This paper gives several fundamental theorems for the stability, uniform stability, asymptotic stability and uniform asymptotic stability. Those theorems allow the derivative of Lyapunov functions to be positive on ce...This paper gives several fundamental theorems for the stability, uniform stability, asymptotic stability and uniform asymptotic stability. Those theorems allow the derivative of Lyapunov functions to be positive on certain sets,relax the restriction about the rate of change of state variable in a system to be bounded in Marachkov's theorem and extend the related results in [4—7].展开更多
We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theore...We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.展开更多
English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited ...English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.展开更多
In this paper, the authors introduce a kind of reducible algebroid functions, that is general algebroid functions and obtain two fundamental theorems of general algebroid functions. At last, as an application, we gene...In this paper, the authors introduce a kind of reducible algebroid functions, that is general algebroid functions and obtain two fundamental theorems of general algebroid functions. At last, as an application, we generalized a theorem of Li Guoping＇s.展开更多
We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf mod...We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it.展开更多
This paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the pro...This paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein’s principle of local realism, the inequality pertains to a linear combination of four polarization products <em>on the same pair of photons</em> arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity<em> s</em>(<span style="white-space:nowrap;"><em>λ</em></span>) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[<em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em>λ</em><span style="white-space:normal;">)</span>] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that <span style="white-space:normal;">E[</span><em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em style="white-space:normal;">λ</em><span style="white-space:normal;">)</span><span style="white-space:normal;">]</span> has four distinct representations as the sum of <em>three</em> expectations of polarization products plus the expectation of a fourth which is restricted to equal a function value determined by the other three. Analysis using Bruno de Finetti’s fundamental theorem of prevision (FTP) yields only a bound for <em>E</em>(<em>s</em>) within <span style="white-space:nowrap;">(1.1213,2]</span> , surely not <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" />at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint<em> P</em><sub>++</sub> probabilities actually motivated by quantum theory at the four stipulated angle settings, as it passes through 3-dimensional space. Bell’s inequality is satisfied everywhere within the convex hull of extreme distributions cohering with quantum theoretic specifications, even while in keeping with local realism. Aspect’s proposed “estimation” of <em>E</em>(<em>s</em>) near to <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" style="white-space:normal;" />is based on polarization products from different photon pairs that do not have embedded within them the functional relations inhering in the relevant gedankenexperiment. When one actively embeds the restrictions into Aspect’s estimation procedure, it yields an estimate of 1.7667, although this is not and cannot be definitive. While my analysis supports the subjectivist construction of probability as clarifying issues relevant to the interpretation of quantum theory, the error resolved herein is purely mathematical. It pertains to the reconsideration of Bell violation irrespective of one’s attitude toward the meaning of probability.展开更多
I provide a critical reassessment of David Mermin’s influential and misleading parable, “Quantum Mysteries for Anyone”, identifying its errors and resolving them with a complete analysis of the quantum experiment i...I provide a critical reassessment of David Mermin’s influential and misleading parable, “Quantum Mysteries for Anyone”, identifying its errors and resolving them with a complete analysis of the quantum experiment it is meant to portray. Accessible to popular readership and requiring no knowledge of quantum physics at all, his exposition describes the curious behaviour of a machine that is designed to parody the empirical results of quantum experiments monitoring the spins of a pair of electrons under various conditions. The mysteries are said to unfold from contradictory results produced by a signal process that is proposed to explain them. I find that these results derive from a mathematical error of neglect, coupled with a confusion of two distinct types of experiments under consideration. One of these, a gedankenexperiment, provides the context in which the fabled defiance of Bell’s inequality is thought to emerge. The errors are corrected by the recognition of functional relations embedded within the experimental conditions that have been long unnoticed. A Monte Carlo simulation of results in accord with the actual abstemious claims of quantum theory supports probability values that Mermin decries as unwarranted. However, the distribution it suggests is not definitive, in accord with the expressed agnostic position of quantum theory regarding measurements that cannot be executed. Bounding quantum probabilities are computed for the results of the gedankenexperiment relevant to Bell’s inequality which inspired the parable. The problem is embedded in a 3 × 3 design of Stern-Gerlach magnet orientations at two observation stations. Computational resolution on the basis of Bruno de Finetti’s fundamental theorem of probability requires the evaluation of a battery of three paired linear programming problems. Though technicalities are ornate, the message is clear. There are no mysteries of quantum mechanics that derive from mistaken understandings of Bell’s inequality… for anyone.展开更多
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For ...A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.展开更多
We prove an L∞ version of the Yan theorem and deduce from it a necessary condition for the absence of free lunches in a model of financial markets, in which asset prices are a continuous R^d valued process and only s...We prove an L∞ version of the Yan theorem and deduce from it a necessary condition for the absence of free lunches in a model of financial markets, in which asset prices are a continuous R^d valued process and only simple investment strategies are admissible. Our proof is based on a new separation theorem for convex sets of finitely additive measures.展开更多
This brief note presents an easily conceivable inequality for analytic functions. As an application, a function-theoretic proposition involving the Fundamental Theorem of Algebra （FTA） is deduced immediately.
We give a comparison of two no-arbitrage conditions for the fundamental theorem of asset pricing. The first condition is named as the no free lunch with vanishing risk condition and the second the no good deal conditi...We give a comparison of two no-arbitrage conditions for the fundamental theorem of asset pricing. The first condition is named as the no free lunch with vanishing risk condition and the second the no good deal condition. We aim to derive a relationship between these two conditions.展开更多
Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in whi...Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separately defined two important concepts in mathematical terms： thefitness in the former and the （non- adiabatic） entropy production in the latter. Both of them play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth, thus sustains continuous natural selection. As a form of Price＇s equation, the fitness increases with （i） natural selection through variations and （ii） a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state —— ＂the red queen＂ scenario. At the same time the growth keeps the proportions of subpopulations away from the ＂intrinsic＂ switching equilibrium of individual cells, thus leads to a continuous entropy production. A covariance, between the per capita growth rate and the ＂chemical potential＂ of subpopulation, counteracts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.展开更多
This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σR,this is a large family of manifolds including the usual space forms R^m,S^m and H^m.We give the fundamental the...This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σR,this is a large family of manifolds including the usual space forms R^m,S^m and H^m.We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space R^n×σR,which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.展开更多
In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|...In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.展开更多
文摘This paper gives several fundamental theorems for the stability, uniform stability, asymptotic stability and uniform asymptotic stability. Those theorems allow the derivative of Lyapunov functions to be positive on certain sets,relax the restriction about the rate of change of state variable in a system to be bounded in Marachkov's theorem and extend the related results in [4—7].
文摘We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.
文摘English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.
基金supported by the National Natural Science Foundation of China(11201083 and 11501127)Foundation for Distinguished Young Talents in Higher Education of Guangdong Province(2014KQNCX068)
文摘In this paper, the authors introduce a kind of reducible algebroid functions, that is general algebroid functions and obtain two fundamental theorems of general algebroid functions. At last, as an application, we generalized a theorem of Li Guoping＇s.
文摘We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it.
文摘This paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein’s principle of local realism, the inequality pertains to a linear combination of four polarization products <em>on the same pair of photons</em> arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity<em> s</em>(<span style="white-space:nowrap;"><em>λ</em></span>) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[<em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em>λ</em><span style="white-space:normal;">)</span>] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that <span style="white-space:normal;">E[</span><em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em style="white-space:normal;">λ</em><span style="white-space:normal;">)</span><span style="white-space:normal;">]</span> has four distinct representations as the sum of <em>three</em> expectations of polarization products plus the expectation of a fourth which is restricted to equal a function value determined by the other three. Analysis using Bruno de Finetti’s fundamental theorem of prevision (FTP) yields only a bound for <em>E</em>(<em>s</em>) within <span style="white-space:nowrap;">(1.1213,2]</span> , surely not <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" />at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint<em> P</em><sub>++</sub> probabilities actually motivated by quantum theory at the four stipulated angle settings, as it passes through 3-dimensional space. Bell’s inequality is satisfied everywhere within the convex hull of extreme distributions cohering with quantum theoretic specifications, even while in keeping with local realism. Aspect’s proposed “estimation” of <em>E</em>(<em>s</em>) near to <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" style="white-space:normal;" />is based on polarization products from different photon pairs that do not have embedded within them the functional relations inhering in the relevant gedankenexperiment. When one actively embeds the restrictions into Aspect’s estimation procedure, it yields an estimate of 1.7667, although this is not and cannot be definitive. While my analysis supports the subjectivist construction of probability as clarifying issues relevant to the interpretation of quantum theory, the error resolved herein is purely mathematical. It pertains to the reconsideration of Bell violation irrespective of one’s attitude toward the meaning of probability.
文摘I provide a critical reassessment of David Mermin’s influential and misleading parable, “Quantum Mysteries for Anyone”, identifying its errors and resolving them with a complete analysis of the quantum experiment it is meant to portray. Accessible to popular readership and requiring no knowledge of quantum physics at all, his exposition describes the curious behaviour of a machine that is designed to parody the empirical results of quantum experiments monitoring the spins of a pair of electrons under various conditions. The mysteries are said to unfold from contradictory results produced by a signal process that is proposed to explain them. I find that these results derive from a mathematical error of neglect, coupled with a confusion of two distinct types of experiments under consideration. One of these, a gedankenexperiment, provides the context in which the fabled defiance of Bell’s inequality is thought to emerge. The errors are corrected by the recognition of functional relations embedded within the experimental conditions that have been long unnoticed. A Monte Carlo simulation of results in accord with the actual abstemious claims of quantum theory supports probability values that Mermin decries as unwarranted. However, the distribution it suggests is not definitive, in accord with the expressed agnostic position of quantum theory regarding measurements that cannot be executed. Bounding quantum probabilities are computed for the results of the gedankenexperiment relevant to Bell’s inequality which inspired the parable. The problem is embedded in a 3 × 3 design of Stern-Gerlach magnet orientations at two observation stations. Computational resolution on the basis of Bruno de Finetti’s fundamental theorem of probability requires the evaluation of a battery of three paired linear programming problems. Though technicalities are ornate, the message is clear. There are no mysteries of quantum mechanics that derive from mistaken understandings of Bell’s inequality… for anyone.
基金Research supported by the NSFC (10231010)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaNatural Science Foundation of Zhejiang Province (101037).
文摘A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.
文摘We prove an L∞ version of the Yan theorem and deduce from it a necessary condition for the absence of free lunches in a model of financial markets, in which asset prices are a continuous R^d valued process and only simple investment strategies are admissible. Our proof is based on a new separation theorem for convex sets of finitely additive measures.
文摘This brief note presents an easily conceivable inequality for analytic functions. As an application, a function-theoretic proposition involving the Fundamental Theorem of Algebra （FTA） is deduced immediately.
文摘We give a comparison of two no-arbitrage conditions for the fundamental theorem of asset pricing. The first condition is named as the no free lunch with vanishing risk condition and the second the no good deal condition. We aim to derive a relationship between these two conditions.
文摘Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separately defined two important concepts in mathematical terms： thefitness in the former and the （non- adiabatic） entropy production in the latter. Both of them play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth, thus sustains continuous natural selection. As a form of Price＇s equation, the fitness increases with （i） natural selection through variations and （ii） a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state —— ＂the red queen＂ scenario. At the same time the growth keeps the proportions of subpopulations away from the ＂intrinsic＂ switching equilibrium of individual cells, thus leads to a continuous entropy production. A covariance, between the per capita growth rate and the ＂chemical potential＂ of subpopulation, counteracts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.
基金Supported by National Natural Science Foundation of China(Grant No.10871149)Doctoral Fund of Education of China(Grant No.200804860046)
文摘This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σR,this is a large family of manifolds including the usual space forms R^m,S^m and H^m.We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space R^n×σR,which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.
基金supported by National Natural Science Foundation of China (Grant No. 11271343)
文摘In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.
