A perturbation method in critical point theory and applications
Abbas
Bahri;
Henri
Berestycki
1-32
Abstract: This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type $- \Delta u = {\left\vert u \right\vert^{p - 1}}u + h(x)$ in $\Omega ,\,u = 0$ on $ \partial \Omega$. Here, $ \Omega \subset {{\mathbf{R}}^N}$ is smooth and bounded, and $h \in {L^2}(\Omega )$ is given. We show that there exists ${p_N} > 1$ such that for any $p \in (1,\,{p_N})$ and any $h \in {L^2}(\Omega )$, the preceding equation possesses infinitely many distinct solutions. The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.
On the zeros of Dirichlet $L$-functions. II
Akio
Fujii
33-40
Abstract: Some consequences of the main theorem of On the zeros of Dirichlet $ L$-functions. I, Trans. Amer. Math. Soc. 196 (1974), 225-235 are proved.
The diameter of random graphs
Béla
Bollobás
41-52
Abstract: Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that if $d = d(n) \geqslant 3$ and $m = m(n)$ satisfy $(\log n)/d - 3\,\log \log n \to \infty$, $ {2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \infty$ and ${d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \infty$ then almost every graph with $n$ labelled vertices and $m$ edges has diameter $d$.
The second conjugate algebra of the Fourier algebra of a locally compact group
Anthony To Ming
Lau
53-63
Abstract: Let $G$ be a locally compact group and let $VN(G)$ denote the von Neumann algebra generated by the left translations of $G$ on ${L_2}(G)$. Then $ VN{(G)^{\ast}}$, when regarded as the second conjugate space of the Fourier algebra of $G$, is a Banach algebra with the Arens product. We prove among other things that when $G$ is amenable, $VN{(G)^{\ast}}$ is neither commutative nor semisimple unless $G$ is finite. We study in detail the class of maximal regular left ideals in $VN{(G)^{\ast}}$. We also show that if ${G_1}$ and ${G_2}$ are discrete groups, then ${G_1}$ and ${G_2}$ are isomorphic if and only if $VN{({G_1})^{\ast}}$ and $VN{({G_2})^{\ast}}$ are isometric order isomorphic.
Anneaux de valuation discr\`ete complets non commutatifs
Robert
Vidal
65-81
Abstract: On construit, grace au concept de bimodule au sens de M. Artin, une bonne généralisation non commutative de la notion d'anneau de valuation discrète complet; puis on étudie la validité du théorème de structure de I. S. Cohen en égale caractéristique. Lorsque l'anneau de valuation possède un corps de représentants qui est un corps local (commutatif) à corps résiduel de caractéristique nulle et algébriquement clos, on donne une classification des déviations de commutativité engendrées par une dérivation continue de ce corps local. Enfin, on propose une méthode générale de construction d'anneaux sans corps de Cohen en égale caractéristique et l'article se termine par des problèmes ouverts dans cette théorie.
Obstructions to deforming a space curve
Daniel J.
Curtin
83-94
Abstract: Mumford described a curve, $\gamma$, in $ {{\mathbf{P}}^3}$ that has obstructed infinitesimal deformations (in fact the Hilbert scheme of the curve is generically nonreduced). This paper studies ${{\mathbf{P}}^3}$ over parameter spaces of the form $ \operatorname{Spec} (k[t]/({t^n})),\,n = 2,\,3,\, \ldots $. Given a deformation of $ \gamma$ over $ \operatorname{Spec} (k[t]/({t^n}))$ one attempts to extend it to a deformation of $\gamma$ over $\operatorname{Spec} (k[t]/({t^{n + 1}}))$. If it will not extend, this deformation is said to be obstructed at the nth order. I show that on a generic version of Mumford's curve, an infinitesimal deformation (i.e., a deformation over $ \operatorname{Spec} (k[t]/({t^2}))$) is either obstructed at the second order, or at no order, in which case we say it is unobstructed.
Algebraic extensions of power series rings
Jimmy T.
Arnold
95-110
Abstract: Let $D$ and $J$ be integral domains such that $D \subset J$ and $J[[X]]$ is not algebraic over $D[[X]]$. Is it necessarily the case that there exists an integral domain $R$ such that $D[[X]] \subset R \subseteq J[[X]]$ and $ R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]$? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if $D$ satisfies any one of the conditions (1) $D$ is Noetherian, (2) $D$ is integrally closed, (3) the quotient field $ K$ of $D$ is countably generated as a ring over $D$, or (4) $D$ has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that $J[[X]]$ is algebraic over $D[[X]]$ if and only if it is integral over $D[[X]]$ and necessary and sufficient conditions are given on $D$ and $J$ in order that this occur. Finally if, for every positive integer $n$, $ D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]$ implies that $R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]$, then it is shown that $ J[[{X_1}, \ldots ,{X_n}]]$ is algebraic over $D[[{X_1}, \ldots ,{X_n}]]$ for every $ n$.
When is a linear functional multiplicative?
M.
Roitman;
Y.
Sternfeld
111-124
Abstract: We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazko: If $ \varphi$ is a linear functional on an algebra with unit $A$ such that $\varphi (1) = 1$ and $ \varphi (u) \ne 0$ for any invertible $u$ in $A$, then $\varphi$ is multiplicative, provided the spectrum of each element in $A$ is bounded. We present also other conditions which may replace the assumptions on $A$ in the theorem above.
Algebraic determination of fiberwise PL involutions
Hayon
Kim;
Jehpill
Kim;
Kyung Whan
Kwun
125-131
Abstract: Some fiberwise PL involutions on fibered $3$-manifolds induce the obvious automorphism of the fundamental group. It is shown that this expected behavior of the fundamental group in turn characterizes such fiberwise involutions.
The local Kronecker-Weber theorem
Jonathan
Lubin
133-138
Abstract: The extension of a local field generated by adjoining the torsion points on a suitable formal group is essentially the maximal abelian extension of the field. This fact is proven by appealing to the functorial properties of the Herbrand transition function of higher ramification theory.
Equivariant cofibrations and nilpotency
Robert H.
Lewis
139-155
Abstract: Let $f:B \to Y$ be a cofibration whose cofiber is a Moore space. We give necessary and sufficient conditions for $f$ to be induced by a map of the desuspension of the cofiber into $B$. These conditions are especially simple if $ B$ and $Y$ are nilpotent. We obtain some results on the existence of equivariant Moore spaces, and use them to construct examples of noninduced cofibrations between nilpotent spaces. Our machinery also leads to a cell structure proof of the characterization of pre-nilpotent spaces due to Dror and Dwyer [7], and to a simple proof, for finite fundamental group, of the result of Brown and Kahn [4] that homotopy dimension equals simple cohomological dimension in nilpotent spaces.
Essential spectra of elementary operators
L. A.
Fialkow
157-174
Abstract: This paper describes the essential spectrum and index function of the operator $X \to AXB$, where $A$, $B$, and $X$ are Hilbert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.
Whitney stratified chains and cochains
R. Mark
Goresky
175-196
Abstract: This paper contains the technical constructions necessary for a "geometric cycle" definition of cohomology and homology in the context of Whitney stratifications. Cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles.
Derivatives and Lebesgue points via homeomorphic changes of scale
Don L.
Hancock
197-218
Abstract: Let $I$ be a closed interval, and suppose $\mathcal{K}$, $ \mathcal{H}$, and $ \Lambda$ denote, respectively, the class of homeomorphisms of $ I$ onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on $I$ for which each point is a Lebesgue point. Maximoff proved that $\Lambda \circ \mathcal{K}$ is exactly the class of Darboux Baire $1$ functions, where $\Lambda \circ \mathcal{K} = \{ f \circ k:f \in \Lambda ,k \in \mathcal{K}\}$. The present paper is devoted primarily to a study of $\mathcal{H} \circ \Lambda = \{ h \circ f:f \in \Lambda ,h \in \mathcal{H}\}$. The characterizations of this class which are obtained show that a function is a member of $\mathcal{H} \circ \Lambda $ if and only if, in addition to the obvious requirement of approximate continuity, it satisfies certain growth and density-like conditions. In particular, any approximately continuous function with countably many non-Lebesgue points belongs to $\mathcal{H} \circ \Lambda $. It is also established that $ \mathcal{H} \circ \Lambda$ is a uniformly closed algebra properly containing the smallest algebra generated from $ \Lambda$, and a characterization of the latter algebra is provided.
On analytic diameters and analytic centers of compact sets
Shōji
Kobayashi;
Nobuyuki
Suita
219-228
Abstract: In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker.
Invariant connections and Yang-Mills solutions
Mitsuhiro
Itoh
229-236
Abstract: A condition on the self-duality and the stability of Yang-Mills solutions are discussed. The canonical invariant $ G$-connections on $ {S^4}$ and ${P_2}({\mathbf{C}})$ are considered as Yang-Mills solutions. The non-self-duality of the connections requires the injectivity of the isotropy homomorphisms. We construct examples of non-self-dual connections on $ G$-vector bundles ($ G$ is a compact simple group). Under a certain property of the isotropy homomorphism, these canonical connections are not weakly stable.
The complexification and differential structure of a locally compact group
Kelly
McKennon
237-258
Abstract: The concept of a complexification of a locally compact group is defined and its connections with the differential structure developed. To provide an interpretation in terms of irreducible representations of separable, Type I groups, a duality theorem and Bochner theorem are presented.
A characterization of best $\Phi $-approximants
D.
Landers;
L.
Rogge
259-264
Abstract: Let $T$ be an operator from an Orlicz space ${L_\Phi }$ into itself. It is shown in this paper that four algebraic conditions and one integration condition assure that $T$ is the best $\Phi$-approximator, given a suitable $ \sigma$-lattice.
Refinement properties and extensions of filters in Boolean algebras
Bohuslav
Balcar;
Petr
Simon;
Peter
Vojtáš
265-283
Abstract: We consider the question, under what conditions a given family $ A$ in a Boolean algebra $\mathcal{B}$ has a disjoint refinement. Of course, $A$ cannot have a disjoint refinement if $ A$ is a dense subset of an atomless $ \mathcal{B}$, or if $\mathcal{B}$ is complete and $A$ generates an ultrafilter on $\mathcal{B}$. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base.
A Phragm\'en-Lindel\"of theorem conjectured by D. J. Newman
W. H. J.
Fuchs
285-293
Abstract: Let $D$ be a region of the complex plane, $\infty \in \partial D$. If $ f(z)$ is holomorphic in $ D$, write $ M(r) = {\sup _{\vert z\vert = r,\,z \in D}}\vert f(z)\vert$. Theorem 1. If $f(z)$ is holomorphic in $D$ and $ \lim {\sup _{z \to \zeta ,\,z \in D}}\vert f(z)\vert \leqslant 1$ for $\zeta \in \partial D$, $\zeta \ne \infty$, then one of the following holds (a) $ \vert f(z)\vert < 1(z \in D)$, (b)$f(z)$ has a pole at $\infty$, (c) $\log \,M(r)/\log r \to \infty$ as $r \to \infty$. If $M(r)/r \to 0(r \to \infty )$, then (a) must hold.
The Hilbert transform and maximal function for approximately homogeneous curves
David A.
Weinberg
295-306
Abstract: Let $ {\mathcal{H}_\gamma }f(x) = {\text{p}}{\text{.v}}{\text{.}}\int_{ - 1}^1 {f(x - \gamma (t))dt/t}$ and ${\mathfrak{M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert dt}$. It is proved that for $f \in \mathcal{S}({{\mathbf{R}}^n})$, the Schwartz class, and for an approximately homogeneous curve $\gamma (t) \in {{\mathbf{R}}^n}$, $ {\left\Vert {{\mathcal{H}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$, ${\left\Vert {{\mathfrak{M}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$. A homogeneous curve is one which satisfies a differential equation $0 < t < \infty$, where $A$ is a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curve $ \gamma (t)$ has the form $ {\gamma _1}(t) + {\gamma _2}(t)$, where $ {\gamma _2}(t)$ is a carefully specified "error", such that $\gamma _2^{(j)}$ is also restricted for $j = 2, \ldots ,n + 1$. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.
Markov processes with Lipschitz semigroups
Richard
Bass
307-320
Abstract: For $f$ a function on a metric space, let $\displaystyle \operatorname{Lip} f = \mathop {\sup }\limits_{x \ne y} \vert f(x) - f(y)\vert/d(x,\,y),$ and say that a semigroup $ {P_t}$ is Lipschitz if $\operatorname{Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname{Lip} f$ for all $f$, $t$, where $K$ is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology. Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.
Prime knots and tangles
W. B. Raymond
Lickorish
321-332
Abstract: A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored.
Erratum to: ``$\sp{\ast} $-valuations and ordered $\sp{\ast} $-fields''
S. S.
Holland
333