An example of an irregular random measure

Summary We consider the set of random measures which consists of measurable maps from [0, 1] to the set of measures on . As it is the dual space ofL ₁ ([0, 1];C()), we can equip this space with the weak^* topology. We construct a special random measure , which appears as the weak^* limit of a sequence of Dirac random measures , where (X _n ) _n is a bounded sequence inL _p [0, 1], (1p<2). The special form of this random measure, which oscillates randomly between twoq-stable standard measures on with different normalizations (p<q<2) allows us to prove two properties of (X _n ) _n is equivalent to the unit vector basis ofl _q and has no almost symmetric subsequence.     

On the Number of Solutions of Polynomial Systems

We consider systems of homogenous polynomial equations of degreedin a projective space ^mover a finite field _q. We attempt to determine the maximum possible number of solutions of such systems. The complete answer for the caser= 2,d<q− 1 is given, as well as new conjectures about the general case. We also prove a bound on the number of points of an algebraic set of given codimension and degree. We also discuss an application of our results to coding theory, namely to the problem of computing generalized Hamming weights forq-ary projective Reed–Muller codes.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant     

On the Structure of Spaces of Polyanalytic Functions

Suppose that A_mL_p(D,) is the space of all m-analytic functions on the disk D={z:|z| < 1} which are pth power integrable over area with the weight (1-|z|²), > -1. In the paper, we introduce subspaces A_kL_p ⁰(D,), k=1,2,...,m, of the space A _mL_p(D,) and prove that A _mL_p(D,) is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.     

On the inverse of the generator of a bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let A be the generator of a uniformly bounded C ₀-semigroup in a Banach space B, and let A have a densely defined inverse A ^-1 . We present sufficient conditions on the resolvent (A-I ^-1, Re > 0, under which A ^-1 is also the generator of a uniformly bounded C ₀-semigroup.Dedicated to V. B. Lidskii on the occasion of his eightieth birthdayTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 6–12, 2004Original Russian Text Copyright © by A. M. Gomilko     

Approximation of convex bodies by simplices

This paper deals with the following kind of approximation of a convex bodyQ in Euclidean space E ⁿ by simplices: which is the smallest positive numberh _S(Q) such thatS ₁ Q S ₂ for a simplexS ₁ and its homothetic copyS ₂ of ratioh _S(Q). It is shown that ifS ₀ is a simplex of maximal volume contained inQ, then a homothetic copy ofS ₀ of ratio 13/3 containsQ.     

A family of metrics on the moduli space of BPST-instantons

A family {g ^s} _s0 of Riemannian metrics on the moduli space of BPST-instantons is considered. Explicit expressions of g ^s are given and conclusions concerning the geometry of the instanton space are deduced.     

Witten deformation of analytic torsion and the spectral sequence of a filtration

LetF be a flat vector bundle over a compact Riemannian manifoldM and letf :M be a Morse function. Letg ^F be a smooth Euclidean metric onF, letg _t ^F =e ^–2tf g ^F , and let ^RS (t) be the Ray-Singer analytic torsion ofF associated with the metricg _t ^F . Assuming that f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ^RS (t) fort+ of the forma ₀+a ₁ t+blog(t/)+o(1), where the coefficientb is a half-integer depending only on the Betti numbers ofF. In the case where all the critical values off are rational, we calculate the coefficientsa ₀ anda ₁ explicitly in terms of the spectral sequence of a filtration associated with the Morse function. These results are obtained as applications of a theorem by Bismut and Zhang.The research was supported by grant No. 449/94-1 from the Israel Academy of Sciences and Humanities.     

Note on function spaces with the topology of pointwise convergence

The note contains two examples of function spaces C _p (X) endowed with the pointwise topology. The first example is C _p (M), M being a planar continuum, such that C _p (M) ^m is uniformly homeomorphic to C _p (M) ⁿ if and only if m = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space C _p (X), where X is a monolithic perfectly normal compact space all linearly orderable closed subspaces of which are metrizable. This example is obtained under the additional set-theoretical axiom . This solves a problem of Arhangelskiĭ.     

A remark on normal derivations of Hilbert-Schmidt type

LetB (H) denote the algebra of operators on the separable Hilbert spaceH. LetC ₂ denote the (Hilbert) space of Hilbert-Schmidt operators onH, with norm .₂ defined by S ₂ ² =(S,S)=tr(SS ^*). GivenA, B B (H), define the derivationC (A, B):B(H)B(H) byC(A, B)X=AX-XB. We show that C(A,B)X+S ₂ ² =C(A,B)X ₂ ² +S ₂ ² holds for allXB(H) and for everySC ₂ such thatC(A, B)S=0 if and only if reducesA, ker S reducesB, andA | S and B| ker S are unitarily equivalent normal operators. We also show that ifA, BB(H) are contractions andR(A, B)B(H)B(H) is defined byR(A, B)X=AXB-X, thenSC ₂ andR(A, B)S=0 imply R(A,B)X+S ₂ ² =R(A,B)X ₂ ² +S ₂ ² for allXB(H).     

On the Prandtl Equation

The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis ⁺ = [0, ) is proved in the Sobolev space W _p ¹ and Bessel potential spaces H _p ^s under certain restrictions on p and s.     

Rate of expansion of an inhomogeneous branching process of brownian particles

Summary Let X be the (B ⁰, {q _n (x)})-branching diffusion where B ⁰is the exp -subprocess of BM(R¹) and q _n (x) is the probability that a particle dying at x produces n offspring, q ₀ q ₁0. Put m(x) = nq _n (x). We assume q _n , n2, m and k are all continuous (but m is not necessarily bounded). If k(x)m(x)0 as ¦x¦, then we prove that R _t /t( ₂/2)^1/2, as t, a.s. and in mean (of any order) where R _t is the position of the rightmost particle at time t and ₀ is the largest eigenvalue of (1/2)d ²/dx ² + Q, Q(x) = k(x)(m(x)–1).This work was supported in part by a grant from the National Science Foundation # MCS-8201470.     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

On the typicality of linear Hamiltonian systems with exponential separateness

Linear Hamiltonian systems of ordinary differential equations,are considered, where f ^t is a continuous action of the group R on a complete metric space , A is an element of the space S _H ,consisting of the continuous bounded mappings of into the set of all pseudosymmetric matrices and endowed with the uniform convergence metric. By ₁ (A, x) _2n (A, x) we denote the Lyapunov exponents of such systems. The typicality (in the Baire sense) in the space S _H × is proved for those pairs (A, x) for which one has the alternative: either _k (A, x) = _k+1 (A, x) or the linear subspace of the solutions of the corresponding system with exponents less than _k (A, x) is exponentially separated from any of its algebraic complements in the space of all the solutions of the system. From here, in particular, there follows the typicality of the formulated alternative for linear Hamiltonian systems with continuous quasiperiodic coefficients (with the same frequency module) and also for linear Hamiltonian systems with continuous almost periodic coefficients. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 137–181, 1992.     

On the negative cyclic homology of <Emphasis Type="Italic">shc</Emphasis>-algebras

Let be a field of characteristic and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC _*⁻ A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC _*⁻ A is isomorphic as a graded algebra to the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S ²ⁿ when p = 2.      

The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

This paper is devoted to a study of the properties of the equationA ^*FA–F=–G, where FL() is unknown, AL(), GL() is positive and is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemx _k+1=Ax _k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rⁿ is considered.This work was supported in part by the Polish Academy of Sciences under the contract Problem Miedzyresortowy I.1, Grupa Tematyczna 3 This paper was written while the author was with the Instytut Automatyki, the same university.     

A Criterion of Sign Definiteness for Quadratic and Quasi-Quadratic Forms in a Cone

A new criterion of sign definiteness is given for quadratic and quasi-quadratic forms in a cone in space ⁿ . The basic results are proved for the nonnegative cone K ₊ ⁿ : x 0, x ⁿ . A change of variables reduces the case of an arbitrary octant of space ⁿ to K ₊ ⁿ .