales should provide help, this should be focused on criteria used for selection and informed rather than ad hoc input.
* *Business focused*. Having ABM programmes aligned to the business can help have the broader organization aligned to making ABM a success.
* *Programme design.* Programmes should be designed as a response to specific business challenges or need and thus add value. Content should be provided at different touch points for the intended target, off- and online.
* *Technology enabled.* Marketing integrating with CRM will help with reporting and tracking performance on a regular basis. Digital audience applications and social media applications can further complement ABM.
* *Communications.* How ABM is communicated to sales makes a difference; the purpose behind the activities needs to be emphasized.
## Challenges with ABM
Despite the benefits of ABM, there are challenges that can hinder its effectiveness. These include lack of sales buy-in where sales are not fully aligned to the ABM activity; this could be due to poor communication or lack of agreement on target customer and type. Other factors can be resource based; for instance the number of accounts delivered to the business can exceed the capacity of specific sales people to cope with the volumes assigned to them.
# Is ABM right for you?
ABM is suited to companies that have either relatively few accounts they wish to target or where they have a target group of accounts with similar characteristics. The latter could be an industry sector or a department within a company from a specific customer segment. Organizations should have some form of tracking and monitoring to carry out ABM effectively; this means having a form of CRM in-house or help from third-party companies.
ABM is not right for companies that are looking to market to a wider or mass audience or where the audience is so broad no segment or characteristics can be drawn. ABM relies on the ability to target so platforms that have targeting capabilities should be used. For example, LinkedIn can target customers by size, type and stakeholder.
The success of ABM lies in the personalized experience it can offer to customers, allowing the creation of relevant content according to buyer stages. ABM is a powerful tool for serving the right message and content to the right customer at the right time. Figure 21.3 shows how ABM leverages content and content marketing to engage customers.
Figure 21.3 ABM and the buying cycle
ABM and the buying cycle
# Measuring ABM
ABM can be measured according to one or more of these categories:
* *Awareness.* Based on serving display adverts to targeted named people at the account, or monitoring according to named people’s web or digital engagement. This allows the vendor to understand which themes are important to the account and potentially allows sales and marketing to improve their efforts.
* *Consideration/evaluation.* Based on the middle of the purchase journey and the evaluation stage the account may be in, this could be response rates or feedback on different tactics, eg e-mail response or white paper activity.
* *Account purchase activity.* This is the frequency and scale of purchase, and what was purchased.
* *Account penetration.* A measure of the extent to which the vendor is engaged in the customer account, the pipeline the customer delivers and the percentage share-of-wallet it has with the customer account.
ABM measurement ensures every campaign or activity scoped against the account is tied to the account in measuring and monitoring.
**EXAMPLE** Business success through ABM – RingCentral
RingCentral, a leading provider of software as a service (SaaS) for business communications, recently found great success through implementing an ABM strategy. The challenge was that around a third of its inbound leads were missing key fields or had inaccurate data and thus could not be aligned to sales teams; also, 20 per cent of inbound leads couldn’t be matched to accounts. By using Leadspace Enrichment, RingCentral was able to add 80 new types of data in real time to leads, which in turn allowed it to qualify and route leads to sales. The results were an increased opportunity for conversion by over three times and the average sales price per customer rose by around 7.2 per cent. An associated benefit was a reduction in sales time to qualify leads.
# References and further reading
Alterra Group (2011) Account-based Marketing: An approach on the rise in professional services, available at: http://alterra-group.com/wp-content/uploads/2011/04/AlterraGroupABMTopLevelResults.pdf (accessed 10 February 2017)
Event Marketer (2016) Event Track 2016 Content Edition – experiential marketing content benchmarking report, EventMarketer.com, available at: http://www.eventmarketer.com/wp-content/uploads/2016/05/2016EventTrackExecSummary.pdf (accessed 10 February 2017)
Newman, D (2016) Why B2B CMOs Need to Know About Account Based Marketing, *Forbes*, 29 April, available at: https://www.forbes.com/sites/danielnewman/2016/04/29/why-b2b-cmos-need-to-know-about-account-based-marketing/#26b58e465ab9 (accessed 10 February 2017)
# Lead generation
This chapter will give you an understanding of:
* lead generation challenges
* types of leads
* leads and the buying cycle
* lead generation process
* lead generation tactics and strategies
* lead nurturing approaches
# Introduction
Lead generation is the bread and butter of any B2B marketer and the challenge as always is extracting better leads and getting better value from them. Lead generation includes anything that constitutes an opportunity to sell to a customer, whether in the short or long term. They can be described as outbound or inbound; early buying cycle, mid or late buying cycle leads; as well as top, mid or bottom of funnel.
## Difference between demand generation and lead generation
Sometimes lead generation is confused with demand generation. Demand generation covers all marketing activities that create awareness about an offering, company or industry and includes a mix of inbound and outbound marketing. Lead generation is a subset of demand generation. Demand generation is a marketing operation that serves to create a demand for or interest in your product. Lead generation, on the other hand, is used to collect specific information about potential clients, turning them into sales leads.
## Trends in lead generation
During the 1990s and earlier, B2B marketers used to rely on outbound telemarketing for generating leads. These leads were typically internal or external third-party sales with accompanying scripts for calling a list of customer contact names. Today, companies doing well with lead generation are leveraging digital a lot more: they are connecting both off- and online and incorporating new automation platforms to better track leads.
Lead generation has recently been influenced by changes in buyers’ journeys, by changes in how customers access and use information, and due to the increased availability of information. To be successful with lead generation today it’s important for marketers to take on board such changes as well as to capitalize on the breadth of lead generation vehicles and content formats that can be used across the customer decision journey. Five years ago social media was rarely cited as a means of generating leads; it has now matured in such a way that it can be very effective in helping marketers identify and capture new business.
# Challenges impacting lead generation
Below are the main challenges marketers and sales need to deal with regarding lead generation:
**Getting the right quality of leads**
Where the lead quality is not meeting sales requirements and leads are handed over before being ready for sales, this can lead to poor follow-up or impact sales productivity as sales are taking more time than they should in qualifying and nurturing the lead.
**Time to load**
If the time between lead capture and loading the lead for sales to follow up becomes unacceptably long, the chances are that the lead will be no longer viable.
**Late sales follow-up**
Where a lead is loaded on time but sales don’t follow up in a reasonable amount of time, the lead can go cold.
**Lack of a lead nurture process**
Not having a specific process in place to engage, qualify, follow up and move the prospect further through the funnel can mean that good potential leads can amount to nothing.
**Misaligned sales expectations**
If sales expectations regarding leads are not set correctly, for example on volume, timing or type of leads, this can quickly result in misalignment of lead execution and opportunity detection.
**Resourcing**
If marketing has misaligned leads and sales resources this can mean poor lead execution. For example, where lots of leads are loaded at any given time and sales are not equipped to follow up, a high percentage of leads could be abandoned.
# Types of leads
Leads can be categorized in many ways. In B2B the language used in defining leads is critical for talking the same language to internal and external parties. Here are the main categories of leads:
* *Marketing-ready leads (MRL)* are leads ready for marketing; unless it has been agreed to do so they should not be released to sales teams.
* *Marketing accepted leads (MAL)* can be the same as an MRL but are defined as those leads accepted by marketing; there can be a filter system only allowing through leads that adhere to a set of criteria.
* *Sales-ready leads (SRL)* are deemed ready for sales and for sales follow-up; depending on company’s internal policies they can be the same as BANT leads.
* *BANT qualified leads* are ready to be passed to sales per the BANT set of criteria (see below).
BANT criteria are used for assessing the quality of prospects and tracking them through the sales qualification process:
**B**: Budgeted, where the leads have been budgeted.
**A**: Authority, where the lead is tagged with contact name, details.
**N**: Need based, where need and requirements are associated with the lead.
**T**: Timescale, regarding the lead in terms of the opportunity.
# Lead generation and the customer buying cycle
In simple terms, we can think of lead type and status according to the marketing funnel; more appropriately should be mapped to the customer buying cycle.
The first are leads that relate to the initial stages of the buyer journey; they can be regarded as early buying cycle leads (EBL). Customers in this stage are aware but not yet in the evaluation phase. The next set of leads relate to customers who are aware of the company or need and are now evaluating a potential vendor; this should be thought of as mid buying cycle leads or MBL. During this phase customers have probably engaged with the vendor through some form of interaction with the vendor’s content, directly or indirectly. The final stage is almost at the point of purchase or late buying cycle. Internal organization approaches to lead management, handover and lead nurture will determine whether these stages can be managed by sales or marketing, internally or externally (see Figure 22.1).
Figure 22.1 Leads and the buying cycle
Leads and the buying cycle
# The lead generation process
The end-to-end lead generation process is summarized in the following main steps, mapped to customer buying cycle and typical marketing/sales process:
| | | |
| --- | --- | --- |
| Enquiries | early part of buying cycle | marketing driven |
| Lead capture | mid buying cycle | marketing |
| Lead pre-sales qualification | mid buying cycle | marketing |
| Lead handover | mid to close | sales and marketing |
| Lead sales qualification | close of buying cycle | sales |
| Sale | close of buying cycle | sales |
**Enquiries** are customers in the early stage of the buying cycle; they could be quantifying their need or evaluating vendors and alternatives. As business customers tend not to engage vendors directly, this stage of lead generation is typically led by marketing or third parties through content distribution.
**Lead capture**. Can be through multiple lead generation vehicles; eg events, content syndication, download of content.
**Lead pre-sales qualification**. Can be through third-party telemarketing agencies, internal market/pre-sales teams in nurturing, or through marketing automation.
**Lead handover**. During this step the lead is handed over to sales, who are responsible for carrying out the remaining stages of further qualification (if required), opportunity conversion and ultimately closing the sale.
**Lead sales qualification**. During this stage sales qualify and progress the lead.
**Sale**. This is the final stage where the customers make the purchase.
## Lead capture
Lead capture can occur through multiple different marketing vehicles or a combination of them; the main ones are door openers, webinars, e-mail with call-to-actions, events, outbound calls and content syndication. Content downloads are in the form of infographics, access to information, white papers or other, SEO, paid search, PR, social, direct e-mail, advertising, mobile and print advertising.
### Multiple sources and lead quality
As customers will use multiple pieces of content across different channels before engaging a vendor it’s important to consider the mix of lead generation tactics by campaign or customer type. Touch points are where customers interact with a vendor directly or indirectly. Multiple sources will tend to lead to a better output in terms of customer and lead capture.
### Gated content and engagement
Lots of companies still ask interested parties to share their details prior to sharing content; it’s common in B2B marketing. The risk with making content ‘gated’ is that a customer is not prepared to fill in long forms to get something in which he or she has a passing interest. Customers with a definite interest are not likely to be put off. Companies can get round the problem by either asking for very few details or providing automated field filling to speed up the process.
## Lead qualification, handover and scoring
A lead is usually qualified against a set of criteria, according to marketing or sales requirements. The quality of these leads can be very different.
The sales handover within the lead generation process is important and lead management disciplines need to be adhered to. On average, less than a quarter of the leads on a company’s website are ready for sales representatives to follow up; that’s why lead scoring approaches are great, followed by lead nurturing. This allows relationships to be built with qualified prospects who may not be ready to speak with sales. The lead scoring systems allow the marketing department to understand which are the right leads to handover.
Handover can occur in different ways, according to the type of sales person and whether the customer is early or later in the buying cycle. The type of sales person can impact how leads are received; this is not about competence or ability but aligning the right lead with the right sales person. Additionally, where a lead requires a sales specialist but is handed to a generalist, this can quickly kill it off. It’s therefore important to categorize leads by richer details and according to different sales types.
It’s important not to pass leads on too early. If the marketing department feels pressurized to deliver on volume and not quality this can lead to poor quality leads being passed to sales. This can backfire on the business in terms of wasted effort, budget and time for both sales and marketing.
Figure 22.2 shows different views on lead scoring.On the left is scoring leads according to cold, warm and hot, which is about capturing and logging leads according to the stage of the buying cycle. Cold leads are very early leads and hot leads are late in the buying cycle, ready to be handed over to sales. In the box on the right, leads are categorized by propensity to purchase and stage in the buying cycle. The hottest leads are A1, A2 and B1.
Figure 22.2 Lead scoring
Lead scoring
# Lead generation tactics and strategies
Here are some of the main tactics and strategies used in lead generation:
* *Sales handover adaptation.* Typically the lead generation process involves passing leads from a lead generation source, eg a telemarketing agency or content syndication company, to marketing and then to sales who may pass them to a different department of sales. This process can quickly expand to having multiple handover points, which prolongs the time it takes to progress a lead and runs the risk of the lead growing cold. One approach to improving effectiveness is to reduce the number of handover points during the lead generation process.
* *Funnel focus.* Depending on marketing department objectives there may be a focus on improving leads according to different areas of the funnel.
* *Website capture improvement.* Optimizing website pages for receiving leads through aligning content to buyer journeys as well as offering information or content alternatives for customers can help them along their buyer journey.
* *Lead nurture alternatives.* Rather than focus on one method, businesses can develop different ways to nurture different campaign leads through a mix of telemarketing, in-house nurturing, automated nurturing or media-based nurturing.
* *Reducing the cost per lead.* This can be achieved by putting out lead generation to tender, requesting more competitive quotes.
# Lead generation agencies
Lead generation agencies come with greatly varying competencies. If the marketing department isn’t responsible for the agency relationship, it’s important to review lead quality through conversion and close ties with sales.
## How do you select and manage a lead generation agency?
In selecting a lead generation agency the marketing department is now the potential customer. Ways to understand and select lead generation agencies can be through forums, word-of-mouth from contacts in similar roles in the industry, referral from a different type of agency, and from online searches.
Lead generation agencies should be managed on the basis of a clear briefing. The briefing should clearly communicate needs, expected quality and quantity of leads, expected cost per lead and contingency arrangements, for example where lead quality doesn’t meet desired level. Outputs from lead generation agencies can then be monitored against this brief. Pipeline may be a difficult measurement, as the agency won’t know the level or scale of opportunity in terms of revenue before taking on the role.
**TIP** How to manage and correct poor lead conversion
One of the recurring and frequent complaints from marketing departments is the conversion levels of leads. Below are some key steps that have been effective in resolving the issue of lead conversion and improving the quality of conversion:
1. *Map out the process in detail.* The current process that has led to poor results should be mapped out from lead capture through lead qualification and opportunity identification to deal closure. Each point in the process should be labelled accordingly.
2. *Look at conversion points* in the process to understand which metric is underperforming. Lead volume, marketing to sales accepted lead, sales lead to opportunity, pipeline creation or deal closure are all conversion points but it may be that only one of them is underperforming.
3. *Analyse further the conversion point* not meeting its KPIs; eg if the opportunity conversion rate is low, it could be that the quality of the lead is poor.
4. *Find the root cause of the impact.* For example, if opportunity conversion is the issue, is the cause a sales person or group not converting enough? Are they distracted with other demands? Opportunity conversion can also be impacted by timing of handover, type of lead, type of sales person, or experience of sales person.
5. *Assess* possible corrective actions and implement the most appropriate.
6. *Review* corrective action, based on implementation and new metrics.
# Lead nurturing
Lead nurturing is the process of developing relationships with buyers at each stage of the sales process and the buyers’ journey. It can be conducted in a variety of ways, depending on type of customer and customer segment. There are four main mechanisms to lead nurturing, shown in Figure 22.3:
**The telemarketing agency’s role** can be to qualify leads and nurture to a stage in the buyer’s journey. The competence of the agency and how marketing and sales expect to receive opportunities will determine to what stage a lead is nurtured. The benefit of this approach is that organizational resources are not tied up with lead nurturing.
**In-house team**. An organization may assign a person or people the task of qualifying leads or may establish a different team to nurture leads. This approach may be useful where the product or solution in question is complex and thus requires in-house expertise and investment in training. The costs for lead nurture are only the salaries of the in-house team members; another benefit is control over the lead’s progress.
**Media agencies** capture leads and can nurture them through retargeting; such leads may still be passed to telemarketing agencies but be managed by media agencies.
**In-house automation** is managed by businesses that may decide to take captured leads and pass them through a marketing automation technology application such as Eloqua or Marketo to nurture them further.
Figure 22.3 Lead nurture mechanics
Lead nurture mechanics
**EXAMPLE** Crowe Horwath lead nurturing
One example of a lead nurturing programme is that of Crowe Horwath, an accountancy firm, which developed a programme covering a 12–18-month sales cycle. It involved targeting C-suite executives of large financial institutions. The programme included 48 pieces of content aligned to the customer decision cycle, intended for 4,000 executives. Executives were sent monthly e-mails offering free content and including them in a nurture programme. Based on the downloading of three pieces of content or one piece of content towards the end of the decision journey, the leads were regarded as sales-ready. This programme resulted in 33 per cent of invited executives entering the programmes and up to 80 per cent open rate for nurtured e-mails.
# Further reading
Kirkpatrick, D (2015) Lead nurturing via e-mail series and content marketing, B2B lead blog, available at: http://www.B2Bleadblog.com/2015/01/lead-nurturing-via-email.html (accessed 8 February 2017)
# 23
# Modern B2B events marketing
This chapter will give you an understanding of:
* B2B events types and segmentation
* events objectives and events selection
* event marketing and business buying stages
* key success factors for B2B events marketing
* event marketing and performance optimization
* social media and events
# B2B events marketing
A marketing event in B2B is an activity revolving around a themed display or presentation leveraging people engagement, either virtually (via webinars) or physically. Events can occur on- or offline and can be participated in, hosted, or sponsored.
## What are the benefits of using events?
Events can take a lot of time to organize, and are often more costly than other marketing vehicles. They involve a lot of effort in collaborating with agencies, third parties and other functions in a business.
However, they are extremely effective in marketing and ultimately selling a company’s products and services. Where a businesses’ portfolio is broad or complex and requires more than a few minutes to pitch, events can be an effective means of explaining more complex solutions. When done well events are also great incentives for potential customers, whether it’s down to the opportunity to network and meet peers or discover new information. Events also help businesses stand out from the competition.
With more and more information accessible online for business customers during all phases of the buying cycle, events hold the opportunity and advantage of being a different communication channel that offers a more powerful engagement approach than online forums. Events also allow business vendors to engage customers via third-party conferences, collaborative events through sponsorship opportunities as well as market stall type events where vendors can pitch their portfolio to a large throughput of people. Webinars allow vendors to engage customers digitally, again through third parties if early in the buying cycle or directly later in the cycle where customers require more information in the evaluation phase.
Companies that run their own events have the advantage of greater control and influence over the event’s scope, agenda and communication. Participating in a third-party event means a company can engage with customers at a lower cost. According to a study by the Event Marketing Institute (2016), 98 per cent of the respondents said that assuming the product or service promoted was one they were interested in, participating at the event or experience made them more inclined to purchase.
# B2B event types and segmentation
Events come in numerous forms so it makes sense to segment them. They can be divided into virtual events (via the internet), face-to-face (in person) and voice-to-voice (using telephone and screens). They can also be segmented into company’s own and third party, and customer orientation: by vertical, stakeholder or sector segment. In this case third-party events are run by a separate organization that subsequently invites different organizations to participate, rather than being appointed by one particular company. Voice-to-voice or webinars can be great ways to achieve customer reach, eg where customers are dispersed across a geography. The main types of events are as follows:
* Physical trade shows or trade fairs are exhibitions organized around a specific industry and allow companies from that sector to showcase their latest products and services. They tend to be large and include a range of businesses involved in the sector whether suppliers, customers, partners or channel partners.
* Seminars are usually educational in nature and focused on specific topics or themes; they are usually for groups of less than 50 and there to update and educate audiences. Companies sometimes use seminar-type events to engage new customers and to talk about topics related to their service offering.
* Product launch events are geared exclusively to the launch of a new product or service.
* Networking events are set up to allow business people to network with one another; event owners can engage and connect with people.
* Webinars are presentations, lectures, workshops or seminars transmitted over the web using video conferencing software. A key feature of a webinar is its interactive elements: the ability to give, receive and discuss information in real time.
* Workshops are like seminars but much more interactive; they involve audience participation.
* Networking dinners and lunches are designed to bring a small number of people together; they are typically there to allow vendors to engage more intensively with potential customers. They can also serve the purpose of retaining customers and help build or reinforce relationships.
**EXAMPLE** SPE Offshore Europe
One great example of a third-party vertical event is the SPE Offshore Europe, recognized as one of Europe’s leading events covering the oil and gas industry. It takes place once a year in Aberdeen. In 2015 it attracted over 56,000 customers, over 1,500 exhibitors and at the event over 100 countries were represented. The event is the main go-to place for business professionals to learn the latest about oil- and gas-related trends, products and services. From a marketing perspective there were over 60 media association supporters who attended, more than 90,000 database contacts captured, and over 17,000 social media followers. Based on a survey of the 2015 events, over a third of exhibitors said they received orders worth between £50,000 and more than £10 million as a result of the event; at the event over 120,000 leads were generated and 24,000 leads were generated online.
# Events marketing goals
There are multiple goals for events. A goal can be to introduce a company to new potential customers set up by a third party. Events can also serve to introduce existing customers to new portfolios, ie where a company gives the customers the opportunity to see new products or solutions first-hand or for organizations to be able to present and pitch new solutions more effectively.
A goal could be to deepen engagement with existing customers. Events are a great way to demonstrate an extended portfolio and discuss ways to improve business with customers. The events will vary, depending on existing customer relationships, the nature of the business and the number of customers. Events can be used as opportunities to surface and resolve any final questions or queries customers have before they purchase. They may be a way to draw customers away from alternative vendors and convert customers while they are in the consideration phase.
Physical events can help vendors reach customers in a particular location, or can span multiple regions. Vertical trade shows or conferences may be more appropriate for engaging potential customers from a vertical sector; annual sector-specific conferences or exhibitions can be an effective way to engage all involved in that sector.
# Events marketing and business buying stages
Events can support a business throughout a customer’s buying cycle in different ways; they can help customers move from one stage to the next. Some of the types of events and event implementations, shown in Figure 23.1, are:
**Need recognition**. Customers and organizations can stay in touch through themed events and trade shows; business can reach a mass of customers some of which use events to keep themselves up to date. Events also allow customers to discover new ways of doing business and optimizing their business and therefore recognize different needs.
**Need quantification**. Webinars and informal events allow customers to engage and further discuss a challenge, providing them with the necessary tools and information to quantify a need internally.
**Vendor evaluation**. In the vendor evaluation phase, vendors may indirectly target a type of customer through sponsoring or participating in events.
**Vendor selection**. Between consideration and purchase, organizations can take advantage of events by inviting customers who may have downloaded white papers or other content or made contact by phone. The event invitation and attendance gives the vendor the opportunity to promote its offering and value-add.
**Post-purchase**. Events or webinars in different forms can be used to reinforce relationships with customers as well as continue to communicate value propositions; hospitality post-event allows businesses to stay in touch with customers and build stronger relationships.
Figure 23.1 Events and the buyer journey
Events and the buyer journey
# Events marketing problems and success factors
## Problems
Events marketing performance can be affected by multiple aspects. An event agency may not capture all leads and contacts due to insufficient event staff being in place, or missed opportunities at the point of entry. If the event agency doesn’t hand over contacts or leads in a timely fashion, the leads can go cold before the vendor has a chance to follow up.
A company can quickly change priorities or marketing change their priorities or focus; where this is the case captured leads may be neglected. Where event leads are not assigned correctly, the quality of follow-up can be adversely affected. Finally, the customer opportunity may not be correctly captured, resulting in a poor sales follow-up.
## Key success factors
Below are some of the main factors that contribute to the success of events marketing:
**Data capture**. Ensure that information can be captured and is captured about customers before, during and after the event.
**Plan**. Allow time to set up and implement; with short lead-times companies can falter by finding ideal venues booked up for the best dates.
**Allow for customers’ timelines and schedules**. Plan in advance to allow for potential attendees having calendars booked up months in advance. Notifying customers a few weeks prior to the event won’t work.
**Know the events market**. Avoid setting up events on dates where competing industry or trade events are taking place.
**Set realistic attendee targets**. A number of customers will drop off between the registration stage and the date of the event; industry standard drop-off rates are around 50 per cent.
**Venue selection**. The venue needs to suit the type of event, audience, audience location, etc. The right venue can have a major impact on the success of the event and therefore its goals.
**Follow-up**. Provide customer contact information and leads from events within a few days of the event.
Capturing data prior to the event applies to customers that are more likely to attend. Where the event is intended to be split by region, the data can be used to understand customer location and invite according to a maximum distance to the event. Attendee details can be gathered along with comments and opinions on presentations, content and speakers. Audience ratings of the event overall will provide feedback as to what can be changed or adapted should the event be repeated.
**TIP** How to engage customers before and after events
The mistake some companies make is to view an event as a single activity and opportunity to engage customers, thereby missing out on potential opportunities to connect with customers before and after the event. Look at the pre- and post-event activities in Figure 23.2. Organizations can use events to inform, educate and interact with customers, through surveys, through Q&A, through sharing pieces of content or research. This can be done on the company website, social media platforms or through advertising.
Figure 23.2 Events pre- and post-engaging the customer
Events pre- and post-engaging the customer
Following the event, organizations can connect and follow up with customers by sharing related pieces of research or content, or by helping them in their evaluation of current challenges. Post-event, companies can use this interaction to acquire, retain or grow business with the customer or prospect.
**TIP** How to maximize attendance and reduce ‘no-shows’
Organizations can maximize attendance by sending event invitations well in advance, helping customers to plan better around the event. Before sending the invitation organizations or agencies can contact the customer directly to explain more about the event; this is easier where the intended audience is existing customers or where the organization has some form of relationship with them. Reminder e-mails help prompt the customer, though organizations need to be careful not to send too many reminder e-mails too often.
There are two different views on charging event fees; on the one hand they help lock in the audience and therefore they are less likely to not show up; on the other hand, it can put off attendees as they feel the event shouldn’t be charged for.
Some vendors involve customers in the event whether providing a stand or booth or as part of the presentation.
Events can be promoted in different ways to increase exposure or targeting; some of the ways are to use telemarketing agencies, PR and the company’s own websites and social media sites.
# New events technologies
New events technologies and digital applications can help B2B marketers maximize and optimize events. With the help of near field communication (NFC), event organizers can make their events much more interactive. NFC-enabled devices come in different forms such as smartphones, barcode readers or tablets. NFC technology can be activated by chips in attendees’ wristbands or badges. Terminals with NFC readers could encourage attendees to take a picture and automatically share it on social networks, for example. NFC devices are perfect voting and rating tools and could be used to determine the most desired product at trade fairs.
Virtual event platforms allow for virtual participation so customers can attend events without leaving their desk. Webinar platforms allow businesses to provide presentations with one-, two- or three-way communication, eg vendor to customers, customer interaction or between customers. Registration platforms can help you quickly set up a webpage and then invite attendees; set-up can be done in minutes, and the platform can help with reminder e-mails. Some platforms are free.
# Social media and events marketing
Social has become a powerful tool for B2B marketing in the past decade. For events marketing purposes social can be used to maximize the event before, during and after (see Table 23.1):
Table 23.1 Social media and events
| | **Before** | **During** | **After** |
| **Twitter** | Great for spreading news, can be good to promote to Twitter groups or specific members | Most interesting as Twitter is more for real-time occurrences at event, photos | Great to talk generally about event |
| **Facebook** | Nice way to promote through graphics | | Share stories about event |
| **LinkedIn** | Great for advertising and promoting Via InMail most targeted way | Not easy to use in real time | Reports and overviews can be shared on forums, company LinkedIn pages, etc |
**Before the event**
Twitter can be used to promote events and to increase exposure through likes, shares regarding the event and related topics. LinkedIn and Facebook can also be used to highlight an event.
**During the event**
The most commonly used social platform is Twitter. By creating a Twitter handle or address linked to the event, information and comments can be provided and shared in real time. It’s important to build Twitter lists and for vendors to promote sharing by providing regular updates through the event to encourage usage and participation.
**After the event**
After the event LinkedIn and Facebook can be used to share stories or longer articles related to the event. Twitter can be used to direct people to LinkedIn and Facebook or other website landing pages.
# Maximizing event lead capture
Twenty years ago event leads capture was conducted manually, writing on paper and/or collecting business cards. Nowadays leads capture includes badge or barcode scanning during and after the event at exits and entrances. The main ways companies can capture event leads are:
* *Badges with barcode and barcode scanning.* Event personnel use barcode scanners, usually at the entrance to the event.
* *Use of PCs or tablets.* Tablets are used to log in audience details, usually at event entrances or reception.
* *Business card collection and scanning electronically.* Business cards are collected manually and then scanned in or manually inputted into a database.
* *Surveys.* These can help host companies get feedback and at the same time understand attendance, providing a hard copy back to attendees.
There are three considerations when selecting the best way to capture leads:
1. *Number of customers.* Where there are few customers it might be better to collect business cards and manually input them rather than invest in using technology.
2. *Speed and ease of registration for visitors.* Visitors may find data capture a lengthy process and be put off.
3. *Data protection.* The form and process of lead capture must comply with data protection legislation.
ACTIVITIES
Review your current marketing activities in engaging customers face-to-face (physical events) or voice-to-voice (webinars). How are you using events to support customers during the buying journey and where are there gaps?
Look at the key success factor list in this chapter and look at where there is potential for optimization.
# Reference and further reading
Event Marketing Institute (2016) Event track 2016 content edition – experiential marketing content benchmarking report, eventmarketer.com, available at: http://www.eventmarketer.com/wp-content/uploads/2016/05/2016EventTrackExecSummary.pdf (accessed 10 February 2017)
SPE Offshore Europe (2015) Post show review (conference and exhibition), SPE Offshore Europe, available at: http://www.offshore-europe.co.uk/RXUK/RXUK_Offshore-Europe/documents/RXOE8255 per cent 20Post per cent20Show_V6_final.pdf (accessed 10 February 2017)
# 24
# Modern marketing operations
This chapter will give you an understanding of:
* marketing data management
* marketing budget management
* marketing reporting and measurement
* metrics – overview and model
# Marketing operations
Marketing operations serves marketing and the business. It covers a set of activities that support efficiencies within marketing departments in their delivery of value to the business and in demonstrating that value. Marketing operations is the backbone behind marketing departments that allows it to function to its maximum capacity. It also acts as a control group to assess how marketing is performing and to identify areas to correct. The main areas it covers are:
* managing a set of activities that support demand generation;
* managing marketing infrastructure;
* budgeting and planning; financial governance and reporting;
* campaign analysis and reporting;
* customer, market, competitive intelligence, research and insights;
* data management;
* technology, automation and pipeline management.
In large organizations, these may be covered by entire departments and teams, usually centralized to avoid duplication of resources. In smaller companies this can be part of someone’s role or subcontracted to an external company, depending on budget and the scale of work involved.
Figure 24.1 MPM areas
MPM areas
# Marketing data management
Data for any marketer is critical in allowing them serve the business, in marketing, in tracking marketing performance and reviewing effectiveness of marketing. Marketing data can be split into three main areas.
1. *Customer data.* Customer data helps profile customers, identify buying behaviours and understand how to engage them, eg by phone, e-mail or other forms.
2. *Marketing campaign data.* This refers to the tracking of leads by activity and campaign, and helps marketers demonstrate value-add to the business; it can also help track effectiveness of budget spend.
3. *Market data.* Includes market trends, segments, growth, size and competition.
Data can also be split according to the source of the data, for example:
* *First-party data.* Data from proprietary assets or contained in enterprise systems.
* *Second-party data.* Someone else’s first-party data, often shared with another company in a mutually beneficial relationship.
* *Third-party data.* Consolidated from websites and platforms other than own data.
## Capturing data
Capturing data or insights is often seen by a business as one of the great value-adds of a marketing department; the department provides a view of markets, customers, competition and other macro dynamics that may not be seen or captured from ongoing business activities or conversations with existing customers. Marketers need to be able to look at different capture mechanisms, understand how to track data, and monitor, manage and present it.
Customer data can come from different sources such as sales account plans, including customer needs, challenges, upcoming projects, and anything related to the account; account plans can also help marketers understand which messages to deliver to address the latest needs. Sales account plans can be loaded to CRM applications where they may be more readily accessible. Surveys are a great way for marketing departments to reach out to potential customers to capture interest. Marketing events and webinars are useful to capture input through networking with customers, as are surveys carried out at events.
From digital traffic, eg customers downloading documents from the company website or syndicated content, information can also be captured. Customer preferences regarding topics and themes can also be tracked through website analytics highlighting most-clicked topics on website pages. Finally, customer data can be purchased from different companies such as Dunn and Bradstreet, Equifax and Experian.
**TIP** The new digital applications for tracking customer behaviours
Social monitoring or analytical tools can help monitor conversations and key words; they can aid marketers in assessing what people think about products, customer service experiences and corporate developments. These tools can also help to monitor competitors, to identify pain points or trends based on chatter themes, and identify influencers. There are many social listening tools; examples are Hootsuite, Radian6, Alterian, Sysomos and Hubspot.
Data management platforms (DMPs) are being increasingly used by B2B marketers; they draw data from internal and external resources and help in personalizing online channels and content for customers. DMPs can be used to drive website and mobile personalization, display targeting, content targeting, e-mail and direct e-mail campaigns and can determine where targeted or frequency of messages drives higher response rates. Examples of DMPs include Bluekai, Mediamath and Turn.
Marketing campaign data can be captured through automation platforms to track response rates on tactics, to track marketing performance, and content click-throughs. Alternative approaches are to use CRM applications to track lead-based marketing activities that eventually require sales follow-up and separately use web-based analytics programs to track digital and online marketing activities.
Market data such as trends and segments can be purchased from market research companies. Alternatively, companies can use their own market research methods such as surveys, forums, and focus groups that potentially leverage sales.
## Data accuracy and data integrity
Marketers constantly wrestle with maintaining accurate data. Data describing companies can become obsolete if they have undergone mergers or acquisitions, or where a company experiences growth moving it from a small business to being categorized as a medium one. At a stakeholder level within customers or prospects, data can be impacted as individuals either change roles within the company or move outside the company.
For small business data accuracy can be maintained by using data houses whose responsibility it is to maintain that accuracy. Smaller organizations can insist that account managers ensure data and profiles are kept up to date. Larger companies can look to sales operations to ensure all existing data is maintained in its CRM systems and via marketing complement data through external data sources. Additionally, software can be used to merge data or capture inaccuracies within it.
‘According to one survey, 55 per cent of respondents had been sent information about an irrelevant product by a business in the previous 12 months,’ says Nigel Turner, VP of information management strategy at Trillium Software (Wickey, 2016). A large proportion of customers are annoyed when businesses get their personal information wrong. This is the consequence of bad data; vendors can lose credibility if they use the wrong names, titles, etc.
## Data protection and marketing solutions
Over the past decade, marketing and data privacy laws have been implemented such that customers now can opt out of e-mails or other forms of communication and thus control what you send to them. This means that marketers need to think more about tailoring content to customers’ needs.
## E-mail marketing and data
The term ‘soft opt-in’ is sometimes used to describe the rule about existing customers. The idea is that if someone bought something recently and provided their details but didn’t request opting out of marketing messages, they are probably happy to receive such messages about similar products. This doesn’t apply to prospective customers: organizations need to provide the customer with the opportunity to opt out of e-mails.
**TIP** How to you improve your contact strategy despite data protection restrictions
Where data protection influences contact strategy to such an extent, organizations can employ different options to engage customers:
* *Use different channels.* Where e-mail is no longer an option, organizations can look to social to engage the same target customers.
* *Request to opt in again.* Using a different route, an organization can offer a customer the chance to opt in to e-mails but based on selected topics or preferred themes.
* *Different contact route.* Where the contact person isn’t receptive, an organization can employ a different contact route through a different department or person.
# Marketing budget management
Many B2B marketers at some point experience the challenge of doing more marketing with the same or lower budget. As a result marketers need to be able to justify every part of their budget while optimizing and improving ROI, as well as looking for creative ways to further optimize their marketing activities.
## Marketing budgeting trends
Some recent trends impacting marketing budgets are the increasing spend on digital marketing. As a result of changing customer behaviours and how they use and access information, allocation of spend to digital channels is increasing.
Programmatic spend is on the increase, is growing more efficient and becoming more focused; it will help maketers target, retarget and engage customers. With increased competition, B2B marketers are looking not only for lead generation but also to position the brand and communicate its value proposition. Sales are demanding better and higher quality leads from marketing. Marketing are being asked to provide nurtured or more qualified leads.
With the need to pre-empt buyers in self-informing and deciding to go for a different vendor, organizations are looking to marketing to invest more in social platforms, social communications and content, so as to capture customers earlier in their buyer journey.
## Defining the marketing budget
There are different ways to calculate the marketing budget and budgets can vary according to the type of marketing, ie consumer or B2B. Typically, consumer marketing budgets can be substantially bigger than those of business marketing. Many businesses allocate a percentage of revenue for marketing, sometimes called the reinvestment rate – the amount of spend the business reinvests in marketing.
Allocation of budget will depend on a number of factors: size of business, growth stage, profit and margins a business is making. Other factors include industry type: in some sectors, B2B marketing is more about sales enablement and support. Marketing-led environments, as in B2C, rely on marketing to do the lion’s share of communicating to customers. Marketing will be expected to support a new strategic focus, for example if the business is to open up new channels or to maximize a product or solutions launch; to meet the temporary peak in marketing activity, budgets may be increased for a while.
## Marketing budget process
Once the level and size of budget have been set, marketers should plan their budget:
1. *Translate business objectives into marketing objectives.* Assuming the business objectives for the upcoming period are already in place, the next step is to define those objectives in marketing terms. If the business objectives are: a) to go for incremental growth, and b) positioning a new product or solution, the marketing objectives could be supporting messaging and communication of a new solution or product, and generating leads for the business.
2. *Prioritize and weight objectives.* Assuming the weighting and importance of a and b above is equal, the budget should be split between the two core areas. If this is not the case, it’s important to prioritize one over the other and agree in principle on a weighting.
3. *Define KPIs by marketing objectives.* Whether for internal briefing purposes or external with agencies, KPIs should be defined. They could be number of opportunities and leads; or objectives linked to customer reach or impressions.
4. *Allocate spend and mix.* Allocate according to priority and weighting of goals and then by marketing vehicle or tactics, considering key tactics to achieve each objective. For example, communication of a new product or solution could be split by social media, PR, advertising, banner, and maybe a launch event.
5. *Implement and review.* Implementation then occurs with time to review. The KPIs in step 3 can be used to see if marketing activity is performing as planned.
6. *Optimize budget.* Depending on ROI and performance against KPIs, budgets can be optimized by goal or tactic.
The budget process is shown in Figure 24.2.
Figure 24.2 Budgeting process
Budgeting process
**TIP** How to optimize marketing budgets
There are many ways to optimize marketing budgets. Costs can be reduced by reviewing the cost of agency fees if the company is using marcom, PR, or telemarketing agencies. The cost of content can be looked at in terms of production or creation cost.
Where ROI is low or contribution to marketing goals is not regarded as effective, marketers can remove some tactics or activities. Marketers should be cautious in defining which activities to remove and in assessing their true performance in terms of relevant KPIs. They may need to allow time for an activity to take effect – acquisition marketing and brand building can take time to implement and demonstrate ROI.
Reducing priorities and objectives is another way to optimize budgets. Marketers can try to do many things, supporting many objectives, and end up with multiple priorities. Reducing priorities, improving focus and aligning marketing resources can lead to improved ROI.
Another possibility is to increase/reallocate the budget. For instance, regional marketing can request more budget from a central marketing function; or the regional marketing department can engage a local partner that will provide additional funding.
Changing the marketing mix to shift from high cost marketing tactics to lower cost ones can help. For example, where a B2B marketing programme contains a lot of video content that eats up a disproportionate amount of the budget, a marketer could consider moving to lower cost marketing channels. Marketing could also look at reducing the number of pieces of content per buying stage or reusing content over a longer period.
# Reporting and analytics
Marketing reporting and analytics provide a consistent way to measure and track business as well as different views of marketing performance. They facilitate marketing in improving performance as well as the business in justifying marketing investment decisions.
## Applications for reporting and analytics
In reporting and tracking performance two main types of applications are available to marketers: marketing automation platforms and CRM systems. Marketing automation platforms are still only used by a minority of marketers according to the Lead Generation Benchmark Report (Demand Metric, 2014), while half use CRM systems.
## Difference between marketing automation and CRM
Marketing automation allows companies to streamline, automate and measure marketing tasks and workflows. It allows marketing departments to increase operational efficiency.
Marketing automation spans several marketing areas including e-mail marketing, campaign management, lead generation, social marketing and much more; it can cover almost every marketing activity and can even integrate elements of CRM.
CRM can be defined as an application for managing all of a business’s interactions with current and prospective customers. Although very much a tool more used by sales, it’s also used by marketing to better track opportunities.
## What to measure
Marketing measurement systems depend on the type of customer segment, business model and marketing. For example, where marketing is supporting a high-touch sales model, measurement may include outbound metrics, use of content by customer journey, number of attendees at events, etc. Where marketing supports a low-touch sales model but high-digital-touch model the measurement may be purely based on digital KPIs.
Marketing measurement systems may also be heavily focused on the bottom of the marketing funnel where marketing’s main role is to support marketing execution, or weighted more to the top of the funnel where the main role is in positioning the brand and communicating its value proposition. In other cases, the marketing role may be a mix of awareness, consideration and purchase-based activities. Table 24.1 shows a summary of the types of marketing and relevant KPIs.
Table 24.1 KPI alignment and goals
| | **Type of marketing** | **Example KPI descriptors** |
| **Hi-touch sales model** | 1. Enablement
2. Outbound
3. Offline
4. Digital | 1. a) Sales participation, b) sales engagement
2. a) Account follow-up, b) opportunity, c) pipeline
3. a) Event leads, b) event opportunities
4. a) Website traffic, b) online downloads, c) online queries, d) digital leads from content syndication |
| **Digital intensive** | 1. Website
2. E-mail
3. Video
4. Other | 1. a) Website traffic, engagement, time on site
2. a) E-mail click-through rate, click-to-open rate
3. a) Video download, engagement
4. a) Digital leads, sales-ready leads, opportunities |
| **Top of funnel** | Awareness
Brand building | 1. PR circulation
2. Online traffic
3. Content engagement
4. Other brand KPIs |
| **Execution intensive** | Marketing execution – bottom of funnel | 1. Sales-ready lead volume
2. Opportunity volume and conversion
3. Pipeline
4. Revenue |
| **Blended** | Awareness
Consideration
Execution (purchase) | Blended mix of the above selecting maximum of one or two key KPIs from each of the areas of the marketing funnel |
**EXAMPLE** Thomson Reuters – marketing automation
Thomson Reuters is a professional services company and is an example of a business that uses marketing automation to improve its targeted messaging. Thomson Reuters relied heavily on e-mail marketing to produce leads, though the quality of the leads wasn’t always high, leading to sales conducting their own lead generation activities.
Through marketing automation, Thomson Reuters could bring together sales and marketing while improving segmentation, targeting and overall lead scoring. This led to 23 per cent more leads provided to sales, and a 17.5 per cent increase in revenue attribution to marketing leads.
ACTIVITIES
Review current marketing performance. What top three KPIs are being used to communicate with the business?
Are all current KPIs relevant to communicate to the business? Which ones need to be removed and which ones need to be added?
Is the current measurement system reflecting the work that marketing delivers for the business or are certain marketing efforts being ignored? Why?
# References and further reading
Demand Metric (2014) Lead generation: fueling the revenue engine, available at: https://www.salesfusion360.com/uploads/1/Kayla/Docs/Demand%20Metric%20-%20lead_generation_benchmark_report.pdf (accessed 8 February 2017)
Oracle (2016) Thomson Reuters uses content to start conversations that boost conversions, Oracle Marketing Cloud, available at: https://www.oracle.com/marketingcloud/content/documents/casestudies/thomson-reuters-customer-success-oracle.pdf (accessed 8 February 2017)
Wickey, W (2016) Why data is the (not so) secret ingredient to marketing and sales alignment, available at: http://www.business2community.com/b2b-marketing/data-not-secret-ingredient-marketing-sales-alignment-01716196 (accessed 8 February 2017)
INDEX
account-based marketing (ABM) 135
ABM process 238–40
and the buying cycle 243
benefits of 237
campaign-based ABM 237
challenges 241
definition 236
display-based ABM 240
features of 236
key applications 239
key success factors 241
measuring ABM 242, 244
methods 240–41
outbound ABM 240
product/solution ABM 237
programmatic ABM 237
RingCentral example 244
segment-based ABM 237
setting up and implementing 238–40
strategic ABM 237
suitability for different organizations 242, 243
types of 237
use of social media 240
vertical account ABM 237
acquisition accounts 46, 59
acquisition channel partner marketing 213, 214, 216–17
acquisition customers, types of 47
acquisition marketing, definition 47
acquisition marketing measurement 55–57
benefit of customer acquisition 57
cost of customer acquisition 56
net new customers 55–56
net new revenue 55–56
return on investment (ROI) 56
acquisition marketing process 47–51
analysis 47, 48–49
customer acquisition 48, 51
education 48, 50
follow-up 48, 50
preparation 47, 48, 49–50
prospect engagement 48, 50
ad serving (advertising display) 135
advertising, digital (online) 129, 135
advocacy programmes 55, 113
AdWords (Google) 135
Agent 239
alliance channel marketing 214
Alterian 269
Amazon ratings 55
Amex Open Forum 142
analytics *see* reporting and analytics
Answers 154
application age 2
Ask 154
Atos UK 233–34
attribution models for marketing spend 125–26
audiences, B2B and B2C marketing compared 3
audit of digital marketing 122–23
augmented product 84
awareness-based marketing, through channel partners 202–03
B2B marketing
audiences 3
buyer research findings 4
channels 3
comparison with B2C marketing 2–3
defining the customer 1
definition 1
emotional factor 3
relationship marketing 4
routes to market 185–87
segmentation 3
transactional marketing 4
types of 3–4
B2B marketing strategy 9
B2C (consumer) marketing 2–3
B2P marketing 95
banner digital advertising 135
BANT qualified leads 247
big data 110
blogs/bloggers 96, 113, 123
social blogs 154, 155
specialist blogs 154, 155
Bluekai 269
brand, definition 166–67
brand building 214
B2B and B2C compared 166–67
B2B goals and programmes 169–70
challenges 168–69
definition 166–67
emotional aspects of B2B 167
brand building (*Continued*)
link to business performance 167
reasons for investing in 167
stages in the process 171–77
brand building and positioning programmes 29, 30
brand consistency, importance of 171
brand loyalty 170
brand managers 82
brand positioning 29, 30, 169, 171, 173–74
brand touch points
B2B touch points 167–68
importance of brand consistency 171
brand trackers 172–73
brand voice 168
Brandz 167
budget management 271–75
business, changing shape of 2
business buyer triggers
definition 42
examples 42
identifying and exploiting 43, 144
business buyer types 39–40
business buying behaviours
buying stages 35–36
changes to 41–42
customer behaviour 41–42
identifying 37
importance of 35–37
making use of information about 37
role of sales staff 41–42
business buying influencers
external 39
internal 38–39
business buying situations 40–41
modified re-purchase 40, 41
new purchase 41
repeat purchases 40, 41
business clusters 6
business environment, dynamic change in 1–2
business forums 16
business market
segmentation 5–6
SIC system of classification 6
size of segments 5
business marketing *see* B2B marketing; marketing
business website *see* website
buyer personas 144
buying stages 35–36
need quantification and research 36
need recognition 35, 36
post-purchase 35, 36
purchase 36
vendor review 36
C-suite
broader view of the company 72
definition 70
early influence in the decision cycle 73
gatekeepers 76–79
importance of 72–73
personas and persona creation 74
C-suite marketing
advantages of 72–73
challenges 73–74
contact strategies and tactics 76–79
definition 70
gaining loyalty 79
key success factors 74–76
measuring the impact of 80–81
C-suite roles 70–71
trends and emerging roles 71–72
CDO (chief digital officer) 71–72
CEO role (chief executive officer) 70–71
CFO role (finance) 71
channel content syndication 209
channel marketing budget management 222–23
channel marketing operations 213, 214
channel partner content syndication 199
channel partner marketing
B2B 182
B2C 182
benefits for B2B marketers 182
channel selection 185–87
definition 181
key trends 185
shift to solutions marketing approach 185
channel partner marketing strategy 212–14
customer acquisition 213, 214, 218–19
goals influencing strategy selection 213
integration marketing 213, 214, 219
lifecycle marketing 215
strategy types aligned to goals 213, 214
channel partners
acquisition 213, 214, 216–17
benefits for B2B marketers 182
changing landscape 184–85
collaboration with 29, 30
contact strategy 220, 221
definition 181–82
enablement 195–97, 213
IT channel partners 183–84
marketing alignment 187–89
sharing and promoting content 150
types of 183–84
types of B2B marketing 187–89
channel partners, marketing through 201–11
also marketing to customers 202
awareness-based marketing 202–03
definition 201–02
demand generation 203, 213
gaining channel partner mindshare 207–09
lead generation 202–03, 204–07, 209
partner programmes 208–09
selecting and sharing marketing 203–04
social media 209
through-partner marketing applications 209–11
types of marketing 202–03
channel partners, marketing to 190–99
challenges 197–98
creating one marketing team with 199
digital communications with 191–93
e-mail marketing to 192
offline communications with 191, 193–94
optimizing communications with 198–99
selecting channel marketing communications 194
sharing ideas and information 199
use of social media 197
vendor portals 192–93
channel planning and control 220, 222
channel resellers 183–84
direct engagement with 185–87
channels to market 15, 16
B2B and B2C marketing compared 3
multi-channel marketing 8, 9
CIO role (IT) 71
cloud-based services 184
cloud business 7
CMO role (marketing) 71
collaborative channel partner marketing 213, 214
collaborative marketing 16, 29, 30, 185
communication channels 15, 16
inclusion of digital channels 108
integration of the different channels 138
communications
and customer retention 63
brand consistency 171
related to the customer lifecycle 66–67
communities, role in the new B2B marketing mix 16
competitive advantage, SWOT analysis 26–27
competitor analysis 25, 26
gap analysis 26–27
consultants 182
consumer (B2C) marketing 2–3
contact strategy, privacy and data protection issues 270–71
content 15, 16
Amex Open Forum example 142
B2B compared to B2C 141
digital and non-digital forms 141
distribution mechanisms 148–50
features of great content 142
impact on vendor selection 140–41
planning 145–46
syndication 148, 149
transformation through 29, 30
types of 141
content amplification 150
content audit 145–46
content creation
process 142–51
stumbling blocks 147–48
content curation 149–50
content distribution
earned success 123
YouTube 112
content diversification 120
content licensing 148
content management systems 239
content mapping 145–46
content marketing 8, 9, 107–08, 110–11
benefits for B2B marketers 140–41
definition 140
effects of good quality content 140–41
measuring effectiveness 150–51
Sage example 151
Xerox example 151
COO role (operations) 71
core product 84
core trends 7–9
content marketing 8, 9
customer centricity 7, 9
digital marketing 8, 9
multi-channel marketing 8, 9
optimization of marketing execution 8, 9
corporate resellers, marketing alignment 188
Corporate Social Responsibility (CSR), and business marketing strategy 30–31
cost, in the new B2B marketing mix 16
cost-per-action (CPA) 7
cost-per-click (CPC) 7
cost-per-mile (CPM) 7
CRM systems 24, 112, 122, 241, 269, 270, 275
cross-device marketing 109–10
crowdsourcing 160–61
Crowe Horwath, lead nurturing 254
CSO role (sales) 71
customer acquisition
definition 44
via channel partners 213, 214, 218–19
customer advocacy 62, 65
customer buying behaviour, tracking 8
customer buying cycle, lead generation 248
customer buying models, effects of digital marketing 113
customer-centric strategies 7, 9, 29, 30
product marketing 85
solutions marketing 84, 85
customer champion 62
customer churn 59
customer classification, REAP model 45–46
customer lifecycle 28, 29, 44–45
and social media marketing 158–62
communication related to 66–67
marketing activities at different stages 66–67
customer loyalty
communication factor 63
definition 60
justifying investment in 60
leveraging customer advocacy for marketing 65
measurement 67–68
process 60–62
types of 62
ways to increase 63–65
customer relationships, use of SWOT analysis 25
customer research 26
customer satisfaction
and marketing 65–66
factors influencing 65–66
Net Promoter Score 51–52
customer solutions marketing 15, 16
customers
after acquisition 58
buyer personas 144
changes in buying behaviour 41–42
digital monitoring tools 15
identifying core revenue customers 58–59
reasons why they lapse 52
segmenting for re-acquisition 52–53, 54–55
tracking their media consumption 8
D&B (application) 239
DABs 183
data management 268–71
data management platforms (DMPs) 239, 269
data protection and privacy issues 270–71
Datapipe 214
Dell
influencer marketing example 95
SMAC (social media and communities) programme 162–63
social crowdsourcing forum 160–61
solutions marketing programme 91–92
Dell Channel Europe 197
Deloitte, sponsorship of the London 2012 Olympics 170
demand generation
distinction from lead generation 245–46
through channel partners 203
Demandbase 135
differentiator review 26–27
digital advertising 129, 135
ad serving (advertising display) 135
retargeting 135
digital age 2
digital analytical tools 269
digital application economy 7
digital applications, driving forces 109–10
digital channels, definition 128
digital channels to market 15, 16
digital marketing 8, 9
accessibility 109–10
attribution models for marketing spend 125–26
audit using the value chain 122–23
availability of information 110
challenges 120
changes in B2B marketing 111–12
changing customer buying models 113
changing media and media consumption 111–12
cross-device marketing 109–10
driving forces 109–10
effects on marketing funnels 113
effects on the sales role 112–13
marketing automation 112
measurement 123–24
opportunities 120–21
recent evolution 107–08
relationship (enterprise) marketing 107–08
role of social media 111–12
shift to mobile devices 109–10
transactional marketing 107–08
use of metrics 123–24
use of video 111
value chain 122–23
versatility of information 110–11
digital marketing channels 108
business website 129, 130
definition 128
digital advertising 135
digital video 137–38
e-mail 131–33
integration of different communication channels 138
main types 128–29
mobile marketing 133–34
online PR 134
SEO and search marketing 131
social media 133
webcasts 136–37
webinars 136–37
digital marketing mix 29, 30
digital marketing strategy
mix of paid, earned and owned media channels 123
objectives 115–20
strategy and planning framework 115–20
digital monitoring tools 15
digital tools and technologies
challenges 120
core area of use 126
opportunities 120–21
technology selection 126
distributors 182, 183, 184
direct engagement with 186–87
marketing alignment 188
door openers and hooks 54
Dunn and Bradstreet 269
e-commerce 7
e-mail marketing 129
avoiding nuisance e-mails 132
B2B marketing 131–33
brand consistency 171
customer opportunity to opt in or out 271
preference systems 132
retargeting 132, 133
rich content e-mails 131–32, 133
services in the cloud 7
social e-mail 154, 155
‘soft opt-in’ for existing customers 271
to channel partners 192
transactional e-mails 131–32, 133
use in enterprise (relationship) marketing 132
e-newsletters 192
e-shots 192
earned media channels 123
Ebuyer 183
EIMS Channel Energizer programme 210–11
emotional factor, B2B and B2C marketing compared 3
employee advocacy programmes 149, 162–63, 169
end-to-end solutions 15
enterprise marketing 204
digital marketing 107–08
using e-mail 132
Equifax 269
Equinix 214
European Union, NACE industrial classification system 6
events, for channel partners 193
events marketing
and the customer buying cycle 259–61
B2B event types and segmentation 257–58
B2B marketing events 256
benefits of using events 256–57
digital technologies 264
engaging customers before and after events 262, 263
goals for events 258–59
key success factors 261–62
maximizing attendance and reducing ‘no shows’ 262, 264
maximizing event leads capture 265–66
new events technologies 264
problems 261
social media and 264–65
use of near field communications (NFC) technology 264
virtual event platforms 264
Everstring 135, 239
expansion accounts 46, 59
expansion channel partner marketing 213, 214
Experian 269
extended product 84
external marketing audit 25–26
Facebook 133, 154, 155, 197, 240, 265
gap analysis of competitors 26–27
geo-targeting of mobile customers 134
Glassdoor ratings site 55
globalization 1, 2
Glocal model 1
Google AdWords 135
Google Plus 154, 155
go-to-market (GTM) models 49, 55
government-incentivized clusters 6
Grainger, CSR programme 31
greenfield customers 47
hardware marketing 83–84
high-tech cluster 6
Hootsuite 269
Hoovers 239
Howard, Catherine 233–34
HP Partner Ready 197
Hubspot 269
IBM, influencer marketing 103–04
inactive customers 47
inbound marketing 8
independent software vendors (ISVs) 183–84
industrial marketing *see* B2B marketing
industry sectors, SIC system of classification 6
influencer marketing
content amplification 150
content curation 149–50
customer target set 97, 98
definition 94
Dell example 95
engaging influencers 102
goals of 97, 98
IBM example 103–04
identifying and selecting influencers 99–101
identifying influencers 94
incentivizing influencers 102
influencer stakeholder map 99
influencer types 96
KPIs 102–03
long-term 103
measuring results 102–03
reasons for using 95–96
role of social media 95–96
steps in the process 97–103
types of 95
infographics 110, 111, 123, 145
Insight 183
integration marketing 89–91, 213, 214, 219
internal marketing review 26
internationalization 1
internet age 2
interruptive marketing 8
IT channel partners 183–84
key account marketing *see* account-based marketing
Klout score 100, 101
know-how-based clusters 6
Kotler, Philip 17
KPIs (key performance indicators) 102–03, 124, 232
defining 274
selection and alignment 276
lapsed customers 47
Lattice 239
lead capture, at events 265–66
lead generation
BANT qualified leads 247
challenges for sales and marketing 246–47
customer buying cycle 248
definition 245
distinction from demand generation 245–46
importance for B2B marketers 245
improving poor lead conversion 253
lead scoring 250–51
marketing-accepted leads (MAL) 247
marketing-ready leads (MRL) 247
process 248–51
sales-ready leads (SRL) 247
tactics and strategies 252
through channel partners 202–03, 204–07, 209
trends in 246
types of leads 247
lead generation agencies 252–53
lead nurturing 254–55
Leadspace 239
Leadspace Enrichment 244
LinkedIn 95, 110, 111, 113, 120, 133, 154, 155, 197, 199, 239, 240, 242, 265
low-cost manufacturing clusters 6
mail order resellers 183
managed service providers (MSPs) 182, 184
market presence building 214
marketing
leveraging customer advocacy 65
preparation for acquisition marketing 50
marketing-accepted leads (MAL) 247
marketing audits 25–26
marketing automation platforms 24, 112, 209, 239, 254, 270, 275, 277
marketing budget management 271–75
budgeting process 272–75
defining the marketing budget 272
optimizing budgets 274–75
trends in 271–72
marketing data management 268–71
data accuracy 270
data capture 269–70
data integrity 270
data protection and privacy issues 270–71
e-mail marketing and data 271
types of data 268
marketing era for business 2
marketing execution, optimization 8, 9, 29, 30
marketing funnels 28, 113, 120, 248, 252
marketing mix
4Ps (price, product, promotion, place) 13–15
5Cs of the new B2B marketing mix 15–16
adaptation over time 14–15
and customer satisfaction 65
definition 13
how to determine 16–17, 18
new B2B marketing mix 15–16
new marketing mix 9
portfolio management 83
process 17, 18
relevance for marketers 13–14
role of social media 153–54
shift away from the 4Ps 14–15
trade-offs 18–20
marketing operations 213, 214
activities 267–68
areas covered by 267–68
budget management 271–75
data management 268–71
reporting and analytics 275–77
marketing planning
and the B2B marketing strategy 22–24
evaluation 23, 24
goal setting 23
implementation 23, 24
marketing audit 23–24
marketing plan creation 23, 24
marketing strategy 23, 24
situation analysis 23–24
traditional and new B2B processes 23–24
marketing-ready leads (MRL) 247
marketing strategy 9
B2B strategies 29–31
CSR example 30–31
formulation 26–27
media channels, paid, earned and owned 123
Mediamath 269
micro-resellers 183
marketing alignment 188
Microsoft, withdrawal of support for Windows XP 42
mobile applications 134
mobile devices, digital marketing 109–10
mobile display ads 134
mobile marketing 129, 133–34
B2B mobile marketing tactics 134
mobile-optimized websites 133
modified re-purchase 40, 41
multi-channel marketing 8, 9
NACE industrial classification system (EU) 6
near field communications (NFC) 264
Net Promoter Scores 51–52, 65, 67
new B2B marketing mix 15–16
channels of communication 15, 16
channels to market 15, 16
communities 16
content 15, 16
cost 16
customer solutions marketing 15, 16
how to determine 16–17, 18
trade-offs 18–20
new purchase 41
omnichannel approach 15, 16
online advertising 7, 129
online PR marketing 129, 134
online resellers 182, 183
marketing alignment 189
optimizing paid search 131
optimizing SEO 131
Oracle, channel marketing handbook 208
owned media channels 123
paid media channels 123
paid search, optimizing 131
partners, collaboration with 29, 30 *see also* channel partners
partnerships, role in brand building 169
payment methods 7
PESTEL analysis 25–26
pop-up digital advertising 135
portals for channel partners 192–93
portfolio management 83
portfolio marketing 82–83
distinction from solutions marketing 87
portfolio of products, purchase of 41
preferred accounts 36, 59
printed brochures and catalogues, for channel partners 193–94
privacy and data protection issues 270–71
product marketing 82, 83–84
customer-centric 85
product focused 85
shift away from products 84–85
product offerings 82–83
production era for business 2
prospects (potential customers) 44, 47
QR (Quick Response) codes 134
Quora 154
Radian6 269
range of products, purchase of 41
re-acquisition customers 47
re-acquisition marketing process 51–55
customer segmenting for re-acquisition 52–53
door openers and hooks 54
identifying customer satisfaction 51–52
Net Promoter Score reports 51–52
reasons why customers lapse 52
strategies and tactics 53–55
REAP model 45–46, 58
referral programme 65
referrals 150
registration platforms for events 264
relationship B2B marketing 4, 188, 189, 204
digital marketing 107–08
using e-mail 132
relationship era for business 2
repeat customers 62
repeat purchases 40, 41
reporting and analytics 275–77
applications for 275, 277
difference between marketing automation and CRM 275
KPIs 276
what to measure 276
reputational marketing 129
retargeting in digital advertising 135
retention accounts 46, 59
retention marketing 58–59
communication factor 63
identifying core revenue customers 58–59
importance for revenue and profits 59
segmenting customers 59
return on investment (ROI) 24, 56
RingCentral 244
Sage, content marketing 151
sales
changes in the selling process 41–42
effects of digital marketing 112–13
role in acquisition marketing 49–50
role in customer satisfaction 65
role in social media marketing 155–58
sales and marketing alignment
alignment areas 231–32
alignment radar 230
Atos UK example 233–34
benefits of 228–29
changing sales role 228
establishing 230–31
importance in B2B marketing 227–28
organizational obstacles to 229–30
sales integration initiative 232–33
sales-ready leads (SRL) 247
SAP, Our Business Runs Better programme 220
Schneider Electric/Invensys ‘Better Together’ campaign (case study) 164
search engines 123 *see also* SEO
segmentation 52–53, 54–55
B2B and B2C marketing compared 3
by customer buying power (share-of-wallet, SoW) 59
self-amplification 149
SEO (search engine optimization) 129
and search marketing 129, 131
benefits of good quality content 141
service offerings 82–83
services marketing 83–84
share-of-wallet (SoW) 59
SIC (Standard Industrial Classification) system 6
SlideShare 110, 145
SMAC (social media and communities) programme (Dell) 162–63
SMART goals 24
smartphones, digital marketing 109–10
SMS 134
social amplification programmes 169
social analytical tools 269
social blogs 154, 155
social business forums 154, 155
social commerce 133
social display advertising 154, 155
social e-mail 133, 154, 155
social knowledge channels 154
social listening 15, 133, 163
social media 111
and marketing through channel partners 209
benefits for business 153–54
crowdsourcing 160–61
definition 153
evolution of 153
marketing channels 129
role in business 120
role in events marketing 264–65
role in influencer marketing 95–96
role in sales 113
role in the marketing mix 153–54
role of sales and marketing 155–58
Schneider Electric/Invensys case study 164
social media advocacy programmes 162–63
social media channels 133, 154–55
channel selection 163–64
social media communities 16
social media marketing
and the customer lifecycle 158–62
enablers 157–58
process 156–57
social media policy, brand consistency 171
social monitoring 163, 269
social networks 154, 155
social platforms 133
social syndication through channel partners 209
social video 154, 155
software marketing 83–84
solutions marketing
business solutions 88
challenges 89–90
customer-centric model 84, 85
dealing with competition 92, 93
Dell example 91–92
distinction from portfolio marketing 87
enabling solutions integration 90
integration marketing 89–91
mechanisms for creating solutions 88
shift in channel partner marketing 185
specialized solutions 88
technology solutions 88
types of solution 88
updating solutions 92–93
verticalized solutions 88
SPE Offshore Europe event 258
Spiceworks 154, 155
sponsorship 169, 170
spot customer 62
spot purchase 41
stories to capture customer attention 55
strategic champion customer 62, 63
SWOT analysis 24, 25, 26
Sysomos 269
systems integrators (SIs) 182, 184
marketing alignment 189
TELS framework 150
Thomson Reuters, marketing automation 277
through-partner marketing applications 209–11
TIPSS model 162
touch points before sales are engaged 50
transactional marketing 188, 189
B2B marketing 4
digital marketing 107–08
true solutions 15
Turn 269
Turner, Nigel 270
Twitter 95, 99, 100, 111, 113, 133, 154, 155, 197, 199, 265
value-added resellers (VARs) 181–82, 183, 184
direct engagement with 186
marketing alignment 188
value chain for digital marketing 122–23
value propositions, creating 85–87
vehicles (digital marketing channels) 108
vendor portals for channel partners 192–93
vertical marketing 188
video
increase in use of 111
use of 123
video marketing channels 129, 137–38
Vimeo 154, 155
virtual event platforms 264
webcasts 136–37
webinar platforms for events 264
webinars 129, 136–37
for channel partners 193
website
access for channel partners 192–93
brand consistency 171
development for different customer types 130
marketing channel 129, 130
measuring effectiveness 130
mobile-optimized sites 133
vendor portals for channel partners 192–93
website heat maps 151
Wikipedia 154
Xerox, content marketing 151
Yahoo! 154
YouTube 95, 111, 112, 129, 137–38, 154, 155, 197
“A long-overdue resource for anyone working in the B2B environment or wishing to enter into it. Simon Hall’s development of practical and relevant B2B marketing models makes this essential reading for marketers and other professionals.”
**Dawn Southgate, Head of Knowledge, The Chartered Institute of Marketing**
“If you read any B2B marketing publication this year, this should be it! Simon Hall provides us with a refreshing view, relevant content and truly practical advice, touching on many areas pertinent in the B2B marketing world right now and for the future. Absolutely fantastic insight.”
**Catherine Howard, UK and Ireland Marketing Director, Atos**
*“Innovative B2B Marketing* provides the guidance that all B2B marketers need to help them navigate for success. I will be keeping this book very close to hand.”
**Richard Robinson, Chair, B2B Marketing Council, The IDM**
“How do you build best-in-class B2B strategy in today’s digital economy? How are great B2B brands built? How do you select the right lead nurturing agency? *Innovative B2B Marketing* is a rich treasure trove of B2B strategic wisdom.”
**Shenda Loughnane, Global Chief Strategy Officer, iProspect**
***Innovative B2B Marketing*** is a clear, practical guide that demystifies modern aspects of B2B marketing, including marketing models, processes and thought leadership. New customer buying habits, the digital era and a changing industry landscape (influenced by the application economy) have all had a great impact, and as a consequence marketing professionals are facing a shift away from traditional practices. The focus of ***Innovative B2B Marketing*** is to cut through the noise and make sense of the new models, strategies and operations that have recently emerged in the B2B marketing sphere.
Simon Hall, a former CMO with over 20 years’ senior-level experience as one of the leading voices in B2B marketing who has worked with the CIM, the IDM and other major associations, brings together a wealth of insight and information sourced from his own first-hand experience. Featuring real-life examples from diverse sectors, plus topical discussion points and problems from key B2B marketing forums and associations, this book provides marketers with new approaches, models and solutions to help deal with any B2B marketing challenge.
Supporting tools and templates can be downloaded from koganpage.com/Innovative-B2B.
**Simon Hall** is a marketing innovator with extensive experience in technology and services marketing. He has served as UK Chief Marketing Officer for Dell and has held many senior roles at Acer, Microsoft and Toshiba. In 2016, Simon founded NextGen Marketing Solutions. He is a course director with the CIM, where he runs public courses as well as in-company training, a B2B council member with the IDM, the Global Thought Leader for B2B Marketing at the Internal Consulting Group and a member of both the CMO Council and European CMO Council.
**Kogan Page**
London
New York
New Delhi
**www.koganpage.com** | |
🕑 Reading time: 1 minute
Cladding is an envelope that protects the exterior of a structure by acting as a shield to the dwelling unit/structure from environmental elements. A cladding should be able to tame inclement weather conditions like torrential rains and heavy winds, preventing them from infiltrating a building structure. Besides, it may also add to the aesthetics of the structure by hiding the unattractive substrate or the underlying stronger material.
Cladding installation systems
A cladding system can be installed using various methods. The installation system to be employed is based on different kinds of materials that have been utilized for the construction of the cladding system.
There are three basic ways to install a cladding system, which are listed as follows:
1. Attached system
In the attached system, the large panels forming the exterior cladding are directly attached to the structural frame of a building having one or more stories or bays. The attached system is used to erect the precast concrete and the steel-stud frames having exterior finishes. During installation, the crane lifts the panel in place, being fixed onto the anchorage that is fastened at the edge of the slab.
By using the attached system, one can eliminate the time consumed by on-site construction and make the construction process much more easier. In addition to this, the attached system also insulates the exterior wall and the structural frame is guarded against the ill-effects of the changing weather.
On the flip side, the time needed for designing, drawing approvals, and manufacturing of the panels may push a project behind schedule thus causing a delay. The design has to be thoroughly examined to avoid problems after installation.
2. Curtain Wall System
The curtain wall system is generally incorporated in large and multi-storeyed buildings. This system is similar to the attached system, with the only difference being- the panels here are not attached to the slab edges with anchors, but are attached to the structural frame using clip angles.
Typically, the curtain wall system comprises a lightweight frame onto which opaque or glazed panels are fixed. Metal and glass are the most commonly used cladding panels in this system.
The material used in the curtain wall system is usually light, providing an added advantage during handling and assembly. Furthermore, a standard combination of the materials and the system is set for the design and manufacturing of panels.
This, in turn, proves to be an added advantage by reducing the time required for approving designs and manufacturing. The manufacturer then designs the joints and connections based on the material, system, and structural frame properties specified by the structural engineers and architects.
3. Infill System
The infill system is easier to differentiate from other systems as the structural frame is exposed in this case. The cladding panels are placed between the columns of the structural frame, spanning from one level of the slab to another. The cast-in-place concrete panel is mostly used as a cladding material in this system. Glass and precast concrete are some other materials that can be effectively used in the infill system.
The cladding panels of the infill system can be installed from the inside of the building, reducing the risks of injury to workers working on scaffolding. This system has weak heat-insulating ability because of a significant heat exchange between the floor slabs due to the exposed structural frames. The heat also causes expansion and contraction of the structural frame.
These reasons leave very little room for error during design and assembly. The designer must also consider the behavior of the precast concrete panel and the structural frame as the structural elements will be subjected to non-uniform loads over the years.
FAQs
Cladding is an envelope to protect the exterior of a structure.
The types of cladding installation systems are the attached system, curtain wall system, and infill system.
In the infill installation system, the structural frame is exposed, and the cast-in-place concrete panel is mostly used.
READ MORE: Type of Wall Claddings
READ MORE: How to Install Natural Stone Cladding by Wet Method? | https://theconstructor.org/building/building-material/cladding-installation-systems/45936/ |
:
Turkey Brine Recipe
Turkey Brine Recipe
Roasting, grilling or deep-frying methods can be used for Turkey Brine recipe. Learn how to make/prepare Turkey Brine by following this easy recipe.
Turkey Brine Recipe
Difficulty
Average
Recipe Type
Non Veg.
Ingredients:
12 to 14 lb Turkey, not pre-basted
1-1/2 Cups kosher salt
1 Cup sugar
2 Bay leaves, broken into pieces
4 Juniper berries, smashed
4 Allspice berries, whole
1 Head of garlic, peel and separate cloves
1/3 Cup fresh chopped thyme
Water as required
How to make Turkey Brine:
In a large bowl combine salt, sugar and 4 quarts of water.
Keep stirring, till the sugar and salt are dissolved.
Add the remaining ingredients to it.
Put the turkey in the stockpot and mix the brining solution with it.
Add sufficient cold water to this so that the turkey is fully submerged.
Now cover the container and place it into the refrigerator or another cold storage area.
Allow the turkey to stand in the solution for 10 to 12 hours.
After the required soaking time, take out it from the container and thoroughly rinse under the cold water.
Brining solution should be discarded as it cannot be reused.
The turkey can be cooked using several cooking methods, such as roasting, grilling and deep-frying.
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Kelley Armstrong (born 14 December 1968) is a Canadian writer, primarily of fantasy novels since 2001.
She has published thirty-one fantasy novels to date, thirteen in her Women of the Otherworld series, four in her Cainsville series, two in her Rockton series, three in her Darkest Powers series, three in her Darkness Rising trilogy and three in the Age of Legends series, and three stand-alone teen thrillers. She has also published three middle-grade fantasy novels in the Blackwell Pages trilogy, with co-author Melissa Marr. As well, she is the author of three crime novels, the Nadia Stafford trilogy. She has also written several serial novellas and short stories for the Otherworld series, some of which are available free from her website. She likes programming.
Kelley Armstrong was born on 14 December 1968, the oldest of four siblings in a "typical middle-class family" in Sudbury, Ontario.
After graduating with a degree in psychology from The University of Western Ontario, Armstrong then switched to studying computer programming at Fanshawe College so she would have time to write.
Her first novel Bitten was sold in 1999, and it was released in 2001. Following her first success she has written a total of 13 novels and a number of novellas in the world of the Women of the Otherworld series, and her first crime novel, Exit Strategy, was released July 2007. Armstrong has been a full-time writer and parent since 2002.
Her novel No Humans Involved was a New York Times bestseller in the hardback fiction category on 20 May 2007. Also, her YA novel The Awakening was a No. 1 New York Times bestseller in the Children's Chapter books category on 17 May 2009.
Armstrong's Women of the Otherworld series is part of a recently popular contemporary fantasy subgenre of the fantasy genre that superimposes supernatural characters upon a backdrop of contemporary North American life, with strong romantic elements. Within that subgenre, she is notable for including many types of supernatural characters, including witches, sorcerers, werewolves, necromancers, ghosts, shamans, demons and vampires, rather than limiting herself primarily to a single type of supernatural creature. Most of her works have a mystery genre plot, with leading characters investigating some novel situation or unsolved question.
In the Otherworld novels, most supernatural powers are either hereditary, or arise from the act of an existing supernatural of the same type. The Otherworld, while it has overarching conflicts and plotlines that span multiple novels is not an epic battle between good and evil. The novels are largely episodic with the continuing plotlines primarily involving the developing lives of the main characters.
Her contemporary fantasy writings share genre similarities with writers Charlaine Harris, Laurell K Hamilton and Kim Harrison. | https://alchetron.com/Kelley-Armstrong |
Basal Metabolic Rate (BMR)
Basal Metabolic Rate (BMR) is the amount of energy expended while at rest in a neutrally temperate environment, in the post-absorptive state (meaning that the digestive system is inactive, which requires about twelve hours of fasting).
The release of energy in this state is sufficient only for the functioning of the vital organs, such as the heart, lungs, brain and the rest of the nervous system, liver, kidneys, sex organs, muscles and skin. BMR decreases with age and with the loss of lean body mass. Increasing muscle mass increases BMR.
There are different ways to calculate BMR and we use the Katch-McArdle Formula (Resting Daily Energy Expenditure):
P = 370 + (21.6 x LBM)
where LBM is the lean body mass in kg. | https://www.activ8rlives.com/support/data-collected/body-composition/basal-metabolic-rate-bmr |
---
abstract: 'In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray-Schauder Topological degree.'
address:
- |
Claudianor O. Alves\
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP:58429-900, Campina Grande - PB, Brazil.
- |
Abdelkrim Moussaoui\
Biology Department, A. Mira Bejaia University, Targa Ouzemour, 06000 Bejaia, Algeria.
author:
- 'Claudianor O. Alves'
- Abdelkrim Moussaoui
title: Positive solutions for a class of quasilinear singular elliptic systems
---
[^1]
[^2]
Introduction {#S1}
============
We consider the following system of quasilinear elliptic equations:$$\left\{
\begin{array}{ll}
-\Delta _{p}u=u^{\alpha _{1}}v^{\beta _{1}} & \text{in }\Omega , \\
-\Delta _{q}v=u^{\alpha _{2}}v^{\beta _{2}} & \text{in }\Omega , \\
u,v>0 & \text{in }\Omega , \\
u,v=0 & \text{on }\partial \Omega ,%
\end{array}%
\right. \tag{$P$} \label{p}$$where $\Omega $ is a bounded domain in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{N}$ $\left( N\geq 2\right) $ with $C^{1,\alpha }$ boundary $\partial
\Omega $, $\alpha \in (0,1)$, $\Delta _{p}$ and $\Delta _{q},$ $1<p,q<N,$ are the $p$-Laplacian and $q$-Laplacian operators, respectively, that is, $%
\Delta _{p}u=div\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) $ and $\Delta _{q}v=div\left( \left\vert \nabla v\right\vert ^{q-2}\nabla
v\right) .$ We consider the system (\[p\]) in a singular case by assuming that$$\left\{
\begin{array}{c}
-1<\alpha _{1}<0<\beta _{1}<\min \{p-1,\frac{q^{\ast }}{p^{\ast }}%
(p-1-\alpha _{1})\} \\
-1<\beta _{2}<0<\alpha _{2}<\min \{q-1,\frac{p^{\ast }}{q^{\ast }}(q-1-\beta
_{2})\}.%
\end{array}%
\right. \label{h1}$$In this case, system (\[p\]) is cooperative, that is, for $u$ (resp. $v$) fixed the right term in the first (resp. second) equation of (\[p\]) is increasing in $v$ (resp. $u$).
The study of singular elliptic problems is greatly justified because they arise in several physical situations such as fluid mechanics pseudoplastics flow, chemical heterogeneous catalysts, non-Newtonian fluids, biological pattern formation and so on. In Fulks & Maybee [@FM], the reader can find a very nice physical illustration of a practical problem which leads to singular problem.
With respect to singular system it is worth to cite, among others, the important Gierer-Meinhardt system which is the stationary counterpart of a parabolic system proposed by Gierer-Meinhardt (see [@GM1; @D]) which occurs in the study of morphogenesis on experiments on hydra, an animal of a few millimeters in length.
Besides the importance of the physical application above mentioned, we would like to mention that from a mathematical point of view the singular problems are also interesting because to solve some of them are necessary nontrivial mathematical techniques, which involve Topological degree, Bifurcation theory, Fixed point theorems, sub and supersolution Method, Pseudomonotone Operator theory and Variational Methods. Here, it is impossible to cite all papers in the literature which use the above techniques, however the reader can find the applications of the above mentioned methods in Alves & Moussaoui [@CM], Hai [@Hai], Ghergu & Radulescu [@GR], Giacomoni, Hernandez & Moussaoui [@GHM], Giacomoni, Hernandez & Sauvy [@GHS], Hernandez, Mancebo & Vega, [@HMV], Khodja & Moussaoui [KM]{}, Zhang [@Z1], Zhang & Yu [@ZY], Diaz, Morel & Oswald [DMO]{}, Alves, Corrêa & Gonçalves [@ACG], Crandall & Rabinowitz [@CR], Taliaferro [@T], Lunning & Perry [@LP], Motreanu & Moussaoui [@MM2; @MM3; @MM], Moussaoui, Khodja & Tas [@MKT], Agarwall and O’Regan [@AO], Stuart [@ST] and their references.
After a review bibliography, we did not find any paper where the existence of multiple solutions have been considered for a singular system. Motivated by this fact, we prove in the present paper the existence of at least two positive solutions for system $(P)$. Our main result has the following statement:
\[T2\] Under assumption (\[h1\]) problem (\[p\]) possesses at least two (positive) solutions in $C^{1,\gamma }(\overline{\Omega })\times
C^{1,\gamma }(\overline{\Omega }),$ for certain $\gamma \in (0,1)$.
In the proof of the above theorem, we will use sub and supersolution method combined with Leray-Schauder Topological degree. However, before proving that theorem it was necessary to get some informations about the regularity of the solutions. To this end, the below result was crucial in our approach.
\[T1\] Assume (\[h1\]) holds. Then, system (\[p\]) has a positive solution $\left( u,v\right) $ in $C^{1,\gamma }(\overline{\Omega })\times
C^{1,\gamma }(\overline{\Omega })$ for some $\gamma \in (0,1)$. Moreover, there exist a sub-supersolution $\left( \underline{u},\underline{v}\right) ,(%
\overline{u},\overline{v})\in C^{1}(\overline{\Omega })\times C^{1}(%
\overline{\Omega })$ for (\[p\]) such that$$\underline{u}(x)\leq u(x)\leq \overline{u}(x)\text{ and }\underline{v}%
(x)\leq v(x)\leq \overline{v}(x)\text{ for all }x\in \overline{\Omega }.
\label{c}$$
In the present paper, a solution of (\[p\]) is understood in the weak sense, that is, a pair $(u,v)\in W_{0}^{1,p}(\Omega )\times
W_{0}^{1,q}(\Omega )$, with $u,v$ positive a.e. in $\Omega ,$ satisfying$$\left\{
\begin{array}{cc}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla \varphi \ dx & =\int_{\Omega
}u^{\alpha _{1}}v^{\beta _{1}}\varphi \ dx, \\
\int_{\Omega }|\nabla v|^{q-2}\nabla v\nabla \psi \ dx & =\int_{\Omega
}u^{\alpha _{2}}v^{\beta _{2}}\psi \ dx,%
\end{array}%
\right. \label{7}$$for all $(\varphi ,\psi )\in W_{0}^{1,p}(\Omega )\times W_{0}^{1,q}(\Omega )$.
The proof of Theorem \[T1\] is done in Section \[S2\]. The main technica[l ]{}difficulty consists in the presence[ ]{}of singular terms in system (\[p\]) under condition (\[h1\]). Our approach is based on the sub-supersolution method in its version for systems [@CLM section 5.5]. However, this method cannot be directly implemented due to the presence of singular terms in system (\[p\]). Applying the sub-supersolution method in conjunction with the regularity result in [@Hai] under hypothesis ([h1]{}), we prove the existence of a (positive) solution $(u,v)\in C^{1,\gamma
}(\overline{\Omega })\times C^{1,\gamma }(\overline{\Omega }),$ for certain $%
\gamma \in (0,1),$ of problem (\[p\]).
The proof of Theorem \[T2\] is done in Section \[S3\]. It is based on topological degree theory with suitable truncations. Here, it suffices to show the existence of a second (positive) solution for problem (\[p\]). The first one is given by Theorem \[T1\] which is located in a rectangle formed by the sub-supersolutions. However, due to the singular terms in system (\[p\]), the degree theory cannot be directly implemented. To handle this difficulty, the degree calculation is applied for the regularized problem (\[pr\]) for $\varepsilon >0$. Under assumption ([h1]{}), Theorem \[T1\] ensures the existence of a smooth solution for ([p]{}). This gives rise to the possible existence a constant $R>0$ such that all solutions $(u,v)$ with $C^{1,\gamma }$-regularity satisfy $\left\Vert
u\right\Vert _{C^{1,\gamma }},\left\Vert v\right\Vert _{C^{1,\gamma }}<R.$ On the basis of this, we show that the degree of an operator corresponding to system (\[pr\]) on a larger set is $0$. Another hand, we show that the degree of an operator corresponding to the system (\[pr\]) is $1$ on an appropriate set. This leads to the existence of a second solution for ([pr]{}) by using the excision property of Leray-Schauder degree. Then the existence of a second solution for (\[p\]) is derived by passing to the limit as $\varepsilon \rightarrow 0$.
In what follows, we denote by $\phi _{1,p}$ and $\phi _{1,q}$ the normalized positive eigenfunctions associated with the principal eigenvalues $\lambda
_{1,p}$ and $\lambda _{1,q}$ of $-\Delta _{p}$ and $-\Delta _{q}$, respectively: $$\begin{array}{c}
-\Delta _{p}\phi _{1,p}=\lambda _{1,p}\left\vert \phi _{1,p}\right\vert
^{p-2}\phi _{1,p}\text{ \ in }\Omega ,\text{ \ }\phi _{1,p}=0\text{ \ on }%
\partial \Omega ,\text{ \ }\int_{\Omega }\phi _{1,p}^{p}=1%
\end{array}
\label{6}$$and$$\begin{array}{c}
-\Delta _{q}\phi _{1,q}=\lambda _{1,q}\left\vert \phi _{1,q}\right\vert
^{q-2}\phi _{1,q}\text{ \ in }\Omega ,\text{ \ }\phi _{1,q}=0\text{ \ on }%
\partial \Omega ,\text{ \ }\int_{\Omega }\phi _{1,q}^{q}=1.%
\end{array}
\label{8}$$
The strong maximum principle ensures the existence of positive constants $%
l_{1}$ and $l_{2}$ such that$$l_{1}\phi _{1,p}(x)\leq \phi _{1,q}(x)\leq l_{2}\phi _{1,p}(x)\text{ for all
}x\in \Omega . \label{5}$$For a later use we recall that there exists a constant $l>0$ such that$$\phi _{1,p}(x),\phi _{1,q}(x)\geq ld(x)\text{ for all }x\in \Omega ,
\label{67}$$where $d(x):=dist(x,\partial \Omega )$ (see, e.g., [@GST]). Moreover, since $\phi_{1,p}$ and $\phi_{1,q}$ belongs to $C^{1}(\overline{\Omega})$, there is $M>0$ such that $$\begin{array}{c}
M=\underset{x \in \overline{\Omega} }{\max }\{|\phi _{1,p}(x)|+|\phi
_{1,q}(x)|\}.%
\end{array}
\label{23}$$
Proof of Theorem \[T1\]: Existence of the first solution {#S2}
========================================================
Let us define $w_{1}$ and $w_{2}$ as the unique weak solutions of the problems $$\left\{
\begin{array}{ll}
-\Delta _{p}w_{1}=w_{1}^{\alpha _{1}} & \text{in }\Omega , \\
w_{1}>0 & \text{in }\Omega , \\
w_{1}=0 & \text{on }\partial \Omega%
\end{array}%
\right. \text{ \ and \ }\left\{
\begin{array}{ll}
-\Delta _{q}w_{2}=w_{2}^{\beta _{2}} & \text{in }\Omega , \\
w_{2}>0 & \text{in }\Omega , \\
w_{2}=0 & \text{on }\partial \Omega ,%
\end{array}%
\right. \label{20}$$respectively, which are known to satisfy $$c_{2}\phi _{1,p}(x)\leq w_{1}(x)\leq c_{3}\phi _{1,p}(x)\text{ \ and \ }%
c_{2}^{\prime }\phi _{1,q}(x)\leq w_{2}(x)\leq c_{3}^{\prime }\phi _{1,q}(x),
\label{21}$$with positive constants $c_{2},c_{3},c_{2}^{\prime },c_{3}^{\prime }$ (see [@GST]). Consider $\xi _{1},\xi _{2}\in C^{1}\left( \overline{\Omega }%
\right) $ the solutions of the homogeneous Dirichlet problems:$$\left\{
\begin{array}{ll}
-\Delta _{p}\xi _{1}(x)=\phi _{1,p}^{\alpha _{1}}(x) & \text{ in }\Omega ,
\\
\xi _{1}=0 & \text{ on }\partial \Omega%
\end{array}%
\right. ,\text{ }\left\{
\begin{array}{ll}
-\Delta _{q}\xi _{2}(x)=\phi _{1,q}^{\beta _{2}}(x) & \text{ in }\Omega , \\
\xi _{2}=0 & \text{\ on }\partial \Omega .%
\end{array}%
\right. \label{12}$$The Hardy–Sobolev inequality (see, e.g., [@AC Lemma 2.3]) guarantees that the right-hand side of (\[12\]) belongs to $W^{-1,p^{\prime }}(\Omega
)$ and $W^{-1,q^{\prime }}(\Omega )$, respectively. Consequently, the Minty–Browder theorem (see [@B Theorem V.15]) implies the existence of unique $\xi _{1}$ and $\xi _{2}$ in (\[12\]). Moreover, (\[20\]), ([21]{}), the monotonicity of the operators $-\Delta _{p}$ and $-\Delta _{q}$ yield $$c_{0}\phi _{1,p}(x)\leq \xi _{1}(x)\leq c_{1}\phi _{1,p}(x)\text{ and }%
c_{0}^{\prime }\phi _{1,q}(x)\leq \xi _{2}(x)\leq c_{1}^{\prime }\phi
_{1,q}(x)\text{ in }\Omega , \label{36}$$for some positive constants $c_{0},c_{1},c_{0}^{\prime },c_{1}^{\prime }$. Let $z_{1}$ and $z_{2}$ satisfy$$-\Delta _{p}z_{1}(x)=h_{1}(x),\text{ }z_{1}=0\text{ \ on }\partial \Omega ,
\label{1}$$and$$-\Delta _{q}z_{2}(x)=h_{2}(x),\text{ }z_{2}=0\text{ \ on }\partial \Omega
\text{,} \label{2}$$where $$h_{1}(x)=\left\{
\begin{array}{ll}
\phi _{1,p}^{\alpha _{1}}(x) & \text{in \ }\Omega \backslash \overline{%
\Omega }_{\delta }, \\
-\phi _{1,p}^{\alpha _{1}}(x) & \text{in \ }\Omega _{\delta },%
\end{array}%
\right. \label{h1*}$$
$$h_{2}(x)=\left\{
\begin{array}{ll}
\phi _{1,q}^{\beta _{2}}(x) & \text{in \ }\Omega \backslash \overline{\Omega
}_{\delta }, \\
-\phi _{1,q}^{\beta _{2}}(x) & \text{in \ }\Omega _{\delta }%
\end{array}%
\right. \label{h2*}$$
and $$\Omega _{\delta }=\left\{ x\in \Omega :d(x)<\delta \right\} ,$$with a fixed $\delta >0$ sufficiently small and $d(x)=d\left( x,\partial
\Omega \right) $.
The Hardy-Sobolev inequality together with the Minty-Browder theorem imply the existence and uniqueness of $z_{1}$ and $z_{2}$ in (\[1\]) and (\[2\]). Moreover, (\[1\]) and (\[2\]), the monotonicity of the operators $%
-\Delta _{p}$ and $-\Delta _{q} $ and [@Hai Corollary 3.1] imply that $$\begin{array}{l}
\frac{c_{0}}{2}\phi _{1,p}(x)\leq z_{1}(x)\leq c_{1}\phi _{1,p}(x)\text{ and
}\frac{c_{0}^{\prime }}{2}\phi _{1,q}(x)\leq z_{2}(x)\leq c_{1}^{\prime
}\phi _{1,q}(x)\text{ in }\Omega .%
\end{array}
\label{c2}$$
Next, our goal is to show the existence of sub and supersolution for $(P)$.
**Existence of subsolution:**
For a constant $C>0$, we have $$\begin{array}{l}
-C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)<0\leq (C^{-1}z_{1}(x))^{\alpha
_{1}}(C^{-1}z_{2}(x))^{\beta _{1}},\text{ }x\in \Omega _{\delta }%
\end{array}
\label{18}$$and$$\begin{array}{c}
-C^{-(q-1)}\phi _{1,q}^{\beta _{2}}(x)<0\leq (C^{-1}z_{1}(x))^{\alpha
_{2}}(C^{-1}z_{2}(x))^{\beta _{2}},\text{ }x\in \Omega _{\delta }.%
\end{array}
\label{19}$$Let $\mu >0$ be a constant such that$$\begin{array}{c}
\phi _{1}\left( x\right) ,\phi _{2}\left( x\right) \geq \mu \text{ in }%
\Omega \backslash \overline{\Omega }_{\delta }.%
\end{array}
\label{256}$$Then, since $\alpha _{1}<0<\beta _{1}$, (\[c2\]) and (\[256\]) lead to $$\begin{array}{l}
C^{\alpha _{1}+\beta _{1}-(p-1)}\phi _{1,p}^{\alpha
_{1}}(x)(z_{1}(x))^{-\alpha _{1}}\leq C^{\alpha _{1}+\beta _{1}-(p-1)}\phi
_{1,p}^{\alpha _{1}}(x)(c_{1}\phi _{1,p}(x))^{-\alpha _{1}} \\
=C^{\alpha _{1}+\beta _{1}-(p-1)}(Mc_{1})^{-\alpha _{1}}<(c_{0}^{\prime }\mu
)^{\beta _{1}}\leq (c_{0}^{\prime }\phi _{1,q}(x))^{\beta _{1}} \\
\leq (z_{2}\left( x\right) )^{\beta _{1}},\ \ \text{for all }x\in \Omega
\backslash \overline{\Omega }_{\delta },%
\end{array}
\label{28}$$provided $C>0$ large enough. This is equivalent to$$\begin{array}{l}
C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)<(C^{-1}z_{1}(x))^{\alpha
_{1}}(C^{-1}z_{2}\left( x\right) )^{\beta _{1}},\ \ \text{for all }x\in
\Omega \backslash \overline{\Omega }_{\delta }.%
\end{array}
\label{29*}$$Similarly, $$\begin{array}{c}
C^{-(q-1)}\phi _{1,q}^{\beta _{2}}(x)<\left( C^{-1}z_{1}\left( x\right)
\right) ^{\alpha _{2}}(C^{-1}z_{2}(x))^{\beta _{2}}\ \ \text{for all }x\in
\Omega \backslash \overline{\Omega }_{\delta },%
\end{array}
\label{29}$$for $C>0$ large enough. The pair $$\begin{array}{c}
\left( \underline{u},\underline{v}\right) =C^{-1}\left( z_{1},z_{2}\right) .%
\end{array}
\label{30}$$is a subsolution for $(P)$, Indeed, a direct computation shows that $$\begin{array}{c}
\int_{\Omega }\left\vert \nabla \underline{u}\right\vert ^{p-2}\nabla
\underline{u}\nabla \varphi \text{ }dx=C^{-(p-1)}\int_{\Omega \backslash
\Omega _{\delta }}\phi _{1,p}^{\alpha _{1}}\varphi \text{ }%
dx-C^{-(p-1)}\int_{\Omega _{\delta }}\phi _{1,p}^{\alpha _{1}}\varphi \text{
}dx%
\end{array}
\label{*}$$and$$\begin{array}{c}
\int_{\Omega }\left\vert \nabla \underline{v}\right\vert ^{q-2}\nabla
\underline{v}\nabla \psi =C^{-(q-1)}\int_{\Omega \backslash \Omega _{\delta
}}\phi _{1,q}^{\beta _{2}}\psi \text{ }dx-C^{-(q-1)}\int_{\Omega _{\delta
}}\phi _{1,q}^{\beta _{2}}\psi \text{ }dx,%
\end{array}
\label{**}$$where $\left( \varphi ,\psi \right) \in W_{0}^{1,p}\left( \Omega \right)
\times W_{0}^{1,q}\left( \Omega \right) $ with $\varphi ,\psi \geq 0$. Combining (\[\*\]), (\[\*\*\]), (\[18\]), (\[19\]), (\[28\]) and ([29]{}), it is readily seen that$$\begin{array}{c}
\int_{\Omega }\left\vert \nabla \underline{u}\right\vert ^{p-2}\nabla
\underline{u}\nabla \varphi \leq \int_{\Omega }\underline{u}^{\alpha _{1}}%
\underline{v}^{\beta _{1}}\varphi%
\end{array}%$$and$$\begin{array}{c}
\int_{\Omega }\left\vert \nabla \underline{v}\right\vert ^{q-2}\nabla
\underline{v}\nabla \psi \leq \int_{\Omega }\underline{u}^{\alpha _{2}}%
\underline{v}^{\beta _{2}}\psi ,%
\end{array}%$$for all $\left( \varphi ,\psi \right) \in W_{0}^{1,p}\left( \Omega \right)
\times W_{0}^{1,q}\left( \Omega \right) $ with $\varphi ,\psi \geq 0$. This proves that $\left( \underline{u},\underline{v}\right) $ is a subsolution for $(P)$.
**Existence of supersolution:**
Next, we prove that $$(\overline{u},\overline{v})=C(\xi _{1},\xi _{2}) \label{32}$$is a supersolution for problem (\[p\]) for $C>0$ large enough. Obviously, we have $\left( \overline{u},\overline{v}\right) \geq \left( \underline{u},%
\underline{v}\right) $ in $\overline{\Omega }$ for $C$ large enough. Taking into account (\[12\]), (\[36\]), (\[23\]) and (\[h1\]) we derive that in $\overline{\Omega }$ one has the estimates $$\begin{array}{l}
\overline{u}^{-\alpha _{1}}\overline{v}^{-\beta _{1}}(-\Delta _{p}\overline{u%
})=C^{p-1-\alpha _{1}-\beta _{1}}\xi _{2}^{-\beta _{1}}{\geq }C^{p-1-\alpha
_{1}-\beta _{1}}(c_{1}^{\prime }\phi _{1,q}(x))^{-\beta _{1}} \\
\geq C^{p-1-\alpha _{1}-\beta _{1}}(c_{1}^{\prime }M)^{-\beta _{1}}\geq 1%
\text{ in }\overline{\Omega }%
\end{array}%$$and$$\overline{u}^{-\alpha _{2}}\overline{v}^{-\beta _{2}}(-\Delta _{q}\overline{v%
}){\geq }C^{q-1-\alpha _{2}-\beta _{2}}(c_{1}M)^{-\alpha _{2}}\geq 1\text{
in }\overline{\Omega },$$provided that $C>0$ is sufficiently large. Consequently, it turns out that$$\int_{\Omega }\left\vert \nabla \overline{u}\right\vert ^{p-2}\nabla
\overline{u}\nabla \varphi \text{ }dx\geq \int_{\Omega }\overline{u}^{\alpha
_{1}}\overline{v}^{\beta _{1}}\varphi \text{ }dx \label{33}$$and $$\int_{\Omega }\left\vert \nabla \overline{v}\right\vert ^{q-2}\nabla
\overline{v}\nabla \psi \text{ }dx\geq \int_{\Omega }\overline{u}^{\alpha
_{2}}\overline{v}^{\beta _{2}}\psi \text{ }dx, \label{34}$$for all $\left( \varphi ,\psi \right) \in W_{0}^{1,p}\left( \Omega \right)
\times W_{0}^{1,q}\left( \Omega \right) .$
**Proof of Theorem \[T1\] (conclusion):**
Using (\[h1\]), (\[67\]), (\[c\]), (\[30\]), (\[32\]), (\[c2\]) and (\[36\]), we get$$u^{\alpha _{1}}v^{\beta _{1}}\leq \underline{u}^{\alpha _{1}}\overline{v}%
^{\beta _{1}}\leq C_{1}d(x)^{\alpha _{1}}\text{ for all }x\in \Omega$$and$$u^{\alpha _{2}}v^{\beta _{2}}\leq \overline{u}^{\alpha _{2}}\underline{v}%
^{\beta _{2}}\leq C_{2}d(x)^{\beta _{2}}\text{ for all }x\in \Omega ,$$where $C_{1}$ and $C_{2}$ are positive constants. Then, owing to [@KM Theorem 2] we deduce that there exists a solution $(u,v)\in C^{1,\gamma
}(\overline{\Omega })\times C^{1,\gamma }(\overline{\Omega }),$ for some $%
\gamma \in (0,1),$ of problem (\[p\]) within $\left[ \underline{u},%
\overline{u}\right] \times \left[ \underline{v},\overline{v}\right] $. This complete the proof.
Proof of Theorem \[T2\] {#S3}
=======================
According to Theorem \[T1\] we know that problem (\[p\]) possesses a (positive) solution $(u,v)$ in $C^{1,\gamma }(\overline{\Omega })\times
C^{1,\gamma }(\overline{\Omega }),$ located in the rectangle $[\underline{u},%
\overline{u}]\times \lbrack \underline{v},\overline{v}]$ for certain $\gamma
\in (0,1)$. Thus, to prove Theorem \[T2\] it suffices to show the existence of a second solution for problem (\[p\]).
Before starting the proof of Theorem \[T2\], we would like point out that by Theorem \[T1\] the set of solutions $(u,v)$ in $C^{1,\gamma }(\overline{%
\Omega })\times C^{1,\gamma }(\overline{\Omega }),$ $\gamma \in (0,1),$ for problem (\[p\]) is not empty. Then, without any loss of generality, we may assume that there is a constant $R>0$ such that all solutions $(u,v)$ with $%
C^{1,\gamma }$-regularity satisfy $$\left\Vert u\right\Vert _{C^{1,\gamma }(\overline{\Omega })},\left\Vert
v\right\Vert _{C^{1,\gamma }\overline{\Omega }}<R. \label{15}$$ Otherwise, there are infinity solutions with $C^{1,\gamma }$-regularity and the proof of Theorem \[T2\] is completed.
Hereafter, we denote $$B_{R}(0)=\left\{ (u,v)\in C^{1}(\overline{\Omega })\times C^{1}(\overline{%
\Omega })\,:\,\Vert u\Vert _{C^{1}}+\Vert v\Vert _{C^{1}}<R\right\} ,$$$$\mathcal{O}_{R}=\left\{ (u,v)\in B_{R}(0)\,:\,\underline{u}\ll u\ll R\,\,\,%
\mbox{and}\,\,\,\underline{v}\ll v\ll R\right\}$$and$$\mathcal{\hat{O}}=\left\{ (u,v)\in B_{R}(0)\,:\,\underline{u}\ll u\ll \hat{u}%
\,\,\,\mbox{and}\,\,\,\underline{v}\ll v\ll \hat{v}\right\} ,$$where $$(\hat{u},\hat{v})=\Lambda (w_{1},w_{2}) \label{3}$$with $w_{1},w_{2}$ fixed in (\[20\]) and $\Lambda >0$ is a constant which will be chosen later on. A simple computation gives that $\mathcal{O}_{R}$ and $\mathcal{\hat{O}}$ are open sets in $C^{1}(\overline{\Omega })\times
C^{1}(\overline{\Omega })$.
In what follows, we will assume without loss of generality that $$R>\max \{\Vert \underline{u}\Vert _{\infty },\Vert \overline{u}\Vert
_{\infty },\Vert \underline{v}\Vert _{\infty },\Vert \overline{v}\Vert
_{\infty },\Vert \hat{u}\Vert _{\infty },\Vert \hat{v}\Vert _{\infty }\}.$$In the sequel, we use the notation $u_{1}\ll u_{2}$ when $u_{1},u_{2}\in
C^{1}(\overline{\Omega })$ satisfy: $$\begin{array}{c}
u_{1}(x)<u_{2}(x)\,\,\,\forall x\in \Omega \,\,\,\mbox{and}\,\,\,\frac{%
\partial u_{2}}{\partial \nu }<\frac{\partial u_{1}}{\partial \nu }\,\,\,%
\mbox{on}\,\,\,\partial \Omega ,%
\end{array}%$$where $\nu $ is the outward normal to $\partial \Omega $.
The next proposition is useful for proving our second main result.
\[P1\]Assume (\[h1\]) holds. Then all solutions $(u,v)$ of (\[p\]) within $[\underline{u},\overline{u}]\times \lbrack \underline{v},\overline{v}%
]$ verifies $$\begin{array}{c}
u(x)\ll \hat{u}(x)\text{ \ and \ }v(x)\ll \hat{v}(x)\text{ \ in }\Omega .%
\end{array}
\label{c3}$$
From (\[32\]), (\[c3\]), (\[h1\]), (\[36\]), (\[67\]), (\[20\]) and (\[21\]), it follows that$$\begin{array}{l}
-\Delta _{p}u=u^{\alpha _{1}}v^{\beta _{1}}\leq \underline{u}^{\alpha _{1}}%
\overline{v}^{\beta _{1}}\leq (C^{-1}\frac{c_{0}}{2}\phi _{1,p})^{\alpha
_{1}}(Cc_{1}^{\prime }\phi _{1,q})^{\beta _{1}} \\
\leq C^{-\alpha _{1}+\beta _{1}}(\frac{c_{0}}{2})^{\alpha
_{1}}(c_{1}^{\prime }M)^{\beta _{1}}\phi _{1,p}^{\alpha _{1}}\leq C^{-\alpha
_{1}+\beta _{1}}(\frac{c_{0}}{2})^{\alpha _{1}}(c_{1}^{\prime }M)^{\beta
_{1}}(c_{3}w_{1})^{\alpha _{1}} \\
< \Lambda ^{p-1}w_{1}^{\alpha _{1}}=-\Delta _{p}(\Lambda w_{1})=-\Delta _{p}%
\hat{u}\text{ in }\Omega ,%
\end{array}
\label{69}$$provided that $\Lambda $ is large enough. Proceeding in the same way with the second equation in (\[p\]) results in$$\begin{array}{c}
-\Delta _{q}v < -\Delta _{q}(\Lambda \xi _{2})=-\Delta _{q}\hat{v}\text{\ \
in }\Omega ,%
\end{array}%$$for $\Lambda $ large enough. Consequently, the strong comparison principle found in [@AR Proposition 2.6] leads to the conclusion. This ends the proof.
An auxiliary problem
--------------------
In this subsection, we will use the Topological degree to get the second solution. However, the singular terms in system (\[p\]) prevents the degree calculation to be well defined. To overcome this difficulty, we disturb system (\[p\]) by introducing a parameter $\varepsilon \in (0,1)$. This gives rise to a regularized system for (\[p\]) defined for $%
\varepsilon >0$ as follows:$$\left\{
\begin{array}{ll}
-\Delta _{p}u=\left( u+\varepsilon \right) ^{\alpha _{1}}v^{\beta _{1}} &
\text{ in }\Omega , \\
-\Delta _{q}v=u^{\alpha _{2}}\left( v+\varepsilon \right) ^{\beta _{2}} &
\text{ in }\Omega , \\
u(x),v(x)>0 & \text{ in } \Omega, \\
u,v=0 & \text{ on }\partial \Omega .%
\end{array}%
\right. \tag{$P_{r}$} \label{pr}$$We apply the degree theory for the regularized problem (\[pr\]). This leads to find a positive solution for (\[pr\]) lying outside of the set $%
\mathcal{\hat{O}}$. Then the existence of a second solution of (\[p\]) is obtain by passing to the limit in (\[pr\]) as $\varepsilon \rightarrow 0$. The proof comprises four steps.
\[R3\]It is very important to observe that the same reasoning exploited in the proof of Theorem \[T1\] and Proposition \[P1\] furnishes that problem (\[pr\]) has a (positive) solution $(u_{\varepsilon
},v_{\varepsilon })\in C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma }(%
\overline{\Omega }),$ $\gamma \in (0,1)$, within $\left[ \underline{u},%
\overline{u}\right] \times \left[ \underline{v},\overline{v}\right] ,$ where functions $(\underline{u},\underline{v})$ and $(\overline{u},\overline{v})$ are sub-supersolutions of (\[pr\]) and $(u_{\varepsilon },v_{\varepsilon
}) $ verifies$$\begin{array}{c}
u_{\varepsilon }(x)\ll \hat{u}(x)\text{ \ and \ }v_{\varepsilon }(x)\ll \hat{%
v}(x)\text{ \ in }\Omega \text{,}%
\end{array}%$$for all $\varepsilon \in (0,1)$.
**Topological degree: The first estimate.**
We transform the problem (\[pr\]) to one with helpful monotonicity properties. To this end, let us introduce the functions $$\widetilde{\phi }=\left\{
\begin{array}{l}
R\text{ \ if }\phi \geq R \\
\phi \text{ \ if }\underline{u}\leq \phi \leq R \\
\underline{u}\text{ \ if }\phi \leq \underline{u}%
\end{array}%
\right. ,\text{ \ }\widetilde{\varphi }=\left\{
\begin{array}{l}
R\text{ \ if }\varphi \geq R \\
\phi \text{ \ if }\underline{v}\leq \varphi \leq R \\
\underline{v}\text{ \ if }\varphi \leq \underline{v},%
\end{array}%
\right. \label{5**}$$where $(\underline{u},\underline{v})$ and $R$ are given by (\[30\]) and (\[15\]), respectively. Define the operators$$\begin{array}{c}
T_{p,\varepsilon }(u)=-\Delta _{p}u+\rho \max \{(\underline{u}+\varepsilon
)^{\alpha _{1}-1}R^{\beta _{1}},u^{p-1}\}, \\
T_{q,\varepsilon }(v)=-\Delta _{q}v+\rho \max \{R^{\alpha _{2}}(\underline{v}%
+\varepsilon )^{\beta _{2}-1},v^{q-1}\},%
\end{array}%$$for $t\in \lbrack 0,1]$, $\varepsilon \in (0,1)$ and a constant $\rho >0$. We shall study the homotopy class of problem$$\left\{
\begin{array}{ll}
T_{p,\varepsilon }(u)={f_{1,\varepsilon ,t}(x,}\widetilde{{u}}{,}\widetilde{{%
v}}{)} & \text{ in }\Omega , \\
T_{q,\varepsilon }(v)={f_{2,\varepsilon ,t}(x,\widetilde{{u}}{,}\widetilde{{v%
}})} & \text{ in }\Omega , \\
u,v>0\text{ \ in }\Omega , & \\
u,v=0\text{ \ on }\partial \Omega , &
\end{array}%
\right. \tag{$P_{f}$} \label{50}$$where functions ${f_{1,\varepsilon ,t}}$ and ${f_{2,\varepsilon ,t}}$ are defined as follows:$$\begin{array}{c}
{f_{1,\varepsilon ,t}(x,\widetilde{{u}}{,}\widetilde{{v}})}=t(\widetilde{u}%
+\varepsilon )^{\alpha _{1}}\widetilde{v}^{\beta _{1}}+m(1-t)\widetilde{u}%
^{p-1} \\
+\rho \max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},%
\widetilde{u}^{p-1}\},%
\end{array}
\label{15**}$$$$\begin{array}{c}
{f_{2,\varepsilon ,t}(x,\widetilde{{u}}{,}\widetilde{{v}})}=t\widetilde{u}%
^{\alpha _{2}}(\widetilde{v}+\varepsilon )^{\beta _{2}}+m(1-t)\widetilde{v}%
^{q-1} \\
+\rho \max \{R^{\alpha _{2}}(\underline{v}+\varepsilon )^{\beta _{2}-1},%
\widetilde{v}^{q-1}\},%
\end{array}
\label{15*}$$with a constant $m>\max \{\lambda _{1,p},\lambda _{1,q}\}$. In the sequel, we fix the constant $\rho >0$ in (\[50\]) sufficiently large so that the following inequalities are satisfied: $$\begin{array}{c}
t\alpha _{1}(s_{1}+\varepsilon )^{\alpha _{1}-1}s_{2}^{\beta _{1}}+\rho \max
\{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta
_{1}},(p-1)s_{1}^{p-2}\}\geq 0%
\end{array}%$$and $$\begin{array}{c}
t\beta _{2}(s_{2}+\varepsilon )^{\beta _{2}-1}s_{1}^{\alpha _{2}}+\rho \max
\{R^{\alpha _{2}}(\underline{v}+\varepsilon )^{\beta
_{2}-1},(q-1)s_{2}^{q-2}\}\geq 0,%
\end{array}%$$uniformly in $x\in \Omega ,$ for $(s_{1},s_{2})\in \lbrack \underline{u}%
,R]\times \lbrack \underline{v},R],$ $\varepsilon \in (0,1)$. By the above choice of $\rho $, the term in the right-hand side of first (resp. second) equation in (\[50\]) increases as $u$ (resp. $v$) increases, for all $%
\varepsilon >0$ small.
The next result is crucial in our approach, because it establishes an important prior estimate for system (\[50\]). Moreover, it is also shown that the solutions of problem (\[50\]) cannot occur outside the rectangle formed by the subsolution $(\underline{u},\underline{v})$ and the constant $%
R $.
\[P2\]Assume (\[h1\]) holds. If $(u,v)$ is a solution of (\[50\]), then $(u,v)$ belongs to $C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma
}(\overline{\Omega })$ for some $\gamma \in (0,1)$ and satisfies$$\left\Vert u\right\Vert _{C^{1,\gamma }(\overline{\Omega })},\left\Vert
v\right\Vert _{C^{1,\gamma }(\overline{\Omega })}<R. \label{31}$$Moreover, it holds$$\begin{array}{c}
\underline{u}(x)\ll u(x)\text{ \ and \ }\underline{v}(x)\ll v(x)\text{ \ in }%
\Omega ,\text{ \ }\forall t\in \lbrack 0,1].%
\end{array}
\label{16}$$
First, by Moser’s iterations technique, we prove the boundedness for solutions of (\[50\]) in $L^{\infty }(\Omega )\times L^{\infty }(\Omega )$. Assuming (\[16\]) holds, it follows that$$\begin{array}{l}
\max \{(\underline{u}+\varepsilon )^{\alpha _{1}}R^{\beta
_{1}},u^{p-1}\}-\max \{(\underline{u}+\varepsilon )^{\alpha _{1}}R^{\beta
_{1}},\widetilde{u}^{p-1}\}\geq 0\text{ in }\Omega%
\end{array}
\label{68}$$and$$\begin{array}{l}
\max \{R^{\alpha _{2}}(\underline{v}+\varepsilon )^{\beta
_{2}},v^{q-1}\}-\max \{R^{\alpha _{2}}(\underline{v}+\varepsilon )^{\beta
_{2}},\widetilde{v}^{q-1}\}\geq 0\text{ in }\Omega .%
\end{array}
\label{68*}$$Then, $$\left\{
\begin{array}{l}
-\Delta _{p}u\leq \widetilde{u}^{\alpha _{1}}\widetilde{v}^{\beta _{1}}+m%
\widetilde{u}^{p-1}\text{ in }\Omega , \\
-\Delta _{q}v\leq \widetilde{u}^{\alpha _{2}}\widetilde{v}^{\beta _{2}}+m%
\widetilde{v}^{q-1}\text{ in }\Omega , \\
u,v>0\text{ in }\Omega , \\
u,v=0\text{ on }\partial \Omega .%
\end{array}%
\right. \label{500}$$Given a constant $A\in (0,R]$, define on $\Omega $ the functions$$\begin{array}{l}
u_{A}=\min \{u(x),A\}\text{ \ and \ }v_{A}=\min \{v(x),A\}.%
\end{array}%$$Acting on (\[50\]) with $$\begin{array}{c}
\left( \varphi ,\psi \right) =\left( u_{A}^{k_{1}p+1},v_{A}^{\overline{k}%
_{1}q+1}\right) ,%
\end{array}%$$where $$\begin{array}{c}
\left( k_{1}+1\right) p=p^{\ast }\text{ and }\left( \overline{k}%
_{1}+1\right) q=q^{\ast },%
\end{array}%$$and integrating over $\Omega $ we get $$\begin{array}{l}
\left( k_{1}p+1\right) \int_{\Omega }\left\vert \nabla u_{A}\right\vert
^{p}u_{A}^{k_{1}p}\text{ }dx\leq \int_{\Omega }(\widetilde{u}^{\alpha _{1}}%
\widetilde{v}^{\beta _{1}}+m\widetilde{u}^{p-1})u_{A}^{k_{1}p}\text{ }dx%
\end{array}
\label{36*}$$and$$\begin{array}{c}
\left( \overline{k}_{1}q+1\right) \int_{\Omega }\left\vert \nabla
v_{A}\right\vert ^{q}v_{A}^{\overline{k}_{1}q}\text{ }dx\leq \int_{\Omega }(%
\widetilde{u}^{\alpha _{2}}\widetilde{v}^{\beta _{2}}+m\widetilde{v}%
^{q-1})v_{A}^{\overline{k}_{1}q+1}\text{ }dx.%
\end{array}
\label{37}$$By the Sobolev embedding theorem, the left-hand sides of (\[36\*\]) and (\[37\]) are estimated from below as follows $$\begin{array}{l}
(k_{1}p+1)\int_{\Omega }|\nabla u_{A}|^{p}u_{A}^{k_{1}p}=\frac{k_{1}p+1}{%
\left( k_{1}+1\right) ^{p}}\int_{\Omega }|\nabla u_{A}^{k_{1}+1}|^{p}\geq
C_{1}\frac{\left( k_{1}p+1\right) }{\left( k_{1}+1\right) ^{p}}\left\Vert
u_{A}\right\Vert _{\left( k_{1}+1\right) p^{\ast }}^{p^{\ast }}%
\end{array}
\label{l6}$$and $$\begin{array}{l}
\left( \overline{k}_{1}q+1\right) \int_{\Omega }\left\vert \nabla
v_{A}\right\vert ^{q}v_{A}^{\overline{k}_{1}q}=\frac{\left( \overline{k}%
_{1}q+1\right) }{\left( \overline{k}_{1}+1\right) ^{q}}\int_{\Omega }|\nabla
v_{A}^{\overline{k}_{1}+1}|^{q}\geq C_{1}^{\prime }\frac{\left( \overline{k}%
_{1}q+1\right) }{\left( \overline{k}_{1}+1\right) ^{q}}\left\Vert
v_{A}\right\Vert _{\left( \overline{k}_{1}+1\right) q^{\ast }}^{q^{\ast }},%
\end{array}
\label{l7}$$where $C_{1}$ and $C_{1}^{\prime }$ are some positive constants. By noticing that $k_{1}p+1+\alpha _{1}>0$ and $\overline{k}_{1}q+1+\beta _{2}>0$ it turns out that$$\begin{array}{l}
\int_{\Omega }(\widetilde{u}^{\alpha _{1}}\widetilde{v}^{\beta _{1}}+m%
\widetilde{u}^{p-1})u_{A}^{k_{1}p+1}\text{ }dx\leq \int_{\Omega
}u_{A}^{\alpha _{1}+k_{1}p+1}v^{\beta _{1}}\text{ }dx+m\int_{\Omega
}u^{(k_{1}+1)p}\text{ }dx \\
\leq \int_{\Omega }u^{\alpha _{1}+k_{1}p+1}v^{\beta _{1}}\text{ }%
dx+m\int_{\Omega }u^{(k_{1}+1)p}\text{ }dx%
\end{array}
\label{60}$$and$$\begin{array}{l}
\int_{\Omega }(\widetilde{u}^{\alpha _{2}}\widetilde{v}^{\beta _{2}}+m%
\widetilde{v}^{q-1})v_{A}^{\overline{k}_{1}q+1}\text{ }dx\leq \int_{\Omega
}u^{\alpha _{2}}v_{A}^{\overline{k}_{1}q+1+\beta _{2}}\text{ }%
dx+m\int_{\Omega }v^{(\overline{k}_{1}+1)q}\text{ }dx \\
\leq \int_{\Omega }u^{\alpha _{2}}v^{\overline{k}_{1}q+1+\beta _{2}}\text{ }%
dx+m\int_{\Omega }v^{(\overline{k}_{1}+1)q}\text{ }dx.%
\end{array}
\label{60*}$$
Then, following the quite similar argument as in [@MM], we obtain that $%
(u,v)\in L^{\infty }(\Omega )\times L^{\infty }(\Omega )$ and there exists a constant $L>0,$ independent of $R$, such that $\left\Vert u\right\Vert
_{\infty },\left\Vert v\right\Vert _{\infty }\leq L$. Furthermore, from ([30]{}) and (\[c2\]), it holds$$\begin{array}{l}
\widetilde{u}^{\alpha _{1}}\widetilde{v}^{\beta _{1}}+m\widetilde{u}%
^{p-1}\leq \widetilde{u}^{\alpha _{1}}(\widetilde{v}^{\beta _{1}}+m%
\widetilde{u}^{p-1-\alpha _{1}}) \\
\leq \underline{u}^{\alpha _{1}}(\left\Vert v\right\Vert _{\infty }^{\beta
_{1}}+m\left\Vert u\right\Vert _{\infty }^{p-1-\alpha _{1}}) \\
\leq (C^{-1}\frac{c_{0}}{2}\phi _{1,p})^{\alpha _{1}}(L^{\beta
_{1}}+mL^{p-1-\alpha _{1}})\leq C_{1}d(x)^{\alpha _{1}}\text{ \ in }\Omega%
\end{array}
\label{70}$$and$$\begin{array}{l}
\widetilde{u}^{\alpha _{2}}\widetilde{v}^{\beta _{2}}+m\widetilde{v}%
^{q-1}\leq \underline{v}^{\beta _{2}}(\left\Vert u\right\Vert _{\infty
}^{\alpha _{2}}+m\left\Vert v\right\Vert _{\infty }^{q-1-\beta _{2}})\leq
C_{2}d(x)^{\beta _{2}}\text{ \ in }\Omega ,%
\end{array}
\label{70*}$$with positive constants $C_{1}$ and $C_{2}$. Thus, on the basis of (\[68\]), (\[68\*\]), (\[70\]), (\[70\*\]) and (\[500\]), the nonlinear regularity theory found in [@Hai] guarantees that the solutions $(u,v)$ of (\[50\]) belong to $C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma
}(\overline{\Omega })$ for some $\gamma \in (0,1)$ and satisfy (\[31\]).
Now, let us prove (\[16\]). We only show the first inequality in (\[16\]) because the second one can be justified similarly. To this end, we set the functions $f,g:\Omega \rightarrow \mathbb{R}$ given by $$f(x)=C^{-(p-1)}h_{1}(x)+\rho \max \{(\underline{u}+\varepsilon )^{\alpha
_{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\}$$and $$g(x)={f_{1,\varepsilon ,t}}({x,}\widetilde{u},\widetilde{v}).$$By Remark \[R3\], the strict inequalities in (\[18\]), (\[29\*\]) and the monotonicity of ${f_{1,\varepsilon ,t}}$ imply$$\begin{array}{l}
f(x)=-C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\} \\
<t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t){m}\underline{u}^{p-1}+\rho \max \{(\underline{u}{+\varepsilon )}%
^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\} \\
={f_{1,\varepsilon ,t}}({x,}\underline{u},\underline{v})\leq {%
f_{1,\varepsilon ,t}}({x,}\widetilde{u},\widetilde{v})=g(x)\text{ \ in }%
\Omega _{\delta }%
\end{array}
\label{10}$$and$$\begin{array}{l}
f(x)=C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\} \\
<(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta _{1}}+\rho
\max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{%
u}^{p-1}\}\text{ \ in }\Omega \backslash \overline{\Omega }_{\delta },%
\end{array}
\label{13**}$$for all $t\in \lbrack 0,1]$ and for all $\varepsilon \in (0,1)$. On another hand, by (\[c2\]), (\[h1\]), (\[30\]), (\[256\]) and (\[23\]), we obtain$$\begin{array}{l}
(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}=(t+1-t)(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}%
^{\beta _{1}} \\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)(C^{-1}\frac{c_{0}}{2}\phi _{1,p})^{\alpha
_{1}}(C^{-1}c_{1}^{\prime }\phi _{1,q})^{\beta _{1}} \\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)(C^{-1}\frac{c_{0}}{2}\mu )^{\alpha _{1}}(C^{-1}c_{1}^{\prime
}M)^{\beta _{1}} \\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)m(C^{-1}\frac{c_{0}}{2}\mu )^{p-1} \\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t){m}\underline{u}^{p-1}\text{ \ in }\Omega \backslash \overline{%
\Omega }_{\delta },%
\end{array}
\label{13***}$$provided that $m>0$ sufficiently large, for all $t\in \lbrack 0,1]$ and all $%
\varepsilon \in (0,1)$. Combining (\[13\*\*\]) with (\[13\*\*\*\]) and using the monotonicity of ${f_{1,\varepsilon ,t}}$, one gets$$\begin{array}{l}
f(x)=C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\} \\
<{f_{1,\varepsilon ,t}}({x,}\underline{u},\underline{v})\leq {%
f_{1,\varepsilon ,t}}({x,}\widetilde{u},\widetilde{v})=g(x)\text{ \ in }%
\Omega \backslash \overline{\Omega }_{\delta }%
\end{array}
\label{10*}$$for all $t\in \lbrack 0,1]$ and all $\varepsilon \in (0,1)$. Consequently, it follows from (\[10\]) and (\[10\*\]) that for each compact set $%
K\subset \subset \Omega ,$ there is a constant $\tau =\tau (K)>0$ such that$$\begin{array}{l}
f(x)+\tau =-C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\}+\tau \\
\leq {f_{1,\varepsilon ,t}}({x,}\widetilde{u},\widetilde{v})=g(x)\text{ \
a.e. in }K\cap \Omega _{\delta }%
\end{array}%$$and $$\begin{array}{l}
f(x)+\tau =C^{-(p-1)}\phi _{1,p}^{\alpha _{1}}(x)+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\}+\tau \\
\leq {f_{1,\varepsilon ,t}}({x,}\widetilde{u},\widetilde{v})=g(x)\text{ \
a.e. in }K\cap \Omega \backslash \overline{\Omega }_{\delta },%
\end{array}%$$for all $t\in \lbrack 0,1]$ and all $\varepsilon \in (0,1)$. Hence, given a compact set $k\subset \subset \Omega $, there is $\tau >0$ such that $$f(x)+\tau \leq g(x),\quad \forall x\in K$$and so, $f\prec g$ and $f,g\in L_{loc}^{\infty }(\Omega )$. Thereby, by the strong comparison principle (see Appendix, Proposition \[P0\]), we infer that $$u(x)\gg \underline{u}(x),\quad \forall x\in \Omega .$$The proof of the second inequality in (\[16\]) is carried out in a similar way. This complete the proof.
\[P6\]Under the assumption (\[h1\]) problem (\[50\]) has no solutions for $t=0$.
Arguing by contradiction, let $(u^{\ast },v^{\ast })\in C^{1,\gamma }(%
\overline{\Omega })\times C^{1,\gamma }(\overline{\Omega }),$ for certain $%
\gamma \in (0,1)$, be a nontrivial (positive) solution of (\[50\]) with $$(u^{\ast },v^{\ast })\in \mathcal{O}_{R}\text{ \ and }t=0. \label{49}$$From (\[c2\]) and (\[30\]) $$\begin{array}{l}
\underline{u}(x)=C^{-1}z_{1}(x)\geq C^{-1}\frac{c_{0}}{2}\phi _{1,p}(x)\text{
in }\Omega \text{.}%
\end{array}%$$In the sequel, we fix $u_{1}=C^{-1}\frac{c_{0}}{2}\phi _{1,p}$ and take $%
\lambda _{\delta }=\lambda _{1,p}+\delta $ for $\delta >0$. Let $u_{2}\in
C_{0}^{1}(\overline{\Omega })$ be the solution of the problem$$\left\{
\begin{array}{l}
-\Delta _{p}u_{2}=\lambda _{\delta }u_{1}^{p-1}\text{in }\Omega , \\
u_{2}=0\text{ on }\partial \Omega .%
\end{array}%
\right.$$Then for $\delta >0$ small and $m$ large enough, we have $$-\Delta _{p}u_{2}=\lambda _{\delta }u_{1}^{p-1}\leq m\widetilde{u}%
^{p-1}=-\Delta _{p}u^{\ast }$$and $$-\Delta _{p}u_{1}=\lambda _{1,p}u_{1}^{p-1}\leq \lambda _{\delta
}u_{1}^{p-1}=-\Delta _{p}u_{2}.$$By the weak comparison principle we get$$u_{1}\leq u_{2}\leq u^{\ast }\text{ in }\Omega \text{.}$$Now let us consider the solutions of the problems$$\left\{
\begin{array}{ll}
-\Delta _{p}u_{n}=\lambda _{\delta }u_{n-1}^{p-1} & \text{ in }\Omega , \\
u_{n}=0 & \text{ on }\partial \Omega .%
\end{array}%
\right.$$We obtain an increasing sequence $\{u_{n}\}$ such that$$u_{1}\leq u_{n-1}\leq u_{n}\leq u^{\ast }\text{ in }\Omega \text{.}$$Passing to the limit we get a positive solution $u\in W_{0}^{1,p}(\Omega )$ for problem$$\left\{
\begin{array}{ll}
-\Delta _{p}u=\lambda _{\delta }u^{p-1} & \text{ in }\Omega , \\
u=0 & \text{ on }\partial \Omega ,%
\end{array}%
\right.$$which is impossible for $\delta >0$ small enough because the first eigenvalue for $p$-Laplacian is isolate. Hence, problem (\[50\]) has no solutions for $t=0$.
Define the homotopy $\mathcal{H}_{\varepsilon }$ on $\left[ 0,1\right]
\times C^{1}(\overline{\Omega })\times C^{1}(\overline{\Omega })$ by$$\mathcal{H}_{\varepsilon }(t,u,v)=I(u,v)-\left(
\begin{array}{cc}
T_{p,\varepsilon }^{-1} & 0 \\
0 & T_{q,\varepsilon }^{-1}%
\end{array}%
\right) \times \left(
\begin{array}{l}
{f_{1,\varepsilon ,t}}({x,}\widetilde{{u}}{,}\widetilde{{v}}) \\
\multicolumn{1}{c}{f{_{2,\varepsilon ,t}}({x,}\widetilde{{u}}{,}\widetilde{{v%
}})}%
\end{array}%
\right) .$$According to Lemma \[L1\] (see Appendix) and because functions ${%
f_{\varepsilon ,t}}$ and ${g_{\varepsilon ,t}}$ belong to $C(\overline{%
\Omega })$ for all $x\in \overline{\Omega }$ and all $\varepsilon \in (0,1),$ $\mathcal{H}_{\varepsilon }$ is well defined. Furthermore, $\mathcal{H}%
_{\varepsilon }:\left[ 0,1\right] \times C^{1}(\overline{\Omega })\times
C^{1}(\overline{\Omega })\rightarrow C(\overline{\Omega })\times C(\overline{%
\Omega })$ is completely continuous for all $\varepsilon \in (0,1)$. This is due to the compactness of the operators $T_{p,\varepsilon
}^{-1},T_{q,\varepsilon }^{-1}:C(\overline{\Omega })\rightarrow C^{1}(%
\overline{\Omega }),$ for all $\varepsilon \in (0,1)$, see appendix for more details. Hence, $(u,v)\in \mathcal{O}_{R}$ is a solution for (\[pr\]) if, and only if, $$\begin{array}{c}
(u,v)\in \mathcal{O}_{R}\,\,\,\mbox{and}\,\,\,\mathcal{H}_{\varepsilon
}(1,u,v)=0.%
\end{array}%$$
From the previous Proposition \[P2\] and since $R$ is the a strict a priori bound, it is clear that solutions of (\[50\]) must lie in $\mathcal{%
O}_{R}$. Thus, the fact that problem (\[50\]) has no solutions for $t=0$ (see proposition \[P6\]) implies that $$\deg \left( \mathcal{H}_{\varepsilon }(0,\cdot ,\cdot ),\mathcal{O}%
_{R},0\right) =0\text{\ \ for all }\varepsilon \in (0,1).$$Consequently, from the homotopy invariance property, it follows that $$\begin{array}{c}
\deg \left( \mathcal{H}_{\varepsilon }(1,\cdot ,\cdot ),\mathcal{O}%
_{R},0\right) =\deg \left( \mathcal{H}_{\varepsilon }(0,\cdot ,\cdot ),%
\mathcal{O}_{R},0\right) =0\text{ for all }\varepsilon \in (0,1).%
\end{array}
\label{35}$$
**Topological degree: The second estimate.**
We show that the degree of an operator corresponding to the system (\[pr\]) is $1$ on the set $\mathcal{\hat{O}}$. To this end, we modify the problem to ensure that solutions cannot occur outside of the rectangle formed by $(%
\underline{u},\underline{v})$ and $(\hat{u},\hat{v})$. Set $$\widetilde{u}=\left\{
\begin{array}{l}
\hat{u}\text{ if }u\geq \hat{u} \\
u\text{ if }\underline{u}\leq u\leq \hat{u} \\
\underline{u}\text{ if }u\leq \underline{u}%
\end{array}%
\right. ,\text{ \ }\widetilde{v}=\left\{
\begin{array}{l}
\hat{v}\text{ if }v\geq \hat{v} \\
v\text{ if }\underline{v}\leq v\leq \hat{v} \\
\underline{v}\text{ if }v\leq \underline{v},%
\end{array}%
\right. \label{5*}$$and let us define the truncation problem$$\left\{
\begin{array}{ll}
T_{p,\varepsilon }(u)=g_{1,\varepsilon ,t}(x,u,v) & \text{in }\Omega , \\
T_{q,\varepsilon }(v)=g_{2,\varepsilon ,t}(x,u,v) & \text{in }\Omega , \\
u,v>0\text{ in }\Omega , & \\
u,v=0\text{ \ on }\partial \Omega , &
\end{array}%
\right. \tag{$P_{g}$} \label{2.7}$$with$$\begin{array}{c}
g_{1,\varepsilon ,t}(x,u,v)=t(\widetilde{u}+\varepsilon )^{\alpha _{1}}%
\widetilde{v}^{\beta _{1}}+(1-t)\eta (\phi _{1,p}+\varepsilon )^{\alpha _{1}}
\\
+\rho \max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},%
\widetilde{u}^{p-1}\},%
\end{array}%$$$$\begin{array}{c}
g_{2,\varepsilon ,t}(x,u,v)=t\widetilde{u}^{\alpha _{2}}(\widetilde{v}%
+\varepsilon )^{\beta _{2}}+(1-t)\eta (\phi _{1,q}+\varepsilon )^{\beta _{2}}
\\
+\rho \max \{R^{\alpha _{2}}(\underline{v}+\varepsilon )^{\beta _{2}-1},%
\widetilde{v}^{q-1}\},%
\end{array}%$$with a constant $\eta >0$. The constant $\rho >0$ is chosen sufficiently large so that the following inequalities are satisfy:[$$\begin{array}{c}
\alpha _{1}(s_{1}{+\varepsilon )}^{\alpha _{1}-1}s_{2}^{\beta _{1}}+\rho
\max \{(\underline{u}{+\varepsilon )}^{\alpha _{1}-1}R^{\beta
_{1}},(p-1)s_{1}^{p-2}\}\geq 0,%
\end{array}%$$]{} uniformly in $x\in \Omega $, for $(s_{1},s_{2})\in \lbrack \underline{u},%
\hat{u}]\times \lbrack \underline{v},\hat{v}],$ for $\varepsilon \in (0,1),$ and[$$\begin{array}{c}
\beta _{2}s_{1}^{\alpha _{2}}(s_{2}{+\varepsilon )}^{\beta _{2}-1}+\rho \max
\{R^{\alpha _{2}}(\underline{v}{+\varepsilon )}^{\beta
_{2}-1},(q-1)s_{2}^{q-2}\}\geq 0,\text{ }%
\end{array}%$$]{} uniformly in $x\in \Omega ,$ for $(s_{1},s_{2})\in \lbrack \underline{u},%
\hat{u}]\times \lbrack \underline{v},\hat{v}],$ for $\varepsilon \in (0,1)$.
We state the following result regarding truncation system (\[2.7\]).
\[P3\]Under condition (\[h1\]) every solution $(u,v)$ of (\[2.7\]) is in $C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma }(\overline{%
\Omega })$ for certain $\gamma \in (0,1),$ with $\left\Vert u\right\Vert
_{C^{1,\gamma }},\left\Vert v\right\Vert _{C^{1,\gamma }}<R$ and satisfies $$\begin{array}{c}
\underline{u}(x)\ll u(x)\ll \hat{u}(x)\text{ and }\underline{v}(x)\ll
v(x)\ll \hat{v}(x), \quad \forall x\in \Omega .%
\end{array}
\label{1*}$$
A quite similar argument as in the proof of Proposition \[P2\] provides that all solutions of (\[2.7\]) are in $C^{1,\gamma }(\overline{\Omega }%
)\times C^{1,\gamma }(\overline{\Omega })$ for certain $\gamma \in (0,1)$.
Let us prove (\[1\*\]). We only show the first part of inequalities in ([1\*]{}) because the second part can be justified similarly. To this end, we set the functions $f,\tilde{g}:\Omega \rightarrow \mathbb{R}$ given by $$f(x)=C^{-(p-1)}h_{1}(x)+\rho \max \{(\underline{u}+\varepsilon )^{\alpha
_{1}-1}R^{\beta _{1}},\underline{u}^{p-1}\}$$and $$\tilde{g}(x)=g_{1,\varepsilon ,t}({x,}\widetilde{u},\widetilde{v}).$$From Remark \[R3\], (\[c2\]) and (\[23\]), for all $\varepsilon \in
(0,1)$ and for all $t\in \lbrack 0,1]$, that$$\begin{array}{l}
(t+1-t)(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta _{1}}
\\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)(C^{-1}\frac{c_{0}}{2}\phi _{1,p}+\varepsilon )^{\alpha
_{1}}(C^{-1}c_{1}^{\prime }\phi _{1,q})^{\beta _{1}} \\
\leq t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)(C^{-1}\frac{c_{0}}{2}\phi _{1,p})^{\alpha
_{1}}(C^{-1}c_{1}^{\prime }M)^{\beta _{1}} \\
t(\underline{u}{+\varepsilon )}^{\alpha _{1}}\underline{v}^{\beta
_{1}}+(1-t)\eta (\phi _{1,p}+\varepsilon )^{\alpha _{1}}\text{ in }\Omega
\backslash \overline{\Omega }_{\delta }%
\end{array}
\label{14}$$provided that $\eta >0$ is sufficiently large. Then, following the quite similar argument which proves (\[16\]) in Proposition \[P2\], we obtain for each compact set $K\subset \Omega ,$ there is a constant $\tau =\tau
(K)>0$ such that $$f(x)+\tau \leq \tilde{g}(x)\quad \mbox{a.e in }\quad \Omega .$$Hence, $f\prec \tilde{g}$ and $f,\tilde{g}\in L_{loc}^{\infty }(\Omega )$. Thereby, by the strong comparison principle (see Proposition \[P0\] in Appendix) we infer that $$u(x)\gg \underline{u}(x)\quad \forall x\in \Omega .$$
Let us define the homotopy $\mathcal{N}_{\varepsilon }$ on $\left[ 0,1\right]
\times C^{1}(\overline{\Omega })\times C^{1}(\overline{\Omega })$ by$$\mathcal{N}_{\varepsilon }(t,u,v)=I(u,v)-\left(
g_{1,\varepsilon ,t}({x,}u,v) \\
\multicolumn{1}{c}{g_{2,\varepsilon ,t}({x,}u,v)}%
\end{array}%
\right) . \label{17}$$Clearly, Lemma \[L1\] together with Proposition \[P4\] (see Appendix) imply that $\mathcal{N}_{\varepsilon }$ is well defined and completely continuous homotopy for all $\varepsilon \in (0,1)$ and all $t\in \lbrack
0,1]$. Moreover, $(u,v)\in \mathcal{\hat{O}}$ is a solution of system ([pr]{}) if, and only if, $$\begin{array}{c}
(u,v)\in \mathcal{\hat{O}}\,\,\,\mbox{and}\,\,\,\mathcal{N}_{\varepsilon
}(1,u,v)=0\text{ for all }\varepsilon \in (0,1).%
\end{array}%$$
In view of Proposition \[P3\] and from the definition of function $\hat{u}
$ and $\hat{v}$ it follows that all solutions of (\[2.7\]) are also solutions of (\[pr\]). Moreover, these solutions must be in the set $%
\mathcal{\hat{O}}$. Moreover, for $t=0$ in (\[17\]), Minty-Browder Theorem together with Hardy-Sobolev Inequality and [@Hai Lemma 3.1] ensure that problems $$\left\{
\begin{array}{ll}
-\Delta _{p}u=\eta (\phi _{1,p}+\varepsilon )^{\alpha _{1}} & \text{in }%
\Omega \\
u=0 & \text{on }\partial \Omega%
\end{array}%
\right. \text{ \ and \ }\left\{
\begin{array}{ll}
-\Delta _{q}v=\eta (\phi _{1,q}+\varepsilon )^{\beta _{2}} & \text{in }\Omega
\\
v=0 & \text{on }\partial \Omega ,%
\end{array}%
\right.$$admit unique positive solutions $\grave{u}_{\varepsilon }$ and $\grave{v}%
_{\varepsilon }$ in $C^{1,\gamma }(\overline{\Omega })$ for certain $\gamma
\in (0,1)$ and for $\varepsilon \in (0,1)$, respectively. Then, the homotopy invariance property of the degree gives $$\begin{array}{ll}
\deg (\mathcal{N}_{\varepsilon }(1,\cdot ,\cdot ),\mathcal{\hat{O}},0) &
=\deg (\mathcal{N}_{\varepsilon }(0,\cdot ,\cdot ),\mathcal{\hat{O}},0) \\
& =\deg (\mathcal{N}_{\varepsilon }(0,\cdot ,\cdot ),B_{R}(0)),0) \\
& =1.%
\end{array}
\label{55}$$Since $$\mathcal{H}_{\varepsilon }(1,\cdot ,\cdot )=\mathcal{N}_{\varepsilon
}(1,\cdot ,\cdot )\,\,\,\text{in}\,\,\,\mathcal{\hat{O}},$$it follows that $$\begin{array}{c}
\deg (\mathcal{H}_{\varepsilon }(1,\cdot ,\cdot ),\mathcal{\hat{O}},0)=1,%
\end{array}
\label{56}$$for all $\varepsilon \in (0,1)$.
**Topological degree: The third estimate.**
Herafter, we will assume that $$\mathcal{H}_{\varepsilon }(1,u,v) \not=0 \,\,\,\, \forall (u,v) \in \partial
\mathcal{\hat{O}},$$ otherwise we will have a solution $(\breve{u}_{\varepsilon },\breve{v}%
_{\varepsilon }) \in \partial \mathcal{\hat{O}}$, which is different from the solution $(u,v)$ in Theorem \[T1\], because $(u,v) \in \mathcal{\hat{O}%
}$. Here, we have used that $\mathcal{\hat{O}}$ is an open set, then $(u,v)
\notin \partial \mathcal{\hat{O}}$.
By (\[55\]), (\[56\]) and (\[35\]), we deduce from the excision property of Leray-Schauder degree that$$\begin{array}{c}
\deg (\mathcal{H}_{\varepsilon }(1,\cdot ,\cdot ),\mathcal{O}_{R}\backslash
\overline{\mathcal{\hat{O}}},0)={-1}%
\end{array}%$$and thus problem (\[pr\]) has a solution $(\breve{u}_{\varepsilon },\breve{%
v}_{\varepsilon })\in C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma }(%
\overline{\Omega })$ for some $\gamma \in (0,1)$ with $$(\breve{u}_{\varepsilon },\breve{v}_{\varepsilon })\in \mathcal{O}%
_{R}\backslash \overline{\mathcal{\hat{O}}} \label{61}$$In view of remark (\[R3\]), $(\breve{u}_{\varepsilon },\breve{v}%
_{\varepsilon })$ is necessarily another solution for problem (\[pr\]).
**Proof of Theorem \[T2\]:**
Set $\varepsilon =\frac{1}{n}$ with any positive integer $n\geq 1$. From (\[61\]) with $\varepsilon =\frac{1}{n}$, we know that there exist $(\breve{%
u}_{n},\breve{v}_{n}):=(\breve{u}_{\frac{1}{n}},\breve{v}_{\frac{1}{n}})$ bounded in $C^{1,\gamma }(\overline{\Omega })\times C^{1,\gamma }(\overline{%
\Omega })$ for some $\gamma \in (0,1)$ such that$$\left\{
\begin{array}{l}
-\Delta _{p}\breve{u}_{n}=\left( \breve{u}_{n}+\frac{1}{n}\right) ^{\alpha
_{1}}\breve{v}_{n}^{\beta _{1}}\text{ in }\Omega , \\
-\Delta _{q}\breve{v}_{n}=\breve{u}_{n}^{\alpha _{2}}\left( \breve{v}_{n}+%
\frac{1}{n}\right) ^{\beta _{2}}\text{ in }\Omega , \\
\breve{u}_{n}=\breve{v}_{n}=0\text{ on }\partial \Omega ,%
\end{array}%
\right. \label{122*}$$satisfying$$\begin{array}{c}
(\breve{u}_{n},\breve{v}_{n})\in \mathcal{O}_{R}\setminus \overline{\mathcal{%
\hat{O}}}\,\,\,\forall n\in \mathbb{N}.%
\end{array}
\label{62}$$Employing Arzelà-Ascoli’s theorem, we may pass to the limit in $C^{1}(%
\overline{\Omega })\times C^{1}(\overline{\Omega })$ and the limit functions $(\breve{u},\breve{v})\in C^{1}(\overline{\Omega })\times C^{1}(\overline{%
\Omega })$ satisfy (\[p\]) with $$(\breve{u},\breve{v})\in \mathcal{O}_{R}\setminus \overline{\mathcal{\hat{O}}%
} \label{63}$$Finally, on account of (\[63\]) and Proposition \[P1\], we achieve that $%
(\breve{u},\breve{v})$ is a second solution of problem (\[p\]). This complete the proof of Theorem \[T2\].
Appendix
========
In this section, we establish a version of the strong comparison principle for the operators $T_{p,\varepsilon }$ and $T_{q,\varepsilon }$ introduced in Section \[S3\] and we study the compactness of the inverse of these operators. We only prove the strong comparison principle for the operator $%
T_{p,\varepsilon }$ and the compactness of $T_{p,\varepsilon }^{-1}$ because for $T_{q,\varepsilon }$ and $T_{q,\varepsilon }^{-1}$ the proof can be justified similarly.
**1. Strong comparison principle.**
\[P0\]Let $u_{1},u_{2}\in C^{1,\beta }(\overline{\Omega }),$ $\beta \in
(0,1),$ be the solutions of the problems$$\left\{
\begin{array}{ll}
T_{p,\varepsilon }(u_{1})={f(x)} & \text{in }\Omega , \\
u_{1}=0 & \text{on }\partial \Omega ,%
\end{array}%
\right. \text{ and\ }\left\{
\begin{array}{ll}
T_{p,\varepsilon }(u_{2})={g(x)} & \text{in }\Omega , \\
u_{2}=0 & \text{on }\partial \Omega ,%
\end{array}%
\right.$$where $$T_{p,\varepsilon }(u)=-\Delta _{p}u+\rho \max \{(\underline{u}+\varepsilon
)^{\alpha _{1}-1}R^{\beta _{1}},\left\vert u\right\vert ^{p-2}u\},$$for some $\varepsilon \in (0,1)$ and $f,g\in L_{loc}^{\infty }(\Omega )$. If $f\prec g$, that is, for each compact set $K\subset \Omega $, there is $\tau
=\tau (K)>0$ such that $$f(x)+\tau \leq g(x)\quad \mbox{a.e in}\quad K,$$then $u_{1}\ll u_{2}$.
The proof is very similar to those of Proposition 2.6 in [@AR], it is sufficient to observe that that for all $a,b,c,d\in \mathbb{R}$ the following inequality holds: $$\begin{array}{c}
|\max \{a,b\}-\max \{c,d\}|\leq \max \{\left\vert a-c\right\vert ,\left\vert
b-d\right\vert \},%
\end{array}
\label{DES}$$which leads to $$\begin{array}{l}
|\max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta
_{1}},\left\vert u_{1}\right\vert ^{p-2}u_{1}\}-\max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert u_{2}\right\vert
^{p-2}u_{2}\}| \\
\leq \left\vert \left\vert u_{1}\right\vert ^{p-2}u_{1}-\left\vert
u_{2}\right\vert ^{p-2}u_{2}\right\vert .%
\end{array}%$$The last inequality is a key point in the arguments found in [@AR].
**2. Compactness of $T_{p,\varepsilon }$.**
Let us consider the Dirichlet problem$$\left\{
\begin{array}{ll}
T_{p,\varepsilon }(u)={f(x)} & \text{in }\Omega , \\
u=0\text{ \ } & \text{on }\partial \Omega ,%
\end{array}%
\right. \label{1***}$$where $\Omega $ is a bounded domain in $%
^{N},$ ${f\in }W^{-1,p^{\prime }}(\Omega )$ and $T_{p,\varepsilon
}:W_{0}^{1,p}(\Omega )\rightarrow W^{-1,p^{\prime }}(\Omega )$ is the operator defined as follows:$$\begin{array}{c}
T_{p,\varepsilon }(u)=-\Delta _{p}u+\rho \max \{(\underline{u}+\varepsilon
)^{\alpha _{1}-1}R^{\beta _{1}},\left\vert u\right\vert ^{p-2}u\}%
\end{array}%$$for all $\varepsilon \in (0,\varepsilon _{0})$.
A solution of (\[1\*\*\*\]) is understood in the weak sense, that is $u\in
W_{0}^{1,p}(\Omega )$ satisfying$$\begin{array}{c}
\int_{\Omega }\left( |\nabla u|^{p-2}\nabla u\nabla \varphi +\rho \max \{(%
\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert
u\right\vert ^{p-2}u\}\varphi \right) \ dx=\int_{\Omega }f\left( x\right)
\varphi \ dx%
\end{array}
\label{7***}$$for all $\varphi \in W_{0}^{1,p}(\Omega )$.
\[L1\]Problem (\[1\*\*\*\]) possesses a unique solution $u_{\varepsilon }$ in $W_{0}^{1,p}(\Omega )$ for all $\varepsilon \in (0,\varepsilon _{0}).$ Moreover, if $f \in L^{\infty}(\Omega)$ the solution $u_{\varepsilon }$ belongs to $C^{1,\gamma }(\Omega ), $ for certain $\gamma \in (0,1),$ and satisfies$$\left\Vert u_{\varepsilon }\right\Vert _{C^{1,\gamma }}<\overline{R},
\label{2***}$$where $\overline{R}$ is a positive constant, which depends of $\|f\|_\infty$.
To prove the lemma we apply Minty-Browder Theorem. To do so, we prove that the operator $T_{p,\varepsilon }$ is continuous, strict monotone and coercive for all $\varepsilon \in (0,\varepsilon _{0})$. Let us show that $%
T_{p,\varepsilon }$ is a continuous operator. For $\{u_{n}\}\subset
W_{0}^{1,p}(\Omega )$ with $u_{n}\rightarrow u$ in $W_{0}^{1,p}(\Omega )$, we have $$\begin{array}{l}
\left\Vert T_{p,\varepsilon }(u_{n})-T_{p,\varepsilon }(u)\right\Vert
_{W^{-1,p^{\prime }}(\Omega )}=\underset{\varphi \in W_{0}^{1,p}(\Omega
),\left\Vert \varphi \right\Vert _{1,p}\leq 1}{\sup }\left\vert \left\langle
T_{p,\varepsilon }(u_{n})-T_{p,\varepsilon }(u),\varphi \right\rangle
\right\vert \\
\\
\leq \int_{\Omega }\left\vert \left\langle \left( |\nabla u_{n}|^{p-2}\nabla
u_{n}-|\nabla u|^{p-2}\nabla u\right) ,\nabla \varphi \right\rangle
\right\vert \text{ }dx \\
\\
+\rho \int_{\Omega }\left\vert \max \{(\underline{u}+\varepsilon )^{\alpha
_{1}-1}R^{\beta _{1}},\left\vert u_{n}\right\vert ^{p-2}u_{n}\}-\max \{(%
\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert
u\right\vert ^{p-2}u\}\right\vert \left\vert \varphi \right\vert dx.%
\end{array}%$$Then if $p\geq 2$, using [@GM Lemma $5.3$] together with Hölder’s inequality and (\[DES\]), we derive $$\begin{array}{l}
\left\Vert T_{p,\varepsilon }(u_{n})-T_{p,\varepsilon }(u)\right\Vert
_{W^{-1,p^{\prime }}(\Omega )}\leq c_{p}\left\Vert |\nabla u|+|\nabla
u|\right\Vert _{p}^{p^{\prime }(p-2)}\left\Vert u_{n}-u\right\Vert
_{1,p}^{p^{\prime }} \\
\\
+\rho \underset{\varphi \in W_{0}^{1,p}(\Omega ),\left\Vert \varphi
\right\Vert _{1,p}\leq 1}{\sup }\int_{\Omega }\left\vert \max \{0,\left\vert
u_{n}\right\vert ^{p-2}u_{n}-\left\vert u\right\vert ^{p-2}u\}\right\vert
\left\vert \varphi \right\vert dx \\
\\
\leq C(\left\Vert u_{n}\right\Vert _{1,p}+\left\Vert u\right\Vert
_{1,p})^{p^{\prime }(p-2)}\left\Vert u_{n}-u\right\Vert _{1,p}^{p^{\prime
}}+\rho \left\Vert \left\vert u_{n}\right\vert ^{p-2}u_{n}-\left\vert
u\right\vert ^{p-2}u\right\Vert _{p^{\prime }},%
\end{array}
\label{3***}$$with some constant $C>0.$ If $1<p<2$ [@GM Lemma $5.4$] and Hölder’s inequality imply that$$\begin{array}{l}
\left\Vert T_{p,\varepsilon }(u_{n})-T_{p,\varepsilon }(u)\right\Vert
_{W^{-1,p^{\prime }}(\Omega )} \\
\leq c_{p}\left\Vert u_{n}-u\right\Vert _{1,p}+\rho \left\Vert \left\vert
u_{n}\right\vert ^{p-2}u_{n}-\left\vert u\right\vert ^{p-2}u\right\Vert
_{p^{\prime }}.%
\end{array}
\label{6***}$$Consequently, the operator $L_{p,\varepsilon }$ is continuous for all $%
\varepsilon \in (0,\varepsilon _{0})$.
Now we claim that $L_{p,\varepsilon }$** **is strict monotone and coercive. Indeed, let $u_{1},u_{2}\in W_{0}^{1,p}(\Omega )$. We note that the integral $$\begin{array}{c}
\int_{\Omega }\left( \max \{(\underline{u}+\varepsilon )^{\alpha
_{1}}R^{\beta _{1}},\left\vert u_{1}\right\vert ^{p-2}u_{1}\}-\max \{(%
\underline{u}+\varepsilon )^{\alpha _{1}}R^{\beta _{1}},\left\vert
u_{2}\right\vert ^{p-2}u_{2}\}\right) (u_{1}-u_{2})dx%
\end{array}%$$is positive because $$\left( \max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta
_{1}},\left\vert u_{1}\right\vert ^{p-2}u_{1}\}-\max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert u_{2}\right\vert
^{p-2}u_{2}\}\right) (u_{1}-u_{2})\geq 0\text{ \ in }\Omega .$$Then for all $\varepsilon \in (0,\varepsilon _{0})$ we have$$\begin{array}{l}
\left\langle T_{p,\varepsilon }(u_{1})-T_{p,\varepsilon
}(u_{2}),u_{1}-u_{2}\right\rangle =\int_{\Omega }\left\langle \left( |\nabla
u_{1}|^{p-2}\nabla u_{1}-|\nabla u_{2}|^{p-2}\nabla u_{2}\right) ,\nabla
(u_{1}-u_{2})\right\rangle \text{ }dx \\
\\
+\rho \int_{\Omega }\left( \max \{(\underline{u}+\varepsilon )^{\alpha
_{1}-1}R^{\beta _{1}},\left\vert u_{1}\right\vert ^{p-2}u_{1}\}-\max \{(%
\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert
u_{2}\right\vert ^{p-2}u_{2}\}\right) (u_{1}-u_{2})dx \\
\\
\geq \int_{\Omega }\left\langle \left( |\nabla u_{1}|^{p-2}\nabla
u_{1}-|\nabla u_{2}|^{p-2}\nabla u_{2}\right) ,\nabla
(u_{1}-u_{2})\right\rangle \text{ }dx%
\end{array}%$$and the claim follows due to the strict monotonicity of $-\Delta _{p}$ in $%
W_{0}^{1,p}(\Omega )$. The coercivity of the operator $T_{1,\varepsilon }$ can be proved easily using the coercivity of $-\Delta _{p}$. Now we are able to apply the Minty-Browder theorem which guarantees the existence of a unique solution for problem (\[1\*\*\*\]) in $W_{0}^{1,p}(\Omega )$.
Next we show that solutions $u_{\varepsilon }$ of (\[1\*\*\*\]) are in $%
C^{1,\gamma }(\overline{\Omega }),$ for certain $\gamma \in (0,1)$ for all $%
\varepsilon \in (0,\varepsilon _{0})$. The proof is based on Moser’s iterations technique combined with nonlinear regularity theory (see [@L]).
For $M>0$, define on $\Omega $ the function $u_{\varepsilon ,M}\left(
x\right) =\min \left( u_{\varepsilon }\left( x\right) ,M\right) .$ We act on (\[7\*\*\*\]) with $\varphi =u_{\varepsilon ,M}^{k_{1}p+1}$ where $$\begin{array}{c}
\left( k_{1}+1\right) p=p^{\ast }%
\end{array}
\label{50***}$$which gives $$\begin{array}{l}
\int_{\Omega }\left( \left( k_{1}p+1\right) \left\vert \nabla u_{\varepsilon
,M}\right\vert ^{p}u_{\varepsilon ,M}^{k_{1}p}+\rho \max \{(\underline{u}%
+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert u_{\varepsilon
}\right\vert ^{p-2}u_{\varepsilon }\}u_{\varepsilon ,M}^{k_{1}p+1}\right)
\text{ }dx \\
=\int_{\Omega }f(x)u_{\varepsilon ,M}^{k_{1}p+1}\text{ }dx%
\end{array}
\label{36***}$$By the Sobolev embedding theorem, the left-hand side of (\[36\*\*\*\]) is estimated from below as follows $$\begin{array}{l}
\geq \int_{\Omega }\left( (k_{1}p+1)|\nabla u_{\varepsilon
,M}|^{p}u_{\varepsilon ,M}^{k_{1}p}+\rho \left\vert u_{\varepsilon
}\right\vert ^{p-2}u_{\varepsilon }\text{ }u_{\varepsilon
,M}^{k_{1}p+1}\right) \\
\geq \int_{\Omega }\left( (k_{1}p+1)|\nabla u_{\varepsilon
,M}|^{p}u_{\varepsilon ,M}^{k_{1}p}+\rho u_{\varepsilon
,M}^{(k_{1}+1)p}\right) \\
=\frac{k_{1}p+1}{\left( k_{1}+1\right) ^{p}}\int_{\Omega }\left\vert \nabla
u_{\varepsilon ,M}^{k_{1}+1}\right\vert ^{p}+\rho \left\Vert u_{\varepsilon
,M}\right\Vert _{p^{\ast }}^{p^{\ast }}\geq C_{1}\frac{\left(
k_{1}p+1\right) }{\left( k_{1}+1\right) ^{p}}\left\Vert u_{\varepsilon
,M}\right\Vert _{(k_{1}+1)p^{\ast }}^{p^{\ast }}%
\end{array}
\label{l6***}$$where $C_{1}$ is some positive constant. From (\[50\*\*\*\])$,$ the right-hand side of (\[36\*\*\*\]) is estimated from above by$$\begin{array}{l}
\int_{\Omega }f(x)u_{\varepsilon ,M}^{k_{1}p+1}\leq \left\Vert f\right\Vert
_{\infty }\int_{\Omega }u_{\varepsilon }^{k_{1}p+1}\leq \left\Vert
f\right\Vert _{\infty }\left\Vert u_{\varepsilon }\right\Vert _{p^{\ast
}}^{k_{1}p+1}.%
\end{array}
\label{8***}$$Following the same arguments as in [@MM] we obtain that $u_{\varepsilon
}\in L^{\infty }(\Omega )$ for all $\varepsilon \in (0,\varepsilon _{0})$. Then from the nonlinear regularity theory (see [@L]) we infer that $%
u_{\varepsilon }\in C^{1,\gamma }(\overline{\Omega }),$ for certain $\gamma
\in (0,1)$ and $\left\Vert u_{\varepsilon }\right\Vert _{C^{1,\gamma }}<%
\overline{R}$ for a large constant $\overline{R}>0$ and for all $\varepsilon
\in (0,\varepsilon _{0})$.
Lemma \[L1\] ensures that the inverse operator $$T_{p,\varepsilon }^{-1}:C(\overline{\Omega })\rightarrow C^{1}(\overline{%
\Omega })$$is well defined for all $\varepsilon \in (0,\varepsilon _{0})$. The next proposition gives some properties regarding $T_{p,\varepsilon }^{-1}.$
\[P4\]The operator $T_{p,\varepsilon }^{-1}$ is continuous and compact for all $\varepsilon \in (0,\varepsilon _{0})$.
First, let us show that $T_{p,\varepsilon }^{-1}$ is a continuous operator. So let $f_{n}\rightarrow f$ in $C(\overline{\Omega })$. Denoting $%
u_{n}=T_{p,\varepsilon }^{-1}(f_{n})$ reads as$$\begin{array}{c}
\int_{\Omega }\left( |\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \varphi +\rho
\max \{(\underline{u}+\varepsilon )^{\alpha _{1}-1}R^{\beta _{1}},\left\vert
u_{n}\right\vert ^{p-2}u_{n}\}\varphi \right) \ dx=\int_{\Omega }f_{n}\left(
x\right) \varphi \ dx%
\end{array}
\label{4***}$$for all $\varphi \in W_{0}^{1,p}(\Omega )$. Since by (\[2\*\*\*\]) the sequence $\{u_{n}\}$ is bounded in $W_{0}^{1,p}(\Omega )$, along a relabeled subsequence there holds $$u_{n}\rightharpoonup u\text{ with some }u\in W_{0}^{1,p}(\Omega ).
\label{5***}$$Setting $\varphi =u_{n}-u$ in (\[4\*\*\*\]). Then Lebesgue’s dominated convergence theorem ensures$$\begin{array}{c}
\underset{n\rightarrow \infty }{\lim }\left\langle -\Delta
_{p}u_{n},u_{n}-u\right\rangle =0.%
\end{array}%$$The $S_{+}$ property of $-\Delta _{p}$ on $W_{0}^{1,p}(\Omega )$ along with (\[5\*\*\*\]) implies $u_{n}\rightarrow u$ in $W_{0}^{1,p}(\Omega )$. Furthermore, the boundedness of the sequence $\{u_{n}\}$ in $C^{1,\gamma }(%
\overline{\Omega })$ and since the embedding $C^{1,\gamma }(\overline{\Omega
})\subset C^{1}(\overline{\Omega })$ is compact, it turns out that along a relabeled subsequence, one has the fact that $u_{n}\rightarrow u$ in $C^{1}(%
\overline{\Omega })$. Finally, (\[4\*\*\*\]) result in $u=T_{p,\varepsilon
}^{-1}(f)$, proving that $T_{p,\varepsilon }^{-1}$ is continuous operator.
Next, we show that $T_{p,\varepsilon }^{-1}(C(\overline{\Omega }))$ is a relatively compact subset of $C^{1}(\overline{\Omega })$. Let $%
u_{n}=T_{p,\varepsilon }^{-1}(f_{n})$ with $f_{n}\in C(\overline{\Omega })$ for all $n$. Following the same reasoning as before, we find $u\in C^{1}(%
\overline{\Omega })$ such that, along a relabeled subsequence, $%
u_{n}\rightarrow u$ in $C^{1}(\overline{\Omega })$, thereby the relative compactness of $T_{p,\varepsilon }^{-1}$ is proven.
The work was accomplished while the second author was visiting the University Federal of Campina Grande with CNPq-Brazil fellowship N$%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ 402792/2015-7. He thanks for hospitality.
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[^1]: C.O. Alves was partially supported by CNPq/Brazil 304036/2013-7 and INCT-MAT
[^2]: A. Moussaoui was supported by CNPq/Brazil 402792/2015-7.
| |
CROSS REFERENCE TO RELATED APPLICATIONS
This application is a national stage entry of international application no. PCT/GB2019/052106, filed Jul. 26, 2019, which is based on and claims the benefit of foreign priority under 35 U.S.C. 119 to GB 1812284.6, filed Jul. 27, 2018. This entire contents of the above-referenced applications are herein expressly incorporated by reference.
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Recently, inkjet printheads have been developed that are capable of depositing ink directly onto ceramic tiles, with high reliability and throughput. This allows the patterns on the tiles to be customized to a customer's exact specifications, as well as reducing the need for a full range of tiles to be kept in stock.
In other applications, droplet ejection heads may be used to form elements such as colour filters in LCD or OLED displays used in flat-screen television manufacturing.
Droplet ejection heads and their components continue to evolve and specialise so as to be suitable for new and/or increasingly challenging applications.
SUMMARY
Aspects of the invention are set out in the appended independent claims, while particular embodiments of the invention are set out in the appended dependent claims.
a mount for receiving an actuator component that provides one or more rows of fluid chambers, each chamber being provided with a respective at least one actuating element and a respective at least one nozzle, the at least one actuating element for each chamber being actuable to eject a droplet of fluid in an ejection direction through the corresponding at least one nozzle, each row extending in a row direction;
a manifold chamber, which extends from a first end to a second end, and widens from said first end to said second end, the second end providing fluidic connection, in parallel, to at least a group of chambers within said one or more rows and being located adjacent said mount; and
at least one port, each port opening into the manifold chamber at the first end thereof;
wherein at least one portion between the first end and second end of the manifold chamber is shaped as a hyperbolic acoustic horn.
The following disclosure describes, in one aspect, a manifold component for a droplet ejection head, the manifold component comprising:
The following disclosure describes, in another aspect, a manifold component for a droplet ejection head, the manifold component comprising one or more manifold chambers and at least one port; wherein a transitional portion connects one of said at least one ports to the second portion of said one or more manifold chambers and wherein said transitional portion comprises a change in cross-sectional shape to blend from the cross-sectional area of said one port to that of said second portion of said one or more manifold chambers.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will now be described with reference to the drawings, which are representational only and are not to scale, and in which:
1
FIG. A
is a cross-sectional view of a manifold component according to a first embodiment of the disclosure;
1
FIG. B
1
FIG. A
is an end view of the manifold component shown in ;
2
FIG. A
is a cross-sectional view of a manifold component according to another embodiment;
2
FIG. B
2
FIG. A
is an end view of the manifold component shown in ;
2
FIG. C
2
2
FIGS. A and B
is a side view of the manifold component shown in ;
3
FIG. A
is a cross-sectional view of a manifold component according to another embodiment which has multiple horn-shaped portions;
3
FIG. B
3
FIG. A
is an end view of the manifold component shown in ;
4
FIG.
is a manifold component according to another embodiment;
5
FIG. A
is a cross-sectional view of a manifold component according to yet another embodiment which has multiple horn-shaped portions;
5
FIG. B
5
FIG. A
is an end view of the manifold component shown in ;
5
FIG. C
5
5
FIGS. A and B
is a side view of the manifold component shown in ;
6
FIG. A
is the fluidic path in a manifold component according to a first test design;
6
FIG. B
is the fluidic path in a manifold component according to another embodiment with multiple horn-shaped portions;
6
FIG. C
6
6
FIGS. A and B
compares the calculated performance of the manifold components in ;
7
FIG. A
is a cut-away three-dimensional view of the fluidic path in a through-flow enabled manifold component according to another test design;
7
FIG. B
(a) is a cut-away three-dimensional view of the fluidic path in a manifold component according to another embodiment that is through-flow enabled and has multiple horn-shaped portions;
7
FIG. B
7
FIG. B
(b) is a cross-sectional view of the manifold component depicted in (a);
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FIG. B
7
FIG. B
750
(c) is a cross-sectional view of the manifold component depicted in (a) and (b);
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FIG. B
7
FIG. B
(d) is a cut-away three-dimensional view of a detail of the fluidic path depicted in (a);
7
FIG. C
7
FIG. A
7
FIG. B
compares the calculated coefficient of reflection across the frequency range for a manifold component as per and a manifold component as per ;
8
FIG.
is an extract from a print sample produced using drop-on-demand inkjet printheads;
9
FIG.
7
FIG. A
7
FIG. B
(A-C) are graphs comparing drop velocity data produced using a printhead comprising a test manifold component as per and a printhead comprising a horn-shaped manifold component as per ; and
10
FIG.
is a schematic diagram depicting a method of designing a horn-shaped portion for a manifold component according to an embodiment.
DETAILED DESCRIPTION OF THE DRAWINGS
Embodiments of the disclosure in general relate to a manifold component for a droplet ejection head.
1
FIG.
1
FIG. A
1
FIG. B
1
FIG. A
1
FIG. A-B
1
1
FIGS. A and B
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1
FIGS. A and B
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Turning now to , shown is a manifold component according to a first example embodiment. More particularly, is a cross-sectional view of a manifold component according to a first embodiment of the disclosure; is an end view of the manifold component shown in . The example embodiment shown in relates in general to a manifold component for a droplet ejection head. The manifold component has a mount for receiving an actuator component that provides one or more rows of fluid chambers (not shown). Each such chamber is provided with a respective at least one actuating element, for example a piezoelectric actuating element, and a respective at least one nozzle. In operation each actuating element is actuable to eject a droplet of fluid in an ejection direction through the corresponding nozzle. Each of the rows of fluid chambers extends in a row direction , indicated with respective arrows in . In the particular example embodiment of , the mount is a flat receiving surface.
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FIG. A
1
FIG. A
1
FIG.
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As may be seen from , a manifold chamber is provided within the manifold component . The manifold chamber extends from a first end to a second end , widening from the first end to the second end . Fluid flowing within the manifold chamber during operation may be described as “fanning out” as it approaches the second end . The second end provides a fluidic connection, in parallel, to at least a group of chambers within the one or more rows in the actuator component , the second end being located adjacent to said mount . As may also be seen from , the manifold component further includes a port which opens into the manifold chamber at the first end thereof. In operation the port may supply fluid to the first end of the manifold chamber such that the port may be said to be an inlet port and the manifold chamber may be said to be an inlet manifold chamber. In operation the fluid then passes through the inlet manifold chamber from its first end to its second end . For the manifold chamber , the entire portion between the first end and the second end thereof is shaped as a hyperbolic acoustic horn so as to assist in transferring acoustic waves away from the corresponding group of chambers in the actuator component . This may be referred to as the ‘horn-shaped portion’. As shown representationally in , the cross-sectional area of the manifold chamber may increase in a hyperbolic fashion from the first end to the second end so as to form a hyperbolic acoustic horn therein. The cross-sectional area of the hyperbolic horn-shaped portion of the manifold chamber increasing in a hyperbolic fashion may, in operation, result in low levels of acoustic cross-talk between the fluid chambers of the actuator component . This may occur because a manifold chamber with a portion shaped as a hyperbolic acoustic horn may assist in transmitting acoustic waves (generated when one or more actuating elements are actuated) out of the manifold chamber and into the fluidic supply. This may in turn improve the drop velocity and volume profile of the droplets of fluid ejected from the nozzles.
1
FIG.
505
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It has been calculated that when using a manifold component similar to that described in relation to an improved acoustic performance may be expected. Furthermore, experiment-based tests for alternative embodiments (described later) have shown that an improved print quality may result. This may be explained as follows. When an actuating element is actuated to eject a droplet of fluid in an ejection direction , acoustic waves pass from the actuator component back into the manifold chamber . Reflected waves may return to the actuator component and influence the behaviour of the fluid in other fluid chambers in the actuator row, leading to non-uniform drop velocity and non-uniform drop volume and causing defects in the appearance of the printed image. Transmitting acoustic waves out of the manifold chamber and into the rest of the fluidic system via the port has now been shown to improve the consistency of the drop velocity and volume of droplets ejected from individual fluid chambers, and hence improve the appearance of the printed image or product.
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FIG.
1
FIG.
It should be understood that an acoustic horn, similar to that shown in the embodiment of the manifold component depicted in , includes a region whose cross-sectional area increases according to a hyperbolic equation connecting two cross-sectional areas A and A, where A is the smaller area and A is the larger area. In the manifold component depicted in , A is the cross-sectional area at the first end , proximate the port and A is the cross-sectional area at the second end , proximate the mount . In the manifold component the source of the acoustic waves of interest is one or more of the fluid chambers in the actuator component . In operation therefore acoustic waves would emanate from any fluid chamber where an actuating element is actuated and into the manifold chamber , e.g. travelling through the area A towards the area A.
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FIG. A
The changing cross-sectional area of the manifold chamber , indicated by arrow in , can be represented by equation (1) where A(x) is the calculated area of area at a given position as x increases along the central path from the area A to towards the area A and where x=0 is at the location of the area A. Such regions of increasing cross-sectional area linking two different cross-sectional areas A and A are generally referred to as hyperbolic horns, also called hyperbolic-exponential or hypex horns.
A
x
A
h
x/x
T
T
h
x/x
T
x
T=c
f
c
Hypex horns are a general family of horns given by the wavefront area expansion:
()=2[cos (_)+_ sin (_)]{circumflex over ( )}2 (1)
where T is a parameter which sets the shape of the horn. For most practical applications 0≤T≤1.
x_T is a reference distance given as:
_/(2π_) (2)
where c is the speed of sound in the fluid and f c is the cutoff frequency. The cutoff frequency is the frequency below which most of the energy is reflected and above which most of the energy is transmitted.
Such equations may be used to design hypex horns as per any of the embodiments described herein. Depending on the desired operational capabilities of a droplet ejection head, it may be understood that different operating frequency ranges and acceptable cutoff frequencies may be required and may be designed for and chosen accordingly.
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FIG.
The manifold component having a cross-sectional area changing in a hyperbolic fashion may therefore be described as follows. The manifold chamber has a central path , which extends centrally through the manifold chamber from the centre of the area A proximate the first end to the centre of the area A proximate the second end . At any given point along the central path the changing cross-sectional area is indicated by arrow and is the area at right-angles to the central path . In the embodiment shown in the central path is parallel to the ejection direction , but this is in no way an essential feature. In the portion shaped as a hyperbolic acoustic horn the cross-sectional area of the area varies according to a hyperbolic function of distance from the area A along the central path . In other words, the portion of the manifold chamber which is shaped as a hyperbolic acoustic horn has a central path , which extends centrally through the manifold chamber, from the centre of the first end of the manifold chamber to the centre of the second end , the areas of cross-sections taken perpendicular to the central path vary approximately according to a hyperbolic function of distance from said first end along the central path .
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55
In some embodiments the cross-sectional area of the horn-shaped portion of the manifold chamber increases according to an exponential function. As those skilled in the art will appreciate, a cross-sectional area increasing in an exponential fashion is a special case (with T=1) of a cross-sectional area increasing in a hyperbolic or hypex fashion and the manifold chamber operates as an exponential acoustic horn.
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It may be understood that the constraints of manufacturing an actual product and the imposition of manufacturing tolerances means that the manifold chamber may not have an exactly mathematically true hypex (hyperbolic profile or exponential profile) for the horn-shaped portion. A horn-shaped portion with a profile that is close to or substantially hyperbolic may still provide improvements in ejection performance when used in the manifold chamber . The terms “hyperbolic fashion”, “hyperbolic horn-shaped portion” and the like, including simply “horn-shaped”, may therefore be understood to encompass a profile that is substantially hyperbolic. For example instead of having a smooth wall profile, a stepped horn-shaped portion consisting of multiple stacked cross-sections, each section having a discrete height in the ejection direction may provide improvements in ejection performance when used in the manifold chamber . Such a profile may occur when using 3D printing, for example, to build up the manifold chamber by depositing multiple layers. Moulded components may tolerate a degree of shrinkage and warpage during manufacture that alters the profile of the manifold chamber , for example, whilst still improving ejection performance. In general a certain amount of noise on the equation (1) may be tolerated when generating the profile for the walls of the manifold chamber whilst still providing acceptable droplet ejection performance.
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1
52
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It may be understood that in other embodiments the hyperbolic horn-shaped portion may not extend through the entire manifold chamber , in which case the area A will be proximate the first end of the manifold chamber , but not necessarily coincident with it, and the area A will be proximate the second end of the manifold chamber , but not necessarily coincident with it.
1
FIG. B
1
FIG. A
150
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510
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500
500
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As may be seen from , in this embodiment the actuator component is elongate such that its length in the row direction is much greater than its depth in the depth direction . In some embodiments it may therefore be desirable that the second end of the manifold chamber is defined by an opening which is elongate parallel to the row direction (for example, such that the opening has a long axis which extends in the row direction and a short axis that extends in the depth direction). This may enable ready fluidic connection from the manifold chamber to the actuator component . In some embodiments, the opening of the manifold chamber at the second end may have a point on each end of the long axis that defines the same angle (see ) between the plane of the opening and the wall of the manifold chamber .
1
FIG.
50
55
50
Furthermore, as can be also be seen from , in this embodiment the manifold component and the manifold chamber are likewise elongate, though this is by no means essential. Such an arrangement may be suitable when, for example, the manifold component is part of an arrangement of multiple manifold components, for example so as to supply different colours for printing onto paper or fabric, or to enable dense printing of a single colour, as this shape enables close packing of multiple manifold components.
1
FIG.
150
55
52
In the particular embodiment depicted in , the actuator component is rectangular and the manifold chamber has an opening that is rectangular in cross-section at its second end . However this is in no way limiting, and other arrangements of fluid chamber rows and shapes of actuator and manifold components are envisaged.
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It may be readily understood that if the actuator component is very long and slender then it may be expedient to use an array of manifold chambers arranged adjacent to each other in the row direction so that, in operation, each acts as an inlet manifold chamber and supplies fluid to a portion of the actuator component . In such an arrangement each individual manifold chamber would feed at least one group of fluid chambers within the one or more rows of fluid chambers in the actuator component . In some embodiments the second end of the manifold chamber may provide a fluidic connection, in parallel, to a corresponding one of said one or more rows of fluid chambers.
2
FIG.
2
FIG. A
2
FIG. B
2
FIG. A
2
FIG. C
2
2
FIGS. A and B
2
FIG. A-C
1
FIG.
250
250
250
250
Turning now to , shown is a manifold component according to another example embodiment. More particularly, is a cross-sectional view of a manifold component , is an end view of the manifold component shown in , and is a side view of the manifold component shown in . The embodiment shown in is in many respects similar to that seen in and thus, where appropriate, like reference numerals have been used.
1
FIG.
2
FIG.
2
FIG. A
1
FIG.
250
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1
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20
Unlike the embodiment shown in , the manifold component depicted in does not have a horn-shaped profile over the entire length of the manifold chamber . Instead, as seen in , the manifold chamber includes a horn-shaped portion and an additional (non-horn-shaped) portion located between the first end and the horn-shaped portion . The horn-shaped portion is the portion of the manifold chamber which has a cross-sectional area that increases in a hyperbolic fashion. The horn-shaped portion starts at the area A which is offset from the first end of the manifold chamber by a distance . The horn-shaped portion may be described as commencing at the area A proximate the first end and finishing at the area A proximate the second end . It should be appreciated that in embodiments where the hyperbolic portion starts at the first end , such as that depicted in , the area A and the first end coincide, and there is no additional portion .
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1
FIG.
The portion of the manifold chamber that increases in a hyperbolic fashion may in part be limited by the physical constraints of the wider droplet ejection head design, but in some embodiments the cross-sectional area of the at least one manifold chamber may increase in a hyperbolic fashion over a majority of the distance between the first end and the second end , i.e. the hyperbolic horn-shaped portion extends at least the majority of the distance between the first end and the second end of the corresponding manifold chamber . More particularly the cross-sectional area may increase in a hyperbolic fashion over a distance that is a fraction between 0.6 and 0.9 of the distance between the first end and the second end , i.e. the hyperbolic horn-shaped portion may extend between 0.6 and 0.9 times the distance between the first end and the second end of the corresponding manifold chamber . In still other embodiments the cross-sectional area of the manifold chamber may increase in a hyperbolic fashion over the entirety of the distance along the manifold chamber between the first end and the second end , i.e. the entire manifold chamber is a hyperbolic horn (as shown in the embodiment depicted in ). It should be understood that the term “distance” in this context refers to the distance between the first end and the second end along the central path of the manifold chamber . It may further be understood that in the embodiments described herein the positions of A and A within the manifold chamber depend on the extent and position relative to the first end of the hyperbolic portion.
1
FIG.
2
FIG.
2
FIG. B
1
FIG.
2
FIG. A
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Like the embodiment illustrated in , the manifold component in also has an actuator component that is elongate such that its length in the row direction is greater than its depth in the depth direction (see ). The manifold component also has a manifold chamber with an elongate opening parallel to the row direction at the second end . The manifold component differs from the manifold component in in that the cross-sectional area is elongate in the row direction over the entire hyperbolic horn-shaped portion . In this case the cross-sectional area may be defined as the area where the area is elongate. As seen in the angle may be measured at opposing (short) ends of the elongate area in the row direction . The area is a portion of a plane that intersects with the bounding walls of the manifold chamber at any given point between the area A and area A such that the angle between the walls of the manifold chamber and the plane of the cross-section is equal at the point of intersection on opposing ends of the manifold chamber . In other words, the manifold chamber has at least two points on opposing ends of the elongate area at least at the second end that define the same angle to the wall of said manifold chamber .
2
FIG.
2
2
FIGS. A and B
2
FIG. A
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1
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1
As can be seen from the depth in the depth direction of the hyperbolic horn-shaped portion remains generally constant. The depth direction is perpendicular to the row direction and the ejection direction , as seen in . Since the depth of the horn-shaped portion is generally constant, the hyperbolic change in cross-sectional area is largely due to increasing the width in the row direction of the horn-shaped portion . As seen in , the central path has a section, central path , which runs centrally through the horn-shaped portion from the area A to the area A. At each point on the central path the width is measured in a direction that is normal to the central path at that point and that is perpendicular to the depth direction . The widths vary generally according to a hyperbolic function with distance along the central path from the area A to the area A.
2
FIG. A
1
FIG.
2
2
FIGS. B and C
1
FIG.
2
FIG.
1
FIG.
2
FIG. C
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As can further be seen from , as for the embodiment in there is a port located on surface on the opposite side of the manifold component to the mount in the ejection direction . The port in this embodiment has a circular cross-sectional area so as to enable ready connection to a fluidic supply system. The first end of the manifold chamber likewise has a circular cross-sectional area. As previously discussed the second portion of the manifold chamber may have a generally constant depth in the depth direction and hence it may also be elongate along the entire central path . This means that the change in cross-sectional shape from circular to elongate occurs in the portion of the manifold chamber and the portion is not a hyperbolic acoustic horn. In some embodiments the portion may be limited in its extent such that the manifold chamber may, for at least the majority of its extent thereof in the ejection direction , have a generally constant depth in the depth direction , which is perpendicular to the row direction and to the ejection direction . As may be seen from , another significant difference to the embodiment in is that the manifold component in has two manifold chambers, manifold chambers and , whereas the manifold component in has only a single manifold chamber . As may be seen from manifold chamber is offset from manifold chamber such that the two are adjacent to each other in the depth direction . Furthermore in this embodiment the geometric shape of the manifold chamber is the same as that of the manifold chamber , but these are by no means essential features and other arrangements and geometries of manifold chamber are envisaged. In this implementation therefore, the portion (which may be referred to herein as a “transitional portion”) and the portion of the manifold chamber are the same geometric shape as the portion and the portion respectively of the manifold chamber . Also, the horn-shaped portion starts at the area A which is offset from the first end of the manifold chamber by a distance and ends at the second end of the manifold chamber .
2
FIG. C
1
FIG.
2
2
FIGS. A-C
1
FIG.
2
2
FIGS. A-C
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As can further be seen from , in contrast to the embodiment shown in , the manifold component of includes two ports and , whereas the manifold component in has only a single port . In the particular embodiment shown, these are located on surface , however it should be understood that this location is by no means essential. In operation, when the ports , are fluidically connected to a suitable fluid supply, the manifold component shown in may be operated in a so-called “through-flow” mode such that droplet fluid may, in operation, flow continuously from the port via the manifold chamber , the actuator component and the manifold chamber to the port , with port , manifold chamber , actuator component , manifold chamber and port being fluidically connected, in series, in that order. In operation, a portion of the fluid flowing through selected fluid chambers in the actuator component may be ejected from the respective nozzles for those fluid chambers, whilst the remainder of the fluid continues through the individual fluid chambers and via the manifold chamber to the port . In such embodiments the manifold chamber is configured as an inlet manifold chamber where the corresponding port is configured as an inlet port, in operation supplying fluid to the first end of the inlet manifold chamber . The second end of the inlet manifold chamber in operation supplies fluid in parallel to each chamber within the corresponding group of fluid chambers in the actuator component . Furthermore, in such embodiments the manifold chamber is configured as an outlet manifold chamber with the corresponding port being configured as an outlet port that in operation receives fluid from the first end of the outlet manifold chamber in question. The second end of the outlet manifold chamber , in operation, receives fluid in parallel from each chamber within the corresponding group of fluid chambers of the actuator component .
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In alternative arrangements, in operation, fluid may be supplied to the actuator component from both ports and , whereby the droplet ejection head may be considered to operate in a non through-flow mode and the manifold chambers and are both inlet manifold chambers and the ports , are both operating as inlet ports.
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FIG.
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In the embodiment depicted in , the central path is a straight line that is parallel to the ejection direction owing to the geometry of the design depicted. In other embodiments, the central path may not be a straight line but may follow a curved or serpentine path or any other path as defined by the shape of the manifold chamber , . Manifold chambers may be shaped in such a manner as a result of, for example, physical constraints elsewhere in the droplet ejection head, or to enable a ready connection to a fluidic supply. In such cases it may be appropriate, for example, to offset the port and/or the port from the centre of the manifold component in the array direction , or even to locate the ports on one of the sides , of the manifold component . In these cases, the central path may follow a different route, for example at an angle to the ejection direction , depending on the shape of the manifold chamber , . In some embodiments the central path , which runs centrally through the manifold chamber in question, from the centre of the first end to the centre of the second end , may not be parallel to the ejection direction along some of its length. It may be desirable to ensure that the central path is running generally parallel to the ejection direction at the second end of the manifold chamber in question, so as to improve fluidic performance by providing fluid flowing in a favourable direction to the actuator component . As may be readily understood, other shapes of manifold chamber , , and hence central path , are envisaged.
3
FIG.
3
FIG. A
3
FIG. B
3
FIG. A
3
FIG. A
3
FIG. A
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350
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x,y,z
Turning now to , shown is a manifold component according to another example embodiment. More particularly, is a cross-sectional view of a manifold component and is an end view of the manifold component shown in . Considering it is clear that the manifold component in this embodiment differs from the preceding two embodiments in that it comprises a plurality of hyperbolic horn-shaped portions (), arranged side-by-side in an array. Such a design may be suitable where the actuator component is long in the row direction , for example, and where a single horn-shaped portion to cover the entire actuator component in the row direction may be too large in the ejection direction to be practical. Using a plurality of hyperbolic horn-shaped portions allows the height of the manifold chamber to be reduced and enables a more compact droplet ejection head to be manufactured. For example, it may be seen from that the height of the manifold chamber in the ejection direction from its first end to its second end is comparatively equal to but less than the extent of the actuator component in the row direction , giving a desirably compact arrangement.
3
FIG. A
2
FIG.
3
FIG.
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x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
As can be seen from , the hyperbolic horn-shaped portions () are arranged side-by-side in an array whereby they are adjacent to each other in the row direction . The horn-shaped portions () each have a central path () respectively that splits from the central path and respective areas () that are perpendicular to the respective central paths (). It may be understood that the areas () of the horn-shaped portions () increase in a substantially hyperbolic fashion along the respective central paths () from the area A() to the areas A(). As in the embodiment shown in , the embodiment in includes hyperbolic portions that don't commence at the first end of the manifold chamber . Suitable spatial offsets () allow for the central path splitting and forming the respective central paths (). It should be understood that, owing to design constraints, the offsets () may not be the same as each other but rather may be determined according to the shape, location, orientation, etc. of the horn-shaped portions () and/or their respective central paths () and/or the path to each of their respective areas A() from the first end . It may be understood that in practice the hyperbolic equation for each horn-shaped profile may more readily be determined by setting x=0 individually for each, located in the centre of their respective area A().
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1
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315
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
x,y,z
i,ii
i,ii
x,y,z
x,y,z
x,y,z
The cross sectional area of each horn-shaped portion () may be defined as follows. The hyperbolic acoustic horns may each have a corresponding central path (), which extends centrally through portions (), from the centre of the area A() to the centre of the area A(). At each point along the central paths () there is defined a corresponding cross-sectional area, area (), which is the area lying within a plane perpendicular to the central path () and bounded by the walls of the hyperbolic acoustic horn, which may be part of the inner surface of the manifold chamber and one side of one of the plurality of walls () or two sides of two opposing walls amongst the plurality of walls (). The cross-sectional area of the area () varies approximately according to a hyperbolic function of distance from the Area A() along the central path ().
3
FIG. A
3
FIG. A
30
70
55
30
70
55
70
51
52
55
30
701
703
701
703
500
701
703
701
703
701
703
70
70
55
701
701
30
702
702
30
703
703
30
30
70
30
350
70
x,y
i
y,z
ii
i,ii
x,y,z
i
ii
i,ii
i
ii
i
ii
i
ii
x
i
ii
y
i
ii
z
x,y,z
i,ii
x,y,z
i,ii
As may also be seen from , neighbouring horn-shaped portions () are separated by a corresponding wall () located within the manifold chamber , and likewise neighbouring horn-shaped portions () are separated by a corresponding wall () also located within the manifold chamber . As is apparent from , each of the walls () extends over only part of the distance between the first end and the second end of the manifold chamber . The horn-shaped portions () comprise first (-)() and second (-)() side surfaces, which are spaced apart in said row direction , said side surfaces -() being substantially concave. The first (-)() and second (-)() side surfaces are formed from amongst the edges of the walls () and () and the sides of the manifold chamber ; such that first side surface () and second side surface () bound horn-shaped portion (), first side surface () and second side surface () bound horn-shaped portion (), and first side surface () and second side surface () bound horn-shaped portion (). It may be understood that the constraints of manufacturing an actual product and the imposition of manufacturing tolerances means that the horn-shaped portions () may not have a mathematically true hyperbolic or exponential profile. For example, while the walls () are depicted as having sharp or pointed ends, this is purely representational and it should be understood that, in practice, the ends might, for example, be blunted or chamfered in order to facilitate manufacture. A profile or shape that is close to or substantially hyperbolic or that changes in a hyperbolic fashion may still, for example, provide desirable print performance when used in the horn-shaped portions () of a manifold component for a drop-on-demand printhead. For example, for manufacturing reasons a design constraint may be to limit the walls () such that they may not be less than a certain thickness, for example 400 micrometers, and fillets or other smoothing features may be necessary at the tips of the walls.
3
FIG. A
70
30
505
20
i,ii
x,y,z
It should be appreciated that, while the particular embodiment shown in includes two walls () and three horn-shaped portions () this is by no means essential and alternative embodiments may comprise any suitable number of horn-shaped portions and corresponding dividing walls. In other embodiments it may be desirable for fluidic reasons to have the plurality of horn-shaped portions staggered, by altering the positions of the walls in the ejection direction , or other suitable arrangements to aid smooth fluidic flow from the first portion into the plurality of horn-shaped portions.
3
FIG. A
70
30
2
1
350
70
30
350
i,ii
x,y,z
x,y,z
x,y,z
i,ii
x,y,z
In the embodiment depicted in the walls () are elongate and curved, and shaped appropriately to provide a suitable hyperbolic profile to the horn-shaped-portions () between the areas A() and A(). A manifold component featuring such walls () might, for example, be manufactured using 3D printing techniques since it may be easier to manufacture such slender internal features using this method as compared to conventional casting, molding or machining techniques. A 3D printed component may also be easy to make without seams and fluid-tight, reducing leakage problems in a droplet ejection head. However, while the use of slender walls is described to partition the horn-shaped manifolds (), other embodiments may instead use much wider walls or other physical features and the manufacturing technique could comprise forming several separate components, for example, and joining them together in any suitable fashion so as to form a single, fluid-tight manifold component .
3
FIG. A
30
52
55
500
30
150
52
500
30
70
201
30
100
70
x,y,z
x,y,z
x,y,z
i,ii
x,y,z
i,ii
As may be seen from , each of the horn-shaped portions () is positioned so as to overlap with a section of the second end of the manifold chamber in the row direction . In operation each horn-shaped portion () preferentially provides a fluidic connection, in parallel, to a respective group of chambers within the one or more rows in the actuator component . In this embodiment the horn-shaped portions divide the second end into three equal sections. For example, if there are 300 fluid chambers in the row direction , each of the horn-shaped portions () will, in operation, largely supply fluid to a group of 100 fluid chambers most closely adjacent to its position. Since the walls () do not extend into the slot , there is the possibility of some fluid intermixing therein, so it may be understood that in operation the number of fluid chambers each horn-shaped portion () supplies may not be precisely and there may be some overlap near the wall () positions. It should be further understood that 300 fluid chambers is merely an example; in some embodiments there may be fewer or far greater numbers of fluid chambers.
52
55
120
81
350
500
55
120
150
30
3
FIG. A
1
2
FIGS. and
3
FIG. A
x,y,z
It should be understood that in other embodiments there may be any number of horn-shaped portions. It may further be understood that in a design with a plurality of horn-shaped portions, that these are not necessarily identical. For example the horn-shaped portions could be of different sizes so as to divide the second end into equal or unequal sections and that therefore the horn-shaped portions may supply equal or unequal sized groups of fluid chambers. As another example, a plurality of non-identical horn-shaped portions may be used to account for asymmetry in the manifold chamber . For example, as seen in , the port is located on the surface at a position offset from the centre of the manifold component in the row direction (unlike the embodiments in ). This is by no means essential, but may be suitable in some embodiments for ease of connection to other components such as a fluidic supply. As a result, as may also be seen in , the manifold chamber connecting the port to the actuator component is asymmetric and the horn-shaped portions () are shaped accordingly such that they are not identical.
350
55
55
20
1
120
30
150
250
120
150
20
1
350
30
500
3
FIG.
1
FIG.
2
FIG.
2
FIG.
3
FIG.
2
FIG.
3
FIG.
x,y,z
The manifold component in is similar to the embodiment depicted in in that it includes a single manifold chamber . However, it is similar to the embodiment in in that it includes a manifold chamber which is divided into a (non-horn-shaped) portion () proximate the port and a horn-shaped portion proximate the actuator component . As for the manifold component in , the embodiment depicted in includes a change in cross-sectional shape, from circular to match the port , to elongate to match the actuator component within the portion (). Furthermore, as for the embodiment depicted in , the manifold component of includes horn-shaped portions () that have an elongate cross-sectional area in the row direction .
3
FIG. A
350
100
200
80
200
505
150
80
150
350
150
100
150
200
150
100
150
It can be further seen in that the manifold component comprises two parts that have been joined together, first manifold section and second manifold section , with the mount now located on the lower surface of the second manifold section in the ejection direction . The actuator component is mounted on the mount . This is in no way an essential feature, but may be useful in some embodiments to aid in securely connecting the actuator component to the manifold component , or for improving the longevity of the actuator component , or for improving the assembly process. For example, if the first manifold section is made from a material such as a resin or a thermosetting plastic or a plastic/fibre composite material for ease of manufacture or cost reasons, it may have different thermal properties to the actuator component which may be manufactured largely from a silicon or piezoceramic material. The second manifold section may be made of a material such as a ceramic or a metal that more closely matches the thermal properties of the actuator component than the first manifold section , and may reduce stresses induced in the actuator component during assembly or operation.
3
FIG. A
1
2
FIGS. and
200
20
2
30
52
55
150
200
150
x,y,z
also shows that the second manifold section has a slot () therein, fluidically connecting the horn-shaped portions () to the second end of the manifold chamber and hence to the actuator component . It may be understood that such a second manifold section is in no way an essential feature and in many embodiments suitable thermal matching may be encompassed within a single manifold component as depicted in and/or within the actuator component .
3
FIG. A
1
52
20
2
30
52
55
150
505
52
500
150
x,y,z
x,y,z
As may also be seen from , the areas A() and the second end do not coincide. The slot () is a non-hyperbolic portion connecting the horn-shaped portions () to the second end of the manifold chamber . In operation the slot may allow for some fluidic mixing between the fluid exiting the horn-shaped portions and entering the actuator component and may also act as a flow straightener, in operation aligning and directing the fluid flow so that it is more closely parallel with the ejection direction at the second end . It may also act to flatten the velocity profile along the row direction such that the fluid supplied to the actuator component is at a more uniform velocity.
4
FIG.
4
FIG.
3
FIG.
450
55
30
1
30
2
30
3
70
71
72
30
150
500
505
55
51
55
505
500
51
500
51
i
ii
i
vi
i
xii
i
ii
i
iii
i
vi
shows a manifold component according to another embodiment. Specifically, depicts a manifold component with a manifold chamber comprising a hierarchical arrangement of a plurality of hyperbolic horn-shaped portions ()(-), ()(-) and ()(-) which are divided by a plurality of walls (-), (-) and (-); where, unlike the embodiment depicted in , not all of the walls extend over the entirety of the portion . Such a design may be suitable, for example, where the actuator component is long in the row direction , and there is a requirement for multiple horn-shaped portions owing to space constraints in the ejection direction . Another reason to have a hierarchical arrangement of a plurality of hyperbolic horn-shaped portions may be when there is insufficient room at the apex of the manifold chamber proximate the first end to fit the plurality of walls owing to manufacturing constraints such as a minimum wall thickness. Introducing increasing numbers of walls as the manifold chamber widens in the ejection direction may overcome this constraint. Another reason to stagger the introduction of the walls may be where, for example, the fluidic design requires a minimum gap in the row direction between the walls proximate the first end . This may be desired in order to ensure that there is smooth unhindered fluidic flow into the horn-shaped portions but where there are space constraints on the length in the array direction of the first end .
4
FIG.
70
30
1
30
2
30
3
71
72
51
52
55
i,ii
i
ii
i
vi
i
xii
i
iii
i
vi
It can be seen from that some of the walls () extend through all three of the hierarchical portions ()(-), ()(-) and ()(-), some through two of the hierarchical portions (walls (-)) and the remainder are only in the final hierarchical portion (walls (-)). Such an arrangement may make design and manufacture easier, but is by no means essential. In some embodiments different arrangements of walls may be used to separate the hierarchical portions, for example an arrangement whereby each wall extends only part of the distance from the first end to the second end of the corresponding manifold chamber is envisaged.
450
30
1
51
55
30
3
52
55
51
52
55
30
1
30
3
30
1
30
2
30
3
52
55
30
1
30
2
30
3
55
500
55
4
FIG.
i
ii
i
xii
i
ii
i
vi
i
xii
i
ii
i
vi
i
xii
A manifold component as depicted in comprising a plurality of said arrays of side-by-side horn-shaped portions, may include an initial array of side-by-side horn-shaped portions ()(-), which is proximate the first end of the inlet manifold chamber , and a final array of horn-shaped portions ()(-), which is proximate the second end of the inlet manifold chamber , said arrays being arranged consecutively from the first end to the second end of the manifold chamber , with the number of horn-shaped portions in each array increasing progressively from said initial array () to said final array (). Furthermore the plurality of horn-shaped portions ()(-), ()(-) and ()(-) is arranged hierarchically, such that a horn-shaped passageway in a given one of said arrays is fluidically connected to two or more horn-shaped passageways in the consecutive array nearer the second end of the manifold chamber . At each hierarchical portion ()(-), ()(-) or ()(-) in the manifold chamber , neighbouring (in the row direction ) horn-shaped portions in the plurality of arrays are separated by a corresponding wall, located within the manifold chamber in question.
4
FIG.
30
3
52
52
500
500
500
52
55
150
150
500
30
3
52
55
i
xii
i
xii
The final stage in the hierarchical arrangement depicted in , horn-shaped portion ()(-) is divided into twelve portions at the ends of the walls proximate the second end . Furthermore it may be preferable for acoustic reasons not to have the width proximate the second end of any individual horn-shaped portion greater than one twelfth of the overall width at that point in the row direction . In other words, the width, in the row direction , of each horn-shaped portion is less than 1/12 of the width, in the row direction , of the second end of the manifold chamber . This may improve acoustic performance by rejecting the first lateral resonance frequency. It may be understood that twelve horn-shaped portions is due to the length of the actuator component and the speed of sound c in a typical fluid for an droplet ejection head for inkjet printing. The number of horn-shaped portions desired may differ depending on the length of the actuator component in the row direction and the speed of sound in the ejection fluid being used. Furthermore it may be preferable for acoustic reasons that all of the horn-shaped portions ()(-) are of equal length in the row direction at the end proximate the second end of the manifold chamber .
5
FIG.
5
FIG. A
5
FIG. B
5
FIG. A
5
FIG. C
5
5
FIGS. A and B
5
FIG. A
3
FIG.
5
FIG. C
5
FIG. A
5
FIG. A
550
550
550
550
550
55
60
55
510
60
55
35
500
60
60
55
510
60
a,b
a
a
x,y,z
b
a
a,b
Turning now to , shown is a manifold component according to another example embodiment. More particularly, is a cross-sectional view of a manifold component , is an end view of the manifold component shown in and is a side view of the manifold component shown in . It can be seen from that the manifold component has a manifold chamber similar to the embodiment shown in . It can also be seen from that there are two further manifold chambers (), partially overlapping the manifold chamber in the depth direction . It can be seen from that the manifold chamber () is a reversed geometrical copy of the manifold chamber and has a plurality of horn-shaped portions ()() arranged side-by-side, or adjacent to each other, in the row direction . Although not shown in , the second manifold chamber () is identical to the first manifold chamber () and located on the opposite side of the manifold chamber to it in the depth direction . Throughout the following description like reference numerals are used for the component parts of the two manifold chambers () with (a) or (b) appended accordingly.
5
FIG.
2
FIG.
2
FIG.
5
FIG.
120
55
60
220
55
60
60
55
a,b
a,b
a,b
The embodiment depicted in is an arrangement, similar to that shown in , which allows so-called “through-flow” of fluid when connected to a suitable fluidic supply. In operation therefore, the port can operate as an inlet port, the manifold chamber can act as an inlet manifold chamber and the manifold chambers () can operate as outlet manifold chambers with the ports () operating as outlet ports. The main difference, as compared to the embodiment depicted in which has one inlet manifold chamber and one outlet manifold chamber , is that there is a ratio of two outlet manifold chambers () to one inlet chamber in the embodiment of .
5
FIG.
150
500
505
In the arrangement shown in , the actuator component has two rows of fluid chambers (not shown) extending parallel to each other in the row direction . As before each fluid chamber in a row may be provided with at least one respective actuating element and at least one respective nozzle, each actuating element being actuable to eject a droplet of fluid in an ejection direction through the corresponding at least one of the nozzles. This example arrangement would therefore have at least two rows of nozzles, each row corresponding to a particular row of fluid chambers.
550
120
55
150
60
220
60
220
550
5
FIG.
5
FIG.
a
a
b
b
In operation in through-flow mode the manifold component depicted in can allow fluid to pass from the inlet port via the inlet manifold chamber to the actuator component where the fluid path will split such that some of the fluid will pass into the first row of fluid chambers, via individual inlets to each fluid chamber therein, while the other part of the fluid will pass through the second row of fluid chambers, via individual inlets to each fluid chamber in the other row. Part of the fluid passing into the chambers may be ejected in the form of droplets, while the remainder will exit the chambers via respective fluid chamber outlets. The fluid chamber outlets of the first row are fluidically connected to the outlet manifold chamber () and hence to the outlet port (). The fluid chamber outlets of the second row are fluidically connected to the outlet manifold chamber () and hence to the outlet port (). When operating the embodiment depicted in in through-flow mode it may be preferable that the fluid split is balanced such that half the fluid follows one path through the manifold component and half the fluid follows the other path.
220
550
120
55
150
52
55
60
60
a,b
a
b
5
FIG.
5
FIG.
In operation, a portion of the fluid passing through any individual fluid chamber may be ejected depending on the drive signals supplied by wiring (not shown) to the actuating element(s). The outlet ports () may be connected in some manner to a single fluidic outlet path to remove the fluid from the manifold component , or they may be separately connected to individual fluidic outlet paths. Since in the example shown in there is a single port and a single inlet manifold chamber it is apparent that this arrangement will, in operation, supply a single fluid type to both the rows of fluid chambers and so both rows of nozzles will eject the same fluid type. This arrangement may allow for close packing of nozzle rows within the actuator component and may be appropriate, for example, where there are space constraints and/or where a high nozzle density is required to form a high resolution droplet ejection head. In the embodiment depicted in , the second end of the inlet manifold chamber provides a fluidic connection, in parallel, to two rows of fluid chambers, whilst each of the outlet manifold chambers () and () provides a fluidic connection, in parallel, to one row of fluid chambers.
5
FIG. A
3
FIG.
550
100
200
200
100
150
200
20
2
30
52
55
150
25
2
25
2
20
2
510
62
60
35
x,y,z
a
b
a,b
a,b
a,b
x,y,z
As can be seen from , like the embodiment depicted in , the manifold component comprises first and second manifold sections , . The second manifold section located between the first manifold section and the actuator component . The second manifold section has three slots, one slot () fluidically connects the horn-shaped portions () to the second end of the inlet manifold chamber and then to the actuator component . Two further slots ()() and ()(), one either side of slot () in the depth direction , fluidically connect the second ends () of the outlet manifold chambers () to the horn-shaped portions ()().
3
FIG.
30
52
500
150
35
150
35
500
75
25
2
x,y,z
a,b
x,y,z
a,b
x,y,z
a,b
i,ii
a,b
As previously discussed with regard to the embodiment depicted in , each inlet horn-shaped portion (), is located so as to cover a portion of the second end in the row direction such that each provides a fluidic connection, in parallel, to a respective group of chambers within the one or more rows in the actuator component . Similarly the outlet horn-shaped portions ()() are located so as to provide fluidic connection, in parallel, to respective groups of chambers within the one or more rows in the actuator component . In operation each outlet horn-shaped portion ()() will largely be receiving fluid from a respective group of fluid chambers adjacent to it in the row direction . However, since the walls ()() do not extend into the slots ()() there is the possibility of some fluid intermixing therein.
150
120
220
a,b
In alternative arrangements, fluid may be supplied to the actuator component from all three ports and (), such that the droplet ejection head may be considered to operate in a non through-flow mode.
6
FIGS. A-C
6
FIG. A
6
FIG. B
6
FIG. C
6
FIG. B
3
FIG.
2
3
FIGS. and
6
FIG. B
10
650
10
650
650
55
55
30
120
150
20
1
s
z
Attention is now directed to , in which: is the inlet-only fluidic path in a manifold component according to a first test design; is the fluidic path in a manifold component according to a further embodiment; and is a graph that compares the calculated performance of the manifold components and . As can be seen in , the manifold component is an embodiment similar to that in where there the manifold chamber is an inlet manifold chamber and there are a plurality of horn-shaped portions (-). Such embodiments might be described as including a multicellular acoustic horn, or might be described as multicellular ‘horned’ manifolds. Also, as for the manifold components in , the embodiment depicted in comprises a change in cross-sectional shape, from circular to match the port , to elongate to match the actuator component within the portion ().
6
6
FIGS. A and B
55
55
10
650
52
52
55
55
52
52
55
55
As may be seen by comparing , whilst the test design and the horned manifold component differ in that the former is not a hyperbolic acoustic horn and the latter has multiple acoustic horns, both the test design and the horned manifold component were designed to share certain features. The manifold chambers ′ and in the test manifold component and the horned manifold component both have a rectangular second end ′, of the manifold chambers ′, and the inlet port opens at the same location relative to the second ends ′, of the manifold chambers ′, .
6
FIG. C
6
6
FIGS. A and B
6
FIG. B
6
FIG. A
10
650
52
52
150
Attention is now directed to , which is a graph showing the coefficient of reflection for acoustic pressure waves for the manifold components , illustrated in as the frequency of ejection is varied. The coefficient of reflection was calculated using Finite Element analysis to investigate the response of a horned manifold as per and a test manifold as per . The calculations were performed using the rigid piston assumption, to perform a frequency sweep from 0 to 100 kHz. The rigid piston was located at the second ends , ′ at a position analogous to that of the actuator component .
6
FIG. C
6
FIG. C
2
1
10
650
650
10
As can be seen in , a coefficient of reflection of 0 corresponds to no reflection, where all acoustic waves are transmitted out of the manifold component through the cross-sectional area A. A coefficient of reflection of 1 means that there is no transmission and all of the acoustic waves are reflected back to the cross-sectional area A. For a droplet ejection head design that utilises one or other of the manifold components (test) and (horned), the frequency range considered is 0 to 100 kHz, where 0-100 kHz is the droplet ejection frequency (100 kHz is the upper frequency limit for the droplet ejection head of the present embodiment). Preferably, a manifold component for a droplet ejection head would have a coefficient of reflection as low as possible over the considered frequency range 0 to 100 kHz. It may be seen from , that for the horned manifold component the coefficient of reflection is reduced across a substantial part of the considered range as compared to the test manifold component . It may be understood that for droplet ejection heads with different frequency conditions/requirements, an improved manifold component may be designed for a different upper frequency limit than 100 kHz.
7
FIG.
7
FIG. A
7
FIG. B
7
FIG. B
7
FIG. B
7
FIG. B
7
FIG. B
7
FIG. C
7
FIG. A
7
FIG. B
6
FIG. C
110
750
55
750
20
2
60
750
25
2
a
a
Considering now , is a cut-away three-dimensional view of the fluidic path in a through-flow enabled manifold component according to a second test design. This may be referred to as a ‘test’ manifold component. (a) is a cut-away three-dimensional view of the fluidic path in a manifold component according to another embodiment that is through-flow enabled and has multiple horn-shaped portions. This may be referred to as a ‘horned’ manifold component. (b) is a cross-sectional view through an inlet manifold chamber in the manifold component depicted in (a), also including the slot (). (c) is a cross-sectional view through an outlet manifold chamber () in the manifold component depicted in (a), also including the slot ()(). compares the calculated coefficient of reflection across the frequency range for a test manifold component as per and a horned manifold component as per . The calculations were performed in a similar manner to those described above with regards to .
7
FIG. B
5
FIG.
5
FIG.
7
FIG. B
750
60
55
60
60
750
60
220
25
60
220
30
35
505
a
a
b
a,b
a,b
x,y,z
a,b
x,y,z
Turning now to (a), the manifold component illustrated therein is similar to the embodiment illustrated in in that it has multiple horn-shaped portions and is through-flow enabled. It differs from the embodiment in in that for ease of connection to a fluidic supply the outlet manifold chamber () is not an identical reflection of the inlet manifold chamber . The outlet manifold chambers () and () in the manifold component are generally identical to each other. It can further be seen from (a) that the outlet manifold chambers () are connected to a single port and that the transitional portion acts to merge the fluid exiting both outlet manifold chambers () before connecting to the port . The plurality of horn-shaped portions () and ()() may have a cross-sectional area that increases in a hyperbolic fashion over at least a portion of the distance in the ejection direction .
7
FIG. B
750
750
80
150
505
500
750
55
60
60
51
61
61
52
62
62
51
61
61
52
62
62
52
62
62
55
60
60
80
120
220
55
51
60
60
61
61
55
60
60
30
35
52
62
62
55
60
60
55
60
60
500
120
220
30
35
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
x,y,z
a,b
x,y,z
a
b
a
b
a
b
x,y,z
a,b
x,y,z
The example embodiment shown in (a) relates in general to a manifold component for a droplet ejection head. The manifold component comprises a mount for receiving an actuator component that provides one or more rows of fluid chambers, each chamber being provided with a respective at least one actuating element and a respective at least one nozzle, the at least one actuating element for each chamber being actuable to eject a droplet of fluid in an ejection direction through the corresponding at least one nozzle, each row extending in a row direction . The manifold component has manifold chambers , (), () that extend from respective first ends , (), () to respective second ends , (), (), and widens from said first ends , (), () to said second ends , (), (). The second ends , (), () of the manifold chambers , (), () provide fluidic connection, in parallel, to at least a group of chambers within said one or more rows and are located adjacent to the mount . There are ports , , the former of which opens into the manifold chamber at the first end and the latter of which opens into the manifold chambers (), () at the first ends (), (). The manifold chambers , (), () comprise a plurality of horn-shaped passageways (), ()() the cross-sectional area of each of which decreases, at a decreasing rate, with distance from the second ends , (), () of the manifold chambers , (), (). The horn-shaped passageways within each respective manifold chamber , (), () are arranged side-by-side in an array which extends generally in the row direction . The ports , are fluidically connected in parallel with their respective horn-shaped passageways () and ()().
7
FIG. B
2
FIG.
7
FIG. B
30
701
703
701
703
500
701
703
250
55
7
20
1
120
150
120
55
510
20
1
x,y,z
i
ii
i,ii
As can be seen from (b) the horn-shaped passageways () comprise first (-)() and second (-)() side surfaces, which are spaced apart in said row direction , said side surfaces -() being substantially concave. As for the manifold component in , the manifold chamber depicted in (a) and B(b) includes a change in cross-sectional shape within the portion (), in this implementation from circular to match the port , to elongate (in this instance rectangular) to match the actuator component (not shown). The port is also offset from the manifold chamber in the depth direction so the portion () also comprises shaping in the depth direction to connect the two.
7
FIG. B
35
711
713
711
713
500
711
713
a
x,y,z
i
ii
i,ii
Likewise as can be seen from (c) the horn-shaped passageways ()() comprise first (-)() and second (-)() side surfaces, which are spaced apart in said row direction , said side surfaces -() being substantially concave.
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Although not shown, it may be understood that manifold chamber () is similarly configured. One or more of the horn-shaped passageways (), ()() may have a hyperbolic profile. In some embodiments all of the horn-shaped passageways (), ()() may be shaped as a hyperbolic acoustic horn, whereby such horn-shaped passageways (), ()() may be described as hyperbolic horn-shaped portions () ()().
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FIG. B
7
FIG. B
The manifold chambers (), () depicted in (a) (and () depicted in (c)) also include a change in cross-sectional shape from elongate to match the actuator component (not shown), to circular to match the port in the transitional portion . Also, as previously mentioned, the outlet manifold chambers () are connected to a single port and that the transitional portion acts to merge the fluid exiting both outlet manifold chambers () before connecting to the port .
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FIG. B
7
FIG. B
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(d) is a detailed view of the fluidic path depicted in (a) depicting the portions and in greater detail. It can be seen that the inlet port is offset from the portion such that the portion further comprises a blended change in position in the depth direction to connect the inlet port to the rectangular cross-sectional area .
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FIG. B
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Considering (d) further it can be seen that the transitional portion comprises two arms () and (), one per outlet manifold chamber ()() which blend from rectangular cross-sectional areas () and () to oval cross-sectional areas () and () and then merge to form a single passage () which connects to outlet port via a circular cross-sectional area . The transitional portion has a blended cowl-like shape which may improve the fluid flow therein and which may also help to reduce acoustic crosstalk by assisting in transmitting acoustic waves out of the manifold chambers () and () and into the fluidic supply. It may be understood that this is merely one implementation and other combinations of cross-sectional shapes and areas and blended regions may be combined in any suitable manner to provide the transitional portion with a blended cowl-like form, for example by sweeping a cross-sectional shape or shapes and/or a range of cross-sectional areas along suitable paths or trajectories. In some implementations the manifold chamber may have an elongate cross-sectional area. The transitional portion may connect a number of cross-sectional areas, both at its ends where it is connectable to the port and the manifold chamber, and along the length of the transitional portion . In some implementations the cross-sectional shapes of the blended cowl-like form of the transitional portion are chosen from a list comprising elongate, rectangular, oval, and circular. In some implementations such blended cowl-like forms may be formed from a 3D printed material. Such a transitional portion may suitably be used in implementations with at least two, or more, manifold chambers, where at least two of said manifold chambers are connected to a single port, which may be an outlet port , wherein the transitional portion connects the port to the at least two manifold chambers ()(). In such an implementation the transitional portion comprises at least one passage () and further comprises an arm ()() per manifold chamber ()(). Further manifold chambers can be connected using a suitable number of additional arms, one per manifold chamber, where the arms may be merged together using any suitable number of connecting passages. It may be understood that such a transitional portion may be used for two or more inlet manifold chambers or two or more outlet manifold chambers to connect to an inlet or outlet port respectively. Further such a transitional portion may suitably be used in implementations where the two or more manifold chambers comprise one or more horn-shaped passageways, or in other implementations where the two or more manifold chambers do not comprise horn-shaped passageways.
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FIG. C
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FIG. A
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FIG. B
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FIG. C
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FIG. A
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FIG. B
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FIG. C
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FIG. A
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FIG. B
compares the calculated coefficient of reflection across the frequency range of 0-100 kHz for a test manifold component as per and a horned manifold component as per . (a) compares calculated coefficients of reflection across the frequency range for the inlet manifold chambers for the test () and horned () manifolds. (b) compares calculated coefficients of reflection across the frequency range for an outlet manifold chamber in the test () and horned () manifolds. It can be seen that the coefficient of reflection is largely reduced for the horned manifold in both the inlet and outlet chambers as compared to the test manifold.
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FIG.
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FIG. A
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FIG. B
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a
FIG. ()
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FIG. ()
Considering now , shown are respective print samples produced using a droplet ejection head comprising a test manifold component similar to , and a horned manifold component similar to that in . The print direction is along the vertical. The heads were operating in through-flow mode at a droplet frequency of 110 kHz. The samples were printed using magenta ink. It can be seen that, from top to bottom of the sample, the greyscale was increased successively per printed block. It may clearly be seen that the horned manifold component () produced an improvement in the quality of the print test sample as compared to the test manifold component (). It is believed that the defects in the test manifold component print sample are due to acoustic crosstalk.
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FIG.
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FIG. A
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FIG. B
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FIG. B
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FIG. C
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(A-C) are graphs comparing drop velocity data produced using a printhead comprising a test manifold component as per and a printhead comprising a horned manifold component as per . The data was collected using a commonly available brand of droplet measurement and analysis tool (a JetXpert™ Dropwatcher by ImageXpert®). The results compare the drop velocity data for droplets ejected from one row of 360 nozzles in the actuator component at droplet ejection frequency (and the drive signal supplied to the actuators in the fluid chambers) of 5 kHz (), 20 kHz () and 40 kHz (). It can clearly be seen that at the frequencies measured, the drop velocity is more consistent in the row direction for the horned manifold as opposed to the test manifold. At the higher frequencies considerable waviness and variability in the drop velocity profile in the row direction can be seen for the test manifold as compared to the horned manifold.
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FIG.
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FIG.
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FIG.
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is a schematic diagram depicting a method of designing a horn-shaped portion for a manifold component according to an embodiment as described herein. As shown in , the method involves determining an initial shape for the manifold chamber , according to which the manifold chamber extends, along an initial, straight-line path , from a first end to a second end , with there being a continuum of cross-sections A(x) perpendicular to the initial path , the areas of said cross-sections increasing from A to A with increasing distance from the first end , such that there is at least a portion of the manifold chamber for which the areas of the cross-sections A(x) increase in a hyperbolic fashion between the first end and the second end . The next step involves deforming said initial path to produce a modified path ′, with each cross-section A(x) being moved with a corresponding point on the initial path , thus providing a modified shape, manifold chamber ′ with cross sections A′, A(x)′ which have the same cross-sectional areas as cross sections A and A(x). It may be seen that cross-section A at the second end remains in its initial position. The deforming step depicted in is such that the cross-sections A′, A(x)′ and A remain substantially parallel, though it should be understood that this may not be essential in all embodiments. Furthermore the modified path ′ may be a straight-line path; and the deforming step may be such that the cross-sections A′, A(x)′ and A remain substantially parallel to one another and angled with respect to said modified path ′. In other embodiments it may be understood that other deforming steps may be implemented, for example using a non straight-line variant of path ′, or some other form of translation or rotation of the initial path .
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FIGS. and
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It should be generally understood that for reasons of space constraint it may be desirable to have a manifold component as per any of the embodiments described herein where the extent of each manifold chamber , in the ejection direction is less than or equal to 2 times the extent in the row direction ; and in some embodiments the extent of each manifold chamber in the row direction is less than or equal to 2 times the extent in the ejection direction . In some embodiments it may be preferable that the extent of each manifold chamber in the row direction is approximately equal to the extent in the ejection direction , as for those shown in . It may be understood that where there are space constraints, using a multi-cellular horn with a plurality of horn-shaped portions as depicted in and elsewhere may enable a suitably compact design.
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FIG. A
It may be understood that in some embodiments the mount may for example comprise a flat receiving surface as in to which the actuator component may be attached by glue. Alternatively the mount may have more complex arrangements of mounting surfaces and connecting elements and the use of fixing devices such as screws or pins or push fits or slide fits or glue to enable the actuator component to be securely attached to any of the manifold components as described herein. The fluid chambers have been described as being in a row of fluid chambers; however, it should be understood that the row is not necessarily a straight line, and that fluid chambers can be staggered within the row.
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In some embodiments the first portion may comprise a hyperbolic acoustic horn as well as a change in cross-sectional shape to blend from the cross-sectional area of the port to one that suits the actuator component . It may therefore be understood that an offset may not be an essential feature in such embodiments. It should further be understood that the offset is not necessarily a distance in a straight line in the ejection direction , it depends on the shape of the manifold chamber , the route that the central path takes and where the portion that forms a hyperbolic acoustic horn occurs. It may be understood that the shape of the first portion may therefore depend on the cross-sectional shape(s) of the port and the actuator component.
It may further be understood that manifold components may comprise a plurality of manifold chambers as described herein and arranged in any manner that is suitable for the application in question. The manifold components may comprise a plurality of inlet manifolds and/or a plurality of outlet manifolds. Some or all of the features described herein may be combined in any suitable manner to form a manifold component.
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FIGS. A and B
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FIG. A
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It may further be understood that where there are two or more manifold components of the same type (as depicted in ) these may all have their own individual port (as depicted in ) or share a common port (as depicted in ). In the latter case, as depicted in , the transitional portion may divide the fluid path into a suitable number of arms to connect to the respective manifold chambers as well as to blend the cross-sectional shape and/or area of the fluid path from that of the common port to one that suits the actuator components . It should be understood that such an arrangement would work whether the manifold chambers are acting as inlet manifolds or outlet manifolds.
It should be understood that manifold components as described herein are suitable for inclusion in a wide variety of droplet ejection heads. In particular, manifold components as described herein are suitable for inclusion in droplet ejection heads having various applications.
In this regard, it should be appreciated that, depending on the particular application, a variety of fluids may be ejected by droplet ejection heads.
For instance, certain heads may be configured to eject ink, for example onto a sheet of paper or card, or other receiving media, such as ceramic tiles or shaped articles (e.g. cans, bottles etc.) Ink droplets may, for example, be deposited so as to form an image, as is the case in inkjet printing applications (where the droplet ejection head may be termed an inkjet printhead or, in particular examples, a drop-on-demand inkjet printhead).
Alternatively, droplet ejection heads may eject droplets of fluid that may be used to build structures, for example electrically active fluids may be deposited onto receiving media such as a circuit board so as to enable prototyping or manufacture of electrical devices. In examples, polymer containing fluids or molten polymer may be deposited in successive layers so as to produce a 3D object (as in 3D printing). In still other applications, droplet ejection heads might be adapted to deposit droplets of solution containing biological or chemical material onto a receiving medium such as a microassay. Droplet ejection heads suitable for such alternative fluids may be generally similar in construction to inkjet printheads—as may the manifold component therein—potentially with some adaptations made to handle the specific fluid in question.
Furthermore, it should be noted that droplet ejection heads may be arranged so as to eject droplets onto suitable receiving media, and may therefore be termed droplet deposition heads.
For instance, as mentioned above, the receiving media could be sheets of paper or card, ceramic tiles, shaped articles (e.g. cans, bottles etc.), circuit boards, or microassays.
Nonetheless, it is by no means essential that droplet ejection heads as described herein are arranged as droplet deposition heads, ejecting droplets onto receiving media. In some applications, it may be relatively unimportant where the ejected droplets land; for instance, in particular examples droplet ejection heads may be utilised to produce a mist of ejected droplets. Moreover, similar head constructions may, in some cases, be used whether or not the ejected droplets land on receiving media. Accordingly, the more general term “droplet ejection head” is (where appropriate) used in the above disclosure.
Manifold components as described in the above disclosure may be suitable for drop-on-demand inkjet printheads. In such heads, the pattern of droplets ejected varies in dependence upon the input data provided to the head.
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A droplet ejection head may comprise a portion for a manifold component as described herein to connect the main portion of the manifold component to a port and an actuator component fixed at the mount .
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A droplet ejection head may comprise a manifold component as described in any of the above embodiments and an actuator component fixed at the mount .
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A droplet ejection head may comprise a manifold component as described in any of the above embodiments and an actuator component fixed at the mount , wherein each group of chambers comprises at least 100 chambers. | |
Q:
Finding the adjoint of an operator
This is from my homework, I'm totally lost as to how to proceed.
Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by
$(Tf)(x) = \int^x_0 f(s) \ ds$
What is the adjoint of $T$?
This operator doesn't seem to be an orthogonal projection, nor is it self-adjoint. How does one find the adjoint of an operator in general? Thanks in advance!
A:
Using the fact that
$$ \langle Tf , g \rangle=\langle f , T^{*}g \rangle, $$
we have
$$ \langle Tf, g\rangle = \int_{0}^{1} (Tf)(t)g(t)\,dt =\int_{0}^{1} \int_{0}^{t} f(\tau)\,d \tau\, g(t)\, dt = \int_{0}^{1} f(\tau)\, \left(\int_{\tau}^{1} g(t) \,dt\right)\, d \tau $$ $$ = \langle f, T^{*}g\rangle $$
From the last integral, we can see that the adjoint is given by
$$ (T^{*}f) (x) = \int_{x}^{1} f(s)\, ds $$
A:
We can find adjoint for kernel operators, that is, operators given by
$$T(f)(x)=\int_{[0,1]}K(x,y)f(y)dy,$$
with $K$ satisfying good conditions. We should have
$$\int_{[0,1]}T^*(f)(x)\overline{g(x)}dx=\int_{[0,1]}f(x)\overline{T(g)(x)}dx.$$
Since
$$\int_{[0,1]^2}f(x)\overline{K(x,y)g(y)}dxdy=\int_{[0,1]}\left(\int_{[0,1]}\widetilde K(y,x)f(x)dx\right)\overline{g(y)}dy,$$
where $\widetilde K(x,y)=\overline{K(y,x)}$. Since it's true for any $g$, we have
$$T^*(f)(x)=\int_{[0,1]}\widetilde K(x,y)f(y)dy.$$
| |
4091 has 2 divisors, whose sum is σ = 4092. Its totient is φ = 4090.
The previous prime is 4079. The next prime is 4093. The reversal of 4091 is 1904.
Added to its reverse (1904) it gives a triangular number (5995 = T109).
It is a strong prime.
It is a cyclic number.
It is not a de Polignac number, because 4091 - 26 = 4027 is a prime.
Together with 4093, it forms a pair of twin primes.
It is a Chen prime.
It is equal to p563 and since 4091 and 563 have the same sum of digits, it is a Honaker prime.
It is a plaindrome in base 12.
It is a nialpdrome in base 8 and base 16.
It is an inconsummate number, since it does not exist a number n which divided by its sum of digits gives 4091.
It is not a weakly prime, because it can be changed into another prime (4093) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (11) of ones.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 2045 + 2046.
It is an arithmetic number, because the mean of its divisors is an integer number (2046).
24091 is an apocalyptic number.
4091 is a deficient number, since it is larger than the sum of its proper divisors (1).
4091 is an equidigital number, since it uses as much as digits as its factorization.
4091 is an odious number, because the sum of its binary digits is odd.
The product of its (nonzero) digits is 36, while the sum is 14.
The square root of 4091 is about 63.9609255718. The cubic root of 4091 is about 15.9934869324.
Adding to 4091 its reverse (1904), we get a palindrome (5995).
Subtracting from 4091 its reverse (1904), we obtain a 7-th power (2187 = 37).
It can be divided in two parts, 40 and 91, that added together give a palindrome (131).
The spelling of 4091 in words is "four thousand, ninety-one". | https://www.numbersaplenty.com/4091 |
The season’s biggest storm is expected to hit Southern California tomorrow, and, once again, it could spell trouble for neighborhoods scarred by recent wildfires.
“This is the big one,” says weather service meteorologist Eric Boldt. “This is the largest storm of the winter.”
The Los Angeles Fire Department is bracing for flash floods, issuing voluntary evacuations for most areas impacted by the La Tuna Canyon fire, including portions of Sun Valley, Sunland, and Tujunga. Mandatory evacuations were ordered for Kagel Canyon, Lopez Canyon, and Little Tujunga, areas ravaged by the Creek Fire, plus several blocks of La Tuna Canyon Road.
Voluntary and mandatory evacuation orders go into effect at 6 p.m., when the heaviest rain is forecasted to arrive.
“Rainfall rates up to 0.6 inches per hour are possible late this evening with rates possibly increasing to 0.75 inches per hour or higher at times Thursday,” according to the National Weather Service. “Rainfall of this intensity can produce dangerous mud and debris flows near recent burn areas.”
The Weather Service says Los Angeles County will see the biggest downpours on Thursday, with rain dissipating Friday. Santa Barbara and Ventura counties, which were devastated by the Thomas Fire, will get the brunt of the rain today.
La Tuna Canyon is closed due to the rain and possible debris flow #stormwatch @ladailynews pic.twitter.com/ImTVFgBhcf
— Hans Gutknecht (@HansGutknecht) March 21, 2018
Radar observed rainfall through 6am this morning. More rain on the way. #CAwx #LAweather #CAstorm #LArain #SoCal pic.twitter.com/d5H4jHKYiX
— NWS Los Angeles (@NWSLosAngeles) March 21, 2018
The latest forecast calls for two to five inches of rain across Southern California’s coasts and valleys, and five to 10 inches of rain in the foothills and mountains. The weather service says LA County’s rainfall will likely be on the lower end of that range.
Still, the high rainfall totals could cause problems in areas that were recently burned.
Areas recovering from the La Tuna Canyon fire or the Creek fire are covered in ash and other material that doesn’t absorb water. Heavy rain runs off recently burned soil “as it would run off of pavement,” says the weather service. That can lead to issues with debris flow and mudslides.
How rapidly the rain falls is another factor in the damage it can do. Forecasters say there’s a 20 percent chance that an inch of rain could be falling per hour at the peak of the storm.
”If the 2-4 inches of rain falls gently over the whole 72 hour period, not a lot is going to happen,” Andrew Rorke, a senior forecaster with the weather service, tells KPCC. But if the rain falls in big surges, that could be a problem, “and unfortunately we are forecasting a couple big bursts of rain.”
According to the weather service, Downtown Los Angeles has seen just 3.4 inches of rain since October 1, the start of the wet season. That’s roughly a quarter of the 12.6 normal rainfall total. | https://keithscaduto.com/major-storm-will-slam-la-this-is-the-big-one/ |
Brighten plain bed linen with this how to make a bed runner craft project to produce a colourful patchwork runner that gives well-worn or charity-shop jumpers a new lease of life.
You will need
- five pure-wool jumpers
- rectangular or square paper template, to fit the back and front panels of the knits
- tape measure, tailors' chalk, scissors
- sewing machine
- backing fabric, optional(we used velvet)
Method
- Draw around the template on your jumpers with tailors' chalk. Cut out these shapes, aligning the panels, where you can, with the design and direction of the knitting.
- When you have cut out ten pieces, lay them out on a large surface and decide on the order in which they should be sewn.
- Starting with the first two panels, place them right sides together and pin along one edge. Stitch to join, removing the pins as you go. Repeat with the remaining pieces of fabric, until you have one long strip that will cover the foot of your bed, and allow for overhang.
- If you wish to back your runner, lay it, right side down, on top of the right side of your backing fabric. Pin it in place around the edges, then cut the backing fabric so it is the same size as the knitted runner.
- Sew these together along the two long edges, and one short edge, removing the pins as you go. Turn the resulting tube the right way out. Fold the remaining raw edges inward and top-stitch to close the opening and finish off the runner. | https://www.countryliving.com/uk/create/craft/how-to/a131/how-to-make-a-bed-runner/ |
Lancaster woman marries US man on Zoom - despite never meeting in real life
A Lancaster woman has married an American man via Zoom - despite the transatlantic couple NEVER meeting in real life.
Cleaner Ayse, 26, and Darrin, 24, from Detroit, met online during the coronavirus pandemic, after Ayse joined a Facebook group aimed at pairing up pen pals from around the world.
She started talking to Kenda, 56, who said her son Darrin was a similar age and would love to chat to someone from the UK – despite being based in Michigan, US.
The two began chatting as friends in July 2020 but were soon inseparable and on the phone to each other every night, even with the five-hour time difference.
After becoming an official item, Ayse began planning a trip to the States to meet her boyfriend for the first time in July – but travel restrictions to the US made this impossible.
The couple continued to share weekly date nights on video calls and on May 21 2021 Darrin surprised Ayse even further by proposing.
Knowing it would be a long time before they’d get to meet for the first time, the couple made the brave decision to get married via Zoom, in a legal ceremony officiated in Utah, US.
On August 19 this year, Ayse and Darrin officially became husband and wife, supported by a small number of friends and family via video call.
They are still yet to meet face-to-face but speak multiple times a day and are eagerly waiting for Ayse’s visa application to be accepted so they can fully enjoy life together as a married couple.
Ayse said: “I never imagined this would happen to me in a million years.
“I don’t think anyone expected this when we started talking last year. But we’re married and it’s all completely legal and official – I still can’t believe it.
“It’s been really hard not being able to meet properly, but in the long run it will make us stronger because we’ve had to go through all of this to be together.
“We just really love each other.”
On how the couple met, having linked up with carer Kendra, she added: “It’s funny that I met my mother-in-law before I met my husband.
“We got on so well, so when she suggested speaking to her son, I thought it was a great idea – although I was a bit nervous at first because I’d seen photos of him and he looked lovely.”
Kenda passed Ayse her son Darrin’s number, and she sent a message to introduce herself.
Darrin responded and the pair began chatting frequently, before he suggested they call each other on the phone.
Ayse downloaded a mobile app which allowed her to call the US for free, before calling her pen pal for the first time.
“I worried it might be awkward, but we talked for hours,” Ayse said.
“We have so much in common, including both being terrified of thunder and lightning, and both having owl tattoos with blue eyes.
“I’m not a flirty person, but it came so naturally when I was chatting to Darrin – we got on so well straight away.”
The pair began chatting every day - via phone, video call and text - and each night would fall asleep while on the phone to each other so they felt closer.
Due to the five-hour time difference, it meant Ayse was falling asleep 2am - a sacrifice she says was ‘completely worth it’.
“It was during the pandemic so I wasn’t working as I’d been furloughed and just altered my sleep pattern, so I was sleeping at the same time Darrin was,” she said.
“We’d be on video call and just be chatting when we fall asleep – then usually when I wake up, he’s still there sleeping.
“It’s the closest we can get and might sound strange but it’s a huge comfort.”
Darrin asked Ayse to be his girlfriend in November 2020 after a few months of chatting every day, and the couple began planning for Ayse to visit the US for them to meet for the first time.
Hoping the travel restrictions in place due to Covid19 would be lifted by the summer, Ayse booked flights to Detroit for July 23 2021, for herself, and mum, Elaine, 67, and dad Nigel, 74, both retired.
Unfortunately, the US was still restricting travel from the UK, which meant Ayse and her family weren’t able to fly directly.
Hoping to avoid the ban on UK arrivals but determined to meet her boyfriend in person for the first time, Ayse flew to Mexico for 15 days before catching an onward flight to Detroit.
In another heart-breaking turn of events, Ayse was then turned away at US immigration, because the immigration office claimed she ‘didn’t have enough money in her bank to support herself during her stay’ and ‘didn’t have strong enough ties to the UK’.
“I think they thought that if they let me into the US then I’d never leave because of Darrin,” Ayse said.
“His address was listed as where I was staying on my tourist visa and they asked my relationship to him, so when I told them he was my boyfriend and they saw I’d been in Mexico for 15 days I guess they thought I’d try to outstay my visa.
“I would never have done that, and it was so heart-breaking to be so close to him in the same city, but not be allowed through the border.
“I was put on the next flight home and cried the whole way back.”
Just days after she returned to her home in Lancaster, Darrin suggested a ‘special date night’ where they both got dressed up to spend some virtual time together.
“When he answered the call he seemed so nervous which was unlike him – but I soon understood why when he got down on one knee and asked me to marry him,” Ayse said.
“I couldn’t believe it. He said he’d been planning to propose when I came to Detroit but couldn’t wait any longer to ask.
“Despite never meeting, there was no doubt in my mind that I wanted to marry Darrin.”
The couple’s engagement was fully supported by both of their families.
“I didn’t tell my parents straight away when we first started talking, but they twigged there was something going on when I was on the phone all the time,” Ayse said.
“They had reservations about it to start with – but I suggested they chat to Darrin on Skype and they saw straight away how crazy we both are about each other.
“When Darrin proposed he spoke to my dad beforehand to ask his permission, which I thought was really sweet and know my dad really appreciated.
“In our heads, as long as our family and friends are supportive, that’s all we need.”
The couple initially planned to tie the knot when they finally got the chance to meet for the first time, but after reading about a new law passed that made international virtual wedding ceremonies legal in the state of Utah, US, they decided to go ahead straight away.
Ayse and Darrin tied the knot on August 19, in an online ceremony officiated by the state of Utah, and are now officially husband and wife.
Although not a conventional wedding, Ayse wore white and was accompanied by her proud parents, and Darrin was joined by his mum and wore a smart burgundy suit to deliver his vows.
“We’ll make sure we celebrate together when we can – but for now, this was perfect,” Ayse said.
“I still can’t believe it – I never expected any of this but you’ve got to take what life throws at you.
“I miss Darrin every day but I know we’ll be together one day and it’ll all be worth it.”
Despite travel restrictions being lifted to the US from the UK this week, Ayse has to wait for another visa to be accepted before she can book her flights.
“We don’t know when I’ll be able to get over there but are trying to be as positive as we can while we wait,” Ayse said.
"Looking back it seems like so long ago we started as pen pals and now we’re battling against the world together," Darrin said.
"As we haven’t met in person yet we haven’t been able to have a physical relationship, which means that our marriage is based off of more than that.
"This hasn’t affected our relationship like it could do others, as it’s not something we see as the most important aspect, and we value other things, such as communication more. | https://www.lep.co.uk/news/people/lancaster-woman-marries-us-man-on-zoom-despite-never-meeting-in-real-life-3463545 |
Just Add Magic: Mystery City Season 2
Just Add Magic: Mystery City Season 2 is a continuation of the Just Add Magic series. The series follows the adventures of a group of young kids who use their magical powers to solve mysteries. In Season 2, the kids travel to Mystery City, a place full of magical mysteries. They must use their powers to solve mysteries and save the city from destruction.
The detectives found many clues in the mystery city. They found evidence that the city was once a thriving metropolis, but now it is abandoned and overrun by monsters. They also found evidence that the city was once home to a powerful magic user, but he mysteriously disappeared. The detectives are now trying to find out what happened to the city and the magic user.
In the second season of Just Add Magic, new characters will be introduced who helps the main characters solve various mysteries. These characters include a genius inventor named Simon; a spunky young witch named Amanda; and a mystical unicorn named Melody. The new season will also see the return of some of the show’s favorite characters, including Alex the Magician; Simon’s big sister, Theresa; and the magical creatures known as the Wiggles.
Detectives are looking into the disappearances of several children in the city. They have a suspect, but they need to find concrete proof to arrest him. The detectives are also looking for any clues that may lead them to the children’s safe return.
Just Add Magic is an animated show about three magical girls who are trying to save their school from being closed down. The second season finale is going to be very exciting because it is going to reveal some secrets about the show’s characters. Among other things, it is going to show who the real villain is and what happened to the other magical girls.
Detective Kincaid and Detective Mills are investigating a murder that appears to be connected to the dark underworld of the city. They have been working on the case for weeks, and they have a lot of evidence linking the murderer to a powerful figure in the city. They are getting close to putting together the pieces of the puzzle, but they still don’t have a clear idea of what the mastermind’s plan is. They know that he is planning to execute a large-scale crime, but they don’t know what it is. They are close to finding out, but they are not sure if they can stop the mastermind in time.
The detectives will use a new gadget called a Geiger counter to determine if there is any radiation in the scene. They will also use a new forensic tool to analyze the DNA of the victim and the perpetrator.
The detectives have been working hard to catch the mastermind for months, but they’ve been unable to find any evidence that connects him to the crime. They have a few leads, but they’re not sure if they’re strong enough to convict him. The detectives are about to give up hope when they get a lead that could finally bring them closer to catching the mastermind. They go to the home of the mastermind’s girlfriend to question her. She tells them that the mastermind is planning to kill someone important and that he’s been planning the murder for weeks. The detectives work hard to find the mastermind and arrest him before he can carry out his plan. They’re successful and the mastermind is sent to prison.
The winner of the Just Add Magic mystery city contest will be a student who can come up with the best plan to solve the mystery city challenges.
Between seasons, the Just Add Magic world will continue to be plagued by magical creatures that have escaped from the magical prison. The magical creatures have begun to spread chaos and destruction throughout the city. The Just Add Magic team is still trying to find a way to imprison the magical creatures once and for all. Meanwhile, the Just Add Magic team is also trying to find a new magical girl to replace Candace.
Ten-year-old Sophie is a budding artist and a huge fan of Just Add Magic, the popular children’s television show. When she returns home one day to find that the show has been canceled, Sophie is devastated. She spends hours online trying to find out what happened to the characters and the show, but no avail. One day, Sophie’s older sister tells her about a website called The Portal that allows users to enter the show’s world. Sophie is skeptical at first but decides to try it out. After logging in, she finds herself in the show’s world and meets up with her favorite characters. Sophie is overjoyed to discover that the show is being re-aired, and she and her sister help to solve the new mysteries. Sophie is even more excited when she learns that Just Add Magic: Mystery City Season 2 is coming back to TV soon!
Just Add Magic: Mystery City Season 2 was an amazing continuation of the first season. The storyline was intriguing and kept me engaged the whole way through. The characters were lovable and I was rooting for them all the way. The settings were vivid and realistic, and the magic was spectacular. I would highly recommend this series to anyone looking for a well-done magical adventure.
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The utility model belongs to the technical field of jack contact element main bodies, particularly relates to an improved jack contact element and a composite hole sleeve structure, and aims to solve the problem of poor conductivity of the conventional jack contact element main body, and provides the following scheme: the improved jack contact element and composite hole sleeve structure comprises a jack contact element main body, the jack contact element main body comprises two contact piece root parts and a simply supported beam, a plurality of metal sheets are arranged on the sides, far away from each other, of the roots of the two contact pieces, the ends, close to each other, of the roots of the two contact pieces are fixedly connected with the simply supported beam, the ends, far away from each other, of the roots of the two contact pieces are fixedly connected with the metal sheets, and the simply supported beam comprises an upper portion, a middle portion and a lower portion. The simply supported beam becomes a rotary hyperboloid, the projection of the axial section of the spiral cylinder wall is a rotary hyperboloid, the rotary hyperboloid is uniformly distributed in a circle, the conductive sectional area of the jack contact main body is large, the load is large, and the weight of the jack contact main body is reduced. | |
Starting at age five, students can take tests that will separate them from their peers. If a kindergarten student scores exceptionally, he is ushered into a new cohort of students who also scored well. This is the Gifted program.
Gifted used to be classified as a form of special education because of the unique needs of the students. Students within this program have particular learning styles and certain behavioral patterns that differ from those of conventional learners and often require additional forethought. However, there is no single characteristic that classifies a student as Gifted.
Physics teacher Mr. Phil Heier states, “In Physics, it’s a student who looks at things differently. They’re able to, instead of just solving a problem just by rote, they are able to go about it a number of different ways.”
Gifted is also not synonymous with academic. Many students who have been tested into the Gifted program tend not to pay attention and complete the work as it comes easily to them.
Others, Heier says, “who fit the county or state definition of Gifted are not truly Gifted, they are just really good students, and there is nothing wrong with that. But some of my worst students are gifted. They view things differently; they’re not focused, academically.”
These may be the students in the back of the room who chirp up to ask questions but may not be seen as brilliant because their test scores may not reflect their abilities.
Students’ Giftedness may also differ from subject to subject with some Gifted across the board while others only excel in one or another.
He continues on to mention students who are extremely grade-conscious who ask simply, “‘show me how to do the problem,’” and they will get it done. “Those typically are not Gifted,” Heier states. “Then there are students who are academically driven and Gifted because not only do they want to do it well, but they want to do it well three different ways.”
Because there is no definitive description of a Gifted Student, school systems attempt to best meet students’ educational needs using an assessment that measures students’ academic abilities.
Head of the Gifted department at Parkview, Tara Finco explains the Gifted test as assessing creativity, motivation, and mental achievement/ability. “They have to meet eligibility in three out of those four areas if they’re going to be considered in the Gifted Program.”
However, students who score exceptionally in the mental categories can enter the program without excelling as highly in the other two.
Gifted classes at Parkview are a mixture of students who have tested into the Gifted program and high-achieving honors level students who have shown dedication or passion for a subject. These classes tend to move at a faster pace and incorporate more individual work. Gifted students can often be identified by their desire for academic rigor and freedom.
“Some Gifted students take regular courses their senior year thinking it’ll be a nice, easy class–it drives them crazy because they’re not being challenged,” Heier says.
Finco believes all students should be given a level of free reign in their schooling. “Ideally, all learning would be project-based,” she ventures.
Although it begins at the classroom level, division between students classified as “Gifted” and “average” often spills over into social interactions and personal mentalities.
In his classes, Heier has noticed traces of this attitude, “For some students, there is a bit of, actually not a bit, a lot of ego. Certainly embrace giftedness, but that doesn’t make you any better than the hard working honors student.”
Sania Chandrani finds words blissful. Her bookshelf is cluttered with tattered journals filled with poetic ramblings and drawings/doodles, well-loved books... | https://phspantera.com/special-feature/2014/12/07/gifted-learning-different-not-better/ |
---
abstract: 'This short note solves the following problem: Given a map $\pi:X_\Sigma\rightarrow X_\sigma$ of normal toric varieties corresponding to a coherent subdivision $\Sigma$ of a cone $\sigma$, find an ideal sheaf ${\mathcalI}$ on $X_\sigma$ such that $\pi$ is the blowup of ${\mathcalI}$.'
address: |
Mathematics Department\
402 Murchie Science Building\
303 East Kearsley Street\
Flint, MI 48502-1950
author:
- Howard M Thompson
title: Pulling Subdivisions of Cones and Blowups of Monomial Ideals on Affine Toric Varieties
---
[^1]
Introduction
============
Thompson [@MR3448804] takes advantage of the following theorem to find identify a well-controlled log resolution of a monomial space curve.
(Gonz[á]{}lez P[é]{}rez & Teissier [@MR1892938 Theorem 3.1]) \[t:GT\] Let $\sigma\subset N_{\mathbb{R}}$ be strictly convex rational polyhedral cone, let $\phi:{\mathsf{S}}_\sigma\rightarrow{\mathsf{S}}$ be a surjective homomorphism of pointed affine semigroups, let $\phi_{\mathbb{R}}:M_{\mathbb{R}}\rightarrow{\mathbb{R}}{\mathsf{S}}$ be the induced linear map, let $\ell=\ker(\phi_{\mathbb{R}})^\perp\subset N_{\mathbb{R}}$, let $\Bbbk$ be a field, and let $Z=\operatorname{Spec}\left(\Bbbk[{\mathsf{S}}]\right)\subset X_\sigma$. If $\Sigma$ is any subdivision of $\sigma$ containing the cone $\tau=\sigma\cap\ell$ and $N_\tau={\mathbb{Z}}(N\cap\tau)$, then
(1) The strict transform of $Z$ by the morphism induced by the subdivision $\pi_\Sigma:X_\Sigma\rightarrow X_\sigma$ is contained in $X_\tau$, it is isomorphic to $X_{\tau,N_\tau}$ and the restriction $\pi_\Sigma|_{X_{\tau,N_\tau}}:X_{\tau,N_\tau}\rightarrow Z$ is the normalization map.
(2) The morphism $\pi_\Sigma$ is a partial embedded resolution of $Z\subset X_\sigma$. (That is, any toric desingularization of $X_\Sigma$ provides an embedded resolution of $Z\subset X_\sigma$).
In , we will construct a minimal subdivision as in the theorem by defining pulling subdivision for cones and present a simple example. In , we briefly describe how to produce an ideal from a coherent subdivision.
Our general reference on toric varieties is Cox, Little & Schenck [@MR2810322] and our general reference on subdivisions is De Loera, Rambau & Santos [@MR2743368].
Pulling Subdivision {#s:pull}
===================
Let $\sigma,\tau\subset N_{\mathbb{R}}$ be strictly convex rational polyhedral cones such that $\tau$ is a subset of $\sigma$. We do not assume $\tau$ is a subcone of $\sigma$.
The following construction defines the *pulling subdivision* $\operatorname{pull}_\tau\sigma$ of $\tau$ in $\sigma$. Fix a (rational) hyperplane $H$ not containing the origin such that $H\cap\rho$ is nonempty for each ray $\rho\in\sigma(1)$. Let $$\mathscr{A}=\{H\cap\rho\mid\rho\in\sigma(1)\}\text{ and }\mathscr{B}=\{H\cap\rho\mid\rho\in\tau(1)\}.$$ be (necessarily finite) point sets in $N_{\mathbb{R}}\times{\mathbb{R}}$, Consider the polytope $\operatorname{Conv}((\mathscr{A}\times\{0\})\cup(\mathscr{B}\times\{1\}))$ corresponding to the height function $\omega:\mathscr{A}\rightarrow{\mathbb{R}}$ that is $1$ on $\mathscr{B}$ and $0$ on $\mathscr{A}\setminus\mathscr{B}$. By construction, the projection of the upper hull of this polytope onto the first factor is a coherent subdivision of $P=H\cap\sigma$. This subdivision of $H\cap\sigma$, $\operatorname{pull}_Q P$, is (essentially) the result of pulling $Q=H\cap\tau$ as in Section 2.2 of Haase & Zharkov [@HZ2002]. Let $\operatorname{pull}_\tau\sigma$ be the fan consisting of the cones over the faces of $\operatorname{pull}_Q P$.
Note that the height function $\omega:\mathscr{A}\rightarrow{\mathbb{R}}$ that is $1$ on $\mathscr{B}$ and $0$ on $\mathscr{A}\setminus\mathscr{B}$ extends to a support function $\varphi:|\operatorname{pull}_\tau\sigma|\rightarrow{\mathbb{R}}$ given by rational vectors $\{{\mathbf}{u}_\rho\}_{\rho\in\operatorname{pull}_\tau\sigma}\subset{\mathbb{Q}}M$. So, some multiple of $\varphi$ is a support function that is integral with respect to the lattice $N$. Let this multiple be given by the set $\{{\mathbf}{m}_\rho\}_{\rho\in\operatorname{pull}_\tau\sigma}$. This set is the Cartier data for some Cartier divisor as in Cox, Little & Schenck [@MR2810322 Theorem 4.2.8].
Let ${\mathbf}{n}_1=\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^\mathsf{T}$, let ${\mathbf}{n}_2=\begin{bmatrix} 0 & 1 & 0 \end{bmatrix}^\mathsf{T}$, let ${\mathbf}{n}_3=\begin{bmatrix}0 & 0 & 1 \end{bmatrix}^\mathsf{T}$, let ${\mathbf}{n}_4=\begin{bmatrix} 2 & 1 & 0 \end{bmatrix}^\mathsf{T}$, let ${\mathbf}{n}_5=\begin{bmatrix}0 & 1 & 2 \end{bmatrix}^\mathsf{T}$, let $\sigma={\mathbb{R}}_{\geq0}{\mathbf}{n}_1+{\mathbb{R}}_{\geq0}{\mathbf}{n}_2+{\mathbb{R}}_{\geq0}{\mathbf}{n}_3$ be the positive orthant in ${\mathbb{R}}^3$, let $\tau={\mathbb{R}}_{\geq0}{\mathbf}{n}_4+{\mathbb{R}}_{\geq0}{\mathbf}{n}_5$, and pick $\begin{bmatrix} 1 & 1 & 1 \end{bmatrix}v=1$ for $H$. Then, $\mathscr{A}=\{{\mathbf}{n}_1,{\mathbf}{n}_2,{\mathbf}{n}_3\}$ and $\mathscr{B}=\left\{\frac{1}{3}{\mathbf}{n}_4,\frac{1}{3}{\mathbf}{n}_5\right\}$.
(N1) at (0,0) [$({\mathbf}{n}_1,0)$]{}; (N2) at (15,0) [$({\mathbf}{n}_2,0)$]{}; (N3) at (5,5) [$({\mathbf}{n}_3,0)$]{}; (V4) at (5,3) [$\left(\frac{1}{3}{\mathbf}{n}_4,1\right)$]{}; (V5) at (8.333,6.333) [$\left(\frac{1}{3}{\mathbf}{n}_5,1\right)$]{}; (N1) – (N2); (N1) – (N3); (N2) – (N3); (N1) – (V4); (V4) – (N2); (N2) – (V5); (V5) – (N3); (V4) – (V5); (0,0) – (15,0) – (5,5) – (0,0); (0,0) – (5,5) – (8.333,6.333) – (5,3) – (0,0); (15,0) – (5,3) – (8.333,6.333) – (15,0);
In the figure, the lower facet $\operatorname{Conv}(\mathscr{A}\times\{0\})$ is green, the upper facets containing $\operatorname{Conv}(\mathscr{B}\times\{1\})$ are blue and red, and the other two (unshaded) facets are vertical. The projections of the two vertical facets of the polyhedron to the green facet are not maximal cells of the subdivision. The projections of the blue and red facets onto the green facets yield the maximal cells of $\operatorname{pull}_Q P$.
According to a Macaulay2 [@M2] calculation, the $3$-dimensional polytope in ${\mathbb{R}}^4$ with vertex set $(\mathscr{A}\times\{0\})\cup(\mathscr{B}\times\{1\})$ is given by the halfspaces $$\begin{matrix*}[r]
w_1 & + w_2 & -3w_3 & & \leq 1 & \\
-3w_1 & + w_2 & +w_3 & & \leq 1 & \\
& & & -w_4 & \leq 0 & \\
-3w_1 & +3w_2 & -3w_3 & +4w_4 & \leq 3 & \qquad (1) \\
3w_1 & -9w_2 & +3w_3 & +4w_4 & \leq 3 & \qquad (2)
\end{matrix*}$$
The triangle with vertex set $\left\{{\mathbf}{n}_2,\frac{1}{3}{\mathbf}{n}_4,\frac{1}{3}{\mathbf}{n}_5\right\}$ is the projection of the red facet supported on the hyperplane defining the halfspace $(1)$ and the trapezoid with vertex set $\left\{{\mathbf}{n}_1,{\mathbf}{n}_3,\frac{1}{3}{\mathbf}{n}_4,\frac{1}{3}{\mathbf}{n}_5\right\}$ is the projection of the blue facet supported on the hyperplane defining the halfspace $(2)$. Thus, the maximal cones of the fan $\operatorname{pull}_\tau\sigma$ are $\sigma_1={\mathbb{R}}_{\geq0}{\mathbf}{n}_2+{\mathbb{R}}_{\geq0}{\mathbf}{n}_4+{\mathbb{R}}_{\geq0}{\mathbf}{n}_5$ and $\sigma_2={\mathbb{R}}_{\geq0}{\mathbf}{n}_1+{\mathbb{R}}_{\geq0}{\mathbf}{n}_3+{\mathbb{R}}_{\geq0}{\mathbf}{n}_4+{\mathbb{R}}_{\geq0}{\mathbf}{n}_5$. Note that this fan is not refined by nor a refinement of the star subdivision along either of the rays of $\tau$. Here the support function $\varphi$ extending the height function defining $\operatorname{pull}_\tau\sigma$ is given by ${\mathbf}{u}_{\sigma_1}=\begin{bmatrix} \frac{3}{2} & 0 & \frac{3}{2} \end{bmatrix}$ and ${\mathbf}{u}_{\sigma_2}=\begin{bmatrix} 0 & 3 & 0 \end{bmatrix}$. That is, $$\varphi:|\operatorname{pull}_\tau\sigma|\rightarrow{\mathbb{R}}; \qquad {\mathbf}{v}\mapsto\min\{\langle{\mathbf}{u}_1,{\mathbf}{v}\rangle,\langle{\mathbf}{u}_2,{\mathbf}{v}\rangle\}.$$ So, $2\varphi$ is given by ${\mathbf}{m}_{\sigma_1}=\begin{bmatrix} 3 & 0 & 3 \end{bmatrix}$ and ${\mathbf}{m}_{\sigma_2}=\begin{bmatrix} 0 & 6 & 0 \end{bmatrix}$. Here, $\operatorname{pull}_\tau\sigma$ is the (inward) normal fan of the polyhedron $\operatorname{Conv}(({\mathbf}{m}_{\sigma_1}+{\mathsf{S}}_\sigma)\cup({\mathbf}{m}_{\sigma_2}+{\mathsf{S}}_\sigma))$. Let $x_i=\chi^{{\mathbf}{e}_i}$ where ${\mathbf}{e}_i$ is the $i$th standard basis element of $M_{\mathbb{R}}$ and let $I=\overline{\left(x_1^3x_3^3,x_2^6\right)}$ in $\Bbbk[{\mathsf{S}}_\sigma]=\Bbbk[x_1,x_2,x_3]$. In this case, $\operatorname{pull}_\tau\sigma$ is the fan of the blowup of $I$.
More generally, if $\operatorname{Newt}(I)=\operatorname{Conv}\{{\mathbf}{m}\mid\chi^{{\mathbf}{m}}\in I\}$ is the Newton polyhedron of a torus invariant ideal $I\subset\Bbbk[{\mathsf{S}}_\sigma]$ for some strictly convex rational polyhedral cone $\sigma$, then the dual fan of $\operatorname{Newt}(I)$ is the fan of the normalized blowup of $I$. See Thompson [@hT2003].
From Coherent Subdivisions to Ideals {#s:ideals}
====================================
Now, if $\Sigma$ is a fan that is a coherent subdivision of a fan $\Delta$ in $N_{\mathbb{R}}$, then we would like to find an ideal sheaf ${\mathcalI}$ on $X_\Delta$ such that the map of toric varieties $\pi_\Sigma:X_\Sigma\rightarrow X_\Delta$ induced by the subdivision is the blowup of ${\mathcalI}$. To achieve this, we want a support function $\varphi$ that is integral with respect to the lattice $N$ with Cartier data $\{{\mathbf}{m}_\sigma\}_{\sigma\in\Sigma}$ (as in Cox, Little & Schenck [@MR2810322 Theorem 4.2.8]) such that whenever $\tau\in\Delta$ with $\sigma\subset\tau$, ${\mathbf}{m}_\sigma\in{\mathsf{S}}_\tau$. Solve the system of linear inequalities for the subset $\{{\mathbf}{m}_\sigma\}_{\sigma\in\Sigma_{max}}$. Namely,
1. $\langle{\mathbf}{m}_{\sigma_1},{\mathbf}{v}_\rho\rangle=\langle{\mathbf}{m}_{\sigma_2},{\mathbf}{v}_\rho\rangle$ whenever ${\mathbf}{v}_\rho$ is the primitive vector on a ray $\rho\subset(\sigma_1\cap\sigma_2)$;
2. $\langle{\mathbf}{m}_{\sigma_1},{\mathbf}{v}_\rho\rangle<\langle{\mathbf}{m}_{\sigma_2},{\mathbf}{v}_\rho\rangle$ whenever ${\mathbf}{v}_\rho$ is the primitive vector on a ray $\rho\subset(\sigma_1\setminus\sigma_2)$.
Choose the torus invariant sheaf ${\mathcalI}$ on $X_\Delta$ such that, for each $\tau\in\Delta_{max}$, $$\Gamma(U_\tau,{\mathcalI})=\overline{(\chi^{{\mathbf}{m}_\sigma}\mid\sigma\subset\tau)}\subset\Bbbk[{\mathsf{S}}_\tau]$$
[^1]:
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BACKGROUND OF THE INVENTION
Derwent Abstract 85-000957/01 of German Application No. 3,321, 969 discloses 1-pyrimidyl-4-substituted piperazine derivative which possess a broad variety of CNS activity including anxiolytic and antidepressant properties. Netherlands Pat. No. 7,017,031 discloses 8- (heteroarylpiperazinylalkyl)-8-azaspiro[4,5]decane-7,9-diones as tranquilizers. U.S. Pat. No. 4,640,921 (Derwent Abstract 87-049798/07 discloses the use of the buspirones of the Netherlands patent in the treatment of sexual dysfunction in anxious patients. The anxiolytic activity of buspirone-like compounds has been attributed to their selective activation of a serotonin (5-hydroxytryptamine; 5-HT) subtype receptor designated the 5-HT.sub.1A receptor. U.S. Pat. No. 4,202,898 discloses the treatment of anxiety and depression with aromatically substituted piperazine derivatives. 5-HT.sub.2 antagonists, such as Ritanserin, lack 5-HT.sub.1A affinity but demonstrated clinical efficacy as anxiolytic-antidepressant agents (Barone et al., Drug Clin. Pharm., 20, 770, 1986).
DESCRIPTION OF THE INVENTION
In accordance with this invention there is provided a group of novel compounds possessing anxiolytic, antidepressant and in some instances antipsychotic activity. The compounds of this invention are of the following structural formula: ##STR4## wherein R.sup.1 is 1- adamantyl, 3- methyl-1-adamantyl, 9-fluorenyl or 1-fluorenyl;
n is 0 or 1;
m is 1, 2, 3, 4 or 5; and
X is ##STR5## where R.sup.2 is phenyl, benzyl, pyridinyl, pyrimidinyl, pyrazinyl, or substituted phenyl or benzyl in which the substituent is alkyl of 1 to 6 carbon atoms, alkoxy of 1 to 6 carbon atoms, halo, cyano, nitro or trifluoromethyl, ##STR6## where R.sup.3 is hydrogen, alkyl of 1 to 6 carbon atoms, alkoxy of 1 to 6 carbon atoms, halo, cyano or nitro;
or a pharmaceutically acceptable salt thereof.
The pharmaceutically acceptable salts are conveniently derived by conventional means from such acids as hydrochloric, hydrobromic, sulfuric, phosphoric, methane sulfonic, nitric, p-toluene sulfonic, acetic, citric, maleic, succinic acid, and the like. The halogens embraced by the term halo are chlorine, bromine, iodine and fluorine, preferably chlorine, bromine or fluorine.
The compounds of this invention may be prepared by a variety of synthetic routes using conventional methods. For instance, 1- adamantanecarboxylic acid halide, 3-methyl-1-adamantanecarboxylic halide, 1-fluorenylcarboxylic acid halide or 9-fluorenylcarboxylic acid halide may be conveniently reacted with the appropriately substituted ##STR7## in CH.sub.2 Cl.sub.2 and the presence of a suitable base, such as triethylamine, to obtain the desired intermediate or final product or, alternatively, the acid halides may be reacted with the desired hydroxyhaloalkane -HO-(CH.sub.2).sub.m -Br- in CH.sub.2 Cl.sub.2 and the presence of a suitable base, such as triethylamine, followed by reacting the intermediate bromoalkylester with the desired aromatically or heteroaromatically substituted piperazine or piperidine intermediate.
The following examples illustrate, without limitation, the specific methods employed in production of a representative number of compounds embraced by this invention.
EXAMPLE 1
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 3-[4-(3- chlorophenyl)-1-piperazinyl]propyl ester
To a stirred solution of 3-bromopropanol (5.56 g, 0.04 mol) in 50 mL of methylene chloride, adamantane-1-carboxylic acid chloride (5.6 g, 0.02 mol) and triethylamine (4 g, 0.04 mol) were added. The reaction mixture was stirred at room temperature overnight, the methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 12 g of the intermediate adamantane-1- carboxylic acid bromopropyl ester. The title compound was prepared by stirring 4-(3-chlorophenyl)-1-piperazine (2.7 g, 0.011 mol), adamantane-1- carboxylic acid bromopropyl ester (3.5 g, 0.011 mol) and triethylamine (2. 5 g, 0.025 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na. sub.2 SO. sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f =0.6 in 30% methanol/ethylacetate) was separated and converted to the dihydrochloride salt with ethanolic HCl (1 g, 20% yield), mp. 220° -222° C.
Elemental Analysis for C.sub.24 H.sub.33 N.sub.2 CLO.sub.2.HCl: Calc'd: C, 58.84; H, 7.15, N, 5.72. Found: C, 59.00; H, 7.03; N, 5.55.
EXAMPLE 2
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 2-[4-(2- pyrimidinyl)-1-piperazinyl]ethyl ester
To a stirred solution of [4-(2-pyrimidinyl)piperazino]ethanol (3. 7 g, 0.017 mol) in 50 mL of methylene chloride, adamantane-1- carboxylic acid chloride (5 g, 0.018 mol) and triethylamine (2 g, 0.02 mol) were added. Stirring was continued at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous sodium sulfate and evaporated under reduced pressure. The remaining residue was subjected to preparative HPLC as in Example 1. In repeated preparations, the residue was dissolved in ethylacetate (10 mL) and subjected to flash chromatography using a 9 inch column of silica gel with ethylacetate as the eluent. The title compound (TLC R.sub.f =0.63 in 30% methanol/ethylacetate system) was separated and converted to the hydrochloride salt with ethanolic HCl (3 g, 43% yield), mp. 232°- 235° C.
Elemental Analysis for C.sub.21 H.sub.30 N.sub.4 O.sub.2.HCl Calc'd: C, 61.92; H, 7.63; N, 13.76. Found: C, 61.38; H, 7.51; N, 14.10.
EXAMPLE 3
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 3-[4-(2- methoxyphenyl)-1-piperazinyl]propyl ester
To a stirred solution of [4-(2-methoxyphenyl)piperazino]ethanol (3.7 g, 0.015 mol) in 50 mL of methylene chloride, adamantane-1- carboxylic acid chloride (5 g, 0.018 mol) and triethylamine (2 g, 0.02 mol) werre added. Stirring was continued at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous sodium sulfate and evaporated under reduced pressure. The remaining residue was subjected to preparative HPLC (as in Example 1). In repeated preparations, the residue was dissolved in ethylacetate (10 mL) and subjected to flash chromatography using a 9 inch column of silica gel and ethylacetate as the eluent. The title compound (TLC R.sub.f =0.65 in 30% methanol/ethylacetate system) was separated and converted to the dihydrochloride salt with ethanolic HCl (4 g, 55% yield); mp. 212° - 213° C.
Elemental Analysis for C.sub.25 H.sub.36 N.sub.2 O.sub.3.2 HCl: Calc'd: C, 61.85; H, 7.83; N, 5.77. Found: C, 61.55; H, 7.95; N, 5.61.
EXAMPLE 4
3-Methyltricyclo[3.3.1.1.sup.3,7 ]decane-1-acetic acid 2-[4-(2- pyrimidinyl)-1-piperazinyl]ethyl ester
To a stirred solution of [4-(2-pyrimidinyl)piperazino]ethanol (3. 73 g, 0.017 mol) in 50 mL of methylene chloride, 3-methyladamantane-1- acetic acid chloride (4 g, 0.018 mol) and triethylamine (5 g, 0.05 mol) were added. Stirring was continued at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous sodium sulfate and evaporated under reduced pressure. The remaining residue was subjected to preparative HPLC (as in Example 1) using ethylacetate as the eluent to afford the title compound as the free base, which was converted to the dihydrochloride salt with ethanolic HCl (2.9 g, 33% yield); mp 202. degree.-206° C.
Elemental Analysis for C.sub.23 H.sub.34 N.sub.4 O.sub.2.2 HCl. H. sub. 2 O: Calc'd: C, 56.43; H, 7.82; N, 11.44; Cl, 14.40. Found: C, 56. 08; H, 7.53; N, 11.39; Cl, 14.32.
EXAMPLE 5
9H-Fluorene-1-carboxylic acid 2-[4-(2-methoxyphenyl)-1- piperazinyl] ethyl ester
To a stirred solution of 2-bromoethanol (3.3 g, 0.026 mol) in 50 mL of methylene chloride, fluorene-1-carboxylic acid chloride (6.0 g, 0. 026 mol) and triethylamine (5 g, 0.05 mol) were added. The reaction mixture was stirred at room temperature overnight, the methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 7 g of the intermediate fluorene-1- carboxylic acid bromoethyl ester. The title compound was prepared by stirring 4-(2-methoxyphenyl)-1-piperazine (2.1 g, 0.011 mol), fluorene-1- carboxylic acid bromoethyl ester (3.5 g, 0.011 mol) and triethylamine (1. 5 g, 0.015 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na. sub.2 SO.sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC using ethylacetate as the eluent (as in Example 1) and the desired product was separated and converted to the dihydrochloride salt with ethanolic HCl (2 g, 35% yield), mp. 213°-215° C.
Elemental Analysis for C.sub.27 H.sub.28 N.sub.2 O.sub.3.2 HCl. H. sub. 2 O: Calc'd: C, 62.43; H, 6.16; N, 5.39. Found: C, 62.73; H, 6.24; N, 5. 30.
EXAMPLE 6
3-Methyltricyclo[3.3.1.1.sup.3,7 ]decane-1-acetic acid 3-[4-[3- (trifluoromethyl)phenyl]-1-piperazinyl]propyl ester
To a stirred solution of [4-(3-trifluoromethylphenyl)piperazino] - propanol (4.3 g, 0.015 mol) in 50 mL of methylene chloride, 3- methyladamantane-1-acetic acid chloride (3.4 g, 0.015 mol) and triethylamine (2 g, 0.02 mol) were added. Stirring was continued at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous sodium sulfate and evaporated under reduced pressure. The remaining residue was subjected to preparative HPLC (as in Example 1) using ethylacetate as the eluent to afford the title compound as the free base which was converted to the hydrochloride salt with ethanolic HCl (2.9 g, 38% yield); mp. 177°-180° C.
Elemental Analysis for C.sub.27 H.sub.37 F.sub.3 N.sub.2 O.sub. 2. HCl: Calc'd: C, 62.97; H, 7.39; N, 5.44. Found: C, 62.70; H, 6.96; N, 5. 29.
EXAMPLE 7
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 3-[4-(2- pyrimidinyl)-1-piperazinyl]propyl ester
To a stirred solution of [4-(2-pyrimidinyl)piperazino]propanol (3. 9 g, 0.017 mol) in 50 mL of methylene chloride, adamantane-1- carboxylic acid chloride (5 g, 0.017 mol) and triethylamine (2 g, 0.02 mol) were added. Stirring was continued at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous sodium sulfate and evaporated under reduced pressure. The remaining residue was subjected to preparative HPLC (as in Example 1). In repeated preparations, the residue was dissolved in ethylacetate (10 mL) and subjected to flash chromatography using a 9 inch column of silica gel and ethylacetate as the eluent. The title compound (TLC R.sub.f =0.67 in 30% methanol/ethylacetate system) was separated and converted to the dihydrochloride salt with ethanolic HCl (3 g, 38% yield); mp. 209° - 211° C.
Elemental Analysis for C.sub.22 H.sub.32 N.sub.4 O.sub.2.2 HCl. 1/2 H. sub.2 O: Calc'd: C, 56.65; H, 7.51; N, 12.01. Found: C, 56.51; H, 7.37; N, 12.04.
EXAMPLE 8
9H-Fluorene-1-carboxylic acid 2-[4-(3-chlorophenyl)-1- piperazinyl] ethyl ester
To a stirred solution of 2-bromoethanol (3.3 g, 0.026 mol) in 50 mL of methylene chloride, fluorene-1-carboxylic acid chloride (6 g, 0. 026 mol) and triethylamine (5 g, 0.05 mol) were added. The reaction mixture was stirred at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 7 g of the intermediate fluorene-1- carboxylic acid bromoethyl ester. The title compound was prepared by stirring 4-(3- chlorophenyl)piperazine (2.6 g, 0.011 mol), fluorene-1- carboxylic acid bromoethyl ester (3.5 g, 0.011 mol) and triethylamine (2. 5 g, 0.025 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na. sub.2 SO.sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f =0.68 in 30% methanol/ethylacetate system) was separated and converted to the dihydrochloride salt with ethanolic HCl (1.8 g, 31% yield), mp. 215. degree.-217° C.
Elemental Analysis for C.sub.26 H.sub.25 N.sub.2 ClO.sub.2.2 HCl. H. sub.2 O: Calc'd: C, 59.60; H, 5.54; N, 5.34. Found: C, 59.72; H, 5.56; N, 4.98.
EXAMPLE 9
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 3-[4-[bis(4- fluorophenyl)methylene]-1-piperidinyl]propyl ester
To a stirred solution of 3-bromopropanol (5.5 g, 0.04 mol) in 50 mL of methylene chloride, adamantane-1-carboxylic acid chloride (5.6 g, 0.02 mol) and triethylamine (4 g, 0.04 mol) were added. The reaction mixture was stirred at room temperature overnight, the methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 12 g of the intermediate adamantane-1- carboxylic acid bromopropyl ester. The title compound was prepared by stirring 4-[bis(4-fluorophenyl)-methylene]piperidine (2 g, 0. 07 mol), adamantane-1-carboxylic acid bromo-propyl ester (2.1 g, 0.007 mol) and triethylamine (1 g, 0.01 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na.sub.2 SO.sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f =0.7 in 30% ethanol/ethylacetate system) was separated and converted to the hydrochloride salt (0.6 g, 16% yield), mp. 267°-269° C.
Elemental Analysis for C.sub.32 H.sub.37 NF.sub.2 O.sub.2.HCl: Calc'd: C, 70.91; H, 7.02; N, 2.58. Found: C, 71.29; H, 7.11; N, 2.30.
EXAMPLE 10
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid [2-[4-(2- cyanophenyl)-1-piperazinyl]ethyl]ester
2-Fluorobenzonitrile (6.0 g, 50 mmoles) and 1-(2-hydroxyethyl)- piperazine (6.5 g, 50 mmoles) were combined in 500 mL of dimethylformamide and heated at 80° C. under N.sub.2 for 24 hours. After cooling, the solvent was removed in vacuum and the crude 1-(2- hydroxyethyl)-4-(2-cyanophenyl)piperazine thus produced was used without purification. 2.3 Grams (10 mmole) of the crude 1-(2-hydroxyethyl)-4-(2- cyanophenyl)piperazine was dissolved in 100 mL of CH.sub.2 Cl.sub.2 and 1. 3 g (10 mmole) of diisopropyl ethylamine was added, followed by addition of 2.0 g (10 mmole) of 1-adamantanecarbonyl chloride. The mixture was stirred at room temperature for 3 hours, washed with saturated NaHCO.sub. 3 solution, saturated brine, dried over Na.sub.2 SO. sub.4, filtered and concentrated in vacuum. The residue was column chromatographed on 100 g of silica gel with CHCl.sub.3 and the product thus obtained was crystallized from isopropanol with the addition of 4N isopropanolic HCl to give 1.1 g of title compound as the monohydrochloride salt, mp. 230. degree.-231° C.
Elemental Analysis for C.sub.24 H.sub.31 N.sub.3 O.sub.2.HCl: Calc'd: C, 67.04; H, 7.50; N, 9.77. Found: C, 66.95; H, 7.28; N, 9.62.
EXAMPLE 11
Tricyclo[3.3.1.1.sup.3,7 ]decane-1-carboxylic acid 2-[4-[bis(4- fluorophenyl)methylene]-1-piperidinyl]ethyl ester
To a stirred solution of 2-bromoethanol (4.9 g., 0.04 mol) in 50 mL of methylene chloride, adamantane-1-carboxylic acid chloride (5.6 g. , 0.02 mol) and triethylamine (4 g., 0.04 mol) were added. The reaction mixture was stirred at room temperature overnight and the methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 12 g. of the intermediate adamantane-1- carboxylic acid bromoethyl ester. The title compound was prepared by stirring 4-[bis(4-fluorophenyl)-methylene]piperidine (2 g., 0. 07 mol), adamantane-1-carboxylic acid bromoethyl ester (2.0 g., 0.007 mol) and triethylamine (1 g., 0.01 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na. sub.2 SO.sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f = 0.75 in 30% ethanol/ethylacetate system) was separated and converted to the hydrochloride salt with ethanolic HCl; mp. 249°-254° C.
Elemental Analysis for C.sub.31 H.sub.35 F.sub.2 NO.sub.2.HCl: Calc'd: C, 70.50; H, 6.87; N, 2.65. Found: C, 70.56; H, 7.07; N, 2.80.
EXAMPLE 12
Tricyclo[3.3.1.1.sup.3,7 ]decane-3-methyl-1-acetic acid 2-[4- [bis(4- fluorophenyl)methylene]-1-piperidinyl]ethyl ester
To a stirred solution of 2-bromoethanol (4.9 g, 0.04 mol) in 50 mL of methylene chloride, 3-methyladamantane-1-acetic acid chloride (6.1 g., 0. 02 mol) and triethylamine (4 g., 0.04 mol) were added. The reaction mixture was stirred at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 12 g. of the intermediate adamantane-1- acetic acid bromoethyl ester. The title compound was prepared by stirring 4-[bis(4- fluorophenyl)methylene]piperidine (2 g., 0. 07 mol), adamantane- 1-acetic acid bromoethyl ester (2.2 g., 0.007 mol) and triethylamine (1 g. , 0.01 mol) in 50 mL of CH.sub.2 Cl.sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na. sub.2 SO.sub.4) and removed under reduced pressure. The residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f =0.7 in 30% ethanol/ethylacetate system) was separated and converted to the hydrochloride salt with ethanolic HCL, mp. 208°- 209° C.
Elemental Analysis for C.sub.33 H.sub.39 F.sub.2 NO.sub.2.HCl: Calc'd: C, 71.26; H, 6.83; N, 2.52. Found: C, 71.62; H, 7.12; N, 2.34.
EXAMPLE 13
Tricyclo[3.3.1.1.sup.3,7 ]decane-3-methyl-1-acetic acid 2-[4-(4- fluorophenyl)hydroxymethyl]-1-piperidinyl]ethyl ester
To a stirred solution of 2-bromoethanol (3.3 g, 0.026 mol) in 50 mL of methylene chloride, 3-methyladamantane-1-acetic acid chloride (6. 1 g, 0. 02 mol) and triethylamine (5 g., 0.05 mol) were added. The reaction mixture was stirred at room temperature overnight. The methylene chloride was washed with water, dried over anhydrous Na.sub.2 SO.sub.4 and removed under reduced pressure to afford 7 g. of the intermediate 3- methyladamantane-1-acetic acid bromoethyl ester. The title compound was prepared by stirring 4-[4-fluorophenyl]hydroxymethyl piperidine (4.0 g., 0.016 mol), 3-methyladamantane-1-acetic acid bromoethyl ester (4.6 g., 0. 016 mol) and triethylamine (2.5 g., 0.025 mol) in 50 mL of CH.sub.2 Cl. sub.2 for 24 hours. The methylene chloride was washed with water, dried (anhydrous Na.sub.2 SO.sub.4) and removed under reduced pressure. The remaining residue was subjected to preparative HPLC over silica gel using ethylacetate as the eluent and the desired product (TLC R.sub.f =0. 75 in 30% methanol/ethylacetate system) was separated and converted to the hydrochloride salt with ethanolic HCl, mp. 179°-184° C.
Elemental Analysis for C.sub.25 H.sub.34 FNO.sub.3.HCl.5/2 H. sub. 2 O: Calc'd: C, 60.41; H, 8.11; N, 2.82. Found: C, 60.35; H, 7.14; N, 2. 89.
The compounds of this invention are antidepressant, anxiolytic agents useful in the treatment of depression and/or anxiety as singular, primary mental problem as well as secondary, attending problems such as sexual dysfunction, senile dementia, and the like. Some of the compounds possess sufficient dopaminergic activity to be useful in treating psychoses such as schizophrenia or paranoia. Examples of compounds with sufficient limbic D.sub.2 (dopamine) receptor affinity to be considered to have an antipsychotic parameter are those demonstrating about 80% or more inhibition of .sup.3 H-spiroperidol binding to limbic brain tissue at 1 . mu.M concentration of the test compound. The D.sub.2 receptor affinity of representative compounds of this invention was determined by a modification of the test procedure of Fields et al., Brain Res. 136, 578 (1977) and Yamamura et al., eds., Neurotransmitter Receptor Binding, Raven Press, N.Y. (1978) as discussed in U.S. Pat. No. 4,636,563. The percentage reduction of .sup.3 H-spirohaloperidol binding at 1 &mgr;M concentration of test compound is reported, infra, and the inhibition constant (Ki) for the specific test compound is reported where available. Buspirone exhibits an Ki of 78 nM (84% inhibition) of .sup.3 H- spirohaloperidol binding in this standard test procedure.
The serotoninergic properties of the compounds of this invention were established by the procedure of Hall et al., J. Neurochem. 44, 1685- 1696 (1985) by demonstrating that representative compounds exemplified herein displace .sup.3 H-8-OH DPAT (dipropylaminotetralin) from the 5-HT. sub.1A serotonin receptor subtype. The results of this standard pharmacological procedure are reported, infra, as the percent inhibition at 1 nM concentration of test compound or by providing the inhibition constant Ki for the specific test compound where that calculation has been made from appropriate IC.sub.50 values. Buspirone exhibits an Ki of 10 nM (97% inhibition) in this test procedure.
5-HT.sub.2 inhibition of .sup.3 H-spiroperidol is determined in an analogous manner, employing rat brain cortex homogenate as the receptor tissue, following a modification of Fields et al., ibid; Yamamura et al., ibid; and Creese et al., Eu. J. Pharmacol. 49, 20 (1978).
______________________________________
Receptor Binding
Ki(nM)
or % Inhibition
at 1 &mgr;M
Compound 5HT.sub.1A 5HT.sub.2 D.sub.2
______________________________________
Example 1 80% 67%
Example 2 8.5 nM 26%
Example 3 100% 90%
Example 4 72% 39%
Example 5 100%
Example 7 69% 33%
Example 8 53%
Example 10 90%
Example 11 95% 98% 100%
Example 12 63% 36%
Buspirone 10 nM 78 nM
(97%) (84%)
______________________________________
In qualitatively evaluating the above data, high affinity values for 5- HT.sub.1A receptors correlate (by analogy with buspirone) with anxiolytic- antidepressant activity, while lower values reflect a lesser activity. High affinity values for D.sub.2 receptor binding (greater than 80%) begin to show some antipsychotic activity and high affinity for 5-HT.sub. 2 receptor sites indicate anxiolytic and/or antidepressant activity.
Hence, the compounds of this invention are antidepressant, anxiolytic agents useful in the treatment of depression and in alleviating anxiety and in the case of the products of Examples 3, 10 and 12 they have some meaningful antipsychotic activity which is useful in the treatment of psychoses such as paranoia and schizophrenia. As such, they may be administered to a patient in need thereof, either neat or with a conventional pharmaceutical carrier. The pharmaceutical carrier may be solid or liquid as suitable for oral or parenteral administration.
A solid carrier can include one or more substances which may also act as flavoring agents, lubricants, solubilisers, suspending agents, fillers, glidants, compression aids, binders or tablet- disintegrating agents; it can also be an encapsulating material. In powders the carrier is a finely divided solid which is in admixture with the finely divided active ingredient. In tablets the active ingredient is mixed with a carrier having the necessary compression properties in suitable proportions and compacted in the shape and size desired. Suitable solid carriers include, for example, calcium phosphate, magnesium stearate, talc, sugars, lactose, dextrin, starch, gelatin, cellulose, methyl cellulose, sodium carboxymethyl cellulose, polyvinylpyrrolidine, low melting waxes and ion exchange resins.
Liquid carriers are used in preparing solutions, suspensions, emulsions, syrups, elixirs and pressurized compositions. The active ingredient can be dissolved or suspended in a pharmaceutically acceptable liquid carrier such as water, an organic solvent, a mixture of both or pharmaceutically acceptable oils or fats. The liquid carrier can contain other suitable pharmaceutical additives such as solubilisers, emulsifiers, buffers, preservatives, sweeteners, flavoring agents, suspending agents, thickening agents, colors, viscosity regulators, stabilisers of osmo- regulators. Suitable examples of liquid carriers for oral and parenteral administration include water (particularly containing additives as above, e.g. cellulose derivatives, preferably sodium carboxymethyl cellulose solution), alcohols (including monohydric alcohols and polyhydric alcohols, e.g., glycols) and their derivatives, and oils (e.g. fractionated coconut oil and arachis oil). For parenteral administration the carrier can also be an oil ester such as ethyl oleate and isopropyl myristate. Sterile liquid carriers are used in sterile liquid form compositions for parenteral administration. The liquid carrier for pressurized compositions can be halogenated hydrocarbon or other pharmaceutically acceptable propellent.
Liquid pharmaceutical compositions which are sterile solutions or suspensions can be utilized by, for example, intramuscular, intraperitoneal or subcutaneous injection. Sterile solutions can also be administered intravenously. When the compound is orally active it can be administered orally either in liquid or solid composition form.
Preferably the pharmaceutical composition is in unit dosage form, e.g. as tablets or capsules. In such form, the composition is subdivided in unit dose containing appropriate quantities of the active ingredient; the unit dosage forms can be packaged compositions, for example packeted powders, vials, ampoules, prefilled syringes or sachets containing liquids. The unit dosage form can be, for example, a capsule or table itself, or it can be the appropriate number of any such compositions in package form.
The dosage to be used in the treatment of anxiety, depression or psychoses must be subjectively determined by the attending physician. The variables involved include the specific state of depression, anxiety or psychoses and the size, age and response pattern of the patient. | |
Receive a single block of file data from an open SCP download channel.
SCP download channel must have been initialized already by calling SCP Init Receive.vi.
Note
This is a low-level VI whose usage is not typical. For quick 1-step downloads, use SCP EZ Receive.vi.
It is not necessary to call this VI when using the “EZ” VIs.
Inputs¶
LabSSH Session¶
|Required:||Yes|
SSH session refnum.
Bytes to Read¶
|Required:||No|
|Default:||32768|
Read up to this amount of bytes from SCP download channel. If requested number of bytes are not available, this functions reads what is available and returns immediately without blocking. See Byte Count for the actual amount of bytes read.
error in¶
|Required:||No|
|Default:||No Error|
Incoming error from previous VI call. Is passed through to error out.
Warning
LabVIEW will not execute Call Library Function Node if there is an error present on the wire. Ensure that there is no error prior to calling this VI – unexpected behavior will occur.
Timeout msec¶
|Required:||No|
|Default:||5000|
Amount of time in milliseconds to wait for a response from remote host before a timeout error occurs.
Outputs¶
LabSSH Session Out¶
Duplicate SSH session refnum.
Data¶
1-D array of byte data, which represents the file being downloaded. It is up to the programmer to process this data (e.g. writing to local file).
Byte Count¶
Total number of bytes transferred.
error out¶
Error (if any) resulting from this VI call. | https://doc.labwerx.net/labssh/v2.6.1/api-reference/scp/scp-receive.html |
We have many pet lovers at West Bend and this month we’ve covered several tips on how to care for cats and dogs. But what if your child brought home a bearded dragon from school? Would you know how to care for it?
That’s exactly what happened to an associate here at West Bend. Here’s her story, along with a West Bend Pet Show and Tell and a wrap-up of May’s Pet Month.
Rachel
In my daughter’s third grade classroom, the class pet is not your typical guinea pig or hamster. It’s a bearded dragon named Fireball. The kids love being the Pet Vet for a week during the school year. This includes feeding and playing with Fireball.
The teacher, however, was having trouble finding a family to take Fireball during spring break so we reluctantly volunteered. When the teacher showed up with three different heat lamps, a giant aquarium, and bags of food, vitamins, and cleaning supplies, I got a bit nervous. Knowing nothing about bearded dragons, we followed the instructions to the letter. We got a little squeamish giving him the mealworms, but we managed.
Taking care of Fireball wasn’t without surprises, either. We were scared to death when we noticed a heat lamp had burned out after returning from a weekend out of town! We know Fireball is cold blooded and doesn’t make his own body heat so we were quite worried about him until he slowly began moving again under the new heat lamp bulb.
Imagine our surprise the next morning when we found several oval, white sacks in the aquarium. A text from the teacher assured us he had done this before; she thought it was some sort of sac he occasionally threw up. That night I spent a little time watching Fireball to see if I could see one of these sacks coming up. In a short time, I witnessed something quite amazing. These sacks weren’t a sign of gastric distress after all. Fireball was a girl and she was laying eggs! When my daughter brought Fireball back to school, she revealed the astonishing news to the class and they all agreed she should be renamed...Firebella.
Rachel’s Pet Care Tip:Be sure you know the gender of your pet.
Rachel
My favorite thing about Pogo is he just loves to hang out with us. He thinks he’s a human.
Rachel’s Pet Care Tip:Keep his nails trimmed so he doesn’t scratch the floor!
Kevin
This cat is very social with immediate family. You need to pay attention to her when you first get home from work, or after you’ve have gone awhile, otherwise she just will not leave you alone! A few treats as a distraction sometimes does the trick, but she wants attention just as much.
We always have two areas of fresh water in the house for both Mia and our dog, Belle, to share. It’s not uncommon to see both of them at or near the same bowl of water; Mia has no fear of Belle. In fact, Mia is sort of the boss between the two of them.
In warmer weather, we let Mia and Belle out on our deck with us (not alone). Mia is exclusively a house cat, however, in nice weather she does like to be outside. She usually sits right next to Belle. Mia doesn’t stray toward the edge of the deck, nor has she left the deck to chase anything... at least not yet! We’ve learned Mia doesn’t need to be on a leash or tether.
Kevin’s Pet Care Tip:Before letting your cat outside without a leash, understand what your cat’s outdoor instincts are.
Elizabeth
My favorite thing about Mia is that she’s always so happy to see people, whether it’s a stranger or a family member. She smiles and wiggles her butt to the side when she greets you.
Elizabeth’s Pet Care Tip:
If you can, take your dog outside to play for a couple hours every day. Many dogs have lots of energy and love to run. Mia loves to chase birds, butterflies etc. If she doesn’t get outside to release some of her energy, she bugs us all day and night to play ball or with one of her other toys.
Mandy
Mesa is an all-black German shepherd. She can look intimidating, but she’s the nicest dog. However, if you happen to be a rabbit or a cat, you better run fast!
Now that it's finally getting nicer outside, Mesa is shedding like crazy, so we try to brush her more often. I walk her every day and she likes to chase tennis balls.
Mandy’s Pet Care Tip:
Pets that shed a lot should be brushed often so they don’t drive themselves crazy from scratching. And it can make a difference in the hair you find around your house!
Latte
My favorite thing about Cocoa is his personality. He’s a love-bug, sweet, silly, and has a great desire to please (only second to his great desire to eat!).
Latte's Pet Care Tip:
My pet care tip for horses is geared toward spring, when they love to go out and eat lots of fresh green grass. Spring grass is very rich, so start them out on pasture slowly - a few minutes each day and build on that. Each horse will have a different tolerance, so it’s wise to ask your vet for advice.
Scott
Growing up, I had a few pets that included Snoopy the goldfish, Speedy the turtle, and Bogie the ferret. We bought Bogie when I was in grade school. I was so excited because he was the first pet I could actually play with.
One of Bogie’s favorite activities, other than escaping from his cardboard box, was playing fetch. We had a soft stuffed red tomato with a green stem on it. Wherever I threw it, Bogie was sure to retrieve it, waddling back with the tomato dangling from his month by the green stem.
Originally, we started feeding him dry cat food, but we quickly learned that he liked canned cat food much better. When it came to treats, he enjoyed milk, raisins, and peanut butter.
Scott’s Pet Care Tips:
If you have a pet ferret, wash their bedding weekly, carefully trim their nails (they bleed if you cut them short), and don’t bathe them too often.
Wendy
Chloe is my 12-year-old rat terrier/bichon frise mix. Chloe hates cameras so I had to hide my phone by my purse to take this picture at my son’s Lacrosse game. She loves to ride in the car and go anywhere with her family. Two years ago, we were told Chloe had only days to live due to a giant bladder stone that filled her abdomen. We did some research and found a natural herb that’s broken up the stone. Two years later, she’s still with us, although her abdomen sounds like a sack of marbles!
Wendy’s Pet Care Tips:
Good nutrition, a lot of TLC, and some creativity are some of the best tips a pet owner can have!
If you haven’t checked out the previous pet care blog posts, you may want to. They focus on:
- Finding the perfect pet for you and your family;
- Five pet care tips from a veterinarian; and
- Pet fun facts (infographic).
Do you have any pet care tips you’d like to share? I’d love to hear them; please share them in the box below. | https://www.thesilverlining.com/westbendcares/blog/bid/198129/So-your-child-brought-home-a-bearded-dragon |
Cats are peculiar and fascinating creatures. Some of their habits, such as your kitty’s soft purrs, are easy for us to understand as affectionate. Other behavioral characteristics of cats are far more confusing.
If you’ve caught your cat intently staring, you might have felt like they were trying to communicate. Perhaps you’ve been woken up to your cat eyeballing you while sitting on your chest. Or, you may have felt their gaze boring into your back.
Whether you’re worried about their well being, curious about your cat’s behavior, or just plain creeped out. You’re not alone in wondering what it means when a cat stares at you. Do these wide-eyed, soul-searing stares mean something specific?
Let’s review what’s known about the topic.
Contents
- 1 What Does it Mean When a Cat Stares at You? Kitty Communication
- 2 5 Reasons Why Your Cat Is Staring At You
- 3 Is Cat-Staring Aggressive?
- 4 Reading Your Cats Body Language
- 5 Cat-Staring Requires Context
What Does it Mean When a Cat Stares at You? Kitty Communication
Have you ever felt like you’re being manipulated by your cat? Cats are highly intelligent creatures. They’re able to pick up a lot of our behaviors that we might not be conscious of.
In one study, it was discovered that cats use a “solicitation purr” to manipulate us into feeding them. This purr is more high-pitched. Humans respond to this sound in the same way we react to a crying baby.
There are many ways that cats verbally communicate. Verbal signs such as meowing, yowling, purring, and hissing, are all ways for cats to express their feelings to us and other cats.
Remember that cats will adapt to their specific environment, which includes you and your behavior. So there are many unique ways in which your cat may have learned to get your attention. One method may be a soul-piercing stare.
5 Reasons Why Your Cat Is Staring At You
Remember that every feline is unique and full of character. You know your cat best, but here are five general reasons why a cat may be eyeballing you.
1. They’re Hungry
This is one of the most likely reasons that your cat is staring at you. Many cats are food-oriented creatures. So it’s not usual for your kitty to try and coax some food out of you outside of their designated mealtimes.
Other times, your cat might be trying to remind you that it’s time for food. Meowing and solicitation purring (as mentioned above) are some ways that cats ask for food. You also might find them sitting near their food bowl, staring at you longingly.
2. They’re Trying To Get Your Attention
We all know that cats can be absolute divas – especially the “only children.” When their Royal Majesties request your attention, they very well expect it. And promptly at that.
While it may often be related to food, there are other reasons why kitty might want some attention. Oftentimes, my own cat simply wants to chat. And possibly coax some food out of me.
Some cats demand attention with long whining meows, others use cute little kitten purrs. Maybe, a soft touch of their paws to get you to notice them.
Other times, you’ll happen upon them staring you into submission. This may be their way of saying, “Hey – look here! Bask in my glory!”
3. They’re Showing Affection
Despite many misconceptions that cats are cold and heartless, they can be wonderfully affectionate to their owners. As well as very protective. This is communicated by a cat rubbing against you to mark you as theirs.
Even though staring is considered rude in humans, it’s a way for cats to let you know that they love you. If you catch your cat staring at you in between soft blinks, this is a probable sign of your cat just taking the time out of their day to adore you.
4. They’re Agitated or Angry
Cats don’t enjoy being ordered around. And there’s nothing worse than being on your cat’s bad side. If you’re forcing a cat to be bathed, groomed, or participate in any other “unauthorized” activity, chances are they’re not going to forget about it any time soon.
If you see your cat staring at you afterward, they’re probably sulking. In this case, staring at you would be their way of communicating that they’re not happy with you.
5. They’re Scared or in Pain
Oftentimes, sick and even terminally ill cats tend to hide away to protect themselves in their vulnerable state. Cat’s are usually quite private, and enjoy having a space of their own.
⇒ Thinking about getting your favourite feline a new collar? Check out my posts on 6 Stylish Leather Cat Collars, 6 Spooky Halloween Cat Collars, 8 Fun Christmas Cat Collar options, 7 Best Cat Tracking Collars, 4 Best Cameras for Cat Collars, 6 Best Flea Collars for Cats choices and 5 Best Designer Cat Collars.
However, in a few instances, cats may come to you when they’re nervous or experiencing discomfort. If you’re the one in your household that spends the most time with your cat, they may identify you as the human they can trust most and may come to you when they’re afraid.
If your cat is staring at you but doesn’t want food or attention, then there may be a problem they think you can solve. Check their fur, mouth, and in between their paws to see if something is causing them discomfort.
⇒ Getting a new kitty? Check out my guide to How to Look after a Kitten, 6 Best Kitten Wet Food Options and 8 Best Kitten Dry Food Options
Is Cat-Staring Aggressive?
One of the many ways in which cats communicate with each other is body language. And there are many ways in which different body-parts can indicate a range of emotions. Staring alone is not enough on its own to signal aggression.
Sometimes, cats will stare at each other out of curiosity, simply enjoying a new object of attention. Nonetheless, staring between cats can also be a form of territorial aggression.
If you’ve ever seen a cat-fight unfold, you likely noticed intense and unblinking eye contact before an escalation to your cat growling, hissing, and snarling.
No one enjoys a catfight. And many owners discourage cat-staring between cats in an attempt to prevent a brawl. However, direct eye-contact is not the only indicator of territorial and aggressive displays.
Reading Your Cats Body Language
This doesn’t necessarily mean that your cats’ staring at you is aggressive or confrontational. When it comes to our quirky kitties, context is key. Keep your eye on the following physical cues, which are often used in cat communication:
- Position of the head and ears
- Tail movements
- The ridge of hair along their backs
- General stance
- Whiskers and pupils
Cats communicate with their entire bodies. Here are some signs that your cat is content.
Signs of a Relaxed & Happy Cat
Signs that your cat is relaxed and content include making themselves small and unthreatening. Often this includes tucking their paws underneath themselves while lying down. Slightly forward-facing ears, purring, and loving soft-soft blinks are also indicators that your cat is feeling at home.
A relaxed cat has ears that are slightly angled outward, but facing forward. Additionally, your cat’s tail is very expressive. When greeting you, your cat will show its friendliness with an upright tail that’s curled just at the end.
⇒ An entertained cat is a happy cat. Check out my posts on 9 Best Treats for Cats, Ultimate Guide to Gifts for Cats, 6 Best Toys for Cats, 27 Beautiful Cat Ornaments, 14 Best Cat Chew Toys, Clothing for Cats, and Best Costumes for Pet Cats.
Their pupils may dilate as well when they’re entertained or suddenly excited. Look out for this the next time you’re playing with your cat Their eyes may also suddenly dilate when they become happy or excited.
⇒ Find Cat Toys in the US
⇒ Find Cat Toys in the UK
Cat-Staring Requires Context
We’ve all heard that the eyes are the windows to the soul. However, in the case of cats, it’s their whole bodies that are used to convey emotion. As we’ve discussed in this article, there are many varying reasons why cats may be boring their eyes into your soul.
Food is right at the top of the list of possibilities, probably because food is often high on a cat’s list of priorities. However, your cat may be staring at you to convey its need for food, attention, or even assistance.
Cat’s enjoy a personal routine. If staring is combined with a sudden change in their behavior, then it may be wise to take your kitty for a checkup. If there’s no reason you can think of, your cat may be simply showing you some love.
⇒ Keen to get your cat out and about? Check out my posts on Is Cat Walking Possible?, 7 Best Escape Proof Cat Harness Options, 5 Best Carrier for Cats choices, 14 Best Cat Carrier for Car Travel options and 4 Top Travel Litter Box options.
Please Note: This what does it mean when a cat stares at you post contains affiliate links. That means if you click through on most of the links and end up making a purchase I will receive a small commission. This will not affect the price that you pay. I wanted to make sure that you were aware of this. | https://thediscerningcat.com/what-does-it-mean-when-a-cat-stares-at-you/ |
I am an artist specializing in 3d modeling and texturing techniques with a strong knowledge of scripting and technical aspects of game design. I recently graduated from the Art Institute of San Antonio with a Bachelor of Game Art and Design. I’ve always been a big fan of understanding the inner workings of videogames. When I create art, I’m really interested in developing the process and apply those techniques to future projects.
With five team members, I helped develop a game called Apex Dawn that we presented in association with the Art Institute of San Antonio at PAX South 2019. I was responsible for any design aspect behind the game’s development. I collaborated with team members to create a game design document that served as a guideline for the Apex Dawn’s development process. Then I created documents in Excel and Word for procedures and conventions for game assets to be organized and optimized. Facilitated communication between non-technical members to achieve milestones and goals. Then ensured all technical and performance guidelines of game assets were implemented in game. Also designed and world-built levels to create engaging and challenging experience for the player.
With my knowledge such as Maxscript, C#, Unreal Engines Blueprints, and general design skills I created several systems. For Example, I scripted in the Unreal Engine procedural foliage and game assets, behavior trees, dynamic materials, and player mechanics. I created real-time particle systems in UE4 and 3ds max using physics simulation and particle dynamics for fire, smoke, and destructible structures. Furthermore, I designed 2d vector art with strong iconography for game menus, buttons, and panels and implemented in game engine. Also, I used a combination of Biped, CAT, and custom bone systems to create facial and character rigs with animator friendly controls
Also, on various projects I modeled efficient organic and modular hard surface assets from concepts with High poly details. Textured assets with the PBR workflow using both procedural and hand painted techniques. Furthermore, I drew environment and character concept art with a strong sense of composition, perspective, and design.
In a visual effects internship with White Rhino Studios, I developed Maxscripts and techniques to improve workflow efficiency in the modeling pipeline. Using Maxscript I procedurally generated greebled cities with textures. I used Maxscripts to develop an efficient texturing pipeline between Substance Designer and 3ds Max to make the texturing process faster. Also, in Art and Design Internship in association with a virtual reality company called Ractive. I helped developed enemy artificial intelligence assets that work in a multiplayer VR training experience.
I have a strong knowledge of art creation software design skills that I would like to utilize to help your company. I would appreciate an opportunity for an interview or discuss any employment opportunities. I will follow up immediately by email. Thank you for the consideration. | https://deon-wilson.com/game-design-2/ |
Graeme is lucky enough to be able to keep an eye on two woods: one at each end of Moorsholm, the village he lives in. He also said how pleased he is that ‘his’ woods are not urban fringed, and therefore don’t have the large numbers of visitors some woodlands can have – and don’t have the problems that can sometimes bring.
All the interviews with volunteers so far have produced some lovely tales about their local woods, but perhaps none more poignant than Graeme’s story about his and other volunteers’ creation of a path called Joey’s Way in Hagg Wood in memory of a local boy who died young and loved the wood.
But although there may be less littering, there’s still attention to woodland management needed, and Graeme told me that along with knowing the small community who use the woods, and like all our volunteer wardens, being the local liaison for the area Site Manager, he attends himself to smaller tasks such as moving branches fallen across paths, or repairing step fronts in Hagg Wood, which is very steep sided.
Graeme named his favourite tree as “the archetypal oak… one of three in a field next to Cow Close wood… and it’s just beautiful… a traditional English tree.” He added that in Cow Close itself there was a fallen larch which had continued to grow upward at a 45 degree angle.
“It’s a real character tree,” Graeme added, and we had a little chuckle together about how we loved to just touch trees as we passed them by. A way of saying hello to these woodland ‘characters’.
In Cow Close by Graeme Aldous
Graeme was more general than specific about his other nature favourites, as he said his main joy was in celebrating the diversity of nature. He returned to discussing woodlands – any woodland – and said he often asks himself the question ‘how many growing things are here in this one small wood?’
He did get specific for a moment when he spoke about the simple beauty of a blackbird singing at dawn, then his viewpoint widened again:
“I love it that nature is so big and fantastically wonderful,” he added, “and that there are creatures here who don’t give a damn about me.”
He told me a lovely story about his own apple trees on his smallholding and how he leaves the fallen apples for his hens to eat.
“I watch a creature – a hen, a blackbird, an insect… any creature – eating a fallen apple, and I think, I’m glad I left that apple.”
Well, nature’s creatures may not care too much about Graeme, but how wonderful it is that he cares about them. | https://whittle.woodlandtrust.org.uk/2021/10/21/in-the-spotlight-an-interview-with-volunteer-graeme-aldous/ |
PRIORITY
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
This application claims priority under 35 U.S.C. §119 to an application filed in the Korean Intellectual Property Office on May 30, 2006 and allocated Serial No. 2006-48530, the contents of which are incorporated herein by reference.
1. Field of the Invention
The present invention relates generally to an apparatus and method for storing security keys in a mobile communication terminal, and in particular, to an apparatus and method for encrypting and storing security keys in a mobile communication terminal so as to prevent unauthorized use of the security keys.
2. Description of the Related Art
In general, a Universal Mobile Telecommunications System/Global System for Mobile communication (UMTS/GSM) system provides a personalization function for allowing a specific terminal to use only a specific Subscriber Identity Module (SIM) card. Examples of the personalization are network personalization, network subset personalization, service provider personalization, corporate personalization, and SIM/USIM (Universal SIM) personalization that are defined in the 3rd Generation Partnership Project (3GPP).
The personalization function allows a specific terminal to use only a specific SIM card, by using an International Mobile Station Identity (IMSI), a General IDentifier 1 (GID 1) or a GID 2 that is stored in the SIM card.
When a personalization function is enabled in a mobile communication terminal, the use of a SIM card is restricted. A prestored control key is used to disable the personalization function of the mobile communication terminal, i.e., to remove the restrictions on the use of the SIM card. The control key is randomly generated using the seed of a process program, and the generated control key is stored in the memory of the mobile communication terminal in the form of a decimal number. The seed of the process program is used to randomly generate not only the control key but also security keys that are used for receiving services such as e-mail and mobile banking.
Recently, hackers have been successful in their attempt to compromise the security of the Internet, thus leading to unauthorized use of the security keys (e.g., control keys) of mobile communication terminals and service providers. Security programs such as the 128-bit Advanced Encryption Standard (AES) algorithm, has been developed to prevent the unauthorized use of the security keys.
A conventional mobile communication terminal generates a security key using the conventional security program and stores the generated security key in its memory. However, the conventional security program merely enhances the security effects on the generation of the security key. Therefore, it is still possible to easily obtain the security key by illegally accessing and dumping the mobile terminal memory. This causes the unauthorized use of the security key and the mobile communication terminal.
An object of the present invention is to substantially solve at least the above problems and/or disadvantages and to provide at least the advantages below. Accordingly, an object of the present invention is to provide an apparatus and method for preventing unauthorized use of a security key in a mobile communication terminal.
Another object of the present invention is to provide an apparatus and method for encrypting a security key and storing the encrypted security key in a mobile communication terminal.
A further object of the present invention is to provide an apparatus and method for encrypting a security key by orthogonal Walsh code spreading and storing the encrypted security key in a mobile communication terminal.
According to one aspect of the present invention, there is provided a method for storing a security key in a mobile communication terminal, the method includes generating the security key randomly; encrypting the generated security key in a predetermined encryption scheme; and storing the encrypted security key.
According to another aspect of the present invention, there is provided an apparatus for storing a security key in a mobile communication terminal, the apparatus includes a controller for generating the security key randomly and encrypting the generated security key; and a memory unit for storing the encrypted security key.
According to a further aspect of the present invention, there is provided a method for storing a control key for disabling personalization in a mobile communication terminal with a SIM card, the method includes randomly generating the control key; encrypting the generated control key; and storing the encrypted control key.
According to still another aspect of the present invention, there is provided a method for disabling personalization in a mobile communication terminal with a SIM card, the method includes detecting an encrypted control key that is stored and a character string for Walsh covering; decrypting the encrypted control key using a Walsh code index contained in the character string; comparing the decrypted control key with the control key contained in the character string; and disabling the personalization if the decrypted control key is identical to the control key contained in the character string.
Preferred embodiments of the present invention will be described herein below with reference to the accompanying drawings. In the following description, well-known functions or constructions are not described in detail since they would obscure the invention in unnecessary detail.
The present invention provides an apparatus and method for encrypting and storing security keys in a mobile communication terminal, with the aim to prevent unauthorized use of the security keys by memory dump. Security keys refer to keys that are randomly generated by a process program for the security of the mobile communication terminal. In the following description, a control key for disabling personalization of a SIM card is taken as an example of the security key.
Examples of a mobile communication terminal are a cellular phone, a personal communication system (PCS) terminal, a personal data assistant (PDA), an international mobile telecommunications-2000 (IMT-2000) terminal, and a fourth generation (4G) broadband communication terminal. The following description is made in terms of a general structure of the above communication terminals.
FIG. 1
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Referring to , the mobile communication terminal includes a controller , a SIM card , a memory unit , a display unit , a keypad , a communication module , an audio processor , a speaker, a microphone, and an antenna. The controller includes a control key encryptor .
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Controller controls the overall operation of the mobile communication terminal. For example, controller processes and controls voice communication and data communication. In addition, using control key encryptor , controller randomly generates a control key, encrypts the random control key, and stores the encrypted control key in memory unit .
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SIM card is an attachment to the mobile communication terminal. SIM card includes a microprocessor and a memory chip and stores a variety of user data.
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Memory unit stores the program for controlling the overall operation of the mobile communication terminal, temporary data generated during the operation of the mobile communication terminal, system parameters, and a variety of updatable data such as phone book and short message service (SMS) message. In addition, memory unit stores the encrypted control key. At this point, the control key is stored in the non-volatile area of memory unit .
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Display unit displays status information generated during an operation of the mobile terminal, characters inputted by a user, moving pictures, still pictures, and so on.
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Keypad includes numeric keys of digits 0-9 and a plurality of function keys, such as, a Talk key, Navigation keys (▴/▾//), and a Confirmation key. Keypad provides controller with key input data that corresponds to a key pressed by the user.
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Communication module processes radio-frequency (RF) signals that are transmitted/received through the antenna. Audio processor connected to controller , and the speaker and the microphone connected to audio processor constitute an audio input/output block that is used for voice communication.
FIG. 2
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Referring to , control key encryptor includes a control key generator , a bit converter , and a spreader .
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Using a process program, control key generator generates the control key randomly. The generated control key is an 8-digit number expressed as Equation (1):
Control key=C, C, C, C, C, C, C, C (1)
where Cdenotes the icode of the control key and is a decimal number.
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i,j
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Bit converter converts the generated decimal control key from control key generator into a bit-format expressed as Equation (2) by a bitwise operation:
C: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, bC: b, b, b, b, b, b, b, b (2)
where bdenotes the jbit of the icode of the control key.
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Using an orthogonal Walsh code expressed as Expression (3) below, spreader spreads the bits of the control key into random data expressed as Equation (3) and Equation (4).
Walsh code=[W, W, W, W, W, W, W, W] (3)
where Wdenotes the iWalsh code that is represented in the form of a row vector, and said code has a length of 8.
S: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sS: s, s, s, s, s, s, s, sE: e, e, e, e, e, e, e, e (4)
where Sdenotes diffused random data of the icode of the control key, sdenotes a Walsh-spread random value of the jbit of the icode of the control key, and E denotes a row vector of random data generated by Walsh-spreading the control key.
i,j
i
The scan be calculated using Equation Expression (5) below and the ecan be calculated using Equation (6) below.
<math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>s</mi><mn>00</mn></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>00</mn></msub><mo>·</mo><msub><mi>W</mi><mn>0</mn></msub></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>s</mi><mn>01</mn></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>01</mn></msub><mo>·</mo><msub><mi>W</mi><mn>0</mn></msub></mrow></mrow></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>s</mi><mn>70</mn></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>70</mn></msub><mo>·</mo><msub><mi>W</mi><mn>7</mn></msub></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>s</mi><mn>71</mn></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>71</mn></msub><mo>·</mo><msub><mi>W</mi><mn>7</mn></msub></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math>
i,j
i,j
i
i
0i
1i
7i
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th
th
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e
=s
+s
+ . . . +s
i,j
th
th
where sdenotes a Walsh-spread random value of the jbit of the icode of the control key.
where sdenotes a Walsh-spread random value of the jbit of the icode of the control key, bdenotes the jbit of the icode of the control key, and Wdenotes the iWalsh code
(6)
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Spreader provides memory unit with the random data E generated by Equation (4).
FIG. 3
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Referring to , controller executes a program to generate a control key that is used to disable personalization of a SIM card, in step .
303
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In step , controller generates a random control key. The random control key is a decimal number expressed as Equation (1).
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In step , controller converts the decimal random key into a bit-format expressed as Equation (2).
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In step , controller spreads the bit-format control key using the Walsh code. The orthogonal Walsh code expressed as Equation (3) is used to spread the bit-format control key into random data expressed as Equation (4).
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In step , controller stores the spread random data in memory unit . Thereafter, controller ends the procedure.
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As described above, controller generates the random control key, the orthogonal Walsh code is used to spread the random-control key into the random data, and the random data is stored in memory unit . Therefore, even if memory unit is illegally accessed and dumped to read the stored control key, an accurate control key cannot be detected without the use of accurate Walsh covering.
FIG. 4
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Referring to , controller detects if a SIM card personalization disable mode is selected.
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If the SIM card personalization disable mode is selected, controller detects if an encrypted control key is stored in memory unit , in step .
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In step , controller decrypts the encrypted control key using the Walsh code index expressed as Equation (7):
01234567=C, C, C, C, C, C, C, C (7)
where “01234567” denotes a Walsh code index that is used to perform a Walsh covering operation for decrypting the codes of the control key, and “C” denotes a control key that is to be compared with a control key (C′) decrypted using the Walsh code index.
0
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0i
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b
i′=E·W
th
For example, in order to decrypt the first code Cof the control key, the random data E stored in memory unit is multiplied by the first Walsh code index ‘0’ as expressed by Equation (8) below.
0 (8)
where bdenotes the ibit of the first code of the control key, E denotes the random data (i.e., the encrypted control key) stored in memory unit , and Wdenotes the first Walsh code index. Because the Walsh codes are orthogonal to one another, they have a value of 0 if Walsh codes other than those for Walsh code spreading are used. Therefore, it is possible to decrypt only a value for the corresponding Walsh code index.
0
As shown in Equation (8), the eight bits of the first code of the control key are recovered using the Walsh covering operation. When the recovered eight bits are converted into a decimal number, the first code C′ of the control key is recovered.
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In the above way, all the encrypted control keys are decrypted using the Walsh code index. In step , the decrypted control key (C′, C′, C′, C′, C′, C′, C′, C′) is compared with the control key (C, C, C, C, C, C, C, C). If the decrypted control key is identical to the control key, controller depersonalizing (unlocking) the personalization of the SIM card and then ends the procedure. On the other hand, if the decrypted control key is different from the control key, the controller ends the procedure maintaining the personalization of the SIM card.
As described above, the security keys generated by the process program is encrypted and stored in the mobile communication terminal. Therefore, even if the stored security keys are read by hacking or memory dump, it is impossible to detect the accurate security keys. Accordingly, it is possible to prevent the unauthorized use of the mobile communication terminal.
While the invention has been shown and described with reference to certain preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as further defined by the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings in which:
FIG. 1
is a block diagram of a mobile communication terminal according to the present invention;
FIG. 2
is a block diagram of a control key encryptor according to the present invention;
FIG. 3
is a flowchart illustrating a procedure for encrypting a control key according to the present invention; and
FIG. 4
is a flowchart illustrating a procedure for disabling personalization of a SIM card according to the present invention. | |
Elmhurst Energy Consultancy has over 25 years’ experience providing specialist advice and services for the built environment and energy management sector.
But we are more than a consultancy service. We’re solution providers. Using our unique access to the latest cutting edge technology, industry methodologies and network of energy assessors we are able to deliver bespoke solutions that cater for your project.
We also provide building regulations/ standards compliance for new buildings, covering Part L of Building Regulations in England and Wales (Part F in Northern Ireland and Section 6 Building Standards in Scotland). Our team are able to provide energy calculations for any new-build project: from simple household extensions to large-scale housing developments.
To find out more about our services, please get in contact using the details below.
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To find out more about our services, contact us on: | https://www.elmhurstenergyconsultancy.co.uk/about-us |
We have allocated 20 minutes for oral presentation of contributed papers, which includes questions and transitions between speakers. Authors should prepare slides that convey their message clearly and concisely in 15 minutes to allow time for questions and discussion. It is the responsibility of the session chair to determine how many questions to solicit from the audience to stay within the time limits.
In order to ensure a smooth session, we kindly ask you to follow the following instructions:
The speaker ready room is in the Salon Cluny, level 5, and will have a clone of the PC’s that are in the session rooms. Presenters are encouraged to finalize and check compatibility of their presentations with the venue’s PC before their sessions. | https://2019.ieee-isit.org/PresentationInstructions.asp |
Being able to translate sentences from one language to another is a skill students need to have developed when they sit their GCSEs and A-levels. This can often be hard, as learners need to understand the meaning of tricky expressions such as collocations and idioms. Here's a collection of new and popular resources to help your pupils practice translation in French, Spanish and German.
French
Check out the most popular resources to work on translation in French. You can also find a large collection of GCSE revision resources in our French GCSE collection.
French (Adv) - translation practice with answers
How to tackle French A Levels translations
A2 Year 13 French Translation Practise
This resource provides a worksheet of thirty questions to practise essential grammar items in preparation for the summer exams. It focuses on translation skills which students who will have to translate in their exams will be likely to benefit from. Many different tenses and prepositions are tested in this task so it would be prudent to inform students that the sentences will usually include a specific grammar point (tense, prepositions etc) which are different in French to what they are in English. An answer guide has also been prepared for tutors/ students as appropriate.
French A level Translation Workbook
A workbook (with an answerbook included as a separate document) for A level students doing French and needing translation practice. There are exercises on each theme from English to French and from French to English.
The themes include:
family structures, modern values, friendships, relationships
youth trends, personality and identity
educational and employment opportunities
regional culture and identity in France and French speaking countries
media, art, film and music in the French speaking world
migration and integration
cultural identity and marginalisation
cultural enrichment and celebrating difference
June1940-May 1945
the cultural dimensions in occupied France
1945-1950
Suitable for any exam board
Short translation tasks with mark scheme
Worksheet includes a short version of the AQA mark scheme.
I go through the correct translation on the board then get pupils to make corrections and peer assess before I mark their work.
Using these every 2-3 weeks as a 10-15 minute activity, to assess pupils regularly and keep exam skills ticking over, rather than doing a massive exam at the end of each topic.
Translation Booklet New GCSE Spec. French into English Higher Level
> Each page has two passages to translate.
> Each page is dedicated to one of the new GCSE sub-topics.
> The back of the booklet has suggested answers.
> Extension work can easily be set, asking students to highlight different tenses in different colours, find negatives, identify adjectives etc
> The pages can be printed off one at a time at the end of each topic or saddle stapled to make a nice booklet.
German
Browse through the most sought-after German translation activities. Have a look at our German GCSE collection for more revision resources.
German translation at A2
German GCSE 9 - 1 Specifications Translation Exercises.
This document is in the format of worksheets/ a booklet that can be given to all GCSE students. It is broken down into themes and then into the topics within each theme. For each topic there are 10 translation exercises, as on the Reading Exam. It is contains in total 120 translations of varying lengths and levels of challenge. The translations have been created carefully and are in line with exam board exemplar materials. This could be used by pupils at the end of teaching each topic/ sub-theme in order to test/ check understanding and to practise translation skills or used for wider translation purposes.
A-Level Exam Practice Translation
GCSE German - Authentic Reading Materials / Translation Exercises
There are works by Bertolt Brecht, Max Frisch, Siegfried Lenz, Friedrich Dürrenmatt, Heinrich Böll, Patrick Suskind and Wolfgang Borchert; as well as blogs, adverts from Deutsche Bahn and Telekom; tannoy announcements in train stations; holiday brochures; blogs; news stories; website screenshots; magazine articles; and some past exam paper material from pre-2009 in order that you may still use 2010-2016 papers as past paper materials in lessons.
Towards the end, there is a section of materials from GCSE papers which are entirely foundation or foundation/higher level.
Please note: No copyright infringement intended. If you own any of the material, or would like it to be taken out of the booklet, please get in touch!
Where possible, I have inserted links to the page where the source was found.
German perfect tense translation game
Spanish
Choose from these frequently-downloaded workbooks and improve learners' translation skills in Spanish. Our Spanish GCSE collection also has a range of revision lesson ideas.
GCSE Spanish translation booklet
I’ve made this Spanish translation booklet (also one on here that I made for French) to comprise all areas of the AQA Spanish GCSE spec.
I’ve included sentence level translation and paragraphs for each topic and in both languages.
Hope that they can be of use - any feedback greatly appreciated!
thanks
Spanish GCSE writing booster: complex language revision booklet & translation practice
A four page revision booklet of key complex phrases & structures that pupils should include in their speaking & writing to achieve Grade 9.
This is a definitive list of complex and higher structures that I encourage my pupils to use in their work. The phrases are split into 22 sub-groups across four A4 pages, each with some translation practice too, so as pupils can see how easy it is to manipulate these themselves in a variety of topics. Suggested answers are included and the document can be downloaded in .doc and .pdf.
**The structures are grouped in 22 mini-sections: **
Shiny phrases + infinitives
Opinion phrases
Structures of obligation
Conditional sentences
Imperfect constructions
Before/ after
Past tense phrases
Key past tense verbs
Pasar + gerund
Future tense constructions
Conditional constructions
Past conditional constructions
Subjunctive constructions
Acabo de + infinitive
Soñar con + infintive
Desde hace + present tense
Lo (adjective) es que constructions
Introducing someone else’s opinion
Interesting adjectives
Conjunctions
Adverbs
Opinions
A similar resource is available here in French too: https://www.tes.com/teaching-resource/french-gcse-complex-and-idiomatic-phrases-full-revision-list-with-translation-practice-12173311
Spanish A level Translation Workbook
A workbook (with an answerbook included as a separate document) for A level students doing Spanish and needing translation practice. There are exercises on each theme from English to Spanish and from Spanish to English.
The themes include:
- family structures, modern values, friendships, relationships
- youth trends, personality and identity
- educational and employment opportunities
- regional culture and identity in Spain and Spanish speaking countries
- media, art, film and music in the Spanish speaking world
- migration and integration
- cultural identity and marginalisation
- cultural enrichment and celebrating difference
- el franquismo
- post civil-war Spain, historical and political repercussions
- coming to terms with the past
Suitable for any exam board
A2 Spanish Translation phrases
AQA A2 TRANSLATION (ENGLISH TO SPANISH)
New Spanish GCSE - Theme 1 (Identity and culture): Translations - UPDATED
A very comprehensive set of sentences (suitable for all abilities) and paragraphs (mainly higher) to translate from Spanish into English. The resource includes 95 sentences and six long paragraphs and they cover the following aspects of Theme 1 (Identity and Culture):
1.a. Me, my family and friends (35 sentences)
1.b. Technology in everyday life (20 sentences)
1.c. Free time activities (20 sentences)
1d. Customs and Festivals in Spain (10 sentences)
Finally, the six longer paragraphs include language related to all four categories.
The sentences include references to a variety of tenses (present, preterite, imperfect, present perfect, conditional, future), as well as a wide range of complex structures (negatives, comparatives, conditional clauses…).
More translation resources
Triumphant Translations Worksheet - Module 1 - Holidays
A translation challenge worksheet to support revision.
This includes 5 sets of translations varying from full paragraphs in the style of higher tier to short sentences like the foundation exam.
As well as the translations themselves, there is also additional support in the form of individual word and verb translations.
AQA A Level French year 2 Translation booklet
In this booklet you will find practice for all the topics of year 2.
The answer booklet is also provided.
German GCSE translation practice (translation pitfalls)
These are two revision lessons for Y11 to practise the main difficulties with translations: difference between mag/gern, using connectives and the correct word order, comparative, impersonable phrases, infinitive phrases, using seit. The lesson assumes that the students have learned these grammar topics previously, and this is just revision.
The lessons are designed as carousel activities, where students translate sentences in pairs.
I did it with my Y11 set 2 this year and they enjoyed it and found it helpful.
Translation skills builder GCSE topics
This booklet covers the different GCSE topics for French (AQA). There are activities to practise accurate translation, and they are designed to encourage students not to just Googletranslate! There are activities of different difficulties. | https://www.tes.com/teaching-resources/blog/translation-101-prepare-gcse-and-a-levels-mfl |
Located 15 minutes drive from the train station Coulommiers serving regularly Paris Gare de l’Est, come and discover this great building of 8 rooms, close to all amenities. Offering calm and brightness, in good condition, it has on the ground floor a beautiful living room of 43sqm, a kitchen, a bedroom, a shower and a separate sanitary. On the floor, are a stay cathedral and living room, the whole of a surface of 72sqm. This living room with its stone walls has a high sloping ceiling height of 6 meters while organizing a low ceiling living room to enjoy the radiation of the fireplace. A hallway serves the kitchen, the bathroom with bath and shower, the three bedrooms, one of which has a large dressing room. If the ground floor requires refreshing, taking advantage of the opportunity to extend it for example for a mixed use because 166sqm currently used as a garage could accommodate a professional activity, the floor requires no work. In total you have 250sqm of living space with an extension to 350sqm by landscaping 100sqm in the barn located in the extension of the living room with an oak frame and a magnificent glass roof overlooking the courtyard access to the property and its beautiful garden facing south as for the rooms. The town is quiet, visited in summer for its beautiful lake, facilitates the lives of its inhabitants thanks to the school bus (primary, secondary school, high school); medical office, pharmacies, supermarkets are 7 minutes by car. This property can address a project of primary or secondary residence, or even mixed for entrepreneurs. Rich in its past, this building has lived the history of flax through its winding unit of flax fiber produced in the former factory of the town. | https://www.espaces-atypiques.com/seine-et-marne/en/annonce/big-house-in-the-countryside/ |
1 What’s it all about?
Known variously as urban sprawl, city growth, urban expansion or urban development, urbanisation is a prominent and central feature of the sustainable development paradigm.
The significance of a predominantly urban population has created various interpretations. To the hard-core environmentalist, cities represent an anathema; their inhabitants taking, destroying and polluting; very essence of cities hedonistic and anonymous. And they are expanding, practically everywhere. At some point in 2008 the urban population exceeded the rural for the first time. Internationally, much of the population growth is predicted in urbanised areas of Asia and Africa. However, development agencies and academics consider the potential of urban concentrations to play a large and increasing part in solving global social, economic and environmental problems.
Starting with Rio ’92, we will investigate the UN Conference process towards a sustainable future in order to determine any progress made in our understanding of urbanisation issues within the UN Conferences on Sustainable Development arena.
Despite the rhetoric of governments on the international stage, it seems the complex, multi-faceted problems associated with urbanisation are getting worse as the global south population continues to grow and simultaneously emulate the lifestyle and consumption choices of the global north.
2 Key issues
A review of recent literature reveals the nomenclature of urban issues to be varied. There are the physical classifications, such as Cities with defined boundaries, Urbanised Areas with sometimes more fluid boundaries; and phenomena such as Urbanisation and Urban Sprawl, where changes in land use patterns result in settlements with increasing footprints becoming more complex.
Whilst western and northern countries have urbanised in the early-mid 1900s, southern and eastern countries to urbanise rapidly and sometimes uncontrollably, leading to an array of environmental, economic, social and political problems. Yet today’s practitioners, academics and NGOs challenge the notion of urbanisation as a problem and have begun to identify the benefits in such characteristics as density and critical mass.
Causes
The causes of or reasons behind urbanisation are manifold and complex. The following is a summarised list to indicate the vast range of considerations:
Industrialisation; Population growth from various sources of in-migration and/or refugees including rural, international, economic, environmental, conflict, disease, etc.; land costs, zoning/planning regulations, land ownership, preservation regulations, build costs, infrastructure availability; job loss or crop failure; a ‘critical mass’ including: employment opportunities with centralised commerce locations, business location/proximity, centralised and accessible governments, improved education opportunities, social networks, transport availability with expanding public and private networks, supply and demand of both products and services such as health services; the draw of hedonistic pleasures and anonymity; and design, including physical, urban and policy design that encourages controlled expansion.
‘Problems’, Benefits
Urbanisation may be seen as a ‘negative,’ yet well-managed urbanisation has since been argued to have significant benefits that some consider far outweigh the drawbacks; the challenge is learning how to harness the inherent possibilities. Understanding various perspectives will help provide a better balance to this argument.
Several challenges spring from concentrating great numbers of people in relatively small areas. Populous urban areas are not without their environmental and social problems, but studies suggest that these aren’t primarily as a result of population concentration. Land use intensity and underutilisation in response to urban population density have an obvious and direct relationship on intra-urban transportation intensity (use, policy, energy, land) for instance – the more urban land used translates into policy needs, increased infrastructure, greater vehicle use and energy consumption.
Peter Newman suggests the problem with urban population increases is not migration, densities, consequences of urbanisation, but from unsustainable consumption and its global impact, and the structure of “how people live in cities” (i.e. lifestyles). Unfortunately, modern consumption and lifestyle trends lack sufficient feedbacks from their distant impacts.
Environmental
Early concerns in the global north involved wildlife habitat destruction due to industrial expansion, urban expansion and infrastructure construction such as house-building, road-building, transport networks, power stations, etc. Late industrialisation in the global south is now driving large scale urbanisation. Urban development (particularly in tropical regions), increased consumption (particularly meat products that drive agriculture expansion) and globalisation of agricultural commodities drive forest loss and associated environmental problems.
Furthermore, the consequences of urbanisation processes have effects on urban and peri-urban ecology, biodiversity, ecosystems and their services, as well as the distant effects created by pollution, exported wastes and imported goods and products. Yet low-density land use can result in more environmental damage due to their increased footprint and reliance on motor-vehicles, as well as scale inefficiencies resulting from extensive services infrastructure networks and distant social and health services, in effect the inverse of well-managed, dense settlements.
Economic
Urban economic growth and urbanisation processes are closely interlinked and self-perpetuating. Businesses locate where opportunities and investment migrate, and people move in search of employment and better services. As lifestyle expectations rise, smallholder agriculture no longer provides the kinds of productive existences people seek, despite government programmes to increase its appeal – agrarian living is challenging and unstable.
As centres of investment, decision-making and economic growth, dense cities concentrate wealth, commercial and public services and infrastructure, which aids national progress, according to the UNPF.
Social
Urban settlements have become concentrations of poverty, inequality and anti-social behaviour of all kinds. Social ills include juvenile delinquency, disconnected communities and social support networks. Malnutrition and disease can result from density and separation from sources of sustenance or income (i.e. subsistence economies reduce), thereby an increased reliance on external assistance.
Political
Stringent or ill-conceived regulations, forced evictions in the name of development, and armed conflict are frequently found in urbanised areas. Urban authorities from LDCs and developing countries are challenged in their capacity and ability to monitor, legislate, maintain and police their ever growing constituencies; these civil servants can be overstretched, under-skilled, or corrupt.
Fundamentally, “cities could indeed be helping to save the planet.” Predicting and managing economic, social, political and environmental change will be essential to the success of urban settlements. Rather than ignoring the issues, capitalising on opportunities created in a creative and proactive matter will help ensure successful outcomes, as ‘cities represent the best hope of escaping poverty.’
For a fully referenced version of the entire three-part story, please click here: | https://oneplanet-sustainability.org/2013/01/06/urbanisation-part-i-key-issues/ |
We, programmers, are human beings ( kind of…). And like any human being we make mistakes; a lot!
Some mistakes have huge repercussions and can certainly require lots of time and energy to correct. But if we have the good habit of initializing Git on any project, we can always go back in time and remove/correct the unwanted parts.
With Git, we have some commands that can be used as a time traveler. Those commands allow us to rewrite the history of the commits, change the message of a commit, set the project to a previous state…
But time-traveling always carries the risk of worsening problems or creating new ones, right Barry?
We will see together how to use those commands, how they can be useful, and how they might get us fired.
What is that we call the history?
Now before we go any further we need to know about three concepts that really help when 'playing' with history.
Those concepts are history, HEAD, and Index.
The history
The history on Git lists the various commits made on the project, in chronological order. It keeps track of the different actions with the name of their author and the date.
We have already got a glance of that history with the command git log, in this article.
git log --oneline
As you can see in the picture above, we have, at this stage, a list of commits that tell the history of our "amazing" project.
Rewriting this history, therefore, consists of editing this register in different ways.
The two other concepts are more abstract. But understanding them will help better understand what happens when we edit the history.
HEAD
The HEAD with git represents the current branch or the last commit of the current branch. So when we move from one branch to another, we are changing the pointer for HEAD. And when we made a commit, the HEAD moves to point to that commit.
The Index
The index is where we place files we want to commit. This is the space where we will find all the staged files. Each time we are using the
git add -a command, it will place the last edited files to the Index.
Here, I will be using the same project we used in the get started with Git article. You can find that project on GitHub. You can clone it with the command.
git clone https://github.com/webamadou/git-101.git
git commit --amend
The first case we will go through is the one with a wrong commit message.
The message of a commit is very important. It helps the team understand what was done at each step of the project.
Whenever a mistake is made on the last commit or we feel the need to add more details, we can use the "amend" command to edit the message on the most recent commit.
git commit --amend
That will open the text editor with the message of the last commit. We can then edit that message and save it.
Let's use the log command to list all our commits.
git log --oneline
Now let's change the last commit message with the
--amend command.
git checkout
The checkout command is mostly used to switch from one branch to another. But it also can be used to navigate through commits. Meaning, that can be used to set our project in one of the previous versions. We need to use the id of a commit to achieve that.
With
git log --oneline we can list all the commits and pick the state we want to set our project back to.
On our git_101 project let's set the navbar back to black.
git log --oneline
git checkout 3109a81
As you can see on the picture the id of the commit before we changed the navbar is 3109a81. We then used that id on the checkout command and the navbar is set back to dark.
Use the command
git checkout masterto get back to the 'normal' state.
Another way to use the checkout command is with the name of a file.
git checkout commit_id file_name
That variation of the checkout command will switch back the specified file to a commit without changing the HEAD, meaning without taking the whole project to that commit. The file will be changed to its state in the past, but the HEAD will remain the same.
If this is applied to our project, that means the navbar will be back to dark mode, but the other files of the project will remain unchanged.
With that case, we can then make changes and make a new commit.
git revert
The git revert command is going to undo actions on the given commit. It will create a new commit and will remove all the actions that were made at some point.
It's like you go back in time to delete that email you sent but still keep a log saying you deleted that email. (I know, that sounds crazy.)
This is a useful command when you have a wrong commit that you want to get rid of. But since it will delete actions of the past we need to be sure that there are not parts of our code that are linked or depended on those actions.
If used wrong we might end up messing up the project. Even if it can be used to go deep back in the past, it is mostly a command to undo the last commit since going far into the past commits might trigger fatal errors when dependencies are deleted.
In our project let's remove the README.md file. The command
git log --oneline tells us that the README.md was created at commit dca9335.
We can then use the following command to undo the creation of the README.md
git revert dca9335
That will open the editor to allow us to add the message for the new commit. We can then, add the needed specifications.
And if we do
git log --oneline we will notice the creation of a new commit.
git reset
The Git reset command is certainly among those that have great control over the past actions. However, it can bring big messes upon ou project if used in an inappropriate way.
The basic use of this command is to remove the last staged files.
git reset
Used on it's simplest way, as shown above, the command will simply remove file from the stage area. It can be handy if you added a wrong file in the staged files.
If it's only one file we need to remove from the staged, we can add the name of the file to that command.
git reset filename.extension
Keep in mind this is not going to undo the changes made on the file. It will only remove the file from the index (stage file zone).
Another way to use the reset command is with the id of a commit.
git reset commit_id
Used this way, the command will move the HEAD to the given commit. Meaning, it will rewrite the history of the current branch by removing all commits made after the specified one.
The
git reset command has three options that give different results.
git reset –soft commit_id
The soft option is the default one. It is the one that applies when no option is used.
That will do one action that is to move the HEAD to the commit.
git reset –mixed commit_id
The mixed option will remove the head to the given commit and unstage all files. It acts like the
--soft option, but will also remove files in the Index.
Both the soft and the mixed options are not impacting the last changes we made. The HEAD is moved to the specified commit (or branch) but the last changes we made are not affected.
That is not the case of the next option.
git reset –hard commit_id
That command is powerful and must thus be used wisely.
First, it will move the HEAD to the commit_id. Then it removes all the files in the index. And will also undo all last changes.
THIS ACTION IS IRREVERSIBLE
If we don't provide a commit id (or the name of a branch), the command
git reset --hard will just remove the last changes on the current branch. That might sound risky but it can actually be helpful in some cases.
Let's say while working on functionality we found ourselves surrounded by bugs and are overwhelmed. Or we found out that our approach is completely wrong. Using the
git reset --hard command will take us to the last stable version of our code where we could safely handle the functionality another way.
git rebase
The rebase command is my preferred one.
It is mainly used to put some order on the history line.
What this command does is it applies changes of one branch to another one and will also rewrite the history.
It changes the base of a branch and makes it look like it was created from that base.
An example should be more meaningful.
To better visualize the actions of the rebase command, we can use a Git GUI. The one I use is Gitkraken; you can download it here.
After installation, link Gitkraken with your Github account and open the local git-101 repository with it.
Let's say we created two branches name source_1 and source_2 both based on the master branch.
git branch source_2
git branch source_1
We switch to source_1 to make some edit and a commit.
git checkout source_1
Change the website title.
git commit -am "Edit the title of the website"
Then switch to source_2 and do the same.
git checkout source_2
Change the menu
git commit -m "Save the edit on source_2"
Now we have two branches diverging and both are based on the master branch. Using the rebase command we can integrate all the changes from source_1 into source_2. That will give the illusion that source_2 were created from source_1 thus rewriting the history.
git rebase source_2
This command is mainly used to maintain a linear project history. That is to have a history that follows one timeline with no diversion.
Making mistakes is part of programming. But as long we are using Git efficiently, we can avoid getting ourselves banned from the dev team.
Still, rewriting history is not recommended on Git. This is an option that should be used as a last resort.
Each of the command seen here offers more options. Here are some links that give more details on how to edit the Git history: | https://webamadou.com/rewrite-history-with-git/ |
The idiom ‘crocodile tears’ came from an old myth that crocodiles shed tears while eating their prey and cry out of grief for the prey’s death.
A French writer brought the story of crocodile tears to Europe in the 14th century, and Shakespeare used the term Crocodile Tears in his plays for the first time in English in the 16th century. But when it was tested experimentally, scientists found it not to be true.
If crocodiles stay out of water for a longer time, their eyes become dry, and the tear glands in the eyes also release water to lubricate them and remove debris from them. Hot air from the nose while eating out of water can also bring tears to the eyes. A disease known as Crocodile tear syndrome also derived its name from Crocodile tears because patients shed tears while eating. The major cause behind this syndrome is facial paralysis when nerve fibres destined for a salivary gland are damaged and then by mistake regrow into a tear gland.
Crocodilia an order of many animals:
Crocodile belongs to a large order known as Crocodilia which is the collection of three families given below.;
1. True crocodiles
2. Alligators
3. Gharials and false gharials
Distribution:
They are found on all the continents, except Europe and Antarctica, in tropical (marsh and swamps) areas with natural forests. They are found in rivers, running water, seasonal lakes and ponds. They are also found in the lagoons of the warm oceanic regions.
They are not found in all kinds of rivers, but where there is a large number of fish, birds of prey, and other animals who often come to drink water. They live mostly in brackish stagnant water but can also be found in running and fresh water.
Physical features:
✓A crocodile has a strong sense of smell and can smell both in water and on land. Its eyes resemble to that of the snake with a vertical pupil which controls the amount of light entering the eye, unlike our round eyes. They have a nictating membrane which protects the eye. Besides this, there is a layer known as Tapetum Lucidum behind the retina which makes them able to see in the dark as well. This layer is also found in the eyes of cats and dogs.
✓Just behind its eyes are ears, which are not visible, but function normally.
✓They have 80 polyphyodont teeth, i.e. their teeth are continually replaced. As they do not have lips, teeth are always visible from the outside.
Teeth are sharp and pointed and are used to kill their prey.
✓They have small black spots around their mouths. These are body sensors and they can sense even the slightest ripple in the water.
✓The bumps, spines or scales that appear on the body are called Osteoderm. They are made of Keratin, the same substance that makes up our nails and hair.
Inside the mouth, there is a membrane in the throat that seals in the water so that the water does not flow into the stomach.
✓The claws on their feet help them to make burrows, and the webs between them help in swimming.
Behaviour:
While swimming, the crocodile places its legs back against the sides of the body to reduce energy consumption and open them to change their direction.
They can also walk on land and can run by lifting their bodies above the ground. They are good swimmers and can also climb trees.
You have often seen them sitting on the dry land with their mouths open, because they do not have sweat glands in their body, so they lie outside with their mouths open. Vapours come out of their mouths and hence body temperature is normalized.
Crocodile as a predator:
A crocodile is an ambush predator and waits for its prey to come close to water, then rushes out to attack. It can hold its breath for 2 hours.
It clamps down on prey with its massive jaws, then crushes it with its powerful jaws, and then swallows the whole prey. A crocodile cannot chew its food so it has to swallow small stones and pieces of rocks that help it to grind up food in the stomach.
You have often seen crocodiles catch and flip their prey in water or on land. This is its most powerful attack. Flipping movement helps crocodiles in breaking the bone and tearing the skin of the prey.
What is special about digestive system of a crocodile?
The digestive system of a crocodile consists of two main parts. A muscular part where it keeps the swollen stones and other which has digestive acid and juices. Stones help to break off food into small pieces which are now available for the mechanism of digestive acid. The acid is so strong that it can digest the bones, hair and even nails of the prey.
Their metabolism rate is very slow, which means that energy from food is released very slowly. Animals with such a slow metabolism rate reduce their body movements in order to conserve energy.
Mode of reproduction:
During mating season the male use different tactics to attract female, it vibrates the water with its back, blows into the water with its nose, and splashes the water with its claws.
The female lays lots of eggs, out of which 1 or 2% survive. The hatchlings are about 7 to 10 inches in length and can grow up to 20 feet in size. | https://zeefacts.com/crocodiles-silent-killers-how-they-help-environment-part-1/ |
Class by: Felicity Yau
When I was a kid, my mom taught me how to make homemade Siu Mai steamed dumplings. They’re yummy, healthy and my ultimate comfort food. I love teaching people around the world how to make this traditional dish.
First, we’ll meet at the Shau Kei Wan metro station. The class starts with a wet market tour to buy the fresh ingredients for your dumplings.
Next, we’ll head to my apartment for the cooking session. Armed with fresh local ingredients and my family’s secret recipe, you’ll learn how to make my favorite Siu Mai.
After our hands-on cooking session, we’ll enjoy our meal together! Vegetarian or gluten-free options are available.
11:10 am: Wet market street tour and buy ingredients for Siu Mai dumplings
11:50 am: Prepare the fillings and make handmade dumpling wrappers
12:00 pm: Finish all cooking and enjoy the meal together
1:00 pm: End of the class
I'll be happy to teach you some simple Cantonese during the wet market walk. | http://tourszoom.com/tour/make-organic-siu-mai-and-visit-a-wet-market-hong-kong-vegetarian-gluten-free-options/ |
ess.com/jobs | 2022-09-29 08:06:32
A withdrawal is the process of taking money out of a savings account. Under certain circumstances, the depositor may be required to give the depositor a certain amount of notice before withdrawing the money from the account. Withdrawals can be made when the depositor is ready to use the money to cover expenses or for other purposes.
A withdrawal is an act of taking out money from a savings account, pension, or other account. This can be a one-time event, or it may be a process that takes place over a period of time. If the withdrawal is made early, there may be penalties and fees associated with it. A withdrawal can be made in a fixed or variable amount, or in a lump sum.
Deposits are another common way to deposit funds into a bank account. The term deposit is derived from the Latin word deponere, which means "to deposit." Deposits add funds to a bank account and help you pay bills or stash cash. Deposits happen in a variety of ways, including making an in-person deposit at a bank or making a deposit via a mobile application.
In the United States, a demand deposit account is an account that offers frequent access to cash. Demand deposit accounts, or "demand accounts," require the customer to provide at least seven days' notice before they can make a withdrawal. In some countries, these accounts are known as NOW accounts. These accounts offer greater interest rates than transactional accounts.
The withdrawal and deposit definition differs slightly from a deposit order. The difference between a demand deposit account and a negotiable order withdrawal account is that a demand deposit requires six days' notice to withdraw the money, while a negotiable order requires a seven-day notice. A demand deposit account also requires the owner to give a customer notice before withdrawing their money. There are many other differences between the two types of withdrawals and deposits, but the basic concept of these terms is the same.
When funds are available for withdrawal, the withdrawal or deposit amount is determined by the bank's policy. The balance is the amount of funds that is in the account, less any holds, uncollected funds, or restrictions. In other words, the difference between the balance and the credit limit is the amount of money that can be withdrawn or deposited. A balance transfer may incur a Balance Transfer Fee.
If you have a significant relationship, consider it to be like a bank account. A significant relationship is a two-way street: people make deposits and withdrawals all the time. A relationship that is lacking in deposits can lead to problems, but a fully-filled relationship can be beneficial for both parties. Ultimately, a relationship is a two-way street, and each person is responsible for keeping their end of the bargain.
XM Forex Marvel Trader can be downloaded for free. Before using it, you must verify your residential address and identity. To do this, you must provide a color copy of your passport or official identification document. After verification, you can deposit funds in your account. The minimum deposit required for XM Zero accounts is $100. There are other account types, including VIP Plus. The demo account comes with a 30-day money back guarantee.
The XM platform is rich in educational material, including regular webinars. The XM team encourages social learning and publishes research and technical analysis. It allows clients to choose from four different account types. Micro accounts, for example, only require $5USD as a deposit. Micro accounts are typically limited to 200 open positions. Depending on your location, you can also open an XM Zero account with a small deposit of as little as $1.
If you want to use a proxy server to gain access to blocked websites, you can choose from several free options. These include Http Injector, eProxy, Apk Custom, and KPN Tunnel. However, free proxy servers are very slow, and they generally only last for a day or two.
One of the benefits of using a free http injector is that it is not detected by phone companies. This means you can use it from any country or telecom company in the world. You only need to change the username and password, and remote proxy, and you're set. In our next post, we'll cover the different configuration options and provide links to videos to help you through the process.
While there are some online gaming servers that allow you to stream movies and TV shows without restrictions, not all of them do. If you want to enjoy Netflix without the hassle of restrictions, you should look into a VPS account. VPS accounts can increase your internet speed, and most online gaming servers have them.
Another option is to sign up for a free VPS through a free internet service provider. You can also get a free account from Amazon's AWS platform, which offers a free year subscription. However, you must register with a credit card to use it. You can find several free services online, including Google Cloud and Aruba Cloud.
Some websites offer a free trial of two months. During this period, you can create a VPS and test it out. However, after the trial period, you will have to pay for the service. To be on the safe side, make sure you sign up for a website that offers a 24-hour money back guarantee.
If you're looking for a cheap yet effective VPS service, VulTR is a great option. It has a good speed and is affordable. VulTR's VPS plans range from 2.5 dollars to $60 and are suitable for games, but are not ideal for streaming services.
If you are using Amazon Cloud, you need to create a virtual machine and add an elastic IP. You can then use the login ubuntu to log in. The first login is required with the ubuntu login. Once you've done this, you're ready to launch your server.
A new federal law would make bank accounts subject to deposit and withdrawal reporting over $600 a year. The legislation is intended to combat a widening tax gap, but it's unclear exactly what the new requirements would entail. The IRS already has access to information about certain types of transactions, such as ATM withdrawals and transfers, but this new law will expand the scope of these disclosures.
The Treasury Department recently proposed a rule that would require banks to report deposits and withdrawals over $600 each year. While this new law would affect millions of American accounts, the proposal is only in the early stages, and is subject to revisions. Essentially, it would require financial institutions to report any transaction with a total value of over $600, which could include deposited paychecks and money transferred between finance apps.
The proposal would apply to bank accounts used for business, personal accounts, or traditional bank accounts. However, unlike the original legislation, the proposed regulation would only apply to the annual inflow and outflow of most accounts. While it would be a big step forward, it would do little to monitor individual transactions.
As of this writing, the IRS's proposal would only apply to accounts with deposits or withdrawals that exceed $600 a year. It also would exclude transactions from government benefits and paychecks. As long as the account owner meets the requirements, the banks would have to report their total deposits and withdrawals every year. | http://www.iranisotop.com/whhd/7625.html |
Emerging research suggests that an ability to intentionally adapt may be associated with increased creativity and innovation, improved wellbeing, and stronger interpersonal relationships.
BKindred has partnered with multinational, Fortune 50, ASX100, and educational organisations to introduce the concept of Intentional Adaptability Quotient (IAQ®) to large workforces - a measure of how skilled individuals are in making intentional change in an environment that is evolving at speed.
The feedback has been intriguing: that an ability to intentionally adapt contributes to increased creativity and innovation in the workplace, unlocked potential, improved wellbeing, and stronger relationships.
Can IAQ® predict high performing teams in the workplace and overall wellbeing? We set out to research our findings further. The result is the IAQ® - The New Intelligence white paper, developed in partnership with the Centre For Leadership Advantage (CLA). We invite you to explore your curiosity by diving into our findings, and share the white paper with your colleagues. Download the white paper today. | https://www.bkindred.com/pages/iaq-the-new-intelligence-white-paper |
If you need to make a scarf and you're just a beginner, here's a really fun, easy way to do it!
Steps
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1Start by making a chain approx. 3 feet (0.9 m) long. (approx. 120 stitches). If you want a longer scarf, add more chain stitches until you reach your desired length. Turn and single crochet in the second stitch from the hook.
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2Single crochet (sc) and continue to sc to the end of the chain. Chain 1 and turn.
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3Repeat step 2 until you have reached the desired width of your scarf. Finish off.
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4Add tassels to your scarf.
- Take a piece of yarn approx. 12 inches (30.5 cm) long and fold in half.
- Weave the yarn piece into the end stitches of the scarf by taking the hook and pushing it through the stitch. Then loop the yarn piece onto the hook and pull the hook through the stitch with the looped yarn piece on the hook.
- Once the yarn piece is pulled through the stitch, remove the hook and pull the ends of the yarn piece through the loop on the other side and pull tight to create the tassel. Continue in this manner until all end stitches have a tassel.
- Trim the tassels so that they are even. Repeat step 4 on the other end of the scarf.
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5Finished.Advertisement
Community Q&A
Search
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QuestionHow do I add the tassels?MicheleTop AnswererThere are several ways to make and add tassels or fringe. See Step 4 above as well as the articles How to Make Tassels and Add Fringe to a Crochet or Knit Project.
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QuestionHow do I get a knot out of my wool?MicheleTop AnswererYou can leave it if it's small enough, or cut the knot out and continue as if you were changing colors or adding on another skein of yarn to your project.
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QuestionWhat do I do when the scarf is getting wide in the middle?Community AnswerOn the next line, lessen the amount of string you use - but don`t do this too much, as it`ll scrunch up the scarf.
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QuestionHow long does this take?Community AnswerIt depends on your skill level, as well as the length of the scarf, hook/yarn width, and patience (if you're a beginner).
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QuestionHow do I "turn and single crochet in the second stitch from the hook"?Community AnswerWell, you have to find the farthest 2 loops in the corner first. Then, after finding these, you insert the hook through (not completely) and from there you pull out some extra string. After pulling the extra string out, you move onto the next 2 loops. It`s pretty complicated to explain in words. You might want to ask someone to show you in person, or view instructional crocheting videos on YouTube.
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QuestionWhat size crochet hook for a scarf? A G?Community AnswerThe hook you use for a scarf most likely doesn't matter. Use whatever hook you feel most comfortable with; just make sure it doesn't get too big. If this means G for you, then sure, use a G.
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Tips
- Do your best and don't rush. You're not in a fastest-crochet contest.Thanks!
- Don't spend too long on steps 2 and 3. Only do it until your scarf is the way you want it.Thanks!
- If you can, you might want to loop long strings through holes on the short ends for some scarf-like effects. If you're not classic like that you can make little stars, hearts, circles, triangles, squares, or any other shape you want out of colored pipe cleaners, and loop those through the loops on the ends.Thanks!
- Don't spend too long doing this. Don't forget about other things too.Thanks!
- If you don't want a 4–5 foot (1.2–1.5 m) scarf, make it 2-3 or 3-4 feet.Thanks!
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Things You'll Need
- Crochet hook that is a good width for your yarn, otherwise an inexpensive, nice crochet hook that is at the store.
- Yarn that is your favorite color, or if you are doing this to sell, then lots of yarn in lots of different colors. If you are just giving them out, then ask what color the person/people want(s) and use that color.
- You should keep a tape measure handy for measuring. If you are selling, make two to five of each color, one or more of the 2-5 being 2–3 feet (0.6–0.9 m), one or more being 3-4, and one or more being 4-5.
- Once again, if you are selling, you will probably want a box, bin, basket, or other large container for all the scarves. Please note that if you're making lots to sell, don't expect to be able to make 30 scarves in 5 days.
About This Article
Thanks to all authors for creating a page that has been read 98,990 times.
Did this article help you? | https://m.wikihow.com/Crochet-a-Scarf-Using-Single-Crochet |
In computer software engineering, a unique relationship in computer structures is a kind of security feature. It is defined as a relationship by which one organization owns precisely the same information (RAM) and has access to it solely, while another entity offers exclusive control of a restricted amount of RAM. In other words, special RAM refers to a situation wherever an individual creator holds the whole exclusive legal rights to the MEMORY while an organization holds the legal rights to a area of the memory space. An exclusive relationship can also be among two clients who each have exclusive rights to a certain little bit of memory.
One of these of a individual or special relationship in software anatomist is the so-called private client/end user marriage. In this case, one organization refers to customers independently while the various other develops items based on buyer requirements. In order to develop these items, one spouse implements a clear method, as the other partner submits a requirement. The implementation company implements the technique and modifies it right into a product making use of the customer’s requirements. This scenario is usually repeated between two end-users marry a filipina woman who have got entered into a personal or different relationship with each other.
However , imagine if one of the lovers sees someone casual moving by? Suppose that he or she listens to someone talking on the phone or observing a person moving via room to room. Let’s assume that this person is not all their boss, they may feel the need to listen in depth. They may start out imagining the actual would do in that particular situation. Any time they find someone casual going by simply and all of a sudden they listen to a obtain from that person for their product, the subconscious begins to work on its response. In this case, the relationship remains to be exclusive but the one who started the conversation is now vulnerable and open and the dominant partner are now able to make use of the opportunity to manipulate the vulnerable partner.
Exclusivity is normally associated with long lasting partnerships or perhaps long-term job associations. However , there are other reasons so why an individual or an organization may well enter into a unique relationship. A person entering into a relationship the first time may desire something that is normally scarce in his/her life. So , that individual may enter into a marriage which requires him/her to stop something that they may have now – their independence to choose in order to live life as per to their unique wishes.
One other common the reason why people go into an exclusive romance is if the two people included share a common interest. When ever two people get into a talk or that they discuss a subject of prevalent interest, the conversation inevitably turns into an intimate conversation — at least for the listener. With this scenario, the listener seems at ease and enjoys the event. They begin to trust their spouse because they already know their partner wants to listen to everything that they may have saying. At this stage, each benefit since the listener profits information from the conversation as well as the partner puts on confidence within their relationship because they will know what their particular partner can be thinking.
An excellent dating marriage must be designed on trust and ease and comfort. It takes a lot of efforts to build a special relationship, but it will pay off in the long run. If you find someone that you are feeling comfortable with, you have found your meet – you could have found your true love. | https://dev.rey.cz/explanations-why-people-get-into-an-exclusive-romantic-relationship/ |
>> Download a printable pdf of this datasheet
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The next step is organizing the data into a database for spatial analysis within a GIS application. Subsequent steps may include statistics/geostatistical analysis of the data or development of a custom web application to manage data/documents and allow clients and stakeholders (if appropriate) to access the information. The final step is communicating the information to the client and stakeholders using appropriate visualization tools that will help the viewer understand the findings.
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Our extensive experience in finding, managing and presenting all aspects of information relevant to a case, from historical maps to relational databases, help us identify key information and put it at our clients’ fingertips. This approach to information management distills what can seem like an overwhelming amount of data, documents, and technical analyses into a clear, easy-to-understand narrative.
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Hard to believe it has been three years today, since we lost our bright star and guiding light, dear Katie. Last night at sundown and throughout today, together we light Yahrzeit (or other) candles to honor Katie, and to remind us that she is still with us in our hearts, still being the voice of encouragement and trusted friend.
I had a hard time last year on this occasion fully summing up all that Katie embodied to all of us, as her reach into all of our lives was treasured and immeasurable. Her love of Nate, of her family and friends, of all those she interacted with personally and professionally — was just enormous. She had a huge heart with space for so many in it, to lend an ear to us, to listen so deeply, to give all she could to us, and to be in our corner cheering us on. She was always pushing for the best for all around her, standing with us against the tide at times if needed.
With Katie still present in our hearts, today let us together light a candle for her, and to keep in our minds her message of stoking our inner flames, our little lights, to burn bright. We light our candles to the love that Katie embodied and that still lives on in each of us. We light candles to think of Nate as he grows up with his cousins and family in Germany. We light candles to the collective that knew Katie and celebrates her memory. We light candles to the lessons Katie gave us about living in love over fear, and appreciating both the moments in life and all those around us on the journey. We light candles to the examples Katie left us of giving back to others more than we have received.
Today, Katie, with our candles we remember your light-up-a-room smile and energy, your kind heart, and the deep impressions you made on all of our lives. We now carry you with us every day, as you help us still nurture our own lights. Much like your orchids, you continue to help us keep blooming and shining brightly.
Please join us today in remembering Katie by lighting a candle near and far, and if you wish to share a photo or thoughts with us, you can email them to [email protected] or add comments below.
Here are some of the candles we light for you, Katie…
[Click thumbnails below for full size photos, click arrows to switch between photos, then click white “X” in upper right corner to return… ]
I have a couple of orchids from Katie’s house and feel joy and warmth when they bloom. Reminds me of renewal and the persistence of life, beauty and light. Reminder of the serenity and the smells of Katie’s home, now doing the same still.
Not quite like the live rendition at Katie’s funeral service, but here is “This Little Light of Mine” as was Katie’s message for those of all religious backgrounds… May we all shine on brightly and help others around us shine as well!
.
If you have a fond or fun memory of Katie on this day, or an example of her love and self, please share it with us in the comments below… Whatever connection you had to Katie, we’d love to hear from you. Photos can be sent to: [email protected] and I will update this post with them.
Share this post with the buttons below, or from our Facebook page, if you think others might like to honor and remember Katie today.
Also, if anyone is interested, this weekend I am doing a 2-day 150-200 mile bicycle ride, fundraising to supply more Katie Kits to women cancer patients getting radiation treatment. More info can be found on this blog post.
Love to you all, and shine on, friends!
Ashby
….Check back here later today to see more memorial photos and comments….
(Click Read More if you don’t see the comments below. On mobile site, the comments are below the input boxes.)
Ad-Blockers may hide the social sharing buttons below. | https://thelifeofkatiek.com/2017/07/13/3-years-later-katie-is-in-our-hearts/ |
Q:
How many sequences of length $n$ with integer values from $[1, k]$ are there such that they contain all integers from $[1, k]$
How many sequences of length $n$ with integer values from $[1, k]$ are there such that they contain all integers from $[1, k]$? You can assume that $n \ge k$. For example if $k = 3$ then $112$ is not a correct sequence, because it does not contain all numbers from $1$ to $3$.
A:
Following lulu's hint we obtain:
$$\binom k0k^n-\binom k1 (k-1)^n+\binom k2(k-2)^n-\cdots=\sum_{r=0}^k \binom kr(-1)^r(k-r)^n$$
| |
What do you earn in the professions?
What do you earn in the professions?
Salary by occupation: This is how much Germans earn in their jobsOccupationMedium gross salary (median*) per monthPharmacist4,111 eurosArchitect3,748 eurosDoctormore than 5,400 eurosDoctor’s assistant2,157 euros96 more lines•
How much does a normal earner earn?
What is normal will be felt differently by everyone. In any case, the national average per year is €30,000, which corresponds to €2,500 per month.
How much is good earnings?
In whole numbers: In 2014, the average German employee earned 1756 euros net per month. Only five percent of employees earn more than 5,000 euros gross, i.e. around 3,000 euros net, which can probably be considered good in relation to this.
How much salary at 25?
The average salary across all age groups is EUR 45,839. Managers at 25 earn an average of €73,613, increasing their salary to €133,901 per year by age 60. This results in a salary increase of 82 percent.
How much does a 25 year old make?
If you are under the age of 25, you need at least 1,692 euros to belong to the richer half of Germany. The amount increases sharply as soon as you start working. Young professionals between the ages of 25 and 34 earn about 1,750 euros net per month, in the following ten years it is 1,982 euros.
What should you earn at 40?
The average salary at the age of 40 levels off at around 48,400 euros a year. Up to the age of 60, however, it only increases by a total of around 1,600 euros gross. Only shortly before retirement, at 60, do skilled workers crack the 50,000 euro mark on average.
Is 5000 euros net a lot?
5 Answers Well, for most statistics, twice the net income counts as rich. That’s something around just over 3500 euros net. In tax class 1, 5000 euros net would require a good 9000 euros gross, i.e. 108k pa The average net for single householders is around 1700-1800 euros net per month.
Is 2000 euros net good?
Are 2000 euros net (NOT gross!) for a young professional (21 years old) a lot or 700 euros gross (550 euros net), in the third approx. 1000 euros (800 euros net), in the 1st
How much do you have to earn to live well?
3000€ is a very good income, on average you actually get about 2000€, so 3000€ is definitely very good, and I think you can live on it, it depends on what you are “satisfied with”, if it is not exactly one Panthaus or something similar, you may pay
How much do you have to earn to live alone?
At least 374 plus your warm rent, 400 euros, so about 800 euros (a Harz 4 recipient does not have more) to be able to live alone!
How much money do you need to live on per month?
The statistics portal statista has determined the value that a German household consumes on average per month. The result converted to one person: We need an average of 1240 euros per month per person.
How much do you have to earn to build a house?
Theoretically, a gross salary of 2,000 euros a month is enough to buy your own little house – it just can’t be that expensive. In this case, you could spend a maximum of around 152,000 euros on your own home. The salary would be enough to finance a corresponding loan for 25 years.
How much do you have to earn to get an apartment?
The 30 percent rent rule is more accurate than the 40 percent rent rule. After that, your rent per month should not be higher than 30 percent of your monthly net income. For example, if your income is $2,333 net per month and you take 30 percent of that, you can spend about $700 on your rent.
Can I finance a house without equity?
Buying a house without equity is basically possible – whether that is advisable is another question. Because the financing of your house is much more expensive without your own financial resources.
Visit the rest of the site for more useful and informative articles! | https://sonichours.com/what-do-you-earn-in-the-professions/ |
When I was a little girl, we used to have big family Christmases with lots of people, tons of food, and enough cookies to feed all of Santa’s elves. While there are a few baked goods we continued to make after the Christmas get-togethers stopped, but these weren’t one of them, for some reason.
When I started making these in my own kitchen, 20 years after the last time I’d had them, it was a lot of trial and error, since I didn’t have a family recipe to work from. I found a basic anisette recipe somewhere else, but by the time I was done with it, you’d never recognize it. I tinkered with the dough by tasting lots of batter. Eventually, I got it right, and boy, did the nostalgia come flooding to me.
When I put the cookies in the oven and they started baking, I almost cried. I hadn’t smelled these cookies baking in so long. The fragrance of the anise and the sweet dough brought back so many memories from when I was little, baking in Nana’s kitchen. This Christmas, I’m going to have a party with friends, just like I used to with family. I’ll make the cookies we’ve carried on from my childhood, and finally, I’ll have these.
Ingredients:
For the Cookies:
6 1/2 cups Flour
6 heaping tsp Baking powder
1/2 tsp Salt
1 1/2 sticks Butter (softened)
1 1/4 cup Sugar
6 Eggs (room temperature)
2 tsp Anise extract
1/2 cup Milk (room temperature)
1 tbsp Anise seeds
For the Icing:
1 cup + 1 tbsp Powdered sugar
3 tbsp Milk
—
Preheat the oven to 400 degrees F and line two baking sheets with parchment paper.
In a large bowl, sift together the flour, baking powder, and salt, and set the bowl aside. In another large bowl, cream the butter and sugar. Add the eggs in one at a time, followed by the anise extract. Add in the milk and flour mix, alternating in thirds until everything has been mixed together. Fold in the anise seeds, and let dough sit for an hour, loosely covered.
Once the dough has been left to sit, spread flour over a section of clean counter top and your hands, and turn the dough out onto the counter. Knead the dough until it is no longer sticky, but firm and workable. You may need to add more flour to the dough, gradually, while kneading it to reach this texture.
Once the dough is no longer sticking to your hands, grab a chunk of dough about the size of a ping pong ball, and roll it our into a 1/4 inch thick “snake” of dough that’s about 6″ long . Twist the length of dough over itself twice into to form a loop (see the picture below for an example).
Bake the cookies for 8-10 minutes or until golden on the bottom. I suggest putting one pan in the oven at a time and filling up the remaining one while the one tray of cookies bakes. This recipe yields about 8 dozen cookies, so it’ll be a while.
To make the icing, pour the powdered sugar into a bowl and add the milk. Whisk the two ingredients together until there are no more lumps. Move the cookies onto a dedicated icing dish, to keep things clean. Brush the icing generously onto the cookies and add sprinkles immediately (otherwise they won’t stick). Move the cookies to a rack or a separate plate from the icing plate (otherwise they’ll get stuck) and let the icing set completely before moving the cookies to an airtight container for storage. | http://apartmenteats.com/2017/09/anisette-cookies/ |
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