Diamond, square, and level by level equivalence

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, holds for every regular uncountable cardinal , and below the least supercompact cardinal , holds on a stationary subset of . There are no restrictions in our model on the structure of the class of supercompact cardinals.The author wishes to thank the referee for numerous helpful comments and suggestions, which have considerably improved the presentation of the material contained herein. The author also wishes to thank Andreas Blass, the corresponding editor, for a useful suggestion, and Grigor Sargsyan for a very helpful conversation on the subject matter of this paper.Mathematics Subject Classification (2000): 03E35, 03E55     

Division Algebras that Ramify only on a Plane Quartic Curve with Simply Connected Components

A central division algebra over the field of rational functions in two variables with coefficients over an algebraically closed field ramifies along a divisor on P ². If the ramification divisor of is a quartic curve which is the union of simply connected curves, we show that is a symbol algebra and satisfies the index equals exponent equation.     

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     

Periodic Algebras which are Almost Koszul

The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such almost Koszul algebras is developed and other examples are given.     

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.     

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.     

Minimal Characteristic Algebras for Some Properties of Identities

We consider algebras of a type , without nullary fundamental operation symbols. A structural property p of an identity of type is hereditary if for every set I of identities of type having the property p every consequence of I (derived identity) has the property p. An algebra of type we call characteristic for a hereditary property p if for every variety V of type we have: V if and only if every identity from Id(V) has the property p. In this paper we show minimal characteristic algebras for several hereditary properties, e.g., to be regular, to be normal, etc.1991 Mathematics Subject Classification 08B05     

Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [–^–, ^–], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0⁺, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as the center of the instanton is far from the endpoints of the interval. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit 0⁺ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.     

Products of chains with monochromatic maximal chains and antichains

There is a product of two linear orders of size with the property that every subset or complement thereof contains a maximal chain. Furthermore, for regular , there is a product of two linear orders of size ₊₂ that when colored with fewer than colors always has a monochromatic maximal chain. As a corollary, for every uncountable strong limit cardinal , there is an ordered set of cardinality that must be colored with at least colors before no monochromatic maximal chains are present. Duals of these results show that at least as much is true for maximal antichains.Research supported in part by ONR Grant N00014-91-J-1150.     

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 ).

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

Review of recent results about theκ-centrum of a tree

This paper is concerned with the-centrum of a graph. This concept, related to a particular location problem, generalizes that of the center and that of the median of a graph: the-centrum is the set of points for which the sum of the (weighted) distances from the farthest vertices is minimized. The paper will review some recent results about this problem. In particular, some properties of cardinality, connectivity and, more generally, of the structure of the-centrum of a weighted tree will be presented.     

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 unconditional convergence of Walsh series

. { (n)} — , {_n}, .     

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     

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.     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

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.     

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.     

Diagonal Bases in Orlik-Solomon Type Algebras

To encode an important property of the no broken circuit bases of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called diagonal basis. We prove that this definition extends naturally to a large class of algebras, the so called - algebras. Our definitions make also use of an iterative residue formula based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.AMS Subject Classification: 52C35, 05B35, 14F40.     

Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a finite-dimensional hereditary algebra over a finite field k, () and () be, respectively, the Hall algebra and the composition algebra of , be the isomorphism classes of finite dimensional -modules and I the isomorphism classes of simple -modules. We define and , in , to be the right and left derivations of () respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of -mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u as the combinations of elements u_i in the Hall algebra, where i I and in is any exceptional -module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.