An error analysis for a certain class of iterative methods

We provide local convergence results in affine form for in-exact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first andmth Fréchet-derivative (m ≥ 2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Fréchet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.     

Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem

The implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a (possibly infinite-dimensional) parameter. In this way, results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions. The abstract results are applied to a linear second order elliptic eigenvalue problem with nonlocal unilateral boundary conditions (Schrödinger operator with the potential as the parameter).     

Effective Isometric Embeddings for Certain Hermitian Holomorphic Line Bundles

We consider bihomogeneous polynomials on complex Euclidean spacesthat are positive outside the origin and obtain effective estimateson certain modifications needed to turn them into squares ofnorms of vector-valued polynomials on complex Euclidean space.The corresponding results for hypersurfaces in complex Euclideanspaces are also proved. The results can be considered as Hermitiananalogues of Hilbert's seventeenth problem on representing apositive definite quadratic form on Rⁿ as a sum of squares ofrational functions. They can also be regarded as effective estimateson the power of a Hermitian line bundle required for isometricprojective embedding. Further applications are discussed.     

Companion <Emphasis Type="Italic">d</Emphasis>-algebras

In this paper we develop a theory of companion d-algebras in sufficient detail to demonstrate considerable parallelism with the theory of BCK-algebras as well as obtaining a collection of results of a novel type. Included among the latter are results on certain natural posets associated with companion d-algebras as well as constructions on Bin(X), the collection of binary operations on the set X, which permit construction of new companion d-algebras from companion d-algebras X also in natural ways. Supported by Korea Research Foundation Grant (KRF-2002-041-C00003).     

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

The spherical functions on a real semisimple Lie group (w.r.t. a maximal compact subgroup) are characterized as joint eigenfunctions of certain differential operators on the corresponding complex group. Using this, several results concerning the spherical Fourier transform on the real group are reduced to the corresponding results for the complex group.When the group in question is a normal real form, this leads to new and simpler proofs of such results as the Plancherel formula, the Paley-Wiener theorem and the characterization of the image under the spherical Fourier transform of the L¹- and L²-Schwartz spaces. In these proofs neither any knowledge of Harish-Chandras c-function nor the series expansion for the spherical function are used.For the proof of the main result some analysis of independent interest on pseudo-Riemannian symmetric spaces is developed. Such as a generalized Cartan decomposition and a method of analytic continuation between two “dual” pseudo-Riemannian symmetric spaces.     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

Trees as Brelot spaces

A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many results on the potential theory of trees. By restricting the functions just to the vertices, we obtain several new results on the potential theory of trees considered in the usual sense. We study trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks. Besides the usual case of harmonic functions on trees (the kernel of the Laplace operator), we also consider as “harmonic” the eigenfunctions of the Laplacian relative to a positive eigenvalue showing that these also yield a Brelot structure and creating new classes of functions for the study of potential theory on trees.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

Log-concavity for Bernstein-type operators using stochastic orders

This paper aims to study the preservation of log-concavity for Bernstein-type operators. In particular, attention is focused on positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of a process with independent increments. This class includes the classical Gamma, Szász, and Szász–Durrmeyer operators. With respect to the first and second operators, the results of this paper correct two erroneous counterexamples in [10]. As a main tool in our results we use stochastic order techniques. Our results include, as a particular case, the log-concavity of certain functions related to the incomplete Gamma function.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Asymptotic behaviour of the containment of certain mesh patterns

We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.     

Effects of Young’s modulus on response of railway sleeper

As a main part of a railroad system, sleepers have important duty in conveying the load from rails to the ballast. The different situations in which the sleepers should function necessitate making them from different materials, such as various types of wood, reinforced concrete and even steel. In this work, the effects of Young’s modulus on response of railway sleeper are evaluated. As a main consideration, Winkler’s theorem is used to model the behavior of the elastic foundation. First, the response of a sleeper on a Winkler’s foundation is found. To evaluate the results of the closed form solution, a finite element model is used. Good agreement between the results of the closed form solution and the finite element model proves the validity of the results. In the next stage, the Young’s modulus is considered as a variable and the fundamental diagrams of the beam are plotted based on the variation of Young’s modulus.     

Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.     

FUNCTIONAL IDENTITIES REVISED: THE FRACTIONAL AND THE STRONG DEGREE

ABSTRACT

The concepts of the fractional and the strong degree of an element in a ring are introduced. It is shown that definitive results on functional identities can be obtained in rings which contain elements of appropriate fractional (or strong) degree. This enables us to extend the results on functional identities from prime to semiprime rings, as well as to some rather different classes of rings, such as matrix rings over any unital ring. As an application, commuting maps, Lie derivations and commutativity preserving maps in such rings are discussed.     

On the simplicial <Emphasis Type="Italic">BF</Emphasis> model

This note is a brief introduction to the results on the simplicial BF model obtained by the author in the framework of the program proposed by A. Losev. These results and more comprehensive explanations will be published elsewhere. We regard them as a step toward solving the problem of constructing the combinatorial Chern-Simons theory, proposed by M. Atiyah. We also popularize the algebraic view on the simplicial BF model as a deformation of the de Rham algebra on a manifold in the (yet to be defined) category of “quantum L_∞-algebras.” Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 84–90.     

On the Convergence of Newton-Like Methods Based on M-Fréchet Differentiable Operators and Applications in Radiative Transfer

In this study we approximate a locally unique solution of a nonlinear operator equation in Banach space using Newton-like methods. A complete error analysis of our method is also given. Our new theorem uses Lipschitz or Hölder continuity assumptions on m-Fréchet-differentiable operators where m 2 is a positive integer. A numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results. A second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer.     

On generalized Erd?s spaces

During the last years both Erd?s space and complete Erd?s space were topologically characterized by Dijkstra and van Mill. Applications include results about Erd?s type spaces in ?p-spaces as well as results about Polishable ideals on ω. We present an unifying theorem in terms of sets with a reflexive relation that among other things contains these apparently dissimilar results as special cases.     

An analysis of neighborhood functions on generic solution spaces

The effectiveness of local search algorithms on discrete optimization problems is influenced by the choice of the neighborhood function. A neighborhood function that results in all local minima being global minima is said to have zero L-locals. A polynomially sized neighborhood function with zero L-locals would ensure that at each iteration, a local search algorithm would be able to find an improving solution or conclude that the current solution is a global minimum. This paper presents a recursive relationship for computing the number of neighborhood functions over a generic solution space that results in zero L-locals. Expressions are also given for the number of tree neighborhood functions with zero L-locals. These results provide a first step towards developing expressions that are applicable to discrete optimization problems, as well as providing results that add to the collection of solved graphical enumeration problems.