Proper bayes minimax estimators for a multivariate normal mean with unknown common variance under a convex loss function

Summary Let X ∼ N_p(μ,σ²I_p) and let s/σ² ∼ χ _n ² , independent ofX, where μ and σ² are unknown. This paper considers the estimation of μ (by δ) relative to a convex loss function given by (δ−μ)′[(1−α)I_p/σ²+αQ](δ−μ)/[(1−α)p/σ²+α tr (Q)], whereQ is a knownp×p diagonal matrix and 0≦α≦1. Two classes of minimax estimators are obtained for μ whenp≦3; the first is a new result and the second is a generalization of a result of Strawderman (1973,Ann. Statist.,1, 1189–1194). A proper Bayes estimator is also obtained which is shown to satisfy the conditions of the second class of minimax estimators. The paper concludes by discussing the estimation of μ relative to another convex loss function. This work was supported by the Army, Navy and Air Force under Office of Naval Research Contract No. N00014-80-C-0093. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

The <Emphasis Type="Italic">p</Emphasis>-rank of Ext<Subscript>ℤ</Subscript>(<Emphasis Type="Italic">G</Emphasis>, ℤ) in certain models of <Emphasis Type="Italic">ZFC</Emphasis>

We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext _ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R _Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ˉρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E ₀ of , then defines a smooth l-adic analytic variety, near the trivial lift ρ₀ of ˉρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R _Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ˉρ odd, using ideas of Coleman, Gouvêa and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ₀, provided this lift satisfies the Artin conjecture and E ₀ satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R _Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case. Received: 12 December 1997 / Revised version: 5 February 1998     

The monodromy of a series of hypersurface singularities

Let {f=0} be a hypersurface inC ⁿ⁺¹ with a 1-dimensional singular set Σ. We consider the series of hypersurfaces {f+ɛx ^N=0} wherex is a generic linear form. We derive a formula, which relates the characteristic polynomials of the monodromies off andf+ɛx ^N. Other ingredients in this formula are the horizontal and the vertical monodromies of the transversal (isolated) singularities on each branch of the singular set. We use polar curves and the carrousel method in the proof. The formula is a generalization of the Iomdin formula for the Milnor numbers: μ(f+ɛx ^N )=μ _n (f)−μ _n −1(f)+Ne ₀(Σ)     

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ³ is considered. It is assumed that the permittivity η^ε(x)=η(ε⁻x), ε > 0, is a rapidly oscillating positive matrix function and the permeability μ₀ is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L ₂(ℝ³)-norm with order-sharp remainder estimates. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 3–23, 2007 Original Russian Text Copyright ? by M. Sh. Birman and T. A. Suslina Dedicated to the memory of the great mathematician Mark Grigor’evich Krein Supported by RFBR grants no. 05-01-01076-a, 05-01-02944-YaF-a.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

The condensate 〈(A µ 2 )〉 in commutative and noncommutative theories

We show that the gauge invariance of the operator ∫ dx tr(A _μ ² −2/(gξ)x ^υθ_μυ A ^μ) in a noncommutative gauge theory does not lead to the gauge independence of its vacuum condensate. We obtain the generalized Ward identities for Green’s functions containing the operator lim_Ω→∞(1/Ω)∫_Ω dx tr (A _μ ² ) in commutative and noncommutative gauge theories. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 350–356, September, 2006.     

Universally L 1 good sequences with gaps tending to infinity

We construct a sequence (n _k ) such that n _{k + 1} − n _k → ∞ and for any ergodic dynamical system (X, Σ, μ, T) and f ε L ¹(μ) the averages converge to ∝ _X f dμ for μ almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the nonexistence of such sequences. Research supported by the Hungarian National Foundation for Scientific research T049727.     

On isomorphism and inclusion property for certain £(p,λ) spaces of strong type

Summary We prove isomorphism and inclusion theorems for certain ￡^(p,λ) spaces of strong type introduced by G. Stampacchia. These results are quite analogous to those of S. Campanato and G. N. Meyers[1], [4], F. John and L. Nirenberg[3] and L. C. Piccinini[9]. Entrata in Redazione il 14 luglio 1976.     

Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Let ϕ_t(x), x ∈ ℝ₊ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process μ_t = μ ○ ϕ _t ⁻¹ that is the image of a certain absolutely continuous measure μ under random mapping ϕ_t(·). We prove that the restriction of the Hausdorff measure H ^d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to H ^d−1. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.     

Continuously Removable Sets for Quasiconformal Mappings

Let D be a domain in the n-dimensional Euclidean space Rⁿ, n ≥ 2, and let E be a compact in D. The paper presents conditions on the compact E under which any homeomorphic mapping f = D ∖ E → Rⁿ can be extended to a continuous mapping f = D → Rˉⁿ = Rⁿ ⋃ {∞}. These conditions define the class of NCS-compacts, which, for n = 2, coincides with the class of topologically removable compacts for conformal and quasiconformal mappings. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 213–220.     

Terzo problema al contorno per una classe di equazioni ellittiche del secondo ordine a coefficienti discontinui

Summary The solvability of the boundary value problem Lu=f a.e. in Ω, u ε H²(Ω), ∂u/∂β+μu=0 on ∂Δ is studied, where L is a linear second order elliptic partial differential operator in non-divergence form with discontinuous coefficients, and β is a vector field which forms with the normal axe to ∂Ω an angle smaller than π/2.

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Entrata in Redazione il 6 febbraio 1976.

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Lavoro eseguito nell’ambito del ? Centro di Studio per la Matematica e la Fisica Teorica ? del C.N.R., Genova.     

On a nonhomogeneous Burgers’ equation

In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u _t +uu _x =μu _xx −kx,x ∈R,t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any L^p(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2)ϕ _t −ϕ _xx = −x ² ϕ. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf’s paper. Especially, we observe that the nonhomogeneous Burgers’ equation (1) is nonlinearly unstable.     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

On Cramer Approximations under Violation of Cramer's Condition

Let X ₁, X ₂,... be independent identically distributed random variables with distribution function F, S ₀ = 0, S _n = X ₁ + ⋯ + X _n , and Sˉ _n = max_1⩽k⩽n S _k . We obtain large-deviation theorems for S _n and Sˉ _n under the condition 1 − F(x) = P{X ₁ ⩾ x} = e^−l(x), l(x) = x ^α L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005.     

Automorphisms with only infinite orbits on non-algebraic elements

 This paper generalizes results of F. K?rner from [4] where she established the existence of maximal automorphisms (i.e. automorphisms moving all non-algebraic elements). An ω-maximal automorphism is an automorphism whose powers are maximal automorphisms. We prove that any structure has an elementary extension with an ω-maximal automorphism. We also show the existence of ω-maximal automorphisms in all countable arithmetically saturated structures. Further we describe the pairs of tuples (ˉa,ˉb) for which there is an ω-maximal automorphism mapping ˉa to ˉb. Received: 12 December 2001 / Published online: 10 October 2002 Supported by the ``Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture' Mathematics Subject Classification (2000): Primary: 03C50; Secondary: 03C57 Key words or phrases: Automorphism – Recursively saturated structure     

A generalization of a result of R. Lyons about measures on [0, 1)

Letμ be a probability measure on [0, 1), invariant underS:x ↦px mod 1, and for which almost every ergodic component has positive entropy. Ifq is a real number greater than 1 for which logq/ logp is irrational, andT _n sendsx toq ⁿx mod 1, then for any ε>0 the measureμT _n ⁻¹ will — for a set ofn of positive lower density — be within ε of Lebesgue measure.