Nonexcellence of the Function Field of the Product of Two Conics

Let k₀ be a field, k₀ ≠ 2, and α, β 2-fold Pfister forms over k₀. Denote by [α], [β] the classes of the corresponding quaternion algebras in ₂Brk₀, and by X_α, X_β the corresponding projective k₀-conics. Suppose ([α] + [β]) = 4. We construct a field F over k₀ such that the field extension F(X_α × X_β)/F is not excellent. Moreover, we find a 2-fold Pfister form γ over F such that ([α ] +[β ] + [γ]) = 4 and the homology group of the complex

Equations on real intervals

A (finite or infinite) set ∑ of equations, in operation symbols F_t (t ∈T) and variables x_i, is said to be compatible with iff there exist continuous operations F_t^A on such that the algebra satisfies the equations ∑ (with the variables x_i understood as universally quantified). It is proved that there is no algorithm to decide -compatibility for all finite ∑. If the definition is restricted to C¹ idempotent operations F_t^A , then there does exist an algorithm for compatibility. Received August 9, 2005; accepted in final form February 14, 2006.     

The generalized Rédei-matrix

Let F be a number field with odd class number, and let E be a quadratic extension of F. Our main aim is to prove that the 4-rank of the class group C(E) of E is equal to m − 1 − rank R _E/F , where m is the number of primes of F ramifying in E, R _E/F is the generalized Rédei-matrix of local Hilbert symbols with coefficients in and the rank is the rank over . We determine the generalized Rédei-matrices R _E/F explicitly for biquadratic number fields E. The research is partly supported by NNSF of China (No. 10371054, No. 10771100) and the Morningside Center of Mathematics in Beijing (MCM).     

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A₃(2), A₂(4)) and(B _n (q), C _n (q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H( ) for a split semisimple algebraic groupH defined over , does not determine the groupH up to isomorphism, but it determines the field under some mild conditions. We then put a group structure on the pairs(H ₁,H ₂) of split semisimple groups defined over a fixed field such that the orders of the finite groups H₁( ) and H₂( ) are the same and the groupsH _i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.     

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

On a New Kato Class and Singular Solutions of a Nonlinear Elliptic Equation in Bounded Domains of $$\mathbb{R}^n$$

Using a new form of the 3G-Theorem for the Green function of a bounded domain Ω in , we introduce a new Kato class K(Ω) which contains properly the classical Kato class K_n(Ω). Next, we exploit the properties of this new class, to extend some results about the existence of positive singular solutions of nonlinear differential equations. Mathematics Subject classification (1991): 34B15, 34B27.     

Density of extremal measures in parabolic potential theory

It is shown that, for the heat equation on , d ≥ 1, any convex combination of harmonic (= caloric) measures , where U ₁, . . . , U _k are relatively compact open neighborhoods of a given point x, can be approximated by a sequence of harmonic measures such that each W _n is an open neighborhood of x in . Moreover, it is proven that, for every open set U in containing x, the extremal representing measures for x with respect to the convex cone of potentials on U (these measures are obtained by balayage, with respect to U, of the Dirac measure at x on Borel subsets of U) are dense in the compact convex set of all representing measures. Since essential ingredients for a proof of corresponding results in the classical case (or more general elliptic situations; see Hansen and Netuka in Adv. Math. 218(4):1181–1223, 2008) are not available for the heat equation, an approach heavily relying on the transit character of the hyperplanes , , is developed. In fact, the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on for practically every space–time structure which is given by a sub-Markov semigroup (P _t ) _t>0 on a space X′ such that there are strictly positive continuous densities with respect to a (non-atomic) measure on X′. In particular, this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals. Moreover, the results may be applied to restrictions of the space–time structure on arbitrary open subsets. I. Netuka’s research was supported in part by the project MSM 0021620839 financed by MSMT, by the grant 201/07/0388 of the Grant Agency of the Czech Republic, and by CRC-701, Bielefeld.     

Raynaud vector bundles

We construct vector bundles on a smooth projective curve X having the property that for all sheaves E of slope μ and rank rk on X we have an equivalence: E is a semistable vector bundle . As a byproduct of our construction we obtain effective bounds on r such that the linear system |R·Θ| has base points on U _X (r, r(g − 1)).      

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \mathbbFⁿ \mathbb{F}^n be the n dimensional linear space over the field \mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from \mathbbFⁿ \mathbb{F}^n to itself {A _i } _i∈I such that for every linear subspace V ⊂ \mathbbFⁿ \mathbb{F}^n of dimension dim(V)<n/2 we have

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

Morse index for solutions of the nonlinear Schrödinger equation in a degenerate setting

In this paper, we study the Morse index of the single-peak solutions concentrating at a point P∊ ℝ ^N of the problem

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a ₊, a ₋, c; v, ) that satisfies four hyperbolic difference equations of Askey-Wilson type. Specializing the family of couplings c∊ to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for ₂ F ₁ in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the Askey-Wilson polynomials when the variables v or are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the Askey-Wilson polynomials. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 39A70     

General Construction of Non-Dense Disjoint Iteration Groups on the Circle

Let be a disjoint iteration group on the unit circle , that is a family of homeomorphisms such that F ^v1 ○ F ^v2 = F ^v1+v2 for v ₁, v ₂ ∈ V and each F ^v either is the identity mapping or has no fixed point ((V, +) is a 2-divisible nontrivial Abelian group). Denote by the set of all cluster points of {F ^v (z), v ∈ V} for . In this paper we give a general construction of disjoint iteration groups for which .     

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d. B. Burgstaller was supported by the Austrian Schr?dinger stipend J2471-N12.