Eisenstein series and the Selberg trace formula. II
H.
Jacquet;
D.
Zagier
1-48
Abstract: The integral of the kernel of the trace formula against an Eisenstein series is investigated. The analytic properties of this integral imply the divisibility of the convolution $L$-function attached to a form by the zeta function of the field.
Knapp-Wallach Szeg\H o integrals. II. The higher parabolic rank case
B. E.
Blank
49-59
Abstract: Let $G$ be a connected reductive linear Lie group with compact center and real rank $l$. For each integer $k(1 \leqslant k \leqslant l)$ and each discrete series representation $\pi$ of $G$, an explicit embedding of $\pi$ into a generalized principal series representation induced from a parabolic subgroup of rank $ k$ is given. The existence of such embeddings was proved by W. Schmid. In this paper an explicit integral formula with Szegö kernel is given which provides these mappings.
A dimension formula for Hermitian modular cusp forms of degree two
Min King
Eie
61-72
Abstract: An explicit dimension formula for the vector space of Hermitian modular cusp forms of degree two with respect to the modular group $ {\Gamma _2}({\mathbf{Z}}[i]) = \operatorname{SU} (2,2) \cap {M_4}({\mathbf{Z}}[i])$ is obtained via the Selberg trace formula and its arithmetic properties. Also, a generating function for the graded ring of Hermitian cusp forms of degree two is given.
Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups. I
Luis G.
Casian;
David H.
Collingwood
73-107
Abstract: Let $G$ be a connected semisimple real matrix group. It is now apparent that the representation theory of $G$ is intimately connected with the complex geometry of the flag variety $ \mathcal{B}$. By studying appropriate orbit structures on $\mathcal{B}$, we are naturally led to representation theory in the category of Harish-Chandra modules $ \mathcal{H}\mathcal{C}$, or the representation theory of category $\mathcal{B}$, associated to $\mathcal{H}\mathcal{C}$ and
The Szeg\H o projection: Sobolev estimates in regular domains
Harold P.
Boas
109-132
Abstract: The Szegö projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornæss, and Catlin. It preserves local smoothness near boundary points of finite type.
Wrappings of permutations
Saul
Stahl
133-152
Abstract: A theory of wrappings of permutations is constructed which is analogous to the well-known concept of branched coverings of Riemann surfaces. It is shown that this theory is strong enough to contain combinatorial definitions of such well-known groups as Fuchsian groups of the first kind and triangle groups.
The homology and higher representations of the automorphism group of a Riemann surface
S. A.
Broughton
153-158
Abstract: The representations of the automorphism group of a compact Riemann surface on the first homology group and the spaces of $ q$-differentials are decomposed into irreducibles. As an application it is shown that ${M_{24}}$ is not a Hurwitz group.
A global approach to the Rankin-Selberg convolution for ${\rm GL}(3,{\bf Z})$
Solomon
Friedberg
159-174
Abstract: We discuss the Rankin-Selberg convolution on $\operatorname{GL} (3,{\mathbf{Z}})$ in the `classical' language of symmetric spaces and automorphic forms.
Orthogonal polynomials, measures and recurrences on the unit circle
Paul
Nevai
175-189
Abstract: New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials.
Periodic phenomena in the classical Adams spectral sequence
Mark
Mahowald;
Paul
Shick
191-206
Abstract: We investigate certain periodic phenomena in the classical Adams sepctral sequence which are related to the polynomial generators $ {\nu _n}$ in ${\pi _{\ast}}(\operatorname{BP} )$. We define the notion of a class $a$ in ${\operatorname{Ext} _A}({\mathbf{Z}}/2,{\mathbf{Z}}/2)$ being ${\nu _n}$-periodic or ${\nu _n}$-torsion and prove that classes that are $ {\nu _n}$-torsion are also $ {\nu _k}$-torsion for all $ k$ such that $0 \leqslant k \leqslant n$. This allows us to define a chromatic filtration of ${\operatorname{Ext} _A}({\mathbf{Z}}/2,{\mathbf{Z}}/2)$ paralleling the chromatic filtration of the Novikov spectral sequence ${E_2}$-term given in [13].
Oscillatory phenomena associated to semilinear wave equations in one spatial dimension
T.
Cazenave;
A.
Haraux
207-233
Abstract: Let $g$ be a nonincreasing, odd $ {C^1}$ function and $l > 0$. We establish that for any solution $u \in C({\mathbf{R}};H_0^1(0,l))$ of the equation ${u_{tt}} - {u_{xx}} + g(u) = 0$ and any ${x_0} \in ]0,l[$, the function $t \mapsto u(t,{x_0})$ satisfies the following alternative: either $u(t,{x_0}) = 0,\forall t \in {\mathbf{R}}$, or $ \forall a \in {\mathbf{R}}$, there exist ${t_1}$ and ${t_2}$ in $ [a,a + 2l]$ such that $u({t_1},{x_0}) > 0$ and $u({t_2},{x_0}) < 0$. We study the structure of the set of points satisfying the first possibility. We give analogous results for ${u_x}$ and for some other homogeneous boundary conditions.
Stability results for a diffusion equation with functional drift approximating a chemotaxis model
James M.
Greenberg;
Wolfgang
Alt
235-258
Abstract: A hyperbolic-parabolic "chemotaxis" system modelling aggregation of motile cells by production of a diffusible chemoattractant, is approximated by a scalar diffusion equation for the cell density, where the drift term is an explicit functional of the current density profile. We prove the unique existence and, using the Hopf-Cole transformation, the local stability of an equilibrium, i.e. a steady aggregation state. We also discuss the limiting hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps.
Forbidden intersections
Peter
Frankl;
Vojtěch
Rödl
259-286
Abstract: About ten years ago P. Erdös conjectured that if $\mathcal{F}$ is a family of subsets of $\{ 1,2, \ldots ,n\}$ without $F$, $ \vert\mathcal{F}\vert < {(2 - \varepsilon )^n}$ holds for some positive absolute constant $ \varepsilon$. Here this conjecture is proved in a stronger form (Theorem 1.1), which solves a 250 problem of Erdös. Suppose $\mathcal{C}$ is a code (i.e., a collection of sequences of length $n$) over an alphabet of $q$ elements, where $\tfrac{1} {2} > \delta > 0$ is arbitrary. Suppose further that there are no two codewords at Hamming distance $d$ where $d$ is a fixed integer, $\delta n < d < (1 - \delta )n$, and $ d$ is even if $ q = 2$. Then $ \vert\mathcal{C}\vert < {(q - \varepsilon )^n}$, where $\varepsilon > 0$ depends only on $ q$ and $\delta$. The following conjecture of Erdös and Szemerédi is also proved: If $ \mathcal{F}$ is a family of subsets of $ \{ 1,2, \ldots ,n\}$ not containing a weak $\Delta$-system of size $r$ (cf. Definition 1.8), then $ \vert\mathcal{F}\vert < {(2 - {\varepsilon _r})^n}$, ${\varepsilon _r} > 0$ holds. An old conjecture of Larman and Rogers is established in the following stronger form: Let $ \mathcal{A}$ be a collection of $4n$-dimensional $( \pm 1)$-vectors, $ r \geqslant 2$ is a fixed integer. Suppose that $A$ does not contain $r$ pairwise orthogonal vectors. Then $ \vert\mathcal{A}\vert < {(2 - \varepsilon )^{4n}}$. All these results can be deduced from our most general result (Theorem 1.16) which concerns the intersection pattern of families of partitions. This result has further implications in Euclidean Ramsey theory as well as for isometric embeddings into the Hamming space $H(n,q)$ (cf. Theorem 9.1).
Two-dimensional nonlinear boundary value problems for elliptic equations
Gary M.
Lieberman
287-295
Abstract: Boundary regularity of solutions of the fully nonlinear boundary value problem $\displaystyle F(x,u,Du,{D^2}u) = 0\quad {\text{in}}\;\Omega ,\qquad G(x,u,Du) = 0\quad {\text{on}}\;\partial \Omega$ is discussed for two-dimensional domains $\Omega$. The function $F$ is assumed uniformly elliptic and $G$ is assumed to depend (in a nonvacuous manner) on $Du$. Continuity estimates are proved for first and second derivatives of $u$ under weak hypotheses for smoothness of $ F$, $G$, and $\Omega$.
Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval
Louis
Block;
Ethan M.
Coven
297-306
Abstract: We say that a continuous map $f$ of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all $ {f^k}(x)$, where $k \geqslant 0$ and $x$ is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Markov maps, partly to motivate the proof of the more complicated linear Markov case.
On weakly countably determined Banach spaces
Sophocles
Mercourakis
307-327
Abstract: For a topological space $X$, let ${C_1}(X)$ denote the Banach space of all bounded functions $ f:X \to {\mathbf{R}}$ such that for every $ \varepsilon > 0$ the set $\{ x \in X:\vert f(x)\vert \geqslant \varepsilon \}$ is closed and discrete in $X$, endowed with the supremum norm. The main theorem is the following: Let $L$ be a weakly countably determined subset of a Banach space; then there exist a subset $\Sigma '$ of the Baire space $ \Sigma$, a compact space $ K$, and a bounded linear one-to-one operator ${C_1}(\Sigma ' \times K)$ (resp. ${C_1}(\Sigma \times K)$) (a compact space $ \Omega$ is called Gulko or Talagrand compact if $ C(\Omega )$ is WCD or a weakly $K$-analytic Banach space); the characterization of WCD (resp. weakly $K$-analytic) Banach spaces $E$, using one-to-one operators from ${E^{\ast}}$ into ${C_1}(\Sigma \times K)$); and the existence of equivalent "good" norms on $E$ and $ {E^{\ast}}$ simultaneously.
Area and Hausdorff dimension of Julia sets of entire functions
Curt
McMullen
329-342
Abstract: We show the Julia set of $\lambda \sin (z)$ has positive area and the action of $ \lambda \sin (z)$ on its Julia set is not ergodic; the Julia set of $\lambda \exp (z)$ has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.
Matrices whose powers are $M$-matrices or $Z$-matrices
Shmuel
Friedland;
Daniel
Hershkowitz;
Hans
Schneider
343-366
Abstract: A matrix $ A$ all of whose (positive) powers are $Z$-matrices is called here a $ ZM$-matrix. A matrix is called a $ZMA$-matrix if all powers of $A$ are irreducible $Z$-matrices. We prove that the spectrum of a $ZMA$-matrix is real and only the eigenvalue minimal in absolute value may be negative. By means of an operation called inflation which generalizes the Kronecker product of two matrices, we determine the class of $ZMA$-matrices of order $n$ in terms of the classes of $ZMA$-matrices of smaller orders. We use this result to show that a $ZMA$-matrix is positively diagonally similar to a symmetric matrix. Similar results hold for $ MMA$-matrices which are defined in analogy with $ZMA$-matrices in terms of $M$-matrices, and for $ZMO$-matrices which are defined to be $ZM$-matrices such that all odd powers are irreducible and all even powers reducible. We also prove that a matrix is a $ZMA$-, $ZMO$- or $MMA$-matrix under apparently weaker conditions. If $A$ is a real matrix such that all sufficiently large powers of $A$ are $Z$-matrices, then $A$ is a $ZMA$-matrix if ${A^2}$ is irreducible, $A$ is a $ZMO$-matrix if $A$ is irreducible and ${A^2}$ is reducible, and $A$ is an $MMA$-matrix if $A$ is an irreducible $Z$-matrix and some odd power of $A$ is an $M$-matrix.
Whitney continua of curves
Hisao
Kato
367-381
Abstract: In this paper, we prove several theorems relating shape properties of Whitney continua of curves. In particular, we investigate the fundamental dimension and the shape type of Whitney continua of curves.