The exponent of discrepancy of sparse grids is at least 2.1933

We study bounds on the exponents of sparse grids for L ₂‐discrepancy and average case d‐dimensional integration with respect to the Wiener sheet measure. Our main result is that the minimal exponent of sparse grids for these problems is bounded from below by 2.1933. This shows that sparse grids provide a rather poor exponent since, due to Wasilkowski and Woźniakowski [16], the minimal exponent of L ₂‐discrepancy of arbitrary point sets is at most 1.4778. The proof of the latter, however, is non‐constructive. The best known constructive upper bound is still obtained by a particular sparse grid and equal to 2.4526.... This revised version was published online in June 2006 with corrections to the Cover Date.     

Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

Let {X(t)} _t∈? be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.     

Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses

The Hölder continuity of the solution X _t (x) to a nonlinear stochastic partial differential equation (see (1.2) below) arising from one dimensional superprocesses is obtained. It is proved that the Hölder exponent in time variable is arbitrarily close to 1/4, improving the result of 1/10 in Li et al. (to appear on Probab. Theory Relat. Fields.). The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.     

A note on stable exponents

Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim sup_n→∞P(γⁿ) denotes the exponent of P(γⁿ) = sup{k/ ?γ_o ∈ γ such that γⁿ = γ^k _o We prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.     

On the Average Number of Unitary Factors of Finite Abelian Groups (II)

Let t(G) be the number of unitary factors of finite abelian group G. In this paper we prove T(x)=∑ _|G|≤x t(G) = main terms for any exponent pair (κ1/2 + 2κ), which improves on the exponent 9/25 obtained by Xiaodong Cao and the author. Received December 8, 1998, Revised April 27, 1998, Accepted June 12, 1998     

Critical exponent for non-Newtonian filtration equation with homogeneous Neumann boundary data

This paper is concerned with large time behavior of solutions to the homogeneous Neumann problem of the non-Newtonian filtration equation. It is shown that the critical Fujita exponent for the problem considered is determined not only by the spatial dimension and the nonlinearity exponent, but also by the coefficient k of the first-order term. In fact, we show that there exist two thresholds k_∞ and k₁ on the coefficient k of the first-order term, and the critical Fujita exponent is a finite number when k is between k_∞ and k₁, while the critical exponent does not exist when k⩽k_∞ or k⩾k₁. Copyright © 2007 John Wiley & Sons, Ltd.     

Piecewise-polynomial approximation of functions fromH ℓ((0, 1) d ), 2ℓ=d,and applications to the spectral theory of the Schrödinger operator

For the selfadjoint Schrödinger operator ?Δ?αV on ?² the number of negative eigenvalues is estimated. The estimates obtained are based upon a new result on the weightedL ₂-approximation of functions from the Sobolev spaces in the cases corresponding to the critical exponent in the embedding theorem.     

The Fefferman-Stein-Type Inequality for the Kakeya Maximal Operator II

Let K _δ, 0 < δ≪1, be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity δ. The (so-called) Fefferman-Stein-type inequality: is shown, where C _d and α _d are constants depending only on the dimension d and w is a weight. The result contains the exponent (d−2)/2d which is smaller than the exponent (d−2)(d−1)/d(2d−3) obtained in [7]. Received October 8, 2001, Accepted February 28, 2002     

On asymptotical normality of stochastic procedures with retardation

Generalized Robbins-Monro and Kiefer-Wolfowitz stochastic approximation procedures with retardation are considered from the point of view of the theory of stochastic differential equation with respect to semimartingales. The results on (a.s.)-convergenceL_2: -convergence and asymptotic normality of these algorithms are obtained with the help of the so-called Dolean exponent.     

Primitive digraphs with smallest large exponent

A primitive digraph D on n vertices has large exponent if its exponent, γ(D), satisfies α_n?γ(D)?w_n, where α_n=⌊w_n/2⌋+2 and w_n=(n-1)²+1. It is shown that the minimum number of arcs in a primitive digraph D on n?5 vertices with exponent equal to α_n is either n+1 or n+2. Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α_n and n+2 arcs. These constructions extend to digraphs with some exponents between α_n and w_n. A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α_n and n+1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n+1 or n+2.     

Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

Let {X(t)}_t∈R be a stationary increments Gaussian process satisfying some assumptions. By using the notion of generalized quadratic variation we build a strongly consistent and asymptotically normal estimator of the uniform Hölder exponent of X, over a compact interval. Our estimator is obtained starting from average values of the process over a regular grid.     

Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation

The generalized multiquadric radial basis function (φ_j=[(x-x_j)²+c²]^β) has the exponent β and shape parameter c that play an important role in the accuracy of the approximation. In this study, we present a trigonometric variable shape parameter and exponent strategy and apply it to function interpolations and linear boundary value problems. Several numerical experiments with the uniformly spaced nodes show that the inverse multiquadric radial basis function (β = −0.5) with the trigonometric variable shape parameter c strategy results in the best accuracy for the one-dimensional interpolations; the trigonometric variable shape parameters and exponent strategy produces the best accuracy for the two-dimensional interpolations and linear boundary value problems. For the non-uniformly spaced nodes, the random variable shape parameter c and exponent β strategy produces the best accuracy for the two-dimensional boundary value problem.     

Longest cycles in polyhedral graphs

A sequence of polyhedral graphsG _n is constructed, having only 3-valent and 8-valent vertices and having only 3-gons and 8-gons as faces with the property that the shortness exponent of the sequence as well as the shortness exponent of the sequence of duals is smaller than one.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

Products of Jacobians as Prym-Tyurin varieties

Let X ₁, ..., X _m denote smooth projective curves of genus g _i ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to . We show that the product JX ₁ × ... × JX _m of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n ^m-1. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.      

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UT _n , compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UT _n , for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UT_n^(?), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UT_n^(?) and for any grading on the Jordan algebra UJ _n .It turns out that the graded exponent for UT _n , considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UT _n coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UT_n^(?).     

Global integral gradient bounds for quasilinear equations below or near the natural exponent

We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for \(\mathcal{A}\) -superharmonic functions in the whole space ? ⁿ . This answers a question raised in our earlier work (On Calderón–Zygmund theory for p- and \(\mathcal{A}\) -superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.