Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ? 1 be integers and S = {x₁, … , x_n} be a set of n distinct positive integers. The matrix having the ath power (x_i, x_j)^a of the greatest common divisor of x_i and x_j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S^a). Similarly we can define the ath power LCM matrix [S^a]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S₁ ∪ ? ∪ S_k, where k ? 1 is an integer and all S_i (1 ? i ? k) are divisor chains such that (max(S_i), max(S_j)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (S^a) and power LCM matrices [S^a] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(S^a)∣det(S^b), det[S^a]∣det[S^b] and det(S^a)∣det[S^b] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.     

Overdetermined linear systems satisfying nonnegativity constraints

Let A be a nonnegative m × n matrix, and let b be a nonnegative vector of dimension m. Also, let S be a subspace of $\text{R}$ⁿ such that if P_S is the orthogonal projector onto S, then P_S ? 0. A necessary condition is given for the matrix A to satisfy the following property: For all b ? 0, if min[boxV]b ? Ax[boxV] is attained at x = x₀, then x₀ ? 0 and x₀ ? S. It is also shown that if a nonnegative matrix A has a nonnegative generalized inverse, then any submatrix of A also possesses a nonnegative generalized inverse.     

A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

Travelling waves of auto-catalytic chemical reaction of general order—An elliptic approach

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at=axx−af(b), bt=Dbxx+af(b), with f a degenerate nonlinearity. In the context of an auto-catalytic chemical reaction, a is the density of a chemical species called reactant A, b that of another chemical species B called auto-catalyst, and D=DB/DA>0 is the ratio of diffusion coefficients, DB of B and DA of A, respectively. Such a system also arises from isothermal combustion. The nonlinearity is called degenerate, since f(0)=f^′(0)=0. One case of interest in this article is the propagating wave fronts in an isothermal auto-catalytic chemical reaction of order with 1<n<2, and D≠1 due to different molecular weights and/or sizes of A and B. The resulting nonlinearity is f(b)=bn. Explicit bounds v_∗ and v^∗ that depend on D are derived such that there is a unique travelling wave of every speed v?v^∗ and there does not exist any travelling wave of speed v<v_∗. New to the literature, it is shown that v_∗∝v^∗∝D when D<1. Furthermore, when D>1, it is shown rigorously that there exists a vmin such that there is a travelling wave of speed v if and only if v?vmin. Estimates on vmin improve significantly that of early works. Another case in which two different orders of isothermal auto-catalytic chemical reactions are involved is also studied with interesting new results proved.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x∈[n], and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [n]=A₀∪?∪At−1 and asks for a such that x∈Aa. Carole responds with some b, and if a≠b, then x accumulates a lie (a,b). Carole's string of lies for x must be in the channel C. Paul wins if he determines x within q rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space [n] for which Paul can guarantee finding the distinguished element is as q→∞, where Ek(C) is the number of lie strings in C of maximum length k. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel C.     

Interpolation by Integer-Valued Polynomials

LetRbe a Krull ring with quotient fieldKanda₁,…,a_ninR. If and only if thea_iare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb₁,…,b_ninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(a_i) = b_iandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringR_vwith quotient fieldKanda₁,…,a_ninSare pairwise incongruent mod allM^k_v ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of thea_iamong residue classes of the idealsM^k_v ∩ S) such that for allb₁,…,b_n ∈ R_vthere exists a polynomialf ∈ K[x] of degree at mostdwithf(a_i) = b_iandf(S) ⊆ R_v.     

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Ω?=(−?,?)×S¹, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b?0, near the characteristic set Σ={0}×S¹. We show that the interplay between the order of vanishing of the functions a and b at x=0 plays a role in the Gevrey solvability.     

A Near-Commutativity Property for Rings

We revisit rings with the property that |A ²| ≤ 3 for each 2-subset A. We then investigate commutativity of rings R with the property that for each pair a, b of noncommuting elements of R, there exists an integer n > 1 for which a ⁿ = b ⁿ .     

Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L=∂/∂t+(a+ib)(x)∂/∂x, a,b∈C^∞(S¹;R), defined on two-torus T²≅R²/2πZ². The kernel of transpose operator is described and the solvability near the characteristic set is also studied.     

Absolute Summability Factors and Absolute Tauberian Theorems for Double Series and Sequences

Let A and B be the linear methods of the summability of double series with fields of bounded summability MA _b ^' and B _b ^' , respectively. Let T be certain set of double series. The condition x T is called B _b-Tauberian for A if A _b ^' B _b ^' .Some theorems about summability factors enable one to find new B _b-Tauberian conditions for A from the already known B _b-Tauberian conditions for A.     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

Lower quasicontinuity, joint continuity and related concepts

Let X and Y be topological spaces, let Z be a metric space, and let f:X×Y→Z be a mapping. It is shown that when Y has a countable base B, then under a rather general condition on the set-valued mappings X∋x→fx(B)∈Z², B∈B, there is a residual set R⊂X such that for every (a,b)∈R×Y, f is jointly continuous at (a,b) if (and only if) fa:Y→Z is continuous at b. Several new results are also established when the notion of continuity is replaced by that of quasicontinuity or by that of cliquishness. Our approach allows us to unify and improve various results from the literature.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n, let R₁(A,n), R₂(A,n) and R₃(A,n) denote the number of solutions of a+a^′=n with the additional restriction a,a^′∈A; a,a^′∈A,a<a^′ and a,a^′∈A,a≤a^′ respectively. In this paper, we specially focus on the monotonicity of R₃(A,n). Moreover, we show that there does not exist any set A⊂N such that R₂(A,n) or R₃(A,n) is eventually strictly increasing.