Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

Global stability of a rational difference equation

In this paper, we study the global stability of the difference equation , where the parameters a,a_i(0,∞) for i=0,…,k, x_-k,…, x_-1[0,∞) and x₀(0,∞). We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).     

Proximity problems on line segments spanned by points

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space.     

On partitioning two matroids into common independent subsets

Let M₁ and M₂ be two matroids on the same ground set S. We conjecture that if there do not exist disjoint subsets A₁,A₂,…,A_k+1 of S, such that _isp_M1(A_i)≠Ø, and similarly for M₂, then S is partitioned into k sets, each independent in both M₁ and M₂. This is a possible generalization of König's edge-coloring theorem. We prove the conjecture for the case k=2 and for a regular case, in which both matroids have the same rank d, and S consists of k·d elements. Finally, we prove another special case related to a conjecture of Rota.     

Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and · any norm in . Denote by B the unit ball of · and the class of convex lattice polygons with n vertices and least ·-perimeter. We prove that after suitable normalization, all members of tend to a fixed convex body, as n→∞.     

On oscillatory solutions of certain first order ordinary differential equations

By constructing a class of solutions to the integral inequality for t  t₀ large enough, where 0<A₁a(τ)A₂<+∞ and λ>1, that tend to zero as t→+∞ we address an open problem in the theory of nonlinear oscillations.     

On relations between 1-lines of Adams–Novikov spectral sequences modulo invariant prime ideals

We have a ring homomorphism Θ from the cohomology of the extended Morava stabilizer group G_n with coefficients in F[w^±1] to the cohomology of G_n+1 with coefficients in the graded field F((u_n))[u^±1]. In this note we study the behavior of Θ on H¹. Then it is shown that Θ is injective on H¹ for n1 and for all primes p.     

Countable choice and compactness

We work in set-theory without choice ZF. Denoting by the countable axiom of choice, we show in that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p1 (respectively p=0), and some closed subset F of [0,1]^I which is a bounded subset of ℓ^p(I), we show that (respectively DC, the axiom of Dependent Choices) implies the compactness of F.     

Cauchy–Riemann equations and J-symplectic forms

Let (Σ,j) be a Riemann surface. The almost complex manifolds (M,J) for which the J-holomorphic curves :Σ→M are of variational type, are characterized. This problem is related to the existence of a vertically non-degenerate closed complex 3-form on Σ×M (see Theorem 4.3 below), which determines a family of J-symplectic structures on (M,J) parametrized by Σ.     

On four color monochromatic sets with nondecreasing diameter

Let m and r be positive integers. Define f(m,r) to be the least positive integer N such that for every coloring of the integers 1,…,N with r colors there exist monochromatic subsets B₁ and B₂ (not necessarily of the same color), each having m elements, such that (a) max(B₁)-min(B₁)max(B₂)-min(B₂), and (b) max(B₁)B₂). We improve previous upper bounds to determine that f(m,4)=12m-9.     

A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations

A discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of Fully-discrete time approximate scheme are formulated. These schemes can be symmetric or nonsymmetric. Hp-version error estimates are analyzed for these schemes. Just because of a damping term ·(b(u)u_t) included in Sobolev equation, which is the distinct character different from parabolic equation, special test functions are chosen to deal with this term. Finally, a priori L^∞(H¹) error estimate is derived for the semi-discrete time scheme and similarly, l^∞(H¹) and l²(H¹) for the Fully-discrete time schemes. These results also indicate that spatial rates in H¹ and time truncation errors in L² are optimal.     

Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

Linear time isotonic and unimodal regression in the and norms

We consider L₁-isotonic regression and L_∞ isotonic and unimodal regression. For L₁-isotonic regression, we present a linear time algorithm when the number of outputs are bounded. We extend the algorithm to construct an approximate isotonic regression in linear time when the output range is bounded. We present linear time algorithms for L_∞ isotonic and unimodal regression.     

General preservers of invariant subspace lattices

Let be the space of all bounded linear operators on a Banach space X and let LatA be the lattice of invariant subspaces of the operator . We characterize some maps with one of the following preserving properties: Lat(Φ(A)+Φ(B))=Lat(A+B), or Lat(Φ(A)Φ(B))=Lat(AB), or Lat(Φ(A)Φ(B)+Φ(B)Φ(A))=Lat(AB+BA), or Lat(Φ(A)Φ(B)Φ(A))=Lat(ABA), or Lat([Φ(A),Φ(B)])=Lat([A,B]).     

Mass formula for self-dual codes over Z

We give a mass formula for self-dual codes over Z_p², where p is an odd prime. Using the mass formula, we classify such codes of lengths up to n=8 over the ring Z₉, n=7 over Z₂₅ and n=6 over Z₄₉.     

Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C_R and C_B with disjoint interiors such that the number of red points covered by C_R plus the number of blue points covered by C_B is maximized. We give an algorithm to solve this problem in O(n^8/3log²n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S_R and S_B instead of unit disks can be solved in O(nlogn) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n³logn) time.     

The total chromatic number of regular graphs of even order and high degree

The total chromatic number χ_T(G) of a graph G is the minimum number of colours needed to colour the edges and the vertices of G so that incident or adjacent elements have distinct colours. We show that if G is a regular graph of even order and , thenχ_T(G)Δ(G)+2.     

Characterizations of some classes of relatively regular operators

A bounded linear operator A on a Banach space is called relatively regular, if there is a bounded linear operator B such that ABA=A. In this case B is called a g₁-inverse of A. In this paper we characterize some classes of relatively regular operators A via the set {B₁-B₂:B₁ and B₂ are g₁-inverses of A}.     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

Maximum genus, connectivity and minimal degree of graphs

This paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form:

Let G be a k-edge-connected graph with minimum degree δ, for each positive integer k(3), there exists a non-decreasing function f(δ) such that the maximum genus γ_M(G) of G satisfies the relation γ_M(G)f(δ)β(G), and furthermore that lim_δ→∞f(δ)=1/2, where β(G)=|E(G)|-|V(G)|+1 is the cycle rank of G.

The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors.