On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.     

Latin squares with bounded size of row prefix intersections

A latin square is a matrix of size n×n with entries from the set {1,…,n}, such that each row and each column is a permutation on {1,…,n}. We show how to construct a latin square such that for any two distinct rows, the prefixes of length h of the two rows share at most about h²/n elements. This upper bound is close to optimal when contrasted with a lower bound derived from the Second Johnson bound [6].     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.     

Simultaneous singular value decomposition

We consider the following problem: Given a set of m×n real (or complex) matrices A₁,…,A_N, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P^*A₁Q,…,P^*A_NQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota-Kanno-Kojima-Kojima and Maehara-Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

Ramsey regions

Let (T₁,T₂,…,T_c) be a fixed c-tuple of sets of graphs (i.e. each T_i is a set of graphs). Let R(c,n,(T₁,T₂,…,T_c)) denote the set of all n-tuples, (a₁,a₂,…,a_n), such that every c-coloring of the edges of the complete multipartite graph, K_a1,a2,…,an, forces a monochromatic subgraph of color i from the set T_i (for at least one i). If N denotes the set of non-negative integers, then R(c,n,(T₁,T₂,…,T_c))⊆Nⁿ. We call such a subset of Nⁿ a “Ramsey region”. An application of Ramsey's Theorem shows that R(c,n,(T₁,T₂,…,T_c)) is non-empty for n?0. For a given c-tuple, (T₁,T₂,…,T_c), known results in Ramsey theory help identify values of n for which the associated Ramsey regions are non-empty and help establish specific points that are in such Ramsey regions. In this paper, we develop the basic theory and some of the underlying algebraic structure governing these regions.     

Sparse anti-magic squares and vertex-magic labelings of bipartite graphs

A sparse anti-magic square is an n×n array whose non-zero entries are the consecutive integers 1,…,m for some m?n² and whose row-sums and column-sums form a set of consecutive integers. We derive some basic properties of these arrays and provide constructions for several infinite families of them. Our main interest in these arrays is their application to constructing vertex-magic labelings for bipartite graphs.     

Polynomial images of R

Let R be a real closed field and n?2. We prove that: (1) for every finite subset F of Rⁿ, the semialgebraic set Rⁿ?F is a polynomial image of Rⁿ; and (2) for any independent linear forms l₁,…,l_r of Rⁿ, the semialgebraic set {l₁>0,…,l_r>0}⊂Rⁿ is a polynomial image of Rⁿ.     

Procrustes problems and associated approximation problems for matrices with k-involutory symmetries

We say that a matrix R∈C^n×n is k-involutary if its minimal polynomial is x^k-1 for some k?2, so R^k-1=R^-1 and the eigenvalues of R are 1,ζ,ζ²,…,ζ^k-1, where ζ=e^2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈C^m×m, A∈C^m×n, S∈C^n×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS^-1=ζ^μA, and A is (R,S,α,μ)-symmetric if RAS^-α=ζ^μA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈C^n×t and B∈C^m×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈C^m×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A.     

Convex integer maximization via Graver bases

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R^d and w₁x,…,w_dx are linear forms on Rⁿ,

max{c(w₁x,…,w_dx):Ax=b,x∈Nⁿ}.     

Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are A_kk+1=A_k+1k=-1,k=1,…,N-1, and A_kk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v^(p) given by . We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.     

A note about proving non-Γ under a finite non-microstates free Fisher information assumption

We prove that if X₁,…,Xn (n>1) are self-adjoints in a W^∗-probability space with finite non-microstates free Fisher information, then the von Neumann algebra W^∗(X₁,…,Xn) they generate doesn't have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.     

Integer version of the multipath flow network synthesis problem

We consider the following integer multipath flow network synthesis problem. We are given two positive integers q, n, (1<q<n), and a non-negative, integer, symmetric, n×n matrix R, each non-diagonal element r_ij of which represents the minimum requirement of q-path flow value between nodes i and j in an undirected network on the node set N={1,2,…,n}. We want to construct a simple, undirected network G=[N,E] with integer edge capacities {u_e:e∈E} such that each of these flow requirements can be realized (one at a time) and the sum of all the edge capacities is minimum. We present an O(n³) combinatorial algorithm for the problem and we show that the problem has integer rounding property.     

Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis

Extensions of the Nourdin-Peccati analysis to Rn-valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an Rn-valued centered random variable Z=(Z¹,…,Zn) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix (−DL−1Zi|DZj), i,j=1,2,…,n, where D and L are respectively the derivative operator and the generator of the Ornstein-Uhlenbeck semigroup on Wiener space.     

Matrix inverse problem and its optimal approximation problem for R-skew symmetric matrices

Let R ∈ C^n×n be a nontrivial involution, i.e., R² = I and R ≠ ±I. A matrix A ∈ C^n×n is called R-skew symmetric if RAR = −A. The least-squares solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are firstly derived, then the solvability conditions and the solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are given. The solutions of the corresponding optimal approximation problem with R∗ = R for R-skew symmetric matrices are also derived. At last an algorithm for the optimal approximation problem is given. It can be seen that we extend our previous results [G.X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. 189 (2007) 482-489] and the results proposed by Zhou et al. [F.Z. Zhou, L. Zhang, X.Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589].     

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R?2 and k?3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2×2×2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2×2×2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

The operator formula for monotone triangles - simplified proof and three generalizations

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of −1s in the matrix when prescribing (1,2,…,n) as the bottom row of the monotone triangle.     

Divisors of the number of Latin rectangles

A k×n Latin rectangle on the symbols {1,2,…,n} is called reduced if the first row is (1,2,…,n) and the first column is T(1,2,…,k). Let Rk,n be the number of reduced k×n Latin rectangles and m=⌊n/2⌋. We prove several results giving divisors of Rk,n. For example, (k−1)! divides Rk,n when k?m and m! divides Rk,n when m<k?n. We establish a recurrence which determines the congruence class of for a range of different t. We use this to show that Rk,n≡((−1)^k−1(k−1)!)ⁿ⁻¹. In particular, this means that if n is prime, then Rk,n≡1 for 1?k?n and if n is composite then if and only if k is larger than the greatest prime divisor of n.