Inverse problems in queueing theory and Internet probing

Queueing theory is typically concerned with the solution of direct problems, where the trajectory of the queueing system, and laws thereof, are derived based on a complete specification of the system, its inputs and initial conditions. In this paper we point out the importance of inverse problems in queueing theory, which aim to deduce unknown parameters of the system based on partially observed trajectories. We focus on the class of problems stemming from probing based methods for packet switched telecommunications networks, which have become a central tool in the measurement of the structure and performance of the Internet. We provide a general definition of the inverse problems in this class and map out the key variants: the analytical methods, the statistical methods and the design of experiments. We also contribute to the theory in each of these subdomains. Accordingly, a particular inverse problem based on product-form queueing network theory is tackled in detail, and a number of other examples are given. We also show how this inverse problem viewpoint translates to the design of concrete Internet probing applications.     

Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm–Liouville relations with separate boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Branges? subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa–Holm equation.     

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.     

Direct and inverse theorems of approximation theory for a generalized modulus of smoothness

An asymmetric operator of generalized translation is introduced in this paper. Using this operator, we define a generalized modulus of smoothness and prove direct and inverse theorems of approximation theory for it.     

Inverse scattering theory for one-dimensional Schrödinger operators with steplike finite-gap potentials

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allows unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.     

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L _p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

Group theory via global semigroup theory

Groups are shown to be special homomorphic images of inverse semigroups that are residually finite (actually: every element has only finitely many elements -above). This also leads to a new approach to the Burnside problem. These results extend an earlier paper ([1.], 249–287), but can be read independently. Our goal here is not so much to prove theorems about inverse semigroups as to demonstrate the usefulness of the constructions of [1.], 249–287.     

On Inverse Limits and Direct Limits for Fuzzy Topological Spaces

Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.     

The finite section method and problems in frame theory

The finite section method is a convenient tool for approximation of the inverse of certain operators using finite-dimensional matrix techniques. In this paper we demonstrate that the method is very useful in frame theory: it leads to an efficient approximation of the inverse frame operator and also solves related computational problems in frame theory. In the case of a frame which is localized w.r.t. an orthonormal basis we are able to estimate the rate of approximation. The results are applied to the reproducing kernel frame appearing in the theory for shift-invariant spaces generated by a Riesz basis.     

Numerical methods in the theory of viscoelasticity

Formulas are given for calculating the coefficients of differential operators of defining equations on the basis of given approximations of the relaxation kernels in the form of the sum of exponential curves. As the defining equations it is suggested to use quadrature formulas into which are substituted the relaxation kernels found experimentally without preliminary analytic approximation. A three-dimensional difference problem of the linear isotropic theory of viscoelasticity is formulated. The direct and inverse -transformation establishing the correspondence between the viscoelastic and elastic difference problems is introduced. The specific characteristics of the use of the net, Ritz, finite-element, and variation-difference methods in solving problems of viscoelasticity theory are examined. A method facilitating the arrangement of the information on relaxation kernels in a computer memory is indicated.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 417–428, May–June, 1973.     

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L _p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.     

On the direct and inverse problems in fluid-structural interaction

This paper is concerned with the direct and inverse scattering problems in fluid-structure interaction. The scattering problem in the fluid-structure interaction can be simply described as follows: an acoustic wave propagates in the fluid domain of infinite extent where a bounded elastic body is immersed. The direct problem is to determine the scattered pressure and velocity fields in the fluid domain as well as the displacement fields in the elastic body, while the inverse problem is to reconstruct the shape of the elastic scatterer from a knowledge of the far field pattern of the fluid pressure or from the measured scattered fluid pressure field. As is well known, the inverse problems are generally nonlinear and highly ill-posed. For treating inverse problem of this kind, we reformulate the problem as a nonlinear optimization problem including special regularization terms. The precise formulation of the nonlinear objective functional will depend on the approaches of the direct problem. In this paper, the direct problem is reformulated by introducing an artificial boundary and the corresponding inverse problem will be analyzed. Some of the basic results are summarized without proofs. The latter are available in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The theory of inverse boundary problems for analytic functions and its applications

Questions of the well-posedness of inverse boundary problems on the plane and on Riemann surfaces, results on the univalent solvability of inverse boundary problems and on the corresponding parts of the geometric theory of functions of a complex variable, various inverse problems of hydroaeromechanics, and also inverse problems of pressured and unpressured filtration, mathematical and applied aspects of inverse problems on change of contours are presented.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 67–124, 1980.     

Direct and inverse dynamic problems for SH-waves in porous media

Direct and inverse dynamic problems for the equation of SH-waves in porous media are considered. A singular solution of the direct dynamic problem is constructed. A system of nonlinear Volterra integral equations of the second kind is obtained for the dynamic inverse problems in question. Theorems of uniqueness and theorems of existence in the small for the considered inverse problems are proved. Also, theorems of continuous dependence of solutions of inverse dynamic problems on input data are proved.     

Method for Solving the Sine-Gordon Equation in Laboratory Coordinates

By solving the direct and inverse scattering problems for a rather ‘unconventional’ eigenvalue problem, we can solve the initial value problem for the sine-Gordon equation entirely in laboratory coordinates. This allows us to extend and accurately define the class of initial value problems which can be solved by ‘inverse scattering transforms.’ Simple examples are given to illustrate both the direct and the inverse scattering methods.     

Solvability of the Direct and Inverse Problems for the Nonlinear Schrödinger Equation

In this paper we study rigorous spectral theory and solvability for both the direct and inverse problems of the Dirac operator associated with the nonlinear Schrödinger equation. We review known results and techniques, as well as incorporating new ones, in a comprehensive, unified framework. We identify functional spaces in which both direct and inverse problems are well posed, have a unique solution and the corresponding direct and inverse maps are one to one.Mathematics Subject Classifications (2000) 34A55, 35Q55.     

修正高阶Hermite插值及Hermite-Fejer插值在Lpw空间中逼近的正逆定理

在LPW空间中引入了一种K-泛函并由此建立了一种以第一类Chebyshev多项式的零点为结点的三种修正高阶Hermite-Fejer插值多项式及一种修正的高阶Hermite插值多项式在LPW空间中逼近的正逆定理.文中的结果说明,对于这几种修正高阶多项式插值的逼近问题而言,正定理的解决意味着逆定理的解决.     

Domain theory and mirror properties in inverse semigroups

Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent elements, such as continuity in the sense of domain theory.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]