This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the correspo...This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.展开更多
We introduced the fuzzy axioms of choice,fuzzy Zorn’s lemma and fuzzy well-ordering principle,which are the fuzzy versions of the axioms of choice,Zorn’s lemma and well-ordering principle,and discussed the relations...We introduced the fuzzy axioms of choice,fuzzy Zorn’s lemma and fuzzy well-ordering principle,which are the fuzzy versions of the axioms of choice,Zorn’s lemma and well-ordering principle,and discussed the relations among them.As an application of fuzzy Zorn’s lemma,we got the following results:(1)Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal.(2)Every nonzero ring contained a fuzzy maximal ideal.(3)Introduced the notion of fuzzy nilpotent elements in a ring R,and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R.(4)Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma,we proved the fuzzy Tychonoff Theorem.展开更多
With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harm...With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.展开更多
This note presents a set of new versions of Barbalat's lemma combining with positive (negative) definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple ex...This note presents a set of new versions of Barbalat's lemma combining with positive (negative) definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple example shows the usefulness of our results.展开更多
On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. T...On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.展开更多
Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including ...Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.展开更多
We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem wit...We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.展开更多
New better estimates, which are given in terms of elementary functions, for the function r →(2/π)(1 -r2)κ(r)κ′(r) + logr appearing in Hubner's sharp upper bound for the Hersch-Pfluger distortion functio...New better estimates, which are given in terms of elementary functions, for the function r →(2/π)(1 -r2)κ(r)κ′(r) + logr appearing in Hubner's sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.展开更多
This paper further investigates the synchronization problem of a new chaotic system with known or unknown system parameters. Based on the Lyapunov stability theory,a novel adaptive control law is derived for the synch...This paper further investigates the synchronization problem of a new chaotic system with known or unknown system parameters. Based on the Lyapunov stability theory,a novel adaptive control law is derived for the synchronization of a new chaotic system with known or unknown system parameters.Theoretical analysis and numerical simulations showthe effectiveness and feasibility of the proposed schemes.展开更多
Digital signature is one of the most important cryptographic primitives. We proposed a new digital signature scheme based on Catalano’s trapdoor. Since Catalano’s trapdoor is more efficient than existing trapdoors i...Digital signature is one of the most important cryptographic primitives. We proposed a new digital signature scheme based on Catalano’s trapdoor. Since Catalano’s trapdoor is more efficient than existing trapdoors in number theory, our scheme need not modular exponentiation but several modular multiplications in the signing algorithm. We also proved our scheme is provably secure against adap-tively chosen message attack by using the Forking lemma.展开更多
The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational...The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.展开更多
The problem of indirect adaptive fuzzy and impulsive control for a class of nonlinear systems is investigated. Based on the approximation capability of fuzzy systems, a novel adaptive fuzzy and impulsive control strat...The problem of indirect adaptive fuzzy and impulsive control for a class of nonlinear systems is investigated. Based on the approximation capability of fuzzy systems, a novel adaptive fuzzy and impulsive control strategy with supervisory controller is developed. With the help of a supervisory controller, global stability of the resulting closed-loop system is established in the sense that all signals involved are uniformly bounded. Furthermore, the adaptive compensation term of the upper bound function of the sum of residual and approximation error is adopted to reduce the effects of modeling error. By the generalized Barbalat's lemma, the tracking error between the output of the system and the reference signal is proved to be convergent to zero Asymptotically. Simulation results illustrate the effectiveness of the proposed approach.展开更多
The Kuhn-Tucker conditions have been used to derive many significant results in economics. However, thus far, their derivation has been a little bit troublesome. The author directly derives the Kuhn-Tucker conditions ...The Kuhn-Tucker conditions have been used to derive many significant results in economics. However, thus far, their derivation has been a little bit troublesome. The author directly derives the Kuhn-Tucker conditions by applying a corollary of Farkas’s lemma under the Mangasarian-Fromovitz constraint qualification and shows the boundedness of Lagrange multipliers.展开更多
基金by Dr Kemp from National Mathematics and Science College.
文摘This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.
基金Supported by the National Natural Science Foundation of China(11971384)by the grant of Natural Science Basic Research Program of Shaanxi(Program No.2021JM-137)the Fundamental Research Funds for the Central Universities under grant QTZX2106,China 111 Project(B16037)and OPPO Research Fund.
文摘We introduced the fuzzy axioms of choice,fuzzy Zorn’s lemma and fuzzy well-ordering principle,which are the fuzzy versions of the axioms of choice,Zorn’s lemma and well-ordering principle,and discussed the relations among them.As an application of fuzzy Zorn’s lemma,we got the following results:(1)Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal.(2)Every nonzero ring contained a fuzzy maximal ideal.(3)Introduced the notion of fuzzy nilpotent elements in a ring R,and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R.(4)Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma,we proved the fuzzy Tychonoff Theorem.
文摘With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.
基金supported by National Natural Science Foundation of China(No.60710002,60974044)Program for Changjiang Scholars and Innovative Research Team in University
文摘This note presents a set of new versions of Barbalat's lemma combining with positive (negative) definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple example shows the usefulness of our results.
基金supported by the National Natural Science Foundation of China (90715011, 10672033 and 10590354) the National Key Basic Research and Development Program (2002CB412709) the Australia Research Council through the ARC International Fellowship Offered at University of Newcastle (LX0666274)
文摘On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(2013R1A1A2005402)National Science Foundation(DMS-1109063)
文摘Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.
文摘We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.
基金Supported by National 973 Project of China(2006CB708304)National Natural Science Foundation of China(10771195)
文摘New better estimates, which are given in terms of elementary functions, for the function r →(2/π)(1 -r2)κ(r)κ′(r) + logr appearing in Hubner's sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.
文摘This paper further investigates the synchronization problem of a new chaotic system with known or unknown system parameters. Based on the Lyapunov stability theory,a novel adaptive control law is derived for the synchronization of a new chaotic system with known or unknown system parameters.Theoretical analysis and numerical simulations showthe effectiveness and feasibility of the proposed schemes.
基金Supported by the National Natural Science Foundation of China (No. 60703086)Program for Excellent Talents in Nanjing University of Posts and Telecommunications(No. NY209014)
文摘Digital signature is one of the most important cryptographic primitives. We proposed a new digital signature scheme based on Catalano’s trapdoor. Since Catalano’s trapdoor is more efficient than existing trapdoors in number theory, our scheme need not modular exponentiation but several modular multiplications in the signing algorithm. We also proved our scheme is provably secure against adap-tively chosen message attack by using the Forking lemma.
基金Supported by National Natural Science Foundation of China (11071198 11101347)+2 种基金Postdoctor Foundation of China (2012M510363)the Key Project in Science and Technology Research Plan of the Education Department of Hubei Province (D20112605 D20122501)
文摘The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.
基金Supported by Tianyuan Fund for Mathematics of National Natural Science Foundation of China(A0324615)and Science Founda-tion of Nanjing Normal University(2002SXXXGQ2B20).
基金supported by Natural Science Foundation of Jiangsu Province of China (No. BK2010292, No. BK2010293)Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 09KJB510018, No. 08KJD510008)
文摘The problem of indirect adaptive fuzzy and impulsive control for a class of nonlinear systems is investigated. Based on the approximation capability of fuzzy systems, a novel adaptive fuzzy and impulsive control strategy with supervisory controller is developed. With the help of a supervisory controller, global stability of the resulting closed-loop system is established in the sense that all signals involved are uniformly bounded. Furthermore, the adaptive compensation term of the upper bound function of the sum of residual and approximation error is adopted to reduce the effects of modeling error. By the generalized Barbalat's lemma, the tracking error between the output of the system and the reference signal is proved to be convergent to zero Asymptotically. Simulation results illustrate the effectiveness of the proposed approach.
文摘The Kuhn-Tucker conditions have been used to derive many significant results in economics. However, thus far, their derivation has been a little bit troublesome. The author directly derives the Kuhn-Tucker conditions by applying a corollary of Farkas’s lemma under the Mangasarian-Fromovitz constraint qualification and shows the boundedness of Lagrange multipliers.