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1.
Necessary conditions for existence of a ( v, k,λ) perfect Mendelsohn design (or PMD) are v k and λ v( v − 1) ≡ 0 mod k. When k = 7, this condition is satisfied if v ≡ 0 or 1 mod 7 and v 7 when λ 0 mod 7 and for all v 7 when λ ≡ 0 mod 7. Bennett, Yin and Zhu have investigated the existence problem for k = 7, λ = 1 and λ even; here we provide several improvements on their results and also investigate the situation for λ odd. We reduce the total number of unknown ( v,7,λ)-PMDs to 36,31 for λ = 1 and 5 for λ > 1. In particular, v = 294 is the largest unknown case for λ = 1, and the only unknown cases for λ > 1 are for v = 42, λ [2,3,5,9] and v = 18, λ = 7. 相似文献
2.
Let Rbe a principal ideal ring Rn the ring of n× nmatrices over R, and dk( A) the kth determinantal divisor of Afor 1 ≤ k≤ n, where Ais any element of Rn, It is shown that if A, Bε Rn, det( A) det( B:) ≠ 0, then dk( AB) ≡ 0 mod dk( A) dk( B). If in addition (det( A), det( B)) = 1, then it is also shown that dk( AB) = dk( A) dk( B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants. 相似文献
3.
Let A, B denote the companion matrices of the polynomials xm, xn over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established. 相似文献
4.
We form squares from the product of integers in a short interval [ n, n + tn], where we include n in the product. If p is prime, p| n, and ( 2p) > n, we prove that p is the minimum tn. If no such prime exists, we prove tn √5 n when n> 32. If n = p(2 p − 1) and both p and 2 p − 1 are primes, then tn = 3 p> 3 √ n/2. For n( n + u) a square > n2, we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g2( n) be minimal such that a square can be formed as the product of distinct integers from [ n, g2( n)] so that no pair of consecutive integers is omitted. We prove that g2( n) 3 n − 3, and list or conjecture the values of g2( n) for all n. We describe the generalization to kth powers and conjecture the values for large n. 相似文献
5.
We give bijective proofs, using weighted lattice paths, of two multinomial identities concerning the generalized h-factorial polynomials of order n. [x]nh:=x(x + h)(x +2h)(x + (n − 1)h) . The first-one is the multinomial identity of order s verified by these polynomials. Using this identity (and its proof) as a lemma, we derive the main identity that generalizes previous results of Carlitz (1977), Egorychev (1974) and the classical identity of Banach. 相似文献
6.
For a 1-dependent stationary sequence { Xn} we first show that if u satisfies p1= p1( u)= P( X1> u)0.025 and n>3 is such that 88 np131, then P{max(X1,…,Xn)u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3, where ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1−p2+p3−p4+2p12+3p22−5p1p2)−1 with pk=pk(u)=P{min(X1,…,Xk)>u}, k1 and From this result we deduce, for a stationary T-dependent process with a.s. continuous path { Ys}, a similar, in terms of P{max 0skTYs< u}, k=1,2 formula for P{max 0stYsu}, t>3 T and apply this formula to the process Ys= W( s+1)− W( s), s0, where { W( s)} is the Wiener process. We then obtain numerical estimations of the above probabilities. 相似文献
8.
Given a graph with n nodes and minimum degree δ, we give a polynomial time algorithm that constructs a partition of the nodes of the graph into two sets X and Y such that the sum of the minimum degrees in X and in Y is at least δ and the cardinalities of X and Y differ by at most δ(δ + 1 if n ≠ δ(mod 2)). The existence of such a partition was shown by Sheehan (1988). 相似文献
9.
The thermal equilibrium state of two oppositely charged gases confined to a bounded domain
, m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy ( p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {( p, n) ε L2(Ω) x L 2(Ω) : p, n ≥ 0, Ωp = P, Ωn = N} and that p, n ε C(Ω) ∩ L∞(Ω). The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe Ve and P, N for the variational problem. 相似文献
10.
Let A be a complex n× n matrix. p an equilibrated vectonal norm and x( A) the spectrial abscissa of A. Then, it is known [5] x( A)≤ x(γ p( A)) where γ p is the matricial logarithmic derivative induced by p. We will make use of the above inequality to obtain regions in the plane which contain the zeros of complex polynomials. 相似文献
11.
We present some identities and congruences for the general partition function p r ( n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function. 相似文献
12.
Let { pk} k≥3 be a sequence of nonnegative integers which satisfies 8 + Σ k≥3 ( k-4) pk = 0 and p4 ≥ p3. Then there is a convex 4-valent polytope P in E3 such that P has exactly pk k-gons as faces. The inequality p4 ≥ p3 is the best possible in the sense that for c < 1 there exist sequences that are not 4-realizable that satisfy both 8 + Σ k ≥3 ( k - 4) pk = 0 and p4 > cp3. When Σ k ≥ 5 pk ≠ 1, one can make the stronger statement that the sequence { pk} is 4-reliazable if it satisfies 8 + Σ k ≥ 3 ( k - 4) pk = 0 and p4 ≥ 2Σ k ≥ 5 pk + max{ k ¦ pk ≠ 0}. 相似文献
13.
We consider the following model Hr( n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = ( r − 1) n. Each r-subset of V × ( r−1U) is chosen to be an edge of H ε Hr( n, p) with probability p = p( n), all choices being independent. It is shown that for every 0 < < 1 if P = ( C ln n)/ nr−1 with C = C() sufficiently large, then almost surely every subset V1 V of size | V1 | = (1 − ) n is matchable, that is, there exists a matching M in H such that every vertex of V1 is contained in some edge of M. 相似文献
14.
Let λ be an irreducible character of Sn corresponding to the partition ( r, s) of n. Let A be a positive semidefinite Hermitian n × n matrix. Let dλ( A) and per( A) be the immanants corresponding to λ and to the trivial character of Sn, respectively. A proof of the inequality dλ( A)≤λ( id)per( A) is given. 相似文献
15.
Let n = n1 + n2 + … + nj a partition Π of n. One will say that this partition represents the integer a if there exists a subsum nil + ni2 + … + nil equal to a. The set
(Π) is defined as the set of all integers a represented by Π. Let
be a subset of the set of positive integers. We denote by p(
,n) the number of partitions of n with parts in
, and by
((
,n) the number of distinct sets represented by these partitions. Various estimates for
(
,n) are given. Two cases are more specially studied, when
is the set {1, 2, 4, 8, 16, …} of powers of 2, and when
is the set of all positive integers. Two partitions of n are said to be equivalent if they represent the same integers. We give some estimations for the minimal number of parts of a partition equivalent to a given partition. 相似文献
16.
Let M n be the set of n× n matrices and r a nonnegative integer with r ≤ n. It is known,from Lie groups, that the rank r idempotent matrices in M n form an arcwise connected 2 n ( n- r)-dimensional analytic manifold. This paper provides an elementary proof of this result making it accessible to a larger audience. 相似文献
17.
We study continuous partitioning problems on tree network spaces whose edges and nodes are points in Euclidean spaces. A continuous partition of this space into p connected components is a collection of p subtrees, such that no pair of them intersect at more than one point, and their union is the tree space. An edge-partition is a continuous partition defined by selecting p−1 cut points along the edges of the underlying tree, which is assumed to have n nodes. These cut points induce a partition into p subtrees (connected components). The objective is to minimize (maximize) the maximum (minimum) “size” of the components (the min–max (max–min) problem). When the size is the length of a subtree, the min–max and the max–min partitioning problems are NP-hard. We present O( n2 log(min( p, n))) algorithms for the edge-partitioning versions of the problem. When the size is the diameter, the min–max problems coincide with the continuous p-center problem. We describe O( n log 3 n) and O( n log 2 n) algorithms for the max–min partitioning and edge-partitioning problems, respectively, where the size is the diameter of a component. 相似文献
18.
Let G be a solvable block transitive automorphism group of a 2−( v,5,1) design and suppose that G is not flag transitive. We will prove that - (1) if G is point imprimitive, then v=21, and GZ21:Z6;
- (2) if G is point primitive, then GAΓL(1,v) and v=pa, where p is a prime number with p≡21 (mod 40), and a an odd integer.
相似文献
19.
In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K( p(1), …, p( n)), conditions that are reminiscent of the Erdös-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number of edge-disjoint 1-factors in K( p(1), …, p( n)), settling the problem in the case where this number is greater than δ - p(2), where p(1) p(2) … p( n). 相似文献
20.
We consider scalar-valued matrix functions for n× n matrices A=( aij) defined by Where G is a subgroup of Sn the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n× n matrices if d( AB)= d( A) d( B) AB∈ S. With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1). 相似文献
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