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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 [...]

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, [...]

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

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.

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.

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.

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 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).

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.

A direct derivation of the solution of the discounted exponential linear quadratic Gaussian optimal control problem is presented. Some extensions to partial observations problems are also given. The solution involves time-varying Riccati equations, providing an example of an infinite horizon, constant coefficient, regulator problem with a time-varying optimal control.

This note presents frequency-domain characterizations of exponential stability and stabilizability of neutral systems based on transfer-function matrices and the existence of 'nice' solutions of certain Bezout equations. It turns out that the existence of H infinity-solutions is not sufficient for exponential stabilizability, but that they have to satisfy an additional growth assumption as well. Whilst the proofs of our results are based on an abstract infinite-dimensional representation of the neutral system, we emphasize that the results are expressed in terms of the original parameters of the neutral equation and do not require a reformulation of the system in an abstract state-space form. The sufficiency parts of the results hold even when the delay operator acting on the derivative contains a singular part.

The matrix exponential plays a central role in linear systems and control theory. In this note, we give explicit formulas for computing the exponential of some special matrices.

Proposed algorithms based on the polynomial method for parameter estimation of multiple exponential signals to noise are tested for multiple snapshot unconditional, single and multiple snapshot conditional maximum likelihood estimations. Results show that, although cost-effective fast local convergence are allowed, the advantages of the polynomial method in terms of computational capacity are retained. Therefore, the programs are useful in improving global convergent parameter estimates. The need to modify the programs to eliminate nonconvergence did not surface in all cases studied.

Generalizing the exponential function allows the development of new memory functions to define solid polymeric mechanical behavior. The proposed function or root function is shown to perform well since its incomplete spectrum fM conforms well with the molecular weight distribution (MWD). The MWD curve can be best derived using the general constitutive equations which allow the response of the material in external impulse to be predicted given the rheological parametric values.

We compare the (asymptotic) efficiencies of the sample mean and the sample median in the exponential power family. This single family of densities is an ideal one to illustrate the effect of 'heavy tails' on the two estimators. We find the density for which the mean and median are equally efficient, and observe what happens for limiting cases within the family. Only basic tools are needed in the analysis and statistical development. KEY WORDS: Asymptotic; Efficiency; Sample mean; Sample median., 1. INTRODUCTION In introductory mathematical statistics courses the concept of (asymptotic) efficiency of an estimator is frequently discussed. The efficiency of the ubiquitous estimators, the sample mean [Mathematical Expression Omitted] [...]

Light cones are generated by the exact world line of a photon. This theoretically corresponds to a situation where the distributional function of the wave equation can be applied. The wave equation is an important model equation in mathematical physics and is physically relevant. No solution exists for this wave equation as it excludes several solutions such as 'photon torpedoes' and 'microphotons. ' The hyperbolity of the wave operator and the non-characteristic features also contribute to the null effect of the wave equation.

This paper exhibits exponentially converging observers for a large class of multiinput multioutput (MIMO) nonlinear systems. Potential applications of these observers to mechanical systems and distillation columns are presented.

The amount of money a person was willing to forgo in order to give $75 to another person decreased as a hyperbolic function of the perceived social distance between them. Similar hyperbolic functions have previously been shown to describe both time and probability discounting.

A formal analysis of the Guggenheim and other difference methods shows that the correlation is not a result of the subtraction but of the multiple uses of the original data in computing the differences. The assumption of random error in the raw data is reasonable in most cases, which makes correlation a nonproblem in the direct fitting approaches.

A study was conducted to show the variety of ways in which hyperbolic functions arise in the solution of the barge problem. The result reveals that the barge problem offers an easily understood application of hyperbolic functions and illustrates many properties of these functions and their inverses.

In a recent paper, Fang and Kincaid proposed an open problem about the relationship between the local stability of the unique equilibrium point and the global stability for a Hopfield-type neural network with continuously differentiable and monotonically increasing activation functions. As a partial answer to the question, in the two-neuron case it is proved that for each given specific interconnection weight matrix, a Hopfield-type neural network has a unique equilibrium point which is also locally exponentially stable for any activation functions and for any other network parameters if and only if the network is globally asymptotically stable for any activation functions and for any other network parameters. If the derivatives of the activation functions of the network are bounded, then the network is globally exponentially stable for any activation functions and for any other network parameters. Index Terms--Global stability, Hopfield-type neural network, local exponential stability.

Global exponential stability is a desirable property for dynamic systems. This paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems, convex programming problems with interval constraints, convex programming problems with nonlinear constraints, and monotone variational inequalities. In contrast to the existing results on global exponential stability, the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability. Therefore, the stability results in this paper further demonstrate the superior convergence properties of the existing neural networks for optimization. Index Terms--Global exponential stability, optimization problems, recurrent neural networks.

The freshman exponentiation law, a common mistake made by elementary algebra students, sometimes actually turns out correct. The n-domain, the domain of this rare case, is defined and analyzed. Ad hoc calculations show its characteristics while congruences from elementary group theory are able to prove its range of 2