The Experts below are selected from a list of 102 Experts worldwide ranked by ideXlab platform
K. B. Reid - One of the best experts on this subject based on the ideXlab platform.
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Total relative displacement of vertex permutations of K
Journal of Graph Theory, 2002Co-Authors: K. B. ReidAbstract:Let α denote a permutation of the n vertices of a connected graph G. Define δα(G) to be the number , where the sum is over all the unordered pairs of distinct vertices of G. The number δα(G) is called the total relative displacement of α (in G). So, permutation α is an automorphism of G if and only if δα(G) = 0. Let π(G) denote the smallest positive value of δα(G) among the n! permutations α of the vertices of G. A permutation α for which π(G) = δα(G) has been called a near-automorphism of G [2]. We determine π(K) and describe permutations α of K for which π(K) = δα(K). This is done by transforming the problem into the combinatorial optimization problem of maximizing the sums of the squares of the entries in certain t by t matrices wIth non–negative integer entries in which the sum of the entries in the Ith row and the sum of the entries in the Ith Column each equal to ni,1≤i≤t. We prove that for positive integers, n1≤n2≤…≤nt, where t≥2 and nt≥2, where k0 is the smallest index for which n = n+1. As a special case, we correct the value of π(Km,n), for all m and n at least 2, given by Chartrand, Gavlas, and VanderJagt [2]. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 85–100, 2002
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Total relative displacement of vertex permutations of K n 1 , n 2 , …, n t
Journal of Graph Theory, 2002Co-Authors: K. B. ReidAbstract:Let α denote a permutation of the n vertices of a connected graph G. Define δα(G) to be the number $\sum |d(x,y)-d(\alpha (x),\alpha(y))|$, where the sum is over all the $\left({n \atop 2} \right)$ unordered pairs of distinct vertices of G. The number δα(G) is called the total relative displacement of α (in G). So, permutation α is an automorphism of G if and only if δα(G) = 0. Let π(G) denote the smallest positive value of δα(G) among the ne permutations α of the vertices of G. A permutation α for which π(G) = δα(G) has been called a near-automorphism of G [2]. We determine π(Kn1,n2,…,nt) and describe permutations α of Kn1,n2,…,nt for which π(Kn1,n2,…,nt) = δα(Kn1,n2,…,nt). This is done by transforming the problem into the combinatorial optimization problem of maximizing the sums of the squares of the entries in certain t by t matrices wIth non–negative integer entries in which the sum of the entries in the Ith row and the sum of the entries in the Ith Column each equal to ni,1≤i≤t. We prove that for positive integers, n1≤n2≤…≤nt, where t≥2 and nt≥2,$\pi {(K_{n_1 ,{n}_2 , \ldots ,{n}_t} }) = \left \{\matrix{2{ n}_{ h + 1} - 2\hfill \quad\quad{\rm if} \,\,\, 1 = { n}_1 = { n}_2 = \cdots = {n_h} < { n}_{{ h} + 1}\hfill \cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\le \cdots \le { n}_{t} , {\rm and}\ {t}\ge(h + 1),\hfill\cr \quad\quad\quad\quad\quad\quad{\rm for\ some} \ { h}\ge 2,\hfil \cr 2{ n}_{{ k}_0}\hfill \quad\quad\quad\quad\quad\quad\quad{\rm if} \,\,\, 1 = {n_1} < { n}_2\ {\rm or}\ { n}_1 \ge 2, {n}_{{ k} + 1} = {n}_{ k} + 1,\hfill \cr \;\quad\quad\quad\quad\quad{\rm\ for\ some}\ k, 1\le { k} \le { t} - 1,\ \cr \;\quad\quad\quad\quad{\rm and}\ 2+ { n}_{{ k}_0 } \le { n}_1 + { n}_2 , \cr 2({ n}_1 + { n}_2 - 2)\hfill {\rm otherwise},\hfill\hfill\hfill\hfill\hfill\cr } \right.$ where k0 is the smallest index for which nk0+1 = nk0+1. As a special case, we correct the value of π(Km,n), for all m and n at least 2, given by Chartrand, Gavlas, and VanderJagt [2]. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 85–100, 2002
B. V. Raghavendra Rao - One of the best experts on this subject based on the ideXlab platform.
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Linear Projections of the Vandermonde Polynomial
Theoretical Computer Science, 2019Co-Authors: C. Ramya, B. V. Raghavendra RaoAbstract:Abstract An n-variate Vandermonde polynomial is the determinant of the n × n matrix where the Ith Column is the vector ( 1 , x i , x i 2 , … , x i n − 1 ) T . Vandermonde polynomials play a crucial role in the theory of alternating polynomials and are useful in Lagrangian polynomial interpolation which arise in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. First, we consider the problem of testing if a given polynomial is equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorIthm to test if f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when f is given as a black-box our algorIthm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and obtain a basis for the associated Lie algebra. Finally, we study arIthmetic circuits built over projections of Vandermonde polynomials. We show universality property for some of the models and obtain a lower bounds against sum of projections of Vandermonde determinant.
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Linear Projections of the Vandermonde Polynomial
arXiv: Computational Complexity, 2017Co-Authors: C. Ramya, B. V. Raghavendra RaoAbstract:An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the Ith Column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation as well as in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. Firstly, we consider the problem of testing if a given polynomial is linearly equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorIthm to test if the polynomial f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when the polynomial f is given as a black-box our algorIthm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and show that the associated Lie algebra is simple.
C. Ramya - One of the best experts on this subject based on the ideXlab platform.
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Linear Projections of the Vandermonde Polynomial
Theoretical Computer Science, 2019Co-Authors: C. Ramya, B. V. Raghavendra RaoAbstract:Abstract An n-variate Vandermonde polynomial is the determinant of the n × n matrix where the Ith Column is the vector ( 1 , x i , x i 2 , … , x i n − 1 ) T . Vandermonde polynomials play a crucial role in the theory of alternating polynomials and are useful in Lagrangian polynomial interpolation which arise in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. First, we consider the problem of testing if a given polynomial is equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorIthm to test if f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when f is given as a black-box our algorIthm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and obtain a basis for the associated Lie algebra. Finally, we study arIthmetic circuits built over projections of Vandermonde polynomials. We show universality property for some of the models and obtain a lower bounds against sum of projections of Vandermonde determinant.
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Linear Projections of the Vandermonde Polynomial
arXiv: Computational Complexity, 2017Co-Authors: C. Ramya, B. V. Raghavendra RaoAbstract:An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the Ith Column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation as well as in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. Firstly, we consider the problem of testing if a given polynomial is linearly equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorIthm to test if the polynomial f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when the polynomial f is given as a black-box our algorIthm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and show that the associated Lie algebra is simple.
Zoli Claudio - One of the best experts on this subject based on the ideXlab platform.
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Feasible shared destiny risk distributions
'Springer Science and Business Media LLC', 2019Co-Authors: Zoli Claudio, Gajdos Thibault, Weymark JohnAbstract:Social risk equity is concerned wIth the comparative evaluation of social risk distributions, which are probability distributions over the potential sets of fatalities. In the approach to the evaluation of social risk equity introduced by Gajdos, Weymark, and Zoli (Shared destinies and the measurement of social risk equity, Annals of Operations Research 176:409\u2013424, 2010), the only information about such a distribution that is used in the evaluation is that contained in a shared destiny risk matrix whose entry in the kth row and Ith Column is the probability that person i dies in a group containing k individuals. Such a matrix is admissible if it satisfies a set of restrictions implied by its definition. It is feasible if it can be generated by a social risk distribution. It is shown that admissibility is equivalent to feasibility. Admissibility is much easier to directly verify than feasibility, so this result provides a simple way to identify which matrices to consider when the objective is to socially rank the feasible shared destiny risk matrices
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Feasible Shared Destiny Risk Distributions
Vanderbilt University Department of Economics, 2018Co-Authors: Gajdos Thibault, Weymark John, Zoli ClaudioAbstract:Social risk equity is concerned wIth the comparative evaluation of social risk distributions, which are probability distributions over the potential sets of fatalities. In the approach to the evaluation of social risk equity introduced by Gajdos, Weymark, and Zoli (Shared destinies and the measurement of social risk equity, Annals of Operations Research 176:409-424, 2010), the only information about such a distribution that is used in the evaluation is that contained in a shared destiny risk matrix whose entry in the kth row and Ith Column is the probability that person i dies in a group containing k individuals. Such a matrix is admissible if it satisfies a set of restrictions implied by its definition. It is feasible if it can be generated by a social risk distribution. It is shown that admissibility is equivalent to feasibility. Admissibility is much easier to directly verify than feasibility, so this result provides a simply way to identify which matrices to consider when the objective is to socially rank the feasible shared destiny risk matrices
Weymark John - One of the best experts on this subject based on the ideXlab platform.
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Feasible shared destiny risk distributions
'Springer Science and Business Media LLC', 2019Co-Authors: Zoli Claudio, Gajdos Thibault, Weymark JohnAbstract:Social risk equity is concerned wIth the comparative evaluation of social risk distributions, which are probability distributions over the potential sets of fatalities. In the approach to the evaluation of social risk equity introduced by Gajdos, Weymark, and Zoli (Shared destinies and the measurement of social risk equity, Annals of Operations Research 176:409\u2013424, 2010), the only information about such a distribution that is used in the evaluation is that contained in a shared destiny risk matrix whose entry in the kth row and Ith Column is the probability that person i dies in a group containing k individuals. Such a matrix is admissible if it satisfies a set of restrictions implied by its definition. It is feasible if it can be generated by a social risk distribution. It is shown that admissibility is equivalent to feasibility. Admissibility is much easier to directly verify than feasibility, so this result provides a simple way to identify which matrices to consider when the objective is to socially rank the feasible shared destiny risk matrices
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Feasible Shared Destiny Risk Distributions
Vanderbilt University Department of Economics, 2018Co-Authors: Gajdos Thibault, Weymark John, Zoli ClaudioAbstract:Social risk equity is concerned wIth the comparative evaluation of social risk distributions, which are probability distributions over the potential sets of fatalities. In the approach to the evaluation of social risk equity introduced by Gajdos, Weymark, and Zoli (Shared destinies and the measurement of social risk equity, Annals of Operations Research 176:409-424, 2010), the only information about such a distribution that is used in the evaluation is that contained in a shared destiny risk matrix whose entry in the kth row and Ith Column is the probability that person i dies in a group containing k individuals. Such a matrix is admissible if it satisfies a set of restrictions implied by its definition. It is feasible if it can be generated by a social risk distribution. It is shown that admissibility is equivalent to feasibility. Admissibility is much easier to directly verify than feasibility, so this result provides a simply way to identify which matrices to consider when the objective is to socially rank the feasible shared destiny risk matrices
