Semiclassical geometry and integrability of the AdS/CFT correspondence

We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005.     

Hua operator on vector bundle: Application to AdS/CFT correspondence of Dirac fields

Hua’s theory of harmonic functions on classical domains is generalized to the theory on holomorphic vector bundles over classical domains and further on vector bundles over the real classical domains and quaternion classical domains. In case of the simplest quaternion classical domain there is a relation between Hua operator and Dirac operator, by which an AdS/CFT correspondence of Dirac fields is established.     

Embedding of Hyperbolic Spaces in the Product of Trees

We show that for each n ≥ 2 there is a quasi-isometric embedding of the hyperbolic space Hⁿ in the product Tⁿ=T × ··· × T of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding for any metric tree T and any m ≥ 0.Mathematics Subject classification (2000). 54Exx.Supported by RFFI Grant 02-01-00090, CRDF Grant RMI-2381-ST-02 and SNF Grant 20-668 33.01. Supported by Swiss National Science Foundation.     

Almost Optimal Estimates for Best Approximation by Translates on a Torus

We investigate the best approximation of some classes of functions defined on a $d$-dimensional torus ${\mbox{\smallbf T}}^d$ by the transfer manifold $H_n(\varphi)=\{\xxsum_{i=1}^n b_i\varphi(\cdot -a_i)\}$ in the Hilbert space $L_2({\mbox{\smallbf T}}^d)$. For Sobolev classes of functions $\tilde{W}_2^r$ we obtain two-sided estimates of the deviation ${\rm dist}(\tilde{W}_2^r,H_n(\varphi))$. These results are applied to obtain estimates for radial basis approximation.     

On a multiplicity one property for the length spectra of even dimensional compact hyperbolic spaces

We prove a multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces, i.e., if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.     

Green's function of the Brinkman equation in a 2D anisotropic case

The purpose of this paper is to obtain the Green's functionof the Brinkman equation in a 2D case of hydrodynamic anisotropywith respect to the permeability. The anisotropic nature ofthe permeability is assumed to be not space or time dependent.We use the method of Fourier transform which reduces the computationof the Green's function to the computation of the fundamentalsolution of a fourth-order partial differential equation. Thisresearch work has several applications in engineering and medicineto the motion of bodies in anisotropic porous media.     

The Propagator of a Massive Field of Spin s = 2 in the Anti-de Sitter Space

We construct the propagator of the massive tensor field of the second rank on the Euclidean continuation of the anti-de Sitter (AdS) space. We find the explicit expression for the propagator in the limit where the field takes values at the boundary of the AdS space. We show that the limiting expression yields the correct Green's function and two-point correlation function of the boundary conformal field theory, as predicted by the AdS/CFT correspondence hypothesis. We thus obtain one more piece of evidence in favor of the interpretation of operators of the boundary conformal field theory as certain limits of quantum fields propagating in the AdS space.     

一个二元平均值不等式猜想的新证明

讨论了一类双曲复合函数的单调性、凹性、几何凹性,由此证得《美国数学月刊》11031问题中提出的一个有关二元平均值不等式猜想.     

Solutions with spikes at the boundary for a 2D nonlinear Neumann problem with large exponent

We consider the elliptic equation −Δu+u=0 in a bounded, smooth domain Ω in R², subject to the nonlinear Neumann boundary condition . Here p>1 is a large parameter. We prove that given any integer m?1 there exist at least two families of solutions up developing exactly m peaks ξi∈∂Ω, in the sense that , as p→∞.     

关于特殊序列上的多维除数函数的和

本文研究了多维除数函数d_k(n)在特殊序列[n~c]上的分布。利用指数和方法中一些新成果,证明了:当实数c满足1相似文献   

A Note on the Bloch Function in SCV

51,Intr0ductionThroughoutthispaper,Cwilldenotethecomplexfield,andC"willbethecartesianl)r0cl-uctofncopiesofClherenisanypositiveinterger.TheP0intsofC"arethusordered)I-tul>lesz=(zl,...,z`),whereeachzieC.ThainnerproductinC"is(z,w)=Zzjwj(z,weC"),j=1andtheassociatednormlzI=(z,z)1l2(zeC")makeC"intoann-dimensionalHilbertspacewhoseopenunitballwillbedenotedbyB..ForapairofpointszandwinB,,thehyperdi8tancebetweenzandwisdenotedbyp(z,w)=oplog(the),wllereh(z,w)=(lz-wl= Izwl'~lzl2It`)l2)1/2.['I1-w5l]-'…     

Existence of solutions to (k, n − k − 2) discrete boundary value problems

We provide general existence criteria to solutions for a class of higher-order discrete boundary value problems. Our results supplement as well as improve several recent results established in the literature.     

On the Distribution in Residue Classes of Integers with a Fixed Sum of Digits

We give an asymptotic formula for the distribution of those integers n in a residue class, such that n has a fixed sum of base-g digits, with some uniformity over the choice of the modulus and g. We then use this formula to solve the problem of I. Niven of giving an asymptotic formula for the distribution of those integers n divisible by the sum of their base-g digits. Our results also allow us to give a stronger form of a result of M. Olivier dealing with the distribution of integers with a given gcd with their sum of base-g digits.For our friend Jean-Louis Nicolas on his sixtieth birthdayResearch partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T029759, and “Balaton” French-Hungarian exchange program F-18/00.2000 Mathematics Subject Classification: Primary—11A63     

The Spectra of General Differential Operators in the Direct Sum Spaces

In this paper, the general ordinary quasi-differential expression M _pof n-th order with complex coefficients and its formal adjoint M _p ⁺ on any finite number of intervals I _p =(a _p ,b _p ),p= 1,...,N, are considered in the setting of the direct sums of L _wp ² (a _p ,b _p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.     

Investigation of coupled physical fields in a compound piezoceramic wedge

A composition consisting of two infinite wedges fastened together along their common side is considered. The outer surfaces of the wedges are fixed and coated with grounded electrodes, and an ideal mechanical and electrical contact is assumed along the interface of both wedges. At certain relations of the geometric and stiffness parameters of the wedge components, the coupled electroelastic fields have a power-type singularity (intensified by oscillations) at the wedge vertex as a result of action of mechanical or electrical loads on the compound wedge. Such an effect is not observed in the case of piezopassive compound wedges (under antiplane strain conditions). The present study is dedicated to the investigation of this phenomenon. Closed-form solutions of some electroelastic boundary-value problems are constructed for the case of action of point shearing forces or electric charges.Sumskii State University, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 777–784, November–December, 1999.     

Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions

The Cauchy problem of the Navier-Stokes-Poisson equations in multi-dimensions (n?3) is considered. We obtain the pointwise estimates of the solution when it is a perturbation of the constant state. Then we have the optimal Lp(Rn) (p?1) convergence rate of the solution.     

双曲空间中Laplace-Beltrami方程的一个带位移的边值问题

讨论了双曲空间中Laplace-Beltrami方程的一个带位移的边值问题.首先将双曲空间中的Laplace-Beltrami方程的解转化为Clifford分析中的超正则函数.然后给出了超正则函数的Plemelj公式并讨论了相关奇异积分算子的性质,最后利用积分方程的方法和压缩不动点原理证明了Laplace-Beltrami方程的一个带位移的边值问题的解的存在性和唯一性.