The structure of Mackey functors
Jacques
Thévenaz;
Peter
Webb
1865-1961
Abstract: Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological $ {\text{K}}$-theory of classifying spaces, the algebraic ${\text{K}}$-theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.
Complete quotient Boolean algebras
Akihiro
Kanamori;
Saharon
Shelah
1963-1979
Abstract: For $I$ a proper, countably complete ideal on the power set $ \mathcal{P}(x)$ for some set $X$, can the quotient Boolean algebra $\mathcal{P}(X)/I$ be complete? We first show that, if the cardinality of $X$ is at least $ {\omega _3}$, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal $I$ is a (nontrivial) ideal over a cardinal $\kappa$ which is $ {\kappa ^ + }$-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over $ \kappa = {\omega _1}$ relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: ${2^{{\omega _1}}} = {\omega _3}$ and there is an ideal ideal $I$ over $ {\omega _1}$ such that $ \mathcal{P}({\omega _1})/I$ is complete. (The cardinality assertion implies that there is no ideal over ${\omega _1}$ which is $ {\omega _2}$-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)
The spectrum of the Hodge Laplacian for a degenerating family of hyperbolic three manifolds
Jozef
Dodziuk;
Jeffrey
McGowan
1981-1995
Abstract: We consider a sequence $ ({M_n})_{n = 1}^\infty$ of compact hyperbolic manifolds converging to a complete hyperbolic manifold ${M_0}$ with cusps. The Laplace operator acting on the space of ${L^2}$ differential forms on ${M_0}$ has continuous spectrum filling the half-line $[0,\infty )$. One expects therefore that the spectra of this operator on ${M_n}$ accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accumulation.
The direct decompositions of a group $G$ with $G/G'$ finitely generated
Francis
Oger
1997-2010
Abstract: We consider the class $\mathcal{C}$ which consists of the groups $ M$ with $M/M'$ finitely generated which satisfy the maximal condition on direct factors. It is well known that any $ \mathcal{C}$-group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any $ \mathcal{C}$-group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for $\mathcal{C}$-groups, a slightly different notion of decomposition, that we call $J$-decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of $J$-decompositions, several generalizations of results of R. Hirshon. For instance, we have $ \mathbb{Z} \times G \cong \mathbb{Z} \times H$ for any groups $G$, $H$ which satisfy $M \times G \cong M \times H$ for a $\mathcal{C}$-group $M$.
Interpretation of Lavrentiev phenomenon by relaxation: the higher order case
Marino
Belloni
2011-2023
Abstract: We consider integral functionals of the calculus of variations of the form $u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $F$ with respect to the weak $ W_{{\text{loc}}}^{n,1}(0,1)$ convergence can be written as
Smooth sets for a Borel equivalence relation
Carlos E.
Uzcátegui A.
2025-2039
Abstract: We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from ${\mathbf{Z}}$-actions to actions by arbitrary countable groups. We show that the $\sigma$-ideal of closed smooth sets is $ \Pi _1^1$ non-Borel.
Singular Jacobi forms
Jae-Hyun
Yang
2041-2049
Abstract: We introduce the differential operator ${M_{g,h,\mathcal{M}}}$ characterizing singular Jacobi forms. We also characterize singular Jacobi forms by the weight of the associated rational representation of the general linear group. And we provide eigenfunctions of the differential operator $ {M_{g,h,\mathcal{M}}}$.
Generalized $(t,s)$-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series
Gerhard
Larcher;
Harald
Niederreiter
2051-2073
Abstract: The theory of $ (t,s)$-sequences leads to powerful constructions of low-discrepancy sequences in an $s$-dimensional unit cube. We generalize this theory in order to cover arbitrary sequences constructed by the digital method and, in particular, the Kronecker-type sequences introduced by the second author. We define diophantine approximation constants for formal Laurent series over finite fields and show their connection with the distribution properties of Kronecker-type sequences. The main results include probabilistic theorems on the distribution of sequences constructed by the digital method and on the diophantine approximation character of $s$-tuples of formal Laurent series over finite fields.
$L\sp p$ theory of differential forms on manifolds
Chad
Scott
2075-2096
Abstract: In this paper, we establish a Hodge-type decomposition for the $ {L^p}$ space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an ${L^p}$ estimate which contains, as a special case, the ${L^2}$ result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the $ {L^p}$ boundedness of Green's operator which we use in developing the $ {L^p}$ theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the ${L^p}$ norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the $A$-harmonic equation.
The connection matrix in Morse-Smale flows. II
James F.
Reineck
2097-2110
Abstract: Given a connection matrix for a Morse-Smale flow on a compact manifold, if there are no periodic orbits of equal or adjacent indices related in the partial order, we show that the periodic orbits can be replaced by doubly connected rest points in such a way that the given connection matrix induces the unique connection matrix for the resulting flow. It follows that for this class of flows, all nonuniqueness in the connection matrix is a consequence of the continuation theorem for connection matrices.
A constructive proof of the Poincar\'e-Birkhoff theorem
Yong
Li;
Zheng Hua
Lin
2111-2126
Abstract: In this paper, with the use of the homotopy method, a constructive proof of the Poincaré-Birkhoff theorem is given. This approach provides a global method for finding fixed points of area-preserving maps and periodic solutions of Duffing equations.
Geometry of strictly convex domains and an application to the uniform estimate of the $\overline\partial$-problem
Ten Ging
Chen
2127-2137
Abstract: In this paper, we construct a nice defining function $\rho$ for a bounded smooth strictly convex domain $\Omega$ in ${R^n}$ with explicit gradient and Hessian estimates near the boundary $ \partial \Omega$ of $ \Omega$. From the approach, we deduce that any two normals through $\partial \Omega$ do not intersect in any tubular neighborhood of $ \partial \Omega$ with radius which is less than $ \frac{1} {K}$, where $ K$ is the maximum principal curvature of $ \partial \Omega$. Finally, we apply such $\rho$ to obtain an explicit upper bound of the constant $ {C_\Omega }$ in the Henkin's estimate $ {\left\Vert {{H_\Omega }f} \right\Vert _{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left\Vert f \right\Vert _{{L^\infty }(\Omega )}}$ of the $\partial$-problem on strictly convex domains $ \Omega$ in ${{\mathbf{C}}^n}$.
The regionally proximal relation
Joseph
Auslander;
David B.
Ellis;
Robert
Ellis
2139-2146
Abstract: Sufficient conditions for the regionally proximal relation $ Q(X)$ of a minimal flow to be an equivalence relation are obtained in terms of the group $ \mathcal{G}(X)$ of the flow and various groups which depend only on the acting group $T$.
Algebraic field theory on curved manifolds
Martin
Olesen
2147-2160
Abstract: In this paper we set up an algebraic framework for the study of quantum field theory in a class of manifolds, which includes Minkowski space and the Kruskal spacetime. The formalism provides a unifying framework for studying problems of Bisognano-Wichmann type, e.g., Hawking radiation in black hole geometries. Analogously to flat spacetime, we establish a correspondence between isometries of certain wedge domains of spacetime and the modular structure of the local algebras. Under an ergodic hypothesis, the wedge algebras are shown to be type III factors as expected, and we derive a result concerning factorization of the equilibrium state. This result generalizes a similar one obtained by Sewell in [Ann. Phys. 141 (1982), 201-224]. Finally an example of a quantum field theory satisfying the basic axioms is constructed. The local algebras are field algebras of bosonic free field solutions to the Klein-Gordon equation twisted through a PCT-like conjugation, and we show that this model realizes the abstract properties developed on the axiomatic basis.
Some inequalities of algebraic polynomials with nonnegative coefficients
Weiyu
Chen
2161-2167
Abstract: Let ${S_n}$ be the collection of all algebraic polynomials of degree $ \leqslant n$ with nonnegative coefficients. In this paper we discuss the extremal problem $\omega (x)$ is a positive and integrable function. This problem is solved completely in the cases $\displaystyle ({\text{i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;$ $\displaystyle ({\text{ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;$ $\displaystyle ({\text{iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.$ The second case was solved by Varma for some values of $\alpha$ and by Milovanović completely. We provide a new proof here in this case.
Bounded point evaluation in ${\bf C}\sp n$
R. M.
Range;
M. I.
Stessin
2169-2177
Abstract: A positive Borel measure $\mu$ on a domain $\Omega \in {{\mathbf{C}}^n}$ is said to be in $ \mathcal{R}(\Omega )$, if point evaluations at every $p \in \Omega$ are locally uniformly bounded in $ {L^2}(\mu )$-norm. It is proved that the multiplication of a measure in $ \mathcal{R}(\Omega )$ by a function decreasing no faster than a power of a holomorphic function produces a measure in $\mathcal{R}(\Omega )$. Some applications to classical Hardy and Bergman spaces are given.
Dualit\'e dans le groupe de Grothendieck de la cat\'egorie des repr\'esentations lisses de longueur finie d'un groupe r\'eductif $p$-adique
Anne-Marie
Aubert
2179-2189
Abstract: We define an involution on the Grothendieck ring of the category of finite length smooth representations of a $ p$-adic algebraic group, which is a direct analogue Curtis-Alvis duality for finite groups of Lie type. This involution commutes with taking the contragredient, with parabolic induction and, up a few twists, with truncation. It also preserves the irreducible representations up to sign.
Simultaneous triangularizability, near commutativity and Rota's theorem
A. A.
Jafarian;
H.
Radjavi;
P.
Rosenthal;
A. R.
Sourour
2191-2199
Abstract: In this paper we consider simultaneously triangularizable collections of compact operators and show that similarities of any finite subcollection can be made arbitrarily close to commuting normal operators. As a consequence, we obtain a variant of a theorem of G.-C. Rota. Also, we give some sufficient conditions for simultaneous triangularization of collections of compact operators. Finally, several counterexamples are given.
On the Littlewood-Paley-Stein $g$-function
Stefano
Meda
2201-2212
Abstract: We consider semigroups $({T_t})$, which are contractive on $ {L^p}(M)$ for all $q \in [1,2)$. We give an example (on symmetric spaces of the noncompact type) which shows that the Littlewood-Paley-Stein $ g$-function associated to the infinitesimal generator of $({T_t})$ may be unbounded on ${L^q}(M)$ and on
An access theorem for analytic functions
Marvin
Ortel
2213-2223
Abstract: Suppose that $\mathcal{M}$ is an analytic manifold, ${m_0} \in \mathcal{M},f:\mathcal{M} \to \mathbb{R}$, and $f$ is analytic. Then at least one of the following three statements is true: (1) ${m_0}$ is a local maximum of $f$. (2) There is a continuous path $ \sigma :[0,1] \to \mathcal{M}$ such that $ \sigma (0) = {m_0}$, $f \circ \sigma$ is strictly increasing on $ [0,1]$, and $\sigma (1)$ is a local maximum of $f$. (3) There is a continuous path $ \sigma :[0,1) \to \mathcal{M}$ with these properties: $\sigma (0) = {m_0};f \circ \sigma$ is strictly increasing on $[0,1)$; whenever $K$ is a compact subset of $\mathcal{M}$, there is a corresponding number $d(K) \in [0,1)$ such that $\sigma (t) \notin K$ for all $t \in [d(K),1)$.
A covering cocycle which does not grow linearly
Kathleen M.
Madden
2225-2234
Abstract: A cocycle $h:X \times {Z^m} \to {R^n}$ of a $ {Z^m}$ action on a compact metric space, provides an ${R^n}$ suspension flow (analogous to a flow under a function) on a space ${X_h}$ which may not be Hausdorff or even $ {T_1}$. Linear growth of $ h$ guarantees that $ {X_h}$ is a Hausdorff space; when $m = n$, linear growth is a consequence of $ {X_h}$ being Hausdorff and a covering condition. This paper contains the construction of a cocycle $h:X \times Z \to {R^2}$ which does not grow linearly yet produces a locally compact Hausdorff space with the covering condition. The $Z$ action used in the construction is a substitution minimal set.
Distinct degree factorizations for polynomials over a finite field
Arnold
Knopfmacher;
Richard
Warlimont
2235-2243
Abstract: Let $ {\widetilde{\mathbb{F}}_q}[X]$ denote the multiplicative semigroup of monic polynomials in one indeterminate $X$, over a finite field ${\mathbb{F}_q}$. We determine for each fixed $ q$ and fixed $ n$ the probability that a polynomial of degree $n$ in $ {\mathbb{F}_q}[X]$ has irreducible factors of distinct degrees only. These results are of relevance to various polynomial factorization algorithms.
Conditions on the logarithmic derivative of a function implying boundedness
T. H.
MacGregor;
F.
Rønning
2245-2254
Abstract: In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain $\Omega$. We obtain conditions on $ \Omega$ which imply that the functions are bounded and that their first derivatives belong to ${H^p}$ for some $p \geqslant 1$. For certain domains $\Omega$ the sufficient conditions that we give are also, in some sense, necessary. Examples of domains to which the results apply are given.
Existence of extremal metrics on compact almost homogeneous K\"ahler manifolds with two ends
Daniel
Guan
2255-2262
Abstract: In this note we prove the existence and the uniqueness of extremal metrics in every Kähler class of any compact almost homogeneous Kähler manifold with two ends by considering the scalar curvature equations, those manifolds might not be projective. We also prove that there are extremal metrics in some Kähler classes of a completion of the multicanonical line bundle of a Kähler-Einstein manifold of positive Ricci curvature.
The Connes spectrum of group actions and group gradings for certain quotient rings
James
Osterburg;
Xue
Yao
2263-2275
Abstract: Let $H$ be a finite-dimensional, semisimple Hopf algebra over an algebraically closed field $ K$ where $H$ is either commutative or cocommutative. We let $A$ be an $H$-module algebra which is semiprime right Goldie. We show that the Connes spectrum of $H$ acting on $A$ is the Connes spectrum of $H$ acting on the classical quotient ring of $ A$. In our last section, we define a symmetric quotient ring and show that the Connes spectrum of the ring and its quotient ring are the same. Finally, we apply our results to finite group actions and group gradings.