$p$-adic hyperbolic planes and modular forms
John A.
Rhodes
469-504
Abstract: For $K$ a number field and ${\mathbf{p}}$ a finite prime of $K$, we define a ${\mathbf{p}}$-adic hyperbolic plane and study its geometry under the action of $ G{L_2}({K_{\mathbf{p}}})$. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator ${T_{\mathbf{p}}}$. Letting $S$ be a finite set of primes of a totally real $K$ (including all the infinite ones), a modular group $\Gamma (S)$ is defined. This group acts discontinuously on a product of classical and ${\mathbf{p}}$-adic hyperbolic planes. $ S$-modular forms and their associated Dirichlet series are studied.
Division algebra coproducts of index $n$
Michel
Van den Bergh;
Aidan
Schofield
505-517
Abstract: Given a family of separable finite dimensional extensions $\{ {L_i}\}$ of a field $k$, we construct a division algebra $ {n^2}$ over its center which is freely generated over $k$ by the fields $ \{ {L_i}\}$.
Decompositions for relatively normal lattices
James B.
Hart;
Constantine
Tsinakis
519-548
Abstract: Continuing the work begun in Snodgrass and Tsinakis [26, 27], we develop a family of decomposition theorems for classes of relatively normal lattices. These results subsume and are inspired by known decomposition theorems in order-algebra due to P. Conrad and D. B. McAlister. As direct corollaries of the main results, we obtain corresponding decomposition theorems for classes of partially ordered sets.
Basic constructions in the $K$-theory of homotopy ring spaces
R.
Schwänzl;
R. M.
Vogt
549-584
Abstract: Using the language of category theory and universal algebra we formalize the passage from the permutative category of finitely generated free $R$-modules to the algebraic $K$-theory $KR$ of $R$ and thus make it applicable to homotopy ring spaces. As applications we construct a Waldhausen type of algebraic $K$-theory for arbitrary homotopy ring spaces, show its equivalence with constructions of May and Steiner, prove its Morita invariance and show that the algebraic $K$-theory $KX$ of an $ {E_\infty }$ ring $ X$ is itself an ${E_\infty }$ ring. Finally we investigate the monomial map $ Q(BX_ + ^\ast) \to KX$.
Finite and countable additivity of topological properties in nice spaces
V. V.
Tkachuk
585-601
Abstract: Let $Q \in$ character $\leq \tau$, pseudocharacter $ \leq \tau$, tightness $ \leq \tau$, weight $ \leq \tau$ , $ {P_\tau }$-property, discreteness, Fréchet-Urysohn property, sequentiality, radiality, pseudoradiality, local compactness, $k$-property. If ${X^n} = \cup \{ {X_i}:i \in n\}$, ${X_i} \vdash Q$ for all $i \in n$ then $ X \vdash Q$ (i.e. the property $Q$ is $n$-additive in ${X^n}$ for any $ X \in {T_3}$). Metrizability is $n$-additive in ${X^n}$ provided $X$ is compact or $ c(X) = \omega$. ${\text{ANR}}$-property is closely $n$-additive in ${X^n}$ if $X$ is compact ("closely" means additivity in case $ {X_i}$ is closed in $ {X^n}$). If $Q \in$ metrizability, character $ \leq \tau$, pseudocharacter $\leq \tau$, diagonal number $\leq \tau$ , $i$-weight $\leq \tau$, pseudoweight $\leq \tau$, local compactness then $ Q$ is finitely additive in any topological group.
Characteristic cycles of discrete series for ${\bf R}$-rank one groups
Jen-Tseh
Chang
603-622
Abstract: We determine the characteristic cycles of the discrete series representations for connected $ \mathbb{R}$-rank one linear groups. The computation is made through the moment maps; we determine their fibers and the cohomology in question case by case. The multiplicity of the discrete series, in terms of their Harish-Chandra parameters, is given by recursive formulae; for groups of type $ A$ and $B$ closed formulae are obtained.
On the hyperbolic Kac-Moody Lie algebra $HA\sb 1\sp {(1)}$
Seok-Jin
Kang
623-638
Abstract: In this paper, using a homological theory of graded Lie algebras and the representation theory of $A_1^{(1)}$, we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra $HA_1^{(1)}$ up to level $4$ and deduce some interesting combinatorial identities.
Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups
Aviezri S.
Fraenkel
639-664
Abstract: For a real number $ \alpha$, the floor function $ \left\lfloor \alpha \right\rfloor$ is the integer part of $\alpha$. The sequence $\{ \left\lfloor {m\alpha } \right\rfloor :m = 1,2,3, \ldots \}$ is the Beatty sequence of $\alpha$. Identities are proved which express the sum of the iterated floor functional $ {A^i}$ for $1 \leq i \leq n$, operating on a nonzero algebraic number $\alpha$ of degree $\leq n$, in terms of only $ {A^1} = \left\lfloor {m\alpha } \right\rfloor ,m$ and a bounded term. Applications include discrete chaos (discrete dynamical systems), explicit construction of infinite nonchaotic subsequences of chaotic sequences, discrete order (identities), explicit construction of nontrivial Beatty subsequences, and certain arithmetical semigroups. (Beatty sequences have a large literature in combinatorics. They have also been used in nonperiodic tilings (quasicrystallography), periodic scheduling, computer vision (digital lines), and formal language theory.)
Perverse sheaves and finite-dimensional algebras
K.
Vilonen
665-676
Abstract: We study finite dimensional algebras which arise from categories of perverse sheaves on complex algebraic varieties.
Stability of special instanton bundles on ${\bf P}\sp {2n+1}$
Vincenzo
Ancona;
Giorgio
Ottaviani
677-693
Abstract: We prove that the special instanton bundles of rank $2n$ on ${\mathbb{P}^{2n + 1}}(\mathbb{C})$ with a symplectic structure studied by Spindler and Trautmann are stable in the sense of Mumford-Takemoto. This implies that the generic special instanton bundle is stable. Moreover all instanton bundles on ${\mathbb{P}^5}$ are stable. We get also the stability of other related vector bundles.
Operator semigroups for functional-differential equations with delay
W. M.
Ruess;
W. H.
Summers
695-719
Abstract: We show that a strongly continuous operator semigroup can be associated with the functional differential delay equation (FDE) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {x\prime(t) + ax(t) + Bx(t) \bac... ...{\mathbb{R}^ - }}} = \varphi \in E} & {} \end{array} } \right.$ under local conditions which give wide latitude to those subsets of the state space $ X$ and initial data space $ E$, respectively, where (a) the (generally multivalued) operator $B \subseteq X \times X$ is defined and accretive, and (b) the historyresponsive function $F:D(F) \subseteq E \to X$ is defined and Lipschitz continuous. The associated semigroup is then used to investigate existence and uniqueness of solutions to (FDE). By allowing the domain of the solution semigroup to be restricted according to specific local properties of $B$ and $F$, moreover, our methods automatically lead to assertions on flow invariance. We illustrate our results through applications to the Goodwin oscillator and a single species population model.
The structure of a random graph at the point of the phase transition
Tomasz
Łuczak;
Boris
Pittel;
John C.
Wierman
721-748
Abstract: Consider the random graph models $G(n,\char93 \;{\text{edges}} = M)$ and $ G(n,\operatorname{Prob}({\text{edge}}) = p)$ with $M = M(n) = (1 + \lambda {n^{ - 1/3}})n/2$ and $p = p(n) = (1 + \lambda {n^{ - 1/3}})/n$. For $l \geq - 1$ define an $l$-component of a random graph as a component which has exactly $l$ more edges than vertices. Call an $ l$-component with $ l \geq 1$ a complex component. For both models, we show that when $ \lambda$ is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order ${n^{2/3}}$, and the maximum value of $ l$ is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order ${n^{2/3}}$, and deletion of all suspended trees results in a "smoothed" graph of size of order ${n^{1/3}}$, with the maximum vertex degree $ 3$. The total number of branching vertices, i.e., of degree $3$, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a $3$-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order ${n^{1/3}}$. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960. It is proven that there exists $p(\lambda )$, the limiting planarity probability, with $ 0 < p(\lambda ) < 1$, $p( - \infty ) = 1$, $p(\infty ) = 0$. In particular, $G(n,M)\quad (G(n,p),{\text{resp}}.)$ is almost surely nonplanar iff $(M - n/2){n^{ - 2/3}} \to \infty \;((np - 1){n^{ - 1/3}}) \to \infty ,{\text{resp}}.)$.
Orthogonal polynomials and hypergroups. II. The symmetric case
R.
Lasser
749-770
Abstract: The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on $ {\mathbb{N}_0}$. If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monic recursion formula and random walk polynomials.
A singular representation of $E\sb 6$
B.
Binegar;
R.
Zierau
771-785
Abstract: Algebraic properties of a singular representation of ${{\mathbf{E}}_6}$ are studied. This representation has the Joseph ideal as its annihilator and it remains irreducible when restricted to ${{\mathbf{F}}_4}$.
Approximation from shift-invariant subspaces of $L\sb 2(\bold R\sp d)$
Carl
de Boor;
Ronald A.
DeVore;
Amos
Ron
787-806
Abstract: A complete characterization is given of closed shift-invariant subspaces of $ {L_2}({\mathbb{R}^d})$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
On convergence and closedness of multivalued martingales
Zhen Peng
Wang;
Xing Hong
Xue
807-827
Abstract: In this paper, various convergence theorems and criteria of closedness of multivalued martingales, submartingales, and supermartingales are proved.
Uniqueness theorems for parametrized algebraic curves
Peter
Hall
829-840
Abstract: Let ${L_1}, \ldots ,{L_n}$ be lines in ${\mathbb{P}^2}$ and let $f,g:{\mathbb{P}^1} \to {\mathbb{P}^2}$ be nonconstant algebraic maps. For certain configurations of lines ${L_1}, \ldots ,{L_n}$, the hypothesis that, for $i = 1, \ldots ,n$, the inverse images $ {f^{ - 1}}({L_i})$ and ${g^{ - 1}}({L_i})$ are equal, not necessarily with the same multiplicities, implies that $ f$ is identically equal to $ g$.
Carleson measures on spaces of homogeneous type
Steven C.
Gadbois;
William T.
Sledd
841-862
Abstract: Let $X$ be a space of homogeneous type in the sense of Coifman and Weiss $[{\text{CW}}2]$ and let ${X^ + } = X \times {{\mathbf{R}}^ + }$. A positive function $F$ on ${X^ + }$ is said to have horizontal bounded ratio $({\text{HBR}})$ on ${X^ + }$ if there is a constant ${A_F}$ so that $F(x,t) \leq {A_F}F(y,t)$ whenever $\rho (x,y) < t$. (By Harnack's inequality, a well-known example is any positive harmonic function in the upper half plane.) $ {\text{HBR}}$ is a rich class that is closed under a wide variety of operations and it provides useful majorants for many classes of functions that are encountered in harmonic analysis. We are able to prove theorems in spaces of homogeneous type for functions in $ {\text{HBR}}$ which are analogous to the classical Carleson measure theorems and to extend these results to the functions which they majorize. These results may be applied to obtain generalizations of the original Carleson measure theorem, and of results of Flett's which contain the Hardy-Littlewood theorems on intermediate spaces of analytic functions. Hörmander's generalization of Carleson's theorem is included and it is possible to extend those results to the atomic ${H^p}$ spaces of Coifman and Weiss.
Invariant subspaces of the Dirichlet shift and pseudocontinuations
Stefan
Richter;
Carl
Sundberg
863-879
Abstract: In this paper we study extremal functions for invariant subspaces $\mathcal{M}$ of the Dirichlet shift, i.e., solutions $\varphi$ of the extremal problem $\displaystyle \sup \{ \vert{f^{(n)}}(0)\vert/{\left\Vert f \right\Vert _D}:f \in \mathcal{M},f \ne 0\}$ . Here $n$ is the smallest nonnegative integer such that the sup is positive. It is known that such a function $ \varphi$ generates $\mathcal{M}$. We show that the derivative $(z\varphi )\prime$ has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where $\varphi$ is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.
Local asymptotic distribution of zeros of orthogonal polynomials
Vilmos
Totik;
Joseph L.
Ullman
881-894
Abstract: Converse results, which state a relation (inequality) for measures from that on their logarithmic potentials, are applied to local density of zeros of orthogonal polynomials when the measure of orthogonality is a general one with compact support. It will be shown that if the measure is sufficiently thick on a part of its support, then on that part the density of the zeros will be at least as large as the equilibrium measure of the support. A corresponding upper estimate on the distribution of the zeros will also be proved. All of our estimates are sharp, and they localize several well-known results.
Regularity of algebras related to the Sklyanin algebra
J. T.
Stafford
895-916
Abstract: This paper continues the research of [SS] by finding further examples of (Artin-Schelter) regular rings of dimension four. Unlike the threedimensional case studied in [ATV1, 2], these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data. Indeed, we find a one-parameter family of regular algebras associated to this data.
On invertible bimodules and automorphisms of noncommutative rings
Robert M.
Guralnick;
Susan
Montgomery
917-937
Abstract: In this article, we attempt to generalize the result that for a commutative ring $R$ the outer automorphism group of $ R$-automorphisms of $ {M_n}(R)$ is abelian of exponent $n$. It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.