Hilbert problem 15 required understanding Schubert's book.In this book,reducing to degenerate cases was one of the main methods for enumeration.We found that nonstandard analysis is a suitable tool for making rigo...Hilbert problem 15 required understanding Schubert's book.In this book,reducing to degenerate cases was one of the main methods for enumeration.We found that nonstandard analysis is a suitable tool for making rigorous of Schubert's proofs of some results,which used degeneration method,but are obviously not rigorous.In this paper,we give a rigorous proof for Example 4 in Schubert's book,Chapter 1.§4 according to his idea.This shows that Schubert's intuitive idea is correct,but to make it rigorous a lot of work should be done.展开更多
Considering the complex constraint between operations in nonstandard job shop scheduling problem （NJSSP）, critical path of job manufacturing tree is determined according to priority scheduling function constructed. ...Considering the complex constraint between operations in nonstandard job shop scheduling problem （NJSSP）, critical path of job manufacturing tree is determined according to priority scheduling function constructed. Operations are divided into dependent operations and independent operations with the idea of subsection, and corresponding scheduling strategy is put forward according to operation characteristic in the segment and the complementarities of identical function machines. Forward greedy rule is adopted mainly for dependent operations to make operations arranged in the right position of machine selected, then each operation can be processed as early as possible, and the total processing time of job can be shortened as much as possible. For independent operations optimum scheduling rule is adopted mainly, the inserting position of operations will be determined according to the gap that the processing time of operations is subtracted from idle time of machine, and the operation will be inserted in the position with minimal gap. Experiments show, under the same conditions, the result that operations are scheduled according to the object function constructed, and the scheduling strategy adopted is better than the result that operations are scheduled according to efficiency scheduling algorithm.展开更多
The perturbation to Noether symmetry and adiabatic invariants for dynamical systems with nonstandard Lagrangians are studied.Based on two kinds of nonstandard Lagrangians(i.e.exponential Lagrangians and power-law Lagr...The perturbation to Noether symmetry and adiabatic invariants for dynamical systems with nonstandard Lagrangians are studied.Based on two kinds of nonstandard Lagrangians(i.e.exponential Lagrangians and power-law Lagrangians),the exact invariants of Noether type are given.Based on the definition of highorder adiabatic invariants,the relationship between the perturbation of Noether symmetry and the adiabatic invariants of the system under a small disturbance is studied,and then the corresponding theorems of adiabatic invariants are established.Finally,two examples are given to illustrate the methods and results appear in this paper.展开更多
Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at ti...Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.展开更多
We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random...We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random variables. As an application, we consider a simple calculation of financial time series.展开更多
Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. T...Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.展开更多
In nonstandard enlargement, the separations are characterized by non- standard analysis methods in [0, 1J-topological spaces. Firstly, the monads of fuzzy point in [0, 1]-topological spaces are described with remote-n...In nonstandard enlargement, the separations are characterized by non- standard analysis methods in [0, 1J-topological spaces. Firstly, the monads of fuzzy point in [0, 1]-topological spaces are described with remote-neighborhoods in non- standard enlarged model. Then the nonstandard characterizations of separations in [0, 1]-topological space are given by the monads. At last, relations of these separations are investigated.展开更多
Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-co...Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-complex special solutions which can be made up of hyperbolicfunctions or elliptic functions.展开更多
A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the d...A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.展开更多
In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’...In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.展开更多
This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has ...This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has different singularity （vanishing in finite time）.展开更多
In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique...In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique.These proposed schemes reproduce long time features of the oscillator solution exactly.Their abilities in preserving the symplecticity,the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied.It can be shown that the component { x_n}_(n≥1) of the INSOPT schemes switch signs infinitely many times as n →∞,almost surely.Further,the mean-square convergence order of 1 is obtained for these INSOPT schemes.Finally,numerical experiments illustrate intuitively the results obtained in this paper.展开更多
This paper deals with reaction-diffusion equations involving nonstandard growth conditions, subject to homogeneous Neumann boundary conditions. The complete clas- sification is established for simultaneous and non-sim...This paper deals with reaction-diffusion equations involving nonstandard growth conditions, subject to homogeneous Neumann boundary conditions. The complete clas- sification is established for simultaneous and non-simultaneous quenching under suitable assumptions on initial data. Moreover, quenching sets and quenching rates are obtained.展开更多
In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis fu...In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such,the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs,leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.展开更多
In this paper we develope a theory of new generalized functions by using Nonstandard Analysis which is closely relevant to that of Colombeau's new generalized functions.
The principal stress rotation is one of the most important features of the stress state in a seabed subjected to wave loading. Most prior investigations focused their attention on the cyclic behaviour of soil deposits...The principal stress rotation is one of the most important features of the stress state in a seabed subjected to wave loading. Most prior investigations focused their attention on the cyclic behaviour of soil deposits under the circular rotation stress path based on the analytical solutions for a seabed of infinite thickness. In this paper, the nonstandard elliptical, i.e., non-circular, rotation stress path is shown to be a more common state in the soil sediments of a finite seabed with an alternating changeover in stress due to a travelling regular wave. Then an experimental investigation in a hollow cylinder triaxial-torsional apparatus is conducted into the effect of the nonstandard elliptical stress path on the cyclic strength. A special attention is placed on the difference between the circular rotation stress path and the elliptical rotation stress path. The results and observations show that the shear characteristics for the circular rotation stress path in the literature are not applicable for analyzing the cyclic strength of sand in a finite seabed, and also indicate that due to the influence of three parameters about the size and the shape of a nonstandard ellipse, the cyclic strength under a nonstandard elliptical rotation stress path is evidently more complex and diversified as compared with that under a circular rotation stress path. Especially the influence of the initial phase difference on the cyclic strength is significant.展开更多
In this paper, we investigate the interior regularity including the local boundedness and the interior HSlder continuity of weak solutions for parabolic equations of the p（x, t）-Laplacian type. We improve the Moser ...In this paper, we investigate the interior regularity including the local boundedness and the interior HSlder continuity of weak solutions for parabolic equations of the p（x, t）-Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.展开更多
The purpose of this paper is to study the strict positivity in a nonstandard hull of an internal normed Riesz space. Several characterizations are obtained for an element of the hull to be strictly positive. Particula...The purpose of this paper is to study the strict positivity in a nonstandard hull of an internal normed Riesz space. Several characterizations are obtained for an element of the hull to be strictly positive. Particularly, it is shown that a nonstandard hull of an internal normed Riesz space has strictly positive elements if and only if it has an order unit. This result has an application to the double infillity problem in economics. It follows that the existence of equilibria in a nonstandard double infinity economy is equivalent to the existence of finite equilibria.展开更多
文摘Hilbert problem 15 required understanding Schubert's book.In this book,reducing to degenerate cases was one of the main methods for enumeration.We found that nonstandard analysis is a suitable tool for making rigorous of Schubert's proofs of some results,which used degeneration method,but are obviously not rigorous.In this paper,we give a rigorous proof for Example 4 in Schubert's book,Chapter 1.§4 according to his idea.This shows that Schubert's intuitive idea is correct,but to make it rigorous a lot of work should be done.
基金National Natural Science Foundation of China(No. 50575062)Natural Science Foundation of Heilongjiang Province,China (No. F200608)+2 种基金Key Project of Scientific Research Subsidy of Abroad Scholars of Heilongjiang Provincial Education Department, China (No.1152hq08)Scientific Research Fund of Heilongjiang Provincial Education Department, China (No.10551z0008)Harbin Municipal Key Project of Science and Technology, China (No.2005AA1CG061-11).
文摘Considering the complex constraint between operations in nonstandard job shop scheduling problem （NJSSP）, critical path of job manufacturing tree is determined according to priority scheduling function constructed. Operations are divided into dependent operations and independent operations with the idea of subsection, and corresponding scheduling strategy is put forward according to operation characteristic in the segment and the complementarities of identical function machines. Forward greedy rule is adopted mainly for dependent operations to make operations arranged in the right position of machine selected, then each operation can be processed as early as possible, and the total processing time of job can be shortened as much as possible. For independent operations optimum scheduling rule is adopted mainly, the inserting position of operations will be determined according to the gap that the processing time of operations is subtracted from idle time of machine, and the operation will be inserted in the position with minimal gap. Experiments show, under the same conditions, the result that operations are scheduled according to the object function constructed, and the scheduling strategy adopted is better than the result that operations are scheduled according to efficiency scheduling algorithm.
基金National Natural Science Foundations of China(Nos.11572212,11272227)Innovation Program for Postgraduate of Suzhou University of Science and Technology,China(No.SKCX15_062)
文摘The perturbation to Noether symmetry and adiabatic invariants for dynamical systems with nonstandard Lagrangians are studied.Based on two kinds of nonstandard Lagrangians(i.e.exponential Lagrangians and power-law Lagrangians),the exact invariants of Noether type are given.Based on the definition of highorder adiabatic invariants,the relationship between the perturbation of Noether symmetry and the adiabatic invariants of the system under a small disturbance is studied,and then the corresponding theorems of adiabatic invariants are established.Finally,two examples are given to illustrate the methods and results appear in this paper.
文摘Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.
文摘We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random variables. As an application, we consider a simple calculation of financial time series.
基金Project supported by the National Natural Science Foundation of China （Grant No 10572135）.
文摘Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.
基金The NSF (2007A12) of Shaanxi Provincethe Special Science Research Project (11JK0507) of Shaanxi Provincial Department of Edueation
文摘In nonstandard enlargement, the separations are characterized by non- standard analysis methods in [0, 1J-topological spaces. Firstly, the monads of fuzzy point in [0, 1]-topological spaces are described with remote-neighborhoods in non- standard enlarged model. Then the nonstandard characterizations of separations in [0, 1]-topological space are given by the monads. At last, relations of these separations are investigated.
文摘Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-complex special solutions which can be made up of hyperbolicfunctions or elliptic functions.
文摘A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.
文摘In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.
基金Partially supported by the NSF(11271154)of China the 985 program of Jilin University
文摘This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has different singularity （vanishing in finite time）.
基金National Natural Science Foundation of China(No.11571373)
文摘In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique.These proposed schemes reproduce long time features of the oscillator solution exactly.Their abilities in preserving the symplecticity,the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied.It can be shown that the component { x_n}_(n≥1) of the INSOPT schemes switch signs infinitely many times as n →∞,almost surely.Further,the mean-square convergence order of 1 is obtained for these INSOPT schemes.Finally,numerical experiments illustrate intuitively the results obtained in this paper.
文摘This paper deals with reaction-diffusion equations involving nonstandard growth conditions, subject to homogeneous Neumann boundary conditions. The complete clas- sification is established for simultaneous and non-simultaneous quenching under suitable assumptions on initial data. Moreover, quenching sets and quenching rates are obtained.
文摘In this paper the author proves that the Phragmen Lindelof principle holds for solutions of elliptic equation (1) with nonstandard growth conditions.
基金the China Postdoctoral Science Foundation Funded Project (No.2017M620113)the National Natural Science Foundation of China (Nos.11801120,71773024 and 11771107)+4 种基金the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058)the Natural Science Foundation of Heilongjiang Province of China (No.G2018006)Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059)National Science Foundation of China (No.11371376)the Innovation-Driven Project and Mathematics.
文摘In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such,the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs,leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.
文摘In this paper we develope a theory of new generalized functions by using Nonstandard Analysis which is closely relevant to that of Colombeau's new generalized functions.
基金Project supported by the Natural Science Foundation of China(Grant Nos.51639002,51209033)the Specialized Re-search Fund for the Doctoral Program of Higher Education(Grant No.20120041130002)
文摘The principal stress rotation is one of the most important features of the stress state in a seabed subjected to wave loading. Most prior investigations focused their attention on the cyclic behaviour of soil deposits under the circular rotation stress path based on the analytical solutions for a seabed of infinite thickness. In this paper, the nonstandard elliptical, i.e., non-circular, rotation stress path is shown to be a more common state in the soil sediments of a finite seabed with an alternating changeover in stress due to a travelling regular wave. Then an experimental investigation in a hollow cylinder triaxial-torsional apparatus is conducted into the effect of the nonstandard elliptical stress path on the cyclic strength. A special attention is placed on the difference between the circular rotation stress path and the elliptical rotation stress path. The results and observations show that the shear characteristics for the circular rotation stress path in the literature are not applicable for analyzing the cyclic strength of sand in a finite seabed, and also indicate that due to the influence of three parameters about the size and the shape of a nonstandard ellipse, the cyclic strength under a nonstandard elliptical rotation stress path is evidently more complex and diversified as compared with that under a circular rotation stress path. Especially the influence of the initial phase difference on the cyclic strength is significant.
文摘In this paper, we investigate the interior regularity including the local boundedness and the interior HSlder continuity of weak solutions for parabolic equations of the p（x, t）-Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.
文摘The purpose of this paper is to study the strict positivity in a nonstandard hull of an internal normed Riesz space. Several characterizations are obtained for an element of the hull to be strictly positive. Particularly, it is shown that a nonstandard hull of an internal normed Riesz space has strictly positive elements if and only if it has an order unit. This result has an application to the double infillity problem in economics. It follows that the existence of equilibria in a nonstandard double infinity economy is equivalent to the existence of finite equilibria.
