Dirichlet series and automorphic forms on unitary groups
Tobias
Orloff
431-456
Abstract: In a special case our unitary group takes the form $\displaystyle G = \{ g \in {\text{GL}}(p + 2,{\mathbf{C}}){\vert^t}\bar gRg = R\} .$ Here $\displaystyle R = \left( {\begin{array}{*{20}{c}} S & 0 & 0 0 & 0 & 1 0 & { - 1} & 0 \end{array} } \right)$ is a skew-Hermitian matrix with entries in an imaginary quadratic number field $ K$. We suppose that $- iR$ has signature $ (p + 1,1)$. This group acts naturally on the symmetric domain $\displaystyle D = \left\{ {w \in {{\mathbf{C}}^p},z \in {\mathbf{C}}\vert\operatorname{Im} (z) > - {{\frac{1}{2}}^t}\bar wSw} \right\}.$ If $ \Gamma = G \cap {\text{SL}}(p + 2,{\mathcal{O}_K})$ with ${\mathcal{O}_K}$ the ring of integers in $ K$, then an automorphic form $f(w,z)$ with respect to $\Gamma$ has an expansion ${\Sigma _r}{g_r}(w) \cdot {e^{2\pi irz}}$. The functions ${g_r}(w)$ are theta functions. Given another automorphic form $g(w,z)$ with an expansion ${\Sigma _s}{h_s}(w) \cdot {e^{2\pi isz}}$ we define a Dirichlet series ${\Sigma _r}\langle {g_r},{h_r}\rangle {r^{ - s}}$ . Here $\langle {g_r},{h_r}\rangle $ is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type: $\displaystyle {\int _{{P_\Gamma }\backslash D}}f\bar g\cdot{\left( {\operatorname{Im} (z) + \frac{1} {2}{i^t}\bar wSw} \right)^s}dw\;d\bar w\;dz\;d\bar z$ with ${P_\Gamma } \subseteq \Gamma$ a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over $\Gamma \backslash D$. In the case $p = 1$ our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.
a.c. groups: extensions, maximal subgroups, and automorphisms
Kenneth
Hickin
457-481
Abstract: In $\S1$ we extend the results of [ ${\mathbf{3}}$] on centralizers to r.e. subgroups and show, e.g., that every a.c. group has an $\infty - \omega $-equivalent subgroup of the same power which is embedded maximally in itself; and we pursue a natural typology of maximal subgroups. $\S2$ shows that if $A$ is a countable group of automorphisms of a countable a.c. group $G$ such that $A \supset \operatorname{Inn}\;G$, then there exists $\tau \in \operatorname{Aut}\;G$ such that the $ {\text{HNN}}$ extension ( $ A,\tau :{\tau ^{ - 1}}g\tau = \tau (g)$ for all $g \in \operatorname{Inn}\;G$) is a subgroup of $ \operatorname{Aut}\;G$. We show in $\S3$ that every a.c. group with a countable skeleton has a proper extension to an a.c. group having any skeleton that contains the original one and any f.g. group which contains the countable a.c. group equivalent to the original one as an r.e. subset. This uses Ziegler's construction [ $ {\mathbf{7}}$]. Finally, in $\S4$, also using Ziegler's construction we show that there exists an a.c. group $A$ of any power and having any countable skeleton which has a free subgroup $M$ such that for all $x \in A - M$ and $y \in A$ there exist free generators $a,b,c \in M$ such that $y = {(ax)^b}{(ax)^c}$.
The Euler equation for functionals with linear growth
Gabriele
Anzellotti
483-501
Abstract: We give a first variation formula for functionals of the type $\int_\Omega {f(x,\mu )}$, where $f(x,p):\Omega \times {{\mathbf{R}}^k} \to {\mathbf{R}}$ is of linear growth in $p$ for large $\vert p\vert$ and $\mu$ is a $ {{\mathbf{R}}^k}$-valued measure in $\Omega$. The Euler equation for the minima of various functionals defined on spaces of ${\text{BV}}$ functions is then studied.
Free boundary regularity for surfaces minimizing ${\rm Area}(S)+c\,{\rm Area}(\partial S)$
Edith A.
Cook
503-526
Abstract: In ${{\mathbf{R}}^n}$, fix a hyperplane $Z$ and $ a\;(k - 1)$-dimensional surface $F$ lying to one side of $Z$ with boundary in $Z$. We prove the existence of $S$ and $B$ minimizing $\operatorname{Area}(S) + c\operatorname{Area}(B)$ among all $k$-dimensional $S$ having boundary $F \cup B$, where $B$ is a free boundary constrained to lie in $ Z$. We prove that except possibly on a set of Hausdorff dimension $ k - 2$, $S$ is locally a ${C^{1,\alpha }}$ manifold with ${C^{1,\alpha }}$ boundary $B$ for $ 0 < \alpha < 1/2$. If $k = n - 1$, $ {C^{1,\alpha }}$ is replaced by real analytic.
On a class of Banach spaces of functions associated with the notion of entropy
Boris
Korenblum
527-553
Abstract: A class of function spaces on the circle is introduced which contain all continuous functions of bounded variation but are included in the set of all continuous functions. The corresponding dual spaces consist of certain types of generalized measures. One application of these spaces is a new convergence test for Fourier series which includes both the Dirichlet-Jordan and the Dini-Lipschitz tests.
A geometric approach to the linear Penrose transform
I. B.
Penkov
555-575
Abstract: We show that under a certain cohomological condition the theorem of Witten, Isenberg, Yasskin and Green about the inverse Penrose transform of a (non-self-dual) connection $ \nabla$ (together with Manin's description of its curvature ${F_\nabla }$) is true in a quite general situation. We then present a (multidimensional) version of the Penrose transform of a vector bundle in the language of jets. This gives a coordinate-free interpretation of certain results of Henkin and Manin, coding a number of classical field equations in terms of obstructions to infinitesimal extension of cohomology classes.
On the spectra of compact nilmanifolds
Jeffrey S.
Fox
577-583
Abstract: We show the equivalence of the Howe-Richardson multiplicity formula for compact nilmanifolds and the formula obtained by Corwin and Greenleaf using the Selberg trace formula.
$P$-points with countably many constellations
Ned I.
Rosen
585-596
Abstract: If the continuum hypothesis $ ({\text{CH}})$ is true, then for any $P$ point ultrafilter $D$ (on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of) $P$ points which lie above $ D$, of order type ${\aleph _1}$. In particular, if $ D$ is an ${\text{RK}}$-minimal ultrafilter, then we have $({\text{CH}})$ that there exist $P$-points with countably many constellations.
The dual Weierstrass-Laguerre transform
Deborah Tepper
Haimo
597-613
Abstract: An inversion algorithm is derived for the dual Weierstrass-Laguerre transform $ \int_0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}$, where the function ${g_\alpha }(x,y,t)$ is associated with the source solution of the Laguerre differential heat equation $x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)$. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform $ \int_0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}$ of a nondecreasing function $ \beta$.
Regular linear algebraic monoids
Mohan S.
Putcha
615-626
Abstract: In this paper we study connected regular linear algebraic monoids. If $ \phi :{G_0} \to {\text{GL}}(n,K)$ is a representation of a reductive group $ {G_0}$, then the Zariski closure of $K\phi ({G_0})$ in ${\mathcal{M}_n}(K)$ is a connected regular linear algebraic monoid with zero. In $\S2$ we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are $V$-regular. In $\S3$ we show that when the ideals are linearly ordered, the group of units is nearly simple of type $ {A_l},{B_l},{C_l},{F_4}\;{\text{or}}\;{G_2}$. In $\S4$, conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.
On minimal surfaces in a K\"ahler manifold of constant holomorphic sectional curvature
Jon G.
Wolfson
627-646
Abstract: This paper studies minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in ${\mathbf{C}}{P^n}$, complex projective $ n$-space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in ${\mathbf{C}}{P^n}$ classically there is associated a particular framing of ${{\mathbf{C}}^{n + 1}}$ called the Frenet frame. Each element of the Frenet frame induces a minimal surface in $ {\mathbf{C}}{P^n}$. The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.
Normal subgroups of doubly transitive automorphism groups of chains
Richard N.
Ball;
Manfred
Droste
647-664
Abstract: We characterize the structure of the normal subgroup lattice of $ 2$-transitive automorphism groups $A(\Omega )$ of infinite chains $(\Omega , \leqslant )$ by the structure of the Dedekind completion $(\bar \Omega , \leqslant )$ of the chain $(\Omega , \leqslant )$. As a consequence we obtain various group-theoretical results on the normal subgroups of $A(\Omega )$, including that any proper subnormal subgroup of $ A(\Omega )$ is indeed normal and contained in a maximal proper normal subgroup of $A(\Omega )$, and that $A(\Omega )$ has precisely $5$ normal subgroups if and only if the coterminality of the chain $(\Omega , \leqslant )$ is countable.
Solving semilinear partial differential equations with probabilistic potential theory
Joseph
Glover;
P. J.
McKenna
665-681
Abstract: Techniques of probabilistic potential theory are applied to solve $- Lu + f(u) = \mu$, where $\mu$ is a signed measure, $ f$ a (possibly discontinuous) function and $L$ a second order elliptic or parabolic operator on ${R^d}$ or, more generally, the infinitesimal generator of a Markov process. Also formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.
The de Branges theorem on univalent functions
Carl H.
FitzGerald;
Ch.
Pommerenke
683-690
Abstract: We present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.
A note on a question of Bass
Sankar P.
Dutta
691-700
Abstract: In this paper we study the following question raised by Bass: Is a local ring with a finitely generated module of finite injective dimension Cohen-Macaulay? We prove that the answer is in the affirmative when a certain local cohomology of the ring is either decomposible or cyclic. We also study the above question in some special cases and some of its implications.
Unstable ground state of nonlinear Klein-Gordon equations
Jalal
Shatah
701-710
Abstract: In this paper we prove the instability of the ground state, i.e. least energy steady-state solution of nonlinear Klein-Gordon equations with space dimension $n \geqslant 3$.
Directed sets and cofinal types
Stevo
Todorčević
711-723
Abstract: We show that $1,\omega ,{\omega _1},\omega \times {\omega _1}$ and ${[{\omega _1}]^{ < \omega }}$ are the only cofinal types of directed sets of size ${\aleph _1}$, but that there exist many cofinal types of directed sets of size continuum.
On the group ${\rm SSF}(G),\;G$ a cyclic group of prime order
M.
Maller;
J.
Whitehead
725-733
Abstract: We extend the definition of the obstruction group ${\text{SSF}}(G)$ in the case where $G$ is a cyclic group of prime order. We show that an endomorphism of a free $ZG$-module is a direct summand of a virtual permutation if its characteristic polynomial has the appropriate form. Among these endomorphisms the virtual permutations are detected by ${K_0}$. The main application is in detecting Morse-Smale isotopy classes.
Abelian rank of normal torsion-free finite index subgroups of polyhedral groups
Youn W.
Lee
735-745
Abstract: Suppose that $ P$ is a convex polyhedron in the hyperbolic $3$-space with finite volume and $P$ has integer $( > 1)$ submultiples of $\pi$ as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group $ G$ associated to $ P$ is one, then $ P$ has exactly one ideal vertex of type $(2,2,2,2)$ and $G$ has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.
Homology and cell structure of nilpotent spaces
Robert H.
Lewis
747-760
Abstract: Let $A$ and $X$ denote finitely dominated nilpotent ${\text{CW}}$ complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex. We investigate the set of possible cofibers of maps $A \to X$, and find a severe restriction. When it is met and the fundamental group is finite, $ X$ can be constructed from $ A$ by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes. We show that the Euler characteristic of $X$ must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes. Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.
Variants of the maximal double Hilbert transform
Elena
Prestini
761-771
Abstract: We prove the boundedness on $ {L_p}({T^2})$, $1 < p < \infty$, of two variants of the double Hilbert transform and maximal double Hilbert transform. They have an application to a problem of almost everywhere convergence of double Fourier series.
Changing the depth of an ordered set by decomposition
E. C.
Milner;
K.
Prikry
773-785
Abstract: The depth of a partially ordered set $ \langle P, < \rangle$ is the smallest ordinal $\gamma$ such that $\langle P, < \rangle$ does not embed ${\gamma ^\ast}$. The width of $\langle P, < \rangle$ is the smallest cardinal number $\mu$ such that there is no antichain of size $ \mu + 1$ in $P$. We show that if $\gamma > \omega$ and $\gamma$ is not an infinite successor cardinal, then any partially ordered set of depth $\gamma$ can be decomposed into $ \operatorname{cf}(\vert\gamma \vert)$ parts so that the depth of each part is strictly less than $\gamma$. If $ \gamma = \omega$ or if $ \gamma$ is an infinite successor cardinal, then for any infinite cardinal $ \lambda$ there is a linearly ordered set of depth $\gamma$ such that for any $\lambda $-decomposition one of the parts has the same depth $\gamma$. These results are used to solve an analogous problem about width. It is well known that, for any cardinal $\lambda$, there is a partial order of width $ \omega$ which cannot be split into $\lambda$ parts of finite width. We prove that, for any cardinal $\lambda$ and any infinite cardinal $ \nu$, there is a partial order of width $ {\nu ^ + }$ which cannot be split into $\lambda$ parts of smaller width.
Local vanishing properties of solutions of elliptic and parabolic quasilinear equations
J. Ildefonso
Díaz;
Laurent
Véron
787-814
Abstract: We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation $- \operatorname{div}\;A(x,u,Du) + B(x,u,Du) = 0$ and of the parabolic equation $\partial \psi (u)/\partial t - \operatorname{div}\;\mathcal{A}(t,x,u,Du) + \mathcal{B}(t,x,u,Du) = 0$. The results are obtained without any assumption of monotonicity on $A$, $B$, $ \mathcal{A}$ and $\mathcal{B}$.
The determinant of the Eisenstein matrix and Hilbert class fields
I.
Efrat;
P.
Sarnak
815-824
Abstract: We compute the determinant of the Eisenstein matrix associated to the Hilbert-Blumenthal modular group $ {\text{PSL}_2}({\mathcal{O}_k})$, and express it in terms of the zeta function of the Hilbert class field of $K$.
Propagation of $C\sp \infty$ regularity for fully nonlinear second order strictly hyperbolic equations in two variables
Paul
Godin
825-830
Abstract: It is shown that if $ u$ is a ${C^3}$ solution of a fully nonlinear second order strictly hyperbolic equation in two variables, then $u$ is $ {C^\infty }$ at a point $ m$ as soon as it is ${C^\infty }$ at some point of each of the two bicharacteristic curves through $m$. For semilinear equations, such a result was obtained before by Rauch and Reed if $u \in {C^1}$
Separation in countably paracompact spaces
W. Stephen
Watson
831-842
Abstract: We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces.
On derivations annihilating a maximal abelian subalgebra
Geoffrey L.
Price
843-850
Abstract: Let $\mathcal{A}$ be an $ {\text{AF}}\;{C^\ast}$-algebra, and let $\delta$ be a closed $\ast$-derivation which annihilates the maximal abelian subalgebra $ \mathcal{C}$ of diagonal elements of $ \mathcal{A}$. Then we show that $\delta$ generates an approximately inner $ {C^\ast}$-dynamics on $\mathcal{A}$, and that $\delta$ is a commutative $\ast $-derivation. Any two closed $\ast$-derivations vanishing on $\mathcal{C}$ are shown to be strongly commuting. More generally, if $\delta$ is a semiderivation on $\mathcal{A}$ which vanishes on $\mathcal{C}$, we prove that $\delta$ is a generator of a semigroup of strongly positive contractions of $\mathcal{A}$.
A quasi-invariance theorem for measures on Banach spaces
Denis
Bell
851-855
Abstract: We show that for a measure $\gamma$ on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which $ \gamma$ is the image of a Gaussian measure under a suitably regular map.
Erratum to: ``Localization of equivariant cohomology rings''
J.
Duflot
857-858