关于特定同余式方程解数的余项（英文）

Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫_1~X?~2(x)dx? ε{X~(3-6/m+3+ε)if m ≥ 3,X~(2+ε) if m = 2,for any non-Abelian polynomial f(x) and any ε 0. This result constitutes an improvement upon that of Lü for the error terms on average.     

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫_(-∞)~(+∞)(x~2)~mexp(-x~2){Hr (x)}2dx, ∫_0~∞exp(-x~2)H_(2k)(x)H_(2s+1)(x)dx,∫_0~∞exp(-x~2)H_(2k)(x)H_(2s)(x)dx and ∫_0~∞exp(-x~2)H_(2k+1)(x)H_(2s+1)(x)dx,are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.     

ON UNIQUENESS OF SOLUTION TO POINTWISE LANDAU PROBLEM ON THE FINITE INTERVAL

1. Introduction Let W_∞~((r)) (β) = {f| f∈W_∞~((r)) [-1,1], ||f||_(C[-1,1]) β, ||f~((r))||_∞ 1}.In this paper, we will consider the following Landau problem:λf~((k))(ξ) + μf~((k-1)) (ξ) →inf, f∈W_∞~((r)) (β), (1.1)where ξ∈[-1,1], 1(?)k(?)r-1, and λ, μ real and not all zero, (if k=1,suppose λ≠0 in addition ). A. Pinkus studied it first. To begin with, we introduce some fundamental definitions anddenotions. The perfect spline f, which satisfies || f~((r))||_∞ = 1 andhas n knots and n+r+1 points of equioscillation in [-1,1], isdenoted by x_(nr), which is refered as Tchebyshev perfect spline. And     

一个加细的反向Jensen不等式

文 [1 ]在函数的凸性理论中 ,给出了一个重要的结论 :设 f ( x)、p( x)为 I上的可积函数 ,而 m≤ f ( x)≤ M,p( x)≥ 0 ,∫Ip( x) dx >0 ,则随连续函数Φ( t) ( m≤ t≤ M)之为下凸或上凸而相应地有Φ∫Ip( x) f ( x) dx∫Ip( x) dx≤或≥∫Ip( x) f ( x) dx∫Ip( x) dx(即 Jensen不等式 )　　为证明其反向不等式 ,引入以下记号 ,并引入严格凸函数的一个几何性质。记 I =[a,b];∫I=∫ba;A( f ( x) ) =∫Ip( x) f ( x) dx∫Ip( x) dx为 f ( x)的加权平均 ,p( x)≥ 0 ,∫Ip( x) dx >0 ,x∈ I。设Φ( x) >0 ,Φ″( x) >0 ,x∈ I,则Φ( …     

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

A NOTE ON MORREY-NIKOL KII INEQUALITY

Let Q_0 be a Cube in R~n and u(x)∈L~p(Q_0).Suppose that∫_Q丨u(x t)-u(x)丨~pdx≤K~p丨t丨~(ap)丨Q丨~(1/β/n)for all parallel subcubes Q in Q_0 and for all t such that the integral makes sense with K≥0,0<α≤1, 0≤β≤n and p≥1.If αp=β,then u(x)is of bounded mean oscillation on Q_0(abbreviated to BMO(Q_0)),i.e.sup QQ_0 1/丨Q丨∫_Q丨u(x)-u_Q丨dx=‖u‖<∞,where u_Q is the mean value of u(x)over Q.     

凸函数的几个Hadamard型不等式

对于凸函数建立了几个新的 Hadamard型不等式 ,比如f ∑nk=1qkak∫A+yA- yg(x) dx ∫A+yA- yf (x) g(x) dx ∑nk=1qkf (ak) ∫A+yA- yg(x) dx和f (pa +qb) p∫ξaf (x) g(x) dx∫ξag(x) dx+q∫bξf (x) g(x) dx∫bξg(x) dx pf (a) +qf (b)等 ,推广了前人所做的工作 .     

Positive solutions for a weighted fractional system

In this article, we study positive solutions to the system{A_αu(x) = C_(n,α)PV∫_(Rn)(a1(x-y)(u(x)-u(y)))/(|x-y|~(n+α))dy = f(u(x), B_βv(x) = C_(n,β)PV ∫_(Rn)(a2(x-y)(v(x)-v(y))/(|x-y|~(n+β))dy = g(u(x),v(x)).To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.     

Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution

Here we consider the following strongly singular integral T_(?,γ,α,β)f (x, t) =∫ _(R~n)e~[i|y|~(-β)]?(y/|y|)/|y|~(n+α)f (x - y, t - γ(|y|))dy,where ? ∈ L~p(S~(n-1)), p 1, n 1, α 0 and γ is convex on(0,∞).We prove that there exists A( p, n) 0 such that if β A( p, n)(1 + α), then T_(?,γ,α,β)is bounded from L~2(R~(n+1)) to itself and the constant is independent of γ. Furthermore,when ? ∈ C~∞(S~(n-1)), we will show that T?,γ,α,βis bounded from L~2(R~(n+1)) to itself only if β 2α and the constant is independent of γ.     

本刊英文版Vol.36(2020),No.4论文摘要（英文）

正A Weighted Trudinger-Moser Inequality on R~N and Its Application to Grushin Operator Jia Jun WANG Qiao Hua YANGAbstract Let x=(x′,x″) with x′∈R~k and x″∈R~(N-k)andΩbe a x′-symmetric and bounded domain in R~N (N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C 0 such that for all x′-symmetric function u∈C_0~∞(Ω) with∫_Ω|▽u(x)|~(N-a)|x′|~(-a)dx≤1,the following uniform inequality holds     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].