On the Vacuum Polarization Density Caused by an External Field

We consider an external potential, – , due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density, as derived in physics literature, after a well-known renormalization procedure, depends decisively on the strength of . For small , more precisely as long as the lowest eigenvalue, e₁(), of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density vanishes. In other words the vacuum stays neutral. But as soon as e₁() dives into the lower continuum the vacuum gets spontaneously charged with charge 2e. Global charge conservation implies that two positrons were emitted out of the vacuum, this is, a large enough external potential can produce electron-positron pairs.We give a rigorous proof of that phenomenon.Communicated by Rafael D. Benguriasubmitted 18/12/03, accepted 08/04/04     

Preassigning the boundary of diametrically-complete sets

In a Minkowski space with unit ballS, a setD() is diametrically complete ifD()={. In this paper a method is given for generating such setsD() which contain an arbitrary setX. This technique is then employed to find necessary and sufficient conditions for a setA to lie on the boundary of some suchD() and bounds are given on the acceptable values of .     

Transformation de Fourier et temps d'occupation browniens

Summary In this paper, we study oscillatory stochastic integrals of the form where is a non zero parameter andg a square integrable function. We study integrability properties of () and its behavior as a function of , using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables () based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.We consider the random measure, image by the Brownian motion of the unbounded measure 1_[0,] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.Finally, under a stronger integrability condition ong, we show the existence of a density for the law of (), using Malliavin's calculus.     

Separation of two (possibly unbounded) components of the spectrum of a linear operator

LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)^–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X ₊X _–, whereX _± are closed subspaces, invariant forA, the spectra of the reduced operatorsA _± are {(A);Im>0} and {(A);Im<0} respectively, and (–A _±)^–1 is bounded forIm0.Finally we give an example of an operator in anL ^p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1.     

Cardinal Invariants of λ-Topologies

We study the basic cardinal-valued invariants of C (X) such as weight, density, network weight, i-weight, and tightness, where C (X) is the space of all continuous real functions on X in the -topology.     

Verma-Type Modules for Quantum Affine Lie Algebras

Let be an untwisted affine Kac–Moody algebra and M_J() a Verma-type module for with J-highest weight P. We construct quantum Verma-type modules M_J^q() over the quantum group , investigate their properties and show that M_J^q() is a true quantum deformation of M_J() in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules. Presented by A. VerschorenMathematics Subject Classifications (2000) 17B37, 17B67, 81R50.The first author is a Regular Associate of the ICTP. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

Exponential estimates for the Wiener sausage

Summary LetC (t) be the Wiener sausage of radius inR ^d up to timet. We obtain bounds on the asymptotics ofE exp (|C (t)|) ast, for all >0.     

Prediction Problems for Square-Transformed Stationary Processes

This paper discusses the prediction problems for square-transformed process, Y _t = X _t ², where X _t is a stationary process with spectral density g(). The square-transformation is important in prediction of the volatility of ARCH models. First, we evaluate the mean square prediction error for square-transformed process when the predictor is constructed from the true spectral density g(). However, it is often that the true structure g() is not completely specified. Hence, we consider the problem of misspecified prediction when a conjectured spectral density f (), , is fitted to g(). Then, constructing the best linear predictor based on f (), we can evaluate the prediction error for square-transformed process. Also, we consider a bias adjusted prediction problem for the above two cases. Furthermore, we may suppose that X _t is a non-Gaussian process. Then, we evaluate the mean square prediction errors when the best linear predictor is constructed by the true spectral density g() and the conjectured spectral density f (), respectively. Since is usually unknown we estimate it by a quasi-MLE . The second-order asymptotic approximations of the mean square errors of the predictors based on g() and f () are given. Finally, we provide some numerical examples, which show some unexpected features.     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

A Result on Ext Over Kac–Moody Algebras

We prove the following result for a not necessarily symmetrizable Kac–Moody algebra: Let x,y W with x y, and let P⁺. If n=l(x)-l(y), then Ext _C() ⁿ (M(x·),L(y·))=1.     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

Vector measure range duality and factorizations of (<Emphasis Type="Italic">D,p</Emphasis>)-summing operators from Banach function spaces

We characterize the relationship between the space L₁() and the dual L₁() of the space L₁(), where (, ) is a dual pair of vector measures with associated spaces of integrable functions L₁() and L₁() respectively. Since the result is rather restrictive, we introduce the notion of range duality in order to obtain factorizations of operators from Banach function spaces that are dominated by the integration map associated to the vector measure . We obtain in this way a generalization of the Grothendieck-Pietsch Theorem for p-summing operators.*The research was partially supported by MCYT DGI project BFM 2001-2670.**The research was partially supported by MCYT DGI project BFM 2000-1111.     

A minimax test for hypotheses on a spectral density

Assume that on the set of the values t = 1, , k there is given a realization of the Gaussian stationary process X_t with spectral density f(), (0, 1). There arises the problem of the minimax testing of the hypothesis H₀: f() = p().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 115–125, 1990.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Boundary Modified Contact Processes

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate _e (where e is for external). Other births occur at rate _i (where i is for internal). Deaths occur at rate 1. The case _e= _i is the well known basic contact process for which there is a critical value _c>1 such that if the birth rate is larger than _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if _e1 then the process always dies out while if _e>1 and if _i is large enough then the process may survive. We also show that if _i< _c the process dies out for all _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.