The conceptions of the knowledge screen generated by S-rough sets are given: f- screen and - screen , and then puts forward - filter theorem, - filter theorem of knowledge. At last, the applications of knowledge separ...The conceptions of the knowledge screen generated by S-rough sets are given: f- screen and - screen , and then puts forward - filter theorem, - filter theorem of knowledge. At last, the applications of knowledge separation are given according to - screen and - screen.展开更多
By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets （singular rough ...By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets （singular rough sets）. S-rough sets are defined by element equivalence class that proposes dynamic characteristic. S-rough sets have dynamic characteristic. By introducing the function equivalence class （law equivalence class） that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets （function singular rough sets）. Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.展开更多
Abstract The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point th...Abstract The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point theorem on the cone.展开更多
The nonlocal boundary value problems for nonlinear elliptic systems in the unbounded domain are considered. Under suitable conditions the existence of solution and comparison theorem for the boundary value problems ar...The nonlocal boundary value problems for nonlinear elliptic systems in the unbounded domain are considered. Under suitable conditions the existence of solution and comparison theorem for the boundary value problems are studied.展开更多
This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's t...This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.展开更多
In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two ca...In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.展开更多
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper est...Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.展开更多
The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain mus...The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the famous growth and covering theorem for univalent functions.This theorem successfully realizes the initial idea of H. Cartan about how to extend geometric function theory from one variable to several complex variables.展开更多
When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be a...When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.展开更多
This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectatio...This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.展开更多
The existence of positive solutions to second-order Neumann BVPs -u' + Mu = f(t, u), u'(0) = u'(1) = 0 and u' + Mu = f(t, u), u'(0) =u'(1) is proved by a simple application of a Fixed Poin...The existence of positive solutions to second-order Neumann BVPs -u' + Mu = f(t, u), u'(0) = u'(1) = 0 and u' + Mu = f(t, u), u'(0) =u'(1) is proved by a simple application of a Fixed Point Theorem in cones due to Krasnoselskii[1,6].展开更多
PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which combines the features of legacy underwater glider and conventional AUV （autonomous underwater vehicle）. It can b...PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which combines the features of legacy underwater glider and conventional AUV （autonomous underwater vehicle）. It can be treated as a multi-rigid-body system with a floating base and a particular hydrodynamic profile. In this paper, theorems on linear and angular momentum are used to establish the dynamic equations of motion of each rigid body and the effect of translational and rotational motion of internal masses on the attitude control are taken into consideration. In addition, due to the unique external shape with fixed wings and deflectable rudders and the dual-drive operation in thrust and glide modes, the approaches of building dynamic model of conventional AUV and hydrodynamic model of submarine are introduced, and the tailored dynamic equations of the hybrid glider are formulated. Moreover, the behaviors of motion in glide and thrust operation are analyzed based on the simulation and the feasibility of the dynamic model is validated by data from lake field trials.展开更多
The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| ...The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.展开更多
The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic Ln is proposed. A limit theorem is obtained, which says that the truth function τ-n induced by truth degrees converges to the integ...The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic Ln is proposed. A limit theorem is obtained, which says that the truth function τ-n induced by truth degrees converges to the integrated truth function τ when n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Lukasiewicz logic and the continuous valued Lukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.展开更多
The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCati...The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCation of a Fixed point Theorem in cones due to Krasnoselskii.展开更多
This paper is on fuzzy decision theory and information security theory which are mutually independent and engrafted. This paper presents both-branch fuzzy decision and problems on decision encryption-authentication, p...This paper is on fuzzy decision theory and information security theory which are mutually independent and engrafted. This paper presents both-branch fuzzy decision and problems on decision encryption-authentication, puts forward two kinds of both-branch fuzzy decision on X: both-branch fuzzy decision on X having bounded domains X = X+∩X- ={x0}, both-branch fuzzy decision on X havingoverlapping domains X* = X+∩X- = {x_1~*,x_2~*,…,x_1~*}. It presents single branch separate-separate out theoremof both-branch fuzzy decision on X, single branch superposition theorem of both-branch fuzzy decision, exclusive-decompose principle of both-branch fuzzy decision. This paper presents encryption-authentication theorem of decision by engrafting information security theory and technique with both-branch fuzzy decision, and provides encryption-decryption and signature-authentication of both-branch fuzzy decision.展开更多
Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and cov...Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.展开更多
Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives...Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives rough law generation model of a-function equivalence class, discussion on law mining and law discovery in systems, and application of law mining and law discovery in communication system. Function S-rough sets is a new theory and method in law mining research.展开更多
By using the generalized quasi-symmetry of the infinitesimal transformation of the transformation group G_r, the Noether’s theory for Birkhoffian systems (including Noether’s theorem and its inverse) has been establ...By using the generalized quasi-symmetry of the infinitesimal transformation of the transformation group G_r, the Noether’s theory for Birkhoffian systems (including Noether’s theorem and its inverse) has been established. An example is given.展开更多
文摘The conceptions of the knowledge screen generated by S-rough sets are given: f- screen and - screen , and then puts forward - filter theorem, - filter theorem of knowledge. At last, the applications of knowledge separation are given according to - screen and - screen.
基金the National Natural Science Foundation of China (Grant No. 60364001)the key Project of Chinese Ministry of Education (Grant No. 206089)the Natural Science Foundation of Shandong Province of China (Grant No. Y2007H02)
文摘By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets （singular rough sets）. S-rough sets are defined by element equivalence class that proposes dynamic characteristic. S-rough sets have dynamic characteristic. By introducing the function equivalence class （law equivalence class） that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets （function singular rough sets）. Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.
文摘Abstract The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point theorem on the cone.
基金The project supported by the National Natural Science Foundation of China
文摘The nonlocal boundary value problems for nonlinear elliptic systems in the unbounded domain are considered. Under suitable conditions the existence of solution and comparison theorem for the boundary value problems are studied.
基金supported by National Basic Research Program of China (Grant No.2007CB814900)(Financial Risk)
文摘This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.
基金supported by the National Natural Science Foundations of China (Grant Nos.11072218 and 11272287)the Natural Science Foundations of Zhejiang Province of China (Grant No.Y6110314)
文摘In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.
基金supported by National Natural Science Foundation of China (Grant No. 11225104)the National Basic Research Program of China (Grant No. 2015CB352302)the Fundamental Research Funds for the Central Universities
文摘Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.
文摘The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the famous growth and covering theorem for univalent functions.This theorem successfully realizes the initial idea of H. Cartan about how to extend geometric function theory from one variable to several complex variables.
文摘When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.
基金Project supported by the National Natural Science Foundation of China(No.10131040).
文摘This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.
文摘The existence of positive solutions to second-order Neumann BVPs -u' + Mu = f(t, u), u'(0) = u'(1) = 0 and u' + Mu = f(t, u), u'(0) =u'(1) is proved by a simple application of a Fixed Point Theorem in cones due to Krasnoselskii[1,6].
基金supported by the National Natural Science Foundation of China(Grant Nos. 50835006 and 51005161)the Science & Technology Support Planning Foundation of Tianjin(Grant No. 09ZCKFGX03000)the Natural Science Foundation of Tianjin(Grant No. 09JCZDJC23400)
文摘PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which combines the features of legacy underwater glider and conventional AUV （autonomous underwater vehicle）. It can be treated as a multi-rigid-body system with a floating base and a particular hydrodynamic profile. In this paper, theorems on linear and angular momentum are used to establish the dynamic equations of motion of each rigid body and the effect of translational and rotational motion of internal masses on the attitude control are taken into consideration. In addition, due to the unique external shape with fixed wings and deflectable rudders and the dual-drive operation in thrust and glide modes, the approaches of building dynamic model of conventional AUV and hydrodynamic model of submarine are introduced, and the tailored dynamic equations of the hybrid glider are formulated. Moreover, the behaviors of motion in glide and thrust operation are analyzed based on the simulation and the feasibility of the dynamic model is validated by data from lake field trials.
基金This work was supported by 973 Project, the National Natural Science Foundation of China (Grant No. 19871081) the Natural Science Foundation of Guangdong Province and Anhui Province.
文摘The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.
基金This work was supported by the National Natural Science Foundation of China(Grant No．10331010)
文摘The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic Ln is proposed. A limit theorem is obtained, which says that the truth function τ-n induced by truth degrees converges to the integrated truth function τ when n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Lukasiewicz logic and the continuous valued Lukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.
文摘The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCation of a Fixed point Theorem in cones due to Krasnoselskii.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 70271048) the Natural Science Foundation of Shandong Province (Y2001G07).
文摘This paper is on fuzzy decision theory and information security theory which are mutually independent and engrafted. This paper presents both-branch fuzzy decision and problems on decision encryption-authentication, puts forward two kinds of both-branch fuzzy decision on X: both-branch fuzzy decision on X having bounded domains X = X+∩X- ={x0}, both-branch fuzzy decision on X havingoverlapping domains X* = X+∩X- = {x_1~*,x_2~*,…,x_1~*}. It presents single branch separate-separate out theoremof both-branch fuzzy decision on X, single branch superposition theorem of both-branch fuzzy decision, exclusive-decompose principle of both-branch fuzzy decision. This paper presents encryption-authentication theorem of decision by engrafting information security theory and technique with both-branch fuzzy decision, and provides encryption-decryption and signature-authentication of both-branch fuzzy decision.
基金Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science (Nos.19540205,200717540138,2007).
文摘Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.
基金This project was supported by Natural Science Foundation of Shandong Province of China (Y2004A04), Natural ScienceFoundation of Fujian of China (Z051049) and Education Foundation of Fujian of China (JA04268),.
文摘Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives rough law generation model of a-function equivalence class, discussion on law mining and law discovery in systems, and application of law mining and law discovery in communication system. Function S-rough sets is a new theory and method in law mining research.
基金Project supported by the National Natural Science Foundation of China
文摘By using the generalized quasi-symmetry of the infinitesimal transformation of the transformation group G_r, the Noether’s theory for Birkhoffian systems (including Noether’s theorem and its inverse) has been established. An example is given.
