Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> linear interpolator for missing values in time series

We propose a minimum mean absolute error linear interpolator (MMAELI), based on theL ₁ approach. A linear functional of the observed time series due to non-normal innovations is derived. The solution equation for the coefficients of this linear functional is established in terms of the innovation series. It is found that information implied in the innovation series is useful for the interpolation of missing values. The MMAELIs of the AR(1) model with innovations following mixed normal andt distributions are studied in detail. The MMAELI also approximates the minimum mean squared error linear interpolator (MMSELI) well in mean squared error but outperforms the MMSELI in mean absolute error. An application to a real series is presented. Extensions to the general ARMA model and other time series models are discussed. This research was supported by a CityU Research Grant and Natural Science Foundation of China.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> spline fits via sliding window process: continuous and discrete cases

The best L ₁ approximation of the Heaviside function and the best ? ₁ approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L ₁ spline fits which are the best L ₁ approximations in an appropriate spline space obtained by the union of L ₁ interpolation splines. We prove here the existence of L ₁ spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.     

<Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript>-continued fraction representation of real numbers and its geometry

We study the geometry of representations of numbers by continued fractions whose elements belong to the set A ₂ = {α₁, α₂} (A ₂-continued fraction representation). It is shown that, for α₁α₂ ≤ 1/2 , every point of a certain segment admits an A ₂-continued fraction representation. Moreover, for α₁α₂ = 1/2, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose A ₂-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its A ₂-continued fraction representation form a homogeneous Markov chain are also investigated.     

(<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

We prove that the conditions of (q₁, 1)- and (1, q₂)-quasimertricity of a distance function ρ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to ρ. It follows that the Box-quasimetric defined with the use of basis vector fields of class C₁ whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of (q₁, q₂)-quasimetrics for which there are no Lipschitz equivalent 1-quasimetrics, which in particular implies another proof of a result by V. Schröder.     

Interpolation of nonlinear operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We show that Peetre’s classical interpolation theorem in weighted L _p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.     

New examples of <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-saturated Banach spaces

For every p ∈ (1, ∞), an isomorphically polyhedral Banach space E _p is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact metric space K. Moreover, E _p admits a quotient isomorphic to ℓ _p .     

A new non-interior continuation method for <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-NCP based on a SSPM-function

In this paper, we consider a new non-interior continuation method for the solution of nonlinear complementarity problem with P ₀-function (P ₀-NCP). The proposed algorithm is based on a smoothing symmetric perturbed minimum function (SSPM-function), and one only needs to solve one system of linear equations and to perform only one Armijo-type line search at each iteration. The method is proved to possess global and local convergence under weaker conditions. Preliminary numerical results indicate that the algorithm is effective.     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

The Almost Contact Metric Hypersurfaces with Small Type Numbers in <Emphasis Type="Italic">W</Emphasis><Subscript>4</Subscript>-manifolds

It is proved that the almost contact metric structures on a hypersurface with type number 1 and on a hypersurface with type number 0 are identical in a W₄-manifold.     

Problem of uniqueness of an element of the best nonsymmetric <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of continuous functions with values in <Emphasis Type="Italic">KB</Emphasis>-spaces

We consider the problem of characterization of subspaces of uniqueness of an element of the best nonsymmetric L ₁-approximation of functions that are continuous on a metric compact set of functions with values in a KB-space. We find classes of test functions that characterize the uniqueness of an element of the best nonsymmetric approximation. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 867–878, July, 2008.     

On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of christoffel symbols of the second kind under infinitesimal deformations of surfaces in the euclidean space <Emphasis Type="Italic">E</Emphasis><Subscript>3</Subscript>

We investigate the problem of reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of Christoffel symbols of the second kind under infinitesimal deformations of surfaces in the Euclidean space E ₃.     

An <Emphasis Type="Italic">L</Emphasis><Subscript>\infty</Subscript>/<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L _\infty norm and in the L ₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

LMI-Based Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Uncertain Neutral Systems with State and Input Delays

Robust stabilization and robust H_∞ control problems for a class of uncertain neutral system with state and input delays are considered. By choosing a Lyapunov-Krasovskii functional, an alternative delay-dependent sufficient condition for the existence of a controller is derived in terms of linear matrix inequalities. Furthermore, a convex optimization problem can be formulated to obtain an H_∞ controller which minimizes an upper H_∞ norm bound of the closed-loop state-input delayed system.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Output Dynamic Observer-Based Control Design of Uncertain Neutral Systems

In this paper, the robust H _∞ control problem of output dynamic observer-based control for a class of uncertain neutral systems is considered. The linear matrix inequality optimization approach is used to design the new H _∞ output dynamic controls. Three classes of H _∞ observer-based controls are proposed. The minimal H _∞-norm bound and the maximal perturbed bound are given. Based on the result of this paper, the constraint of matrix equality is not necessary for designing the H _∞ observer-based controls. A numerical example is given to stress the usefulness of the proposed results. Communicated by C. T. Leondes The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.