This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specific...This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.展开更多
Dear Editor,This letter investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative.By using the Mittag-Leffler function,Laplace transform and the Gronwall-B...Dear Editor,This letter investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative.By using the Mittag-Leffler function,Laplace transform and the Gronwall-Bellman lemma,one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in(0,2).According to this theory,if the nonlinear term satisfies some conditions,then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems.Two examples are provided to illustrate the applications of our result.展开更多
We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separat...We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.展开更多
We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant ...We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.展开更多
The solution of the Riemann Problem (RP) for the one-dimensional (1D) non-linear Shallow Water Equations (SWEs) is known to produce four potential wave patterns for the scenario where the water depth is always positiv...The solution of the Riemann Problem (RP) for the one-dimensional (1D) non-linear Shallow Water Equations (SWEs) is known to produce four potential wave patterns for the scenario where the water depth is always positive. In this paper, we choose four test problems with exact solutions for the 1D SWEs. Each test problem is a RP with one of the four possible wave patterns as its solution. These problems are numerically solved using schemes from the family of Weighted Essentially Non-Oscillatory (WENO) methods. For comparison purposes, we also include results obtained from the Random Choice Method (RCM). This study has three main objectives. Firstly, we outline the procedures for the implementation of the methods employed in this paper. Secondly, we assess the performance of the schemes in conjunction with a second-order Total Variation Diminishing (TVD) flux on a variety of RPs for the 1D SWEs (for both short- and long-time simulations). Thirdly, we investigate if a single method yields optimal outcomes for all test problems. Optimal outcomes refer to numerical solutions devoid of spurious oscillations, exhibiting high resolution of discontinuities, and attaining high-order accuracy in the smooth parts of the solution.展开更多
Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjec...Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).展开更多
This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappin...This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.展开更多
In this paper,we study the convergence of a second-order finite volume approximation of the scalar conservation law.This scheme is based on the generalized Riemann problem(GRP)solver.We first investigate the stability...In this paper,we study the convergence of a second-order finite volume approximation of the scalar conservation law.This scheme is based on the generalized Riemann problem(GRP)solver.We first investigate the stability of the GRP scheme and find that it might be entropy-unstable when the shock wave is generated.By adding an artificial viscosity,we propose a new stabilized GRP scheme.Under the assumption that numerical solutions are uniformly bounded,we prove the consistency and convergence of this new GRP method.展开更多
文摘This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.
基金supported by the Natural Science Foundation of Hebei Province,China(A2015108010,A2015205161)the Science Research Project of Hebei Higher Educa tion Institutions,China(z2012021).
文摘Dear Editor,This letter investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative.By using the Mittag-Leffler function,Laplace transform and the Gronwall-Bellman lemma,one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in(0,2).According to this theory,if the nonlinear term satisfies some conditions,then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems.Two examples are provided to illustrate the applications of our result.
基金supported by the National Natural Science Foundation of China(11871218,12071298)in part by the Science and Technology Commission of Shanghai Municipality(21JC1402500,22DZ2229014)。
文摘We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.
基金This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396The authors gratefully acknowledge the support of the US Department of Energy National Nuclear Security Administration Advanced Simulation and Computing Program.The Los Alamos unlimited release number is LA-UR-19-32257.
文摘We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.
文摘The solution of the Riemann Problem (RP) for the one-dimensional (1D) non-linear Shallow Water Equations (SWEs) is known to produce four potential wave patterns for the scenario where the water depth is always positive. In this paper, we choose four test problems with exact solutions for the 1D SWEs. Each test problem is a RP with one of the four possible wave patterns as its solution. These problems are numerically solved using schemes from the family of Weighted Essentially Non-Oscillatory (WENO) methods. For comparison purposes, we also include results obtained from the Random Choice Method (RCM). This study has three main objectives. Firstly, we outline the procedures for the implementation of the methods employed in this paper. Secondly, we assess the performance of the schemes in conjunction with a second-order Total Variation Diminishing (TVD) flux on a variety of RPs for the 1D SWEs (for both short- and long-time simulations). Thirdly, we investigate if a single method yields optimal outcomes for all test problems. Optimal outcomes refer to numerical solutions devoid of spurious oscillations, exhibiting high resolution of discontinuities, and attaining high-order accuracy in the smooth parts of the solution.
文摘Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).
文摘This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.
基金funded by the Gutenberg Research College and by Chinesisch-Deutschen Zentrum fiur Wissenschaftsforderung(中德科学中心)Sino-German Project No.GZ1465M.L.is grateful to the Mainz Institute of Multiscale Modelling and SPP 2410 Hyperbolic Balance Laws in Fluid Mechanics:Complexity,Scales,Randomness(CoScaRa)for supporting her research.
文摘In this paper,we study the convergence of a second-order finite volume approximation of the scalar conservation law.This scheme is based on the generalized Riemann problem(GRP)solver.We first investigate the stability of the GRP scheme and find that it might be entropy-unstable when the shock wave is generated.By adding an artificial viscosity,we propose a new stabilized GRP scheme.Under the assumption that numerical solutions are uniformly bounded,we prove the consistency and convergence of this new GRP method.
