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1.
For any n-by-n matrix A , we consider the maximum number k=k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B⊕C, we show that k(A)=2 if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C and k(C)=2. For an irreducible matrix A , we can determine exactly when the value of k(A) equals the size of A . These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W(A). 相似文献
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The period annuli of the planar vector field x′=−yF(x,y), y′=xF(x,y), where the set {F(x,y)=0} consists of k different isolated points, is defined by k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1, the provided upper bound is reached. Finally, the case k=2 is also treated. 相似文献
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Bárat and the present author conjectured that, for each tree T , there exists a natural number kT such that the following holds: If G is a kT-edge-connected graph such that |E(T)| divides |E(G)|, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1). 相似文献
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This paper is devoted to a problem of finding the smallest positive integer s(m,n,k) such that (m+1) generic skew-symmetric (k+1)-forms in (n+1) variables as linear combinations of the same s(m,n,k) decomposable skew-symmetric (k+1)-forms. 相似文献
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Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A is defined to be Int(A)={f∈B[x]|f(A)⊆A}. Restricting the coefficients to elements of k , we obtain the commutative ring Intk(A)={f∈k[x]|f(A)⊆A}; this makes Int(A) a left Intk(A)-module. Previous researchers have noted instances when a D-module basis for A is also an Intk(A)-basis for Int(A). We classify all the D-algebras A with this property. Along the way, we prove results regarding Int(A), its localizations at primes of D, and finite residue rings of A. 相似文献
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In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if Ak−1 is the dominant coefficient, then every transcendental solution f(z) of equation
satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z). 相似文献
f(k)+Ak-1 f(k-1)+?+A0 f=0
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An operator convex function on (0,∞) which satisfies the symmetry condition k(x−1)=xk(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. 相似文献
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Peter Frankl Mitsuo Kato Gyula O.H. Katona Norihide Tokushige 《Journal of Combinatorial Theory, Series B》2013
Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k -cliques is always less than cknk, where ck∈(0,1) is the unique root of the equation zk=(1−z)k+kz(1−z)k−1. On the other hand, we construct a coloring in which there are at least cknk−O(nk−1) red k-cliques and at least the same number of blue k-cliques. 相似文献
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In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t -(v,k,Λ) design with t finite, v infinite and k,λ<v is resolvable and, in fact, has α orthogonal resolutions for each α<v. We also show that, while a t -(v,k,Λ) design with t and λ finite, v infinite and k=v may or may not have a resolution, any resolution of such a design must have v parallel classes containing v blocks and at most λ−1 parallel classes containing fewer than v blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v and λ=v and when k=v and λ is infinite, we give various examples of resolvable and non-resolvable t -(v,k,Λ) designs. 相似文献
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This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n,k,δ) such that (n−k)|δ. We also discuss the size of F needed so that the proposed matrices are superregular. 相似文献
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We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(Δ,k) is a period in the sense of Kontsevich and Zagier when k?12. As an illustration, we reduce L(Δ,k) to explicit integrals of hypergeometric and algebraic functions when k∈{12,13,14,15}. 相似文献
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We study the existence of weak solutions to (E) (−Δ)αu+g(u)=ν in a bounded regular domain Ω in RN(N≥2) which vanish in RN?Ω, where (−Δ)α denotes the fractional Laplacian with α∈(0,1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|k−1r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved. 相似文献
17.
László Miklós Lovász Carsten Thomassen Yezhou Wu Cun-Quan Zhang 《Journal of Combinatorial Theory, Series B》2013
The main theorem of this paper provides partial results on some major open problems in graph theory, such as Tutte?s 3-flow conjecture (from the 1970s) that every 4-edge connected graph admits a nowhere-zero 3-flow, the conjecture of Jaeger, Linial, Payan and Tarsi (1992) that every 5-edge-connected graph is Z3-connected, Jaeger?s circular flow conjecture (1984) that for every odd natural number k?3, every (2k−2)-edge-connected graph has a modulo k -orientation, etc. It was proved recently by Thomassen that, for every odd number k?3, every (2k2+k)-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G is Z3-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassen?s method is refined to prove the following: For every odd number k?3, every (3k−3)-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph is Z3-connected and admits therefore a nowhere-zero 3-flow. Note that it was proved by Kochol (2001) that it suffices to prove the 3-flow conjecture for 5-edge-connected graphs. 相似文献
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Let u(t)=−Fx(t) be the optimal control of the open-loop system x′(t)=Ax(t)+Bu(t) in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system x′(t)=(A−BF)x(t). Main attention is given to the case of a skew-Hermitian matrix A. Given an operator A, for a class of cases, we find a matrix B that provides an almost optimal decay rate. 相似文献