New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

The R(m,n) equations u_tt+a(uⁿ)_xx+b(u^m)_xxtt=0 (a, b const) are investigated by using some ansatze. As a result, new exact solitary patterns solutions and solitary wave solutions are obtained. These obtained solutions show that not only the nonlinearly dispersive R(m,m) equations (m≠1) but also the linearly dispersive R(1,n) equations (m=1) possess solitary patterns solutions, which has infinite slopes or cusps and solitary wave solutions.     

Integral approach to compacton solutions of Boussinesq-like B(m,n) equation with fully nonlinear dispersion

There exists much good work in the area of usual solitons, but there appears little in the field of compacton solutions. Only a few mathematical tools were employed so far. Recently, Yan [Chaos, Solitons & Fractals 14 (2002) 1151] extended the decomposition method to seek compacton solutions of B(m,n) equation u_tt=(uⁿ)_xx+(u^m)_xxxx. In this paper we present a different approach, integral approach, to investigate the compacton solutions of the B(m,n) equation. Not only Yan’s results but also many new compacton solutions of the B(m,n) equation are obtained. Our approach is simple and also suitable for studying compacton solutions of some other equations.     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

Exact solutions with solitary patterns for the Zakharov–Kuznetsov equations with fully nonlinear dispersion

In this paper, the nonlinear dispersive Zakharov–Kuznetsov ZK(m, n, k) equations are solved exactly by using the Adomian decomposition method. The two special cases, ZK(2, 2, 2) and ZK(3, 3, 3), are chosen to illustrate the concrete scheme of the decomposition method in ZK(m, n, k) equations. General formulas for the solutions of ZK(m, n, k) equations are established.     

The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

We consider the nonlinear parabolic equation u_t = (k(u)u_x)_x + b(u)_x, where u = u(x, t, x ε R¹, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation u_t = (uⁿu_x)_x + (u^β)_x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

On some properties of the series ∑k=0 kx and the Stirling numbers of the second kind

We partially characterize the rational numbers x and integers n 0 for which the sum ∑_k=0^∞ kⁿx^k assumes integers. We prove that if ∑_k=0^∞ kⁿx^k is an integer for x = 1 − a/b with a, b> 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain for all integers k, 2 k n, and a 3, provided a is odd or divisible by 4, where v_a(m) denotes the exponent of the highest power of a which divides m, for m and a> 1 integers.

New identities are also derived for the Stirling numbers, e.g., we show that ∑_k=0²ⁿk! S(2n, k) , for all integers n 1.     

Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ(A)u_j, j = 0, 1,…, where u₀ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u_n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_k} with some sufficiently large n, where x^(j+1)_m = ψ(A)x^(j)_m, j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

An inequality for M-matrices

If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]^1/n ≥ (det A)^1/n + (det B)^1/n.     

On L(d,1)-labelings of graphs

Given a graph G and a positive integer d, an L(d,1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)−f(v)|d; if u and v are not adjacent but there is a two-edge path between them, then |f(u)−f(v)|1. The L(d,1)-number of G, λ_d(G), is defined as the minimum m such that there is an L(d,1)-labeling f of G with f(V){0,1,2,…,m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497–1514), the L(2,1)-labeling and the L(1,1)-labeling (as d=2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that λ_d(G)Δ²+(d−1)Δ for any graph G with maximum degree Δ. Different lower and upper bounds of λ_d(G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.     

Regular expression searching on compressed text

We present a solution to the problem of regular expression searching on compressed text. The format we choose is the Ziv–Lempel family, specifically the LZ78 and LZW variants. Given a text of length u compressed into length n, and a pattern of length m, we report all the R occurrences of the pattern in the text in O(2^m+mn+Rmlogm) worst case time. On average this drops to O(m²+(n+Rm)logm) or O(m²+n+Ru/n) for most regular expressions. This is the first nontrivial result for this problem. The experimental results show that our compressed search algorithm needs half the time necessary for decompression plus searching, which is currently the only alternative.     

A parallel computing scheme for minimizing a class of large scale functions

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form

where x = (x^T₁,…,x^T_m)^T and x_i R^n_i, ∑^m_{i = 1}n_i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of ƒ(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ƒ(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.     

A note on convergence in Banach spaces of cotype p

Let B be a separable Banach space. The following is one of the results proved in this paper. The Banach space B is of cotype p if and only if

1. d_n, n 1, has no subsequence converging in probability, and

2. ∑_{n 1}|a_n|^p < ∞ whenever ∑_{n 1}a_nd_n converges almost surely are equivalent for every sequence d_n, n 1, of symmetric independent random elements taking values in B.

Scheduling starting times for an active redundant system with non-identical lifetimes

Here we examine an active redundant system with scheduled starting times of the units. We assume availability of n non-identical, non-repairable units for replacement or support. The original unit starts its operation at time s₁ = 0 and each one of the (n − 1) standbys starts its operation at scheduled time s_i (i = 2, …, n) and works in parallel with those already introduced and not failed before s_i. The system is up at times s_i (i = 2, …, n), if and only if, there is at least one unit in operation. Thus, the system has the possibility to work with up to n units, in parallel structure. Unit-lifetimes T_i (i = 1, …, n) are independent with cdf F_i, respectively. The system has to operate without inspection for a fixed period of time c and it stops functioning when all available units fail before c. The probability that the system is functioning for the required period of time c depends on the distribution of the unit-lifetimes and on the scheduling of the starting times s_i. The reliability of the system is evaluated via a recursive relation as a function of the starting times s_i (i = 2, …, n). Maximizing with respect to the starting times we get the optimal ones. Analytical results are presented for some special distributions and moderate values of n.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Possible values for 2

From GCH and P^{m(κ)-hypermeasurable} (1 <m<gw), we construct a model satisfying 2_ⁿ = _a(n) and 2_^ω = _ω+m for a monotone a:ω→ω satisfying a(n)>n.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.