Homogeneous Borel sets of ambiguous class two
Fons
van Engelen
1-39
Abstract: We describe and characterize all homogeneous subsets of the Cantor set which are both an $ {F_{\sigma \delta }}$ and a $ {G_{\delta \sigma }}$; it turns out that there are $ {\omega _1}$ such spaces.
Regular cardinals in models of ${\rm ZF}$
Moti
Gitik
41-68
Abstract: We prove the following Theorem. Suppose $ M$ is a countable model of $ZFC$ and $\kappa$ is an almost huge cardinal in $ M$. Let $ A$ be a subset of $ \kappa$ consisting of nonlimit ordinals. Then there is a model $ {N_A}$ of $ ZF$ such that ${\aleph _\alpha }$ is a regular cardinal in $ {N_A}$ iff $\alpha \in A$ for every $\alpha > 0$.
Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups
Stephen H.
McCleary
69-79
Abstract: It is easy to pose questions about the free lattice-ordered group $ {F_\eta }$ of rank $ \eta > 1$ whose answers$ ^{2}$ are "obvious", but difficult to verify. For example: 1. What is the center of ${F_\eta }$? 2. Is ${F_\eta }$ directly indecomposable? 3. Does $ {F_\eta }$ have a basic element? 4. Is ${F_\eta }$ completely distributive? Question 1 was answered recently by Medvedev, and both $ 1$ and $2$ by Arora and McCleary, using Conrad's representation of ${F_\eta }$ via right orderings of the free group $ {G_\eta }$. Here we answer all four questions by using a completely different tool: The (faithful) representation of $ {F_\eta }$ as an $o{\text{-}}2$-transitive $l$-permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite $ \eta$, and is here extended to all $\eta > 1$. Curiously, the existence of a transitive representation for ${F_\eta }$ implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of ${G_\eta }$ which suffices all by itself to give a faithful representation of ${F_\eta }$. For finite $\eta$, we find that every transitive representation of ${F_\eta }$ can be made from a pathologically $o{\text{-}}2$-transitive representation by blowing up the points to $o$-blocks; and every pathologically $o{\text{-}}2$-transitive representation of $ {F_\eta }$ can be extended to a pathologically $ o{\text{-}}2$-transitive representation of $ {F_{{\omega _0}}}$.
An even better representation for free lattice-ordered groups
Stephen H.
McCleary
81-100
Abstract: The free lattice-ordered group ${F_\eta }$ (of rank $\eta$) has been studied in two ways: via the Conrad representation on the various right orderings of the free group ${G_\eta }$ (sharpened by Kopytov's observation that some one right ordering must by itself give a faithful representation), and via the Glass-McCleary representation as a pathologically $ o{\text{-}}2$-transitive $ l$-permutation group. Each kind of representation yields some results which cannot be obtained from the other. Here we construct a representation giving the best of both worlds--a right ordering $({G_\eta }, \leqslant )$ on which the action of $ {F_\eta }$ is both faithful and pathologically $ o{\text{-}}2$-transitive. This $ ({G_\eta }, \leqslant )$ has no proper convex subgroups. The construction is explicit enough that variations of it can be utilized to get a great deal of information about the root system $ {\mathcal{P}_\eta }$ of prime subgroups of ${F_\eta }$. All ${\mathcal{P}_\eta }$'s with $1 < \eta < \infty $ are $ o$-isomorphic. This common root system $ {\mathcal{P}_f}$ has only four kinds of branches (singleton, three-element, $ {\mathcal{P}_f}$ and $ {\mathcal{P}_{{\omega _0}}}$), each of which occurs ${2^{{\omega _0}}}$ times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of ${\mathcal{P}_f}$.
Subspaces of ${\rm BMO}({\bf R}\sp n)$
Michael
Frazier
101-125
Abstract: We consider subspaces of $ {\text{BMO}}({{\mathbf{R}}^n})$ generated by one singular integral transform. We show that the averages along ${x_j}$-lines of the $j$ th Riesz transform of $g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ or $ g \in {L^\infty }({{\mathbf{R}}^n})$ satisfy a certain strong regularity property. One consquence of this result is that such functions satisfy a uniform doubling condition on a.e. $ {x_j}$-line. We give an example to show, however, that the restrictions to $ {x_j}$-lines of the Riesz transform of $g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ do not necessarily have uniformly bounded ${\text{BMO}}$ norm. Also, for a Calderón-Zygmund singular integral operator $K$ with real and odd kernel, we show that $ K({\text{BMO}_c}) \subseteq \overline {{L^\infty } + K(L_c^\infty )}$, where $L_c^\infty$ and $ {\text{BMO}_c}$ are the spaces of ${L^\infty }$ or ${\text{BMO}}$ functions of compact support, respectively, and the closure is taken in ${\text{BMO}}$ norm.
A proof of Andrews' $q$-Dyson conjecture for $n=4$
Kevin W. J.
Kadell
127-144
Abstract: Andrews' $ q$-Dyson conjecture is that the constant term in a polynomial associated with the root system $ {A_{n - 1}}$ is equal to the $q$-multinomial coefficient. Good used an identity to establish the case $q = 1$, which was originally raised by Dyson. Andrews established his conjecture for $n \leqslant 3$ and Macdonald proved it when $ {a_1} = {a_2} = \cdots = {a_n} = 1,2$ or $\infty$ for all $ n \geqslant 2$. We use a $ q$-analog of Good's identity which involves a remainder term and linear algebra to establish the conjecture for $ n = 4$. The remainder term arises because of an essential problem with the $ q$-Dyson conjecture: the symmetry of the constant term. We give a number of conjectures related to the symmetry.
Good and OK ultrafilters
Alan
Dow
145-160
Abstract: In this paper we extend Kunen's construction of ${\alpha ^ + }$-good ultrafilters on $\mathcal{P}(\alpha )$ to more general algebras, as well as the construction of ${\alpha ^ + }$-OK ultrafilters. In so doing, we prove the existence of $({2^\alpha } \times {\alpha ^ + })$-independent matrices, as defined by Kunen, in these algebras. Some of the topological properties of the Stone spaces of these algebras are then investigated. We find points, for example, in $ U(\alpha )$ which can be regarded as a generalization of weak $P$-points.
On twisted lifting
Yuval Z.
Flicker
161-178
Abstract: If $\sigma$ is a generator of the galois group of a finite cyclic extension $E/F$ of local or global fields, and $\varepsilon$ is a character of $ {C_E}( = {E^ \times }\;{\text{or}}\;{E^ \times }\backslash {{\mathbf{A}}^ \times })$ whose restriction to $ {C_F}$ has order $ n$, then the irreducible admissible or automorphic representations $ \pi$ of ${\text{GL}}(n)$ over $E$ with $^\sigma \pi \cong \pi \otimes \varepsilon$ are determined.
On the relative consistency strength of determinacy hypotheses
Alexander S.
Kechris;
Robert M.
Solovay
179-211
Abstract: For any collection of sets of reals $C$, let $ C{\text{-DET}}$ be the statement that all sets of reals in $C$ are determined. In this paper we study questions of the form: For given $C \subseteq C\prime$, when is $C\prime {\text{-DET}}$ equivalent, equiconsistent or strictly stronger in consistency strength than $C {\text{-DET}}$ (modulo ${\text{ZFC}}$)? We focus especially on classes $ C$ contained in the projective sets.
Bifurcation from a heteroclinic solution in differential delay equations
Hans-Otto
Walther
213-233
Abstract: We study a class of functional differential equations $\dot x(t) = af(x(t - 1))$ with periodic nonlinearity $ f:{\mathbf{R}} \to {\mathbf{R}},0 < f$ in $(A,0)$ and $f < 0$ in $(0,B),f(A) = f(0) = f(B) = 0$ . Such equations describe a state variable on a circle with one attractive rest point (given by the argument $\xi = 0$ of $f$) and with reaction lag $a$ to deviations. We prove that for a certain critical value $a = {a_0}$ there exists a heteroclinic solution going from the equilibrium solution $t \to A$ to the equilibrium $ t \to B$. For $a - {a_0} > 0$, this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle.
Unitary structures on cohomology
C. M.
Patton;
H.
Rossi
235-258
Abstract: Let ${{\mathbf{C}}^{p + q}}$ be endowed with a hermitian form $H$ of signature $(p,q)$. Let ${M_r}$ be the manifold of $r$-dimensional subspaces of ${{\mathbf{C}}^{p + q}}$ on which $H$ is positive-definite and let $ E$ be the determinant bundle of the tautological bundle on ${M_r}$. We show (starting from the oscillator representation of ${\text{SU}}(p,q))$ that there is an invariant subspace of ${H^{r(p - r)}}({M_r},\mathcal{O}(E(p + k)))$ which defines a unitary representation of $ {\text{SU}}(p,q)$. For $W \in {M_p},\operatorname{Gr}(r,W)$ is the subvariety of $r$-dimensional subspaces of $W$. Integration over $\operatorname{Gr}(r,W)$ associates to an $ r(p - r)$-cohomology class $ \alpha$, a function $P(\alpha )$ on ${M_p}$. We show that this map is injective and provides an intertwining operator with representations of $ {\text{SU}}(p,q)$ on spaces of holomorphic functions on Siegel space
Minimal surfaces of constant curvature in $S\sp n$
Robert L.
Bryant
259-271
Abstract: In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in ${S^n}$ of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in ${S^n}$ of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in ${S^n}$ and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into ${S^n}$.
On the boundary behaviour of generalized Poisson integrals on symmetric spaces
Henrik
Schlichtkrull
273-280
Abstract: On a Riemannian symmetric space $X$ of the noncompact type we introduce a generalized Poisson transformation from functions on the minimal boundary to functions on the maximal compactification whose restrictions to $X$ are eigenfunctions of the invariant differential operators. Some continuity- and "Fatou"-theorems are proved.
Analytic uniformly bounded representations of ${\rm SU}(1,n+1)$
Ronald J.
Stanke
281-302
Abstract: By analytically continuing suitably normalized spherical principal series, a family of uniformly bounded representations of $SU(1,n + 1)$, all of which act on the same Hilbert space ${L^2}({{\mathbf{R}}^{2n + 1}})$, is constructed which is parametrized by complex numbers $s$ lying in the strip $- 1 < \operatorname{Re} (s) < 1$. The proper normalization of the principal series representations involves the intertwining operators of equivalent principal series representations. These intertwining operators are first analyzed using Fourier analysis on the Heisenberg group.
Absolutely continuous invariant measures that are maximal
W.
Byers;
A.
Boyarsky
303-314
Abstract: Let $A$ be a certain irreducible $0{\text{-}}1$ matrix and let $\tau$ denote the family of piecewise linear Markov maps on $[0,1]$ which are consistent with $ A$. The main result of this paper characterizes those maps in $\tau$ whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with $ A$, the absolutely continuous invariant measure is not maximal.
On the decomposition numbers of the finite general linear groups
Richard
Dipper
315-344
Abstract: Let $G = {\text{GL}_n}(q)$, $q$ a prime power, and let $r$ be an odd prime not dividing $q$. Let $s$ be a semisimple element of $G$ of order prime to $r$ and assume that $r$ divides. ${q^{\deg (\Lambda )}} - 1$ for all elementary divisors $\Lambda$ of $s$. Relating representations of certain Hecke algebras over symmetric groups with those of $ G$, we derive a full classification of all modular irreducible modules in the $ r$-block ${B_s}$ of $G$ with semisimple part $s$. The decomposition matrix $D$ of ${B_s}$ may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover $ D$ is lower unitriangular. This applies in particular to all $r$-blocks of $G$ if $r$ divides $q - 1$. Thus, in this case, the $ r$-decomposition matrix of $ G$ is lower unitriangular.
Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients
J.
Yeh
345-361
Abstract: Let $B$ be a $2$-parameter Brownian motion on ${\mathbf{R}}_ + ^2$. Consider the nonMarkovian stochastic differential system in $2$-parameter $\displaystyle \left\{ {\begin{array}{*{20}{c}} {dX(z) = \alpha (z,X)\;dB(z) + \... ...text{for}}\;z \in \partial {\mathbf{R}}_ + ^2,} \end{array} } \right.$ i.e., $\displaystyle \left\{ {\begin{array}{*{20}{c}} {X(z) = X(0) + \int_{{R_z}} {\al... ...}}_ + ^2,} {x(0) = \xi ,} & {} \end{array} } \right.$ where $ {R_z} = [0,s] \times [0,t]$ for $z = (s,t) \in {\mathbf{R}}_ + ^2$. An existence theorem for weak solutions of the system is proved in this paper. Under the assumption that $\alpha$ and $\beta$ satisfy a continuity condition and a growth condition and ${\mathbf{E}}[{\xi ^6}] < \infty$, it is shown that there exist a $2$-parameter stochastic process $X$ and a $2$-parameter Brownian motion $B$ on some probability space satisfying the stochastic integral equation above, with $ X(0)$ having the same probability distribution as $\xi$.
Strongly pure subgroups of separable torsion-free abelian groups
Loyiso G.
Nongxa
363-373
Abstract: In this paper we prove that countable strongly pure subgroups of completely decomposable groups are completely decomposable. We also show that strongly pure subgroups of separable torsion-free groups are separable.
Harmonic functions on semidirect extensions of type $H$ nilpotent groups
Ewa
Damek
375-384
Abstract: Let $S = NA$ be a semidirect extension of a Heisenberg type nilpotent group $N$ by the one-parameter group of dilations, equipped with the Riemannian structure, which generalizes this of the symmetric space. Let ${\{ {P_a}(y)\} _{a > 0}}$ be a Poisson kernel on $N$ with respect to the Laplace-Beltrami operator. Then every bounded harmonic function $F$ on $S$ is a Poisson integral $F(yb) = f \ast {P_b}(y)$ of a function $f \in {L^\infty }(N)$. Moreover the harmonic measures $\mu _a^b$ defined by ${P_b} = {P_a} \ast \mu _a^b,b > a$, are radial and have smooth densities. This seems to be of interest also in the case of a symmetric space of rank $ 1$.
Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in ${\bf R}\sp n$. I. $n\geq 4$
A. J.
Tromba
385-413
Abstract: A degree theory applicable to Plateau's problem is developed and the Morse equality for minimal surfaces spanning a contour in $ {{\mathbf{R}}^n},n \geq 4$, is proved.
Spectral properties of elementary operators. II
Lawrence A.
Fialkow
415-429
Abstract: Let $A = ({A_1}, \ldots ,{A_n})$ and $B = ({B_1}, \ldots ,{B_n})$ denote commutative $n$-tuples of operators on a Hilbert space $\mathcal{H}$. Let ${R_{AB}}$ denote the elementary operator on $\mathcal{L}(\mathcal{H})$ defined by $ {R_{AB}}(X) = {A_1}X{B_1} + \cdots + {A_n}X{B_n}$. We obtain new expressions for the essential spectra of ${R_{AB}}$ and $ {R_{AB}}\vert\mathcal{J}$ (the restriction of ${R_{AB}}$ to a norm ideal $\mathcal{J}$ of $ \mathcal{L}(\mathcal{H})$). We also study isolated points of joint spectra defined in the sense of $ {\text{R}}$. Harte.