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# Discrete Mathematics - Science topic

Explore the latest questions and answers in Discrete Mathematics, and find Discrete Mathematics experts.

Questions related to Discrete Mathematics

Is the lattice geometry in semiconductor crystals the same as the one covered in group theory? Is this concept derived from the discrete mathematical group theory?

If possible discuss the space, basis, dimension in the context of discreate mathematics and machine learning.

Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?

If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure

*$\mu(t)$*in the integral representation in the Bernstein--Widder theorem for*$f(x)=*\frac{1}{\arctan x}*$*?These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090

Dear Colleagues

I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.

Best regards

Feng Qi (F. Qi)

Hi

I wanna solve partial differential equation in terms of x and t (spatial and time), As I know one of the most useful way for solving pde is variable separation. well explained examples about mentioned way are wave equation, heat equation, diffusion....

wave equation is Utt=C^2 .Uxx

in other word; derivatives of displacement to time, equals to derivatives of displacement to spatial multiplied by constant or vice versa.

however my equation is not like that and derivatives are multiplied to each other.for example : Uxx=(1+Ux)*Utt

Im wondering how to solve this equation.

I will be thankful to hear any idea.

In section 1.4,The answer of 13th exercise question is not correct.In the question 13(d), the answer is printed as True. But the Domain under consideration is set of all integers.When we apply negative integers 3n > 4n. Example -6 > -8. So the universal quantifier when applied to the domain the set of integers, ithe given proposition will not hold for all integers. Hence the answer must be false.

This Textbook( 8th edition ) has been used by students of most of all universities in this world. So this error must be noticed.Whats your opinion?

See the attachments.

I have two type of resources A and B. The resources are to be distributed (discretely) over k nodes.

the number of resources A is a

the number of resources B is b

resources B should be completely distributed (sum of resources taken by nodes should be b)

resources A may not be completely distributed over the nodes. In fact, we want to reduce the usage of resources A.

Given resources (A or B) to a node enhance the quality of the nodes, where the relation is non-linear.

All nodes should achieve a minimum quality.

What is the type of the problem and how I can find the optimal value?

Which is best book for discrete mathematical structures, for undergraduate teaching?

Hi,

I'm interested in solving a nonconvex optimization problem that contains continuous variables and categorical variables (e.g. materials) available from a catalog.

What are the classical approaches? I've read about:

- metaheuristics: random trial and error ;

- dimensionality reduction: https://www.researchgate.net/publication/322292981 ;

- branch and bound: https://www.researchgate.net/publication/321589074.

Are you aware of other systematic approaches?

Thank you,

Charlie

Can anybody tell me what the current situation in the hierarchical product of graphs is? Are there any articles on this?

Does Faulhaber’s formula for the sum of powers have any useful mathematical or real life applications?

Does generalizing this formula to calculate the multiple sum of power have useful applications?

1- The distance between any two vertices u and v, denoted d(u, v), is the length of a shortest u − v path, also called a u − v geodesic.

2- Suppose G is a (weighted) graph and S a set of vertices in G. Then the Steiner distance for S, denoted by d_G(S), is the smallest weight of a connected subgraph of G containing S. Such a subgraph is necessarily a tree, called a Steiner tree for S. The radius, diameter and average distance have a natural extension. For a given vertex v in a connected (weighted) graph G and integer k (2 ≤ k ≤ n), the k-eccentricity of v, denoted by e_k(v), is the maximum Steiner distance among all k-sets of vertices in G that contain v. The k-radius, rad_k(G), of G is the minimum k-eccentricity of the vertices of G, and the k-diameter, diam_k(G), of G is the maximum k-eccentricity

I am considering to distribute N-kinds of different parts among M-different countries and I wan to know the "most probable" pattern of distribution. My question is in fact ambiguous, because I am not very sure how I can distinguish types or patterns.

Let me give an example. If I were to distribute 3 kinds of parts to 3 countries, the set of all distribution is given by a set

{aaa, aab, aac, aba, abb, abc aca, acb, acc, baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb, ccc}.

The number of elements is of course 33 = 27. I may distinguish three types of patterns:

(1) One country receives all parts:

aaa, bbb, ccc 3 cases

(2) One country receives 2 parts and another country receives 1 part:

aab, aac, aba, abb, aca, acc, baa, bab, bba, bbc, bcb, caa, cac, cbb, cbc, cca, ccb 17 cases

(3) Each county rceives one part respectively:

abc, acb, bac, bca, cab, cba 6 cases

These types may correspond to a partition of integer 3 with the condition that (a) number of summands must not exceed 3 (in general M). In fact, 3 have three partitions:

3, 2+1, 1+1+1

In the above case of 3×3, the number of types was the number of partitions of 3 (which is often noted p(n)). But I have to consider the case when M is smaller than N.

If I am right, the number of "different types" of distributions is the number of partitions of N with the number of summands less than M+1. Let us denote it as

p*(N, M) = p( N | the number of summands must not exceed M. )

N.B. * is added in order to avoid confusion with p(N, M), wwhich is the number of partitions with summands smaller than M+1.

Now,

**my question is the following**:*Which type (a partition among p*(N, M)) has the greatest number of distributions?*

Are there any results already known? If so, would you kindly teach me a paper or a book that explains the results and how to approach to the question?

A typical case that I want to know is N = 100, M = 10. In this simple case, is it most probable that each country receives 10 parts? But, I am also interested to cases when M and N are small, for example when M and N is less than 10.

We don't have a result yet, but what is your opinion on what it may be? For example, P =NP, P!=NP, or P vs. NP is undecidable? Or if you are not sure, it is feasible to simply state, I don't know.

I have problem with drawing a planar connected graph with 81 vertices and maximum degree 4 and diameter 6.

at first I drew a big plus on the paper and tried to put as much vertices as possible on it and give it some branches so that degrees of all vertices is at most 4 and we can walk from every vertex to another one by going over at most 6 edges and try not to crash the edges ,but I could just put 53 vertices in my graph and if I try to add some vertices ,the diameter will be bigger than 6!you can see my attempt in the file.

I also tried to start with drawing some polygons but in this way I could not prevent the edges from having intersection so that the graph could not be planar.

I will really appreciate any help or idea.

Is it theoretically possible, that after discretization by using Talyor Series Expansion, a non-observable nonlinear system will became an observable?

It was proved, that used continuous model of PMSM is non-observable (see attached). I want to know, if resulting discrete system is observable or not. Any comment appreciated. Thanks.

Assume, we found an approximate solution A(D),

where A is a metaheuristic algorithm, D is concrete data of your problem.

How close the approximate solution A(D) to an optimal solution OPT(D)?

For Example, I have a South Carolina map comprising of 5833 grid points as shown below in the picture. How do I interpolate to get data for the unsampled points which are not present in 5833 points but within the South Carolina(red region in the picture) region? Which interpolation technique is best for a South Carolina region of 5833 grid points?

Hello everyone, could some people suggest a good syllabus for graph theory and discrete mathematics for Computer science - Network department, please.

Thank you in advance.

Hi i am planning to investigate the applications of semiring in decision making.i wish to have some clues/clarifications of basic queries as follows:

Q.1 what are the examples of semiring structure's real life situation? how do we describe some real life situation that could be modeled into a semiring structure?

Q.2 how can we use boolean lattices/boolean logic/ boolean search etc to solve certain practical problems in semiring structure to arrive at decision making?

Q.3 can we use graphs/vectors/matrices etc as tools in " Application of semiring in decision making" ?

Q.4 how to link semirings to :

(a) graph theory?

(b) vectors and matrices?

(c) boolean algebra and boolean logic etc ?

In order to get a better conditioned A matrix, the absolute mean of the eigenvalues of the A matrix should be one (all eigenvalues are between -1 and 1, so within the unit circle and the absolute mean is 0.4389). This could be done by scaling the time.

For the following continuous-time state-space model:

dx/dt = Ax(t) + Bu(t)

y = Cx(t)

the state-space model will look like:

dx/dtau = (1/lambda_avg)*Ax(tau/lambda_avg) + (1/lambda_avg)*Bu(tau/lambda_avg)

y = Cx(tau/lambda_avg)

with

lambda_avg, the absolute mean of the eigenvalues of the A matrix

tau, the new timescale

tau = lambda_avg*t

However, I want to scale the time of a discrete-time state-space model in order to get a better conditioned A matrix:

xi+1|k = Axi|k + Bui|k

yk = Cxk

How could I do that in the same way as for the continuous model?

In socialist millionaire problem, two millionaires learns whether their wealth are same or not without revealing detail wealth. It is a step in protocol where Qa = g^x and Qb = g^y, where x and y are wealth, g is base of the discrete logarithm. a) Is it not computational overhead on users if value of x and y are huge? b) If I want to replace x and y as string in place of numbers, then how may I do it?

The musical melody is a structure consisting of a series of two types of entities: tones and pauses. Each tone has two properties: pitch and duration; each pause has one property - duration. According to these properties, they can compare to each other. The result of a comparison can be identity or difference.

Hypothesis: some combination of tones and pauses give us a sense of beauty, others don’t. Let us assume that beauty is proportional to the quantity and variety of the identity relations that the melody structure contains.[1]

Question: how can we determine the quantity and variety of identity relations in a given melody structure if we know that there are:

1. identity relations between individual tones and pauses;

2. identity relations between relations. (example: A and B are different in the same (identical) way as B and C; duration of A is half of the duration of B just like (identically) the duration of C is half of the duration of D; etc.)[2]

3. between groups of tones (and pauses)

And a second question: by which method can we create structures that contain maximum quantity and variety of identity relations?

*********

[1] About the reasons behind this hypothesis seePreprint , part 3.

[2] The structure must be observed throw time. If we play the tones and pauses of a beautiful melody in random time order the beauty will be lost. These types of relations allow us that.

If we have a fourth order polynomial as follows:

f(X)=a*X^4+b*X^3+c*X^2+d*X+e

how to make this equation on the following form:

f(X)=-(m*X^2+n*X+y)^2. I tried to extract the all terms of the second equation and compare the terms coefficients but it didn't work.

What is the tensor type for Green Lagrange strain tensor and 2nd 2nd piola kirchhoff tensor?

Stress and strain are called 2nd order tensor because they follow the transformation rule: sigma`= R * sigma * R' where R is transformation matrix.

Deformation gradient, rotation matrix and 1st piola kirchhoff are called two point tensor because they relate two configuration. For example deformation gradient is: F(iJ) = xi * XJ where * is dyadic or tensor product and XJ and xi are the element before and after deformation.

What about Green Lagrange strain tensor and 2nd piola kirchhoff tensor?

which they do not change with element rotation

I've seen incorrect proofs for some cases of m, and I've seen it claimed to be proven by Tout, Dabboucy, and Howalla, but cannot access their paper.

I am working on the construction of Barnette graphs for given diameter. I would like to know the reason why many cubic 3 connected planar , (not a bipartite) are both non-Hamiltonian and Hamiltonian graphs. I found a unique property of those Hamiltonian graphs. I need the latest results related to my question.

I have a function as follows:

y= a*x^b

a=7e-5

b=-0.755

I attached a simple graph of the function. As it is apparent from the graph the CURVATURE of the function increases from ZERO to a finite value (around x=0.1-0.2) and then it decreases to reach a value of ZERO. I did my best to draw the CURVATURE of this function using the following formula:

K=f"/(1+f'^2)^1.5

However, using this formula I could not reach the predictable trend of the curvature. Do you have any idea what is the problem?

I can work with MATLAB and Excel.

Your help is appreciated in advance.

Kind regards,

Ebrahim

A trajectory is obtained for discrete points, what is the procedure for measuring the smoothness of this trajectory. The answer to this question will help me get a clear picture about the convergence rate of Legendre Pseudospectral method, where the rate of convergence is defined as 1/( N^2m/3−1 ). Here m is defined as the smoothness of the optimal trajectory and N is the number of nodes or points. This rate of convergence formula and further discussions can be found in the paper titled "

**Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems**" written by Wei Kang .Given the presented scatter-plot, it is looking like that there is a relationship between X and Y in my data. Unfortunately, the simple nonlinear curves can not describe this relationship. However, I guessed some equations like Y= aX^b + c and Y= a*exp(b*lnX) that can describe the relationship but it seems that they are not the perfect ones.

I am able to do the analysis in MATLAB, SPSS and Excel if you have any suggestion to solve the problem.

kind regards,

Ebrahim

Pls, anyone with contributions on how i can use DEA to solve Graph Algorithms problems such, Network flow, Project management, Scheduling, Routing.etc

Majorly I need information on how to identify the input and output variables in this kind of problems(where there is no complete knowledge of the I/O ).

I think I can identify my DMUs.

I shall be glad to receive contributions on the appropriate general DEA model approach for solving Combinatorial Optimization problems of these kind.

Thanks

Do you think that the iThenticate/CrossCheck/Similarity Index would cause heavy and serious confusion in mathematics? Even destroy, ruin, damage Mathematics? Our mathematics and mathematicians should follow and inherite symbols, phrases, terminology, notions, notations in previous papers, but now we have to change these to avoid, to escape, to hide, to decrease the iThenticate/CrossCheck/Similarity Index! It’s very ridiculous for mathematics and mathematicians! Mathematics is disappearing! being damaged!

As I would study the singularity problem at the intersection of line heat source, so I wonder if anyone know any analytical solutions for the problem with intersecting line heat source in a finite/infinite domain? Really appreciate it!

The cardinality of a minimal dominating set of a graph is called domination number and the upper domination number is the maximum cardinality of a minimal dominating set. How can they be different?

The Brute force algorithm takes O(n^2) time, is there a faster exact algorithm?

Can you direct me to new research in this subject, or for approximate farthest point AFP?

In

**Theorem 5 of [1]**it is proved that for any graph G if S a convex subset of vertices of G, then the convex hull of the contour set of S equals the original set S: S = co(Ct(S)).As noted by the authors this a similar, more general, property to the classical Minkowski-Krein-Milman property defined in terms of extreme vertices.

Does this theorem hold for hypergraphs as well?

I think so, but I haven't yet found any references. I would appreciate if somone could clarify this.

Thank you.

Kindest regards,

Marcos.

[1] Cáceres, J., Márquez, A., Oellerman, O.R., Puertas, M.L.: Rebuilding convex sets in graphs. Discrete Mathematics 297(1), 26-37 (2005), Elsevier. https://doi.org/10.1016/j.disc.2005.03.020

a comparatively simpler proof of change of variables in Lebesgue multiple integral in euclidean spaces is in serge lang analysis Ii and only one part is valid for Banach spaces?

I need help in understanding the role of (random) sampling in implementation of a control system in Simulink. I need a basic, general example to visualize the role of the sampler in a control system, and the way it can be programmed (to be random/event-triggered etc).

Any help in this regard is very much appreciated

Thank you in advance

Given an undirected(weighted) graph depicted in the attached diagram which is a representation of relationships between different kinds of drink, how best can I assign weight to the edges? Is there a technique that I can use?

Are there techniques to automatically assign weights on weighted graphs or weights on links in concept hierarchy? Assuming the scenario depicted here :

*https://cs.stackexchange.com/questions/90751/weight-assignment-in-graph-theory* is a form of a weighted graph. Are there ways weights can be assigned to each edges?

I am planning to submit a manuscript to the Electronic Notes in Discrete Mathematics, but I failed many times to find its submitting address on its Elsevier website. Where is the submiting address for the Electronic Notes in Discrete Mathematics?

what is the origin and significance of writing imaginary term exp i(kx-wt) in the wave equation ?

Let suppose we say that there is a document D and a List of Sensitive Terms S. all NPs are belongs to D. Now IF are those NPs that belongs to S.

(1) IF = All NPs that belongs S.

Now Let suppose we have list A and B. Where A contains Identifiers and B contain medical finding.

All those NPs that is in A are I and All those NPs that is in B are F.

(2) I = { i E D | i E {a E A }}

(3) F = {f E D | f E {b E B}}

So implies that

(4) IF = I = { i E D | i E {a E A }} | F = {f E D | f E {b E B}}

or

(5) IF = I + F

X and Y are not correlated (0.3); however, when I place X in random forests classifier predicting Y, alongside two (A, B) other (related) variables, X and two other variables (A, B) are significant predictors of Y. Note that the two other (A, B) variables are also not correlated with Y.

How can I interpret this according to statistics and machine learning idea?

Representing one or more variable (A, or B or Y) with respect to another variable (X), where the variables don't have a strong correlation.

Discrete Mathematics is (for example)

- historically the foundation of all mathematics,
- used in applied math (in discretizations for example) to solve many of the most complicated problems,
- producing extremely powerful results, especially in the last 50 years,
- and much more!

But despite these examples that merely scratch the surface, discrete mathematics still does not enjoy equal stature among the various areas of mathematics. Any thoughts? Let's discuss!

Note: I realize there's no simple answer to this question, just looking for conversation.

In the set of prime numbers, the equation p-q = 2 has multiple solutions. There is a hypothesis that this equation has infinitely many solutions. What happens if 2 is replaced by 2k?

I want to build a kind of guess game. I do not know the right name but the concept of the game is: person-1(P-1) thinks a name(of anything) and person-2 will have to predict that name by asking as less questions as possible. For example:

p1: thinks something(Steve jobs)

p2: Are you talking about man?

p1: yes.

p2: Is he/she media guy?

p1: No

P2: is he tech personality?

p1: yes

p2: steve jobs.

p1: yes.

So p2 has to ask 4 questions. It could be even more as number of predictors are infinite. Now I want to model this scenario. My Goal is to reduce the number of question. Note that the number of predictors are limited. So situation is not that broad.

I can think of decision tree. But question is, how can I decide where to split so that length of the brunch will be small.

Any suggestion/reference will be appreciated.

Specifically, i have an interest in studying idempotent semiring with respect to linearly dependent and linearly independent set of vectors.

How can I find out the dependency of two variables from one or multiple equations.

Example:

Set 1 -

x

_{1},af(x

_{1},a)=z_{1}Set 2 -

x

_{2},af(x

_{2},a)=y_{2}f(y

_{2})=z_{2}How highly ( z

_{1 }and x_{1}) and ( z_{2}and x_{2}) are related among themselves and can we quantify it?Which Set's relation/dependency of the variables are stronger in comparison with the other Set ?

How easy will it be to get back x from z from both the Sets ? Which one will be harder and how to express that ?

I am trying to get my nomenclature correct.

I personally use backtracking searches that enumerate a finite (though large) set of multi-digraphs to prove chains either exists or do not exists of a certain length. The multi-digraphs can't contain certain subgraphs as well has having limits on edges and vertices that make the search set finite.

**Take a look at image #1**

Let $\mathbb{G}(\mathbb{R}^n, \mathbb{R}^m)$ the vector space of computational

functions, this means, that all functions in $\mathbb{G}$ are computable by an

imaginary but finite computer.

Then there is a mapping $t:\mathbb{G} \times \mathbb{R}^n \rightarrow \mathbb{N}$

that maps a function and an input value to the processor time requiered

to calculate the result.

Let $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a computable function,

and $f: \mathbb{R} \rightarrow \mathbb{N}; x \mapsto \sup\limits_{\|y\| < x} t[g, y]$.

Then the asymptotial runtime exists, if there is a function $r: \mathbb{R} \rightarrow \mathbb{R}$

continuous with

$$ \lim\limits_{x \rightarrow\infty} \frac{f(x)}{r(x)} = c \in \mathbb{R}$$

Then $\mathcal{O}(f) = \mathcal{O}(r)$.

See the rendered output in link #1.

Assume we have a class of graphs. Now what does this sentence mean?

"each of the graphs in the class, monotonically should make no difference".

If there are k sets of vertices in a graph, with the condition that each vertex in a set should be connected to at least one vertex from each of the other sets, then what is the least number of complete sub-graphs $K_{k}$ in this graph?

I have a fairly large matrix (250*250) in

**symbolic**form in MATLAB. Matrix is**square invertible**with size**multiple of 2**. I have to calculate the first two elements of the first two rows of its inverse**(i.e. first 2x2 block).**Matrix is very large and in symbolic form ,so, Matlab is not able to calculate the whole inverse due to time limitations. I have tried guassian elimination, LU factorization, block wise inverse technique. I have also tried the simple method of cofactors and determinent. In all the cases the problem is the same: very long time in the range of hours. Can anyone suggest some technique?I am preparing for my master thesis in Quantum Image Processing (QImP), i choose to work with [Novel Enhanced Quantum Representation of Digital Images][1] (NEQR).

To convert an image from Classical domain to Quantum domain we need to do a Quantum Image Preparation which in case of NEQR is consists of two steps as shown in the image below:

<a data-flickr-embed="true" href="https://www.flickr.com/photos/144570300@N07/33793456936/in/dateposted-public/" title="NEQR Quantum Image Preparation steps"><img src="https://c1.staticflickr.com/3/2904/33793456936_2d29a8d7e5_z.jpg" width="640" height="373" alt="NEQR Quantum Image Preparation steps"></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>

The second step is the one that set the colors. The paper descripe this step as follow

> It is divided into $2^{2n}$ sub-operations to store the gray-scale

> information for every pixel. For pixel $(Y,X)$, the quantum sub-

> operation $ U_{YX}$ is shown as (8) $$ U_{YX} = \Biggl(I \otimes

> \sum_{j=0}^{2^n -1} \sum_{i=0,ji \neq YX}^{2^n - 1} \lvert ji \rangle

> \langle ji \rvert \Biggr) + \Omega_{YX} \otimes \lvert YX \rangle \langle YX \rvert \tag{8}$$

>

> Where $ \Omega_{YX} $ is a quantum operation as shown in (9), which is

> the value setting operation for pixel $ (Y,X)$: $$ \Omega_{YX} =

> {\displaystyle \bigotimes_{i=0}^{q-1} \Omega_{YX}^{i}} \tag{9}$$

Because $ q $ qubits represent the gray-scale value in NEQR, $ \Omega_{YX}$ is consisted of $ q $ *quantum oracles* as shown in (10):

$$ \Omega_{YX}^{i} : \rvert 0 \rangle \rightarrow \Bigl\rvert 0 \oplus C_{YX}^{i} \Bigr\rangle \tag{10}$$

From (10), if $ C_{YX}^{i}=1, \Omega_{YX} $ is a $ 2n - CNOT $ gate. Otherwise, it is a quantum gate which will do nothing on the quantum state.

My question is, how (10) is a $2n - CNOT $ gate if $ C_{YX}^{i}$ is $1$?

From my understanding $ C_{YX}^{i}$ is a computational basis, that is it is either $\rvert 0 \rangle$ or $\rvert 1 \rangle$ and the tensoring of

$ C_{YX}^{i}$ in (9) will produce a column vector.

Also if i interpret $ \Bigl\rvert 0 \oplus C_{YX}^{i} \Bigr\rangle $ as follow: it is the result of $ 0 \oplus C_{YX}^i$ this is just $C_{YX}^i$ because $ 0 \oplus x$ is just $x$. Where $ \oplus $ is XOR. How this will produce a $2n-CNOT$ gate where it is a 3 qubit gate (its matrix is 8 * 8)

I have a linear relation between a dependent and an independent variable (x and y). I need to prove that x and y are equivalent. To do that I have already considered two ways: 1-verifying reflexivity, symmetry and transitivity; and 2- proving that it is a bijective function. If this is right, I have already done the first step. As a second task I have to extend this relation to fuzzy sets and I only need to prove min-max transitivity at present.

I need to build the adjacency matrix of such relation to demonstrate the rest of the properties and I think I should get a identity matrix representing variables x and y, but I don´t know if there is any theorem about this. That´s to say: If the linear ecuation is bijective, the adjacency matrix of such relationship is necessary an identity matrix?

Intuitively I would say that the number of concepts is bounded by min(2

^{|O|}, 2^{|A|}) where O and A are resp the objects and attributes sets. I get this (most likely wrong) intuition from the observaton that given X_{0}, X_{1}included in O and the Y_{0}, Y_{1}included in A, for any pair of concepts (X_{0},Y_{0}) and (X_{1},Y_{1}) we have X_{0}=X_{1}iff Y_{0}=Y_{1}. Thus there can not be more concepts than the number of object sets neither than the number of attribute sets appearing in the lattice. But the best upper bounds I find in some research papers are much more larger than the one I propose : for example 2^{|O|+|A|}, or 2^{sqrt(|O|.|A|)}. And I really dont understand why... Could someone explain me where is my mistake ?I am analyzing 3D-reconstructed images taken of embryo vasculature. In these images, I am trying to understand the behavior of endothelial cells; comparing controls with mutants. So far I managed to have the X,Y, Z coordinates of these cells. I am trying to figure out how clumpy they are. And compare that between controls and mutants.

So basically I am trying to analyze the data using these coordinates. I am sure that there some sort of formula or plot that can show me how clumped these cells are.

I am wondering if anyone has a good background with this.

Thank you,

Ali

Given a graph G and a finite list L ( v ) ⊆ N for each vertex v ∈ V , the list-coloring problem asks for a list-coloring of G , i.e., a coloring f such that f ( v ) ∈ L ( v ) for every v ∈ V. The list coloring problem is NP-Complete for most of the graph classes. Can anyone please provide the related literature in which the list coloring problem has been proved NP-Complete for the general graph using reduction (from well know NP-Complete problem)?

Let G(N,p) be an Erdos-Renyi graph, where N is the number of vertices, and p is the probability that two distinct vertices form an edge. If I plot 1-b0/N over log(p), then I obtain a curve which looks like a logistic function, where b0 is the number of connected components of G(N,p), and p is in (0,1). If p is not too close to zero, then a logistic function has a very good fit. If this were the true model, then the expected value for b0 would be

E(b0) = N/(1+(pN)^k)

with k = k(N) in (0,1), and at least for p not too close to 0. How can one prove this observation? And what can be said about k(N)? One consequence would be that at the percolation point p = 1/N, one has

E(b0) = N/2.

I was looking for examples of first order sentences written in the language of fields, true in Q (field of rational numbers) and C (field of complex numbers) but false in R (field of real numbers). I found the following recipe to construct such sentences. Let a be a statement true in C but false in R and let b be a statement true in Q but false in R. Then the statement z = a \/ b is of course true in Q and C, but false in R.

Using this method, I found the following z:=

(Ex x^2 = 2) ---> (Au Ev v^2 = u)

which formulated in english sounds as "If 2 has a square-root in the field, then all elements of the field have square roots in the field." Of course, in Q the premise is false, so the implication is true. In C both premise and conclusion are true, so the implication is true. In R, the premise is true and the conclusion false, so the implication is false. Bingo.

However, this example is just constructed and does not really contain too much mathematical enlightment. Do you know more interesting and more substantial (natural) examples? (from both logic and algebraic point of view)

Please can someone give me a reference on Pisot-Dufresnoy-Boyd Algorithm.

Key words: Pisot number

We know that infinite union of CFL is undecidable \cite{BORDIHN2004445}. But what would be a result in case of DCFL?

DEMATLE and Fuzzy ANP are two method of MCDM methods.

MCDM equal with Multi Criteria Decision Making Methods.

please help me.

best regard

Mohammad

A NSWE-path is a path consisting of North, South, East and West steps of length 1 in the plane. Define a weight w for the paths by w(N)=w(E)=1 and w(S)=w(W)=t. Define the height of a path as the y-coordinate of the endpoint. For example the path NEENWSSSEENN has length 12, height 1 and its weight is t^4.

Let B(n,k) be the weight of all non-negative NSEW-paths of length n (i.e. those which never cross the x-axis) with endpoint on height k.

With generating functions it can be shown that for each n the identity

(*) B(n,0)+(1+t)B(n,1)+…+(1+t+…+t^n)B(n,n)=(2+2t)^n

holds. The right-hand side is the weight of all paths of length n.

Is there a combinatorial proof of this identity?

For example B(2,0)=1+3t+t^2 because the non-negative paths of length 2 with height 0 are EE with weight 1, EW+WE+NS with weight 3t, and WW with weight t^2.

B(2,1)=2+2t because the non-negative paths are NE+EN with weight 2 and NW+WN with weight 2t. And B(2,2)=1 because w(NN)=1.

In this case we get the identity

B(2,0)+(1+t)B(2,1)+(1+t+t^2)B(2,2)=(2+2t)^2.

Covariance of two random variable tells how strongly they vary together, but in case of binary transmission..lets assume x1=[1 1 1 1 1] and x2=[1 1 1 1 1] the covariance between them is zero. But therotecically the are highly correlated. I find this contradictory. Is there anything i am missing out here.. Thanks?

logical systems associated with topology are some modal lgical system which is compete, now if we exclue the condiation that intersection of two open set ,may not be copen, then will the logiassociated lmodal logical system complete?

I want to run an exploratory factor analysis (EFA) on a questionnaire consisting of 9 yes/no answered questions. Is it OK to run an EFA on a scale with binary coded questions?

I know the definition of δ-homeomorphism and δg#-homeomorphism . But I need the reference of the journal in which it is published?

How would you calculate inter-coder reliability (Kappa coefficient?) for the following scenario?

10 texts coded across 6 categories (codes) by 3 coders

Mehdi

Or generally:

\sum _{i=1}^n \sum _{j=1}^i \left\lfloor \frac{-j+n+1}{2 i}\right\rfloor

I have a set of axioms. I want to prove each axiom is independent of others. I mean an axiom is not implied by a single axiom, or a combination of axioms. I think we need examples. If so how many?

Is there any simple formula for calculating the Legendre function

P

_{-n}^{-m}(x), n,m: integers, with m>=n ?In the question given above, H is the mean curvature of the immersed real projective space in the Euclidean m-space.