Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,l ^k?1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx ₀<x ₁<...<x _l ofX with δ(x ₀,x ₁)=δ(x ₁,x ₂)=...=δ(x _l?1,x _l), or else there exist elementsy ₀<y ₁<...<y _k ofX with δ(y _i?1,y _i) ∈ δ(y _j?1,y _j) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)^2k . The general question remains open.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

Oberwolfach rectangular table negotiation problem

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x₁,x₂,…,x_k},Y={y₁,y₂,…,y_k} and edges x₁y₁,x₁y₂,x_ky_k−1,x_ky_k, and x_iy_i−1,x_iy_i,x_iy_i+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs K_n,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3).     

On the integer points on some special hyper-elliptic curves over a finite field

If $\text{l}$_r(p) is the least positive integral value of x for which y² ≡ x(x + 1) ? (x + r ? 1)(modp) has a solution, we conjecture that $\text{l}$_r(p) ≤ r² ? r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y² ≡ (x + a₁) ? (x + a_r) is suggested, where the a's are fixed positive integers.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

Chen’s Conjecture and Its Generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that （k ＋l1）^2, （k ＋l2）^2, ..., （k ＋lg）^2 can not be expressed as 2^n＋p^α is at least c（g）x, where p is an odd prime and the constant c（g） depends only on g.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

l-trace k-Sperner families of sets

A family of sets F⊆X² is defined to be l-trace k-Sperner if for any subset Y of X with size l the trace of F on Y (the restriction of F to Y) does not contain any chain of length k+1. In this paper we investigate the maximum size that an l-trace k-Sperner family (with underlying set [n]={1,2,…,n}) can have for various values of k, l and n.     

On a multivariate normal probability minimization problem

Given a 2k component (column) vector z=(x′,y′)′, x′=(x₁,…, x_k), y′= (y₁,…,y_k), having a 2k variate multivariate normal distribution with mean vector zero, and an unknown 2k×2k positive definite correlation matrix Σ, Shafer and Olkin (1983) conjecture that P(x,y)=P{|x₁|<c,…,|x_k|<c,|y₁|<c,…,|y_k|<c} is minimized when Σ has a certain pattern. This conjecture is proved to be correct. A known inequality is generalized.     

Steiner intervals, geodesic intervals, and betweenness

The concept of the k-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping S:V×?×V?2^V such that S(u₁,…,u_k) consists of all vertices in G that lie on some Steiner tree with respect to a multiset W={u₁,…,u_k} of vertices from G. In this paper we obtain, for each k, a characterization of the class of graphs in which every k-Steiner interval S has the so-called union property, which says that S(u₁,…,u_k) coincides with the union of geodesic intervals I(u_i,u_j) between all pairs from W. It turns out that, as soon as k>3, this class coincides with the class of graphs in which the k-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, S satisfies (m), if x₁,…,x_k∈S(u₁,…,u_k) implies S(x₁,…,x_k)⊆S(u₁,…,u_k), and S satisfies (b2) if x∈S(u₁,u₂,…,u_k) implies S(x,u₂,…,u_k)⊆S(u₁,…,u_k). In the case k=3, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval S satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class of graphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.     

The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers

In this paper, we give a systematical study of the local structures and fractal indices of the limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. For a given Pisot number in the interval (1,2), we construct a finite family of non-negative matrices (maybe non-square), such that the corresponding fractal indices can be re-expressed as some limits in terms of products of these non-negative matrices. We are especially interested in the case that the associated Pisot number is a simple Pisot number, i.e., the unique positive root of the polynomial x^k-x^k-1-…-x-1 (k=2,3,…). In this case, the corresponding products of matrices can be decomposed into the products of scalars, based on which the precise formulas of fractal indices, as well as the multifractal formalism, are obtained.     

Finite time extinction of super-Brownian motions with deterministic catalyst

In this paper we consider a super-Brownian motion X with branching mechanism k(x)zα, where k(x) > 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥ //x// -l(0 ≤l < ∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥exp(-l‖x‖)(0≤l < ∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥ 3: it is O(‖x‖-(α+1)) in one dimension, O(‖x‖-2(log‖x‖)-(α+1) ) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2Δu =k(x)uα.     

Walking in circles

Consider the unit circle S¹ with distance function d measured along the circle. We show that for every selection of 2n points x₁,…,x_n,y₁,…,y_n∈S¹ there exists i∈{1,…,n} such that . We also discuss a game theoretic interpretation of this result.     

On the number of coprime integral solutions of y2 = x3 + k

Let N′(k) denote the number of coprime integral solutions x, y of y² = x³ + k. It is shown that lim sup_k→∞N′(k) ≥ 12.     

Simple formulas for lattice paths avoiding certain periodic staircase boundaries

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x=ky when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x=ky is replaced by certain periodic staircase boundaries—but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

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.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.