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Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of [Formula omitted] for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we show that the KL exponent is [Formula omitted] for several important convex optimization models, including some semidefinite-programming-representable functions and some functions that involve [Formula omitted]-cone reducible structures, under conditions such as strict complementarity. Our results are applicable to concrete optimization models such as group-fused Lasso and overlapping group Lasso. In addition, for nonconvex models, we show that the KL exponent of many difference-of-convex functions can be derived from that of their natural majorant functions, and the KL exponent of the Bregman envelope of a function is the same as that of the function itself. Finally, we estimate the KL exponent of the sum of the least squares function and the indicator function of the set of matrices of rank at most k., Author(s): Peiran Yu [sup.1] , Guoyin Li [sup.2] , Ting Kei Pong [sup.1] Author Affiliations: (1) grid.16890.36, 0000 0004 1764 6123, Department of Applied Mathematics, The Hong Kong Polytechnic University, [...]

We initiate the study of congestion games with variable demands in which the players strategically choose both a nonnegative demand and a subset of resources. The players' incentives to use [...]

Nonstandard Lagrangians are generating functions of different equations of motion. They have gained increasing importance in many different fields. In fact, nonstandard Lagrangians date back to 1978, when Arnold entitled them 'nonnatural' in his classic book, Mathematical Methods of Classical Mechanics (Springer, New York. 1978). In applied mathematics, most dynamical equations can be obtained by using generating Lagrangian functions (e.g., power-law and exponential Lagrangians), which has been shown by mathematicians, who have also demonstrated that there is an infinite number of such functions. Besides this interesting field, the topic of fractional calculus of variations has gained growing importance because of its wide application in different fields of science. In this paper, we generalize the fractional actionlike variational approach for the case of a nonstandard exponential Lagrangian. To appreciate this new approach, we explore some of its main consequences in Einstein's general relativity. Some results are revealed and discussed accordingly mainly the transition from general relativity to complex relativity and emergence of a discrete gravitational coupling constant. PACS Nos.: 02.30.Xx, 04.20.-q, 04.20.Fy. Les Lagrangiens non standard sont des fonctions generatrices d'equations differentes du mouvement. Ils sont de plus en plus importants dans differents domaines scientifiques. En fait, les Lagrangiens non standard datent de 1978, lorsque Arnold les a baptises de non naturels dans son livre Methodes Mathematiques de la Mecanique Classique (Springer, New York. 1978). En mathematiques appliquees, la plupart des equations dynamiques peuvent etre obtenues de fonctions generatrices de Lagrange (par exemple, des Lagrangiens en loi de puissance ou exponentiels), ce qui a ete illustre par des mathematiciens qui ont aussi demontre qu'il y a un nombre infini de telles fonctions. A cote de ce champ, le calcul fractionnaire des variations croit en importance, a cause de son large domaine d'applications en sciences. Ici, nous generalisons l'approche variationnelle de type action fractionnelle pour un Lagrangien exponentiel non standard. Pour mieux apprecier cette nouvelle approche, nous explorons certaines de ses consequences majeures en relativite generale (RG) d'Einstein. Quelques resultats sont presentes et nous analysons surtout le passage de la RG a la relativite complexe et une constante de couplage gravitationnel discrete. [Traduit par la Redaction], 1. Introduction The nonstandard Lagrangian (NSL) dates back to 1978 when Arnold entitled them 'non-natural' in his classic book, Mathematical Methods of Classical Mechanics [1]. NSL plays a significant role [...]

The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]\[a.sub.n] for every i [member of] {1, 2, ..., s}. The exponential convolution of aithmetic functions is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The inverse of the constant function with respect to [mid dot] is called the exponential analogue of the Mobius function, which is denoted by p,(e). The aim of this paper is to establish a short interval result for the function [[mu].sup.(e)]. Keywords The exponential divisor function, generalized divisor function, short interval., [section]1. Introduction Let n > 1 be an integer of canonical from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential [...]

An integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]\[a.sub.i] for every i [member of] {1, 2, ..., s}. Let [[tau].sup.(e)](n) denote the number of exponential divisors of n, where [[tau].sup.(e)](1) = 1 by convention. The aim of this paper is to establish a short interval result for -r-th power of the function [[tau].sup.(e)] for any fixed integer r [greater than or equal to] 1. Keywords The exponential divisor function, arithmetic function, short interval., [sections]1. Introduction Let n > 1 be an integer of canonical from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. An integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the exponential [...]

In this paper we shall study the mean value of the exponential divisor function involving a negative r-th power by the convolution method. This gives a comparison with the result of Laszlo Toth. Keywords The exponential divisor function, the general divisor function, Euler product, convolution method., [section] 1. Introduction The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [b.sub.i] | [a.sub.i] for every i [...]

The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every i [member of] 1, 2, ...; s. Let [[tau].sup.(e)](n) denote the number of exponential divisors of n, where [[tau].sup.(e)](1) = 1 by convention. In this paper we shall establish a short interval result for r-th power of the function [[tau].sup.(e)], where r [greater than or equal to] 1 is a fixed integer. Keywords The exponential divisor function, generalized divisor function, short interval., [section] 1. Introduction The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [b.sub.i]|[a.sub.i] for every i [member of] [...]

Let [t.sup.(e)] (n) denote the number of e-squarefree e-divisor of n. The aim of the present paper is to establish an asymptotic formula for the mean value of the function ([t.sup.(e)])r, where r [greater than or equal to] 1 is a fixed integer. Keywords E-squarefree e-divisor function, the generalized divisor function, convolution method., [section] 1. Introduction An integer d = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an e-divisor of n =[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]|[a.sub.i] for [...]

1. Introduction Let F(x) denote a distribution function (d.f.) in [R.sup.d] with F(0+) = 0 and F(x) < 1, [for all]x [member of] [R.sup.d]. Let X, [X.sub.1], [X.sub.2],..., [X.sub.n] denote [...]

Abstract In this note there are introduced two new concepts of e-harmonic numbers, and a new concept of e-perfect number. Their initial study is provided. Keywords Exponential divisors, harmonic numbers, [...]

ABSTRACT. We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. E. Cartan's [...]

We show in this paper that stochastic processes associated with nonlinear parabolic equations and systems allow one to construct a probabilistic representation of a generalized solution to the Cauchy problem. [...]

A combination of moving averages has been shown previously to be more accurate than simple moving averages, under certain conditions, and to be more robust to non-optimal parameter specification. However, the use of the method depends on specification of three parameters: length of greater moving average, length of shorter moving average, and the weighting given to the former. In this paper, expressions are derived for the optimal values of the three parameters, under the conditions of a steady state model. These expressions reduce a three-parameter search to a single-parameter search. An expression is given for the variance of the sampling error of the optimal combination of moving averages and this is shown to be marginally greater than that for exponentially weighted moving averages (EWMA). Similar expressions for optimal parameters and the resultant variance are derived for equally weighted combinations. The sampling variance of the mean of such combinations is shown to be almost identical to the optimal general combination, thus simplifying the use of combinations further. It is demonstrated that equal weight combinations are more robust than EWMA to noise to signal ratios lower than expected, but less robust to noise to signal ratios higher than expected. Keywords: forecasting; time-series; moving averages; combinations of forecasts; exponentially weighted moving averages

Byline: I.I. Guseinov (1), B.A. Mamedov (2) Keywords: exponential type orbitals; Slater type orbitals; overlap integrals; multicenter integrals Abstract: In this study, using complete orthonormal sets of exponential type orbitals (ETOs), a single closed analytical relation is derived for a large number of different expansions of overlap integrals over Slater type orbitals (STOs) with the same screening parameters in terms of Gegenbauer coefficients. The general formula obtained for the overlap integrals is utilized for the evaluation of multicenter nuclear attraction and electron repulsion integrals appearing in the Hartree--Fock--Roothaan equations for molecules. The formulas given in this study for the evaluation of these multicenter integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, scaling parameters of STOs and internuclear distances. Author Affiliation: (1) Department of Physics, Faculty of Arts and Sciences, Onsekiz Mart University, Canakkale, Turkey (2) Department of Physics, Faculty of Arts and Sciences, GaziosmanpaAa University, Tokat, Turkey Article History: Registration Date: 09/10/2004

Byline: J. Vigo-Aguiar (1), T.E. Simos (2) Keywords: symmetric multistep methods; exponential fitted methods; Schrodinger equation Abstract: In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrodinger equation. This application shows the efficiency of the new family of methods. Author Affiliation: (1) Department of Applied Mathematics, University of Salamanca, E37008, Salamanca, Spain (2) Section of Mathematics, Democritus University of Trace, Greece Article History: Registration Date: 09/10/2004

Abstract: The self-adjoint elliptic boundary-value problem in a domain with cylindrical outlets to infinity is considered. The notion of an augmented scattering matrix is introduced on the basis of artificial radiation conditions. Properties of the augmented scattering matrix are studied, and the relationship with the classical scattering matrix is demonstrated. The central point is the possibility of calculating the number of linearly independent solutions of a homogeneous problem with fixed rate of decrease at infinity by analyzing the spectrum of the augmented scattering matrix. This property is applied to the problem on diffraction on a periodic boundary as an example. Bibliography: 21 titles. Article History: Registration Date: 21/10/2004

Byline: B. Dittmar (1), A. Yu. Solynin (2) Abstract: This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205a218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity [delta].sub.h(A,[OMEGA]) of the boundary set Aaa[OMEGA] under a conformal mapping of a domain [OMEGA]a U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality cap.sub.h f(A) [delta] .sub.h (A,[OMEGA] ) is sharp in the class of conformal mappings f:[OMEGA][right arrow] U such that f(a U)=a U. Here cap.sub.h f(A) is the hyperbolic capacity of a compact set f(A)a U. We give some examples demonstrating properties of functions which realize the case of equality in relation (*). Bibliography: 15 titles. Author Affiliation: (1) Fachbereich Mathematik und Informatik Halle, Martin-Luther Universitat zu Halle-Wittenberg, Deutschland (2) St.Petersburg Department of the Steklov Mathematical Institute, Russia Article History: Registration Date: 18/10/2004

Byline: G. Avdelas (1), E. Kefalidis (2), T.E. Simos (1) Keywords: Schrodinger equation; exponential fitting; P-stability; finite difference; multistep methods; scattering problems Abstract: A family of P-stable high algebraic order exponentially-fitted methods for the numerical solution of the Schrodinger equation is developed in this paper. Numerical illustration to the resonance problem of the radial Schrodinger equation indicates that the new proposed methods are generally more efficient than the previously developed exponentially-fitted methods of the same kind. Author Affiliation: (1) Laboratory of Applied Mathematics and Computers, Department of Sciences, Technical University of Crete, Kounoupidiana, GR-731 00 Chania, Crete, Greece (2) Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR-671 00, Xanthi, Greece Article History: Registration Date: 12/10/2004

The dual, cost and revenue interpretation of the hyperbolic efficiency measures is shown to be related to Georgescu-Roegen's notion of return to the dollar.

Byline: O. L. Semenova (1) Abstract: Let G be a purely hyperbolic, finitely generated, nonelementary Fuchsian group of the second kind. If [LAMBDA] is the limit set of the group G, then the function ( dist( x,[LAMBDA] ) ) belongs to the class BMO. This follows from the fact that [LAMBDA] is porous, proved in the paper. Bibliography: 4 titles. Author Affiliation: (1) St.Petersburg State University, Russia Article History: Registration Date: 11/10/2004

Byline: A. A. Balkema (1), C. Kluppelberg (2), S. I. Resnick (3) Author Affiliation: (1) University of Amsterdam, NL-1018TV, Amsterdam, Holland (2) Munich University of Technology, D-80290, Munich, Germany (3) Cornell University, Ithaca, NY, 14853-7501, USA Article History: Registration Date: 20/10/2004

Research examining the acoustics algorithms of the exponentiated gradient with negative and positive weights, the algorithm of gradient descent and the algorithm of the exponentiated gradient with unnormalized negative and positive weights is presented.

Abstract: In this article, we develop and substantiate a two-step, explicit--implicit numerical algorithm for solving boundary-value problems for systems of partial hyperbolic differential equations of the first order. The algorithm is based on finite-difference schemes with the differences taken against the flow. The approximation error of the solution has order two in the time variable and order one in the spatial variables. Article History: Registration Date: 06/10/2004

Research investigating the prediction intervals for monotonic systems with incomplete data is presented. Particular attention is given to analyses of research based on a monotonic system with components which are statistically independent.

Polygons can be arranged through a hyperbolic plane in a rich pattern, such as seen in the art of M. C. Escher. Several propositions for variations in this exercise are offered, along with proofs.

A group-theoretic approach to spline interpolation is presented. The interpolant and its inverse show the same functional form.

Byline: T.E. Simos (1) Abstract: A family of Pastable exponentiallyafitted methods for the numerical solution of the Schrodinger equation is developed in this paper. An application to the resonance problem of the radial Schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentiallyafitted methods of the same kind. Author Affiliation: (1) Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GRa67100, Xanthi, Greece Article History: Registration Date: 14/10/2004

A discontinuous Galerkin method was formulated for the hyperbolic conservation laws without the use of quadrature formulas. The method's accuracy was then demonstrated by its application to both linear and nonlinear test problems in different space dimensions. In addition, operation count was found to be less than that for quadrature-based methods. Further testing of the method demonstrated its suitability to both unstructured and structured grids.

A first-best sharing rule was obtained for Holmstrom and Milgrom's continuous-time principal-agent model with exponential utility. The first-best sharing rule, like the second-best sharing rule, is found to be a linear function of aggregated output. The rule was derived considering the fact that the principal and the agent both possess CARA utility and the problem under consideration is separable.

A modified parallel-in-parallel-out linear-systolic power-sum circuit designed to perform [AB.sup.2] + C computations in the finite field GF([2.sup.m]) is presented, where A.B. and C are arbitrary elements of GF([2.sup.m]). On the basis of the linear-systolic power-sum circuits, a VLSI architecture for exponentiation in GF([2.sup.m]) is developed. Furthermore, two modified architectures that can be used to compute inverses and divisions over GF([2.sup.m]) are proposed. All the architectures are constructed from m 1 linear-systolic power-sum circuits. It should be noted that the presented exponentiator, inverter, and divider are the only such circuits having a throughput of 100%. The latency of the presented pipeline exponentiator, inverter, and divider is m(m - 1) clock cycles. The cycle time (i.e., clock period) of the presented architectures is only two logic gate delays plus a short routing delay. For moderate values of m, say m [less than or equal to] 10. the circuit complexity of the presented circuits is realizable using presently available VLSI technology. The computation time of near two gate delays and 100% throughput enables the greatest computation speed in finite field arithmetic. Index Terms - Division, exponentiation, finite field arithmetic, multiplicative inverse, VLSI architecture.

Four subclasses of the hyperbolic class (HC) of quadratic/bilinear time-frequency representations are examined, namely, the localized-kernel subclass, the affine subclass, the power subclass and the power-warp subclass. One-dimensional kernel functions are used to characterize the subclasses. The properties of regularity and unitarity in the HC are also analyzed.

The homogeneity of uniform scale distributions is analyzed and compared with that of two-sided and one-sided alternatives. The results show that for both two- and one-sided alternatives, each test in the class of constant-size permutations is unbiased. The exact and asymptotic distribution of the likelihood ratio test are derived., 1. INTRODUCTION AND SUMMARY Consider the model where [X.sub.ij], i = 1, 2, ..., k; j = 1, 2, ..., [n.sub.i] are independent uniform random variables with scale parameter [[Theta].sub.i] [...]

The evolution of common generalizations from pairs or triplets of hyperbolic functions is useful in some areas but unsuccessful in other fields. The common generalizations in the area of q-analogues, inequalities, the nonlinear generalization of hyperbolic functions and the common generalizations of two hyperbolic integrals are discussed.

A model that facilitates the joint determination of product durability and order quantity for perishable goods is introduced. The model represents product durability choices in terms of a design parameter that influence the time-to-onset of deterioration (TOD) distribution. Using it, two scenarios were explored under the assumption of deterministic exponential decay. The first pertains to products with an expiry date while the second involves products with a random (TOD).

A finite mixture distribution consists of the super-position of a finite number of component probability densities, and is typically used to model a population composed of two or more subpopulations. Mixture models find utility in situations where there is a difficulty in directly observing the underlying components of the population of interest. This paper examines the method of moments as a general estimation technique for estimating the parameters of the component distributions and their mixing proportions. It is shown that the same basic solution can be applied to any continuous or discrete density from the exponential family with a known common shape parameter. Results of an empirical study of the method are also presented.

Usage of the symbolic program Derive is a versatile methodology, which saves on time in computing the matrix of exponential functions. This program has the potential to substitute matrices into scalar polynomials, perform integration and simplify matrix expressions, thereby providing insightful step-by-step details in symbolic computations.

Both exponential and stochastic stabilities of the Hopfield neural network are analyzed. The results are especially useful for analyzing the stabilities of asymmetric neural networks. A constraint on the connection matrix has been found under which the neural network has a unique and exponentially stable equilibrium. Given any connection matrix, this constraint can be satisfied through the adjustment of the gains of the amplifiers and the resistances in the neural net circuit. A one-to-one and smooth map between input currents and the equilibria of the neural network can be set up. The above results can be applied to the Master/Slave net to prove that the master net can find the best connection matrix for the slave net. For the neural network disturbed by some noise, the stochastic stability of the network is also analyzed. A special asymmetric neural network formed by a closed chain of formal neurons is also studied for its stability and oscillation. Both stable and oscillatory dynamics are obtained in the closed chain network through the adjustment of the gains and resistances of the amplifiers.

We use an economic approach of Mendel to derive new bivariate exponential lifetime distributions. Features distinguishing this approach from the existing ones are (1) it makes use of the principle of indifference: (2) our parameter of interest is a measurable function of observable quantities; (3) the assessment of the probability measure for random lifetimes is performed by assessing that for random lifetime costs with a change of variables; and (4) characterization properties other than the bivariate loss-of-memory property are used to construct distributions. For the infinite population case, our distributions correspond to mixtures of existing bivariate exponential distributions such as the Freund distribution, the Marshall-Olkin distribution, and the Friday-Patil distribution. Moreover, a family of natural conjugate priors for Bayesian Freund(-type) bivariate exponential distributions is discussed. KEY WORDS: Freund distribution; Friday-Patil distribution; [l.sub.1]-isotropy; Marshall-Olkin distribution; Principle of indifference., A Mendelian economic approach is used to generate new bivariate lifetime distributions. The new technique applies the principle of indifference, provides a parameter of interest that can be described functionally from observable variables, allows random lifetime probability measure assessment and develops distributions beyond the bivariate loss-of-memory property. The approach also corresponds to different distributions in the infinite population case., 1. INTRODUCTION The exponential distribution is frequently used as a component lifetime distribution in reliability theory because of its distinctive loss-of-memory property or constant failure rate. In other words, the [...]

Step responses of an exponential line terminated with resistive loads are evaluated by means of time-domain scattering parameters. The results are used to illustrate the interaction between a nonuniform line and a nonlinear load. We find that an exponential line could be used to compensate the nonlinear load effects in high-speed backdriving.