On topological diversity of rectilinear representations of countably infinite graphs

We prove that, for each simple graph G whose set of vertices is countably infinite, there is a family ${\varvec{\mathcal{R}}(\varvec{G})}$ of the cardinality of the continuum of graphs such that (1) each graph ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ is isomorphic to G, all vertices of H are points of the Euclidean space E ³, all edges of H are straight line segments (the ends of each edge are the vertices joined by it), the intersection of any two edges of H is either their common vertex or empty, and any isolated vertex of H does not belong to any edge of H; (2) all sets ${\varvec{\mathcal{B}}(\varvec{H})}$ ( ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ ), where ${\varvec{\mathcal{B}}(\varvec{H})\subset \mathbf{E}^3}$ is the union of all vertices and all edges of H, are pairwise not homeomorphic; moreover, for any graphs ${\varvec{H}_1 \in \varvec{\mathcal{R}}(\varvec{G})}$ and ${\varvec{H}_2 \in \varvec{\mathcal{R}}(\varvec{G})}$ , ${\varvec{H}_1 \ne \varvec{H}_2}$ , and for any finite subsets ${\varvec{S}_i \subset \varvec{\mathcal{B}}(\varvec{H}_i)}$ (i = 1, 2), the sets ${\varvec{\mathcal{B}}(\varvec{H}_1){\setminus} \varvec{S}_1}$ and ${\varvec{\mathcal{B}}(\varvec{H}_2){\setminus} \varvec{S}_2}$ are not homeomorphic.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.     

Cluster Algebras of Type A _{\bf{2}}^{\bf {(1)}}

In this paper we study cluster algebras $\mathcal{A}$ of type $A_2^{(1)}$ . We solve the recurrence relations among the cluster variables (which form a T-system of type $A_2^{(1)}$ ). We solve the recurrence relations among the coefficients of $\mathcal{A}$ (which form a Y-system of type $A_2^{(1)}$ ). In $\mathcal{A}$ there is a natural notion of positivity. We find linear bases B of $\mathcal{A}$ such that positive linear combinations of elements of B coincide with the cone of positive elements. We call these bases atomic bases of $\mathcal{A}$ . These are the analogue of the “canonical bases” found by Sherman and Zelevinsky in type $A_{1}^{(1)}$ . Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of B are parameterized by ?³ via their g-vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\mathcal{A}$ restricts to a bijection between B and ?³. We find explicit recurrence relations to express every element of $\mathcal{A}$ as linear combinations of elements of B.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Invertible Factorization over Multiplier Algebras

Let ${\mathcal{A}}$ denote the multiplier algebra of an E-valued reproducing kernel Hilbert space, ${H_E^2(k)}$ . Then when H ²(k) is nice, we give necessary and sufficient conditions that T > 0 factors as A*A, where A and ${A^{-1} \in \mathcal{A}}$ . Such nice spaces include the Bergman and Hardy spaces on the unit polydisk and unit ball in ${\mathbb{C}^d}$ .     

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in ${\mathfrak{H}}$ , let ${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$ be a boundary triplet for A ^* and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A ₀}, uniquely up to the unitary similarity. Here ${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric ${A \subset A^*}$ and a special boundary triplet ${\widetilde{\Pi}}$ for{A, A} such that the corresponding Weyl function is ${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$ , where B is a non-self-adjoint bounded operator in ${\mathcal{H}}$ . We are interested in the problem whether the result on the unitary similarity remains valid for ${\widetilde{M}(\cdot)}$ in place of M(·). We indicate some sufficient conditions in terms of the operators A ₀ and ${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in ${L^2(\mathbb{R}_+)}$ , we show that ${\widetilde{M}(\cdot)}$ defined in ${\Omega_+ \subset \mathbb{C}_+}$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function ${\widetilde{M}(\cdot)}$ does not determine A _B even up to the similarity.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

Categories of partial algebras for critical points between varieties of algebras

We denote by Con_c A the ${(\vee, 0)}$ -semilattice of all finitely generated congruences of an algebra A. A lifting of a ${(\vee, 0)}$ -semilattice S is an algebra A such that ${S \cong {\rm Con}_{\rm c} A}$ . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of ${(\vee, 0)}$ -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an ${\aleph_0}$ -lifter of P, of infinite cardinality ${\lambda}$ . Let ${\vec{A}}$ be a P-indexed diagram of finite algebras in ${\mathcal{V}}$ . If ${{\rm Con}_{\rm c} \circ \vec{A}}$ has no partial lifting in the category of gamps of ${\mathcal{W}}$ , then there is an algebra ${A \in \mathcal{V}}$ of cardinality ${\lambda}$ such that Con_c A is not isomorphic to Con_c B for any ${B \in \mathcal{W}}$ . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that ${\mathcal{W}}$ is locally finite. Let ${\mathcal{V}}$ be locally finite and let ${\mathcal{W}}$ be congruence-proper (i.e., congruence lattices of infinite members of ${\mathcal{W}}$ are infinite). The following equivalence holds. Every countable ${(\vee, 0)}$ -semilattice with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ if and only if every ${\omega}$ -indexed diagram of finite ${(\vee, 0)}$ -semilattices with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ . Gamps are also applied to the study of congruence-preserving extensions. Let ${\mathcal{V}}$ be a non-semidistributive variety of lattices and let n ≥ 2 be an integer. There is a bounded lattice ${A \in \mathcal{V}}$ of cardinality ${\aleph_1}$ with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.     

The modal logic of {\beta(\mathbb{N})}

Let ${\beta(\mathbb{N})}$ denote the Stone–?ech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast}$ is S4 and that the modal logic of ${\beta(\mathbb{N})}$ is S4.1.2.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .