The reason of Hopf’s and Oleinik’s proofs for countability of shocks being wrong

For the number of complete shock curves of a conservation law with one space variable,Hopf in 1950 for the Burger equation,and Oleinik in 1956 for the general,stated that it is at most countable.In 1979,the present author published an example to show that the statement of Hopf and Oleinik is wrong.But after so long time,the wrong statement for countability still appeared in some publications,which is at least partly due to that some ones felt difficult to understand Hopf and Oleinik’s proofs being wrong.So,pointing out where they went wrong becomes very necessary.

Countability in Mandarin Chinese: Bridging Theory and Experiments

The present study conducts a systematic investigation of the encoding of countability in three grammatical systems in Mandarin Chinese, including the nominal system, the whpronoun system, and the quantifier system. We examine the countability of linguistic expressions from these three grammatical systems, comparing their countability in the presence/absence of a classifier. A common pattern is identified: classifiers determine the countability of linguistic expressions in the three systems. Our account thus represents a syntactic account of the count-mass in Chinese. We support this theoretical account with three sets of experimental data in Mandarin-speaking adults, and show that the data also militate against the opposing semantic account. The present study contributes new data to the investigation of the count-mass issue in Chinese, and the count-mass issue in general.

Linguistic Form and Function: Expression of Countability in Chinese wh-Phrases

The present paper investigates the relationship between form(morpho-syntax)and function(semantic interpretation)as involved in the expression of countability in the Chinese whpronoun system.We focus on three wh-pronouns:duo-shao‘much-little’,duo-shao-ge‘much-little-Classifier’,and ji-ge‘how many-Classifier’.Based on the examination of the interpretation and distribution of these wh-phrases in three distinct linguistic contexts(questions,bare conditionals,and negative statements),we generalize that while duo-shao functions as an indeterminate determiner,duo-shao-ge and ji-ge function as two count determiners.This generalization invites the conclusion that the morpho-syntax of these whphrases,particularly,the presence/absence of a classifi er,determines and shapes their semantic interpretation in countability.From a more general perspective,the present study suggests that a linguistic form determines its own linguistic functions.

Mackey First Countability and Docile Locally Convex Spaces

We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.

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.

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.

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.

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.

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.

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"。

ON THE FINITE-SHEETED COVERING MAP

The main result in this letter is the following Theorem. Let X and Y be path connected, Hausdorff spaces satisfying the first countability axiom, and X a normal space. Suppose that f is a local homeomorphism of X onto Y. Then the following results are equivalent:

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.

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.