In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a se...In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.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
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna...Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.展开更多
In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagr...In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagrange coordinate transformation, the local well-posedness of the solutions for the 1D pressureless Euler-alignment in Besov spaces with 1≤p∞ is established. Next, the ill-posedness of the solutions for this model in Besov spaces with 1≤p and is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces with 1≤p .展开更多
The effect of gravity on the self-similarity of jet shape at late stage of Worthington jet development is investigated by experiment in the study.In addition,the particle image velocimetry(PIV)method is introduced to ...The effect of gravity on the self-similarity of jet shape at late stage of Worthington jet development is investigated by experiment in the study.In addition,the particle image velocimetry(PIV)method is introduced to analyze the development of flow field.There is a linear scaling regarding the axial velocity of the jet and the scaling coefficient increases with the Froude number.展开更多
文摘In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.
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.
文摘Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.
文摘In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagrange coordinate transformation, the local well-posedness of the solutions for the 1D pressureless Euler-alignment in Besov spaces with 1≤p∞ is established. Next, the ill-posedness of the solutions for this model in Besov spaces with 1≤p and is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces with 1≤p .
基金supported by the National Natural Science Founda-tion of China(Grant Nos.12122214,12272382,12293000,12293003,and 12293004)the Youth Innovation Promotion Association CAS(Grant No.2022019)High-level Innovation Research Institute Program of Guangdong Province(Grant Nos.2020B0909010003 and GARA2022002000).
文摘The effect of gravity on the self-similarity of jet shape at late stage of Worthington jet development is investigated by experiment in the study.In addition,the particle image velocimetry(PIV)method is introduced to analyze the development of flow field.There is a linear scaling regarding the axial velocity of the jet and the scaling coefficient increases with the Froude number.
