Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C. Suppose that x ^s d(x)x ^t ∈ Z(R) for all x ∈ {d(f(x ₁, ..., x _n ))|x ₁, ..., x _n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x ₁, ..., x _n ) ^N is central-valued in eRCe, where N = s + t + 1.      

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Представление и приб лижение периодических функц ий многих переменных с заданным смешанным модулем непрепывнос ти

In this paper the author studies classesH _q of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity (t ₁ ...,t _d ). Necessary and sufficient conditions of belonging of a functionf(x ₁, ...,x _d ) to the classH _q are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on (t ₁, ...,t _d ). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesH _q with some special (t ₁, ...,t _d ) are given (Theorems 4 and 5).     

On the Almost Intersections of Transient Brownian Motions

Let {B ₁ ^d (t)} and {B ^d ₂(t)} be independent Brownian motions in R ^d starting from 0 and nx respectively, and let w ^d _i (a,b) ={xR ^d : B ^d _i (t)=x for some t(a,b)}, i=1,2. Asymptotic expressions as n for the probability of dist(w ^d ₁(n ² t ₁, n ² t ₂), w ₂ ^d (0,n ² t ₃))1 with d4, respectively for the probability of dist(w ₁ ⁴(n ² t ₁,n ² t ₂),w ₂ ⁴(0,n ² t ₃))1 are obtained. As an application, an improvement of a result due to M. Aizenman concerning the intersections of Wiener sausages in R ⁴ is presented.     

Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.     

<Emphasis Type="Italic">Φ</Emphasis>-Variation of a bifractional Brownian motion

Let B _H,K = {B _H,K (t)} _t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t _i }_n ⁱ⁼⁰ of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08     

Moduli of continuity for <Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript>-valued Gaussian random fields

This paper establishes the general moduli of continuity for l ^∞-valued Gaussian random fields {X(t):= (X ₁(t),X ₂(t), h.), t ∈ [0, ∞) ^N } indexed by the N-dimensional parameter t:= (t ₁,…,t _N ), under the explicit condition yielding that the covariance function of distinct increments of X _k (t) for fixed k ≧ 1 is positive or nonpositive. Supported by KOSEF-R01-2008-000-11418-0.     

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

An integral strong law of large numbers for processes with independent increments

For a stochastically continuous stochastic process with independent increments overD[0,T], letN(t,ε) be the number of smaple function jumps that occur in the interval [0,t] of sizes less than −ε or greater than ε, where ε>0. LetM(t,ε)=EN(t,ε), and assumeM(t,0+)=∞ for 0<t≦T. If lim_{ε ↓0}(M(t,ε)/M(T,ε)) exists and is positive for eacht∈(0,T], then lim_{ε ↓0}(N(t,ε)/M(T,ε)) for allt∈(0,T] with probability one. The research of Howard G. Tucker was supported in part by the National Science Foundation, Grant No. MCS76-03591A01.     

Small ball probabilities for a Wiener process under weighted sup-norms,with an application to the supremum of bessel local times

LetR be the radial part of ad-dimensional Wiener process, starting from 0. In this paper, small ball probabilities are evaluated for sup_0<11(t ^–p R(t)) and sup _t 0(e ^–1 R(t)), withp[0, 1/2]. Chung's law of the iterated logarithm is established for the supremum of the local times of a two-dimensional Bessel process.     

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

CHAOS EXPANSION OF LOCAL TIME OF FRACTIONAL BROWNIAN MOTIONS

We find the chaos expansion of local time ? _T ^(H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈R ^d . As an application we show that when H ₀ d<1 then ? _T ^(H)(x,·)∈L ²(μ). Here μ denotes the probability law of B ^(H) and H ₀=max{H ₁,…,H _d }. In particular, we show that when d=1 then ? _T ^(H)(x,·)∈L ²(μ) for all H∈(0,1).     

Spectral Conditions for Strong Local Nondeterminism and Exact Hausdorff Measure of Ranges of Gaussian Random Fields

Let X={X(t),t∈ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d _t ) _t∈ℤ of real numbers; if the parameter sequence is constant d _t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)^−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε _t and B(d)ε _t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X _t ^A = A(d)ε _t and X _t ^B = B(d)ε _t by assuming that d = (d _t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε _t ). In the case where the mean , we show that large sample properties of X ^A and X ^B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X _t ^A ) of a randomly fractionally integrated process X _t ^A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d _t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

On the ε-entropy of classes of holomorphic functions

Suppose thatB _R ^d is a ball of radiusR in ℂ ^d and σ is the standard measure on the unit sphere in ℂ ^d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH _R ⁰ (l, p, d) we denote the class of functionsf holomorphic inB _R ^d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH ^p (B _R ^d ). In this paper an asymptotic formula for the ε-entropy of the classH _R ⁰ (l, p, d) in the spacesL ^p (σ), 1≤p<∞, and is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 286–293, August, 2000.     

Formule de Rice en dimension d

Summary Let {x(t): tR ^d} a stochastic process with parameter in R ^d, and u a fixed real number. Denote by C _u, A_u, B_u respectively the random sets {t: x(t)= u}, {t: x(t)}, {t: x(t)>u}. The paper contains two main results for processes with continuously differentiable paths plus some additional requirements: First, a formula for the expectation of Q _T(A_u) and Q _T(B_u), where for a given bounded open set T in R ^d, Q_T(B) denotes the perimeter of B relative to T and second, sufficient conditions on the process, so that it does not have local extrema on the barrier u. The second result can also be used to interpret the first in terms of C _u.     

Results on Local Times of a Class of Multiparameter Gaussian Processes

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant