On the deformation of algebra morphisms and diagrams
M.
Gerstenhaber;
S. D.
Schack
1-50
Abstract: A diagram here is a functor from a poset to the category of associative algebras. Important examples arise from manifolds and sheaves. A diagram $ {\mathbf{A}}$ has functorially associated to it a module theory, a (relative) Yoneda cohomology theory, a Hochschild cohomology theory, a deformation theory, and two associative algebras ${\mathbf{A}}!$ and ${\mathbf{(\char93 A)!}}$. We prove the Yoneda and Hochschild cohomologies of $ {\mathbf{A}}$ to be isomorphic. There are functors from ${\mathbf{A}}$-bimodules to both ${\mathbf{A}}!$-bimodules and ${\mathbf{(\char93 A)!}}$bimodules which, in the most important cases (e.g., when the poset is finite), induce isomorphisms of Yoneda cohomologies. When the poset is finite every deformation of ${\mathbf{(\char93 A)!}}$ is induced by one of ${\mathbf{A}}$; if $ {\mathbf{A}}$ also takes values in commutative algebras then the deformation theories of $ {\mathbf{(\char93 A)!}}$ and ${\mathbf{A}}$ are isomorphic. We conclude the paper with an example of a noncommutative projective variety. This is obtained by deforming a diagram representing projective $2$-space to a diagram of noncommutative algebras.
Weighted iterates and variants of the Hardy-Littlewood maximal operator
M. A.
Leckband;
C. J.
Neugebauer
51-61
Abstract: In a recent paper, M. A. Leckband and C. J. Neugebauer obtained a rearrangement inequality for a generalized maximal operator with respect to two measures. For an application they studied norm bounds for the iterated Hardy-Littlewood maximal operator with respect to two measures. In this paper this theory is further developed and other applications of the rearrangement inequality are obtained.
On the distribution of the principal series in $L\sp{2}(\Gamma \backslash G)$
Roberto J.
Miatello;
Jorge A.
Vargas
63-75
Abstract: Let $G$ be a semisimple Lie group of split rank one with finite center. If $\Gamma \subset G$ is a discrete cocompact subgroup, then ${L^2}(\Gamma \backslash G) = {\Sigma_{\omega \in \mathcal{E}(G)}}{n_\Gamma }(\omega ) \cdot \omega$. For fixed $ \sigma \in \mathcal{E}(M)$, let $P(\sigma )$ denote the classes of irreducible unitary principal series $ {\pi_{\sigma ,iv}}(v \in {\mathcal{U}^{\ast}})$. Let, for $s > 0,{\psi_\sigma }(s) = {\Sigma_{\omega \in P(\sigma )}}{n_\Gamma }(\omega ) \cdot {e^{s{\lambda_\omega }}}$, where $ {\lambda_\omega }$ is the eigenvalue of $\Omega$ (the Casimir element of $G$) on the class $\omega$. In this paper, we determine the singular part of the asymptotic expansion of ${\psi_\sigma }(s)$ as $s \to {0^ + }$ if $\Gamma$ is torsion free, and the first term of the expansion for arbitrary $\Gamma$. As a consequence, if ${N_\sigma }(r) = \Sigma_{\omega \in P(\sigma ),\vert{\lambda_{{\omega }}\vert < r}}{n_\Gamma }(\omega )$ and $G$ is without connected compact normal subgroups, then $\displaystyle {N_\sigma }(r)\;\sim {C_G}\; \cdot\; \vert Z(G) \cap \Gamma \vert... ...\; \cdot\; \dim(\sigma )\; \cdot\; {r^c} \qquad (c = \frac{1} {2}\,\dim \,G/K),$ as $r \to + \infty$. In the course of the proof, we determine the image and kernel of the restriction homomorphism ${i^{\ast}}:R(K) \to R(M)$ between representation rings.
Handle attaching on generic maps
Youn W.
Lee
77-94
Abstract: Using the handle attaching technique along the singular value set of generic maps in the stable range together with the handle subtraction of Haefliger, smooth immersions and embeddings are studied. We generalize Whitney's immersion theorem, and Haefliger and Hirsh's result on embedding and classification of embeddings of $k$-connected ($(k + 1)$-connected for the classification) smooth $ n$-manifolds into $ {{\mathbf{R}}^{2n - k}}$. For example, we obtain the following as a generalization of Whitney's immersion theorem. If $f: {V^n} \to {M^m}, {3n} < {2m}$, is a generic map such that each component of its double point set is either a closed manifold or diffeomorphic to the $(2n - m)$-disk, then $f$ is homotopic to an immersion.
Operators of $P$-variation and the evolution representation problem
M. A.
Freedman
95-112
Abstract: In contrast to a continuous linear semigroup, a continuous linear evolution $U( \cdot )$ may be nondifferentiable or of unbounded variation. In order to study these evolutions we introduce a class of operator-valued functions $A( \cdot )$ which satisfy a generalized bounded variation condition and represent $U$ as the product integral $U = \prod [I + dA]$.
Sign-embeddings of $l\sp{n}\sb{1}$
John
Elton
113-124
Abstract: If $({e_i})_{i = 1}^n$ are vectors in a real Banach space with $\parallel {e_i}\parallel \leqslant 1$ and Average$_{{\varepsilon_1} = \pm 1}\parallel \sum\nolimits_{i = 1}^n {{\varepsilon_i}{e_i}\parallel \geqslant \delta n}$, where $\delta > 0$, then there is a subset $ A \subseteq \{ 1,\ldots,n\}$ of cardinality $ m \geqslant cn$ such that ${({e_i})_{i \in A}}$ is $K$-equivalent to the standard $ l_1^m$ basis, where $ c > 0$ and $K < \infty$ depend only on $\delta$. As a corollary, if $1 < p < \infty$ and $l_1^n$ is $K$-isomorphic to a subspace of ${L_p}(X)$, then $l_1^m(m \geqslant cn)$ is $K^{\prime}$-isomorphic to a subspace of $ X$, where $c > 0$ and $K^{\prime} < \infty$ depend only on $ K$ and $p$.
Certain reflexive sheaves on ${\bf P}\sp{n}\sb{{\bf C}}$ and a problem in approximation theory
Peter F.
Stiller
125-142
Abstract: This paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces.
Stability criteria for Volterra equations
T. A.
Burton;
W. E.
Mahfoud
143-174
Abstract: We consider a system of integro-differential equations of the form (1.1) $\displaystyle x^{\prime} = A(t)x + \int_0^t {C(t,s)x(s)\;ds}$ with $A$ and $C$ being $n \times n$ matrices. Various types of stability are defined and results are obtained showing when one type of stability is equivalent to another type. We also construct a number of Lyapunov functional from which we obtain necessary and sufficient conditions for stability of (1.1). Finally, we prove several results concerning qualitative behavior of solutions of (1.1).
An extension of the theory of completely decomposable torsion-free abelian groups
Fred
Richman
175-185
Abstract: We construct a class of strongly indecomposable finite rank torsion-free groups that includes the rank-one groups, and develop a complete set of invariants for them and their direct sums.
Spherical harmonics and integral geometry on projective spaces
Eric L.
Grinberg
187-203
Abstract: The Radon transform $ R$ on $ {\mathbf{C}}{P^{\text{n}}}$ associates to a point function $f(x)$ the hyperplane function $ Rf(H)$ by integration over the hyperplane $H$. If ${R^t}$ is the dual transform, we can invert $ {R^t}R$ by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason [7] with very simple computations. We view the Radon transform as a $ G$-invariant map between representations of the group of isometries $G = U(n + 1)$ on function spaces attached to $ {\mathbf{C}}{P^n}$. Pulling back to a sphere via a suitable Hopf fibration and using the theory of spherical harmonics, we can decompose these representations into irreducibles. The scalar by which $R$ acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for $R$. The action of ${R^t}R$ is immediately related to the spectrum of $ {\mathbf{C}}{P^n}$. This shows that ${R^t}R$ can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact $2$-point homogeneous spaces: ${\mathbf{R}}{P^n}$, $ {\mathbf{H}}{P^n}$, ${\mathbf{O}}{P^n}$, as well as spheres.
Chaotic difference equations: generic aspects
Hans Willi
Siegberg
205-213
Abstract: It is shown that in the set of all continuous selfmaps of a compact acyclic polyhedron (i.e. the homology groups of the space vanish in all dimensions $> 0$) the chaotic maps form a dense set. The notion of chaos used here is that of Li and Yorke. If this notion is slightly weakened ("almost chaotic") the density result can be improved by the theorem that the set of almost chaotic (continuous) selfmaps of a compact acyclic polyhedron $P$ contains a residual subset of the space of all continuous selfmaps of $P$. Moreover, the topological entropy of such a generic almost chaotic map is shown to be infinite. The basic ingredients of the proofs are from fixed point index theory.
Elementary first integrals of differential equations
M. J.
Prelle;
M. F.
Singer
215-229
Abstract: We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form. This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.
Symbolic dynamics in flows on three-manifolds
John
Franks
231-236
Abstract: This article deals with the problem of what suspended subshifts of finite type can be realized as a basic set of a nonsingular Smale flow on three-dimensional manifolds. It is shown that any suspended subshift can be realized in such a flow on some three-manifold. Also if signs reflecting orientation are included in the matrix of the subshift of finite type then there is an obstruction to the realization on $ {S^3}$ of basic sets corresponding to some matrices.
On compact cohomology theories and Pontrjagin duality
Keith
Johnson
237-247
Abstract: Cohomology theories taking values in the category of topological groups are examined and a representation theorem is established for those whose coefficient groups are compact and locally euclidean. A method for constructing unstable homology operations is developed using this theorem, and application is made to the case of complex $ K$-theory.
Nakayama algebras and graded trees
B.
Rohnes;
S. O.
Smalø
249-256
Abstract: Let $k$ be an algebraically closed field. We show that if $T$ is a finite tree, then there is a grading $ g$ on $T$ such that $(T,g)$ is a representation finite graded tree, and such that the corresponding simply connected $k$-algebra is a Nakayama algebra (i.e. generalized uniserial algebra).
Cohomology detects failures of the axiom of choice
Andreas
Blass
257-269
Abstract: We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set $X$, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every $ X$-indexed family of nonempty sets has a choice function. We also obtain related results when the coefficient groups are required to be abelian or well-orderable. In particular, we show that, if all discrete spaces have trivial first cohomology for all abelian coefficient groups, then the axiom of choice holds.
Finitely additive $F$-processes
Thomas E.
Armstrong
271-295
Abstract: If one replaces random variables by finitely additive measures one obtains instead of an $F$-process a finitely additive $F$-process. Finitely additive $ F$-processes on a decreasing collection of Boolean algebras form a dual base norm ordered Banach space. When the collection is linearly ordered they form a dual Kakutani $L$-space. This $L$-space may be represented as the $ L$-space of all finitely additive bounded measures on the Boolean ring of predictable subsets of the extreme points of the positive face of the unit ball. Of independent interest is the fact that any bounded supermartingale is a decreasing process in contrast to the usual case where only the supermartingales of class $DL$ are decreasing processes.
Graphs with relations, coverings and group-graded algebras
Edward L.
Green
297-310
Abstract: The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced.
Characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets
Wolfgang
Maass
311-336
Abstract: We show that the lattice of supersets of a recursively enumerable (r.e.) set $A$ is effectively isomorphic to the lattice of all r.e. sets if and only if the complement $ \bar A$ of $A$ is infinite and $\{ e\vert{W_e} \cap \bar A\;{\text{finite}}\}\;{\leqslant_{1}}\emptyset''$ (i.e. $\bar A$ is semilow$_{1.5}$). It is obvious that the condition `` $ \bar{A}\;$ semilow$_{1.5}$'' is necessary. For the other direction a certain uniform splitting property (the "outer splitting property") is derived from semilow$_{1.5}$ and this property is used in an extension of Soare's automorphism machinery for the construction of the effective isomorphism. Since this automorphism machinery is quite complicated we give a simplified proof of Soare's Extension Theorem before we add new features to this argument.
Groups and simple languages
Robert H.
Haring-Smith
337-356
Abstract: With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduced word problem is a simple language if and only if each vertex of the presentation's Cayley diagram has only a finite number of simple closed paths passing through it. Furthermore, if the reduced word problem is simple, then the group is a free product of a free group of finite rank and a finite number of finite groups.
The slice map problem for $\sigma $-weakly closed subspaces of von Neumann algebras
Jon
Kraus
357-376
Abstract: A $\sigma $-weakly closed subspace $\mathcal{S}$ of $ B(\mathcal{H})$ is said to have Property $ {S_\sigma }$ if for any $ \sigma$-weakly closed subspace $ \mathcal{T}$ of a von Neumann algebra $\mathcal{N},\{ x \in \mathcal{S}\;\overline \otimes \mathcal{N}:{R_\varphi }(x... ...hi \in B{(\mathcal{H})_{\ast}}\} = \mathcal{S}\,\overline \otimes \,\mathcal{T}$, where $ {R_\varphi }$ is the right slice map associated with $\varphi$. It is shown that semidiscrete von Neumann algebras have Property $ {S_\sigma }$, and various stability properties of the class of $\sigma $-weakly closed subspaces with Property $ {S_\sigma }$ are established. It is also shown that if $(\mathcal{M},G,\alpha )$ is a ${W^{\ast}}$-dynamical system such that $\mathcal{M}$ has Property ${S_\sigma }$ and $G$ is compact abelian, then all of the spectral subspaces associated with $\alpha$ have Property $ {S_\sigma }$. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if $ {\mathcal{L}_1}$ is a commutative subspace lattice with totally atomic core, and $ {\mathcal{L}_2}$ is an arbitrary subspace lattice, then $ {\text{alg}}({\mathcal{L}_{1}} \otimes {\mathcal{L}_2}) = {\text{alg}}\;{\mathcal{L}_{1}}\,\overline \otimes {\text{alg}}\;{\mathcal{L}_2}$.
Hamburger-Noether expansions over rings
Antonio
Campillo
377-388
Abstract: We study Hamburger-Noether expansions over rings, obtaining some applications to equisingular deformation theory and the moduli problem of plane curve singularities, and construct a universal equation for a given equisingularity class.
Group rings which are Azumaya algebras
F. R.
DeMeyer;
G. J.
Janusz
389-395
Abstract: The group ring $ RG$ of a group $ G$ over a ring $ R$ (with identity $ 1(R)$) is a separable algebra over its center if and only if the following conditions hold: (a) $R$ is a separable algebra over its center; (b) the center of $G$ has finite index in $G$: (c) the commutator subgroup $G^{\prime}$ of $G$ has finite order $m$ and $m1(R)$ is invertible in $R$.
Improved Sobolev inequalities
Robert S.
Strichartz
397-409
Abstract: For a function $ f$ defined on ${{\mathbf{R}}^n}$, Sobolev's inequality $\parallel f{\parallel_q} \leqslant c(\parallel \,f\;{\parallel_{p}} + \parallel \nabla f{\parallel_{p}})$, where $1 < p < q < \infty$ and $1/p - 1/q = 1/n$, can be improved if the Fourier transform $\hat f$ is assumed to have support in a set $ A$ which satisfies an estimate $\vert\{ \xi \in A:\vert\vert\xi \vert\vert \leqslant s\} \vert \leqslant c{s^d}$ for some $ d < n$ the improvement being that we can take $ 1/p - 1/q = 1/d$, provided we also assume $ p \leqslant 2 \leqslant q$. Analogous results are proved for other Sobolev inequalities, for embeddings into Lipschitz-Zygmund spaces, and for functions on symmetric spaces whose Fourier expansions are suitably limited. Improved Sobolev inequalities are established locally for solutions of the wave equation. An application to the Radon transform on spheres is given.
Borel games and the Baire property
Kenneth
Schilling;
Robert
Vaught
411-428
Abstract: The Borel game operations are a natural generalization of the operation $($A$)$. It is shown that these operations preserve the property of Baire in all topological spaces. Applications are given to invariant descriptive set theory and the model theory of infinitary logic.
Erratum to: ``Homotopy in functor categories''
Alex
Heller
429