Weights for classical groups
Jian Bei
An
1-42
Abstract: This paper proves the Alperin's weight conjecture for the finite unitary groups when the characteristic r of modular representation is odd. Moreover, this paper proves the conjecture for finite odd dimensional special orthogonal groups and gives a combinatorial way to count the number of weights, block by block, for finite symplectic and even dimensional special orthogonal groups when r and the defining characteristic of the groups are odd.
Second order differentiability of convex functions in Banach spaces
Jonathan M.
Borwein;
Dominikus
Noll
43-81
Abstract: We present a second order differentiability theory for convex functions on Banach spaces.
Separation and coding
Stephen
Watson
83-106
Abstract: We construct a normal collectionwise Hausdorff space which is not collectionwise normal with respect to copies of [0,1]. We do this by developing a general theory of coding properties into topological spaces. We construct a para-Lindelöf regular space in which para-Lindelöf is coded directly rather than $\sigma $-para-Lindelöf and normal. We construct a normal collectionwise Hausdorff space which is not collectionwise normal in which collectionwise Hausdorff is coded directly rather than obtained as a side-effect to countable approximation. We also show that the Martin's axiom example of a normal space which is not collectionwise Hausdorff is really just a kind of "dual" of Bing's space.
Indecomposable generalized Cohen-Macaulay modules
Mihai
Cipu;
Jürgen
Herzog;
Dorin
Popescu
107-136
Abstract: The aim of this paper is to study the indecomposable modules which are Cohen-Macaulay on the punctured spectrum of a local ring, and to describe some of their invariants such as their local cohomology groups and ranks. One of our main concerns is to find indecomposable quasi-Buchsbaum modules of high rank with prescribed cohomology over a regular local ring.
Growth functions for some nonautomatic Baumslag-Solitar groups
Marcus
Brazil
137-154
Abstract: The growth function of a group is a generating function whose coefficients $ {a_n}$ are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form $ \langle a,b\vert{a^{ - 1}}ba = {a^2}\rangle$ based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups $\langle a,b\vert{a^{ - 1}}ba = {a^m}\rangle$ for all positive integers m.
On the generalized Benjamin-Ono equation
Carlos E.
Kenig;
Gustavo
Ponce;
Luis
Vega
155-172
Abstract: We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation $ {\partial _t}u + {u^k}{\partial _x}u - {\partial _x}{D_x}u = 0$, $k \in {\mathbb{Z}^ + }$, in Sobolev spaces ${H^s}(\mathbb{R})$. For small data and higher nonlinearities $(k \geq 2)$ new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrödinger equations (or systems) of the form $ {\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0$.
Vaught's conjecture for varieties
Bradd
Hart;
Sergei
Starchenko;
Matthew
Valeriote
173-196
Abstract: We prove that if $\mathcal{V}$ is a superstable variety or one with few countable models then $\mathcal{V}$ is the varietal product of an affine variety and a combinatorial variety. Vaught's conjecture for varieties is an immediate consequence.
Operations on resolutions and the reverse Adams spectral sequence
David A.
Blanc
197-213
Abstract: We describe certain operations on resolutions in abelian categories, and apply them to calculate part of a reverse Adams spectral sequence, going "from homotopy to homology", for the space $ {\mathbf{K}}(\mathbb{Z}/2,n)$. This calculation is then used to deduce that there is no space whose homotopy groups are the reduction $\bmod \; 2$ of ${\pi _\ast}{{\mathbf{S}}^r}$. As another application of the operations we give a short proof of T. Y. Lin's theorem on the infinite projective dimension of all nonfree $\pi $-modules.
Noncharacteristic embeddings of the $n$-dimensional torus in the $(n+2)$-dimensional torus
David
Miller
215-240
Abstract: We construct certain exotic embeddings of the n-torus $ {T^n}$ in ${T^{n + 2}}$ in the standard homotopy class. We turn an embedding $f:{T^n} \to {T^{n + 2}}$ characteristic if there exists some map $\alpha :{T^{n + 2}} \to {T^{n + 2}}$ in the standard homotopy class with the property that $ \alpha \; \circ \;f:{T^n} \to {T^{n + 2}}$ is the standard coordinate inclusion and $\alpha ({T^{n + 2}} - f({T^n})) \subset {T^{n + 2}} - {T^n}$. We find examples of noncharacteristic embeddings, f, in dimensions $n = 4k + 1$, $n \geq 5$, and show that these examples are not even cobordant to characteristic embeddings. We let G denote the fundamental group of the complement of the standard coordinate inclusion, ${T^{n + 2}} - {T^n}$. Then we can associate to f a real-valued signature function on the set of j-dimensional unitary representations of $ \bar G$, where $ \bar G$ denotes the fundamental group of the localization of ${T^{n + 2}} - {T^n}$ with respect to homology with local coefficients in $\mathbb{Z}[{\mathbb{Z}^{n + 2}}]$. This function is a cobordism invariant which has certain periodicity properties for characteristic embeddings. We verify that this periodicity does not hold for our examples, f, implying that they are not characteristic. Additional results include a proof that the examples, f, become cobordant to characteristic embeddings upon taking the cartesian product with the identity map on a circle.
The $H\sp 2$ corona problem and $\overline\partial\sb b$ in weakly pseudoconvex domains
Mats
Andersson
241-255
Abstract: We derive a Bochner-Kodaira-Nakano-Morrey-Kohn-Hörmander type equality in holomorphic vector bundles and obtain ${L^2}$-estimates for ${\bar \partial _b}$ in a pseudoconvex domain that admits a plurisubharmonic ${C^2}$ defining function. We combine these with the trick in Wolff's proof of the corona theorem and obtain a ${H^2}$-corona theorem in such a domain.
Homology and cohomology of $\Pi$-algebras
W. G.
Dwyer;
D. M.
Kan
257-273
Abstract: We study a type of homological algebra associated to the collection of all homotopy groups of a space (just as the theory of group homology is associated to the fundamental group).
Varieties of commutative semigroups
Andrzej
Kisielewicz
275-306
Abstract: In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.
Wavelets of multiplicity $r$
T. N. T.
Goodman;
S. L.
Lee
307-324
Abstract: A multiresolution approximation ${({V_m})_{m \in {\mathbf{Z}}}}$ of ${L^2}({\mathbf{R}})$ is of multiplicity $r > 0$ if there are r functions ${\phi _1}, \ldots ,{\phi _r}$ whose translates form a Riesz basis for ${V_0}$. In the general theory we derive necessary and sufficient conditions for the translates of ${\phi _1}, \ldots ,{\phi _r},\;{\psi _1}, \ldots ,{\psi _r}$ to form a Riesz basis for ${V_1}$. The resulting reconstruction and decomposition sequences lead to the construction of dual bases for ${V_0}$ and its orthogonal complement ${W_0}$ in ${V_1}$. The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given.
Harmonic diffeomorphisms of the hyperbolic plane
Kazuo
Akutagawa
325-342
Abstract: In this paper, we consider the Dirichlet problem at infinity for harmonic maps between the Poincaré model D of the hyperbolic plane ${\mathbb{H}^2}$, and solve this when given boundary data are ${C^4}$ immersions of $ D(\infty )$, the boundary at infinity of D, to $ D(\infty )$. Also, we present a construction of nonconformal harmonic diffeomorphisms of D, and give a complete description of the boundary behavior, including their first derivatives.
Smooth extensions for finite CW complexes
Guihua
Gong
343-358
Abstract: In this paper, we have completely classified the ${C_n}$-smooth elements of $\operatorname{Ext} (X)$ modulo torsion for X being an arbitrary finite CW complex.
Best uniform approximation by solutions of elliptic differential equations
P. M.
Gauthier;
D.
Zwick
359-374
Abstract: We investigate best uniform approximations to continuous functions on compact subsets of $ {\mathbb{R}^n}$ by solutions of elliptic differential equations and, in particular, by harmonic functions. An axiomatic setting general enough to encompass problems of this kind is given, and in this context we extend necessary and sufficient conditions for best harmonic approximation on precompact Jordan domains to arbitrary compact sets and to more general classes of solutions of linear elliptic differential equations.
A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold
Bruce K.
Driver
375-395
Abstract: The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let $h:[0,1] \to {T_0}M$ be a ${C^1}$ function where M is a compact Riemannian manifold, $o \in M$ is a base point, and ${T_o}M$ is the tangent space to M at $o \in M$. Let $W(M)$ be the space of continuous paths from [0,1] into M, $\nu$ be Wiener measure on $W(M)$ concentrated on paths starting at $ o \in M$, and ${H_s}(\omega )$ denote the stochastic-parallel translation operator along a path $\omega \in W(M)$ up to "time" s. (Note: ${H_s}(\omega )$ is only well defined up to $ \nu$-equivalence.) For $\omega \in W(M)$ let ${X^h}(\omega )$ denote the vector field along $ \omega$ given by $ X_s^h(\omega ) \equiv {H_s}(\omega )h(s)$ for each $ s \in [0,1]$. One should interpret ${X^h}$ as a vector field on $W(M)$. The vector field ${X^h}$ induces a flow ${S^h}(t, \bullet ):W(M) \to W(M)$ which leaves Wiener measure $(\nu )$ quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if $ h(1) = 0$ and the Wiener measure $(\nu )$ is replaced by a pinned Wiener measure $({\nu _e})$. (The measure ${\nu _e}$ is proportional to the measure $ \nu$ conditioned on the set of paths which start at $o \in M$ and end at a fixed end point $ e \in M$.) Also as in [13], one gets an integration by parts formula for the vector-fields ${X^h}$ defined above.
Infinite families of isomorphic nonconjugate finitely generated subgroups
F. E. A.
Johnson
397-406
Abstract: Let $ \langle \;,\;\rangle :L \times L \to \mathbb{Z}$ be a nondegenerate symmetric bilinear form on a finitely generated free abelian group L which splits as an orthogonal direct sum $(L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle )$ in which $ ({L_1},\;\langle \;,\;\rangle )$ has signature (2, 1), $({L_2},\;\langle \;,\;\rangle )$ has signature (n, 1) with $n \geq 2$, and $({L_3},\;\langle \;,\;\rangle )$ is either zero or indefinite with $ {\text{rk}}_\mathbb{Z}({L_3}) \geq 3$. We show that the integral automorphism group $ {\operatorname{Aut} _\mathbb{Z}}(L,\;\langle \;,\;\rangle )$ contains an infinite family of mutually isomorphic finitely generated subgroups ${({\Gamma _\sigma })_{\sigma \in \Sigma }}$, no two of which are conjugate. In the simplest case, when ${L_3} = 0$, the groups ${\Gamma _\sigma }$ are all normal subdirect products in a product of free groups or surface groups. The result can be seen as a failure of the rigidity property for subgroups of infinite covolume within the corresponding Lie group $ {\operatorname{Aut} _\mathbb{Z}}(L{ \otimes _\mathbb{Z}}\mathbb{R},\;\langle \;,\;\rangle \otimes 1)$.
Tur\'an inequalities and zeros of Dirichlet series associated with certain cusp forms
J. B.
Conrey;
A.
Ghosh
407-419
Abstract: The "Turan inequalities" are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros. We give a one-parameter family of generalized Dirichlet series, each with functional equation, for which the Turan inequalities hold for the associated $\xi$-function (normalized so that the critical line is the real axis). For a discrete set of values of the parameter the Dirichlet series has an Euler product and is the L-series associated to a modular form. For these we expect the analogue of the Riemann Hypothesis to hold. For the rest of the values of the parameter we do not expect an analogue of the Riemann Hypothesis. We show for one particular value of the parameter that the Dirichlet series in fact has zeros within the region of absolute convergence.
Abel's theorem for twisted Jacobians
Donu
Arapura;
Kyungho
Oh
421-433
Abstract: A twisted version of the Abel-Jacobi map, associated to a local system with finite monodromy on a smooth projectve complex curve, is introduced. An analogue of Abel's theorem characterizing the kernel of this map is proved. The proof, which is new even in the classical case, involves reinterpreting the Abel-Jacobi map in the language of mixed Hodge structures and their extensions.
Intersection bodies and the Busemann-Petty problem
R. J.
Gardner
435-445
Abstract: It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space $ {\mathbb{E}^d}$ is negative for a given d if and only if certain centrally symmetric convex bodies exist in ${\mathbb{E}^d}$ which are not intersection bodies. It is also shown that a cylinder in ${\mathbb{E}^d}$ is an intersection body if and only if $d \leq 4$, and that suitably smooth axis-convex bodies of revolution are intersection bodies when $ d \leq 4$. These results show that the Busemann-Petty problem has a negative answer for $d \geq 5$ and a positive answer for $d = 3$ and $d = 4$ when the body with smaller sections is a body of revolution.