Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

Strongly non-embeddable metric spaces

Enflo (1969) [4] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu (2002) [3] modified Enflo?s example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space (Z,ζ) which is strongly non-embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non-zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non-zero generalized roundness. Our construction is then adapted to show that the group Zω=ℵ_0⊕Z admits a Cayley graph which may not be coarsely embedded into any metric space of non-zero generalized roundness. Finally, for each p?0 and each locally finite metric space (Z,d), we prove the existence of a Lipschitz injection f:Z→?p.     

Rational power series, sequential codes and periodicity of sequences

Let R be a commutative ring. A power series f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z_m for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in F[[x₁,…,x_n]]. We extend Kronecker’s criterion for rationality in F[[x]] to F[[x₁,…,x_n]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F_p for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction.     

Group actions and Helly's theorem

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encoded in a property that we introduce called “property FAr”, which reduces to Serre's property FA when r=1. The method applies to S-arithmetic groups in higher Q-rank, to simplex reflection groups (including some nonarithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example SLn(Z[x₁,…,xd]), n>2).     

Nilpotency of group ring units symmetric with respect to an involution

Let F be an infinite field of characteristic different from 2. Let G be a torsion group having an involution ∗, and consider the units of the group ring FG that are symmetric with respect to the induced involution. We classify the groups G such that these symmetric units satisfy a nilpotency identity (x₁,…,x_n)=1.     

Locally finite polynomial endomorphisms

We study polynomial endomorphisms F of C^N which are locally finite in the following sense: the vector space generated by r°Fⁿ (n≥0) is finite dimensional for each r∈C[x₁,…,x_N]. We show that such endomorphisms exhibit similar features to linear endomorphisms: they satisfy the Jacobian Conjecture, have vanishing polynomials, admit suitably defined minimal and characteristic polynomials, and the invertible ones admit a Dunford decomposition into “semisimple” and “unipotent” constituents. We also explain a relationship with linear recurrent sequences and derivations. Finally, we give particular attention to the special cases where F is nilpotent and where N=2.     

A note about proving non-Γ under a finite non-microstates free Fisher information assumption

We prove that if X₁,…,Xn (n>1) are self-adjoints in a W^∗-probability space with finite non-microstates free Fisher information, then the von Neumann algebra W^∗(X₁,…,Xn) they generate doesn't have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.     

A Metric Space with Transfinite Asymptotic Dimension 2ω + 1*

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

A generalized Cayley-Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if (AB-BA)C=C(AB-BA) for all A,B,C∈F where A,B,C need not be distinct. Let f_k(x₁,x₂,…,x_p),(k=1,2,…,r), be polynomials in the indeterminates x₁,x₂,…,x_p with coefficients in the complex field C, and let M₁,M₂,…,M_r be n×n matrices over C which are not necessarily distinct. Let and let δ_F(x₁,x₂,…,x_p)=detF(x₁,x₂,…,x_p). In this paper, we prove that, for n×n matrices A₁,A₂,…,A_p over C, if {A₁,A₂,…,A_p,M₁,M₂,…,M_r} is a quasi-commuting family, then F(A₁,A₂,…,A_p)=O implies that δ_F(A₁,A₂,…,A_p)=O.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

A note on properties of locally finitely convergent sequences with respect to median filter

Let F_k be a mapping from R^Z to R^Z, satisfying that for x∈R^Z and n∈Z, F_k(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for k∈{2,3}, if there exists n₀∈Z such that x is locally finitely convergent with respect to F_k on {n₀,…,n₀+k−1}, then x is finitely convergent with respect to F_k. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true.     

Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x₁,…,xk) with xi∈G, there exists a non-empty subsequence (xj₁,…,xjl) and a₁,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x₁,…,xt) with xi∈G, there exist indices j₁,…,jn∈N,1?j₁<?<jn?t, and ?₁,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

On endomorphism-regularity of zero-divisor graphs

The paper studies the following question: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ(R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z₂×Z₂×Z₂; Z₂×Z₄; Z₂×(Z₂[x]/(x²)); F₁×F₂, where F₁,F₂ are fields. In addition, we determine all positive integers n for which Γ(Z_n) has the property.     

Multiplication Operators on Invariant Subspaces of Function Spaces

Let M_φ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator M_z, we derive some spectral properties of the multiplication operator M_φ : F → F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n ∈ {1, 2, …, + ∞}.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

The t-median function on graphs

A median of a sequence π=x₁,x₂,…,x_k of elements of a finite metric space (X,d) is an element x for which is minimum. The function M with domain the set of all finite sequences on X and defined by M(π)={x:x is a median of π} is called the median function on X, and is one of the most studied consensus functions. Based on previous characterizations of median sets M(π), a generalization of the median function is introduced and studied on various graphs and ordered sets. In addition, new results are presented for median graphs.