As the scale of the networks continually expands,the detection of distributed denial of service(DDoS)attacks has become increasingly vital.We propose an intelligent detection model named IGED by using improved general...As the scale of the networks continually expands,the detection of distributed denial of service(DDoS)attacks has become increasingly vital.We propose an intelligent detection model named IGED by using improved generalized entropy and deep neural network(DNN).The initial detection is based on improved generalized entropy to filter out as much normal traffic as possible,thereby reducing data volume.Then the fine detection is based on DNN to perform precise DDoS detection on the filtered suspicious traffic,enhancing the neural network’s generalization capabilities.Experimental results show that the proposed method can efficiently distinguish normal traffic from DDoS traffic.Compared with the benchmark methods,our method reaches 99.9%on low-rate DDoS(LDDoS),flooded DDoS and CICDDoS2019 datasets in terms of both accuracy and efficiency in identifying attack flows while reducing the time by 17%,31%and 8%.展开更多
Among all statements of Second Law, the existence and uniqueness of stable equilibrium, for each given value of energy content and composition of constituents of any system, have been adopted to define thermodynamic e...Among all statements of Second Law, the existence and uniqueness of stable equilibrium, for each given value of energy content and composition of constituents of any system, have been adopted to define thermodynamic entropy by means of the impossibility of Perpetual Motion Machine of the Second Kind (PMM2) which is a consequence of the Second Law. Equality of temperature, chemical potential and pressure in many-particle systems are proved to be necessary conditions for the stable equilibrium. The proofs assume the stable equilibrium and derive, by means of the Highest-Entropy Principle, equality of temperature, chemical potential and pressure as a consequence. A first novelty of the present research is to demonstrate that equality is also a sufficient condition, in addition to necessity, for stable equilibrium implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality of temperature potential and pressure addressed to as generalized potential. The second novelty is that the proof of sufficiency of equality, or necessity of stable equilibrium, is achieved by means of a generalization of entropy property, derived from a generalized definition of exergy, both being state and additive properties accounting for heat, mass and work interactions of the system underpinning the definition of Highest-Generalized-Entropy Principle adopted in the proof.展开更多
Axiomatization of Shannon entropy is a subject that has received lots of attention in the information theory literature.While Shannon entropy is defined on probability distribution,we define a new type of entropy on t...Axiomatization of Shannon entropy is a subject that has received lots of attention in the information theory literature.While Shannon entropy is defined on probability distribution,we define a new type of entropy on the set of partitions of finite subsets of metric spaces,which has a rich algebraic structure as a partially ordered set.We propose an axiomatization of an entropy-like measure of partitions of sets of objects located in metric spaces,and we derive an analytic expression of this new type of entropy referred to as inertial entropy.This approach starts with the notion of inertia of a partition and includes a study of the behavior of the sum of square errors of a partition.In this context,we characterize the chain of partitions produced by the Ward hierarchical clustering method.Starting from inertial entropies of partitions,we introduce conditional entropies which,in turn,generate metrics on partitions of finite sets.These metrics are used as external validation tools for clusterings of labeled data sets.The metric generated by inertial entropy can be used to validate data clustering for labeled data sets.This type of validation aims to determine to what extend labeling of the data coincides with the clustering obtained algorithmically,and we obtain a high degree of consistency of the data labeling with the results of several hierarchical clusterings.展开更多
In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the ...In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Cross Fuzzy Entropy Optimization Methods in the form of MinMinx（F）EntM and MaxMinx（F）EntM are suggested on the basis of primary phase of minimizing cross fuzzy entropy measure for fixed moment vector function and on the definition of the special functional with Minx（F）Ent values of cross fuzzy entropy measure. Next phase for obtaining mentioned distributions consists of optimization of defined functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as （MinMinx（F）Ent）m and （MaxMinx（F）Ent）m distributions.展开更多
This study is connected with new Generalized Maximum Fuzzy Entropy Methods （GMax（F）EntM） in the form of MinMax（F）EntM and MaxMax（F）EntM belonging to us. These methods are based on primary maximizing Max（F）En...This study is connected with new Generalized Maximum Fuzzy Entropy Methods （GMax（F）EntM） in the form of MinMax（F）EntM and MaxMax（F）EntM belonging to us. These methods are based on primary maximizing Max（F）Ent measure for fixed moment vector function in order to obtain the special functional with maximum values of Max（F）Ent measure and secondary optimization of mentioned functional with respect to moment vector functions. Distributions, in other words sets of successive values of estimated membership function closest to （furthest from） the given membership function in the sense of Max（F）Ent measure, obtained by mentioned methods are defined as （MinMax（F）Ent）m which is closest to a given membership function and （MaxMax（F）Ent）m which is furthest from a given membership function. The aim of this study consists of applying MinMax（F）EntM and MaxMax（F）EntM on given wind speed data. Obtained results are realized by using MATLAB programme. The performances of distributions （MinMax（F）En0m and （MaxMax（F）Ent）m generated by using Generalized Maximum Fuzzy Entropy Methods are established by Chi-Square, Root Mean Square Error criterias and Max（F）Ent measure.展开更多
By using the entanglement entropy method, this paper calculates the statistical entropy of the Bose and Fermi fields in thin films, and derives the Bekenstein-Hawking entropy and its correction term on the background ...By using the entanglement entropy method, this paper calculates the statistical entropy of the Bose and Fermi fields in thin films, and derives the Bekenstein-Hawking entropy and its correction term on the background of a rotating and charged black string. Here, the quantum field is entangled with quantum states in the black string and thin film to the event horizon from outside the rotating and charged black string. Taking into account the effect of the generalized uncertainty principle on quantum state density, it removes the difficulty of the divergence of state density near the event horizon in the brick-wall model. These calculations and discussions imply that high density quantum states near the event horizon of a black string are strongly correlated with the quantum states in a black string and that black string entropy is a quantum effect. The ultraviolet cut-off in the brick-wall model is not reasonable. The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon. From the viewpoint of quantum statistical mechanics, the correction value of Bekenstein-Hawking entropy is obtained. This allows the fundamental recognition of the correction value of black string entropy at nonspherical coordinates.展开更多
A new type of superconductive true random number generator (TRNG) based on a negative-inductance superconducting quantum interference device (nSQUID) is proposed. The entropy harnessed to generate random numbers comes...A new type of superconductive true random number generator (TRNG) based on a negative-inductance superconducting quantum interference device (nSQUID) is proposed. The entropy harnessed to generate random numbers comes from the phenomenon of symmetry breaking in the nSQUID. The experimental circuit is fabricated by the Nb-based lift-off process. Low-temperature tests of the circuit verify the basic function of the proposed TRNG. The frequency characteristics of the TRNG have been analyzed by simulation. The generation rate of random numbers is expected to achieve hundreds of megahertz to tens of gigahertz.展开更多
Modified classical Boltzmann entropy as generalized entropy, then proposed Maximum Generalized Entropy Principle fusing physics and biology, and established a new model for biological origin and evolutions based on th...Modified classical Boltzmann entropy as generalized entropy, then proposed Maximum Generalized Entropy Principle fusing physics and biology, and established a new model for biological origin and evolutions based on this principle, finally took protein evolution for an example to analyze. The model provided some reference for biological complexity research.展开更多
A method with the fuzzy entropy for measuring fuzziness to fuzzy problem in rough sets is proposed. A new sort of the fuzzy entropy is given. The calculating formula and the equivalent expression method with the fuzzy...A method with the fuzzy entropy for measuring fuzziness to fuzzy problem in rough sets is proposed. A new sort of the fuzzy entropy is given. The calculating formula and the equivalent expression method with the fuzzy entropy in rough sets based on equivalence relation are provided, and the properties of the fuzzy entropy are proved. The fuzzy entropy based on equivalent relation is extended to generalize the fuzzy entropy based on general binary relation, and the calculating formula and the equivalent expression of the generalized fuzzy entropy are also given. Finally, an example illustrates the way for getting the fuzzy entropy. Results show that the fuzzy entropy can conveniently measure the fuzziness in rough sets.展开更多
Map is one of the communication means created by human being.Cartographers have been making efforts on the comparison of maps to natural languages so as to establish a"cartographic language"or"map langu...Map is one of the communication means created by human being.Cartographers have been making efforts on the comparison of maps to natural languages so as to establish a"cartographic language"or"map language".One of such efforts is to adopt the Shannon’s Information Theory originated in digital communication into cartography so as to establish an entropy-based cartographic communication theory.However,success has been very limited although research work had started as early as the mid-1960 s.It is then found that the bottleneck problem was the lack of appropriate measures for the spatial(configurational)information of(graphic and image)maps,as the classic Shannon entropy is only capable of characterizing statistical information but fails to capture the configurational information of(graphic and image)maps.Fortunately,after over 40-year development,some bottleneck problems have been solved.More precisely,generalized Shannon entropies for metric and thematic information of(graphic)maps have been developed and the first feasible solution for computing the Boltzmann entropy of image maps has been invented,which is capable of measuring the spatial information of not only numerical images but also categorical maps.With such progress,it is now feasible to build the"Information Theory of Cartography".In this paper,a framework for such a theory is proposed and some key issues are identified.For these issues,some have already been tackled while others still need efforts.As a result,a research agenda is set for future action.After all these issues are tackled,the theory will become matured so as to become a theoretic basis of cartography.It is expected that the Information Theory of Cartography will play an increasingly important role in the discipline of cartography because more and more researchers have advocated that information is more fundamental than matter and energy.展开更多
We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the ...We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p（Sd 1）inL q（Sd 1）whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p（Sd 1）inL q（Sd 1）are concerned.展开更多
The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES rel...The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Rényi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.展开更多
Without any other approximations apart from the spectral method which is employed,the energy spectra corresponding to two kinds of'negative temperatures'are simulated with a symmetric trapezium truncation.The ...Without any other approximations apart from the spectral method which is employed,the energy spectra corresponding to two kinds of'negative temperatures'are simulated with a symmetric trapezium truncation.The simulated results with either of the two negative temperatures are reasonably consistent with those from the statistical theory of turbulence.The more usual case for two positive temperatures evolves differently from the theoretical prediction.The viscosity influence on the ergodicity is discussed. It is shown that two--dimensional(2D)ideal flows on thesphere have a less pronounced tendency to be ergodic than those on planar geometry due to the curvature of thespherical surface that weakens the interaction between different parts of the flow,enabling these parts to behave inmore relative isolation. The expressions for the standard deviations from a canonical ensemble for the two differentoptions of coefficients are shown to be proportional to in(N is the total number of independent modes in the system),independent of the initial conditions of the system.展开更多
We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such th...We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.展开更多
Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, · · ·, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-...Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, · · ·, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-1), ξ_k).In this paper the limit properties of F_(m_n,b_n)(ω) and the generalized relative entropy density f_(m_n,b_n)(ω) =-(1/b_n) log p(ξ_(m_n,m_n+b_n)) are discussed, and some theorems on a.s. convergence for(ξ_n)_n=0~∞ and the generalized Shannon-McMillan(AEP) theorem on finite nonhomogeneous Markov chains are obtained.展开更多
In a C1 non-uniformly hyperbolic systems with limit domination, we consider the periodic measures that supported on the Pesin set and keep a distance at least 6 to a hyperbolic ergodic measure μ given before. And the...In a C1 non-uniformly hyperbolic systems with limit domination, we consider the periodic measures that supported on the Pesin set and keep a distance at least 6 to a hyperbolic ergodic measure μ given before. And then, we bound from top the exponential growth rate of such periodic measures by the supremum of measure theoretic entropy on a closed set.展开更多
This paper describes a non-linear information dynamics model for integrated risk assessment of complex disaster system from an evolution perspective. According to the occurrence and evolution of natural disaster syste...This paper describes a non-linear information dynamics model for integrated risk assessment of complex disaster system from an evolution perspective. According to the occurrence and evolution of natural disaster system with complicated and nonlinear characteristics, a non-linear information dynamics mode is introduced based on the maximum flux principle during modeling process to study the integrated risk assessment of complex disaster system. Based on the non-equilibrium statistical mechanics method, a stochastic evolution equation of this system is established. The integrated risk assessment of complex disaster system can be achieved by giving reasonable weights of each evaluation index to stabilize the system. The new model reveals the formation pattern of risk grade and the dynamics law of evolution. Meanwhile, a method is developed to solve the dynamics evolution equations of complex system through the self-organization feature map algorithm. The proposed method has been used in complex disaster integrated risk assessment for 31 provinces, cities and autonomous regions in China mainland. The results have indicated that the model is objective and effective.展开更多
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.U22B2005,62072109)the Natural Science Foundation of Fujian Province(Grant No.2021J01625)the Major Science and Technology Project of Fuzhou(Grant No.2023-ZD-003).
文摘As the scale of the networks continually expands,the detection of distributed denial of service(DDoS)attacks has become increasingly vital.We propose an intelligent detection model named IGED by using improved generalized entropy and deep neural network(DNN).The initial detection is based on improved generalized entropy to filter out as much normal traffic as possible,thereby reducing data volume.Then the fine detection is based on DNN to perform precise DDoS detection on the filtered suspicious traffic,enhancing the neural network’s generalization capabilities.Experimental results show that the proposed method can efficiently distinguish normal traffic from DDoS traffic.Compared with the benchmark methods,our method reaches 99.9%on low-rate DDoS(LDDoS),flooded DDoS and CICDDoS2019 datasets in terms of both accuracy and efficiency in identifying attack flows while reducing the time by 17%,31%and 8%.
文摘Among all statements of Second Law, the existence and uniqueness of stable equilibrium, for each given value of energy content and composition of constituents of any system, have been adopted to define thermodynamic entropy by means of the impossibility of Perpetual Motion Machine of the Second Kind (PMM2) which is a consequence of the Second Law. Equality of temperature, chemical potential and pressure in many-particle systems are proved to be necessary conditions for the stable equilibrium. The proofs assume the stable equilibrium and derive, by means of the Highest-Entropy Principle, equality of temperature, chemical potential and pressure as a consequence. A first novelty of the present research is to demonstrate that equality is also a sufficient condition, in addition to necessity, for stable equilibrium implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality of temperature potential and pressure addressed to as generalized potential. The second novelty is that the proof of sufficiency of equality, or necessity of stable equilibrium, is achieved by means of a generalization of entropy property, derived from a generalized definition of exergy, both being state and additive properties accounting for heat, mass and work interactions of the system underpinning the definition of Highest-Generalized-Entropy Principle adopted in the proof.
文摘Axiomatization of Shannon entropy is a subject that has received lots of attention in the information theory literature.While Shannon entropy is defined on probability distribution,we define a new type of entropy on the set of partitions of finite subsets of metric spaces,which has a rich algebraic structure as a partially ordered set.We propose an axiomatization of an entropy-like measure of partitions of sets of objects located in metric spaces,and we derive an analytic expression of this new type of entropy referred to as inertial entropy.This approach starts with the notion of inertia of a partition and includes a study of the behavior of the sum of square errors of a partition.In this context,we characterize the chain of partitions produced by the Ward hierarchical clustering method.Starting from inertial entropies of partitions,we introduce conditional entropies which,in turn,generate metrics on partitions of finite sets.These metrics are used as external validation tools for clusterings of labeled data sets.The metric generated by inertial entropy can be used to validate data clustering for labeled data sets.This type of validation aims to determine to what extend labeling of the data coincides with the clustering obtained algorithmically,and we obtain a high degree of consistency of the data labeling with the results of several hierarchical clusterings.
文摘In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Cross Fuzzy Entropy Optimization Methods in the form of MinMinx（F）EntM and MaxMinx（F）EntM are suggested on the basis of primary phase of minimizing cross fuzzy entropy measure for fixed moment vector function and on the definition of the special functional with Minx（F）Ent values of cross fuzzy entropy measure. Next phase for obtaining mentioned distributions consists of optimization of defined functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as （MinMinx（F）Ent）m and （MaxMinx（F）Ent）m distributions.
文摘This study is connected with new Generalized Maximum Fuzzy Entropy Methods （GMax（F）EntM） in the form of MinMax（F）EntM and MaxMax（F）EntM belonging to us. These methods are based on primary maximizing Max（F）Ent measure for fixed moment vector function in order to obtain the special functional with maximum values of Max（F）Ent measure and secondary optimization of mentioned functional with respect to moment vector functions. Distributions, in other words sets of successive values of estimated membership function closest to （furthest from） the given membership function in the sense of Max（F）Ent measure, obtained by mentioned methods are defined as （MinMax（F）Ent）m which is closest to a given membership function and （MaxMax（F）Ent）m which is furthest from a given membership function. The aim of this study consists of applying MinMax（F）EntM and MaxMax（F）EntM on given wind speed data. Obtained results are realized by using MATLAB programme. The performances of distributions （MinMax（F）En0m and （MaxMax（F）Ent）m generated by using Generalized Maximum Fuzzy Entropy Methods are established by Chi-Square, Root Mean Square Error criterias and Max（F）Ent measure.
基金supported by the Natural Science Foundation of Shanxi Province,China(Grant No 2006011012)the Doctoral Scientific Research Starting Foundation of Shanxi Datong University,China
文摘By using the entanglement entropy method, this paper calculates the statistical entropy of the Bose and Fermi fields in thin films, and derives the Bekenstein-Hawking entropy and its correction term on the background of a rotating and charged black string. Here, the quantum field is entangled with quantum states in the black string and thin film to the event horizon from outside the rotating and charged black string. Taking into account the effect of the generalized uncertainty principle on quantum state density, it removes the difficulty of the divergence of state density near the event horizon in the brick-wall model. These calculations and discussions imply that high density quantum states near the event horizon of a black string are strongly correlated with the quantum states in a black string and that black string entropy is a quantum effect. The ultraviolet cut-off in the brick-wall model is not reasonable. The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon. From the viewpoint of quantum statistical mechanics, the correction value of Bekenstein-Hawking entropy is obtained. This allows the fundamental recognition of the correction value of black string entropy at nonspherical coordinates.
基金Supported by the State Key Program for Basic Research of China under Grant No 2011CBA00304the National Natural Science Foundation of China under Grant No 60836001the Tsinghua University Initiative Scientific Research Program under Grant No 20131089314
文摘A new type of superconductive true random number generator (TRNG) based on a negative-inductance superconducting quantum interference device (nSQUID) is proposed. The entropy harnessed to generate random numbers comes from the phenomenon of symmetry breaking in the nSQUID. The experimental circuit is fabricated by the Nb-based lift-off process. Low-temperature tests of the circuit verify the basic function of the proposed TRNG. The frequency characteristics of the TRNG have been analyzed by simulation. The generation rate of random numbers is expected to achieve hundreds of megahertz to tens of gigahertz.
基金Supported by National Basic Research Program of China (973 Program) (Grant No. 2007CB714101)~~
文摘Modified classical Boltzmann entropy as generalized entropy, then proposed Maximum Generalized Entropy Principle fusing physics and biology, and established a new model for biological origin and evolutions based on this principle, finally took protein evolution for an example to analyze. The model provided some reference for biological complexity research.
文摘A method with the fuzzy entropy for measuring fuzziness to fuzzy problem in rough sets is proposed. A new sort of the fuzzy entropy is given. The calculating formula and the equivalent expression method with the fuzzy entropy in rough sets based on equivalence relation are provided, and the properties of the fuzzy entropy are proved. The fuzzy entropy based on equivalent relation is extended to generalize the fuzzy entropy based on general binary relation, and the calculating formula and the equivalent expression of the generalized fuzzy entropy are also given. Finally, an example illustrates the way for getting the fuzzy entropy. Results show that the fuzzy entropy can conveniently measure the fuzziness in rough sets.
基金National Natural Science Foundation of China(Nos.41930104,41971330)Hong Kong Research Grants Council General Research Fund(No.152219/18E)。
文摘Map is one of the communication means created by human being.Cartographers have been making efforts on the comparison of maps to natural languages so as to establish a"cartographic language"or"map language".One of such efforts is to adopt the Shannon’s Information Theory originated in digital communication into cartography so as to establish an entropy-based cartographic communication theory.However,success has been very limited although research work had started as early as the mid-1960 s.It is then found that the bottleneck problem was the lack of appropriate measures for the spatial(configurational)information of(graphic and image)maps,as the classic Shannon entropy is only capable of characterizing statistical information but fails to capture the configurational information of(graphic and image)maps.Fortunately,after over 40-year development,some bottleneck problems have been solved.More precisely,generalized Shannon entropies for metric and thematic information of(graphic)maps have been developed and the first feasible solution for computing the Boltzmann entropy of image maps has been invented,which is capable of measuring the spatial information of not only numerical images but also categorical maps.With such progress,it is now feasible to build the"Information Theory of Cartography".In this paper,a framework for such a theory is proposed and some key issues are identified.For these issues,some have already been tackled while others still need efforts.As a result,a research agenda is set for future action.After all these issues are tackled,the theory will become matured so as to become a theoretic basis of cartography.It is expected that the Information Theory of Cartography will play an increasingly important role in the discipline of cartography because more and more researchers have advocated that information is more fundamental than matter and energy.
基金Supported by National Natural Science Foundation of China(Grant Nos.10871132,11271263)Beijing Natural Science Foundation(Grant Nos.1102011,1132001)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20091108110004)
文摘We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p（Sd 1）inL q（Sd 1）whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p（Sd 1）inL q（Sd 1）are concerned.
文摘The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Rényi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.
文摘Without any other approximations apart from the spectral method which is employed,the energy spectra corresponding to two kinds of'negative temperatures'are simulated with a symmetric trapezium truncation.The simulated results with either of the two negative temperatures are reasonably consistent with those from the statistical theory of turbulence.The more usual case for two positive temperatures evolves differently from the theoretical prediction.The viscosity influence on the ergodicity is discussed. It is shown that two--dimensional(2D)ideal flows on thesphere have a less pronounced tendency to be ergodic than those on planar geometry due to the curvature of thespherical surface that weakens the interaction between different parts of the flow,enabling these parts to behave inmore relative isolation. The expressions for the standard deviations from a canonical ensemble for the two differentoptions of coefficients are shown to be proportional to in(N is the total number of independent modes in the system),independent of the initial conditions of the system.
基金supported by National Natural Science Foundation of China (GrantNo. 10901080)supported in part by KRF (Grant No. 2010-0020946)
文摘We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.
基金supported in part by the NNSF of China(No.11571142)the RP of Anhui Provincial Department of Education(No.KJ2017A851)
文摘Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, · · ·, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-1), ξ_k).In this paper the limit properties of F_(m_n,b_n)(ω) and the generalized relative entropy density f_(m_n,b_n)(ω) =-(1/b_n) log p(ξ_(m_n,m_n+b_n)) are discussed, and some theorems on a.s. convergence for(ξ_n)_n=0~∞ and the generalized Shannon-McMillan(AEP) theorem on finite nonhomogeneous Markov chains are obtained.
基金supported by National Natural Science Foundation of China (Grant Nos. 10671006)supported by National Natural Science Foundation of China (Grant Nos. 10671006, 10831003)+1 种基金National Basic Research Program of China (973 Program, 2006CB805903)supported by CAPES (Brazil)
文摘In a C1 non-uniformly hyperbolic systems with limit domination, we consider the periodic measures that supported on the Pesin set and keep a distance at least 6 to a hyperbolic ergodic measure μ given before. And then, we bound from top the exponential growth rate of such periodic measures by the supremum of measure theoretic entropy on a closed set.
基金supported by the National Twelfth Five-year Technology Support Projects of China (Grant Nos. 2009BAJ28B04, 2011BAK07B01,2011BAJ08B03, and 2011BAJ08B05)the National Natural Science Foundation of China (Grant No. 51208017)+1 种基金Beijing Postdoctoral Research Foundation (Grant No. 2012ZZ-17)China Postdoctoral Science Foundation Funded Project (Grant No. 2011M500199)
文摘This paper describes a non-linear information dynamics model for integrated risk assessment of complex disaster system from an evolution perspective. According to the occurrence and evolution of natural disaster system with complicated and nonlinear characteristics, a non-linear information dynamics mode is introduced based on the maximum flux principle during modeling process to study the integrated risk assessment of complex disaster system. Based on the non-equilibrium statistical mechanics method, a stochastic evolution equation of this system is established. The integrated risk assessment of complex disaster system can be achieved by giving reasonable weights of each evaluation index to stabilize the system. The new model reveals the formation pattern of risk grade and the dynamics law of evolution. Meanwhile, a method is developed to solve the dynamics evolution equations of complex system through the self-organization feature map algorithm. The proposed method has been used in complex disaster integrated risk assessment for 31 provinces, cities and autonomous regions in China mainland. The results have indicated that the model is objective and effective.
