Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1)ⁿx⁽²ⁿ⁺¹⁾=f(t,x,…,x⁽²ⁿ⁾) together with the Lidstone boundary conditions x^(2j)(0)=x^(2j)(T)=0, 0?j?n−1, and the next condition x⁽²ⁿ⁾(0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

Factorization of entire solutions of some algebraic differential equations

We investigate the factorization of entire solutions of the following algebraic differential equations:

bn(z)finjn(f^′)+bn−1(z)fin−1jn−1(f^′)+?+b₀(z)fi₀j₀(f^′)=b(z),     

Kolmogorov Vector Fields with Robustly Permanent Subsystems

The following results are proven. All subsystems of a dissipative Kolmogorov vector field ?_i = x_if_i(x) are robustly permanent if and only if the external Lyapunov exponents are positive for every ergodic probability measure μ with support in the boundary of the nonnegative orthant. If the vector field is also totally competitive, its carrying simplex is C¹. Applying these results to dissipative Lotka-Volterra systems, robust permanence of all subsystems is equivalent to every equilibrium x* satisfying f_i(x* > 0 whenever x_i^* = 0. If in addition the Lotka-Volterra system is totally competitive, then its carrying simplex is C¹.     

Pointwise Estimate for Linear Combinations of Bernstein-Kantorovich Operators

For linear combinations of Bernstein-Kantorovich operators K_n, r(f, x), we give an equivalent theorem with ω^2r_?^λ(f, t). The theorem unites the corresponding results of classical and Ditzian-Totik moduli of smoothness.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

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.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

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.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence,uniqueness and stability for solutions of the polynomial-like iterative equation Σni=1λifi(x)=F(x) were obtained in the case of λ 1 = 0.In this paper,we construct C 0 decreasing solutions of the iterative equation in the case that λ 1 can vanish to answer the Leading Coefficient Problem.Moreover,we also give the necessary and sufficiently condition for uniqueness of solutions.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators J_n,αf for functions f measurable and locally bounded on the interval [0, ∞) with α ? 1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of J_n,αf(x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann-Butzer-Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.     

Existence of a positive solution to a system of discrete fractional boundary value problems

We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by -Δ^ν_iy_i(t)=λ_ia_i(t+ν_i-1)f_i(y₁(t+ν₁-1),y₂(t+ν₂-1)), for t∈[0,b]_N0, subject to y_i(ν_i − 2) = ψ_i(y_i) and y_i(ν_i + b) = ?_i(y_i), for i = 1, 2, where ψ_i,?_i:R^b+3→R are given functionals. We also assume that ν_i ∈ (1, 2], for each i. Although we assume that both a_i and f_i(y₁, y₂) are nonnegative for each i, we do not necessarily presume that each ψ_i(y_i) and ?_i(y_i) is nonnegative for each i and each y_i ? 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case.