Self-maps of loop spaces. I
H. E. A.
Campbell;
F. P.
Peterson;
P. S.
Selick
1-39
Abstract: In this paper we study self-maps of $ {\Omega ^k}{S^{m + 1}}$ and show that, except for the cases $m = 1,\,3,\,7$, or $p$ and $m$, if $f$ induces an isomorphism on $ {H_{m + 1 - k}}({\Omega ^k}{S^{m + 1}};\,Z/pZ)$ with $k < m$, then ${f_{(p)}}$ is a homotopy equivalence.
Self-maps of loop spaces. II
H. E. A.
Campbell;
F. R.
Cohen;
F. P.
Peterson;
P. S.
Selick
41-51
Abstract: We study under what conditions on a finite CW complex $X$ is $Q(X)$ atomic.
Derivatives of mappings with applications to nonlinear differential equations
Martin
Schechter
53-69
Abstract: We present a new definition of differentiation for mappings of sets in topological vector spaces. Complete flexibility is allowed in choosing the topology with which the derivative is taken. We determine the largest space on which the derivative can act. Our definition includes all others hitherto given, and the basic theorems of calculus hold for it. Applications are considered here and elsewhere.
A rearranged good $\lambda$ inequality
Richard J.
Bagby;
Douglas S.
Kurtz
71-81
Abstract: Let $Tf$ be a maximal Calderón-Zygmund singular integral, $Mf$ the Hardy-Littlewood maximal function, and $ w$ an ${A_\infty }$ weight. We replace the ``good $ \lambda$'' inequality $\displaystyle w\left( {\{ x:\,Tf(x) > 2\lambda \,{\text{and}}\,Mf(x) \leq \vare... ...a \} } \right) \leq C(\varepsilon )w\left( {\{ x:\,Tf(x) > \lambda \} } \right)$ by the rearrangement inequality $\displaystyle (Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t)$ and show that it gives better estimates for $Tf$. In particular, we obtain best possible weighted ${L^p}$ bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for ${(Tf)^ \ast }$.
Conjugacy problem in ${\rm GL}\sb 2({\bf Z}[\sqrt{-1}])$ and units of quadratic extensions of ${\bf Q}(\sqrt{-1})$
Hironori
Onishi
83-98
Abstract: A highly efficient procedure for deciding if two given elements of $ {\text{G}}{{\text{L}}_2}(\mathbf{Z}[\sqrt { - 1} ])$ are conjugate or not will be presented. It makes use of a continued fraction algorithm in $\mathbf{Z}[\sqrt { - 1} ]$ and gives a fundamental unit of any given quadratic extension of $\mathbf{Q}(\sqrt { - 1} )$.
The balanced-projective dimension of abelian $p$-groups
L.
Fuchs;
P.
Hill
99-112
Abstract: The balanced-projective dimension of every abelian $p$-group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for $\lambda = {\omega _n}$ that every ${p^\lambda }$-high subgroup of a $p$-group $G$ has balanced-projective dimension exactly $n$ whenever $G$ has cardinality $ {\aleph _n}$ but ${p^\lambda }G \ne 0$. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.
The fifth and seventh order mock theta functions
George E.
Andrews
113-134
Abstract: The theory of Bailey chains is extended to yield identities for Hecke type modular forms and related generalizations. The extended results allow us to produce Hecke type series for the fifth and seventh order mock theta functions. New results on the generating function for sums of three squares also follow, and a new proof that every integer is the sum of three triangular numbers is given.
Strange attractors of uniform flows
Ittai
Kan
135-159
Abstract: Consider orbitally stable attractors of those flows on the open solid torus ${D^2} \times {S^1}$ which have uniform velocity in the ${S^1}$ direction (uniform flows). It is found that any such attractor is the frontier of a strictly nested sequence of positively invariant open solid tori. Necessary and sufficient conditions related to these tori are derived for an arbitrary set to be an orbitally stable attractor. When the cross-section of an orbitally stable attractor is a Cantor set, the first return map is found to be conjugate to an irrational rotation on a certain compact abelian group. New examples are constructed of orbitally stable attractors of uniform ${C^\infty }$ flows whose cross-sections have uncountably many components (one of these attractors has positive $3$-dimensional Lebesgue measure).
Multivariate rational approximation
Ronald A.
DeVore;
Xiang Ming
Yu
161-169
Abstract: We estimate the error in approximating a function $f$ by rational functions of degree $ n$ in the norm of $ {L_q}(\Omega ),\,\Omega : = {[0,\,1]^d}$. Among other things, we prove that if $ f$ is in the Sobolev space $W_p^k(\Omega )$ and if $k/d - 1/p + 1/q > 0$, then $f$ can be approximated by rational functions of degree $n$ to an order $ O({n^{ - k/d}})$.
On periodic solutions of superlinear parabolic problems
Maria J.
Esteban
171-189
Abstract: In this paper we study the existence of positive nontrivial periodic solutions of semilinear parabolic problems. Most of the nonlinearities considered are of the superlinear type. Some bifurcation results are proved as well.
Quasilinear evolution equations and parabolic systems
Herbert
Amann
191-227
Abstract: It is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values, provided the boundary conditions are ``time-independent''. Moreover it is shown that, in the autonomous case, these equations generate local semiflows on appropriate Sobolev spaces. Our results apply, in particular, to the case of prescribed boundary values (Dirichlet boundary conditions).
Regularization for $n$th-order linear boundary value problems using $m$th-order differential operators
D. A.
Kouba;
John
Locker
229-255
Abstract: Let $X$ and $Y$ denote real Hilbert spaces, and let $L:\,X \to Y$ be a closed densely-defined linear operator having closed range. Given an element $ y \in Y$, we determine least squares solutions of the linear equation $ Lx = y$ by using the method of regularization. Let $Z$ be a third Hilbert space, and let $T:\,X \to Z$ be a linear operator with $ \mathcal{D}(L) \subseteq \mathcal{D}(T)$. Under suitable conditions on $ L$ and $T$ and for each $\alpha \ne 0$, we show that there exists a unique element $ {x_\alpha } \in \mathcal{D}(L)$ which minimizes the functional ${G_\alpha }(x) = {\left\Vert {Lx - y} \right\Vert^2} + {\alpha ^2}{\left\Vert {Tx} \right\Vert^2}$, and the $ {x_\alpha }$ converge to a least squares solution ${x_0}$ of $Lx = y$ as $ \alpha \to 0$. We apply our results to the special case where $L$ is an $n$th-order differential operator in $X = {L^2}[a,b]$, and we regularize using for $ T$ an $m$th-order differential operator in ${L^2}[a,b]$ with $m \le n$. Using an approximating space of Hermite splines, we construct numerical solutions to $ Lx = y$ by the method of continuous least squares and the method of discrete least squares.
Derivation, $L\sp \Psi$-bounded martingales and covering conditions
Michel
Talagrand
257-291
Abstract: Let $(\Omega ,\,\Sigma ,\,P)$ be a complete probability space. Let ${({\Sigma _t})_{t \in J}}$ be a directed family of sub-$\sigma$-algebras of $\Sigma$. Let $ (\Phi,\,\Psi)$ be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of ${L^\Psi }$-bounded martingales. We do not assume that either $\Phi$ or $\Psi$ satisfy the $ {\Delta _2}$ condition. We show that when $\Phi$ satisfies condition Exp, that is when there exists an $a > 0$ such that $\Phi (u) \leq \operatorname{exp} \,au$ for each $u \ge 0$, the essential convergence of $ {L^\Psi }$-bounded martingales is equivalent to the classical covering condition $ {V_\Phi }$. This covers in particular the classical case $\Psi (t) = t{(\operatorname{log} \,t)^ + }$. The growth condition Exp on $\Phi$ cannot be relaxed. When $ J$ contains a countable cofinite set, we show that the essential convergence of $ {L^\Psi }$-bounded martingales is equivalent to a covering condition $ {D_\Phi }$ (that is weaker than ${V_\Phi }$). When $\Phi$ fails condition Exp, condition $ {D_\Phi }$ is optimal. Roughly speaking, in the case of ${L^1 }$-bounded martingales, condition $ {D_\Phi }$ means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where $J$ does not contain a countable cofinal set and $\Phi$ fails condition Exp. In this case, it seems impossible to characterize the essential convergence of $ {L^\Psi }$-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable ${L^1 }$-bounded martingales, but not all $ {L^1 }$-bounded martingales, converge essentially. Similar results are also established in the derivation setting.
McKay quivers and extended Dynkin diagrams
Maurice
Auslander;
Idun
Reiten
293-301
Abstract: Let $k$ be an algebraically closed field and $G$ a finite nontrivial group whose order is not divisible by the characteristic of $k$. Associated with an $m$-dimensional representation of $ G$ is the McKay quiver, whose vertices correspond to the irreducible representations of $G$. We show that if $m = 2$, then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.
The normal closure of the coproduct of rings over a division ring
Wallace S.
Martindale
303-317
Abstract: Let $R = {R_1}\coprod {R_2}$ be the coproduct of $ \Delta$-rings $ {R_1}$ and ${R_2}$ with 1 over a division ring $\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta$, with at least one of the dimensions ${({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2$, greater than 2. If ${R_1}$ and ${R_2}$ are weakly $1$-finite (i.e., one-sided inverses are two-sided) then it is shown that every $X$-inner automorphism of $ R$ (in the sense of Kharchenko) is inner, unless $ {R_1},\,{R_2}$ satisfy one of the following conditions: (I) each $ {R_i}$ is primary (i.e., $ {R_i} = \Delta + T,\,{T^2} = 0$), (II) one ${R_i}$ is primary and the other is $ 2$-dimensional, (III) char. $\Delta = 2$, one ${R_i}$ is not a domain, and one ${R_i}$ is $2$-dimensional. This generalizes a recent joint result with Lichtman (where each ${R_i}$ was a domain) and an earlier joint result with Montgomery (where each ${R_i}$ was a domain and $\Delta$ was a field).
Approximation theorems for Nash mappings and Nash manifolds
Masahiro
Shiota
319-337
Abstract: Let $0 \leq r < \infty$. A Nash function on ${\mathbf{R}^n}$ is a ${C^r}$ function whose graph is semialgebraic. It is shown that a ${C^r}$ Nash function is approximated by a ${C^\omega}$ Nash one in a strong topology defined in the same way as the usual topology on the space $\mathcal{S}$ of rapidly decreasing ${C^\infty}$ functions. A ${C^r}$ Nash manifold in ${\mathbf{R}^n}$ is a semialgebraic ${C^r}$ manifold. We also prove that a ${C^r}$ Nash manifold for $r \ge 1$ is approximated by a ${C^\omega}$ Nash manifold, from which we can classify all ${C^r}$ Nash manifolds by ${C^r}$ Nash diffeomorphisms.
On the existence and uniqueness of complex structure and spaces with ``few'' operators
Stanisław J.
Szarek
339-353
Abstract: We construct a $ 2n$-dimensional real normed space whose (Banach-Mazur) distance to the set of spaces admitting complex structure is of order $ {n^{1/2}}$, and two complex $n$-dimensional normed spaces which are isometric as real spaces, but whose complex Banach-Mazur distance is of order $n$. Both orders of magnitude are the largest possible. We also construct finite-dimensional spaces with the property that all ``well-bounded'' operators on them are ``rather small'' (in the sense of some ideal norm) perturbations of multiples of identity. We also state some ``metatheorem'', which can be used to produce spaces with various pathological properties.
Reflexivity and order properties of scalar-type spectral operators in locally convex spaces
P. G.
Dodds;
B.
de Pagter;
W.
Ricker
355-380
Abstract: One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space $X$ is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.
The trace of an action and the degree of a map
Daniel Henry
Gottlieb
381-410
Abstract: Two integer invariants of a fibration are defined: the degree, which generalizes the usual notion, and the trace. These numbers represent the smallest transfers for integral homology which can be constructed for the fibrations. Since every action gives rise to a fibration, we have the trace of an action. A list of properties of this trace is developed. This list immediately gives, in a mechanical way, new proofs and generalizations of theorems of Borsuk-Ulam, P. A. Smith, Conner and Floyd, Bredon, W. Browder, and G. Carlsson.
Long time asymptotics of the Korteweg-de Vries equation
Stephanos
Venakides
411-419
Abstract: We study the long time evolution of the solution to the Kortewegde Vries equation with initial data $\upsilon (x)$ which satisfy $\displaystyle \lim \limits_{x \to - \infty } \upsilon (x) = - 1,\qquad \lim \limits_{x \to + \infty } \upsilon (x) = 0$ We show that as $t \to \infty$ the step emits a wavetrain of solitons which asymptotically have twice the amplitude of the initial step. We derive a lower bound of the number of solitons separated at time $ t$ for $t$ large.
Symmetric semicontinuity implies continuity
Jaromir
Uher
421-429
Abstract: The main result of this paper shows that, for any function, symmetric semicontinuity on a measurable set $E$ implies continuity a.e. in $ E$ and, similarly, that symmetric semicontinuity on a set residual in $ R$ implies continuity on a set residual in $R$. These propositions are used to prove more precise versions of the fundamental connections between symmetric and ordinary differentiability.