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.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

Bessel Potentials with Logarithmic Components and Sobolev-Type Embeddings

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L ^n/k (log L) (R ⁿ ), are continuously embedded into L (R ⁿ ) if > 1/p, p n/k. In this paper we replace L ^n/k (log L) (R ⁿ ) by the Lebesgue space L ^n/k (R ⁿ ) and increase the smoothness of the functions involved by a "logarithmic" order > 1/p to obtain the continuous embedding into L (R ⁿ ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brézis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.     

GBRD's: Some new constructions for difference matrices,generalised hadamard matrices and balanced generalised weighing matrices

A new construction is given for difference matrices. The generalized Hadamard matrices GH(q(q – 1)²; EA(q)) are constructed whenq andq – 1 are both prime powers. Other generalised Hadamard matrices are also shown to exist. For example, there exist GH(n; G) forn = 5² 2 3, 2⁶ 3², 11² 2² 3, 17² 2 3², 53² 2 3³, 71² 2² 3², 107² 2² 3³, 149² 5² 2 3,.... Finally, a new construction for the BGW ((q ⁴ – 1)/(q – 1),q ³,q ²(q – 1);q _q-1), and a construction for the new BGW ((q ⁸ – 1)/(q ² – 1),q ⁶,q ⁴(q ² – 1);G) are given, wheneverq is a prime power, andG is a group of orderq + 1.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

Barth type vanishing theorems

Let X be a smooth, projective, d-dimensional subvariety of ⁿ (). Barth's theorem says that H ^q (X, ^p _X )=0 when pq and q+p2d–n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H ^q (X, ^p _X (m)), m (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H ^q (X, ^p _X (1)), for H ^q (X, ^p _X (m)) when m–1, and, if dim(X)=n–2, for H ^q (X, ² _X (m)) and H ^q (X, S ^{k 1} _X (m)). We use standard techniques and some of our previous results.     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Fano'sche Moufang-Ebenen mit Gauss-Figur sind pappos'sch

LetV be a quadrilateral in aMoufang-plane , in which theFano-proposition is valid. Take the pointsP,Q,R respectively in the diagonalsp,q,r ofV and construe the pointsP ^*,Q ^*,R ^* inp,q, r harmonic toP,Q,R with respect to pairs of edges ofV. IfP,Q,R are collinear, so areP ^*,Q ^*,R ^*, if and only if is aPappos-plane. Is V classical, the pointsP ₁ p,Qq,Rr and their harmonic conjugatesP ₁ ^* ,Q ^*,R ^* (construed as above mentioned) lay in a curve of 2^nd order.

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R. Artzy zum 70. Geburtstag zugeeignet     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary The integral _- [C _2n (it)]^–2(1+t ²)^–-1/2 dt is evaluated for > –1/2 whereC _2n is the Gegenbauer polynomial of degree 2n. Letting gives the value _- [H _2n (it)]^–2 e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.     

Convergence in distribution and Skorokhod Convergence for the general theory of processes

Summary This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X ⁿ , ⁿ ),n, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y ⁿ , ⁿ ),n, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X ⁿ , ⁿ ) has the same stochastic properties as (X ⁿ , ⁿ ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y ⁿ , ⁿ ),n, and their limit (Y, ) may be taken as all having the same fixed filtration ⁿ =. In the continuous time case, theY ⁿ , ⁿ may require different filtrations ⁿ , which converge to. To handle this, convergence of filtrations is defined and its theory developed.During part of the time this work was in progress, it was supported by an NSERC operating grant, and the author was an NSERC University Research Fellow. The author wishes to thank the Steklov Mathematical Institute of the Soviet Academy of Sciences for its hospitality while the principle research in this paper was being begun, A.N. Shiryaev and P.C. Greenwood, who made the author's visit there possible, and Ján Miná for his hospitality while that research was being finished. We thank the referee who suggested the results in Sect. 12     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Π r P1-bundle from which a surjective morphism to Π m P1 exists

Let : X Y be a morphism of smooth projective varieties over an algebraically closed field k of characteristic not equal to 2 whose closed fibres are all isomorphic to ^r P¹ and let ': X ^r P¹ be a surjective morphism. This article gives a sufficient condition concerning ' and Y under which X is isomorphic to Y× ^r P¹.     

A “from scratch” proof of a theorem of Rockafellar and Fulkerson

The theorem of this paper is of the same general class as Farkas' Lemma, Stiemke's Theorem, and the Kuhn—Fourier Theorem in the theory of linear inequalities. LetV be a vector subspace ofR ⁿ , and let intervalsI ¹,, I ⁿ of real numbers be prescribed. A necessary and sufficient condition is given for existence of a vector (x ¹ ,, x ⁿ ) inV such thatx ⁱ I ⁱ (i = 1, ,n); this condition involves the elementary vectors (nonzero vectors with minimal support) ofV . The proof of the theorem uses only elementary linear algebra.The author at present holds a Senior Scientist Award of the Alexander von Humboldt Stiftung.     

Minty Variational Inequalities,Increase-Along-Rays Property and Optimization<Superscript>1</Superscript>

Let E be a linear space, let K E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^,K ), which can be considered as a generalization of MVI (f ^,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^K ) and the property of f to increase-along-rays starting at x ^*, fIAR (K,x ^*). We prove that the GMVI (f ^,K ) with radially l.s.c. function f has a solution x ^* ker K if and only if fIAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^,K ) has a solution x ^*K, then x ^* is a global minimizer of the problem min f(x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

A phase transition for the coupled branching process

Summary We consider a particular Markov process _t ^u on ^S ,S= ⁿ . The random variable _t ^u (x) is interpreted as the number of particles atx at timet. The initial distribution of this process is a translation invariant measure withf(x)d<. The evolution is as follows: At rateb(x) a particle is born atx but moves instantaneously toy chosen with probabilityq(x, y). All particles at a site die at ratepd withp[0, 1],d, ⁺ and individual particles die independently from each other at rate (1–p)d. Every particle moves independently of everything else according to a continuous time random walk.We are mainly interested in the caseb=d andn3. The process exhibits a phase transition with respect to the parameterp: Forp<p ^* all weak limit points of ( _t ^µ ) ast still have particle density (x)d. Forp>p ^*, _t ^µ ) converges ast to the measure concentrated on the configuration identically 0. We calculatep ^* as well asp ⁽ⁿ⁾ , the points with the property that the extremal invariant measures have forp>p ⁽ⁿ⁾ infiniten-th moment of (x) and forp<p ⁽ⁿ⁾ finiten-th moment. We show the case 1>p ^*>p⁽²⁾>p⁽³⁾...p ⁽ⁿ⁾ >0, p⁽ⁿ⁾0 occurs for suitable values of the other parameters. Forp<p ⁽²⁾ we prove the system has a one parameter set of extremal invariant measures and we determine their domain of attraction. Part I contains statements of all results but only the proofs of the results about the process for values ofp withp<p ⁽²⁾ and the behaviour of then-th moments andp ⁽ⁿ⁾ .