Limsup results and LIL for partial sum processes of a Gaussian random field

Let {ξ _j ; j ∈ ℤ₊ ^d be a centered stationary Gaussian random field, where ℤ₊ ^d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ^d, having nonnegative integer coordinates. For each j = (j ₁ , ..., jd) in ℤ₊ ^d , we denote |j| = j ₁ ... j _d and for m, n ∈ ℤ₊ ^d , define S(m, n] = Σ _m<j≤n ζ _j , σ²(|n−m|) = ES ² (m, n], S _n = S(0, n] and S ₀ = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa     

A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions P ^α = {P ^x} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d _H < α < ∞, where d _H = log(3 ^d − 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions P ^α are reversible with invariant measures μ = μ^[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3^α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet. Received: 30 September 1999 / Revised version: 15 June 2000 / Published online: 24 January 2000     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998     

A Krasnosel’skii-type theorem for paths of bounded length

Let S be a nonempty closed, simply connected set in the plane, and let α τ; 0. If every three points of 5 see a common point of S via paths of length at most α, then for some point s0 of S, s0 sees each point of S via such a path. That is, S is starshaped via paths of length at most α. Supported in part by NSF grant DMS-9207019     

Mixing properties of the generalized T, T-1-process

Consider a general random walk on ℤ^d together with an i.i.d. random coloring of ℤ^d. TheT, T ^-1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk together with the color of the newly arrived at location. S. Kalikow proved that ifd = 1 and the random walk is simple, then this process is not Bernoulli. We generalize his result by proving that it is not Bernoulli ind = 2, Bernoulli but not Weak Bernoulli ind = 3 and 4, and Weak Bernoulli ind ≥ 5. These properties are related to the intersection behavior of the past and the future of simple random walk. We obtain similar results for general random walks on ℤ^d, leading to an almost complete classification. For example, ind = 1, if a step of sizex has probability proportional to l/|x|^α (x ⊋ 0), then theT, T ^-1-process is not Bernoulli when α ≥2, Bernoulli but not Weak Bernoulli when 3/2 ≤α < 2, and Weak Bernoulli when 1 < α < 3/2. Research partially carried out while a guest of the Department of Mathematics, Chalmers University of Technology, Sweden in January 1996. Research supported by grants from the Swedish Natural Science Research Council and from the Royal Swedish Academy of Sciences.     

On the Kauffman skein modules

 Let k be a subring of the field of rational functions in α, s which contains α ^±1 ,s ^±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S ¹ × S ² is freely generated by the empty link, i.e., . Received: 20 October 2001 / Revised version: 20 March 2002     

An illumination problem for zonoids

Let Z ⊂ E ^d be ad-dimensional zonoid, whered≥3. Boltjanskii and Soltan recently proved that if Z is not a parallelotope, then Z can be illuminated by 3·2 ^d−2 points disjoint from Z. In the present paper we prove a related result. Namely, we show that ifd+1=2 ^p , then Z can be illuminated by 2 ^d /d+1 lines lying outside Z. The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213 and 326-0113.     

The central Campanato spaces and its application

Let 1<p<∞, α≥0 andK be a local field. In this paper, the author introduces the spaces J _p ^α (K) and the central Campanato spaces as well as the S_p,α- and S _p,α ⁺ -type singular integrals. Then, the author investigates the behavior of these singular integrals and their altered operators on these spaces. The research was supported by the NNSF of China.     

Random Partial Orders Defined by Angular Domains

The d-dimensional random partial order is the intersection of d independently and uniformly chosen (with replacement) linear orders on the set [n] = {1, 2, . . . , n}. This is equivalent to picking n points uniformly at random in the d-dimensional unit cube Q_d=[0,1]^dQ_d=[0,1]^d with the coordinate-wise ordering. If d = 2, then this can be rephrased by declaring that for any pair P ₁, P ₂ ∈ Q ₂ we have P ₁ ≺ P ₂ if and only if P ₂ lies in the positive upper quadrant defined by the two axis-parallel lines crossing at P ₁. In this paper we study the random partial order with parameter α (0 ≤ α ≤ π) which is generated by picking n points uniformly at random from Q ₂ equipped with the same partial order as above but with the quadrant replaced by an angular domain of angle α.     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

The Projection Median of a Set of Points in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

The projection median of a finite set of points in ℝ² was introduced by Durocher and Kirkpatrick [Computational Geometry: Theory and Applications, Vol. 42 (5), 364–375, 2009]. They proved that the projection median in ℝ² provides a better approximation of the two-dimensional Euclidean median than the center of mass or the rectilinear median, while maintaining a fixed degree of stability. In this paper we study the projection median of a set of points in ℝ ^d for d≥2. Using results from geometric measure theory we show that the d-dimensional projection median provides a (d/π)B(d/2,1/2)-approximation to the d-dimensional Euclidean median, where B(α,β) denotes the Beta function. We also show that the stability of the d-dimensional projection median is at least \frac1(d/p)B(d/2, 1/2)\frac{1}{(d/\pi)B(d/2, 1/2)}, and its breakdown point is 1/2. Based on the stability bound and the breakdown point, we compare the d-dimensional projection median with the rectilinear median and the center of mass, as a candidate for approximating the d-dimensional Euclidean median. For the special case of d=3, our results imply that the three-dimensional projection median is a (3/2)-approximation of the three-dimensional Euclidean median, which settles a conjecture posed by Durocher.     

Tight Beltrami fields with symmetry

Let M be a compact orientable Seifered fibered 3-manifold without a boundary, and α an S ¹-invariant contact form on M. In a suitable adapted Riemannian metric to α, we provide a bound for the volume Vol(M) and the curvature, which implies the universal tightness of the contact structure ξ = ker α.      

Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that

Riesz Spherical Potentials with External Fields and Minimal Energy Points Separation

In this paper we consider the minimal energy problem on the sphere S ^d for Riesz potentials with external fields. Fundamental existence, uniqueness, and characterization results are derived about the associated equilibrium measure. The discrete problem and the corresponding weighted Fekete points are investigated. As an application we obtain the separation of the minimal s-energy points for d – 2 < s < d. The explicit form of the separation constant is new even for the classical case of s = d – 1. Research supported, in part, by a National Science Foundation Research grant DMS 0532154.     

A Helly-Type Theorem for Hyperplane Transversals to Well-Separated Convex Sets

Let S be a finite collection of compact convex sets in \R ^d . Let D(S) be the largest diameter of any member of S . We say that the collection S is ɛ-separated if, for every 0 < k < d , any k of the sets can be separated from any other d-k of the sets by a hyperplane more than ɛ D(S)/2 away from all d of the sets. We prove that if S is an ɛ -separated collection of at least N(ɛ) compact convex sets in \R ^d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S . The number N(ɛ) depends both on the dimension d and on the separation parameter ɛ . This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one. Received August 10, 2000, and in revised form January 24, 2001. Online publication April 6, 2001.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α _H ∈V ^* such that H=ker(α _H ). For each nonnegative integer m, define the derivation module D ^(m) (?)={θ∈Der _S |θ(α _H )∈Sα ^m _H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D ^(m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m _i (1≤i≤ℓ) (when m is odd). Here m ₁≤···≤m _ℓ are the exponents of G and h=m _ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result. Oblatum 27-XI-2001 & 4-XII-2001?Published online: 18 February 2002     

A Note on Large Deviations for the Stable Marriage of?Poisson and Lebesgue with Random Appetites

Let Ξ⊂ℝ ^d be a set of centers chosen according to a Poisson point process in ℝ ^d . Let ψ be an allocation of ℝ ^d to Ξ in the sense of the Gale–Shapley marriage problem, with the additional feature that every center ξ∈Ξ has an appetite given by a nonnegative random variable α. Generalizing some previous results, we study large deviations for the distance of a typical point x∈ℝ ^d to its center ψ(x)∈Ξ, subject to some restrictions on the moments of α.