Group-universal unary varieties

LetV be a variety of unary algebras and letM(V) be the monoid of all unary polynomials ofV. Then every group appears as the automorphism group of an algebraA∈V if and only if the left ideals ofM(V) do not form an inclusion-ordered chain. The support of the National Research Council of Canada is gratefully acknowledged. Presented by J. Mycielski.     

Complexified real arrangements of hyperplanes

LetV be a finite dimensional vector space over the real or complex numbers. Areal (orcomplex)arrangement A inV is a finite collection of real (or complex) affine hyperplanes. A real arrangement inV can becomplexified to form a complex arrangement in the complex vector spaceA. The (complex)complement of a real arrangementA is defined byM(A)=V⊗ℂ−⋃ _H∈ ^A H⊗ℂ. There are two different finite simplicial complexes which carry the homotopy type ofM(A), one given by M. Salvetti, the other by P. Orlik. In this paper we describe both complexes and exhibit a simplicial homotopy equivalence between them.     

Existence d’un generateur suivant une sous-suite de densite superieure nulle

Ergodic theory: for every dynamical system (X,A,T, μ), totally ergodic and of finite entropy, there exist a sequenceS of integers, of upper density zero, and a partitionQ ofX, such that V _i∈S T ⁻ⁱ Q is the whole σ-algebraA. Furthermore, there is a “universal” sequenceS ⁰ for which this property is true if we restrict ourselves to the class of strongly mixing systems.      

Semigroups containing proximal linear maps

A linear automorphism of a finite dimensional real vector spaceV is calledproximal if it has a unique eigenvalue—counting multiplicities—of maximal modulus. Goldsheid and Margulis have shown that if a subgroupG of GL(V) contains a proximal element then so does every Zariski dense subsemigroupH ofG, providedV considered as aG-module is strongly irreducible. We here show thatH contains a finite subsetM such that for everyg∈GL(V) at least one of the elements γg, γ∈M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version of proximality, reducible representations, several eigenvalues of maximal modulus. Partially supported by NSF grant DMS 9204-720.     

Hyperidentities of lattices and semilattices

Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type <2,2> with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letH ^m (V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andH _n (V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofH ^m (V)∪H _n (V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3]. The research of the author was supported by NSERC of Canada. Presented by W. Taylor.     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Universal coextensions within a variety

A category is universal if it contains a full subcategory isomorphic to the category Γ of all directed graphs without loops and isolated points. LetV be a universal semigroup variety,S a semigroup inV, andV _S = {T εV;S is a homomorphic image ofT} the full subcategory ofV of all coextensions ofS withinV. We establish the universality ofV _S in two cases:(a) ifV is the varietyS of all semigroups andS has an idempotent, and(b) ifV is an arbitrary universal semigroup variety andS has a kernel. The results of this paper had been presented at the Colloquium on Semigroups, Szeged (Hungary), August 1994. Support of the Grant Agency of the Czech Republic under Grant 201/93/950 is gratefully acknowledged.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

Some remarks on the structure of free automata

In this paper we define automata-linearly independence. An automatonM has a basis B iffM is free provided that we assume that the action ofS onX × S is (x,s)a = (x,sa) for alla, s ∈ S andx ∈X. If a semigroupS is PRID, every subautomaton of a freeS-automaton is free.     

Complex algebras of subalgebras

Let V be a variety of algebras. We specify a condition (the so-called generalized entropic property), which is equivalent to the fact that for every algebra A ∈ V, the set of all subalgebras of A is a subuniverse of the complex algebra of the subalgebras of A. The relationship between the generalized entropic property and the entropic law is investigated. Also, for varieties with the generalized entropic property, we consider identities that are satisfied by complex algebras of subalgebras. Dedicated to George Gr?tzer on the occasion of his 70th birthday Supported by INTAS grant No. 03-51-4110. Supported by MŠMTČR (project MSM 0021620839) and by the Grant Agency of the Czech Republic (grant No. 201/05/0002). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 655–686, November–December, 2008.     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

On countable stable structures which are homogeneous for a finite relational language

LetL be a finite relational language andH(L) denote the class of all countable structures which are stable and homogeneous forL in the sense of Fraissé. By convention countable includes finite and any finite structure is stable. A rank functionr :H(L) →ω is introduced and also a notion of dimension for structures inH(L). A canonical way of shrinking structures is defined which reduces their dimensions. The main result of the paper is that anyM ∈H(L) can be shrunk toM′ ∈H(L),M′ ⊆M, such that |M′| is bounded in terms ofr(M), and the isomorphism type ofM overM′ is uniquely determined by the dimensions ofM. Forr<ω we deduce thatH(L, r), the class of allM ∈H(L) withr(M)≦r, is the union of a finite number of classes within each of which the isomorphism type of a structure is completely determined by its dimensions. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death     

Classi localmente filtrali

Riassunto Si dà la definizione di classe ?localmente filtrale?. Si diceche K è una classe localmente filtrale se per ogni n∈ω, per ogni A ₀,...,A _n−1, εK e per ogni famiglia di sotto-insiemi V_i di A_i (i∈n) con V_i finiti, la classe {B ₀,...,B _n−1 delle algebre generate da V₀, ..., V_n−1 è costituita da algebre finite ed è filtrale. Si dimostra che seK è localmente filtrale alloraV _L(K)=IR _L DS(K) e si dà un teorema di caratterizzazione per queste classi.

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Summary We define a ?classe localmente filtrale? as follows: LetK be a class of similar algebras;K is a ?classe localmente filtrale? if for andn ∈ ω and for anyA ₀,...,A _n−1 ink and for any family of finite subsetsV _i ofA _i(i∈n), the class {B ₀,...,B _n−1 of algebras generated byV _o, ...,V _n−1 consists of finite algebras and is ?filtrale?. We show that ifK is ?localmente filtrale? thenV _L (K)=IR _L DS(K) and we give a characterization theorem for these classes.

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Lavoro eseguito nell'ambito dei gruppi di ricerca matematici del C.N.R. per l'anno 1970–71.     

Homomorphisms of extensions of a semigroup within a variety

LetV be a semigroup variety containing all commutative semigroups such that the law of exponents (xy) ⁿ =xⁿyⁿ fails inV for everyn > 1. For every semigroupS V such that the reflection of the semigroup obtained fromS by an adding unity has only one idempotent there exists a semigroupT V extendingS without non-trivial endomorphisms. In more general, the full subcategory ofV formed by all extensions ofS withinV is universal.Presented by B. M. Schein.     

Characterization of non-matrix varieties of associative algebras

A variety of associative algebras is called a non-matrix variety if it does not contain the algebra of 2 × 2 matrices over the base field K. There are some known characterizations of non-matrix varieties. We give some new characterizations in terms of properties of nilelements. Let V be a variety of associative algebras over an infinite field. Then the following conditions are equivalent: (1) V is a non-matrix variety, (2) any finitely generated algebra A ∈ V satisfies an identity of the form [x ₁, x ₂] … [x _2s−1, x _2s ] ≡ 0, (3) let A ∈ V; then for any nilelements a, b ∈ A, the element a + b is again a nilelement. Let E be the Grassmann algebra in countable many generators. We also give similar characterizations for non-matrix varieties over fields of characteristic zero that do not contain E or E ⊗ E.     

Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows

Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.     

(<Emphasis Type="Italic">p, λ</Emphasis>)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK _λ ^p (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK _λ ^p (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK _λ ^p (A) we prove that ⊕ _i≥0 Ext _A ⁱ (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK _λ ^p (A) under certain conditions.     

Matrices with zero trace

LetM _n(F) denote the algebra ofn-square matrices with elements in a fieldF. In this paper we show that ifM ∈M _n(F) has zero trace thenM=AB−BA for certainA, B ∈ M _n(F), withA nilpotent and traceB=0, apart from some exceptional cases whenn=2 or 3. We also determine whenM=MB−BM for someB ∈ M _n(F). The preparation of this paper was supported in part by the U.S. Air Force under contract AFOSR 698-65.