An automatic augmented galerkin method for singular integral equations with Hilbert kernel

In [1, 2], described a Chebyshev series method for the numerical solution of integral equations with three automatic algorithms for computing two regularization parameters,C _f andr. Here we describe a Fourier series expansion method for a class of singular integral equations with Hilbert kernel and constant coefficients with using a new automatic algorithm.     

Solution of the eigenvalue problem for a regular pencil λA0−A1 with singular matrices

One considers the generalized eigenvalue problem (A₀λ?A₁)x=0, (1) when one or both matrices A₀,A₁ are singular and ker A₀ ∩ ker A₁=φ is the empty set. With the aid of the normalized process, the solving of problem (1) reduces to the solving of the eigenvalue problem of a constant matrix of order r=min (r₀,r₁), where r₀,r₁ are the ranks of the matrices A₀,A₁, which are determined at the normalized decomposition of the matrices. One gives an Algol program which performs the presented algorithm and testing examples.     

Symmetrization for singular semilinear elliptic equations

In this paper, we prove some comparison results for the solution to a Dirichlet problem associated with a singular elliptic equation and we study how the summability of such a solution varies depending on the summability of the datum f.     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

GKB-FP: an algorithm for large-scale discrete ill-posed problems

We describe an algorithm for large-scale discrete ill-posed problems, called GKB-FP, which combines the Golub-Kahan bidiagonalization algorithm with Tikhonov regularization in the generated Krylov subspace, with the regularization parameter for the projected problem being chosen by the fixed-point method by Bazán (Inverse Probl. 24(3), 2008). The fixed-point method selects as regularization parameter a fixed-point of the function ‖r _λ ‖₂/‖f _λ ‖₂, where f _λ is the regularized solution and r _λ is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed-points of functions ||r_l^(k)||₂/||f_l^(k)||₂\|r_{\lambda}^{(k)}\|_{2}/\|f_{\lambda}^{(k)}\|_{2}, where f_l^(k)f_{\lambda}^{(k)} approximates f _λ in a k-dimensional Krylov subspace and r_l^(k)r_{\lambda}^{(k)} is the corresponding residual. Based on this and provided the sought fixed-point is reached, we prove that the regularized solutions f_l^(k)f_{\lambda}^{(k)} remain unchanged and therefore completely insensitive to the number of iterations. This and the performance of the method when applied to well-known test problems are illustrated numerically.     

On the posterior distribution of a dirichlet process given randomly right censored observations

The purpose of this note is to show that the conditional distribution of a Dirichlet process P given n independent observations X₁… X_n from P and belonging to measurable sets A₁,… A_n with A₁ ? A₁₊₁ for i=1,… n=1 is a mixture of Dirichlet processes as introduced by Antoniak. It is also shown that this result is applicable in Bayesin decision problems concerning a random survival distribution under Dirichlet process priors.     

Dead cores of singular Dirichlet boundary value problems with ϕ-Laplacian

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable. This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Council of Czech Government MSM 6198959214.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite

Computing a function f(A) of an n-by-n matrix A is a frequently occurring problem in control theory and other applications. In this paper we introduce an effective approach for the determination of matrix function f(A). We propose a new technique which is based on the extension of Newton divided difference and the interpolation technique of Hermite and using the eigenvalues of the given matrix A. The new algorithm is tested on several problems to show the efficiency of the presented method. Finally, the application of this method in control theory is highlighted.     

A geometric computing method for nonlinear optimal regulator problems with singular arcs

A computing algorithm, based on the geometry of certain reachable sets, is presented for fixed terminal time optimal regular problems having differential equations \(\dot x = f(x ,u , t)\) . Admissible controls must be measurable and have values in a setU, which must be compact, but need not be convex. Functionsf(x, u, t) andf _x (x, u, t) must be continuous and Lipschitz inx andu, but existence off _u (x, u, t) or second derivatives is not required. The algorithm is based on taking a sequence of nonlinear steps, each of which linearizes \(\dot x = f(x ,u , t)\) in state only, about a current nominal control and trajectory. Small perturbations are assured by keeping the perturbed controlclose to the nominal control. In each nonlinear step, a regulator problem,linear in state, is solved by a convexity method of Barr and Gilbert (Refs. 1–2), which is undeterred by the possibility of singular arcs. The resulting control function is substituted into the original nonlinear differential equations, producing an improved trajectory. Convergence of the algorithm is not proved, but demonstrated by a computing example, known to be singular. In addition, procedures are described for choosing parameters in the algorithm and for testing for theplausibility of convergence.     

Decomposition of the completer-graph into completer-partiter-graphs

Forn ≥ r ≥ 1, letf _r (n) denote the minimum numberq, such that it is possible to partition all edges of the completer-graph onn vertices intoq completer-partiter-graphs. Graham and Pollak showed thatf ₂(n) =n ? 1. Here we observe thatf ₃(n) =n ? 2 and show that for every fixedr ≥ 2, there are positive constantsc ₁(r) andc ₂(r) such thatc ₁(r) ≤f _r (n)?n ^?[r/2] ≤n ₂(r) for alln ≥ r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.     

Maximal extensions of partial orders preserving antimonotonicity

Let (A, ?? _r ) be a partially ordered set and f : A ?? A be an order reversing (antimonotone) map. We characterize the maximal partial order extensions of r preserving the antimonotonicity of f.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

Optimal adaptive solution of initial-value problems with unknown singularities

The optimal solution of initial-value problems in ODEs is well studied for smooth right-hand side functions. Much less is known about the optimality of algorithms for singular problems. In this paper, we study the (worst case) solution of scalar problems with a right-hand side function having r   continuous bounded derivatives in R$\mathbf{R}$, except for an unknown singular point. We establish the minimal worst case error for such problems (which depends on r similarly as in the smooth case), and define optimal adaptive algorithms. The crucial point is locating an unknown singularity of the solution by properly adapting the grid. We also study lower bounds on the error of an algorithm for classes of singular problems. In the case of a single singularity with nonadaptive information, or in the case of two or more singularities, the error of any algorithm is shown to be independent of r.     

Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds

In this paper, we investigate the Dirichlet eigenvalue problems of second order elliptic operators in divergence form on bounded domains of complete Riemannian manifolds. We discuss the cases of submanifolds immersed in a Euclidean space, Riemannian manifolds admitting spherical eigenmaps, and Riemannian manifolds which admit l functions ${f_\alpha : M \longrightarrow \mathbb{R}}$ such that ${\langle \nabla f_\alpha, \nabla f_\beta \rangle = \delta_{\alpha \beta}}$ and Δf _α = 0, where ? is the gradient operator. Some inequalities for lower order eigenvalues of these problems are established. As applications of these results, we obtain some universal inequalities for lower order eigenvalues of the Dirichlet Laplacian problem. In particular, the universal inequality for eigenvalues of the Laplacian on a unit sphere is optimal.     

A non-steady flow of liquid in a porous pipe with variable permeability

LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent R_λ, on Banach space L. We define related operators P and V so that λR_λf = Pf + λVf + o(λ), as λ → 0⁺. For α, η > 0 and possibly unbounded, linear operator A, we let U_{α, η}(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, U_{α, η}(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)^rf where r =min{j ? 0, PA(VA)^j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.     

Second Order Regularity for the A-Laplace Operator

In this paper we establish second order regularity for the quasilinear elliptic equation Δ _A u =  f, where Δ _A is the so called A-Laplace operator.     

奇异椭圆方程组径向正解的存在性

Abstract. The existence of positive radial solutions to the systems of     

Singular and nonsingular first-order initial value problems

Under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x^′=f(t,x,x^′), x(0)=A, where the scalar function f(t,x,p) may be singular at x=A.