Abstract:
Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. In this note, we give a new proof of CLT for independent identically distributed (i.i.d.) random variables. Our main tool is the viscosity solution theory of partial differential equation (PDE). 1. Introduction Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. The most familiar method to prove CLT is to use characteristic functions. To a mathematician having been already familiar with Fourier analysis, the characteristic function is a natural tool, but to a student of probability or statistics, confronting a proof of CLT for the first time, it may appear as an ingenious but artificial device. Thus, although knowledge of characteristic functions remains indispensable for the study of general limit theorems, there may be some interest in an alternative way of attacking the basic normal approximation theorem. Indeed, due to the importance of CLT, there exist the numerous proofs of CLT such as Stein’s method and Lindeberg’s method. Let us mention the contribution of Lindeberg [1] which used Taylor expansions and careful estimates to prove CLT. For more details of the history of CLT and its proofs, we can see Lindeberg [1], Feller [2, 3], Adams [4], Billingsley [5], Dalang [6], Dudley [7], Nourdin and Peccati [8], Ho and Chen [9], and so on. Recently, motivated by model uncertainties in statistics, finance, and economics, Peng [10, 11] initiated the notion of independent identically distributed random variables and the definition of -normal distribution. He further obtained a new CLT under sublinear expectations. In this note, inspired by the proof of Peng’s CLT, we give a new proof of the classical CLT for independent identically distributed (i.i.d.) random variables. Our proof is short and simple since we borrow the viscosity solution theory of partial differential equation (PDE). 2. Preliminaries In this section, we introduce some basic notations, notions, and propositions that are useful in this paper. Let denote the class of bounded functions satisfying for some depending on ; let denote the class of continuous functions ; let denote the class of bounded and -time continuously differentiable functions with bounded derivatives of all orders less than or equal to on and -time continuously differentiable functions with bounded derivatives of all orders less than or equal to on . Let be a random variable with distribution function , so that, for any , If is any function in , the mathematical

Abstract:
This paper stidies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.

The present study aimed to reveal the role of neuroticism on the relationship between self-esteem and aggressive emotion. We conducted a cross-sectional study in which a battery of self-report questionnaires was used to assess self-esteem, neuroticism and aggressive emotion in 1085 Chinese adolescents (N = 1085, M_{age} = 16.38 years, 753 boys). We found that self-esteem could make a negative prediction of aggressive emotion both in males and females. And also, we found the mediating role of neuroticism was in both males and females, on the relations between self-esteem and aggressive emotion, especially, the moderating role of neuroticism among males in the aspect of relationship between self-esteem and aggressive emotion. In conclusion, neuroticism was of importance for aggressive emotion, which was conducive to interventions. According to these findings, at the same time, implications and limitations were discussed in the content.

Abstract:
Business English (BE) language characteristics and many classroom teaching factors contribute to the anxiety of BE learning. The ecology can be put in use into BE teaching to make the ecological classroom. This research is taken to show relations between the ecological classroom teaching of BE (BEECT) and BE learning anxiety (BELA). The findings are as followed: 1) BEECT is in good order and students’ BELA is to the intermediate degree; 2) BELA has the generally negative correlation to students’ general English proficiency; 3) BEECT has the higher negative correlation to BELA, and they are significantly linearly dependent; 4) as to BELA, significant statistical differences exist between high score group and low score group in the classroom learning efficiency, the teacher-student affective intercommunication, and student-student intercommunication. In order to lower BELA, BE teachers are suggested cultivating learners’ academic English proficiency, strengthening learners’ ambiguity tolerance, and using ecologically multimodal discourses.

Abstract:
Talents are the most important factor of all in the efforts to establish an innovative country. In universities and colleges, the bases for talent cultivation, the turnover of teachers in an explicit or recessive way calls for our urgent attention. This article aims at analyzing the complex causes of college teachers’ turnover from the perspective of institutional economics including Limitation of Incentive Theory, Interests Relative Theory and Principal-Agent Theory. Besides, it puts forward some relevant measures to solve the current problem.

As the requirement of the educational concept of innovation and
entrepreneurship, it is necessary to develop the students’ learning ability
that is vital for students not only at school but also afterwards. In college
English teaching, language learning strategies are beneficial for successfully
learning. Many famous researches have given a definition and classifications of
learning strategies, where exist some similarities and distinctions. This paper
suggests that learning strategies are classified into four strategies at the hierarchical
level: cognitive strategies, metacognitive strategies, affective strategies and
social strategies.

As ITU-R Recommendations is widely
implemented for countries all over the world, the role and status of ITU-R
Recommendations are increasingly prominent in the field of radio engineering.
ITU and ITU-R Study Groups are summarized. Furthermore, the operating mode of
the third study group, and the input documents are interpreted in detail.
Lastly, from both wireless system design and electromagnetic compatibility
analysis perspective, all of 79 P-series Recommendations are analyzed and
classified, and the main contents of each Recommendation are summarized. The
above research promote P-series Recommendations are widely used in China.

Abstract:
The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values of -valued meromorphic functions from the to an infinite-dimensional complex Banach space with a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

Abstract:
Firstly, an inequality for vector-valued meromorphic functions is established which extend a corresponding inequality of Milloux for meromorphic scalar-valued function (1946). As an application, the relationship between the characteristic function of a vector-valued meromorphic function and its derivative is studied, results are obtained to extend some related results for meromorphic scalar-valued function of Weitsman (1969) and Singh and Gopalakrishna (1971). 1. Introduction of Vector-Valued Meromorphic Function In 1980s, Ziegler [1] established Nevanlinna’s theory for the vector-valued meromorphic function in finite dimensional spaces. After Ziegler some works related to vector-valued meromorphic function were done in 1990s [2–4]. In this section, we shall introduce the following fundamental notations and results of vector-valued Nevanlinna theory which were quoted from Ziegler [1]. We denote by the usual dimensional complex Euclidean space with the coordinates , the Hermitian scalar product and the distance Let be complex valued functions of the complex variable , which are meromorphic and not all constant in the Gaussian plane , or in a finite disc Thus in (we put ), a vector-valued meromorphic function is given, which does not reduce to the constant zero vector . The th derivative of are defined by For such a function, the notations “zero,” “pole,” and “multiplicity” are defined as in the scalar case of only one meromorphic function . More explicitly, in the punctured vicinity of each point , the vector function can developed into a Laurent series where the coefficients are vectors In order to introduce the Nevanlinna theory of vector-valued meromorphic function, we will denote by " " the ideal element of the Aleksandrov one-point compactification of (the two real infinities will be denoted by and , resp.). Now, if in the above Laurent expansion, then will be called a pole or an -point of of multiplicity ; in such a point at least one of the meromorphic component functions has a pole of this multiplicity in the ordinary sense of function theory, so that in itself is not defined. If in Laurent expansion, then is called a zero of of multiplicity ; in such a point , all component functions vanish, each with at least this multiplicity. Let or denote the number of poles of in and denote the number of -points of in , counting with multiplicities. Define the volume function associated with vector-valued meromorphic function , and the counting function of finite or infinite -points by respectively. Next, we define Let or denote the number of poles of in