A New Approach to the Representation Theory of the Symmetric Groups. II

The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles. To the memory of D. Coxeter __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 57–98.     

Aspects of strong compactness, measurability, and indestructibility

 We prove three theorems concerning Laver indestructibility, strong compactness, and measurability. We then state some related open questions. Received: 11 December 1999 / Revised version: 17 September 2000 / Published online: 2 September 2002 Mathematics Subject classification (2000): 03E35, 03E55 Keywords or phrases: Measurable cardinal – Strongly compact cardinal – Supercompact cardinal – Indestructibility     

Spectra of degrees of some structures

We argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree. Also it is stated that there exist structures with the spectrum of degrees, {x : x ≰ a} ⋃ {x : x ≰ b}, for any low degrees a and b. Supported by RFBR grant No. 05-01-00605. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 729–744, November–December, 2007.     

Stability analysis with respect to two measures of impulsive systems under structural perturbations

The asymptotic stability with respect to two measures of impulsive systems under structural perturbations is investigated. Conditions of asymptotic (ρ₀, ρ)-stability of the system in terms of the fixed signs of some special matrices are established. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1476–1484, November, 1999.     

Deconvolution: a wavelet frame approach

This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). However, an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers. This paper is to establish the theoretical foundation of this wavelet frame approach. In particular, a proof of convergence, an analysis of the stability of algorithms and a study of the minimization property of solutions are given.     

A generalization of an inequality by N. V. Krylov

In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application to an L ^p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242). We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.      

Interpolation of nonlinear functionals by integral continued fractions

We constructively prove the theorem of existence of an interpolation integral chain fraction for a nonlinear functionalF:Q[0,1]→R ¹. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 364–375, March, 1999.     

An asymptotic formula of the Hellinger integral for renewal processes

Asymptotic formulas of the Hellinger integral are used in the investigation of properties of optimal estimates and statistical criteria. For a certain class of renewal processes, this formula was obtained by the author in [Lith. Math. J.,38(2), 131–143 (1998)]. In this paper, we obtain such a formula for all renewal processes whose intermediate renewal moments have absolutely continuous distributions. We use the traditional representation of the Hellinger integral and the theory of large deviations. Šiauliai University, Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 493–497, October–December, 1999. Translated by V. Mackevičius     

Structure of nonlinear elastic relationships for the highly anisotropic layer of a nonthin shell

Laminated nonthin shells made of nonlinearly elastic fiber composites are considered. The composite material is assumed to be transversely isotropic in planes perpendicular to reinforcement. The asymptotic method and the condition of material stability are applied to analyze the structure of constitutive relations. To introduce a small parameter, the high stiffness in the reinforcement direction of the fiber composite is used. This allows us to obtain simplified constitutive relations containing functions with one or two arguments instead of five as in the initial general case. Kazan State Architectural Building Academy, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 615–628, September–October, 1999.     

The stable admissions polytope

The stable admissions polytope– the convex hull of the stable assignments of the university admissions problem – is described by a set of linear inequalities. It depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been fruitful in the analysis of the stable marriage problem. Received: April 10, 1998 / Accepted: June 3, 1999?Published online January 27, 2000     

The mathematics of eigenvalue optimization

 Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread. Received: December 4, 2002 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words.  eigenvalue optimization – convexity – nonsmooth analysis – duality – semidefinite program – subdifferential – Clarke regular – chain rule – sensitivity – eigenvalue perturbation – partly smooth – spectral function – unitarily invariant norm – hyperbolic polynomial – stability – robust control – pseudospectrum – H _∞ norm Mathematics Subject Classification (2000): 90C30, 15A42, 65F15, 49K40     

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R ³×R ⁺ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp. Received April 1999 – Accepted September 1999     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

Composites in offshore technology in the next century

The advisability of application of advanced composites in deep-water offshore technology is shown using risers and tendons as examples. Numerical estimations of the parameters of multilayered metal-composite risers are obtained. Two new processes of spatial braiding for creating the external jackets of compound tendons are considered. Advanced composites are the only possible choice for exploitation of deposits at depths greater than 1500 m. That is why they are the most promising structural materials for offshore technology in the next century. This report was presented at the Symposium “Composites for the Next Millennium” in honor of Stephen Tsai's 70th birthday, Tours, France, 2–3 July, 1999. Institute of Polymer Mechanics, University of Latvia, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 549–560, September–October, 1999.     

Weyl algebras over quantum groups

The term “Weyl algebras” is proposed for differential algebras associated with dual pairs of Hopf algebras. The principle of complete reducibility for the category of “admissible” modules over Weyl algebras is proved. Comodule structures that connect Weyl algebras with the Drinfeld quantum double are investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 118, No. 2, pp. 190–204, February, 1999.     

Une infinité de structures de contact tendues sur une infinité de variétés

V de dimension trois, tout champ de plans tangents est homotope à une structure de contact, c'est-à-dire – dans le cas d'un champ orientable – au noyau ξ d'une 1-forme α dont le produit extérieur avec dα ne s'annule pas. Plus tard, les travaux de D.Bennequin [Be] puis de Y. Eliashberg [El1] ont conduit à ne retenir comme géométriquement intéressantes que les structures de contact dites tendues, c'est-à-dire telles qu'aucun disque plongé dans V ne soit tangent àξ en tous les points de son bord. En outre, Y. Eliashberg [El2, El3] a prouvé qu'un certain nombre de variétés –à savoir D ³, R ³, S ³, S ²×S ¹ et S ²×R– portent une seule structure de contact tendue, ce qui l'a amenéà conjecturer qu'une variété close orientable ne pouvait admettre qu'un nombre fini de structures tendues à isotopie près [El2, Conjecture 8.6.1]. Cependant, cette conjecture s'est vite révélée fausse sur le tore T ³ [Gi2, Gi3, Ka, Th] et le but du présent article est de montrer que toutes les variétés orientables fibrées en tores au-dessus du cercle (donc une infinité de variétés de dimension trois) portent une infinité de structures de contact tendues deux à deux non conjuguées. Oblatum 14-V-1998 & 11-lX-1998 / Published online: 28 January 1999     

Survival and coexistence in stochastic spatial Lotka–Volterra models

A spatially explicit, stochastic Lotka–Volterra model was introduced by Neuhauser and Pacala in Neuhauser and Pacala (Ann. Appl. Probab. 9, 1226–1259, 1999). A low density limit theorem for this process was proved by the authors in Cox and Perkins (Ann. Probab. 33, 904–947, 2005), showing that certain generalized rescaled Lotka–Volterra models converge to super-Brownian motion with drift. Here we use this convergence result to extend what is known about the parameter regions for the Lotka–Volterra process where (i) survival of one type holds, and (ii) coexistence holds. Supported in part by an NSERC Research grant.     

On the maximum feasible subsystem problem,IISs and IIS-hypergraphs

 We consider the Max FS problem: For a given infeasible linear system A x≤b, determine a feasible subsystem containing as many inequalities as possible. This problem, which is NP-hard and also difficult to approximate, has a number of interesting applications in a wide range of fields. In this paper we examine structural and algorithmic properties of Max FS and of Irreducible Infeasible Subsystems (IISs), which are intrinsically related since one must delete at least one constraint from each IIS to attain feasibility. First we provide a new simplex decomposition characterization of IISs and prove that finding a smallest cardinality IIS is very difficult to approximate. Then we discuss structural properties of IIS-hypergraphs, i.e., hypergraphs in which each edge corresponds to an IIS, and show that recognizing IIS-hypergraphs subsumes the Steinitz problem for polytopes and hence is NP-hard. Finally we investigate rank facets of the Feasible Subsystem polytope whose vertices are incidence vectors of feasible subsystems of a given infeasible system. In particular, using the IIS-hypergraph structural result, we show that only two very specific types of rank inequalities induced by generalized antiwebs (which generalize cliques, odd holes and antiholes to general independence systems) can arise as facets. Received: December 2000 / Accepted: November 2002 Published online: February 14, 2003 RID="⋆" ID="⋆" Part of this work was done while the first two authors were with the School of OR&IE, Cornell University, USA. A preliminary version appeared in the Proceedings of the 10th IPCO conference [7], held in Graz, Austria, June 1999. This work was supported by NSF grant DMS-9527124. Key words. infeasible linear systems – feasible subsystems – Irreducible Infeasible Subsystem (IIS) – IIS-hypergraphs – independence systems – feasible subsystem polytope – rank facets