Low Mach number limit of multidimensional steady flows on the airfoil problem

Given $$\alpha >0$$, we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where $$W^{1,n}_{0,\mathrm{rad}}(B)$$ is the usual Sobolev spaces of radially symmetric functions on B in $${\mathbb {R}}^n$$ with $$n\ge 2$$. Without restricting to the class of functions $$W^{1,n}_{0,\mathrm{rad}}(B)$$, we should emphasize that the above inequalities fail in $$W^{1,n}_{0}(B)$$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.     

A family of polylog-trigonometric integrals

In this paper, we study the sequences

$$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$

where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals

$$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$

With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.

     

Exact expansions of Hankel transforms and related integrals

The Ramanujan Journal - The Hankel transform $$\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x$$ is studied for integer $$n\geqslant -1$$ and positive parameter q. It...     

关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

On Products of $$\mathrm{F}^*(G)$$ -Subnormal Subgroups of Finite Groups

Mathematical Notes - A subgroup $$H$$ of a finite group $$G$$ is said to be $$\mathrm{F}^*(G)$$ -subnormal if it is subnormal in $$H\mathrm{F}^*(G)$$ , where $$\mathrm{F}^*(G)$$ is the generalized...     

On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables

Lithuanian Mathematical Journal - We present upper bounds of the integral $$ {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N0\left({S}_N{X}_1+\dots +{X}_N\right) $$ of...     

Approximating Characteristics of the Nikol’skii–Besov Classes S1,θrBℝd$$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$

Ukrainian Mathematical Journal - We establish the exact-order estimates for the approximation of the classes $$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$ by entire functions of...     

Capacity solution for a perturbed nonlinear coupled system

We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, the model problem we refer to is $$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset \mathbb {R}^N$$, $$N\ge 2$$ and $$\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.     

The yellow cake

In this paper we consider the following property:

For every function there are functions

(for ) such that

We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .

     

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

Trigonometric multiplicative chaos and applications to random distributions

Science China Mathematics - The random trigonometric series $$\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} $$ on the circle $$\mathbb{T}$$ are studied under the...     

Hyperbolic spaces,principal series,and $${\mathrm{O}}(2,\infty )$$O(2,∞)

We prove that there exists no irreducible representation of the identity component of the isometry group $${\mathrm{PO}}(1,n)$$ of the real hyperbolic space of dimension n into the group $${\mathrm{O}}(2,\infty )$$ if $$n\ge 3$$. This is motivated by the existence of irreducible representations (arising from the spherical principal series) of $${\mathrm{PO}}(1,n)^{\circ }$$ into the groups $${\mathrm{O}}(p,\infty )$$ for other values of p.     

New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

The Ramanujan Journal - Recently, Lin introduced two new partition functions $$\hbox {PD}_{\mathrm{t}}(n)$$ and $$\hbox {PDO}_{\mathrm{t}}(n)$$, which count the total number of tagged parts over...     

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Mathematische Zeitschrift - Let $$\pi $$ be an irreducible cuspidal representation of $$\mathrm {GL}_{kn}(\mathbb {F}_q)$$ . Assume that $$\pi = \pi _{\theta }$$ , corresponds to a regular...     

An average of special values of Dirichlet series of Rankin–Selberg type

The Ramanujan Journal - For an integer n and a Dirichlet character $$\xi $$ modulo N, we denote by $$\mathcal {S}_n(N,\xi )$$ the space of cusp forms of weight n with respect to $$\varGamma _0(N)$$...     

Hermite Expansion of the Riemann Zeta Function

Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.     

Estimates of linear combinations of near-exponential functions with positive and negative exponents

Mathematical Notes - The functions $$\begin{gathered} f_n (z) = e^{\lambda _n ^z } [1 + \alpha _n (z)], \hfill \\ \varphi _n (z) = e^{\mu _n ^z } [1 + \beta _n (z)](n = 1.2, ...), \hfill \\...