Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

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].     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

Some results on a heat conduction problem by Myshkis

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u₀>0. The temperature at the origin then decays, and when it reaches u₀, another equal-sized heat impulse is applied at a normalized time τ₁=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ_n, n=2,3,…, when the temperature there has decayed to u₀. This sequence of normalized waiting times τ_n can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91].

A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.     

Iterative methods for the extremal positive definite solution of the matrix equation

In this paper, the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation X+A^*X^-A=Q with the case 0<1 and the minimal positive definite solution of the same matrix equation with the case 1 are proposed. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Numerical examples to illustrate the behavior of the considered algorithms are also given.     

Error analysis of a Collocation method for numerically inverting a Laplace transform in case of real samples

In [S. Cuomo, L. D’Amore, A. Murli, M.R. Rizzardi, Computation of the inverse Laplace transform based on a collocation method which uses only real values, J. Comput. Appl. Math., 198 (1) (2007) 98–115] the authors proposed a Collocation method (C-method) for real inversion of Laplace transforms (Lt), based on the truncated Laguerre expansion of the inverse function:

where σ, b are parameters and c_k, kN, are the MacLaurin coefficients of a function depending on the Lt. The computational kernel of a C-method is the solution of a Vandermonde linear system, where the right hand side is obtained evaluating the Lt on the real axis. The Bjorck Pereira algorithm has been used for solving the Vandermonde linear system, providing a computable componentwise error bound on the solution.

For an inversion problem on discrete data F is known on a pre-assigned set of points (we refer to these points as samples of F) only and the major challenge is to deal with a significative loss of information. A natural approach to overcome this intrinsic difficulty is to construct a suitable fitting model that approximates the given data. In this case, we show that such approach leads to a C-method with perturbed right hand side, and then we use again the Bjorck Pereira algorithm.

Starting from the error introduced by the fitting model, we study its propagation in order to determine the maximum attainable accuracy on f_N. Moreover we derive a computable error bound that allows to get the suitable value of the parameter N that gives the maximum attainable accuracy.     

Analysis of an retrial queue

We consider a single server retrial queueing system in which each customer (primary or retrial customer) has discrete service times taking on value D_j with probability , and . An arriving primary customer who finds the server busy tries later. Moreover, each retrial customer has its own orbit, and the retrial customers try to enter the service independently of each other. We call this retrial queue an M/{D_n}/1 retrial queue. A necessary and sufficient condition for this system stability is given. In the steady state, we derive the joint distribution of the state of the server and the number of customers in the retrial orbits. The explicit expressions of some performance measures are given. In addition, the steady-state distribution of the waiting time is discussed.     

Comparison of Krylov subspace methods on the PageRank problem

PageRank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions. The problem incorporates a parameter that determines the difficulty of the problem. In this paper, the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of . We see that stationary methods are very reliable and more competitive when the problem is well conditioned, that is for small values of . However, for large values of the parameter the problem becomes more difficult and methods such as preconditioned BiCGStab or restarted preconditioned GMRES become competitive with stationary methods in terms of Mflops count as well as in number of iterations necessary to reach convergence.     

Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d^w(u)+d^w(v)2d for every pair of non-adjacent vertices u and vV(G) x,y . Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.     

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→∞.     

Micromagnetic simulations of demagnetization curves for single-phase nanocrystalline PrFeB magnet

The Gauss–Seidel projection method developed in [E. Weinan, X.P. Wang, SIAM J. Numer. Anal. 38 (2000) 1647–1665 and X.P. Wang, C.J. Garcia-Cervera, E. Weinan, J. Comp. Phys. 171 (2001) 357] is used to calculate the demagnetization curves for the single-phase nanocrystalline PrFeB magnet. It is observed that magnetic reversal for PrFeB magnet starts near the grain boundary where the angle between the external magnetic field and easy direction of the magnet can be as large as 90. We also calculated the demagnetization curves, coercivity μ₀H_c and remanence J_r for different temperatures. The numerical results are consistent with the experimental observations.     

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.     

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.     

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.     

On periodic radical groups in which permutability is a transitive relation

A group G is said to be a -group if permutability is a transitive relation in the set of all subgroups of G. Our purpose in this paper is to study -groups in the class of periodic radical groups satisfying min-p for all primes p.     

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.     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

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.     

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.     

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.