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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
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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?
It is just easy to examine such difference via googling., e.g., https://www.sciencedirect.com/topics/physics-and-astronomy/crystal-lattices
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If possible discuss the space, basis, dimension in the context of discreate mathematics and machine learning.
see
A SURVEY OF THE DIFFERENT TYPES OF VECTOR SPACE PARTITIONS
HEDEN, OLOF
Journal:
Discrete Mathematics Algorithms and Applications
Year:
2012
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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
It seems that a correct proof for this question has been announced at arxiv.org/abs/2112.09960v1.
Qi’s conjecture on logarithmically complete monotonicity of the reciprocal of the inverse tangent function
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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)
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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.
Dear Alireza Akbari looks like your equation is a nonlinear PDE, there are tables for those:
However I could not find yours, but don't worry, I tell you a trick we use in MHD.
1. You linearized it, i.e., you solve the PDE as a function of ei(k.r - omega t)
2. You get a complex polinom, but I don't see any parameters in your equation.
3. Anyway you can try an algebraic manipulator such as math or maple and find the roots. However, I find it strange that there is not a parameter, you need it to scan the complex solution.
Best Regards.
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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.
Books are known to have errata.
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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?
Genetic algorithms never find an optimum. I hope you will use better tools than those. You should know that this forum has basically been hi-jacked by meta-heuristics enthusiasts - an inflated array of posts at RG are based on the fact that RG "scholars" do not know that there are better tools. So beware!
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Which is best book for discrete mathematical structures, for undergraduate teaching?
Discrete Mathematical Structures, Bernard Kolman, Robert C. Busby, Sharon Ross
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Hi,
I'm interested in solving a nonconvex optimization problem that contains continuous variables and categorical variables (e.g. materials) available from a catalog.
- metaheuristics: random trial and error ;
Are you aware of other systematic approaches?
Thank you,
Charlie
Z. Nedelkov\'a, C.\ Cromvik, P.\ Lindroth, M.\ Patriksson, \and A.-B.\ Str\"omberg}, {\em A splitting algorithm for simulation-based optimization problems with categorical variables}, {\sf Engineering Optimization}, vol.~51 (2019), pp.~815--831.
It might help!
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Can anybody tell me what the current situation in the hierarchical product of graphs is? Are there any articles on this?
Hello. If you are still interested in the hierarchical product, there has recently been some work on how the domination number of the product graph is related to the domination number of the factor graphs. This is related to an old open conjecture of Vizing. Here's a link to the paper on Science Direct:
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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?
Yes dear Dr Roudy
I just have seem them.
Really good.
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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
REFERENCES:
Buckley, F. and Harary, F. Distance in Graphs. Redwood City, CA: Addison-Wesley, 1990.
Diestel, R. Graph Theory, 3rd ed. New York: Springer-Verlag, p. 8, 1997.
Wilson, R. J. Introduction to Graph Theory, 3rd ed. New York: Longman, p. 30, 1985.
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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.
Thank, Luis Daniel Torres Gonzalez , you for your contribution. My question does not ask the probability distribution. It asks what is the most probable "pattern" when we distribute N-items among M-boxes. I have illustrated the meaning of "pattern" by examples, but it seems it was not sufficient. Please read Romeo Meštrović 's comments above posted in March, 2019.
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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.
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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.
Drawing in MATLAB is simple, see e.g., https://au.mathworks.com/help/matlab/ref/graph.plot.html
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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.
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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)?
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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?
Dear Vishnu,
in which format is the data, which you would like to interpolate, available: NetCDF, ASCII-Text, Excel, ... ?
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Hello everyone, could some people suggest a good syllabus for graph theory and discrete mathematics for Computer science - Network department, please.
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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 ?
You can link semiring with graph theory by giving weights to vertices and edges of a graph G by the elements from a given semiring. Such a study have been initiated in the year 2015 by me with my scholars. The theory developed is called $S$ valued (Semiring valued) graphs. Lot of work is going on in this area, which you can use to study for decision making problems by using shortest path problem or network analysis.
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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?
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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?
Peter Breuer Thank you Dr. Peter for your answer. May you please let us know how may we replace the integer x with a string message?
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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.
Quite true. There is a danger that rationalization of beauty can lead to its destruction. Explaining a joke just makes it not funny. However, curiosity, desire for knowledge, seems to be stronger.
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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.
To get your request you need the following conditions to be satisfied:
with all coefficients a,b,c,d and e are negative values.
The proof ( Using the resultant techniques)
Example:
f(x) = -(x²+3x+1)² = -x⁴ - 6x³ -11x² - 6x -1
a = -1, b = -6, c = -11, d = -6, e = -1
Check the conditions:
(3(-6)²-8(-1)(-11)/(-6)(-6)-16(-1)(-1))= 1
((-6)(-11)-6(-1)(-6))/(-11)(-6)-6(-6)(-1))= 1
((-6)(-6)-16(-1)(-1))/(3(-6)²-8(-11)(-1))= 1
Best regards
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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
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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.
Yes, the graph is prime. It is can be shown easily. There is a conjecture due to Seoud and Youssef in 1999 (in Congrussus Numerantium ) states that every unicyclic graph is prime
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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.
If any one property of Barnatte graph is dropped it is non hamiltonion
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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.
Kind regards,
Ebrahim
Check the power index in the denominator of your expression for the curvature. It should be 3/2 rather than 1/2. Good luck
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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 .
Respected Dr.Xinwei Wang, first of all thank you for suggesting me the paper. It has cleared certain doubts of mine but my initial question about what is meant by smoothness of trajectory generated by joining discrete points still remains unanswered. Is there a way by which the smoothness of the trajectory generated by joining the values of state or control at the Legendre Gauss Lobatto(lgl) points can be quantified? More specifically is there a measure of the same?
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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
I decided to write a short piece, based on the questions and reasoning I saw here. I mean no offense of course, but I think this is an important point: https://medium.com/@dgoldman0/bad-science-and-modeling-data-c2c13c305c8e
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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
DEA is generally applied to assess the relative performance a set of decision making units (DMUs) that consume inputs to produce outputs under a similar production technology. This may be valid whenever the systems under consideration fit within such a structure. In graph related problems such as scheduling, routing, etc., the objectives are completely different. Although there is a flow of material over the network, which may suggest that a node can be assimilated to a DMU, here, we are more concerned with finding an optimal route that satisfies constraints that may be as complex as the practical problem under study. In large scale problems, one may think of DEA for building clusters of nodes so that to reduce the problem size and, hence, related computational cost. I think that this aspect is worth to be investigated.
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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!
Yes! Even standard mathematical symbols and notations are captured in similarity index. The habit of using unconventional symbols and notations just to reduce similarity index is destroying the beauty and taste of mathematics.
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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!
Try, the book, Heat conduction using Green's Functions. J.V. Beck. I guess do you find it there.
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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?
A set D is minimal dominating set of a graph if for any vertex v in D, the set D - v is not a dominating set of the graph, hence not every minimal dominating set is minimum but every minimum dominating set is minimal.
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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?
Thanks Vincent, a hybrid the proposed methods might be better, but there is only one way to find out, Experiments!
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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
Yes.
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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?
A comprehensive study on this topic is Dunford Schwartz, Part I, p. 489-511, see also L. Meziani, Acta Math. Univ. Comenianae 74 (2005), 59-70, available at
[access May 2018]
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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
Hi Samira,
Referring to the Examples 9.3 and 9.4 in Prof. Lewis' book (Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, 2e), the attached MATLAB example (m-file) shows how to simulate a stochastic control system.
Hope this helps!
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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?
Use Structural Equation Modeling (SEM).
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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?
Hi Jesujoba,
AFAIK, this should be accomplished based on previous knowledge (a.k.a background) such as ranking the content using certain aspect; otherwise, there is no meaning or logic behind such process.
HTH.
Samer Sarsam, PhD.
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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?
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what is the origin and significance of writing imaginary term exp i(kx-wt) in the wave equation ?
I think that question is not fully correct... Immagine the simple wave equation
df/dt + u df/dx = 0 (u=constant)
in a periodic domain with some initial function f0(x).
Now, you can see that, according to a Fourier series, a real solution f(x,t) can be writtes as a sum of products of two complex Fk(t)*exp(i*k*x). By substituting it in the equation, you can easily see that Fk(t)=Fk(0)*exp(-i*u*t).
The key is that using the complex nomenclature is nothing but a way to rewrite a real function (consider the conjugate part in the series).
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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
Dear Aryan Khan,
if I understand well, you assume (1),(2),(3) and you are asking, which of (4), (5) characterizes the set IF.
For an answer, I need to know whether both (I and F) are necessary, or only one of them suffices.
As for (1),(2),(3) I have following questions:
1. Is D the basic set, i.e. does D contain all sensitive together with non-sensitive terms?
2. does every element of A belong to D? Using "set notations", I am asking: is A⊆D?
3. similar question for B; do you have B⊆D?
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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.
Dear Rahul,
Many decision tree algorithms rely on mutual information criteria to build a decision tree.
Mutual Information serves to measure how information can one variable tell you about another one.
NOTE that, if 'X' is helpful in predicting 'Y' (based on some information criterion) we cannot assume that 'Y' is also helpful in predicting 'X'.
in these cases, the principal of correlation (or any linear dependency) does not apply.
Unlike correlation coefficients, such as the product moment correlation coefficient, mutual information contains information about all dependence—linear and nonlinear—and not just linear dependence as the correlation coefficient measures.
hope this helps
Best.
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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.
Depending on what you mean by discrete mathematics, it may also
include very classical topics which do have a high status in math. Number theory for me definitely belongs to discrete mathematics. Paul Erdos is the discrete mathematician par excellence, and his role is widely acknowledged in mathematics in general.
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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?
It is called: The twin prime conjecture. You can see this book:
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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.
Maximize the entropy, the information gain
for every question.
Regards,
Joachim
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Specifically, i have an interest in studying idempotent semiring with respect to linearly dependent and linearly independent set of vectors.
This is an important question.
An RG project directly related to this question is
See the full project description.
A research area directly related to vector spaces over semirings is the study of vector spaces over supertropical semirings.   See, e.g.:
SUPERTROPICAL LINEAR ALGEBRA
See Section 2.3, starting on page 5 for the details.
More to the point in the construction of vector spaces over semirings, see
Which semifields are exact?
See Section 2, starting on page 3, for a thorough introduction.
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How can I find out the dependency of two variables from one or multiple equations.
Example:
Set 1 -
x1,a
f(x1,a)=z1
Set 2 -
x2,a
f(x2,a)=y2
f(y2)=z2
How highly ( zand x1 ) and ( z2 and x2 ) 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 ?
Hi, z1 depends on x1 and a. On the other side, z2 depends on y2 thus it depends on x2 and a. If x1 and x2 are independent variables, the only link between z1 and z2 is through a.
To obtain x1 from z1 you have to set the value of a and solve numerically the equation z1=f(x1,a) unless f is sufficiently simple (linear, quadratic, ...) to be inverted analytically.
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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.
Yes the graphs can have parallel edges that both start and end at the same vertices.
By 'chain' I mean an addition chain. That's just a list of integers with the typical properties of an addition chain. We try to prove an addition chain exists of a specific length and if that fails we try a larger length.
By 'certain subgraphs' I mean that certain portions of the graph can't contain certain things expressible as a subgraph of the original graph. For example you may not have 4 parallel edges but you can have 3 or 2 and of course 1 (which has no parallel edges). There are other excluded subgraphs as well. The subgraphs excluded are directed as well and edge direction must match.
Yes by limits there are limits on the number of edges that may have the same source vertex. There are more complicated limits like a source vertex can't have more than one distinct set of parallel edges emanating from it. So for example source vertex A could have two or three parallel edges to vertex B but that would rule out anything but a single edge from A to vertex C.
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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.
Well I think I got your point now.
The notation is the notation we use all the time so I assumed everyone would understand it. The text is a translation from a handwritten German text, so please excuse some mistakes.
I still don't get why I cannot use a real norm. I use simple vectors. Wouldn't it be natural to use some kind of norm? And I do not get why a metric is necessary. (Although the presence of a metric is induced by the norm.)
Thank you for being so kind.
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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".
@Nazanin "A graph peoperty is monoton if every subgraph of a graph with property P, also has the property P"  is called "hereditary property".
As for the original (part of a) sentence: if we join it to another domain of mathematics, it makes sense such as it is, despite imprecise wording: e.g. in a class of exponential functions with a basis a>1...
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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?
Here you have another counterexample, is this what you mean?
I don't understand your question about being $\chi$-colorable. Usually, $\chi(G)$ stands for the chromatic number of G... In the case of the graph I sent, it's 2-colorable.
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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?
Hi, perhaps another solution is to solve two linear systems of the form A.x=b where b is set to (1,0,0...0) and (0,1,0,0....0) respectively, just to obtain the first two columns of the inverse of A.
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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)
Sorry I am not the person you are looking for.
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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?
Dear Mohammed, thank´s for replying!. Sorry if I did not explained properly.
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?
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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 X0, X1 included in O and the Y0 , Y1 included in A, for any pair of concepts (X0,Y0) and (X1,Y1) we have X0=X1 iff Y0=Y1. 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 2sqrt(|O|.|A|). And I really dont understand why... Could someone explain me where is my mistake ?
Your upper bound min(2|O| , 2|A|) is correct for the reason you indicate: "there can not be more concepts than the number of object sets neither than the number of attribute sets." This upper bound is achieved when O = A and every element of O is related by the incidence relation to every other element except itself. In this case, (S, O\S) is a concept for every subset S of O and there are no other concepts.
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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 the centres of the cells you could look at the cumulative distribution of the distances between all pairs of cells.
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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)?
Yes Sir, that is the trivial way of doing this.  Thanks
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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.
See http://snap.stanford.edu/class/cs224w-readings/erdos59random.pdf or http://www.leonidzhukov.net/hse/2016/sna/papers/erdos-1960-10.pdf for a detailed asymptotic probabilistic description of the number of connected components in a certain regime of p's.
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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)
Something algebraic, implicitly talking about ordering:
"for every nonzero number x, x or -x is a square but not both."
This holds in R (it is essentially an axiom of real closed fields) but not in Q or C (x=2 is a counterexample for both). Now you can take the logical negation.
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Please can someone give me a reference on Pisot-Dufresnoy-Boyd Algorithm.
Key words: Pisot number
Dear Hanifa
This paper presents two algorithms on certain computations
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We know that infinite union of CFL is undecidable \cite{BORDIHN2004445}. But what would be a result in case of DCFL?
Dear Jantzen, it is clear that that DCFL is not closed under countably infinite union. Now what about pow(L) = {x^k: x \in L and k \ge 0}, if L is DCFL. What would be the condition on L such that pow(L) is DCFL too?
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DEMATLE and Fuzzy ANP  are two method of MCDM methods.
MCDM equal with Multi Criteria Decision Making Methods.
best regard
Matlab
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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.
In the meantime I have found a combinatorial proof in the literature: Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group,  J. Integer Sequences   8 (2005), Article 05.3.7, proof of identity 1. https://www.researchgate.net/profile/Asamoah_Nkwanta/publications?sorting=newest
Instead of NSWE-paths they use bicolored Motzkin paths, but their proof can easily be translated to the situation of NSWE-paths. So my question has been answered.
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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?
Thanks alot Franklin...This is what i am looking for...you are right on point..But the problem is matlab give zero co-variance value for my example and if we calculate the pearson correlation coefficient it will also be 0. apply matlab function corrcoef(x1,x2) the answer is 0.
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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?
dear subrata this is corresponding to csazar generalized topology of course u can check it ZF or AC system
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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?
Item response theory (IRT) is designed for this. It is used. for example, in educational testing with tests that are binary. It is also called latent trait modeling. Bartholomew et al.'s book (http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470971924.html) describes the relationship between EFA and IRT.
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I know the definition of δ-homeomorphism and δg#-homeomorphism . But I need the reference of the journal in which it is published?

I checked the paper that there exists definitions of -homeomorphism and δg#-homeomorphism.
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How would you calculate inter-coder reliability (Kappa coefficient?) for the following scenario?
10 texts coded across 6 categories (codes) by 3 coders
Mehdi
Dear Prof. Riazi
I am no expert in statistics but I think we can use Krippendorff's alpha coefficient for this case.
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Or generally:
\sum _{i=1}^n \sum _{j=1}^i \left\lfloor \frac{-j+n+1}{2 i}\right\rfloor
I'm assuming from your question that you're looking for a closed form solution.
Let q = Floor[n/2i] and r = n - q*2i (quotient and remainder when n is divided by 2i). Therefore, 0 <= r < 2i.  Since i <= n,
sum_{j=1}^i Floor[(n-j+1)/2i] = iq - d,
where d=0 if i-1 <= r < 2i, and d=i-1-r if 0 <= r < i-1.
This gives a closed form for the inner summation.  Obtaining a closed form over both summations looks like a harder problem, but maybe possible.
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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?
I like the first answer by Peter Breuer. It was also the method used by Alfred Tarski to prove that the 7-axiom theory Q of arithmetic is built of independent axioms. He constructed 7 structures such that in any of them exactly one of the 7 axioms was false and the remaining 6 axioms were true. See "Undecidable theories" by Mostowski, R. M. Robinson and Tarski.
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Is there any simple formula for calculating the Legendre function
P-n-m(x), n,m: integers, with m>=n ?
Frithjof,
I don't know what you consider simple, but I've done what your asking for in the past.  It was harder than I initially thought.  There are many recursion formula available, but they can't just be applied without some thought up front.  Otherwise, you will encounter divisions by zero or other unpleasant things.
First, I believe that, because of your use of minus signs, you're intending that m and n be non-negative integers.
P_{-n}^{-m}(x) = P_{n-1}^{-m} (x) for n=1,2,3,...
Using this, the problem reduces to computing P_n^{-m}(x) for n=0,1,2,...
I will address the case where m can be any non-negative integer (not restricting it to m >= n).  The attached is an excerpt from a document I wrote a while back that will work. Just ignore anything mentioning derivatives. Also, reference [3] is
I hope that helps.
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