Torsion Completeness of Sylow <Emphasis Type="Italic">p</Emphasis>-Groups in Semisimple Group Rings

Let G be an abelian p-group and K be a field of the first kind with respect to p of char K ≠p and of sp(K) = N or NU {0}. Then it is shown that the normed Sylow p-subgroup S(KG) is torsion complete if and only if G is bounded (Theorem 1). An analogous fact is proved for the case when K is of the second kind (Theorem 2). These completely settle a conjecture posed by us in Compt. Rend. Acad. Bulg. Sci. (1993) and are also a supplement to our result in the modular case published in Acta Math. Hungar. (1997).     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

S. G Mikhlin (1908 – 1990)

Our concern is to find a representation theorem for operators in B(c(X), c(Y)) where X and Y are Banach spaces with Y containing an isomorphic copy of c₀. Cass and Gao [1] obtained a representation theorem that always applies if Y does not contain an isomorphic copy of c₀. Maddox [3], Melvin - Melvin [4], and Robinson [5] consider operators in B(c(X), c(Y)) that are given by matrices. In this paper we show that Cass's and Gao's result in [1] can be extended, when Y contains an isomorphic copy of c₀, to certain operators that we call represent able. In addition, we show that when Y contains an isomorphic copy of co there are always operators that fall outside the scope of our representation theorem. Light is also cast on a theorem given in Maddox [3, Theorem 4.2] and [5, Theorem IV].     

Scheme-theoretic generation for ideals generated by forms of the same degree: a nice formula

Given a homogeneous ideal I?K[n ₀,…,n _n] (k an infinite field) and supposing that I is generated by forms of the same degree, we prove a formula to compute the minimal number σ (I) of the scheme-theoretic generations of I.     

Presemiclosed mappings

The purpose of this paper is to investigate the properties of presemiclosed mappings and improve some already established results. Moreover, it is shown that, if the domain space isT ₀, then optimal continuity and homeomorphism coincide (Theorem 4.18).     

A characterization of one class of graphs without 3-claws

In the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative -bilinear multiplication on its tangent sheaf is an F-manifold in the sense of Hertling-Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle T _M ^*, is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties. To the memory of V.A. Iskovskikh     

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π₂X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T _π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T _π-compact provided G/T _πG is π-complete, extending the abelian group result in a natural way.     

Newton functions generating symmetric fields and irreducibility of Schur polynomials

We prove (Theorem 1.1) that if e₀>>e_r>0 are coprime integers, then the Newton functions , i=0,…,r, generate over the field of symmetric rational functions in X₁,…,X_r. This generalizes a previous result of us for r=2. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Theorem 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Theorem 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Theorem 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1.     

Automatic continuity of linear maps on spaces of continuous functions

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map AC(T)C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).     

Modules Satisfying the S-Noetherian Property and S-ACCR

Let A = k[x₁,..., x_n] be a standard graded k-algebra over an infinite field k. We assume A is Cohen-Macaulay and has Krull dimension one. Let e denote the multiplicity of A and r - 1 the postulation number of A. Let I be a homogeneous ideal in A of grade one. Let j(I) denote the smallest degree of a regular form in I. Let l (I) denote the smallest power of (x₁,... , x_n) contained in I. If μ(I) denotes the minimum number of generators of I, then we show μ(I) ≤ e + j(I) - max{r,l(I)}, We then show how Dubreil's Second Theorem follows easily from this inequality     

Geometric characterization of RN-operators

Let X and Y be Banach spaces andtl (x, y). An operator T: X Y is called an RN-operator if it transforms every X-valued. measure ¯m of bounded variation into a Y-valued measure having a derivative with respect to the variation of the measure ¯m. The notions of T-dentability and Ts-dentability of bounded sets in Banach spaces are introduced and in their terms are given conditions equivalent to the condition that T is an RN-operator (Theorem 1). It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace X_o of X the set (T|X_o)*(Y*) is separable (Theorem 2).Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 189–202, August, 1977.     

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL ₃(K),PSp ₄(K) orG ₂(K) for some algebraically closed fieldK. Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field. Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed. We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ₁-categorical polygon have Morley degree 1. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Supported by the Minerva Foundation (Germany). Research Director at the Fund for Scientific Research-Flanders (Belgium).     

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.     

Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well-posed problem (in the Hadamard classical sense) in spaces H^k(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (a_n, ?_n) → (a, ?) in H^k × H^k and if f_n → f in ??²(0, T;H^k), then (v_n, g_n) → (v, g) in ?? (0, T; H^k × H^k). The method followed here (see also [8]) also applies to the non-barotropic case p = p(ρ, s) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural-stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.     

Elementary theories of completely simple semigroups

The connections between first-order formulas over a completely simple semigroupC and corresponding formulas over its structure groupH are found in this paper. For the case of finite sandwich-matrix the criterion of decidability of the elementary theoryT(C) is established in terms of the elementary theory ofH in the enriched signature (Theorem 1). For the general case the criterion is established in terms of two-sorted algebraic systems (Theorem 2). Sufficient conditions in terms ofH for decidability and for undecidability ofT(C) are outlined. Corollaries and examples are presented, among them an example of a completely simple semigroup with a finite structure group and with undecidable elementary theory (Theorem 3).     

Deducing the multidimensional Szemerédi theorem from an infinitary removal lemma

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T ₁, T ₂, …, T _d : ℤ ↷ (X, ∑, μ) ([6]), and so, via the Furstenberg correspondence principle introduced in [5], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed in [1] to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao in [13] for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemerédi’s Theorem set in motion by Furstenberg [5].     

On the edge-coloring problem for a class of 4-regular maps

A (plane) 4-regular map G is called C-simple if it arises as a superposition of simple closed curves (tangencies are not allowed); in this case σ (G) is the smallest integer k such that the curves of G can be colored with k colors in such a way that no two curves of the same color intersect. We prove that if σ (G) ≤ 4, G is edge colorable with 4 colors. Moreover we show that a similar result for maps G with σ(G) ≤ 5 would imply the Four-Color Theorem.     

Espaces de fonctions ponderables

The main theorems solve two problems about weighted spacesC ₀ V(T) of continuous functions on a completely regular spaceT. These problems were first raised by W. H. Summers who solved them for the locally compact case. The first theorem gives necessary and sufficient conditions onT forC ₀ V(T) to be complete for the weighted topology; the second one shows that continuous linear functionals onC ₀ V(T) can be represented by means of measures onT.