Models of set theory with definable ordinals

A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following:1. If T is a consistent completion of ZF+VOD, then T has continuum-many countable nonisomorphic Paris models.2. Every countable model of ZFC has a Paris generic extension.3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal there is a Paris model of ZF of cardinality which has a nontrivial automorphism.4. For a model ZF, is a prime model is a Paris model and satisfies AC is a minimal model. Moreover, Neither implication reverses assuming Con(ZF).Mathematics Subject Classification (2000): 03C62, 03C50, Secondary 03H99     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,), and assume that the moment map :XL ^* is proper. We consider the function ||²:X, and use a version of Morse theory to show that the inclusion map j:^-1(0)X induces a surjection j ^*:H _G ^*(X)H _G ^*(^-1(0)), in analogy with Kirwans surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.     

Moderate Deviations for the overlap parameter in the Hopfield model

We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature is different from the critical inverse temperature _c=1 and the number of patterns M(N) satisfies M(N)/N 0, the overlap parameter multiplied by N, 1/2 < < 1, obeys a moderate deviation principle with speed N^1–2 and a quadratic rate function (i.e. the Gaussian limit for = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N, 1/4 < < 1. If then M(N) satisfies (M(N)⁶ log N M(N)²N⁴ log N)/N 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N^1–4 and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at _c = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N^1/4, e.g. if we multiply the overlap parameter by N^1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If _N converges to _c fast enough, i.e. faster than the non-Gaussian rate function persists, whereas for _N converging to one slower than the moderate deviations principle is given by the Gaussian rate. At the borderline the moderate deviation rate function is the one at criticality plus an additional Gaussian term.Research supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach, Germany).Mathematics Subject Classification (2000): 60F10 (primary), 60K35, 82B44, 82D30 (secondary)     

Characters and a Verlinde-type formula for symmetric Hopf algebras

We study certain aspects of finite-dimensional non-semisimple symmetric Hopf algebras H and their duals H^*. We focus on the set I(H) of characters of projective H-modules which is an ideal of the algebra of cocommutative elements of H^*. This ideal corresponds via a symmetrizing form to the projective center (Higman ideal) of H which turns out to be ΛH, where Λ is an integral of H and is the left adjoint action of H on itself. We describe ΛH via primitive and central primitive idempotents of H. We also show that it is stable under the quantum Fourier transform. Our best results are obtained when H is a factorizable ribbon Hopf algebra over an algebraically closed field of characteristic 0. In this case ΛH is also the image of I(H) under a “translated” Drinfel'd map. We use this fact to prove the existence of a Steinberg-like character. The above ingredients are used to prove a Verlinde-type formula for ΛH.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

We prove that a complete embedded maximal surface in = (³, dx₁² + dx₂²-dx₃²) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x₃=0} in with n+12 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x₃=0}. Moreover, the position of the singular points in ³ and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space of marked graphs with n+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n–1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .Mathematics Subject Classification (2000): 53C50, 58D10, 53C42First and second authors research partially supported by MEC-FEDER grant number MTM2004-00160Second and third authors research partially supported by Consejería de Educación y Ciencia de la Junta de Andalucía and the European Union.     

A class of <Emphasis Type="Italic">p</Emphasis>-adic Galois representations arising from abelian varieties over

Let V be a p-adic representation of the absolute Galois group G of that becomes crystalline over a finite tame extension, and assume p2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module V_p() of an abelian variety defined over . These conditions are stated on the filtered (,G)-module attached to V.Mathematics Subject Classification (2000): 14F30, 11G10, 11F80, 14G20, 14F20     

<Emphasis Type="Italic">p</Emphasis>-adic properties of division polynomials and elliptic divisibility sequences

For a fixed rational point P E (K) on an elliptic curve, we consider the sequence of values (F_n (P))_n1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=p^e so that for all m1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.Mathematics Subject Classification (1991): 11G07, 11D61, 14G20, 14H52The authors research supported by NSA grant H98230-04-1-0064.     

An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Let a sequence of iid. random variables ξ ₁, . . . ,ξ _n be given on a space with distribution μ together with a nice class of functions f(x ₁, . . . ,x _k ) of k variables on the product space For all f ∈ we consider the random integral J _n,k (f) of the function f with respect to the k-fold product of the normalized signed measure where μ _n denotes the empirical measure defined by the random variables ξ ₁, . . . ,ξ _n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too. Supported by the OTKA foundation Nr. 037886     

Exchangeable measures for subshifts

Let Ω be a Borel subset of where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when . We apply the ergodic theory of equivalence relations to study the case , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0,1] (a non-Markovian constraint).     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the inequalities, and we establish necessary and sufficient conditions which guarantee that the inequalities are facet-defining. A separation heuristic for inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. This research has been supported in part by the National Science Foundation under Award number DMII-0121495.     

On Mean Convergence of Fourier-Bessel Series of Negative Order

The expansion of f ∈ L^p(0, 1) Fourier series of Bessel functions of order converges to f in L^p whenever Let be the space of p-integrable functions with respect to the measure t dt and where {s_n}, n = 1, 2, …, is the set of positive zeros of J_v. Then, the expansion of in a Fourier series of functions ψ_n, ?1 < ν < ?½, converges to in whenever      

Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane , . Let be the linear space over of the holomorphic functions f in Ω such that f⁽ⁿ⁾ is bounded in Ω and is continuously extendible to the closure of Ω, n=0,1,2,… . We endow , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

On the disconnection of a discrete cylinder by a random walk

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N ^{2 d} and comparable to the cover time of the slice , but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height . These mechanisms are in contrast with what happens when d=1.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.