Accuracy of approximation in the Poisson theorem in terms of the χ<Superscript>2</Superscript>-distance

We study the limit behavior of the χ²-distance between the distributions of the nth partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying Poisson law. As a consequence in the i.i.d. case we make the multiplicative constant preciser in the available upper bound for the rate of convergence in the Poisson limit theorem.     

Uniform approximation by harmonic functions on compact subsets of ℝ<Superscript>3</Superscript>

We consider uniform approximation by harmonic functions on compact subsets in ℝ³. Under an additional assumption that the approximated function is Dini-continuous, we prove a natural analog of well-known Vitushkin’s uniform approximation lemma for an individual analytic function. Bibliography: 14 titles.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

The Isometry Group of an <Emphasis FontCategory="SansSerif">RCD</Emphasis><Superscript>∗</Superscript> Space is Lie

We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by R C D ^?-spaces and, under extra assumptions, by C D-spaces, C D ^? P-spaces. However, we show that the M C C P-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.     

The Primitive Ideals of some Étale Groupoid <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-Algebras

We consider the Deaconu–Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C ^?-algebra. For a special class of actions we describe the Jacobson topology.     

Asymptotic unitary equivalence and classification of simple amenable <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-algebras

Let C and A be two unital separable amenable simple C ^?-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ? ₁,? ₂:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u _t :t∈[0,∞)} of A such that

$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$

if and only if [? ₁]=[? ₂] in \(KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}\) and a rotation related map \(\overline{R}_{\varphi_{1},\varphi_{2}}\) associated with ? ₁ and ? ₂ is zero.

Applying this result together with a result of W. Winter, we give a classification theorem for a class \({\mathcal{A}}\) of unital separable simple amenable C ^?-algebras which is strictly larger than the class of separable C ^?-algebras with tracial rank zero or one. Tensor products of two C ^?-algebras in \({\mathcal{A}}\) are again in \({\mathcal{A}}\). Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K ₀ is the same as the tracial state space and also some unital simple ASH-algebras whose K ₀-group is ? and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are \({\mathcal{Z}}\)-stable are isomorphic to ones with no dimension growth.     

Lorentzian Sasaki-Einstein metrics on connected sums of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not #kS ² × S ³ are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki η-Einstein and Lorentzian Sasaki-Einstein metrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors \({\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_i}\) such that the D _i are rational curves.     

Moduli of continuity for <Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript>-valued Gaussian random fields

This paper establishes the general moduli of continuity for l ^∞-valued Gaussian random fields {X(t):= (X ₁(t),X ₂(t), h.), t ∈ [0, ∞) ^N } indexed by the N-dimensional parameter t:= (t ₁,…,t _N ), under the explicit condition yielding that the covariance function of distinct increments of X _k (t) for fixed k ≧ 1 is positive or nonpositive. Supported by KOSEF-R01-2008-000-11418-0.     

Biflatness of ℓ<Superscript>1</Superscript>-Semilattice Algebras

We show that if L is a semilattice then the ℓ¹-convolution algebra of L is biflat precisely when L is "uniformly locally finite". Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space ℓ¹(L) equipped with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

Some estimates of the normal approximation for ��-mixing random variables

Let ?? _n be a ??-mixing sequence of real random variables such that $ \mathbb{E}{\xi_n} = 0 $ , and let Y be a standard normal random variable. Write S _n = ?? ₁ + · · · + ?? _n and consider the normalized sums Z _n = S _n /B _n , where $ B_n^2 = \mathbb{E}S_n^2 $ . Assume that a thrice differentiable function $ h:\mathbb{R} \to \mathbb{R} $ satisfies $ {\sup_{x \in \mathbb{R}}}\left| {{h^s}(x)} \right| < \infty $ . We obtain upper bounds for $ {\Delta_n} = \left| {\mathbb{E}h\left( {{Z_n}} \right) - \mathbb{E}h(Y)} \right| $ in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of ?? _n is of order O(n ^?1/2). We note that the ??-mixing coefficients ??(r) are defined between the ??past?? and ??future.?? To prove the results, we apply the Bentkus approach.     

Exact Calculation of Sums of the Lorentz Spaces Λ<Superscript>α</Superscript> and Applications

The norm on the sum of Lorentz spaces endowed with norms equal to the products of the classical norm by some numbers is exactly calculated. The obtained result makes it possible to prove an extrapolation theorem for collections of Lorentz, Lebesgue, and Marcinkiewicz spaces with a sharp constant.     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

On the definition of the Monge-Ampère operator in ℂ<Superscript>2</Superscript>

We show that if u is a plurisubharmonic function defined on an open subset of ² then the Monge-Ampère measure (dd^cu)² can be well defined if and only if u belongs to the Sobolev space W^1,2_loc().Partially supported by KBN Grant #2 P03A 028 19     

On some properties of finite sums of ridge functions defined on convex subsets of ℝ<Superscript> <Emphasis Type="Italic">n</Emphasis> </Superscript>

Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets E of the space Rⁿ. It is shown that under some constraints imposed on the summed functions ?_i, in the case when E is open, the continuity of the sum implies the continuity of all ?_i. In the case when E is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions ?_i in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.     

The Δ<Superscript>0</Superscript><Subscript><Emphasis Type="Italic">α</Emphasis></Subscript>-Computable Enumerations of the Classes of Projective Planes

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K ^c /? admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰ _α -computable isomorphism.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.