Stability of the travelling wave solution of the FitzHugh-Nagumo system
Christopher K. R. T.
Jones
431-469
Abstract: Travelling wave solutions for the FitzHugh-Nagumo equations have been proved to exist, by various authors, close to a certain singular limit of the equations. In this paper it is proved that these waves are stable relative to the full system of partial differential equations; that is, initial values near (in the sup norm) to the travelling wave lead to solutions that decay to some translate of the wave in time. The technique used is the linearised stability criterion; the framework for its use in this context has been given by Evans [6-9]. The search for the spectrum leads to systems of linear ordinary differential equations. The proof uses dynamical systems arguments to analyse these close to the singular limit.
On subalgebras of simple Lie algebras of characteristic $p>0$
B.
Weisfeiler
471-503
Abstract: The main results of the paper are Theorems I.5.1, II.1.3 and III.2.1. Theorem I.5.1 states that if a maximal subalgebra $ M$ of a simple finite-dimensional Lie algebra $G$ has solvable quotients of dimension $\geqslant 2$, then every nilpotent element of $ H$ acts nilpotently on $ G$. Theorem II.1.3 states that if such a simple Lie algebra $G$ contains a maximal subalgebra which is solvable, then $G$ is Zassenbaus-Witt algebra. Theorem III.2.1 states that certain $ {\mathbf{Z}}$-graded finite-dimensional simple Lie algebras are either classical or the difference between the number of nonzero positive and negative homogeneous components is large.
Multi-invariant sets on compact abelian groups
Daniel
Berend
505-535
Abstract: Let $G$ be a finite-dimensional connected compact abelian group. Generalizing previous results, dealing with the case of finite-dimensional tori, a full characterization is given herewith of those commutative semigroups $\Sigma$ of continuous endomorphisms of $ G$ which satisfy the following property: The only infinite closed $ \Sigma$-invariant subset of $G$ is $G$ itself.
Tempered ultradistributions as boundary values of analytic functions
R. S.
Pathak
537-556
Abstract: The spaces ${S_{{a_k}}}$, $ {S^{{b_q}}}$ and $S_{{a_{k}}}^{{b_q}}$ were introduced by I. M. Gel'fand as a generalization of the test function spaces of type $ S$; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space $ (S_{{b_{k}}}^{{a_{q}}})\prime$, which is the space of Fourier transforms of elements in $ (S_{{a_{k}}}^{{b_{q}}})\prime$. This gives rise to a representation of the Fourier transform of an element $U \in (S_{{a_{k}}}^{{b_{q}}})\prime$ having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning $S_{{a_{k}}}^{{b_{q}}}$ and its dual. Finally, in the appendix the equality $ S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}}$ is proved.
Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in ${\bf R}\sp{4}$
Steven R.
Dunbar
557-594
Abstract: We establish the existence of traveling wave solutions for a reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction. The waves are of transition front type, analogous to the solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction-diffusion equation. The waves discussed here are not necessarily monotone. There is a speed ${c^\ast} > 0$ such that for $c > {c^\ast}$ there is a traveling wave moving with speed $c$. The proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a Liapunov function to guarantee the existence of the traveling wave solution. The traveling wave solution is equivalent to a heteroclinic orbit in $4$-dimensional phase space.
Minimal periodic orbits for continuous maps of the interval
Lluís
Alsedà;
Jaume
Llibre;
Rafel
Serra
595-627
Abstract: For continuous maps of the interval into itself, Sarkovskii's Theorem gives the notion of minimal periodic orbit. We complete the characterization of the behavior of minimal periodic orbits. Also, we show for unimodal maps that the min-max essentially describes the behavior of minimal periodic orbits.
The radiance obstruction and parallel forms on affine manifolds
William
Goldman;
Morris W.
Hirsch
629-649
Abstract: A manifold $ M$ is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear $ M$ is called radiant. The obstruction to radiance is a one-dimensional class $ {c_M}$ with coefficients in the flat tangent bundle of $M$. Exterior powers of ${c_M}$ give information on the existence of parallel forms on $M$, especially parallel volume forms. As applications, various kinds of restrictions are found on the holonomy and topology of compact affine manifolds.
The Godbillon-Vey invariant of a transversely homogeneous foliation
Robert
Brooks;
William
Goldman
651-664
Abstract: A real projective foliation is a foliation $ \mathfrak{F}$ with a system of local coordinates transverse to $\mathfrak{F}$ modelled on ${\mathbf{R}}{P^1}$ (so that the coordinate changes are real linear fractional transformations). Given a closed manifold $M$, there is but a finite set of values in ${H^3}(M;{\mathbf{R}})$ which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For $M$ an oriented circle bundle over a surface, this finite set is explicitly computed.
Radial functions and invariant convolution operators
Christopher
Meaney
665-674
Abstract: For $1 < p < 2$ and $n > 1$, let ${A_p}({{\mathbf{R}}^n})$ denote the Figà-Talamanca-Herz algebra, consisting of functions of the form $ ( \ast)$ $\displaystyle \sum\limits_{k = 0}^\infty {{f_k} \ast {g_k}}$ with $\sum\nolimits_k {\vert\vert{f_k}\vert{\vert _p}\cdot\vert\vert{g_k}\vert{\vert _{p\prime}} < \infty }$. We show that if $2n/(n + 1) < p < 2$, then the subalgebra of radial functions in ${A_p}({{\mathbf{R}}^n})$ is strictly larger than the subspace of functions with expansions $( \ast )$ subject to the additional condition that ${f_k}$ and ${g_k}$ are radial for all $k$. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix $G \in [FIA]_{B}^{ - }$ and consider $^B{A_p}(G)$ the subalgebra of $ B$-invariant elements of $ {A_p}(G)$. In particular, we show that the dual of $ ^B{A_p}(G)$ is equal to the space of bounded, right-translation invariant operators on $ {L^{p}}(G)$ which commute with the action of $B$.
Hermitian forms in function theory
Christine R.
Leverenz
675-688
Abstract: Let $f$ and $g$ be analytic in the unit disk $\vert z\vert\; < 1$. We give a new derivation of the positive semidefinite Hermitian form equivalent to $ \vert g(z)\vert \leq \vert f(z)\vert$, for $\vert z \vert < 1$, and use it to derive Hermitian forms for various classes of univalent functions. Sharp coefficient bounds for these classes are obtained from the Hermitian forms. We find the specific functions required to make the Hermitian forms equal to zero.
The transformation of vector-functions, scaling and bifurcation
R. J.
Magnus
689-713
Abstract: Various known methods for studying the bifurcation of zeros of a Banach-space-valued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.
Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element
C. J.
Read
715-725
Abstract: If $A$ is a commutative unital Banach algebra and $G \subset A$ is a collection of nontopological zero divisors, the question arises whether we can find an extension $A\prime$ of $A$ in which every element of $G$ has an inverse. Shilov [1] proved that this was the case if $G$ consisted of a single element, and Arens [2] conjectures that it might be true for any set $ G$. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set $G$ for which no extension $A\prime$ can contain inverses for more than countably many elements of $G$. Bollobás proved that it was possible to find inverses for any countable $G$, and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of $G$ differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of $A$. This answers the question posed by Bollobás in [5].
Triangulations of subanalytic sets and locally subanalytic manifolds
M.
Shiota;
M.
Yokoi
727-750
Abstract: If two polyhedrons are locally subanalytically homeomorphic (that is, the graph is locally subanalytic), they are ${\text{PL}}$ homeomorphic. A locally subanalytic manifold is one whose coordinate transformations are locally subanalytic. It is proved that a locally subanalytic manifold has a unique ${\text{PL}}$ manifold structure. A semialgebraic manifold also is considered.
Asymptotic periodicity of the iterates of Markov operators
A.
Lasota;
T.-Y.
Li;
J. A.
Yorke
751-764
Abstract: We say $P:{L^1} \to {L^1}$ is a Markov operator if (i) $ Pf \geq 0$ for $f \geq 0$ and (ii) $\Vert Pf\Vert = \Vert f\Vert$ if $f \geq 0$. It is shown that any Markov operator $P$ has certain spectral decomposition if, for any $f \in {L^1}$ with $f \geq 0$ and $ \Vert f\Vert = 1$, ${P^n}f \to \mathcal{F}$ when $n \to \infty$, where $ \mathcal{F}$ is a strongly compact subset of ${L^1}$. It follows from this decomposition that $ {P^n}f$ is asymptotically periodic for any $ f \in {L^1}$.
Finite subgroups of formal $A$-modules over ${\germ p}$-adic integer rings
Tetsuo
Nakamura
765-769
Abstract: Let $B \supset A$ be $ \mathfrak{p}$-adic integer rings such that $A/{Z_p}$ is finite and $B/A$ is unramified. Generalizing a result of Fontaine on finite commutative $p$-group schemes, we show that galois homomorphisms of finite subgroups of one-dimensional formal $ A$-modules over $ B$ are given by power series.
Galois theory for cylindric algebras and its applications
Stephen D.
Comer
771-785
Abstract: A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper. Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective.
Behaviour of the velocity of one-dimensional flows in porous media
Juan Luis
Vázquez
787-802
Abstract: The one-dimensional flow of gas of density $u$ through a porous medium obeys the equation $ {u_t} = {({u^m})_{xx}}$, where $ m > 1,x \in {\mathbf{R}}$ and $t > 0$. We prove that the local velocity of the gas, given by $\upsilon = - m{u^{m - 2}}{u_x}$, not only is bounded for $ t \geqslant \tau > 0$ but approaches an $N$-wave profile as $ t \to \infty$. $ N$-waves are the typical asymptotic profiles for some first-order conservation laws, a class of nonlinear hyperbolic equations. The case $ m \leqslant 1$ is also studied: there are solutions with unbounded velocity while others have bounded velocity.
Exact sequences in stable homotopy pair theory
K. A.
Hardie;
A. V.
Jansen
803-816
Abstract: A cylinder-web diagram with associated diagonal sequences is described in stable homotopy pair theory. The diagram may be used to compute stable homotopy pair groups and also stable track groups of two-cell complexes. For the stable Hopf class $\eta$ the stable homotopy pair groups $ {G_k}(\eta ,\eta )(k \leqslant 8)$ are computed together with some of the additive structure of the stable homotopy ring of the complex projective plane.
Graphs of tangles
J. C.
Gómez-Larrañaga
817-830
Abstract: We prove that under necessary conditions a graph of tangles is a prime link. For this we generalize the result that the sum of $ 2$-string prime $ L$-tangles is a prime link. Some applications are found. We explore Property ${\text{L}}$ for tangles in order to prove primeness of knots.
Nonvanishing local cohomology classes
Ira
Moskowitz
831-837
Abstract: We discuss the nonvanishing of a top-dimensional canonical cohomology class of the space $\bar B\mathcal{D}if{f_\omega }\;M$. We treat parallelizable and odd-dimensional stably parallelizable manifolds.
Oriented manifolds that fiber over $S\sp{4}$
Steven M.
Kahn
839-850
Abstract: Necessary and sufficient conditions are given for an oriented manifold $ M$ to fiber up to cobordism over the $4$-sphere ${S^4}$ (i.e. for $M$ to be oriented cobordant to a fiber bundle over ${S^4}$). The result here extends those previously obtained for fiberings over ${S^1}$ and ${S^2}$. In addition, fiberings over products of surfaces are studied with complete solutions (in the sense above) being given in most cases including those of ${S^2} \times {S^2}$ and ${({S^1})^4}$.