The Infinity Tree: Representing Infinities of Real Numbers with Countably Infinite Tree Structures

This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.

On Countably Generated Dimension

In this article,we introduce and study the concept of countably generated dimension,which is a Krull-like dimension extension of the concept of DCC on countably generated submodules.We show that some of the basic results of Krull dimension are true for countably generated dimension.It is shown that an jR-module M has Krull dimension if and only if it has countably generated dimension,and its Krull dimension and countably generated dimension coincide.

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle(see Qiu, 2014), we prove a new version of Ekeland variational principle(briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results(no matter scalar-valued case,or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

Decomposition of Supercritical Linear-Fractional Branching Processes

It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.

Fixed Point of a Countable Family of Uniformly Totally Quasi- <i>&Oslash</i>-Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces with Applications

The purpose of this article is to discuss a modified Halpern-type iteration algorithm for a countable family of uniformly totally quasi- ? -asymptotically nonexpansive multi-valued mappings and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern-type iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Applied Mathematics and Computation, 218, 6489-6497).

Utility Representations of Preferences on General Choice Spaces

The purpose of this paper is to reconsider the utility representation problem of preferences,Sev-eral representation theorems are obtained on general choice spaces.Preferences having continuous utility functions are characterized by their continuities and countable satiation.It is showed that on a pairwise separable choice space,the sufficient and necessary condition for a preference to be represented by a contin-uous utility function is that the preference is continuous and countably satiable.For monotone prefer-ences,we obtain that any space has continuous utility representations.

A Note on the Principal Theorems in the Theory of Functional Analysis

Two theorems are proved. They are with principal significance in functional analysis, for they imply some well known theorems, such as the open mapping theorem, the closed graph theorem and the Banach Steinhaus theorem.

On Spaces with a Locally Countable sn-network

In this paper,spaces with a locally countable sn-network are discussed.It is shown that a space with a locally countable sn-network iff it is an snf-countable space with a locally countable k-network.As its application,almost-open and closed mappings（or finite-to-one and closed mapping） preserve locally countable sn-networks,and a perfect preimage theorem on spaces with a locally countable sn-network is established.

Reflected BSDEs Driven by L&eacute;vy Processes and Countable Brownian Motions

A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.

Poincaré Inequalities on a Countable Set E and Ω={x:0≤x_i≤a, i=1,2,…,n}

In the study of Poincaré inequalities, most of the traditional methods were based on the bounded domain in n dimensional Euclidean space Rn, while the method in this paper is based on a countable set E and accordingly the accurate expressions of Poincaré inequalities JB(=(f-μ(f))nJB)=_B≤cD(f, f) is presented to expand the research and application scope. As to inequalities for Ω={DK(x:0DK)≤x_i≤a, i=1,2,…,n}, the existing studies was usually made for n=2, but such an inequality was not the best. Therefore, the different values of n is discussed in this paper, and accordingly the accurate expressions of Poincaré inequalities is presented.

On Ws-Regular Spaces

In the present, the authors investigate a new type of separation axioms, which they call it w s-regular. The authors obtained some of its basic properties and its characterizations. Also, the authors notice that the axiom of tO s-regularity is weaker than the regularity, stronger than s-regularity and it is independent of w -regularity. However, the authors showed that the w s-regularity and regularity are identical on the class of all locally countable spaces, while the concepts ofw s-regularity and s-regularity are same on the class of anti-locally countable spaces：; furthermore, they proved that the three concepts w s-regularity, s-regularity and w s-regularity are same on the class of extremally disconnected spaces. The authors characterized w s-regular Trspaces by g-open sets, and they proved that the w s-regularity is an open hereditary property and it is also a topologizal property. The w s-closure of subsets of topological spaces are investigated and characterized. The authors used the concepts w s-closure to obtain some characterizations of the w s-regular spaces. Behind those, the authors obtained some properties and characterizations of w -semi open sets.

Learner Challenges

As a Chinese English learner,it is really a hard part to recognize well the usage of he,she,and it in the spoken English.Since in Chinese,the pronunciation between he,she,and it is the same one:"ta".Just the writing is different.However,in English,the meaning of he,she,it is totally different.Sometimes if people choose the wrong one,it is pretty possible to make some misunderstandings since the meaning will be different.

The Ontological Analysis of Count/Mass Distinction of French Nouns and ItsApplication in French Teaching

The countability and uncountability of French nouns are closely related to the discontinuity and homogeneity of the object that the nouns refer to.Discontinuity gives an object a bounded extension in space and time,and separates it from other things,so the object can be counted;while homogeneity makes an object remain his nature and have the same designation even after being cut apart,so it cannot be simply calculated.The discontinuity and homogeneity explain why nouns are countable or not from the perspective of ontology,applied in French teaching,they can help learners to distinguish these two kinds of nouns.

Strong Minkowski Separation and Co-Drop Property

In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.

A Definition and Some Properties of Group-Valued Measure

We give a definition and some primary properties of group-valued measure with a second countable complete Boolean algebra domain, and with a subset of a first countable complete Abelian po-group codomain.

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if l（M）∩y（M） = {0}, where l（M） and y（M） denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right ∑-CS if and only if R is right perfect and right countably ∑-NCS. Recall that a ring R is called right ∑-CS if every direct sum of copies of RR is a CS module. And a ring R is called right countably ∑-NCS if every direct sum of countable copies of RR is an NCS module.

A Version of Ekeland's Variational Principle in Countable Semi-Normed Spaces

In this paper, a now version of Ekeland's variational principle in countable semi-normed spaces is given.

加强政府责任性 提高市民满意度——香港特区“高官问责制”解读

在现代民主社会,提高政府责任性应当是改善政府作风,提高政府合法性程度的重要思路。责任性,即问责性(ac-countability),是指人们应当对其自己的行为负责。在公共管理中,问责性意味着政府及其工作人员为其政策后果负责。现代行政发展的一个重要方向就是"运用法律和道义的双重手段,增大个人及机构的责任性"。

The relations among the three kinds of conditional risk measures

Let（Ω , E, P） be a probability space, F a sub-σ-algebra of E, L^p（E）（1 p ＋∞） the classical function space and LF^p（E） the L^0（F）-module generated by L^p（E）, which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L^∞（E）, L^p（E）（1 p ＋∞） and LF^p（E）（1 p ＋∞） respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L^p（E） and LF^p（E）, namely LF^p（E） = Hcc（L^p（E））, which shows that LF^p（E）is exactly the countable concatenation hull of L^p（E）. Based on the precise relation, we then prove that every L^0（F）-convex L^p（E）-conditional risk measure（1 p ＋∞） can be uniquely extended to an L^0（F）-convex LF^p（E）-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of L^p-conditional risk measures can be incorporated into that of LF^p（E）-conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^0-convex conditional risk measures.

A CLASS OF STRONG LAWS OF LARGE NUMBERS WHICH HOLD FOR ARBITRARY COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

The strong laws of large numbers for countable nonhomogeneous Markov chains have been discussed (cf. [1]—[3] ), where various restrictions were imposed on the Markov chains. The purpose of this report is to give a class of strong laws of large numbers which hold for arbitrary nonhomogeneous Markov chains. As corollaries of the main result, a relation between the relative frequency of occurrence of state couples and the transition probability of arbitrary nonhomogeneous Markov chains is established.