The products of three theta functions and the general cubic theta functions

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.     

SOLUTIONS TO A CLASS OF NONLINEAR WAVE EQUATIONS

By introducing a transformation and applying the trial function approach,many exact solutions to a class of nonlinear wave equations are presented. Among them,some are given for the first time.     

Derivatives of meromorphic functions with multiple zeros and elliptic functions

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple except finitely many and T (r, h) = o{T (r, f )} as r →∞, then f' = h has infinitely many solutions (including poles).     

One-dimensional nonlinear Laplacians under a 3-point boundary condition

We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas involving zeros of a real-valued function. They are shown to be orderpreserving, for some parameter values, and non-singleton valued for others. The operators are shown to be m-dissipative in the space of continuous functions.     

A general vectorial Ekeland’s variational principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.     

Nonparametric regression with interval-censored data

In many medical studies,the prevalence of interval censored data is increasing due to periodic monitoring of the progression status of a disease.In nonparametric regression model,when the response variable is subjected to interval-censoring,the regression function could not be estimated by traditional methods directly.With the censored data,we construct a new response variable which has the same conditional expectation as the original one.Based on the new variable,we get a nearest neighbor estimator of the regression function.It is established that the estimator has strong consistency and asymptotic normality.The relevant simulation reports are given.     

Many Families of Exact Solutions for Nonlinear System of Partial Differential Equations Describing the Dynamics of DNA

The objective of this paper is to apply the improved generalized Riccati equation mapping method to find many families of exact traveling wave solutions for the general nonlinear dynamic system in a new double-chain model of DNA. This model consists of two long elastic homogeneous strands connected with each other by an elastic membrane. Hyperbolic and trigonometric function solutions of this model are obtained. Comparison between our results and the well-known results are given.     

KY FAN TYPE BEST APPROXIMATION THEOREM FOR A CLASS OF FACTORIZABLE MULTIFUNCTIONS

An existence result on Ky Fan type best approximation is proved. For this purpose, a class of factorizable multifunctions and the other one being a demicontinuous, relative almost quasi-convex, onto function on an approximately weakly compact, convex subset of Hausdorff locally convex topological vector space are used. As consequence, this result extends the best approximation results of Basha and Veeramani [8] and many others.     

整函数及其导函数的素性

In this paper, we deal with the factorization of entire functions in the sense of composition, and prove the following.Theorem. For any real number ρ≥1, there exists a transcendental entire function F(z) of order p , such that there are infinitely many prime elements and infinitely many composite elements in the class {F⁽ⁿ⁾ n = 0.1, 2, …}.     

CONTINUOUS-TIME STOCHASTIC APPROXIMATION UNDER MEASUREMENT ERROR BEING PROCESS OF DEPENDENT INCREMENT

§1 Introduction If the accurate value of a function (·) is available at any point then there are many numerical methods recursively estimating the root of the regression equation (·)=. But the measurement is usually disturbed by random errors. In this case how can the root     

What is the integration by partial fractions？

1.What a rational function？
Do you remember what a polynomial function is？If you know a function of the form is a form as following f（x）=anxn＋an-ixn-1＋...＋a1x＋a0,where an,an-i,...a1,a0are real numbers and nis nonnegative integer.The domain of a polynomial function is the set of all real numbers.But here are some     

Function P-sets and Mining-discovery Theorems of Unknown Information Law

Function P-sets are a set pair which are composed of function internal packetsets S F and function outer packet sets S F or （S F , S F ） are function P-sets. Function P-sets have dynamic characteristics and law characteristics. Based on function P-sets, P-information law discovery theorems which has attribute supplementing-deleting are given. Mining-discovery and mining-discovery theorems of unknown information law are proposed. Function P-sets are a new theory and new method to research dynamic information laws for information systems.     

Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

ON ENTIRE FUNCTIONS, BLOCH AND NORMAL FUNCTIONS

New criteria for an analytic function to be Bloch and for a meromorphic function to benormal are given. These criteria generalize the recently introduced area integral conditionsinvolving a Green's function.     

A DIFFERENTIABLE CANTOR-LIKE FUNCTION

the Cantor function is one of the most important and useful exaraples in real function theory. Recently there are a lot of research on differentiable Cantor-like functions, but the methods involved are not elementary. In this note a differentiable Cantor-like function was given by an elementary method.     

A NEW APPROACH TO THE LINEARITY OF TESTING PLANARITY OF GRAPHS

In testing planarity of graphs,there are many criteria.The earliest one as known is the Kuratowski's theorem,then Whitney's,Maclane's, and so forth.Since the early sixties,people have begun researches on algorithms.Up to 1974,Hopcroft and Tarjan found an algorithm with a computing time being a linear function of the order of a graph.This is the linearity concerned here.This paper presents a new approach to the linearity by means of transforming the problem of testing planarity of a graph G into that of finding a spanning tree on another graph H,called an auxiliary graph of G,with the order of H being a linear function of that of G.And moreover,we can also make the size of H be a linear function of that of G.The whole procedure is based on the building up of a theory of linear equations on GF(2)related to G.     

Zero distribution of some difference polynomials

In this paper,suppose that a,c∈C{0},c_j∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-afⁿ(z) and f (z)f (z+c₁)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.     

A new family of differentially 4-uniform permutations over F_(2~(2k)) for odd k

We study the differential uniformity of a class of permutations over F2 n with n even. These permutations are different from the inverse function as the values x~(-1) are modified to be(γx)~(-1) on some cosets of a fixed subgroup γ of F_(2n)~*. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent(CCZ-equivalent) classes as checked by Magma for small numbers n. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.     

ON THE EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR SINGULAR p-LAPLACE EQUATIONS

In this paper N-dimensional singular, p-Laplace equations of the following form are considered, where p ≥N, β > 0, and f : [0,∞)×(0,∞)×[0, ∞)→[0,∞) is a continuous function. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of |x|(p-N)/(p-1) for p > N or log|x| for N = p as |x|→∞.     

THE LIMITING DISTRIBUTIONS OF SOME SUBCLASSES OF THE GENERALIZED NON-CENTRAL t-DISTRIBUTION

A general theorem on the limiting distribution of the generalized t-distribution is obtained,many applications of this theorem to some subclasses of elliptically contoured distributions in cluding multivariate normal and multivariate t distributions are discussed.Further,their limiting distributions by density function are derived.