Das 0-1-Gesetz der terminalen σ-Algebra bei Harrisirrfahrten

Für homogene M.K., bestehend aus einer aperiodischen rekurrenten Klasse, ist bekannt, da\ die terminale -Algebra das 0–1-Gesetz erfüllt. Der erste Teil dieser Arbeit behandelt die Gültigkeit des 0–1-Gesetzes für transiente Ketten, für die eine Folge ₀, ₁, ₂, ₃,... von ZustÄnden existiert mit P(nX_n= _l+1¦X₀= _l) = 1, N. Das herzuleitende Kriterium wird nur von dieser Folge und den Stoppzeiten, von _l nach _l+1 zu gelangen, abhÄngen. Eine Anwendung auf Harrisirrfahrten wird uns dort konkrete Aussagen liefern.     

On the Densities of Certain Lattice Packings by Parallelepipeds

Given any (non-degenerate) n-dimensional lattice L, let (L) denote the supremum of the numbers such that there exists a lattice packing Q + L of density where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density _n taken over all n-dimensional lattices. It is known that 0$$ " align="middle" border="0"> . Here we investigate a sequence of lattices L _n which are known to minimize the function (L) in dimensions n 3 and are likely to provide the minima _n = (L _n ) in certain higher dimensions. We establish the inequality (L _n ) n ^–n/2 which supports the conjecture that lim sup _n ( _n )^{1/(n log n)} is positive.     

On the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.     

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.     

Dirichlet Series Associated with Strongly q-Multiplicative Functions

In this work the Dirichlet series associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case _i=0 ^q–1 f(i) = 0. It is known that in this case the function _f (s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function _f (t) satisfying the integral equation with delayed argument for some nonzero real _f naturally appears in the representation of the function _f (s). In this article we find some asymptotic properties of the function _f (s), prove that _f (s) is an entire function of order 2, and also prove that in the region the function _f (s) has only trivial zeros which are simple.     

Quadrature formulas obtained by variable transformation

Quadrature formulas suitable for evaluation of improper integrals such as are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals by means of variable transformations =sinhu and =tanu.     

A note on extensions of infinitary logic

We show that a strong form of the so called Lindströms Theorem [4] fails to generalize to extensions of L and L : For weakly compact there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to . With an additional set-theoretic assumption, there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to <.We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions.Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726]Research partially supported by grant 40734 of the Academy of Finland.     

On the free subset property at singular cardinals

We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, ₁)holds. Assume is singular. Then there is an inner model with ₁ measurable cardinals.     

Sequential completeness of subspaces of products of two cardinals

Let be a cardinal number with the usual order topology. We prove that all subspaces of ² are weakly sequentially complete and, as a corollary, all subspaces of are sequentially complete. Moreover we show that a subspace of (₁ + 1)² need not be sequentially complete, but note that X = A × B is sequentially complete whenever A and B are subspaces of .     

The GB and GC properties of generalized elliposids

In the paper one investigates the conditions under which the subsets of the form of a Hilbert space, where a0, p>1, =1, ..., possesses the GB or the GC properties.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 104–108, 1983.The author is grateful to V. N. Sudakov for his interest in the paper.     

On unfoldable cardinals, ω-closed cardinals,and the beginning of the inner model hierarchy

Let be a cardinal, and let H be the class of sets of hereditary cardinality less than ; let () > be the height of the smallest transitive admissible set containing every element of {}H. We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H is as long as . (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: TheoremCon(ZFC+²- ¹₁-Determinacy) Con(ZFC+V=K+ a long unfoldable cardinal Con(ZFC+X(X^# exists) + is universally Baire rR(DL(r))), and this is set-generically absolute). We isolate a notion of -closed cardinal which is weaker than an ₁-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let be -closed. Then there is a long unfoldable <.Mathematics Subject Classification (2000): 03E45, 03E15, 03E55, 03E60The author wishes to gratefully acknowledge support from Nato Grant PST.CLG 975324.     

On graphs that can be oriented as diagrams of ordered sets

We study some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set. For each 1, M. Aigner and G. Prins have introduced a notion of a vertex colouring, here called -good colouring, such that a 1-good colouring is the usual concept and graphs that have a 2-good colouring are precisely covering graphs. We present some inequalities for the corresponding chromatic numbers , especially for x ₂. There exist graphs that satisfy these inequalities for =2 but are not covering graphs. We show also that x ₂ cannot be bounded by a function of x=x ₁. A construction of Neetil and Rödl is used to show that x ₂ is not bounded by a function of the girth.     

Interpolation of monotone functions in lattices

We show that for every latticeL ₀ and for every cardinal there is a lattice on which every monotone function can be interpolated by a polynomial on any set of size .Presented by G. Grätzer.Dedicated to '"N     

On ηα-groups and fields

Let :=. The following are known: two -sets of power are isomorphic. Let >0. Two ordered divisible Abelian groups that are -sets of power are isomorphic, two real closed fields that are -sets of power are isomorphic. The following is shown: (1) there exist 2 nonisomorphic ordered Abelian groups (respectively ordered fields) that are -sets of power ; (2) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are -sets having the same order type.     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

On the concept of code-isomorphy

LetF be a finite set of cardinality ¦F¦ =q 2,n 1 an integer and :F ⁿ×Fⁿ₀ theHamming metric. Acode isomorphism C D between two block codesC,D Fⁿ is defined as an isometry which can be extended to an isometry of the whole space Fⁿ. Any permutation S _n of the positions canonically induces a so-calledequivalence map Aut Fⁿ; any system (₁,₂,...,_n) ofn permutations of the character setF induces a so-calledconfiguration Aat Fⁿ. The group Aut Fⁿ of all isometries of Fⁿ turns out to be semidirect product of the configuration group with the symmetric group of degreen. The codeword estimating failure probability of a maximum likelihood codeword estimator for aq-nary symmetric channel does not depend on the transmitted codeword, if the automorphism group of the code acts transitively on the set of codewords. When using a systematic (n, k)-encoder, the symbol decoding failure probability does not depend on the transmitted symbol or on the time of transmission if the configuration group and the automorphism group act transitively on the set of codewords resp. on the set of thek information positions.In memoriam Giuseppe Tallini     

The Busy Period of the M/GI/∞ Queue

An equation for the distribution Z() of the duration T of the busy period in a stationary M/GI/ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re()>₀ for some ₀0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace–Stieltjes transform E(e^–S ). If ₀<0 then E(e^–T ) is analytic in a half-plane (₁,), where ₀₁<0 and ₁ is determined by the distribution of S; then for any 0<s<|₁|.When ₀=0, E(e^–T ) is analytic in (0,), and now more is known about T. Inequalities on the tail () are used to show that for any 1, E(T ) is finite if and only if E(S ) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S ²)=, in which case it has Hurst index equal to [frac12](3–), where is the moment index of S.If also the tail (x)=Pr{Sx} of the service time distribution satisfies the subexponential density condition ₀ ^x (x–u) (u)du/ (x)2E(S) as x, then (x)/ (x)e^E(S), where is the arrival rate.