A parameter free filled function for unconstrained global optimization

The filled function method is considered as an efficient method to find the global minimum of multidimensional functions. A number of filled functions were proposed recently, most of which have one or two adjustable parameters. However, there is no efficient criterion to choose the parameter appropriately. In this paper, we propose a filled function without parameter. And this function includes neither exponential terms nor logarithmic terms so it is superior to the traditional ones. Theories of the filled function are investigated. And an algorithm which does not compute gradients during minimizing the filled function is presented. Moreover, the numerical experiments demonstrate the efficiency of the proposed filled function.     

Hitting, returning and the short correlation function

We consider a stochastic process with the weakest mixing condition: the so called α. For any fixed n-string we prove the following results. (1) The hitting time has approximately exponential law. (2) The return time has approximately a convex combination between a Dirac measure at the origin and an exponential law. In both cases the parameter of the exponential law is λ(A)ℙ(A) where ℙ(A) is the measure of the string and λ(A) is a certain autocorrelation function of the string. We show also that the weight of the convex combination is approximately λ(A). We describe the behavior of this autocorrelation function. Our results hold when the rate of mixing decays polinomially fast with power larger than the golden number.     

Comparison of exponential‐logarithmic and logarithmic‐exponential series

We explain how the field of logarithmic‐exponential series constructed in 20 and 21 embeds as an exponential field in any field of exponential‐logarithmic series constructed in 9 , 6 , and 13 . On the other hand, we explain why no field of exponential‐logarithmic series embeds in the field of logarithmic‐exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non‐isomorphic models of Th$(\mathbb {R}_{\mbox{an, exp}})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.     

Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters

In this paper, we study the multi-parameter Tikhonov regularization method which adds multiple different penalties to exhibit multi-scale features of the solution. An optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters. Some theoretical results of the regularization solution about the dependence on regularization parameters are presented. Then, an a posteriori parameter choice, i.e., the damped Morozov discrepancy principle, is introduced to determine multiple regularization parameters. Five model functions, i.e., two hyperbolic model functions, a linear model function, an exponential model function and a logarithmic model function, are proposed to solve the damped Morozov discrepancy principle. Furthermore, four efficient model function algorithms are developed for finding reasonable multiple regularization parameters, and their convergence properties are also studied. Numerical results of several examples show that the damped discrepancy principle is competitive with the standard one, and the model function algorithms are efficient for choosing regularization parameters.     

Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation

We show that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm. Numerical error analysis and computational test reveal that with essentially higher accuracy it is as fast as FFT-based Weideman’s algorithm at a regular size of the input array and considerably faster at an extended size of the input array. As this exponential series approximation is based only on elementary functions, the algorithm can be implemented utilizing freely available functions from the standard libraries of most programming languages. Due to its simplicity, rapidness, high-accuracy and coverage of the entire complex plane, the algorithm is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring error-function evaluation over extended input arrays.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

指数凸函数的积分不等式及其在Gamma函数中的应用

仿对数凸函数的概念,给出指数凸函数的定义,并证明有关指数凸函数的几个积分不等式,作为应用,得到一个新的Kershaw型双向不等式.     

Decay rates of energy of the 1D damped original nonlinear wave equation

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.     

Exponential decay for a viscoelastically damped timoshenko beam

Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-known that the system is exponentially stable if the kernel in the memory term is sub-exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions: What if the kernel is tested against a different function (say Gamma) other than the exponential function? Would there still be stability? In the affirmative, what kind of decay rate we get? It is proved that for a non-decreasing function “Gamma” whose “logarithmic derivative” is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.     

A improved F‐expansion method and its application to the Zhiber–Shabat equation

Based on the F‐expansion method and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic functions, logarithmic function, and other type of functions for the Zhiber–Shabat equation are derived. Some previous results are extended. Copyright © 2012 John Wiley & Sons, Ltd.     

Linear ordinary differential equations of the second order with unbounded solutions

We consider three families of equations of the form y″ + (1 + φ(x))y = 0, where the coefficient φ(x) satisfies the condition lim _x→+∞ φ(x) = 0. We obtain solutions of these equations in closed form. We show that the maximum absolute values of solutions grow at the rate of a logarithmic function, a power-law function, and even an exponential function as x → ∞.     

Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

In this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Im z|<2π. Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions,multiple Euler sums and log‐sine integrals

The Gamma function and its n th logarithmic derivatives (that is, the polygamma or the psi‐functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi‐functions and to evaluate a class of log‐sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi‐function expressions for some known parameterized multiple sums (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces

Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).     

Estimates, decay properties, and computation of the dual function for Gabor frames

We present a simple proof of Ron and Shen's frame bounds estimates for Gabor frames. The proof is based on the Heil and Walnut's representation of the frame operator and shows that it can be decomposed into a continuous family of infinite matrices. The estimates then follow from a simple application of Gershgorin's theorem to each matrix. Next, we show that, if the window function has exponential decay, also the dual function has some exponential decay. Then, we describe a numerical method to compute the dual function and give an estimate of the error. Finally, we consider the spline of order 2; we investigate numerically the region of the time-frequency plane where it generates a frame and we compute the dual function for some values of the parameters.     

On irrationality measures of the values of Gauss hypergeometric function

The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the case of logarithmic function.     

非均匀调和函数的两类边值问题与积分表达式

本文给出非均匀指数函数的定义及性质,并且进一步引入了非均匀三角函数、非均匀双曲函数和非均匀对数函数.最后利用非均匀指数函数表达形式和非均匀解析函数的Cauchy积分理论,建立了非均匀泊松积分公式和非均匀施瓦茨积分公式,获得了非均匀调和函数在两类特殊边界上的狄利克雷问题和诺伊曼问题解的显示表达式.