On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit

We consider the time evolved states of the free motion _t (q, v)=(q+tv,v),q,v ^d , starting in some non-equilibrium state and look at the associated processX _t of fluctuations of the actual number _t/ () of particles of the realization in with velocities inB at timet/ around its mean as 0 (i.e., in the hydrodynamic limit). It is shown that under natural conditions on the initial state , especially a mixing condition in the space variables, for eacht the laws of the fluctuations become Gaussian in the hydrodynamic limit in the following sense: as 0, where denotes weak convergence and is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution is also given by the free motion in the sense that On the other hand we shall see that ast, whereP _z× is the Poisson process with intensity measurez·×, i.e., the equilibrium state for the free motion with particle densityz and velocity distribution . In the hydrodynamic limit this behaviour corresponds to the ergodic theorem for the fluctuation process: ast. Here is a centered Gaussian state describing the equilibrium fluctuations, i.e., the fluctuations ofP _z× . Thus we prove the central limit theorem for the ideal gas: fluctuations are Gaussian even in non-equilibrium. The proofs rest on an adaption of the method of moments for sequences of generalized fields.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

Characterizations of Translation Generalized Quadrangles

If x is a regular point of the generalizedquadrangle of order (s,t), s 1 t, then x defines a dual net . If contains a line L of regularpoints and if for at least one point x on Lthe automorphism group of the dual net satisfies certain transitivityproperties, then is a translation generalized quadrangle. Thisresult has many applications. We give one example. Ifs=t 1, then is a dual affine plane. Let be a generalizedquadrangle of orders,s odd and s 1, which contains a lineL of regular points. If for at least one pointx on L the plane is Desarguesian, then is isomorphic to the classical generalizedquadrangleW(s).     

Extremal solutions for nonlinear parabolic problems with discontinuities

This paper examines nonlinear parabolic initial-boundary value problems with a discontinuous forcing term, which is locally of bounded variation. Assuming that there exist an upper solution and a lower solution , we prove the existence of a maximal and of a minimal solution within the order interval [,] L ^P (P xZ). Our approach is based on a Jordan-type decomposition for the discontinuous forcing term and on a fixed point theorem for nondecreasing maps in ordered Banach spaces.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

Rates of convergence in the operator-stable limit theorem

Suppose that the ^d -valued random vector is strictly operator-stable in the sense that , the characteristic function of , satisfies for everyt<0, for some invertible linear operatorB on ^d . Suppose also that for the i.i.d. random vectors {X _i } in ^d , . In the present paper, we study the rates of convergence of this operator-stable limit theorem in terms of several probability metrics. A new type of ideal metrics suitable for this rate-of-convergence problem is introduced.This research was partially supported by NSF, Grant DMS-9103452 and NATO, Grant CRG900798.     

Projection in a Space of Solenoidal Vector Fields

Let R ³ be a bounded domain, 0$$ " align="middle" border="0"> , a family of extending subdomains, and =(x) a positive function in be a space of -solenoidal vector fields, 0$$ " align="middle" border="0"> , a family of subspaces, G orthogonal projectors in onto . A unitary transformation that diagonalizes the family of projectors {G} is constructed: it takes to the operator of multiplication by the independent variable. The isometry of this transformation is proved with the help of the operator Riccati equation for the NeumanntoDirichlet mapping. Bibliography: 8 titles.     

Irrationalité de Valeurs Associées a l'Exponentielle de Carlitz

Lete and be the Carlitz-module analogues of their usual counterparts. We have proved in [4]-that these elements of are algebraically independent over whenq3. We study here the remaining caseq=2 and prove among other things that 1,e, are linearly independent over .

     

The Bellman equation for time-optimal control of noncontrollable,nonlinear systems

For a general nonlinear system and closed target set we study the value functions and of the control problems of reaching and, respectively, its interior, in minimum time. Under no controllability assumptions on the system, we characterize them as, respectively, the minimal viscosity supersolution and the maximal viscosity subsolution of the Bellman equation with appropriate boundary conditions. Then we prove that is the unique upper semicontinuous complete solution of such a boundary value problem, which means in particular that the (completed) graph of contains the graph of any solution, as well as all the limits of reasonable approximating sequences. We give some applications to verifications theorems and to the stability of the minimum time function with respect to general perturbations.The authors are partially supported by the Italian National Projects Equazioni di evoluzione e applicazioni fisico-matematiche and Equazioni differenziali e calcolo delle variazioni, respectively.     

On Rounding Error Sums Related to the Circle Problem. ii

For a positive real parameter t, real numbers , , and , we consider sums , where is the rounding error function, i.e.\ . Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant such that for all , and all . Further, we give applications concerning the circle problem with linear, polynomial, and general weight.     

Numerical stability of the cyclic Richardson iteration

Summary The success of the cyclic Richardson iteration depends on the proper ordering of the acceleration parameters. We give a rigorous error analysis to show that, with the proper ordering, the relative error in the iterative method, when properly terminated, is not larger than the error incurred in stable direct methods such as Cholesky factorization. For both the computed approximation tou=L ^–1f satisfies cond (L)u2^–t and this bound is attainable. We also show that the residual norm is bounded by L cond . This bound is attainable for a small cycle lengthN. Our analysis suggests that for a larger cycle lengthN the residuals are bounded by . We construct a theoretical example in which this bound is attainable. However we observed in all numerical tests that ultimately the residual norms were of order . We explain why in practice even the factor is never encountered. Therefore the residual stopping criterion for the Richardson iteration appears to be very reliable and the method itself appears to be stable.The author gratefully acknowledges partial support from ONR Contract N00014-85-K-0180On leave from the University of Krakow, Poland, during the spring semester 1989     

On the Binomial Asymptotics of an Entire Dirichlet Series

Let M() be the maximum modulus and let () be the maximum term of an entire Dirichlet series with nonnegative exponents _n increasing to . We establish a condition for _n under which the relations

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (, ) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d() such that

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

A structure theorem for weak buildings of spherical type

Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building , there is canonically associated a thick building whose Weyl group W( ) can be considered as a reflection subgroup of the Weyl group W() of . One can reconstruct from together with the embedding W( ) W(). Conversely, if is any thick building and W any reflection group containing W( ) as a reflection subgroup, there exists a weak building with Weyl group W and associated thick building .     

A companion ideal of a multiplication module

Let M _R be a faithful multiplication module, where R is a commutative ring. As defined by Anderson, this ideal has proved to be useful in studying multiplication modules. First of all a cancellation law involving M and the ideals contained in is proved. Among various applications given, the following result is proved:: There exists a canonical isomorphism from onto such that for any ( Hom _R(M,M), x ( M, a ( (M), (xa) = x.(()(a). As an application of this later result it is proved that M is quasi-injective if and only if (M) is quasi-injective.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.