Classification of topologically conjugate affine mappings

We consider affine mappings from ℝ ⁿ into ℝ ⁿ , n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine mappings from ℝ ⁿ into ℝ ⁿ , n > 1, having at least one fixed point and the nonperiodic linear part. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009.     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

Percolation in invariant Poisson graphs with?i.i.d.?degrees

Let each point of a homogeneous Poisson process in ℝ ^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution μ on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution μ. Leaving aside degenerate cases, we prove that for any μ there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on μ for the absence and presence of infinite components.     

Dense analytic subspaces in fractalL 2-spaces

We show that for certain self-similar measures μ with support in the interval 0≤x≤1, the analytic functions {e ^i2πnx :n=0,1,2, …} contain an orthonormal basis inL ² (μ). Moreover, we identify subsetsP ⊂ ℕ₀ = {0,1,2,...} such that the functions {e _n :n ∈ P} form an orthonormal basis forL ² (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝ ^ν , obtained from a given expansivev-by-v matrix and a finite set of translation vectors. We show that the correspondingL ² (μ) has an orthonormal basis of exponentialse ^i2πλ·x , indexed by vectors λ in ℝ ^ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system. Work supported by the National Science Foundation.     

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ ^d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ² and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.     

On the theory of mappings with finite area distortion

It is shown that mappings in ℝⁿ with finite distortion of area in all dimensions 1 ≤ k ≤ n − 1 satisfy certain modulus inequalities in terms of inner and outer dilatations of the mappings; in particular, generalizations of the well-known Poletskii inequality for quasiregular mappings are proved. The theory developed is applicable, for example, to the class of finitely bi-Lipschitz mappings, which is a natural generalization of the bi-Lipschitz mappings, as well as isometries and quasi-isometries in ℝⁿ.     

The Existence and Uniqueness of the Solution for Neutral Stochastic Functional Differential Equations with Infinite Delay in Abstract Space

The main aim of this paper is to prove the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay, which the initial data belong to the phase space ℬ((−∞,0];ℝ ^d ). The vital work of this paper is to extend the initial function space of the paper (Wei and Wang, J. Math. Anal. Appl. 331:516–531, 2007) and give some examples to show that the phase space ℬ((−∞,0];ℝ ^d ) exists. In addition, this paper builds a Banach space ℳ²((−∞,T],ℝ ^d ) with a new norm in order to discuss the existence and uniqueness of the solution for such equations with infinite delay.     

Deriving hydrodynamic equations for lattice systems

We study the dynamics of lattice systems in ℤ^d, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ₀^ɛ, ɛ > 0}. The measures μ₀^ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ⁻¹ ) and inhomogeneous under shifts of the order ɛ⁻¹ . Moreover, correlations of the measures μ₀^ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝ^d, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛ^κ at points close to [r/ɛ] ∈ ℤ^d. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.     

Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Convergence of Sobolev spaces on varying manifolds

We deal with variational problems on varying manifolds in ℝⁿ. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝⁿ. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μ_h -a.e., being {μ_h} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (P_{μ h} H(μ_h)), where P_{μ h} are the projectors onto the tangent spaces T_{μ h}. When μ_h belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary.     

Absolute continuities of exit measures for superdiffusions

Suppose X= X_t, X_T, P_μis a superdiffusion in ℝ^d with general branching mechanism ψ and general branching rate functionA. We discuss conditions onA to guarantee that the exit measure X_TL of the superdiffusionX from bounded smooth domains in ℝ^d have absolutely continuous states.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Finite quasihypermetric spaces

Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (X) → ℝ by I(μ) = ∫ _X ∫ _X d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L ¹-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].      

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

We show that whenever the q-dimensional Minkowski content of a subset A ⊂ ℝ ^d exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ℝ ^d , d ⩾ 3.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.     

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

 We consider the elliptic operator P(D)+V in ℝ ^d , d≥2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2ℓ with a homogeneous convex symbol P(ξ), and V is a real periodic function in L ^∞(ℝ ^d ). We show that the number of gaps in the spectrum of P(D)+V is finite if 4ℓ>d+1. If in addition, V is smooth and the convex hypersurface {ξℝ ^d :P(ξ)=1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8ℓ>d+3 and 9≥d≥2, or 4ℓ>d−3 and d≥10. Received: 10 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 35J10 Research supported in part by NSF Grant DMS-9732894.     

Duality Questions for Operators,Spectrum and Measures

We explore spectral duality in the context of measures in ℝ ⁿ , starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases of complex exponentials in L ²(Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in ℝ ⁿ , formulated first by Jorgensen and Pedersen.