Solutions to the polynomial Dirac equations on unbounded domains in Clifford analysis

In this paper we study polynomial Dirac equation p(??)f = 0 including (?? ? λ)f = 0 with complex parameter λ and ??^kf = 0(k?1) as special cases over unbounded subdomains of ?^{n + 1}. Using the Clifford calculus, we obtain the integral representation theorems for solutions to the equations satisfying certain decay conditions at infinity over unbounded subdomains of ?^{n + 1}. Copyright © 2010 John Wiley & Sons, Ltd.     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

A Drazin inverse for rectangular matrices

The definition of the Drazin inverse of a square matrix with complex elements is extended to rectangular matrices by showing that for any B and W,m by n and n by m, respectively, there exists a unique matrix, X, such that (BW)^k=(BW)^k+1XW for some positive integer k, XWBWX = X, and BWX = XWB. Various expressions satisfied by B, W,X and related matrices are developed.     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

Edge disjoint cycles in graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ? 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ? 2δ², then for any 3 ? l₁ ? l₂ ? ? ? l_k ? n, 1 = k = [(δ - 1)/2], such graph has K edge disjoint cycles with lengths l₁, l₂…l_k, respectively. In particular, when l₁ = l₂ = ? = l_k = n and k = [(δ - 1)/2], the graph contains [(δ - 1)/2] edge disjoint hamiltonian cycles.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.     

Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces

Suppose thatX andY are real Banach spaces,U ⊂X is an open bounded set star-shaped with respect to some point,n, k ∈ ℕ,k <n, andMn, k (U,Y) is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for anyM ≥M _n,k (U, Y) there exists a constantB > 0 such that for any functionf ∈C ⁿ (U, Y) the following inequality holds:

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Analysis of Algorithms for Orthogonalizing Products of Unitary Matrices

We consider the problem of computing U_k = Q_kU_k−1(where U₀ is given) in finite precision (ϵ_M = machine precision) where U₀ and theQ_i are known to be unitary. The problem is that Û_k, the computed product may not be unitary, so one applies an O(n²) orthogonalizing step after each multiplication to (a) prevent Û_k from drifing too far from the set of untary matrices (b) prevent Û_k from drifting too far from U_k the true product. Our main results are 1. Scaling the rows to have unit length after each multiplication (the cheaptest of the algorithms considered) is usually as good as any other method with respect to either of the criteria (a) or (b). 2. A new orthogonalization algorithm that guarantees the distance of Û_k (k = 1, 2, …) to the set of unitary matrices is bounded by n^3.5ϵ_M for any choice of Q_i.     

Accuracy of suboptimal solutions to kernel principal component analysis

For Principal Component Analysis in Reproducing Kernel Hilbert Spaces (KPCA), optimization over sets containing only linear combinations of all n-tuples of kernel functions is investigated, where n is a positive integer smaller than the number of data. Upper bounds on the accuracy in approximating the optimal solution, achievable without restrictions on the number of kernel functions, are derived. The rates of decrease of the upper bounds for increasing number n of kernel functions are given by the summation of two terms, one proportional to n ^−1/2 and the other to n ⁻¹, and depend on the maximum eigenvalue of the Gram matrix of the kernel with respect to the data. Primal and dual formulations of KPCA are considered. The estimates provide insights into the effectiveness of sparse KPCA techniques, aimed at reducing the computational costs of expansions in terms of kernel units.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Constructions of Small Complete Caps in Binary Projective Spaces

In the binary projective spaces PG(n,2) k-caps are called large if k > 2^n-1 and smallif k ≤ 2^n-1. In this paper we propose new constructions producing infinite families of small binary complete caps.AMS Classification: 51E21, 51E22, 94B05     

Computation of certain infinite series of the form for arbitrary real-valued k

We provide a general method from which one may obtain more arbitrary closed form solutions for infinite series of the form ∑f(n)n^k, for and sufficiently well-behaved f(n). We illustrate the methods for the special case of f(n)=aⁿ/n!, obtaining generalizations to the Bell numbers and polynomials.     

On isometries of matrix spaces

Let A _kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = R n = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].     

Direct construction methods for incomplete perfect Mendelsohn designs with block size four

Let v, k, λ, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD, is a triple (X,Y,B) where X is a v-set (of points), Y is an n-subset of X, and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a,b) E (X × X)\(Y × Y) appears t-apart in exactly λ blocks of B and no ordered pair (a,b) E Y × Y appears in any block of B for any t, where 1 ≤ t ≤ k − 1. In this article, we introduce an effective and easy way to construct IPMDs for k = 4 and even v − n, and use it to construct some small examples for λ = 1 and 2. Obviously, these results will play an important role to completely solve the existence of (v,n,4,λ)-IPMDs. Furthermore, we also use this method to construct some small examples for HPMDs. © 1996 John Wiley & Sons, Inc.     

Stirling Number Asymptotics from Recursion Equations Using the Ray Method

A technique for computing asymptotic expansions of combinatorial quantities from their recursion relations is presented. It is applied to the Stirling numbers of the first and second kinds, s(n, k) and S(n, k), for n ? 1 and three ranges of k: (i) k = O(1), (ii) n ? k = O(1), (iii) k ? 1, 0 < k / n < 1, The technique uses asymptotic methods of applied mathematics such as the ray method and the method of matched asymptotic expansions.     

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

We say that X = [x_ij] is centrosymmetric if x_ij = x_{n ? j + 1, n ? i + 1}, 1?i, j?n. In this paper, we present an efficient algorithm for minimizing ∥AXB ? C∥ where ∥·∥ is the Frobenius norm, A∈?^{m × n}, B∈?^{n × s}, C∈?^{m × s} and X∈?^{n × n} is centrosymmetric with a specified central submatrix [x_ij]_{p?i, j?n ? p}. Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable in any case, and we give results of numerical experiments that support this claim. Copyright © 2011 John Wiley & Sons, Ltd.     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

Stability of a Pivoting Strategy for Parallel Gaussian Elimination

Gaussian elimination with partial pivoting achieved by adding the pivot row to the kth row at step k, was introduced by Onaga and Takechi in 1986 as means for reducing communications in parallel implementations. In this paper it is shown that the growth factor of this partial pivoting algorithm is bounded above by _n <#60; 3 ^n–1, as compared to 2 ^n–1 for the standard partial pivoting. This bound _n, close to 3 ^n–2, is attainable for class of near-singular matrices. Moreover, for the same matrices the growth factor is small under partial pivoting.This revised version was published online in October 2005 with corrections to the Cover Date.