On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Beitrag zur Reduktion des Entscheidungsproblems auf Klassen von Hornformeln mit kurzen Alternationen

Ohne Zusammenfassung Zusatz bei der Korrektur: Ein vollständiger und korrekter Beweis für die Entscheidbarkeit der eingangs angeführten Aanderaaschen Klasse ((0, ), (, , ...)) erscheint demnächst im JSL (S.O. Aanderaa/H.R.Lewis: Prefix classes of Krom formulas). Ebendort wird auch die Reduktionstypeneigenschaft für ((0, ), (0, 0, )) und ((0, )), (0, 0, )) nachgewiesen, während ((0, ), (, )) sich als entscheidbar herausgestellt hat (s. E. Börger: Eine entscheidbare Klasse von Kromformeln. ZMLG 19 (1973), 117–120.) Der Kromsche Reduktionstyp konnte mittlerweile einerseits zu ((0, ), (0, 4)) verschärft werden (s. D. Rödding, E. Börger: The undecidability of (0, 4)-formulae with binary disjunctions, vorgetragen auf dem Logic Coll. Bristol 1973, ein abstract erscheint im JSL), andererseits kündigt H.R.Lewis die Reduktionstypeneigenschaft für ((0, ), (0, 1)) an (s. H.R.Lewis: Krom formulas with one dyadic predicate letter. Notices AMS 20, 5 (1973) A-500, abstr. no. 73T-E78.)Dieser Aufsatz geht aus der Dissertation [2] hervor, die dem Fachbereich Mathematik der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster im Sommersemester 1971 vorgelegt worden ist. Die Ergebnisse stammen aus dem Wintersemester 1970/71. Eine Ankündigung der hauptsächlichen Resultate ist in den Notices of the American Mathematical Society 19, 2 (1972) A-333 unter der abstract no. * 72T-E24 erschienen.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

A modified sinc quadrature rule for functions with poles near the arc of integration

Sinc function approach is used to obtain a quadrature rule for estimating integrals of functions with poles near the are of integration. Special treatment is given to integration over the intervals (–, ), (0, ), and (–1, 1). It is shown that the error of the quadrature rule converges to zero at the rateO(exp(–cN)) asN , whereN is the number of nodes used, and wherec is a positive constant which is independent ofN.     

Interpolation of generalized analytical functions of complex variable

A general approach is proposed to the interpolation of x -analytical functions of a complex variable with an arbitrary –,+[Basis x -analytical functions whose imaginary pan is a polynomial in x, and y are obtained in explicit form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 3–9, 1986.     

Smoothness of Nonlinear Median-Interpolation Subdivision

We give a refined analysis of the Hölder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The synthesis part of this algorithm can be interpreted as a nonlinear triadic subdivision scheme where new points are inserted based on local quadratic polynomial median interpolation and imputation. We introduce the analogon of the Donoho–Yu scheme for dyadic refinement, and show that its limit functions are in C for >log₄(128/31)=1.0229.... In the triadic case, we improve the lower bound of >log₂(135/121)=0.0997... previously obtained in [6] to >log₃(135/53)=0.8510.... These lower bounds are relatively close to the anticipated upper bounds of log₂(16/7)=1.1982... in the dyadic, respectivly 1 in the triadic cases, and have been obtained by deriving recursive inequalities for the norm of second rather than first order differences of the sequences arising in the subdivision process.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Nonlinear potential theory and PDEs

We consider equations like -div(|u| ^p–2u)=, where is a nonnegative Radon measure and 1u and the measure are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.     

Durch graduierte Lie-Algebren definierte Paar-Algebren

Pair algebras which have a non degenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible are characterised by pairs (C,), where C is a n×n matrix satisfying certain conditions and is a sequence of n integers equal to 0 or 1. They occur as pair algebras of type (S(C,)_–1,S(C,)₁), xuy=[[x,u],y], where (S(C,)_r)_r is the gradation induced by . in the Kac-Moody algebraS(C). If C is an affin Cartan matrix (as in the case of Lie triple systems), there exists a finite dimensional simple Lie algebrag and a Aut (g), ord =m< such that the pair algebra is isomorphic to the pair algebra (g _–1,g ₁), xuy=[[x,u],y] (product ing), whereg _i. is the eigenspace of of eigenvalue ⁱ, a primitive m-th root of unity.     

Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

This paper deals with the following Dirichlet problem Lu = 1A ( Au – qu = – f ( , u ) on ] 0, [ , u, ( 0 ) = 0, u ( ) = 0, where ] 0, + ], q 0 is continuous on [ 0, [ × ] 0, + [ ] 0, + [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ). Moreover, we study the behaviour of this solution in a neighbourhood of . Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.     

A characterization of nonlinear φ-accretive operators

Let E be a Banach space and E(–,] a proper lower semi-continuous convex function. The main purpose of this paper is to characterize those m-accretive operators AE x E that are also -accretive. This is done by using the semigroup S generated by -A, and by first establishing a necessary and sufficient condition for to be a Lyapunov function for S. We also obtain similar results for accretive operators that are not necessarily m-accretive, and deduce invariance and order-preserving criteria for nonlinear semigroups.This research was partially supported by the National Science Foundation under Grant MCS-8102086.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Some zero-one laws for additive functionals of Markov processes

Summary Let (X _t,P ^x ) be anm-symmetric Markov process with a strictly positive transition density. Consider the additive functionalA _t : = ₀ ^t f (X _s ) wheref:E[0, ] is a universally measurable function on the state spaceE. Among others, we prove thatP ^x (A _t <)=1, for somexE and somet>0, already impliesP ^x (A _t <)=1, for quasi everyxE and allt>0. The latter is also equivalent toP ^x (A _t <)>0, for quasi everyxE and allt>0, and to the analytic condition , for a sequence of finely open Borel setsF _n such thatEF _n is polar. In the special cases of Brownian motion and Bessel process, these results were obtained earlier by H.J. Engelbert, W. Schmidt, X.-X. Xue and the authors.     

Une application de Corps des Normes

Let k be a perfect field of characteristic p > 0, K₀ = Frac(W(k)), a uniformizer in K₀ and _n K ₀ (n N) such that ₀ = and _n+1 ^p = _n. We write K = _nN K₀ (_n), H = Gal (K₀/ K and G = Gal(K₀/ K₀). The main result of this paper is that the functor restriction of the Galois action from the category of crystalline representations of G with Hodge–Tate weights in an interval of length p - 2 to the category of p-adic representations of H is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H: one thanks to the norm field of K, the other thanks to some categories of filtered modules with divided powers previously introduced by the author.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

Inverse problem of the scattering of plane waves for a class of layered media

The direct and inverse problems of the scattering of plane waves in a layered, inhomogeneous medium are considered in the paper. In the appropriate variables the wave equation of the problem has the formu (z,)=Q(Z)u _ZZ(Z,), – < Z, <, Q(Z)|_Z<01. A special feature of the case considered, in contrast to those studied earlier, is that Q(Z)|_Z0 may change sign; because of this, the equation of the problem is, in general, an equation of mixed type. The correct formulation of the direct problem for such an equation and the study of the properties of its solution form a necessary step in the investigation. For a very broad class of media including cases of Q(z) of variable sign (Q(z) can change sign by a jump a finite number of times without vanishing anywhere) a procedure is developed for solving the corresponding inverse problem of determining Q(z) on the basis of the scattering datau(0,)|_(–,). This procedure makes it possible to recover Q(z) for all z[0,). The solution of the inverse problem is unique in this class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 78, pp. 30–53, 1978.The author thanks his scientific supervisor A. S. Blagoveshchenskii for his constant attention and assistance in the work.     

Sample path behaviour in connection with generalized arcsine laws

Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA ⁺=(A ⁺ (t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law.