SU(d)-Biinvariant Random Walks on SL\,{\left( {d,\,\mathbb{C}} \right)} and their Euclidean Counterparts

We establish a deformation isomorphism between the algebras of -biinvariant compactly supported measures on and -conjugation invariant measures on the Euclidean space of all Hermitian -matrices with trace . This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from on the other, which has recently been observed by Klyachko [13]. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded -biinvariant measures on and a certain (non-invariant) subalgebra of the bounded signed measures on . We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of -biinvariant random walks on in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.Margit Rösler was partially supported by the Netherlands Organisation for Scientific Research (NWO), project nr. B 61-544.     

Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition

We investigate the spectral singularities and the eigenvalues of the boundary value problem $$\begin{gathered} y'' + \left[ {\lambda - Q\left( x \right)} \right]^2 y = 0,x \in R_ + = [0,\infty ), \hfill \\ \quad \int\limits_0^\infty {K\left( x \right)y\left( x \right)dx + \alpha y'\left( 0 \right) - \beta y\left( 0 \right) = 0,} \hfill \\ \end{gathered}$$ where Q and K are complex valued functions, K∈L ²(R ₊), α,β∈C with |α|+|β|≠0 and λ is a spectral parameter.     

Multiple rational trigonometric sums and multiple integrals

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$\left| {\sum {_{x_{1, \ldots ,} x_r = 1^{\exp \left( {{{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} \mathord{\left/ {\vphantom {{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} q}} \right. \kern-\nulldelimiterspace} q}} \right)} }^q } } \right| \ll q^{{{r - 1} \mathord{\left/ {\vphantom {{r - 1} {n + \varepsilon }}} \right. \kern-\nulldelimiterspace} {n + \varepsilon }}} ,$$ where $$\begin{gathered} f\left( {x_{1, \ldots ,} x_r } \right) = \sum {_{0 \leqslant t_1 , \ldots ,t_r \leqslant n^a t_1 , \ldots ,t_r x_1^{t_1 } \ldots x_r^{t_r } ,} } \hfill \\ a_{0, \ldots ,0} = 0,\left( {a_{0, \ldots ,0,1} , \ldots ,a_{n, \ldots ,n,} q} \right) = 1 \hfill \\ \end{gathered} $$ , and an estimate of the modulus of a multiple trigonometric integral.     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

非线性方程组的符号矩阵，正则分裂与单调包含（Ⅰ）

As a continuation of part I of the paper under the same title, we developgeneral monotonic enclosure methods for the couple systems of the splitting equations {x=G([x]a,[x]b,[y]c) y=G([y]a,[y]b,[x]c),which models the system of equations associated with hybrid and aaynchronotts monotonicity as well as convexity. The resulting algorithms and convergence theorems generalize and unify various known methods and monotonic enclosure theorents established by other authors.     

On the absence of positive eigenvalues of Schrödinger operators with rough potentials

We prove the absence of positive eigenvalues of Schrödinger operators $ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough potentials $V$. To describe our class of potentials fix an exponent $q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$. For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$ (or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and$\begin{equation*}$$\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$$\end{equation*}$Under these assumptions we prove that the operator $H$ does not admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form$\begin{equation*}$$||W_m u||_{l^a(L^{p(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$$\end{equation*}$for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$, where $p(q)$ and $p(q)$ are dual exponents with the property that $1/p(q)-1/p(q)=1/q$.     

BOUNDARY VALUE PROBLEM FOR A GENERALIZED LIENARD EQUATION

By coincidence degree, the existence of solution to the boundary value problem of a generalized Liénard equation

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

We consider an eigenvalue problem for a system on [0, 1]: $$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}} {{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array} } \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) = \lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi ^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array} } \right.$$ with constants $$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p₂₁, p₂₂ are known, we prove a uniqueness theorem and provide a reconstruction formula for p₁₁ and p₁₂ from the spectral characteristics consisting of one spectrum and the associated norming constants.     

一类非线性$m$-点边值问题的正解

应用锥理论和不动点指数方法,在与相应线性算子的第一特征值相关的条件下,得到了下述非线性二阶常微分方程m-点边值问题{u"(t) a(t)u' b(t)u h(t)f(u(t))=0,0＜t＜1,u'(0)=0,u(1)=m-2∑i=1αiu(ξi).的正解,改进了相关文献中的结论.     

Some improved Caffarelli-Kohn-Nirenberg inequalities

For 1 < p < N and we obtain the following improved Hardy-Sobolev Inequalities where 1 < q < p and if , if , for some positive constant . Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16]. Received: 2 February 2004, Accepted: 12 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35P05, 35R05, 46E30, 46E35 Partially supported by Project BFM2001-0183     

Symmetry of Sections in Fields of Formal Power Series and a Non-Standard Real Line

Let be a field of formal power series with real coefficients, whose supports are well ordered subsets of an Abelian group of cardinality strictly less than . For , we give criteria of a section being symmetric and of a symmetric section being Dedekind. It is proved that an -saturated non-standard real line is isomorphic to some field of the form . For , some consequences are inferred regarding symmetric sections, and the cofinality of banks of the sections.     

Spectral Asymptotics of the Harmonic Oscillator Perturbed by Bounded Potentials

Consider the operator in where q is a real function with q′ and bounded. The spectrum of T is purely discrete and consists of simple eigenvalues. We determine their asymptotics and we extend these results for complex q.Communicated by Bernard Helffersubmitted 23/04/04, accepted 26/10/04     

An inequality associated with the uncertainty principle

We prove \(\left\| F \right\|_{2,\Omega } \leqslant c({\rm T} \Omega )\left\| f \right\|_{A{}_T} \) , whereF is the Fourier transform off,||F||_2,Ω is theL ²-norm ofF on \([ - \Omega ,\Omega ],\left\| f \right\|_{A{}_T} \) is the absolutely convergent Fourier series norm for 2T-periodic functions, and $$c(T\Omega ) = (\frac{1}{\pi }\int\limits_{ - T\Omega }^{T\Omega } {\frac{{\sin ^2 \gamma }}{{\gamma ^2 }}d\gamma } )^{1/2} $$ Analogous inequalities, depending on prolate spheroidal wave functions, are more difficult to prove and their constants are less explicit.     

A New Bound on the Number of Designs with Classical Affine Parameters

The hyperplanes in the affine geometry AG(d, q) yield an affineresolvable design with parameters $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ . Jungnickel [3]proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parametersfor d ≥ 3. The bound of Jungnickel was improved recently [5] by a factor of $q^{\frac{{d^2 + d - 6}}{2}} (q - 1)^{d - 2}$ for any d ≥ 4. In this paper, a construction of $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ designs based on group divisible designs is given that yieldsat least $$\frac{{\left( {q^{d - 1} + q^{d - 2} + \cdots + 1} \right)!\left( {q - 1} \right)}}{{\left| {{\text{P}}\Gamma {\text{L(}}d,q{\text{)}}} \right|\left| {{\text{A}}\Gamma {\text{L(}}d,q{\text{)}}} \right|}}$$ non-isomorphic designs for any d ≥ 3. This new bound improves the bound of[5] by a factor of $$\frac{1}{{q^d }}\mathop \Pi \limits_{i = 1}^{(q^{d - 1} - q)/(q - 1)} (q^{d - 1} + i).$$ For any given q and d, It was previously known [2,11] that there are at least 8071non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is atleast 245,100,000.     

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

Oscillatory Behavior of Second-Order Nonlinear Differential Equations with a Nonpositive Neutral Term

We shall present new oscillation criteria of second-order nonlinear differential equations with a nonpositive neutral term of the form:

$$\begin{aligned} \left( (a(t)\left( \left( x(t)-p(t)x(\sigma (t) )^{\prime } \right) ^{\gamma } \right) ^{\prime }+q(t)x^{\beta }(\tau (t))=0,\right. \end{aligned}$$

with positive coefficients. The obtained results answer an open problem raised in Li et al. [Adv Differ Equ 35:7, 2015, Remark 4.3 (P2)]. Examples are given to illustrate the main results.

     

Sharp Inequalities Involving $$(n!)^{1/n}$$

We prove that, for all integers \(n\ge 1\),

$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$

and

$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$

with the best possible constants

$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$

     

四阶奇异边值问题的正解

研究了一类四阶奇异边值问题正解的存在性,在f和g满足比超线性和次线性条件更广泛的极限条件下,利用锥压缩和拉伸不动点定理获得了正解的存在性结果,推广和包含了一些已知结果.