In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hoppin...In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hopping integrals which are the physical origin of cohesion of atoms. For thesimple case of s-valent system, the inversion of the many-body potentials are discussed in detail by using the lattice inversion method.展开更多
In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach,which include Webb's[1]and Shaw's[2]formulae and some new one as special c...In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach,which include Webb's[1]and Shaw's[2]formulae and some new one as special cases.We also give the quantitative estimations for the general formulae.展开更多
For any positive integer m,letℤ_(m)be the additive group of residue classes modulo m.For A⊆ℤ_(m)and n∈ℤ_(m),let the representation function R_(A)(n)denote the number of solutions of the equation n=a=a'with unorde...For any positive integer m,letℤ_(m)be the additive group of residue classes modulo m.For A⊆ℤ_(m)and n∈ℤ_(m),let the representation function R_(A)(n)denote the number of solutions of the equation n=a=a'with unordered pairs(a,a')∈A×A.Let m=2^(α)M>2,whereαis a positive integer and M is a positive odd integer.In this paper,the author proves that if M≥3,then there exist two distinct sets A,B⊆ℤ_(m)with∣A∪B∣=m−2,A∩B=∅andB≠m/2+A such that R_(A)(n)=R(B)(n)for all n∈ℤ_(m).The author also proves that if M=1 and A,B⊆ℤ_(m)with∣A∪B∣=m−2 and A∩B=∅,then R_(A)(n)=R(B)(n)for all n∈ℤ_(m)if and only if B=m/2+A.展开更多
Since 2008, the author of this paper has conducted historiographic research on the visual history of science in the West since the mid-twentieth century. The findings show that the cognitive functions of visual scient...Since 2008, the author of this paper has conducted historiographic research on the visual history of science in the West since the mid-twentieth century. The findings show that the cognitive functions of visual scientific representations in the history of science are connected with theories of knowledge development in dialectical materialist epistemology and theories on children's cognitive features at different ages in developmental psychology, as well as the stage-specific curriculum objectives outlined in the Compulsory Education Science Curriculum Standards(2022 Edition). These insights provide essential inspiration and theoretical support for the establishment of the twin-theme logical structure in the Primary School Science Textbooks(Daxiang Edition)—core competencies as the warp and cognitive development as the weft—and for the intentional cultivation of students' cognitive abilities using scientific images across different learning stages and textbooks.展开更多
This paper introduces a new concept of ＂State Representation Methodology （SRM）＂ which is a kind of bridge condition assessment method for structural health monitoring system （SHM）. There are many methods for sys...This paper introduces a new concept of ＂State Representation Methodology （SRM）＂ which is a kind of bridge condition assessment method for structural health monitoring system （SHM）. There are many methods for system identification from the simplicity comparison of damage index to the complicated statistical pattern recognition algorithms in SHM. In these methods, modal analysis and parameters identification or many defined indices are common-used for extracting the dynamic or static characteristics of a system. However, there is a common problem： due to the complexity of a large size system with high-order nonlinear characteristics and severe environment interference, it is impossible to extract and quantify exactly these modal parameters or system parameters or indices as the feature vectors of a system in damage detection in an easy way. The SRM considered a more general theory for the non-parametric description of system state.展开更多
Let k1 ,k2 be nonzero integers with （k1, k2） = 1 and k1kk ≠-1. In this paper, we prove that there is a set A Z such that every integer can be represented uniquely in the form n = k1a1 ＋ k2a2, a1,a2 ∈ A.
This paper presents a combined strategy to solve the trajectory online optimization problem for unmanned combat aerial vehicle (UCAV). Firstly, as trajectory directly optimizing is quite time costing, an online trajec...This paper presents a combined strategy to solve the trajectory online optimization problem for unmanned combat aerial vehicle (UCAV). Firstly, as trajectory directly optimizing is quite time costing, an online trajectory functional representation method is proposed. Considering the practical requirement of online trajectory, the 4-order polynomial function is used to represent the trajectory, and which can be determined by two independent parameters with the trajectory terminal conditions; thus, the trajectory online optimization problem is converted into the optimization of the two parameters, which largely lowers the complexity of the optimization problem. Furthermore, the scopes of the two parameters have been assessed into small ranges using the golden section ratio method. Secondly, a multi-population rotation strategy differential evolution approach (MPRDE) is designed to optimize the two parameters; in which, 'current-to-best/1/bin', 'current-to-rand/1/bin' and 'rand/2/bin' strategies with fixed parameter settings are designed, these strategies are rotationally used by three subpopulations. Thirdly, the rolling optimization method is applied to model the online trajectory optimization process. Finally, simulation results demonstrate the efficiency and real-time calculation capability of the designed combined strategy for UCAV trajectory online optimizing under dynamic and complicated environments.展开更多
The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mist...The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.展开更多
For any positive integers k,kand any set A■N,let R(A,n)be the number of solutions of the equation n=ka+kawith a,a∈A.Let■=NA.Yang and Chen proved that if kand kare two integers with k>k≥2 and(k,k)=1,then there d...For any positive integers k,kand any set A■N,let R(A,n)be the number of solutions of the equation n=ka+kawith a,a∈A.Let■=NA.Yang and Chen proved that if kand kare two integers with k>k≥2 and(k,k)=1,then there does not exist any set A■N such that R(A,n)=R(■,n)for all sufficiently large integers n.For two integers k>1 and t≥1,define f(t)to be the number of sets A■N such that R(A,n)=R(■,n)holds for all integers n≥t.Yang and Chen proved that f(t)is finite and limlog fk(t)/t=log 2.They also asked if it is true that for any integers k,l>1 there exists t(k,l)such that f(t)=fl(t)for all integers t≥t.In this paper,we give the exact formula of f(t)when t≤k,which implies that f(t)=f(t)for all integers t≤min{k,l}.展开更多
The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than ba...The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.展开更多
Let k,l≥2 be positive integers.Let A be an infinite set of nonnegative integers.For n∈N,let r1,k,...,k^(l-1)(A,n)denote the number of solutions of n=a0+ka1+···+k^(l-1)al-1,a0,...,al-1∈A.In this paper...Let k,l≥2 be positive integers.Let A be an infinite set of nonnegative integers.For n∈N,let r1,k,...,k^(l-1)(A,n)denote the number of solutions of n=a0+ka1+···+k^(l-1)al-1,a0,...,al-1∈A.In this paper,we show that r1,k,...,kl-1(A,n)=1 for all n≥0 if and only if A is the set of all nonnegative integers such that all its digits in its kl-adic expansion are smaller than k.This result partially answers a question of Sarkozy and Sos on representation for multivariate linear forms.展开更多
The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the ...The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.展开更多
Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A...Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A, n） = 1 for all n ∈ Z. Let f ： Z-→ N0∪ {∞} be a function such that f-1（0） is finite. In this paper, we generalize Xiong＇s result and prove that there exist uncountably many sets A Z such that Rk1,k2（A, n） = f（n） for all n ∈ Z.展开更多
This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice o...This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.展开更多
The draping behavior of fabric is simulated by using four node quadrilateral thin plate elements with finite rotation. The finite element formulation is based on the total Lagrangian approach. An exact representatio...The draping behavior of fabric is simulated by using four node quadrilateral thin plate elements with finite rotation. The finite element formulation is based on the total Lagrangian approach. An exact representation of finite rotation is introduced. The strain energy function accounting for the material symmetry is obtained by the tensor representation theory. To avoid shear locking, the assumed strain technique for transverse shear is adopted. The conjugate gradient method with a proposed line search algorithm is employed to minimize energy and reach the final shape of fabric. The draping behavior of a rectangular piece of fabric over a rectangular table is simulated. (Author abstract) 9 Refs.展开更多
For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be th...For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.展开更多
This paper proposes an approach for functional knowledge representation based on problem reduction,which represents the organization of problem-solving activities in two levels:reduction and reasoning.The former makes...This paper proposes an approach for functional knowledge representation based on problem reduction,which represents the organization of problem-solving activities in two levels:reduction and reasoning.The former makes the functional plans for problem-solving while the latter constructs functional units, called handlers,for executing subproblems designated by these plans.This approach emphasizes that the representation of domain knowledge should be closely combined with(rather than separated from)its use therefore provides a set of reasoning-level primitives to construct handlers and formulate the control strate- gies for executing them.As reduction-level primitives,handlers are used to construct handler-associative networks,which become the executable representation of problem-reduction graphs,in order to realize the problem-solving methods suited to domain features.Besides,handlers and their control slots can be used to focus the attention of knowledge acquisition and reasoning control.展开更多
Assume that B is a separable real Banach space and X(t) is a diffusion process on B. In thispaper, we will establish the representation theorem of martingale additive functionals of X(t).
In cryptology, it is an important topic to study the best affine approach of functions. The best affine approach of Boolean functions has been discussed in ref. [1] by using the Walsh spectrum, of which the key proble...In cryptology, it is an important topic to study the best affine approach of functions. The best affine approach of Boolean functions has been discussed in ref. [1] by using the Walsh spectrum, of which the key problem is how to represent the correspondence of Boolean functions by using Walsh spectrum. For the multi-valued logical functions so far, the spectral representation of their correspondence has not been presented yet. This let-展开更多
文摘In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hopping integrals which are the physical origin of cohesion of atoms. For thesimple case of s-valent system, the inversion of the many-body potentials are discussed in detail by using the lattice inversion method.
文摘In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach,which include Webb's[1]and Shaw's[2]formulae and some new one as special cases.We also give the quantitative estimations for the general formulae.
基金supported by the National Natural Science Foundation of China(No.12371003).
文摘For any positive integer m,letℤ_(m)be the additive group of residue classes modulo m.For A⊆ℤ_(m)and n∈ℤ_(m),let the representation function R_(A)(n)denote the number of solutions of the equation n=a=a'with unordered pairs(a,a')∈A×A.Let m=2^(α)M>2,whereαis a positive integer and M is a positive odd integer.In this paper,the author proves that if M≥3,then there exist two distinct sets A,B⊆ℤ_(m)with∣A∪B∣=m−2,A∩B=∅andB≠m/2+A such that R_(A)(n)=R(B)(n)for all n∈ℤ_(m).The author also proves that if M=1 and A,B⊆ℤ_(m)with∣A∪B∣=m−2 and A∩B=∅,then R_(A)(n)=R(B)(n)for all n∈ℤ_(m)if and only if B=m/2+A.
基金supported by a general research programme‘Research on the Compilation System of Primary School Science Textbooks with Marxist Epistemology as a Main Theme’under the 2023 Henan Province Education Sciences Plan launched by the Education Department of Henan Province(grant no.2023YB0610)
文摘Since 2008, the author of this paper has conducted historiographic research on the visual history of science in the West since the mid-twentieth century. The findings show that the cognitive functions of visual scientific representations in the history of science are connected with theories of knowledge development in dialectical materialist epistemology and theories on children's cognitive features at different ages in developmental psychology, as well as the stage-specific curriculum objectives outlined in the Compulsory Education Science Curriculum Standards(2022 Edition). These insights provide essential inspiration and theoretical support for the establishment of the twin-theme logical structure in the Primary School Science Textbooks(Daxiang Edition)—core competencies as the warp and cognitive development as the weft—and for the intentional cultivation of students' cognitive abilities using scientific images across different learning stages and textbooks.
文摘This paper introduces a new concept of ＂State Representation Methodology （SRM）＂ which is a kind of bridge condition assessment method for structural health monitoring system （SHM）. There are many methods for system identification from the simplicity comparison of damage index to the complicated statistical pattern recognition algorithms in SHM. In these methods, modal analysis and parameters identification or many defined indices are common-used for extracting the dynamic or static characteristics of a system. However, there is a common problem： due to the complexity of a large size system with high-order nonlinear characteristics and severe environment interference, it is impossible to extract and quantify exactly these modal parameters or system parameters or indices as the feature vectors of a system in damage detection in an easy way. The SRM considered a more general theory for the non-parametric description of system state.
基金Supported by the National Natural Science Foundation of China(Grant No.10901002)the Natural Science Foundation of Anhui Province(Grant No.1208085QA02)
文摘Let k1 ,k2 be nonzero integers with （k1, k2） = 1 and k1kk ≠-1. In this paper, we prove that there is a set A Z such that every integer can be represented uniquely in the form n = k1a1 ＋ k2a2, a1,a2 ∈ A.
基金supported by the National Natural Science Foundation of China(61601505)the Aeronautical Science Foundation of China(20155196022)the Shaanxi Natural Science Foundation of China(2016JQ6050)
文摘This paper presents a combined strategy to solve the trajectory online optimization problem for unmanned combat aerial vehicle (UCAV). Firstly, as trajectory directly optimizing is quite time costing, an online trajectory functional representation method is proposed. Considering the practical requirement of online trajectory, the 4-order polynomial function is used to represent the trajectory, and which can be determined by two independent parameters with the trajectory terminal conditions; thus, the trajectory online optimization problem is converted into the optimization of the two parameters, which largely lowers the complexity of the optimization problem. Furthermore, the scopes of the two parameters have been assessed into small ranges using the golden section ratio method. Secondly, a multi-population rotation strategy differential evolution approach (MPRDE) is designed to optimize the two parameters; in which, 'current-to-best/1/bin', 'current-to-rand/1/bin' and 'rand/2/bin' strategies with fixed parameter settings are designed, these strategies are rotationally used by three subpopulations. Thirdly, the rolling optimization method is applied to model the online trajectory optimization process. Finally, simulation results demonstrate the efficiency and real-time calculation capability of the designed combined strategy for UCAV trajectory online optimizing under dynamic and complicated environments.
文摘The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.
基金Supported by the National Natural Science Foundation of China(Grant No.12101009)the Natural Science Foundation of Anhui Province(Grant No.2108085QA02)。
文摘For any positive integers k,kand any set A■N,let R(A,n)be the number of solutions of the equation n=ka+kawith a,a∈A.Let■=NA.Yang and Chen proved that if kand kare two integers with k>k≥2 and(k,k)=1,then there does not exist any set A■N such that R(A,n)=R(■,n)for all sufficiently large integers n.For two integers k>1 and t≥1,define f(t)to be the number of sets A■N such that R(A,n)=R(■,n)holds for all integers n≥t.Yang and Chen proved that f(t)is finite and limlog fk(t)/t=log 2.They also asked if it is true that for any integers k,l>1 there exists t(k,l)such that f(t)=fl(t)for all integers t≥t.In this paper,we give the exact formula of f(t)when t≤k,which implies that f(t)=f(t)for all integers t≤min{k,l}.
基金Supported by the 973program-2006CB303102SGST 09DZ 2272900NSFC No.11026089
文摘The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.
基金Supported by the National Natural Science Foundation of China(Grant No.11971033)Top Talents Project of Anhui Department of Education(Grant No.gxbj ZD05)the Natural Science Foundation of Anhui Province(Grant No.2008085QA06)。
文摘Let k,l≥2 be positive integers.Let A be an infinite set of nonnegative integers.For n∈N,let r1,k,...,k^(l-1)(A,n)denote the number of solutions of n=a0+ka1+···+k^(l-1)al-1,a0,...,al-1∈A.In this paper,we show that r1,k,...,kl-1(A,n)=1 for all n≥0 if and only if A is the set of all nonnegative integers such that all its digits in its kl-adic expansion are smaller than k.This result partially answers a question of Sarkozy and Sos on representation for multivariate linear forms.
基金Supported by the National Natural Science Foundation of China(11171298)the Zhejiang Natural Science Foundation of China(Y6110425)
文摘The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.
基金Supported by the National Natural Science Foundation of China(Grant No.11471017)
文摘Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A, n） = 1 for all n ∈ Z. Let f ： Z-→ N0∪ {∞} be a function such that f-1（0） is finite. In this paper, we generalize Xiong＇s result and prove that there exist uncountably many sets A Z such that Rk1,k2（A, n） = f（n） for all n ∈ Z.
文摘This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.
文摘The draping behavior of fabric is simulated by using four node quadrilateral thin plate elements with finite rotation. The finite element formulation is based on the total Lagrangian approach. An exact representation of finite rotation is introduced. The strain energy function accounting for the material symmetry is obtained by the tensor representation theory. To avoid shear locking, the assumed strain technique for transverse shear is adopted. The conjugate gradient method with a proposed line search algorithm is employed to minimize energy and reach the final shape of fabric. The draping behavior of a rectangular piece of fabric over a rectangular table is simulated. (Author abstract) 9 Refs.
基金supported by National Natural Science Foundation of China (Grant No.11071121)
文摘For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.
基金This research was supported by National High-tech Program(863 Program)of P.R.China.
文摘This paper proposes an approach for functional knowledge representation based on problem reduction,which represents the organization of problem-solving activities in two levels:reduction and reasoning.The former makes the functional plans for problem-solving while the latter constructs functional units, called handlers,for executing subproblems designated by these plans.This approach emphasizes that the representation of domain knowledge should be closely combined with(rather than separated from)its use therefore provides a set of reasoning-level primitives to construct handlers and formulate the control strate- gies for executing them.As reduction-level primitives,handlers are used to construct handler-associative networks,which become the executable representation of problem-reduction graphs,in order to realize the problem-solving methods suited to domain features.Besides,handlers and their control slots can be used to focus the attention of knowledge acquisition and reasoning control.
文摘Assume that B is a separable real Banach space and X(t) is a diffusion process on B. In thispaper, we will establish the representation theorem of martingale additive functionals of X(t).
文摘In cryptology, it is an important topic to study the best affine approach of functions. The best affine approach of Boolean functions has been discussed in ref. [1] by using the Walsh spectrum, of which the key problem is how to represent the correspondence of Boolean functions by using Walsh spectrum. For the multi-valued logical functions so far, the spectral representation of their correspondence has not been presented yet. This let-
