Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent

It is proved that commensurable hyperbolic groups are bi-Lipschitz equivalent. Therefore, subgroups of finite index in an arbitrary hyperbolic group also share this property. In addition, it is shown that any two separated nets Γ¹ and Γ² in the hyperbolic space Hⁿ of dimension n≥2 are bi-Lipschitz-equivalent. These results answer the questions posed in [1]. Supported by RFFR grant No. 96-01-01781. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 259–272, May–June, 1997.     

Error estimates for the discretization of elliptic control problems with pointwise control and state constraints

A family of elliptic optimal control problems with pointwise constraints on control and state is considered. We are interested in approximation of the optimal solution by a finite element discretization of the involved partial differential equations. The discretization error for a problem with mixed state constraints is estimated in the semidiscrete case and in the fully discrete scheme with the convergence of order h|ln h| and h ^1/2, respectively. However, considering the unregularized continuous problem and the discrete regularized version, and choosing suitable relation between the regularization parameter and the mesh size, i.e., ε∼h ², a convergence order arbitrary close to 1, i.e., h ^1−β is obtained. Therefore, we benefit from tuning the involved parameters.     

Eigenvalue distributions from impacts on a ring

We consider the collision dynamics produced by three beads with masses (m ₁, m ₂, m ₃) sliding without friction on a ring, where the masses are scaled so that m ₁ = 1/ɛ, m ₂ = 1, m ₃ = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ ² ∼ αɛ ^β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ ² ∼ log(ɛ ^β).     

Covering properties which,under weak diamond principles,constrain the extents of separable spaces

We show that separable, locally compact spaces with property (a) necessarily have countable extent — i.e., have no uncountable closed, discrete subspaces — if the effective weak diamond principle ⋄(ω,ω,<) holds. If the stronger, non-effective, diamond principle Φ(ω,ω,<) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ^ω ₁ ω.     

Structure and Classification of Superconformal Nets

We study the general structure of Fermi conformal nets of von Neumann algebras on S ¹ and consider a class of topological representations, the general representations, that we characterize as Neveu–Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by taking the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2. Sebastino Carpi: Supported by MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962 Yasuyuki Kawahigashi: Supported in part by the Grants-in-Aid for Scientific Research, JSPS. Submitted: March 3, 2008. Accepted: May 5, 2008.     

Scrambling non-uniform nets

In this paper, we consider Owen’s scrambling of an (m−1, m, d)-net in base b which consists of d copies of a (0, m, 1)-net in base b, and derive an exact formula for the gain coefficients of these nets. This formula leads us to a necessary and sufficient condition for scrambled (m − 1, m, d)-nets to have smaller variance than simple Monte Carlo methods for the class of L ₂ functions on [0, 1] ^d . Secondly, from the viewpoint of the Latin hypercube scrambling, we compare scrambled non-uniform nets with scrambled uniform nets. An important consequence is that in the case of base two, many more gain coefficients are equal to zero in scrambled (m − 1, m, d)-nets than in scrambled Sobol’ points for practical size of samples and dimensions.      

The best Lipschitz constants of Bernstein polynomials and Bezier nets over a given triangle

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃Lip_Aα, then the corresponding Bernstein Bezier net f_n∃Lip _Asec ^αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net f_n∃Lip _Bα, then its elevation Bezier net Ef_n∃Lip _Bα; and (3) If f(P)∃Lip _Aα, then the corresponding Bernstein polynomials B_n(f;P)∃Lip _Asec ^αφα, and the constant Asec^αφ is best in some sense. Supported by NSF and SF of National Educational Committee     

An A ∞ structure on simplicial complexes

We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular, on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential d and the exterior product ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called A _∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)ⁿ = 0 of the second degree, n = 2. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 3–37, July, 2008.     

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.     

Indefinite higher Riesz transforms

Stein’s higher Riesz transforms are translation invariant operators on L ²(R ⁿ ) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature (p,q). We prove that these operators extend to L ^r -bounded operators for 1<r<∞ if the parameter of the discrete series representations is generic.     

Lattice-embeddingL p into Orlicz spaces

Given 0<α≤p≤β<∞, we construct Orlicz function spacesL ^F [0, 1] with Boyd indicesα andβ such thatL ^p is lattice isomorphic to a sublattice ofL ^F [0, 1]. Forp>2 this shows the existence of (non-trivial) separable r.i. spaces on [0, 1] containing an isomorphic copy ofL ^p . The discrete case of Orlicz spaces ℓ ^F (I) containing an isomorphic copy of ℓ ^p (Γ) for uncountable sets Γ ⊂I is also considered. Supported in part by DGICYT, grant PB91-0377.     

Applications of operator semigroups to Fourier analysis

Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T ⁿ /n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T ⁿ f,f〉|², |〈T (n)f,f〉|² are computed (withn∈ℤ⁺ orn∈ℤ⁺). Specializing the Hilbert space to beL ²(T,μ) (discrete case) orL ²(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of (asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT ^M , and ℝ ^M are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context. Partially supported by an NSF grant     

Sharp determinants

We introduce a sharp trace Tr ^# ℳ and a sharp determinant Det ^# (1−z ℳ) for an algebra of operators ℳ acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of ℳ. The relationship with weighted zeta functions of interval maps and Milnor–Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators. Oblatum 8-V-1995 & IX-1995     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

On the Equicontinuity Region of Discrete Subgroups of <Emphasis Type="Italic">PU</Emphasis>(1,<Emphasis Type="Italic">n</Emphasis>)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ_ℂ ⁿ preserving the unit ball ℍ_ℂ ⁿ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ_ℂ ⁿ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ_ℂ ⁿ which are tangent to ∂ℍ_ℂ ⁿ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ_ℂ ⁿ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.     

On the possibility of exact reciprocal transformations for one-soliton solutions to equations of the Lobachevsky class

Problems on reciprocal transformation of solutions to equations of Λ²-class (equations related to special coordinate nets on the Lobachevsky plane Λ²) are discussed. A method of construction of solutions to one analytic differential equation of Λ²-class by a given solution of another analytic differential equation of this class is proposed. The reciprocal transformation of one-soliton solutions of the sine-Gordon equation and one-soliton solutions of the modified Korteweg-de Vries equation (MKdV) is obtained. This result confirms the possibility of construction of such transition. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

A completion problem for finite affine planes

A partial affine plane (PAP) of ordern is ann ²-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn ²+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn ²+n−√n can be completed and that forn≠3 any PAP withn ²+n−2 lines can be completed.     

Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space with the largest time step Δt ₁; subsequent approximations are generated on a succession of higher-dimensional spaces with small time step Δt _j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H ¹-norm and L ²-norm are investigated, i.e., m _j∼m _j−1 ^3/2 , Δt _j∼Δt _j−1 ^3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method. Mathematics subject classifications (2000) 35L70, 65N30, 76D06 Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City University of Hong Kong Research Project 7001093, NSF of China 50323001.