A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This p...A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This paper discusses how a finite-state Turing machine could, in a countably infinite number of state transitions, write all the infinite paths in the infinity tree to a countably infinite tape. Hence it is argued that the real numbers in the interval [0, 1] are countably infinite in a non-Cantorian theory of infinity based on Turing machines using countably infinite space and time. In this theory, Cantor’s Continuum Hypothesis can also be proved. And in this theory, it follows that the power set of the natural numbers P(ℕ) is countably infinite, which contradicts the claim of Cantor’s Theorem for the natural numbers. However, this paper does not claim there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, this paper considers the situation as a paradox, resulting from different choices about how to represent and count the continuum of real numbers.展开更多
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 au...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.展开更多
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...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.展开更多
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-pronou...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.展开更多
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 ...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.展开更多
In the paper [properties defined with semi-continuous functions and some related spaces＇, Houston J. Math., 2015, 41（3）： 1097-1106] properties （UL）m^wl, （UL）mK and （UL）m were defined and it was shown that sp...In the paper [properties defined with semi-continuous functions and some related spaces＇, Houston J. Math., 2015, 41（3）： 1097-1106] properties （UL）m^wl, （UL）mK and （UL）m were defined and it was shown that spaces having these properties coincide with countably paracompact spaces, countably mesocompact spaces and countably metacompact spaces, respectively. In this paper, we continue with the study on the relationship between properties defined with real-valued functions and some covering properties. Some characterizations of countably compact spaces and pseudo-compact spaces in terms of real-valued functions are obtained.展开更多
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 scal...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.展开更多
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 c...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).展开更多
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 c...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.展开更多
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 theo...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.展开更多
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 applicatio...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.展开更多
In this paper, the concept of countable compactness degree and the concept of Lindelof property degree are defined in L-fuzzy topological spaces by means of implication operator → Many properties of them are discussed.
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 exis...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.展开更多
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 w...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.展开更多
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 accord...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.展开更多
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 i...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.展开更多
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 a...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.展开更多
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...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.展开更多
文摘A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This paper discusses how a finite-state Turing machine could, in a countably infinite number of state transitions, write all the infinite paths in the infinity tree to a countably infinite tape. Hence it is argued that the real numbers in the interval [0, 1] are countably infinite in a non-Cantorian theory of infinity based on Turing machines using countably infinite space and time. In this theory, Cantor’s Continuum Hypothesis can also be proved. And in this theory, it follows that the power set of the natural numbers P(ℕ) is countably infinite, which contradicts the claim of Cantor’s Theorem for the natural numbers. However, this paper does not claim there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, this paper considers the situation as a paradox, resulting from different choices about how to represent and count the continuum of real numbers.
基金partially supported by National Basic Research Program of China (Grant No.2011CB302400)National Natural Science Foundation of China (Grant No. 10771206)
文摘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.
基金funded by National Social Science Fund of China (18BYY176)Post-doctoral Science Foundation of China (2014M561702)+2 种基金the Distinguished Post-doctoral Science Foundation of China (2015T80574)the Young Scholars Social Science Foundation of China’s Education Ministry (15YJC740031)the Philosophy and Social Science Foundation of Jiangsu Province (2014SJB519)
文摘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.
基金funded by the Post-doctoral Science Foundation of China(2014M561702)the Distinguished Post-doctoral Science Foundation of China(2015T80574)+1 种基金the Young Scholars Social Science Foundation of China’s Education Ministry(15YJC740031)the Philosophy and Social Science Foundation of Jiangsu Province(2014SJB519)
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1140126211271178)
文摘In the paper [properties defined with semi-continuous functions and some related spaces＇, Houston J. Math., 2015, 41（3）： 1097-1106] properties （UL）m^wl, （UL）mK and （UL）m were defined and it was shown that spaces having these properties coincide with countably paracompact spaces, countably mesocompact spaces and countably metacompact spaces, respectively. In this paper, we continue with the study on the relationship between properties defined with real-valued functions and some covering properties. Some characterizations of countably compact spaces and pseudo-compact spaces in terms of real-valued functions are obtained.
基金supported by the National Natural Science Foundation of China(11471236,11561049)
文摘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.
文摘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).
基金This work is supported by the natural science foundation.
文摘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.
文摘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.
基金Supported by the NNSF of China(1097118510971186)Supported by NSF of Fujian Province(2008F5066)
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant No.11471297)
文摘In this paper, the concept of countable compactness degree and the concept of Lindelof property degree are defined in L-fuzzy topological spaces by means of implication operator → Many properties of them are discussed.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
