Absolute graphs with prescribed endomorphism monoid

We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid $\mathop{\rm End}\nolimits (\Gamma)We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid and M is absolute, i.e. holds in any generic extension of the given universe of set theory. This is true if and only if |M|,|Γ| are smaller than the first Erdős cardinal (which is known to be strongly inaccessible). We will encode Shelah’s absolutely rigid family of trees (Isr. J. Math. 42(3), 177–226, 1982) into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see G?bel and Pokutta (Shelah’s absolutely rigid trees and absolutely rigid fields, in preparation). This work was supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development and a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

Roots of C*-algebra elements under numerical range condition

Let be a C*-algebra with unit 1. For each a ∈ , the C*-algebra numerical range is defined by V(a) = {φ(a): φ ∈ , φ ≥ 0,φ(1) = 1}. In a 2003 paper Li, Rodman and Spitkovsky have found the ω-th roots of elements in C*-algebra under a numerical range condition, when ω ∈ [1,∞). In this paper, we will give a short proof of the above result in the case of ω is a positive integer number. We also give a simple proof for ω-th root of an element a ∈ , when ω ∈ [1,∞) and V(a)∩ {z ∈ ℂ: z ≤ 0} = . The first author was supported by the Shiraz university Research Council Grant No. 86-GRSC-32.     

Normal Subgroups of B u Aut(Ω)

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B _u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B _u Aut(Ω) has normal subgroups, and there is a certain ideal of the lattice of subsets of (), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to . To Bernhard Banaschewski on the occasion of his 80th birthday.     

Unitarily covariant maps in approximately finite-dimensional <Emphasis Type="Italic">C</Emphasis>*-algebras

We consider maps defined on a real space Asa of all self-adjoint elements of a C*-algebra A commuting with the conjugation by unitaries: F(u* au) = u* F(a)u for any a ∈ A _sa, u ∈ (A). In the case where A is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized to the class of uniformly hyperfinite C*-algebras and to the algebra of all compact operators in a Hilbert space. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 213–227, 2007.     

Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ _ν,n , the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation . Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004.     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.     

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

Second-Order Optimality Conditions for Constrained Domain Optimization

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫ _Ω g(x) dx=1, a domain is sought which maximizes either , fixed x ₀∈Ω, or ℱ(Ω)=∫ _Ω F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.     

Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Dirichlet Problem at Infinity for $\mathcal A$-Harmonic Functions

We study the Dirichlet problem at infinity for -harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x ₀∈M(∞) to be -regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x ₀ in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.      

A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝ^N, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|^p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂ν_a related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝ^N, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L¹(ℝ^N), g∈L¹(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

Hyperbolic Distortion of Conformal Maps at Corners

We consider conformal self-maps φ of the unit disk onto simply connected domains. We assume φ is continuous in a neighborhood of a point , with φ(ζ) of modulus one, and that has a corner at φ(ζ). We prove that the modulus of the hyperbolic derivative of φ tends to a limit along certain simple curves in the disk that end at ζ non-tangentially. Moreover, we prove that the value of this limit depends only on the geometry of the corner and on the angle of approach to ζ. Our proof is based on a constructive approximation of the domain by more special domains. This research in its different stages was supported partially by MEC grants MTM2006-14449-C02-02 and MTM2006-26627-E (Acciones Complementarias), Spain; and also by “Ingenio Mathematica (i-MATH)” CSD2006-00032 (Consolider—Ingenio 2010), from MCyT, Spain; as well as by the MEC/Fulbright Fellowship 2007-0752.     

Versal deformations and versality in central extensions of Jacobi schemes

Let be the scheme of the laws defined by the Jacobi identities on with a field. A deformation of , parametrized by a local ring A, is a local morphism from the local ring of at ϕ _m to A. The problem of classifying all the deformation equivalence classes of a Lie algebra with given base is solved by “versal” deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕ _m = R ⋉ φ _n in and its nilpotent radical φ _n in the R-invariant scheme with reductive part R, under some conditions. So the versal deformations of ϕ _m in are deduced from those of φ _n in , which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras. Supported by the EC project Liegrits MCRTN 505078.     

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.      

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation