Coproducts in the Category <Emphasis Type="Bold">M</Emphasis>κ<Emphasis Type="Bold">Frm</Emphasis>

The purpose of this paper is to study coproducts in the category MFrm (resp. M_cFrm), of metric -frames and uniform (resp. contractive) -frame maps. First, by applying the same technic that was used to find coproducts in Frm, we construct coproducts in the category Frm of -frames and -frame maps. Then, we define a metric diameter on the coproduct in Frm of a family of metric -frames and show that coproduct in Frm preserves metrizability.Mathematics Subject Classifications (2000) 06B23, 06D22, 18A30.     

Der Beweis eines Satzes von G. Choodnovsky

1. If is a weakly compact cardinal then ( ⁺)( ). 2. If is measurable andU a normal ultrafilter then ( ⁺)(U ).

---

---

Diese Arbeit ist ein Teil der Dissertation des Autors. Teilweise gefördert von der DFG.     

Quasi-coproducts and accessible categories with wide pullbacks

We establish a 2-categorical duality involving the 2-category A of all -accessible categories with wide pullbacks, also known as locally -polypresentable categories, and of functors preserving -filtered colimits and wide pullbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt (A, Set)=A The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Mad families, forcing and the Suslin Hypothesis

Let be a regular cardinal and P a partial ordering preserving the regularity of . If P is (-Baire and) of density , then there is a mad family on killed in all generic extensions (if and) only if below each pP there exists a -sized antichain. In this case a mad family on is killed (if and) only if there exists an injection from onto a dense subset of Ult(P) mapping the elements of onto nowhere dense sets. If 2^<=, then in each generic extension of V, in which is the minimal cardinal obtaining new subsets, some mad family on is killed or an independent subset of appears. Also, the -Suslin Hypothesis holds iff there exists a mad family on which is killed in each generic extension containing new subsets of and preserving P() for <.Mathematics Subject Classification (2000): 03E35, 03E40, 03E65     

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.     

λ-Automorphisms of a Riemannian Foliation II

In Pak (Ann. Global Anal. Geom.13 (1995), 281–288), the notion of -automorphisms of a harmonic Riemannian foliation was extended to a general Riemannian foliation. By applying the characterization obtained in Pak of a -automorphism to be transversal Killing, we here consider the problem that a transversal conformal or projective field is transversal Killing. A special condition, namely _{B B} = 0, is discussed.     

Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP

A large-step infeasible path-following method is proposed for solving general linear complementarity problems with sufficient matrices. If the problem has a solution, the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. The algorithm generates points in a large neighborhood of the central path. Each iteration requires only one matrix factorization and at most three (asymptotically only two) backsolves. It has been recently proved that any sufficient matrix is a P _*()-matrix for some 0. The computational complexity of the algorithm depends on as well as on a feasibility measure of the starting point. If the starting point is feasible or close to being feasible, then the iteration complexity is . Otherwise, for arbitrary positive and large enough starting points, the iteration complexity is O((1 + )² nL). We note that, while computational complexity depends on , the algorithm itself does not.     

Some remarks on the behaviour of the solution in dynamic processes for rate-type models

This work examines a dynamic problem in the study of semilinear rate-type models for which the plastic rate of deformation depends also on a parameter . The continuous dependence of the solution with respect to is obtained and the problem of finite time stability is also discussed. In the case when is interpreted as the absolute temperature, the dynamic problem is studied in the context of a Cattaneo-type heat law and also using the classical Fourier law. In the case when is interpreted as an internal state variable an existence and uniqueness result is obtained using a fixed point method and the finite time stability is also investigated.     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

Random time change and an integral representation for marked stopping times

Summary Consider the set of all possible distributions of triples (, , ), such that is a finite stopping time with associated mark in some fixed Polish space, while is the compensator random measure of (, ). We prove that is convex, and that the extreme points of are the distributions obtained when the underlying filtration is the one induced by (, ). Moreover, every element of has a corresponding unique integral representation. The proof is based on the peculiar fact that EV _, =0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT _, isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of , a suitably discounted version of , is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.Research supported by NSF grant DMS-8703804     

Spinc Structures and Scalar Curvature Estimates

In this note, we look at estimates for the scalar curvature of a compact, connected Riemannian manifold Mwhich are related to spin ^c Dirac operators.We show that one may not enlarge a Kähler metric with positiveRicci curvature without making smaller somewhere on M.More generally, if f: N M is an area-nonincreasing map of a certain topological type,then the scalar curvature k of Ncannot be everywhere larger than f.If k f, then N is isometric to M × F, where F possesses a parallel untwisted spinor.We also give explicit upper bounds for min for arbitrary Riemannian metrics on certainsubmanifolds of complex projective space.In certain cases, these estimates are sharp:we give examples where equality is obtained.     

Über das Modell einer Nicht-Newtonschen Flüssigkeit mit Relaxationseigenschaften

This paper deals with the equations ·u=o,u – u – u+u · u+p=f, describing a Non-Newtonian liquid with velocityu, pressure p under the external forcef. The difference to the Navier-Stokes equations consists in the fact that it is assumed here that the external forcef only has an effect on the volume-element of the liquid after the relaxation-time /.     

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

Stochastic Loewner evolution in doubly connected domains

This paper introduces the annulus SLE processes in doubly connected domains. Annulus SLE₆ has the same law as stopped radial SLE₆, up to a time-change. For 6, some weak equivalence relation exists between annulus SLE and radial SLE. Annulus SLE₂ is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE₂ satisfies the reversibility property. We also consider the disc SLE process defined as a limiting case of the annulus SLEs. Disc SLE₆ has the same law as stopped full plane SLE₆, up to a time-change. Disc SLE₂ is the scaling limit of loop-erased random walk, and is the reversal of radial SLE₂.     

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.     

Estimation of an Exponential Quantile under a General Loss and an Alternative Estimator under Quadratic Loss

Estimation of the quantile + of an exponential distribution with parameters (, ) is considered under an arbitrary strictly convex loss function. For obeying a certain condition, the inadmissibility of the best affine equivariant procedure is established by exhibiting a better estimator. The LINEX loss is studied in detail. For quadratic loss, sufficient conditions are given for a scale equivariant estimator to dominate the best affine equivariant one and, when exceeds a lower bound specified below, a new minimax estimator is identified.     

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.     

Some remarks on indestructibility and Hamkins’ lottery preparation

In this paper, we first prove several general theorems about strongness, supercompactness, and indestructibility, along the way giving some new applications of Hamkins lottery preparation forcing to indestructibility. We then show that it is consistent, relative to the existence of cardinals < so that is supercompact and is inaccessible, for the least strongly compact cardinal to be the least strong cardinal and to have its strongness, but not its strong compactness, indestructible under -strategically closed forcing. Mathematics Subject Classification (2000):03E35, 03E55     

On covering designs with block size 5 and index 5

LetV be a finite set of order . A (, , ) covering design of index and block size is a collection of -element subsets, called blocks, such that every 2-subset ofV occurs in at least blocks. The covering problem is to determine the minimum number of blocks, (, , ), in a covering design. It is well known that , where [x] is the smallest integer satisfyingx[X]. It is shown here that (, 5, 5)=(, 5, 5) for all positive integers 5 with the possible exception of =24, 28, 56, 104, 124, 144, 164, 184.     

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.