Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

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.     

Non-orientable fundamental surfaces in lens spaces

We give a concrete example of an infinite sequence of (pn,qn)-lens spaces L(pn,qn) with natural triangulations T(pn,qn) with pn tetrahedra such that L(pn,qn) contains a certain non-orientable closed surface which is fundamental with respect to T(pn,qn) and of minimal crosscap number among all closed non-orientable surfaces in L(pn,qn) and has n−2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn,qn). Actually, we can set p₀=0, q₀=1, pk+1=3pk+2qk and qk+1=pk+qk.     

The minimum spanning strong subdigraph problem is fixed parameter tractable

A digraph D is strong if it contains a directed path from x to y for every choice of vertices x,y in D. We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strong subdigraph of a strong digraph. It is easy to see that every strong digraph D on n vertices contains a spanning strong subdigraph on at most 2n−2 arcs. By reformulating the MSSS problem into the equivalent problem of finding the largest positive integer k≤n−2 so that D contains a spanning strong subdigraph with at most 2n−2−k arcs, we obtain a problem which we prove is fixed parameter tractable. Namely, we prove that there exists an O(f(k)n^c) algorithm for deciding whether a given strong digraph D on n vertices contains a spanning strong subdigraph with at most 2n−2−k arcs.We furthermore prove that if k≥1 and D has no cut vertex then it has a kernel of order at most (2k−1)². We finally discuss related problems and conjectures.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

Oscillations of the system of linear nonautonomous difference equations

We consider the linear nonautonomous system of difference equations x_n+1−x_n+P(n)x_n−k=0, n=0,1,2,… , where k∈Z, P(n)∈R^rxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system.     

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.     

On the Diophantine Equation x+q=y

In this paper it has been proved that if q is an odd prime, q?7 (mod 8), n is an odd integer ?5, n is not a multiple of 3 and (h,n)=1, where h is the class number of the filed Q(√−q), then the diophantine equation x²+q^2k+1=yⁿ has exactly two families of solutions (q,n,k,x,y).     

Degrees of the Elliptic Teichmüller Lift

The purpose of this paper is to find upper bounds for the degrees, or equivalently, for the order of the poles at O, of the coordinate functions of the elliptic Teichmüller lift of an ordinary elliptic curve over a perfect field of characteristic p. We prove the following bounds:ord₀(x_n)?−(n+2)pⁿ+npⁿ⁻¹, ord₀(y_n)?−(n+3)pⁿ+npⁿ⁻¹. Also, we prove that the bound for x_n is not the exact order if, and only if, p divides (n+1), and the bound for y_n is not the exact order if, and only if, p divides (n+1)(n+2)/2. Finally, we give an algorithm to compute the reduction modulo p³ of the canonical lift for p≠2,3.     

Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n^−2r) and the gap between the spectral subspaces are of the orders O(n^−r) in L₂-norm and O(n^1/2−r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n^−2r) in both L₂-norm and infinity norm. We illustrate our results with numerical examples.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).     

Reciprocal relations between cyclotomic fields

We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ?n(c−ζm) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ?m(c−ζn) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ?n is the n-th cyclotomic polynomial, ζm a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers q(1−ζp)−1 and p(1−ζq)−1, where p and q are prime numbers.     

An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order

We prove that each OBDD (ordered binary decision diagram) for the middle bit of n-bit integer multiplication for one of the variable orders which so far achieve the smallest OBDD sizes with respect to asymptotic order of growth, namely the pairwise ascending order x₀,y₀,…,x_n−1,y_n−1, requires a size of Ω(2^(6/5)n). This is asymptotically optimal due to a bound of the same order by Amano and Maruoka (2007) [1].     

Multi-bump solutions to Carrier's problem

We study the existence of well-known singularly perturbed BVP problem ε²y″=1−y²−2b(1−x²)y, y(−1)=y(1)=0 introduced by G.F. Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O(ε|lnε|), while only single spikes are allowed near the boundaries x=±1.     

Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl m_α(λ) function for the differential equation d²y/dx²+[λ−q(x)]y=0. The test potential q(x)=x², for which the relevant m_α(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the m_α(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m_α(λ) function for these potentials to illustrate the computational effectiveness of the method used.