On the isoperimetric problem in the Heisenberg group ℍ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

It has been recently conjectured that, in the context of the Heisenberg group ℍⁿ endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of ℍⁿ, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn–Minkowski inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F⊂ℍⁿ of class C² can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed. Mathematics Subject Classification (2000) 28A75, 22E25, 49Q20     

Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x∈G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they show in Math. Ann. 321, 479–531, 2001 and J. Geom. Anal. 13, 421–466, 2003 that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace. The second author was partially supported by NSF grant DMS-0701515.     

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L¹ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.     

Connected components of sets of finite perimeter and applications to image processing

This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in ℝ ^N , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called M-connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathematical basis to a large class of denoising filters acting on connected components of level sets. We introduce a natural domain for these filters, the space WBV(Ω) of functions of weakly bounded variation in Ω, and show that these filters are also well behaved in the classical Sobolev and BV spaces. Received July 27, 1999 / final version received June 8, 2000?Published online November 8, 2000     

Localization Properties of the Chalker–Coddington Model

The Chalker–Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove first that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly, that this implies spectral localization. Thirdly, we prove a Thouless formula and compute the mean Lyapunov exponent, which is independent of M.     

Classification of topologies on finite sets using graphs

With the use of directed graphs, we study topologies on finite sets. On this basis, we propose a new classification of these topologies. Some properties of T ₀-topologies on finite sets are proved. In particular, we prove the existence, in T ₀-topologies, of open sets containing any number of elements that does not exceed the cardinality of the set itself. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 992–996, July, 2008.     

Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

In this paper, we study exhaustions, referred to as p-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as p-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δp, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of p-restricted Patterson measure, and show that these measures are non-atomic, δp-harmonic, δp-subconformal on special sets and δp-conformal on very special sets. Furthermore, we obtain the results that each p-restriction of our Kleinian group is of δp-divergence type and that the Hausdorff dimension of the p-restricted limit set is equal to δp.     

Tight sets,weighted <Emphasis Type="Italic">m</Emphasis>-covers,weighted <Emphasis Type="Italic">m</Emphasis>-ovoids,and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q⁻(5,3), to 56-caps contained in this elliptic quadric Q⁻(5,3).      

Hardy Spaces, Singular Integrals and The Geometry of Euclidean Domains of Locally Finite Perimeter

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO’s), such as the Riesz transforms as well as Cauchy–Clifford and harmonic double-layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO’s.     

Universal bounds on the selfaveraging of random diffraction measures

 We consider diffraction at random point scatterers on general discrete point sets in ℝ^ν, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence of the random scattering measures applied to an observable towards its mean, when the finite volume tends to infinity. We give an explicit universal large deviation upper bound that is exponential in the number of scatterers. The rate is given in terms of a universal function that depends on the point set only through the minimal distance between points, and on the observable only through a suitable Sobolev-norm. Our proof uses a cluster expansion and also provides a central limit theorem. Received: 10 October 2001 / Revised version: 26 January 2003 / Published online: 15 April 2003 Work supported by the DFG Mathematics Subject Classification (2000): 78A45, 82B44, 60F10, 82B20 Key words or phrases: Diffraction theory – Random scatterers – Random point sets – Quasicrystals – Large deviations – Cluster expansions     

Decomposition of test sets in stochastic integer programming

 We study Graver test sets for linear two-stage stochastic integer programs and show that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. We present a finite algorithm to compute the building blocks directly, without prior knowledge of test set vectors. Once computed, building blocks can be employed to solve the stochastic program by a simple augmentation scheme, again without explicit knowledge of test set vectors. Finally, we report preliminary computational experience. Received: March 14, 2002 / Accepted: March 27, 2002 Published online: September 27, 2002 Key words. test sets – stochastic integer programming – decomposition methods Mathematics Subject Classification (2000): 90C15, 90C10, 13P10     

CM-triviality and generic structures

 We show that any relational generic structure whose theory has finite closure and amalgamation over closed sets is stable CM-trivial with weak elimination of imaginaries. Received: 21 December 2001 / Published online: 5 November 2002 Mathematics Subject Classification (2000): 03C45 Key words or phrases: CM-triviality – Generic structures – Stability     

Toward a possible solution of the generalized continuum problem

A natural property of finite sets is extrapolated to the infinite and is couched in terms of the ABS-axiom in the language of ZF. It is shown that the generalized continuum hypothesis enters the picture of ZF+ABS-theory. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 409–418, July–August, 1999.     

A notion of rank in set theory without choice

 Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory. Received: 22 September 2000 / Revised version: 14 May 2002 Published online: 19 December 2002 The research of the first author was supported by the SERC. Mathematics Subject Classification (2000): 03E25 Key words or phrases: Rank – Degree – Amorphous     

On finite <Emphasis Type="Italic">X</Emphasis>-semilattices of unions

Using a characteristic family of sets, a characteristic mapping, and basis sources of an X-semilattice of unions D, we characterize the class Σ(X, m) consisting of all finite X-semilattices of unions that are isomorphic to a semilattice D given in advance. For a finite set X, the number of elements in the considered class is found. Commutative semigroups of idempotents are known to play a significant role in semigroup theory (see [25, 26]). Moreover, any commutative idempotent semigroup is isomorphic to some X-semilattice of unions (see [26]), whereas X-semilattices play an especially important role in studying many abstract properties of complete semigroups of binary relations (see [1–4, 7–24]). __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 27, Algebra and Geometry, 2005.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Localization of invariant compacta of autonomous systems

A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with desired properties, e.g., contain certain solutions of the system of differential equations. Such problems include construction of positive invariant sets that contain certain separatrices of the Lorenz system [1], analysis of asymptotic behavior of solutions of the Lorenz system and determination of sets that contain the Lorenz attractor [2–5, 14], as well as determination of sets containing all periodic trajectories [6–13], separatrices, and other trajectories [10, 11]. Such sets may be naturally called localizing sets and it is obviously interesting to study methods and results that produce exact or nearly exact localizing sets for each phase space structure. In this article we focus on localization of the invariant compact sets in the phase space of a differential equation system, specifically the problem of finding phase space subsets that contain all the invariant compacta of the system. Invariant compact sets are equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets and their finite unions. These sets and their properties largely determine the phase space structure and the qualitative behavior of solutions of the differential equation system.     

Recognizability of finite simple groups <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) and <Emphasis Type="Italic">U</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) by spectrum

It is proved that finite simple groups L₄(2^m), m ⩾ 2, and U₄(2^m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8. Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry of Education of China (Project for Retaining Foreign Expert). Supported by NSF of Chongqing (CSTC: 2005BB8096). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008.