Impossibility of extending Pólya’s theorem to “forms” with arbitrary real exponents

Pólya proved that if a form (homogeneous polynomial) with real coefficients is positive on the nonnegative orthant (except at the origin), then it is the quotient of two real forms with no negative coefficients. While Pólya’s theorem extends, easily, from ordinary real forms to “generalized” real forms with arbitrary rational exponents, we show that it does not extend to generalized real forms with arbitrary real (possibly irrational) exponents.     

Effective Pólya semi-positivity for non-negative polynomials on the simplex

We consider homogeneous polynomials f∈R[_x1,…,_xn]$f\in \mathbb{R}[x_1,\dots ,x_n]$ which are non-negative on the standard simplex in Rⁿ${\mathbb{R}}^n$, and we obtain sufficient conditions for such an f   to be Pólya semi-positive, that is, all the coefficients of (_x1+?+_xn)^Nf$(x_1+?+x_n{)}^{N}f$ are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N   under such conditions. Moreover, we also show that any Pólya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Pólya semi-positive polynomials in the low dimensional case when n?3$n?3$ as well as the case (in any dimension) when the zero set of the polynomial in the simplex consists of a finite number of points. We also discuss an application to the representations of non-homogeneous polynomials which are non-negative on a general simplex.     

Sums of products of Kronecker's double series

Closed expressions are obtained for sums of products of Kronecker's double series of the form , where the summation ranges over all nonnegative integers j₁,…,jN with j₁+?+jN=n. Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.     

Degree complete graphs

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s₁,s₂,…,s_n) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then

(Π)     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x₁,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ?>0 such that if X=(X₁,…,Xg) is a g-tuple of n×n symmetric matrices satisfying

     

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n₁,…,nm, nm+1=n₁, and any nonnegative integer r, the expression

     

On the Bateman-Horn Conjecture

Let r be a positive integer and f₁,…,f_r be distinct polynomials in Z[X]. If f₁(n),…,f_r(n) are all prime for infinitely many n, then it is necessary that the polynomials f_i are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f₁(n)···f_r(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363-367) proposed a conjectural asymptotic estimate on the number of positive integers n?x such that f₁(n),…,f_r(n) are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r?2. We consider the Bateman-Horn conjecture for the polynomials in any partition {f₁,…,f_s}, {f_s+1,…,f_r} with a linear change of variables. Our main result is as follows: If the Bateman-Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman-Horn conjecture for f₁,…,f_r is equivalent to a plausible error term conjecture for the minor arcs in the circle method.     

On the geometry of generalized quadrics

Let {f₀,…,f_n;g₀,…,g_n} be a sequence of homogeneous polynomials in 2n+2 variables with no common zeros in P²ⁿ⁺¹ and suppose that the degrees of the polynomials are such that is a homogeneous polynomial. We shall refer to the hypersurface X defined by Q as a generalized quadric. In this note, we prove that generalized quadrics in for n≥1 are reduced.     

Integer valued polynomials over a number field

A number field is called a Pólya field if the module of integer valued polynomials over that field is generated by (f_i) _i=0 over the ring of integers, with deg(f_i)=i, i=0, 1, 2,... In this paper bounds on the class numbers and on the number of ramified primes in Pólya fields are derived.     

The Erd?s-Ginzburg-Ziv theorem for finite solvable groups

Let G be a non-cyclic finite solvable group of order n, and let S=(g₁,…,g_k) be a sequence of k elements (repetition allowed) in G. In this paper we prove that if , then there exist some distinct indices i₁,i₂,…,i_n such that the product g_i1g_i2?g_in=1. This result substantially improves the Erd?s-Ginzburg-Ziv theorem and other existing results.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

A recurrence formula for leaping convergents of non-regular continued fractions

Given a continued fraction [a₀;a₁,a₂,…], p_n/q_n=[a₀;a₁,…,a_n] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent p_rn+i/q_rn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p_3n+1/q_3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p_3n/q_3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown.     

Weak cycle partition involving degree sum conditions

Let G be a graph of order n and k a positive integer. A set of subgraphs H={H₁,H₂,…,H_k} is called a k-degenerated cycle partition (abbreviated to k-DCP) of G if H₁,…,H_k are vertex disjoint subgraphs of G such that and for all i, 1≤i≤k, H_i is a cycle or K₁ or K₂. If, in addition, for all i, 1≤i≤k, H_i is a cycle or K₁, then H is called a k-weak cycle partition (abbreviated to k-WCP) of G. It has been shown by Enomoto and Li that if |G|=n≥k and if the degree sum of any pair of nonadjacent vertices is at least n−k+1, then G has a k-DCP, except G≅C₅ and k=2. We prove that if G is a graph of order n≥k+12 that has a k-DCP and if the degree sum of any pair of nonadjacent vertices is at least , then either G has a k-WCP or k=2 and G is a subgraph of K₂∪K_n−2∪{e}, where e is an edge connecting V(K₂) and V(K_n−2). By using this, we improve Enomoto and Li’s result for n≥max{k+12,10k−9}.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

Pósa-condition and nowhere-zero 3-flows

Let G be a simple graph on n vertices and π(G)=(d₁,d₂,…,d_n) be the degree sequence of G, where n≥3 and d₁≤d₂≤?≤d_n. The classical Pósa’s theorem states that if d_m≥m+1 for and d_m+1≥m+1 for n being odd and , then G is Hamiltonian, which implies that G admits a nowhere-zero 4-flow. In this paper, we further show that if G satisfies the Pósa-condition that d_m≥m+1 for and d_m+1≥m+1 for n being odd and , then G has no nowhere-zero 3-flow if and only if G is one of seven completely described graphs.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.