A Hausdorff‐like moment problem and the inversion of the Laplace transform

We consider the problem of finding u ∈ L ²(I ), I = (0, 1), satisfying ∫_I u (x )x dx = μ _k , where k = 0, 1, 2, …, (α _k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ _kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A best approximation for the solution of one‐dimensional variable‐coefficient Burgers' equation

In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W_(3,2). It is proved that the approximation w_n(x,t) converges to the exact solution u(x,t) for any initial function w₀(x,t) ε W_(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of w_n(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation u_t(x,t)=(k(u(x,t))u_x(x,t))_x, with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[?]:?? →C¹[0,T], Ψ[?]:??→C¹[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[?] and Ψ[?] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))u_x(0,t) or/and h(t):=k(u(1,t))u_x(1,t), the values k(ψ₀) and k(ψ₁) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values k_u(ψ₀) and k_u(ψ₁) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C¹[0,T], Ψ[?]:??→C¹[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.     

Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Conservative finite difference scheme for a singularly perturbed elliptic reaction-diffusion equation: Approximation of solutions and derivatives

A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter ɛ², where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem. Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ɛ-uniformly at a rate of O(N ₁⁻²ln² N ₁ + N ₂⁻²), where N ₁ + 1 and N ₂ + 1 are the number of mesh points on the x ₁-axis and the minimal number of mesh points on a unit interval of the x ₂-axis respectively. The normalized difference derivatives ɛ ^k (∂ ^k /∂x ₁ ^k )u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer, and the derivatives along the boundary layer (∂ ^k /∂ x ₂ ^k )u(x) (k = 1, 2) converge ɛ-uniformly at the same rate.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

The least degree of identities in the subspace M 1 (m,k) (F) of the matrix superalgebra M (m,k)(F)

We determine the least degree of identities in the subspace M ₁ ^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over a field F for arbitrary m and k. For the subspace M ₁ ^(m,k) (F) (k > 1) we obtain concrete minimal identities and generalize some results by Chang and Domokos.     

Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M _i (u) be the number of solutions of f = u in (R/P ⁱ ) ⁿ for and. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f.     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

Parallel Multiplication in GF(2k) using Polynomial Residue Arithmetic

We present a novel method of parallelization of the multiplication operation in GF(2^k) for an arbitrary value of k and arbitrary irreducible polynomial n(x) generating the field. The parallel algorithm is based on polynomial residue arithmetic, and requires that we find L pairwise relatively prime modulim _i(x) such that the degree of the product polynomialM(x)=m ₁(x)m ₂(x)··· m_L(x) is at least 2k. The parallel algorithm receives the residue representations of the input operands (elements of the field) and produces the result in its residue form, however, it is guaranteed that the degree of this polynomial is less than k and it is properly reduced by the generating polynomial n(x), i.e., it is an element of the field. In order to perform the reductions, we also describe a new table lookup based polynomial reduction method.