On partitions of functions on finite sets

Let D and R be finite sets with cardinality n and m respectivelyR ^D be the set of all functions from D into R, and G and H be permutation groups acting on D and R respectively. Two functions f and g in R ^D are said to be related if there exists a σ in G and a τ in H with f(σd) = τg(d) for every d in D. Since the relation is an equivalence relation, R ^D is partitioned into disjoint classes. Roughly, by using the cycle indices of G and H, de Bruijn's theorem determines the number of equivalence classes, and Pólya's theorem, with H being the identity group, gives the function counting series, Pólya-de Bruijn's theorem has many applications (for instance, see Pólya and Read [6]). The theorem and its applications, basically, centered around the partitions of functions. Here, we present an algorithm to determine which functions in R ^D belong to the same equivalent class. Our algorithm does not use the cycle indices of G and H (to compute the cycle index of a given group, in general, is difficult), but it uses the generators of G and H, and the m-nary numbers to code the functions in R ^D . Our algorithm also gives the function counting series and the number of equivalence classes. An important application is that for each positive integer n, we use our algorithm and the symmetric group S _n to determine all isomorphic and nonisomorphic graphs and directed graphs with n vertices.     

Generalized hypergeometric functions associated with k-uniformly convex functions

For a certain linear operator which is defined by means of the Hadamard product (or convolution) with a generalized hypergeometric function, the authors aim at investigating various mapping and inclusion properties involving such subclasses of analytic and univalent functions as (for example) k-uniformly convex functions and k-starlike functions. Relevant connections of the definitions and results presented in this paper with those in several earlier works on the subject are also pointed out.     

一种新的二元有理插值及其性质

构造了一种带参数的仅基于函数值的分子为双四次、分母为双二次的二元有理插值样条函数,插值函数具有简洁的显式表示。插值函数中含有4个参数,当这些参数满足一定条件时,插值曲面在插值区域上C1光滑。由于插值函数中含有参数,这样可以在插值数据不变的情况下通过对参数的选择进行插值曲面的局部修改。最后讨论了插值函数的一些性质。     

Some integrals and orthogonal properties for a class of hypergeometric functions

We present three integrals and three orthogonal properties for a class of hypergeometric functions. We employ orthogonal properties to generate a theory concerning infinite series expansions involving our hypergeometric functions.     

Similarity measures,penalty functions,and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.     

Remarks on some series expansions associated with certain products of the incomplete Gamma functions

In several recent works, some interesting generalizations of the first-order Volterra-type integro-differential equation governing the unsaturated behavior of the free electron laser (FEL) were introduced and investigated by making use of fractional calculus (that is, calculus of integrals and derivatives of an arbitrary real or complex order). Among other things, it is observed here that an expansion formula for the confluent hypergeometric function in a series of the product of two entire (integral) incomplete Gamma functions does not hold true as asserted and applied in these earlier works. Some necessary corrections and possible remedies are also pointed out.     

Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established.     

The cut sets,decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets

In this paper,the cut sets,decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail.First,new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced,which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets.Second,based on these new cut sets,the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established.Each kind ...     

双时间尺度下的设备随机退化建模与剩余寿命预测方法

基于退化建模的剩余寿命预测（Remaining useful life,RUL）是当前可靠性领域研究的热点.现有的退化模型都是针对单个时间尺度下的退化设备,缺少对设备性能变化与多个时间尺度相关的退化建模与剩余寿命预测方法.鉴于此,本文基于Wiener过程提出了一种双时间尺度随机退化建模与剩余寿命预测方法,用随机比例系数描述不同时间尺度之间的不确定关系,推导出丫首达时间意义下设备的双时间尺度剩余寿命分布,讨论了其与基于单时间尺度退化模型得到的剩余寿命分布之间的关系,并给出了基于历史退化数据的未知参数极大似然估计方法.最后,将所提方法应用到惯性平台关键器件陀螺仪的退化建模与剩余寿命预测中,验证了方法的有效性.     

A version of a theorem of R. Datko for stability in average

In this note we obtain a version of the well-known theorem of R. Datko for the notion of the exponential stability in average. We consider both cocycles over flows as well as cocycles over maps.     

Uncertainty and negation—Information theoretic applications

The negation of a probability distribution developed by Yager and Liu deals with the uncertainty when the probability of a particular event is equally distributed among the other alternatives. In this case, intuitively the uncertainty associated with that particular event should be equally distributed among the uncertainty associated with the other alternatives. However, for verifying this, we need to develop uncertainty measures that will measure the uncertainty associated with the negation of a probability distribution. In the present study, we have developed measures analogous to the well known Shannon entropy function and the Kullback Leibler’s divergence for determining the uncertainty related with the negation and studied its properties. Finally, we have derived the Baye’s thoerem associated with the negation of probabilities of a sequence of events.     

Rigorous cubical approximation and persistent homology of continuous functions

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.     

Comments on: A locally constrained radial basis function for registration and warping of images

In a recent paper, Siddiqui et al. introduced a kernel function to be used as a radial basis function (RBF) in image registration tasks. This function is mainly designed so that the resulting deformation is fairly distributed inside its support. The important property of positive definiteness is checked in the paper erroneously, so that the conclusions inferred are wrong. In this communication, we discuss this point and some other methodological errors in the formulation. In addition, we provide some insights into the importance of positive definiteness, concluding that this property may not be critical, or may even be worthless, in certain interpolation problems.     

Analytical and numerical results onM-variable generalized Bessel functions

Recently, some multivariable special functions have been obtained by generalizing functions of Bessel type. Here, we continue the treatment of these functions starting fromJ _n (x, y; i), which is of noticeable practical interest. Finally, we consider the cases of functionsJ _n(x₁, x₂,..., x_M) and the related modified version,I _n(x₁, x₂,..., x_M), with two significant physical applications. Calculations of multivariable generalized Bessel functions are discussed and numerical results are given forJ _n(x₁,x₂;i), withn=0, 1, in a region of interest.     

Sufficient conditions for strongly Carathéodory functions

Recently, by applying the principle of subordination for analytic functions in the open unit disk U, Kim and Cho [I. H. Kim, N. E. Cho, Sufficient conditions for Carathéodory functions, Comput. Math. Appl. 59 (2010), 2067–2073] considered several sufficient conditions for a family of Carathéodory functions. The main purpose of this paper is to investigate some (presumably new) sufficient conditions for the class of strongly Carathéodory functions in U. One illustrative example and several corollaries of the main results presented here are also considered.     

Explicit construction of quadratic Lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. Part I: Continuous-time theory

The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati-like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure-Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust stability. In the case of the Popov criterion, the Lyapunov function is a parameter-dependent quadratic Lyapunov function.