Let R be a Dedekind domain with quotient field $K$. That every maximal order in a finite dimensional central simple $K$-algebra $A$, (the algebra of nxn matrices over $D$), where $D$ is separable over $K$, is Morita equivalent to every maximal order in $D$ is a well known linear result. Hahn defined the notion of Hermitian Morita equivalence (HME) for algebras with anti-structure, generalizing previous work by Frohlich and McEvett. The question this paper investigates is the hermitian analogue of the above linear result. Specifically, when are maximal orders with anti-structure in $A$, HME to maximal orders with anti-structure in $D$ in the sense of Hahn? Two sets of necessary and sufficient conditions are obtained with an application which provides the hermitian analogue under some conditions.

We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras, we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structures with respect to two opposite orders, that they have strong exact Borel and $Δ$-subalgebras and the corresponding triangular decompositions.

We present further properties of the boundary spectrum $S_{\partial }(a) = \{\lambda: \lambda-a \in\partial S\}$ of $a$, where $\partial S$ denotes the topological boundary of the set $S$ of all noninvertible elements of a Banach algebra $A$, and where $a$ is an element of $A$. In particular, we investigate the conditions under which it is true that $S_{\partial}(f(a))=f(S_{\partial}(a))$, where $f$ is a complex valued function which is analytic on a neighbourhood of the spectrum of $a$. We also consider continuity properties of the boundary spectrum.

We study the Cheeger constants of certain infinite families of arithmetic hyperbolic three-manifolds, as well as certain graphs associated to these manifolds. We derive computable bounds on the Cheeger constants, and therefore bounds on the first eigenvalue of the Laplacian, by adapting discrete methods due to Brooks, Perry and Petersen. We then modify probabilistic methods due to Brooks and Zuk to obtain sharper, asymptotic bounds. A consequence is that the Cheeger constants are quite small, implying that Cheeger’s inequality is generally insufficient to prove Selberg’s eigenvalue conjecture.

Let $(n_k)_{k \geq1}$ be a lacunary sequence, i.e., a sequence of positive integers satisfying the Hadamard gap condition $n_{k+1}/ n_k \ge q >1, k \geq1$. By a classical result of Philipp (Acta Arith. 26 (1975) 241-251), the discrepancy $D_N$ of $(n_k x)_{k\ge1}$ mod 1 satisfies the law of the iterated logarithm, i.e., we have $1/(4 \sqrt{2}) \leq\limsup_{N \to\infty} N D_N(n_k x) (2 N \log\log N)^{-1/2} \leq C_q$ for almost all $x \in(0,1)$, where $C_q$ is a constant depending on $q$. Recently, Fukuyama computed the exact value of the $\limsup$ for $n_k=\theta^k$, where $\theta>1$, not necessarily an integer, and the author showed that for a large class of lacunary sequences the value of the $\limsup$ is the same as in the case of { i.i.d.} random variables. In the sublacunary case,\break the situation is much more complicated. Using methods of Berkes, Philipp and Tichy, we prove an exact law of the iterated logarithm for a large class of sublacunary growing sequences $(n_k)_{k \geq1}$, characterized in terms of the number of solutions of certain Diophantine equations, and show that the value of the $\limsup$ is the same as in the case of { i.i.d.} random variables.

This paper introduces the notion of the maximal pure spectrum of an Abelian group—this is the set of isomorphism classes of maximal proper pure subgroups—and focuses on the situation in which this spectrum is small. The converse situation is also examined i.e., given a collection of isomorphism classes of groups, can one find an Abelian group having precisely this collection as its maximal pure spectrum. Finally, it is shown that in some familiar situations, the answers to these questions may be undecidable.

We construct a sequence of compact embedded minimal disks in the unit ball in Euclidean 3-space whose boundaries are in the boundary of the ball and where the curvatures blow up at every point of a line segment of the vertical axis, extending from the origin. We further study the transversal structure of the minimal limit lamination and find removable singularities along the line segment and a non-removable singularity at the origin. This extends a result of Colding and Minicozzi where they constructed a sequence with curvatures blowing up only at the center of the ball, Dean’s construction of a sequence with curvatures blowing up at a prescribed discrete set of points, and the classical case of the sequence of re-scaled helicoids with curvatures blowing up along the entire vertical axis.

In this article, some dynamical properties for continuous linear operators are studied. We investigate that neither normal operators nor compact operators can be Li–Yorke chaotic. In addition, we show that a small compact perturbation of the unit operator could be distributionally chaotic.

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,\ldots , T_l$ be $l$ commuting invertible measure preserving transformations of $X$. We show that if $T_1^{c_1} \ldots T_l^{c_l}$ is ergodic for each $(c_1,\ldots ,c_l)\neq(0,\ldots,0)$, then the averages $\frac{1}{|\Phi_N|}\sum_{u\in\Phi_N}\prod _{i=1}^r T_1^{p_{i1}(u)}\ldots T_l^{p_{il}(u)}f_i$ converge in $L^2(\mu)$ for all polynomials $p_{ij} : \mathbb {Z}^d\to\mathbb{Z}$, all $f_i\in L^{\infty}(\mu)$, and all Følner sequences $\{\Phi_N\}_{N=1}^{\infty}$ in $\mathbb{Z}^d$.

By employing work concerning Selberg’s Kloosterman zeta-function, we carry out the decomposition of a special value of a nonanalytic Poincaré series of nonpositive even weight, with a nonsingular multiplier system, on the full modular group. The summands that emerge are connected meaningfully to each other as well as to classical expressions for Eichler integrals and modular forms.

Let $A$ and $B$ be unital $C^*$-algebras such that at least one of them is of real rank zero. We investigate surjective linear maps from $A$ to $B$ preserving the conorm, the (von Neumann) regularity, the generalized spectrum, and their essential versions. As a consequence, we recover results of Mbekhta, and Mbekhta and Šemrl for $\mathcal L(H)$ when $H$ is an infinite-dimensional complex Hilbert space.

We give a complete description of the Dirichlet and Neumann spectra of the Laplacian on a class of homogeneous hierarchical fractals introduced by Hambly. These fractals are finitely ramified but not self-similar. We use the method of spectral decimation. As applications, we show that these spectra always have infinitely many large spectral gaps, allowing for nice convergence results for eigenfunction expansions, and under certain restrictions we give a computer-assisted proof that the set of ratios of eigenvalues has gaps, implying the existence of quasielliptic PDE’s on the product of two such fractals. The computer programs used in this paper and more detailed explanations of the algorithms can be found at www.math.cornell.edu/˜sld32/ FractalAnalysis.html.

CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities of compact submanifolds.