Crystallographic number systems

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion $$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$ for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

Let $\mathcal P _\lambda $ be a homogeneous Poisson point process of rate $\lambda $ in the Clifford torus $T^2\subset \mathbb E ^4$ . Let $(f_0, f_1, f_2, f_3)$ be the $f$ -vector of conv $\,\mathcal P _\lambda $ and let $\bar{v}$ be the mean valence of a vertex of the convex hull. Asymptotic expressions for $\mathsf E \, f_1$ , $\mathsf E \, f_2$ , $\mathsf E \, f_3$ and $\mathsf E \, \bar{v}$ as $\lambda \rightarrow \infty $ are proved in this paper.     

A Hilbert Module Approach to the Haagerup Property

We develop a Hilbert module version of the Haagerup property for general C*-algebras ${{\mathcal{A} \subseteq \mathcal{B}}}$ . We show that if ${\alpha : \Gamma \curvearrowright \mathcal{A}}$ is an action of a countable discrete group Γ on a unital C*-algebra ${\mathcal{A}}$ , then the reduced C*-algebra crossed product ${\Gamma \ltimes _{\alpha, r} \mathcal{A}}$ has the Hilbert ${\mathcal{A}}$ -module Haagerup property if and only if the action α has the Haagerup property. We are particularly interested in the case when ${\mathcal{A} = C(X)}$ is a unital commutative C*-algebra. We compare the Haagerup property of such an action ${\alpha: \Gamma \curvearrowright C(X)}$ with the two special cases when (1) Γ has the Haagerup property and (2) Γ is coarsely embeddable into a Hilbert space. We also prove a contractive Schur mutiplier characterization for groups coarsely embeddable into a Hilbert space, and a uniformly bounded Schur multiplier characterization for exact groups.     

Uniqueness of the Group Measure Space Decomposition for Popa’s {\mathcal{HT}} Factors

We prove that if ${\Gamma\curvearrowright (X, \mu)}$ is a free ergodic rigid (in the sense of Popa in Ann Math 163:809–889, 2006) probability measure preserving action of a group Γ with positive first ${\ell^2}$ -Betti number, then the II₁ factor ${L^{\infty}(X)\rtimes\Gamma}$ has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many ${\mathcal{HT}}$ factors, including the II₁ factors associated with the usual actions ${\Gamma\curvearrowright \mathbb{T^2}}$ and ${\Gamma\curvearrowright}$ ${{\rm SL}_2(\mathbb R)/{\rm SL}_2(\mathbb Z)}$ , where Γ is a non-amenable subgroup of ${{\rm SL}_2(\mathbb Z)}$ , have a unique group measure space decomposition.     

Flat bundles on G/Γ and stability

Let G be a connected complex Lie group and Γ a cocompact lattice in G. Let H be a connected reductive complex affine algebraic group and \({\rho\, : \Gamma\, \longrightarrow H}\) a homomorphism such that \({\rho(\Gamma)}\) is not contained in some proper parabolic subgroup of H. Let \({E^\rho_H}\) be the holomorphic principal H–bundle on G/Γ associated to ρ. We prove that \({E^\rho_H}\) is polystable. If ρ satisfies the further condition that \({\rho(\Gamma)}\) is contained in a compact subgroup of H, then we prove that \({E^\rho_H}\) is stable.     

A graph associated to a lattice

In this paper, we associate a simple graph to a lattice $\mathcal L $ , in which the vertex set is being the set of all elements of $\mathcal L $ , and two distinct vertices $x$ and $y$ are adjacent if $x\vee y\in S$ , when $S$ is a multiplicatively closed subset of $\mathcal L $ . We denote this graph by $\Gamma _S(\mathcal L )$ . We study some properties of $\Gamma _S(\mathcal L )$ . Moreover, we investigate the planarity of $\Gamma _S(\mathcal L )$ , whenever $S$ is a saturated multiplicatively closed subset of $\mathcal L $ .     

Hilbert transform of a wavelet system: Boundedness of orthogonal projections in the uniform norm

Let ?? be a wavelet associated with a scaling function ??, and let $\bar \psi$ be the Hilbert transform of ??. Under some natural conditions on the smoothness and decay of ??, we show that the orthogonal projections onto the spaces generated by $\bar \psi$ are bounded in the uniform norm.     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

Volume invariant and maximal representations of discrete subgroups of Lie groups

Let $\Gamma $ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma $ into $G$ , which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma $ into $G$ .     

Extreme points of the set of Banach limits

Let ${\mathbb{N}}$ be the set of all natural numbers and ${\ell_\infty=\ell_\infty (\mathbb{N})}$ be the Banach space of all bounded sequences x = (x ₁, x ₂ . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let ${\ell_\infty^*}$ be its Banach dual. Let ${\mathfrak{B} \subset \ell_\infty^*}$ be the set of all normalised positive translation invariant functionals (Banach limits) on ? _∞ and let ${ext(\mathfrak{B})}$ be the set of all extreme points of ${\mathfrak{B}}$ . We prove that an arbitrary sequence (B _j ) _{j ≥ 1}, of distinct points from the set ${ext(\mathfrak{B})}$ is 1-equivalent to the unit vector basis of the space ? ₁ of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of ${ext(\mathfrak{B})}$ does not contain a Cesáro-invariant Banach limit.     

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 ${a \in G}$ .     

Heegner cycles and higher weight specializations of big Heegner points

Let ${\mathbf{{f}}}$ be a $p$ -ordinary Hida family of tame level $N$ , and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis relative to $N$ . By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level $\Gamma _0(N)\cap \Gamma _1(p^s)$ , Howard has constructed a canonical class $\mathfrak{Z }$ in the cohomology of a self-dual twist of the big Galois representation associated to ${\mathbf{{f}}}$ . If a $p$ -ordinary eigenform $f$ on $\Gamma _0(N)$ of weight $k>2$ is the specialization of ${\mathbf{{f}}}$ at $\nu $ , one thus obtains from $\mathfrak{Z }_{\nu }$ a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes $\mathfrak{Z }_{\nu }$ to the étale Abel-Jacobi images of Heegner cycles when $p$ splits in $K$ .     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

Domination in Bipartite Graphs and in Their Complements

The domatic numbers of a graph G and of its complement $\bar G$ were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones: Characterize bipartite graphs G having $d\left( G \right) = d\left( {\bar G} \right)$ Further, we will present a partial solution to the problem: Is it true that if G is a graph satisfying $d\left( G \right) = d\left( {\bar G} \right)$ then $\gamma \left( G \right) = \gamma \left( {\bar G} \right)$ ? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement     

MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

Let G be a simple algebraic group defined over ?. Let e be a nilpotent element in $ \mathfrak{g} $ = Lie(G) and denote by U ( $ \mathfrak{g} $ , e) the finite W-algebra associated with the pair ( $ \mathfrak{g} $ , e). It is known that the component group Γ of the centraliser of e in G acts on the set ? of all one-dimensional representations of U ( $ \mathfrak{g} $ , e). In this paper we prove that the fixed point set ?^Γ is non-empty. As a corollary, all finite W-algebras associated with $ \mathfrak{g} $ admit one-dimensional representations. In the case of rigid nilpotent elements in exceptional Lie algebras we find irreducible highest weight $ \mathfrak{g} $ -modules whose annihilators in U ( $ \mathfrak{g} $ ) come from one-dimensional representations of U ( $ \mathfrak{g} $ , e) via Skryabin’s equivalence. As a consequence, we show that for any nilpotent orbit $ \mathcal{O} $ in $ \mathfrak{g} $ there exists a multiplicity-free (and hence completely prime) primitive ideal of U ( $ \mathfrak{g} $ ) whose associated variety equals the Zariski closure of $ \mathcal{O} $ in $ \mathfrak{g} $ .     

A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations

Let ${f:\Omega \rightarrow \mathbb{R}}$ be a smooth function on a domain   ${\Omega \subset \mathbb{C}^n}$ with its Hessian matrix ${\left( \frac{\partial^2 f}{\partial z^i \partial\bar{z}^j}\right)}$ positive Hermitian. In this paper, we investigate a class of partial differential equations $$\Delta \ln \det (f_{i\bar{j}}) = \beta \;\| \text{grad} \ln \det (f_{i\bar{j}}) \|^2, $$ where Δ and ${\| \cdot \|}$ are the Laplacian and tensor norm, respectively, with respect to the metric ${G = \sum f_{i\bar{j}} \,dz^i \otimes d\bar{z}^j}$ , and β > 1 is some real constant depending on the dimension n. We prove that the above PDEs have a Bernstein property when the metric G is complete, provided that ${\det (f_{i\bar{j}})}$ and the Ricci curvature are bounded.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .