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1.
When is a monic polynomial the characteristic polynomial of a symmetric matrix over an integral domain D? Known necessary conditions are shown to be insufficient when D is the field of 2-adic numbers and when D is the rational integers. The latter counterexamples lead to totally real cubic extensions of the rationals whose difierents are not narrowly equivalent to squares. Furthermorex3-4x+1 is the characteristic polynomial of a rational symmetric matrix and is the characteristic polynomial of an integral symmetric p-adic matrix for every prime p, but is not the characteristic polynomial of a rational integral symmetric matrix. 相似文献
2.
Edward A. Bender 《Linear and Multilinear Algebra》1974,2(1):55-63
When is a monic polynomial the characteristic polynomial of a symmetric matrix over a field? A theorem on determinants of matrices commuting with a given matrix provides a simple necessary condition. The condition is sufficient over local number fields except for certain irreducible quartic polynomials over the 2-adic rational numbers. 相似文献
3.
M. Shahryari 《Linear algebra and its applications》2010,433(7):1410-1421
In this article, we introduce the notion of a relative symmetric polynomial with respect to a permutation group and an irreducible character and we give answers for some natural questions about their structures. In order to study symmetric polynomials with respect to linear characters, we define the concept of relative Vandermonde polynomial. Finally, we present some interesting research problems concerning relative symmetric polynomials. 相似文献
4.
V. M. Zolotarev 《Journal of Mathematical Sciences》1987,38(5):2262-2272
Let {Q(n)(x1,...,xn)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {Xn1,...,Xnn}, n k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Yn=Q(n)(Xn1, ... Xnn), n k One obtains limit theorems for the sequence Yn, under very general assumptions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 170–188, 1986. 相似文献
5.
The first section surveys recent results on the permanental polynomial of a square matrix A, i.e., per(xI – A). The second section concerns the permanental polynomial of the adjacency matrix of a graph. The final section is an introduction to the permanental polynomial of the Laplacian matrix of a graph. An appendix lists some of these latter polynomials. 相似文献
6.
Martin Kochol 《Journal of Graph Theory》2002,40(3):137-146
The tension polynomial FG(k) of a graph G, evaluating the number of nowhere‐zero ?k‐tensions in G, is the nontrivial divisor of the chromatic polynomial χG(k) of G, in that χG(k) = kc(G)FG(k), where c(G) denotes the number of components of G. We introduce the integral tension polynomial IG(k), which evaluates the number of nowhere‐zero integral tensions in G with absolute values smaller than k. We show that 2r(G)FG(k)≥IG(k)≥ (r(G)+1)FG(k), where r(G)=|V(G)|?c(G), and, for every k>1, FG(k+1)≥ FG(k)˙k / (k?1) and IG(k+1)≥IG(k)˙k/(k?1). © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 137–146, 2002 相似文献
7.
Hong Zhang 《Journal of Graph Theory》1992,16(1):1-5
The class of self-complementary symmetric graphs is characterized using the classification of finite simple group. 相似文献
8.
《Discrete Mathematics》2004,274(1-3):173-185
An orientation of a graph is acyclic (totally cyclic) if and only if it is a “positive orientation” of a nowhere-zero integral tension (flow). We unify the notions of tension and flow and introduce the so-called tension-flows so that every orientation of a graph is a positive orientation of a nowhere-zero integral tension-flow. Furthermore, we introduce an (integral) tension-flow polynomial, which generalizes the (integral) tension and (integral) flow polynomials. For every graph G, the tension-flow polynomial FG(k1,k2) on G and the Tutte polynomial TG(k1,k2) on G satisfy FG(k1,k2)⩽TG(k1−1,k2−1). We also characterize the graphs for which the inequality is sharp. 相似文献
9.
It is proved that a cyclically (k ? 1)(2n ? 1)-edge-connected edge transitive k-regular graph with even order is n-extendable, where k ≥ 3 and k ? 1 ≥ n ≥ ?(k + 1)/2?. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc. 相似文献
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Phung Van Manh 《Numerical Algorithms》2017,76(3):709-725
We study the Hermite interpolation problem on the spaces of symmetric bivariate polynomials. We show that the multipoint Berzolari-Radon sets solve the problem. We also give a Newton formula for the interpolation polynomial and use it to prove a continuity property of the interpolation polynomial with respect to the interpolation points. 相似文献
14.
Karin Erdmann 《Linear algebra and its applications》2011,434(12):2475-2496
In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type. 相似文献
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Shijun Yang 《Journal of Mathematical Analysis and Applications》2010,368(2):438-6363
In this paper, we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in Milovanovi? and Cvetkovi? [J. Math. Anal. Appl. 311 (2005) 191], thereby improving on their observations. 相似文献
17.
Italo Simonelli 《Discrete Mathematics》2008,308(11):2228-2239
Let F(n,e) be the collection of all simple graphs with n vertices and e edges, and for G∈F(n,e) let P(G;λ) be the chromatic polynomial of G. A graph G∈F(n,e) is said to be optimal if another graph H∈F(n,e) does not exist with P(H;λ)?P(G;λ) for all λ, with strict inequality holding for some λ. In this paper we derive necessary conditions for bipartite graphs to be optimal, and show that, contrarily to the case of lower bounds, one can find values of n and e for which optimal graphs are not unique. We also derive necessary conditions for bipartite graphs to have the greatest number of cycles of length 4. 相似文献
18.
Summary The asymptotic behaviour of elementary symmetric polynomials S
n
(k)
of order k, based on n independent and identically distributed random variables X
1,..., X
n,is investigated for the case that both k and n get large. If
, then the distribution function of a suitably normalised S
n
(k)
is shown to converge to a standard normal limit. The speed of this convergence to normality is of order
, provided
and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k at the rate n
1/2 then a non-normal weak limit appears, provided the X
i's are positive and S
n
(k)
is standardised appropriately. On the other hand, if k at a rate faster than n
1/2 then it is shown that for positive X
j'sthere exists no linear norming which causes S
n
(k)
to converge weakly to a nondegenerate weak limit. 相似文献
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Bryan Ek 《Journal of Difference Equations and Applications》2019,25(2):262-293
The q-binomial coefficients were conjectured to be unimodal as early as the 1850's but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an ‘elementary’ proof using linear algebra. Finally, in 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyse how these perturbations affect the final polynomial and analyse some specific cases. 相似文献