An integral version of the Brown-Gitler spectrum
Don H.
Shimamoto
383-421
Abstract: In this paper, certain spectra ${B_1}(k)$ are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra $ B(k)$. The bulk of our work emphasizes the similarities between $ {B_1}(k)$ and $ B(k)$, shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of ${B_1}(k)$, we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime $2$. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for ${\Omega ^2}({S^3}\left\langle 3 \right\rangle )$ and other related spaces.
On cyclic trigonal Riemann surfaces. I
Robert D. M.
Accola
423-449
Abstract: Definition. Call the Riemann surfaces for the equation ${y^3} = P(x)$ cyclic trigonal. For one case of genus $4$ ($2$ distinct $g_3^1$'s) and all genera greater than $4$, cyclic trigonal Riemann surfaces are characterized by the vanishing properties of the theta function at certain $(1/6)$-periods of the Jacobian. Also for trigonal Riemann surfaces of genera $5$, $6$, and $7$, homogeneous theta relations are derived using the fact that Prym varieties for trigonal Riemann surfaces are Jacobians.
Hecke modular forms and the Kac-Peterson identities
George E.
Andrews
451-458
Abstract: The identity of certain Hecke modular forms with well-known infinite products is derived in an elementary manner. New identities and applications are discussed.
The equivariant Dolbeault lemma
Wilfried
Schmid;
Joseph A.
Wolf
459-463
Abstract: A form of the Dolbeault Lemma is obtained for circular domains $D \subset {{\mathbf{C}}^n}$, which is equivariant for the subgroup of $ \operatorname{GL} (n;{\mathbf{C}})$ that stabilizes $D$.
Dichotomies and asymptotic behaviour for linear differential systems
James S.
Muldowney
465-484
Abstract: Sufficient conditions that a system of differential equations
On stable blocks of Auslander-algebras
Christine
Riedtmann
485-505
Abstract: The Auslander-algebra ${E_\Lambda }$ of an algebra $\Lambda$ of finite representation type is the endomorphism algebra of the direct sum $M = \oplus {M_i}$ of one copy of each indecomposable $ \Lambda$-module. A stable block of $ {E_\Lambda }$ is a connected direct factor of the residue algebra of ${E_\Lambda }$ modulo the two-sided ideal generated by the projections of $M$ to the ${M_i}$'s that are not stable under $DTr$. This paper describes the stable blocks whose quiver is a stable translation-quiver of class ${A_n}$ or ${D_n}$.
On pairs of recursively enumerable degrees
Klaus
Ambos-Spies
507-531
Abstract: Lachlan and Yates proved that some, but not all, pairs of incomparable recursively enumerable (r.e.) degrees have an infimum. We answer some questions which arose from this situation. We show that not every nonzero incomplete r.e. degree is half of a pair of incomparable r.e. degrees which have an infimum, whereas every such degree is half of a pair without infimum. Further, we prove that every nonzero r.e. degree can be split into a pair of r.e. degrees which have no infimum, and every interval of r.e. degrees contains such a pair of degrees.
Pure states on some group-invariant $C\sp{\ast} $-algebras
Geoffrey L.
Price
533-562
Abstract: Let $\mathfrak{A}$ be a UHF algebra of Glimm type ${n^\infty }$, i.e., $ \mathfrak{A} = \otimes _{k \geqslant 1}^{\ast}{N_k}$, where $N = {N_1} = {N_2} = \cdots $ are $n \times n$ matrix algebras. We define an AF-subalgebra $ {\mathfrak{A}^G}$ of $\mathfrak{A}$, consisting of those elements of $\mathfrak{A}$ invariant under a group of automorphisms $\{ {\alpha _g}:g \in G = \operatorname{SU} (n)\}$ of product type. $ {\mathfrak{A}^G}$ is shown to be generated by an embedding of $S(\infty )$, the discrete group of finite permutations on countably many symbols. Let $\omega$ be a pure product state on $\mathfrak{A}$, $ {\omega ^G}$ its restriction to $ {\mathfrak{A}^G}$. Let $ e \in N$ be a one-dimensional projection with corresponding projections ${e^k} \in {N_k}$. Then if both (i) ${\Sigma _{k \geqslant 1}}\omega ({e^k}) = \infty$, and (ii) $ 0 < {\Sigma _{k \geqslant 1}}\omega ({e^k})[1 - \omega ({e^k})] < \infty$ hold, ${\omega ^G}$ is not pure. ${\omega ^G}$ is shown to be pure if there exist orthogonal one-dimensional projections $\{ {p_i}:1 \leqslant i \leqslant n\}$ of $N$ with corresponding projections $p_i^k \in {N_k}$ such that $\omega (p_i^k) = 0$ or $1$, $1 \leqslant i \leqslant n,\,k \geqslant 1$, and $0 < {\Sigma _{k \geqslant 1}}\omega (p_i^k) < \infty$ for at most one $i$.
Tangential equivalence of group actions
Sławomir
Kwasik
563-573
Abstract: We consider the problem of tangential equivalence of group actions on manifolds. In particular we discuss a conjecture of B. Mazur and its modifications. The negative answer to this conjecture is presented. On the other hand we prove that the "isovariant" version of this conjecture, as well as the modified one, remains true. As an application some results on the tangential equivalence of ${Z_p}$-actions on homotopy spheres are obtained.
An infinite graph of girth $12$
Asia Ivić
Weiss
575-588
Abstract: From the regular hyperbolic honeycomb $ \{ 3,6,3\}$ we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them.
Eisenstein series of weight ${3\over 2}$. II
Ting Yi
Pei
589-603
Abstract: In a previous paper we proved that for some special levels, in the space of elliptic modular forms with weight $3/2$ the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by the Eisenstein series. In this paper we prove that this fact is true for any level.
Subordination-preserving integral operators
Sanford S.
Miller;
Petru T.
Mocanu;
Maxwell O.
Reade
605-615
Abstract: Let $\beta$ and $\gamma$ be complex numbers and let $ H$ be the space of functions regular in the unit disc. Subordination of functions $ f$, $g \in H$ is denoted by $f \prec g$. Let $K \subset H$ and let the operator $A:K \to H$ be defined by $F = A(f)$, where $\displaystyle F(z) = {\left[ {\frac{1} {{{z^\gamma }}}\int_0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right]^{1/\beta }}.$ The authors determine conditions under which $\displaystyle f \prec g \Rightarrow A(f) \prec A(g),$ and then use this result to obtain new distortion theorems for some classes of regular functions.
A partial order on the regions of ${\bf R}\sp{n}$ dissected by hyperplanes
Paul H.
Edelman
617-631
Abstract: We study a partial order on the regions of $ {{\mathbf{R}}^n}$ dissected by hyperplanes. This includes a computation of the Möbius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs.
Structural stability of equivariant vector fields on two-manifolds
G. L.
dos Reis
633-643
Abstract: A class of vector fields on two-dimensional manifolds equivariant under the action of a compact Lie group is defined. Properties of openness, structural ability, and density are proved.
Tameness of pairs of nuclear power series spaces and related topics
Kaisa
Nyberg
645-660
Abstract: The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios $ {\alpha _m}/{\beta _n},n,m \in {\mathbf{N}}$, is bounded, (ii) Every operator from $ {\Lambda _\infty }(\beta )$ to $ {\Lambda _1}(\alpha )$ is compact, (iii) The pair $({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha ))$ is tame, i.e., for every operator $ T$ from ${\Lambda _\infty }(\beta )$ to ${\Lambda _1}(\alpha )$ there is a positive integer $ a$ such that for every $k \in {\mathbf{N}}$ there is a constant ${C_k}$ such that $\vert\vert Tx\vert{\vert _k} \leqslant {C_k}\vert x{\vert _{ak}}$ for every $x \in {\Lambda _\infty }(\beta )$. (iv) Every short exact sequence $ 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0$, where $ \tau = 1$ or $ \infty$, splits. (v) ${\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta )$ has a regular basis, (vi) ${\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta )$ has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.
On the arithmetic and homology of algebras of linear type
J.
Herzog;
A.
Simis;
W. V.
Vasconcelos
661-683
Abstract: Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.
Commuting analytic functions
Carl C.
Cowen
685-695
Abstract: Let $f$ and $g$ (not conformal automorphisms of the unit disk) be analytic mappings of the unit disk into itself. We say $f$ and $g$ commute if $f \circ g = g \circ f$. This paper characterizes those functions $g$ that commute with a given function $f$. Several corollaries of this characterization give qualitative information about $g$ given similar information about $ f$, and examples are given in each case to show the limitations of the conclusions. Some of the qualitative properties considered are univalence, fixed point sets, and whether two such $ g$ must commute with each other.
Contraction operators quasisimilar to a unilateral shift
V. T.
Alexander
697-703
Abstract: Let ${U_n}$ denote the unilaterial shift of finite multiplicity $n$. It is shown that a contraction operator $ T$ is quasisimilar to $ {U_n}$ if and only if $ T$ is of Class $ {C_1}$., the canonical isometry $V$ associated with $T$ is pure and $T$ is $n$-cyclic with analytically independent vectors. For this, the notions of operators of analytic type and analytic independence of vectors are introduced. A characterization of the cyclic vectors of the Backward Shift is also presented.
On certain elementary extensions of models of set theory
Ali
Enayat
705-715
Abstract: In $\S1$ we study two canonical methods of producing models of $ \operatorname{ZFC}$ with no elementary end extensions. $\S2$ is devoted to certain "completeness" theorems dealing with elementary extensions, e.g., using ${\diamondsuit _{{\omega _1}}}$ we show that for a consistent $T \supseteq \operatorname{ZFC}$ the property "Every model $ \mathfrak{A}$ of $ T$ has an elementary extension fixing $ {\omega ^\mathfrak{A}}$" is equivalent to $T\vdash$ "There exists an uncountable measurable cardinal". We also give characterizations of $ T\vdash$ "$\kappa$ is weakly compact" and $ T\vdash$ "$\kappa$ is measurable" in terms of elementary extensions.
Contraction semigroups for diffusion with drift
R.
Seeley
717-728
Abstract: Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian $\Delta$ on a manifold $M$; earlier, Yau gave related results for $L = \Delta + V$ for a vector field $V$. The present paper considers $L = \Delta - V - c$, with $c$ a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of $ {C_0}$ (functions vanishing at $\infty$) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on $ {L^p}(\mu ),\;1 \leqslant p < \infty$, for an arbitrary smooth density $ \mu$ on $M$. The conditions concern $L\rho /\rho$, where $\rho$ is a smooth function, $\rho \to \infty$ as $ x \to \infty$. They variously extend, strengthen, and complement the previous results mentioned above.
Some sequence spaces and absolute almost convergence
G.
Das;
B.
Kuttner;
S.
Nanda
729-739
Abstract: The object of this paper is to introduce a new concept of absolute almost convergence which emerges naturally as an absolute analogue of almost convergence, in the same way as convergence leads to absolute convergence.
Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems
A. K.
Aziz;
M.
Schneider
741-752
Abstract: We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem $\displaystyle u{\vert _{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0$ and the Gellerstedt-Neumann problem $\displaystyle ({d_n}u = k(y){u_x}dy - {u_y}dx{\vert _{{\Gamma _0}}} = 0,\qquad u{\vert _{{\Gamma _1} \cup {\Gamma _2}}} = 0)$ for the equation of mixed type $\displaystyle L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname{const} < 0$ in a region $G$ bounded by a piecewise smooth curve ${\Gamma _0}$ lying in the half-plane $y > 0$ and intersecting the line $y = 0$ at the points $ A( - 1,0)$ and $ B(1,0)$. For $ y < 0$, $G$ is bounded by the characteristic curves ${\gamma _1}(x < 0)$ and ${\gamma _2}(x > 0)$ of (1) through the origin and the characteristics $ {\Gamma _1}$ and ${\Gamma _2}$ through $A$ and $B$ which intersect $ {\gamma _1}$ and ${\gamma _2}$ at the points $P$ and $Q$, respectively. Using a variation of the energy integral method, we give sufficient conditions for the existence of weak and semistrong solutions of the boundary value problems (Theorems 4.1, 4.2, 5.1).
Concatenations applied to analytic hypoellipticity of operators with double characteristics
Kil Hyun
Kwon
753-763
Abstract: We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic.