共查询到20条相似文献,搜索用时 15 毫秒
1.
Zhi-Hong Sun 《Discrete Mathematics》2008,308(1):71-112
Let {Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for , where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for and the sum , where h(d) is the class number of the quadratic field of discriminant d and p-regular functions are those functions f such that are rational p-integers and for n=1,2,3,… . We also establish many congruences for Euler numbers. 相似文献
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Jeremy M. Greene J. William Helton Victor Vinnikov 《Journal of Functional Analysis》2011,261(11):3390-3417
We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1,x2,…,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y)
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Xiongping Dai 《Journal of Differential Equations》2011,250(9):3584-3629
Let {A1,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R+ satisfying tj−tj−1?Δ and
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Mark S. MacLean 《Discrete Mathematics》2008,308(7):1230-1259
We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers bi,ci, distance matrices Ai, and eigenvalues θ0>θ1>?>θD. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of MatX(C) generated by , where A=A1 and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where Ei denotes the primitive idempotent of A associated with θi and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W. 相似文献
7.
Rostislav Caha 《Discrete Mathematics》2007,307(16):2053-2066
A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π1(n) such that for every hypercube Q with dim(Q)?π1(n) there exists a family of pairwise vertex-disjoint paths Pi between Ai and Bi for i=1,2,…,n with if and only if {Ai,Bi∣i=1,2,…,n} is a balanced set. 相似文献
8.
Noga Alon Michal Feldman Ariel D. Procaccia Moshe Tennenholtz 《Discrete Mathematics》2010,310(23):3432-3435
Consider the unit circle S1 with distance function d measured along the circle. We show that for every selection of 2n points x1,…,xn,y1,…,yn∈S1 there exists i∈{1,…,n} such that . We also discuss a game theoretic interpretation of this result. 相似文献
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An overlarge set of , denoted by , is a collection {(X?{x},Bx):x∈X}, where X is a (v+1)-set, each (X?{x},Bx) is a and {Bx:x∈X} forms a partition of all triples on X. In this paper, we give a tripling construction for overlarge sets of KTS. Our main result is that: If there exists an with a special property, then there exists an . It is obtained that there exists an for u=22n−1−1 or u=qn, where prime power q≡7 (mod 12) and m≥0,n≥1. 相似文献
10.
Let G be a vertex-disjoint union of directed cycles in the complete directed graph Dt, let |E(G)| be the number of directed edges of G and suppose or if t=5, and if t=6. It is proved in this paper that for each positive integer t, there exist -decompositions for Dt−G if and only if . 相似文献
11.
We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ1,θ2,…,θN−1} is an N-spectrum for A(x) if the θj are all distinct and
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Let Ω⊂{0,1}N be a nonempty closed set with N={0,1,2,…}. For N={N0<N1<N2<?}⊂N and ω∈{0,1}N, define ω[N]∈{0,1}N by and
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We consider isometric embedding of trees into the infinite graph Zm whose vertices are the m-dimensional lattice points where two vertices a=(a1,a2,…,am) and b=(b1,b2,…,bm) are adjacent if and only if |ai-bi|?1 for 1?i?m. Linial, London, and Rabinovich have shown that this can be done with , where t is the number of leaves. In this note, we sketch a proof that . 相似文献
14.
Dejan Brcanov 《Discrete Mathematics》2010,310(19):2550-2554
Let and be two n-tuples of nonnegative integers. An all-4-kings n-partite tournament T(V1,V2,…Vn) is said to have a -property if there exists an n-partite tournament T1(W1,W2,…,Wn) such that for each i∈{1,…,n}:
- (1)
- Vi⊆Wi;
- (2)
- exactly ti 4-kings of Vi are not 4-kings in T1;
- (3)
- exactly ci 4-kings of Wi are not vertices of Vi.
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Yasuo Teranishi 《Discrete Mathematics》2002,257(1):183-189
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Let {f0,…,fn;g0,…,gn} be a sequence of homogeneous polynomials in 2n+2 variables with no common zeros in P2n+1 and suppose that the degrees of the polynomials are such that is a homogeneous polynomial. We shall refer to the hypersurface X defined by Q as a generalized quadric. In this note, we prove that generalized quadrics in for n≥1 are reduced. 相似文献
19.
Sanguo Zhu 《Journal of Mathematical Analysis and Applications》2008,344(1):583-591
Let be a family of similitudes on R1 satisfying the strong separation condition and ν the self-similar measure associated with and a probability vector (t1,…,tN). Let μ be the attracting measure of a condensation system associated with ν, and a probability vector (p0,p1,…,pN). We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets. 相似文献
20.
Wang Qin 《Discrete Mathematics》2005,294(3):303-309
A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to K1,3. The kth power of a graph G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are , and Tr, r?2, where Tr is the graph with 4r vertices ui,vi,xi,yi, 1?i?r, such that for each i with 1?i?r, {ui,vi,xi,yi} is a clique of Tr and . We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K4×K2, and . 相似文献