Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Finite-dimensional attainable sets for stochastic control systems

Letx _t ^u () be a stochastic control system on the probability space (, ,P) intoR ⁿ. We say that the pointxR ⁿ is (, ) attainable at timet if there exists an admissible controlu such thatP _xo{x _t ^u ()S (x)}, wherex ₀()=x ₀, 0, 10, andS (x) is the closed Euclidean -ball inR ⁿ centered atx. We define the attainable setA (t) to be the set of all pointsxR ⁿ which are (, ) attainable at timet. For a large class of stochastic control systems, it is shown thatA (t) is compact for eacht and continuous as a function oft in an appropriate metric. From this, the existence of stochastic time-optional controls is established for a large class of nonlinear stochastic differential equations.This research was supported by the National Research Council of Canada, Grant No. A-9072.     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.     

On a theorem of G. Pólya

f(z), :f(n)=0 (n=0, ±1, ±2, ...). ((n)} L _p ,p>1, .     

Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

We show in this paper that iff is a quadratic infinitely many times renormalizable polynomial of sufficient high combinatorial type, then: HD (J(f))= inf{: -conformal measure for f} We use Lyubich's construction of the principal nest ([Lyu97]) in order to prove this result.Partially supported by CNP q-Brazil grant # 300534/96-5     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

L 2(R n )-extremal problems on the basis of incomplete information

. ( ), R ⁿ L ₂(R ²).

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The author is supported by the National Natural Science Found of China.     

On the stochastic independence properties of hard-core distributions

A probability measurep on the set of matchings in a graph (or, more generally 2-bounded hypergraph) ishard-core if for some : [0,), the probabilityp(M) ofM is proportional to . We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP () the matching polytope of , and >0, if the vector ofmarginals, (Pr(AM):A an edge of ), is in (1–) MP (), then the weights (A) are bounded by someA(). This eventually implies, for example, that under the same assumption, with fixed, as the distance betweenA, B tends to infinity.Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]–[13]).Supported in part by NSFThis work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.     

Factorization of One-Generated Formations

A large class of one-generated factorable foliated formations including composition, bicanonical, and other formations is described. V. A. Vedernikov suggested to study foliated formations with directions such that , where and are also some directions, from a unified point of view. We study one-generated factorable foliated formations with directions from the interval [,], where and are the directions of the bicanonical and composition formations, respectively. In particular, all irreducible factorizations of one-generated composition formations are studied.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Inversion and structure formulae for a generalized Laplace transform

, D_v(.) , .     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

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On the bestL p multipoint local approximation

P (f) — , f L ^p - , k . f 02k–2 P (f) 0.     

Convergence proofs and error estimates for an iterative method for conformal mapping

Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC ²⁺, the order of convergence is at least 1+. If is inC ^l+1+ withl1, 0<<1, then the iteration converges inC ^l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N ^1–l–) if is inC ^l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported.     

On the minimum modulus of a multiple Dirichlet series with monotone coefficients

We establish conditions under which the relation M(x, F) (x, F) m(x, F) holds except for a small set, as ¦x¦ + for an entire function F(z) of several complex variables z (p2) represented by a Dirichlet series, where M(x, F) = sup{¦F(x+iy¦: y ^p}, m(x, F) = inf{¦F(x+iy)¦: y ^p} (x, F) being the maximal term of the Dirichlet series, and x ^p.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 21–25.     

A generalization of second variation

f p- , l p . p=1 . . p - , f -.