查看更多>>摘要：This is a survey of local and global classification results concerning Dupin hy-persurfaces in Sn(or Rn)that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of Sn(or Rn),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.
查看更多>>摘要：We consider the singular Riemann problem for the rectilinear isentropic com-pressible 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 unique-ness 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.
查看更多>>摘要：In this article,we study Kähler metrics on a certain line bundle over some com-pact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.
查看更多>>摘要：We find the exact forms of meromorphic solutions of the nonlinear differential equations fn+q(z)eQ(z)f(k)=p1eα1z+p2eα2z,n ≥ 3,k ≥ 1,where q,Q are nonzero polynomials,Q(≡)Const.,and P1,p2,α1,α2 are nonzero constants with α1 ≠ α2.Compared with previous results on the equation p(z)f3+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f(k)perturbed by multiplying an exponential function will affect the structure of its solutions.
查看更多>>摘要：In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R3 × T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L2-L∞ decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.
查看更多>>摘要：In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson's transformation for-mula.In terms of application,by specializing certain parameters in the two transformations,four Rogers-Ramanujan type identities associated with moduli 20 are obtained.
查看更多>>摘要：We study the Nadaraya-Watson estimators for the drift function of two-sided reflected stochastic differential equations.The estimates,based on either the continuously observed process or the discretely observed process,are considered.Under certain conditions,we prove the strong consistency and the asymptotic normality of the two estimators.Our method is also suitable for one-sided reflected stochastic differential equations.Simulation results demonstrate that the performance of our estimator is superior to that of the estimator proposed by Cholaquidis et al.(Stat Sin,2021,31:29-51).Several real data sets of the currency exchange rate are used to illustrate our proposed methodology.
查看更多>>摘要：This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.
查看更多>>摘要：We study the closed range property and the strict singularity of integration operators acting on the spaces F(p,pα-2,s).We completely characterize the closed range property of the Volterra companion operator Ig on F(p,pα-2,s),which generalizes the existing results and answers a question raised in[A.Anderson,Integral Equations Operator Theory,69(2011),no.1,87-99].For the Volterra operator Jg,we show that,for 0＜α ≤ 1,Jg never has a closed range on F(p,pα-2,s).We then prove that the notions of compactness,weak compactness and strict singularity coincide in the case of Jg acting on F(p,p-2,s).
查看更多>>摘要：In this note,we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional cur-vature.Like the Michale-Simon Sobolev inequality,this inequality contains a term involving the mean curvature.
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