The error in spatial truncation for systems of parabolic conservation laws
Hung Ju
Kuo
433-465
Abstract: In this paper we investigate the behavior of the solution of \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = D{u_{xx}} - f{{(u)}_x},} ... ...in {L^\infty },\qquad u(t, \pm L) = {u^ \pm },} \end{array} \end{displaymath} where $t \geqslant 0$ and $x \in [ - L,L]$. Solutions of this equation are considered to be approximations to the solutions of the corresponding parabolic conservation laws. We obtain decay results on the norms of the difference between the solution for $L$ infinite and the solution when $L$ is finite.
Regular coverings of homology $3$-spheres by homology $3$-spheres
E.
Luft;
D.
Sjerve
467-481
Abstract: We study $ 3$-manifolds that are homology $3$-spheres and which admit nontrivial regular coverings by homology $3$-spheres. Our main theorem establishes a relationship between such coverings and the canonical covering of the $3$-sphere ${S^3}$ onto the dodecahedral space $ {D^3}$. We also give methods for constructing irreducible sufficiently large homology $3$-spheres $ \tilde M,\;M$ together with a degree $1$ map $ h:M \to {D^3}$ such that $ \tilde M$ is the covering space of $M$ induced from the universal covering ${S^3} \to {D^3}$ by means of the degree $ 1$ map $h:M \to {D^3}$. Finally, we show that if $p:\tilde M \to M$ is a nontrivial regular covering and $\tilde M,\;M$ are homology spheres with $ M$ Seifert fibered, then $\tilde M = {S^3}$ and $M = {D^3}$.
Unramified class field theory for orders
Peter
Stevenhagen
483-500
Abstract: The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field $ K$, is canonically isomorphic to the Galois group of the maximal totally unramified abelian extension of $K$ over $K$, is generalized and proved for all infinite commutative rings with unit that, like rings of integers, are connected and finitely generated as a module over ${\mathbf{Z}}$. Modulo their nilradical, these rings are exactly the connected orders in products of number fields.
A criterion for the boundedness of singular integrals on hypersurfaces
Stephen W.
Semmes
501-513
Abstract: This paper gives geometric conditions on a hypersurface in ${{\mathbf{R}}^n}$ so that certain singular integrals on that hypersurface define bounded operators on $ {L^2}$. These singular integrals include the Cauchy integral operator in the sense of Clifford analysis and in particular the double layer potential. For curves in the plane, this condition is more general than the chord-arc condition but less general than the Ahlfors-David condition. The main tool is the $T(b)$ theorem [DJS].
A $K$-theoretic invariant for dynamical systems
Yiu Tung
Poon
515-533
Abstract: Let $(X,T)$ be a zero-dimensional dynamical system. We consider the quotient group $G = C(X,Z)/B(X,T)$, where $C(X,Z)$ is the group of continuous integer-valued functions on $X$ and $B(X,T)$ is the subgroup of functions of the form $f - f \circ T$. We show that if $(X,T)$ is topologically transitive, then there is a natural order on $G$ which makes $G$ an ordered group. This order structure gives a new invariant for the classification of dynamical systems. We prove that for each $n$, the number of fixed points of $ {T^n}$ is an invariant of the ordered group $G$. Then we show how $G$ can be computed as a direct limit of finite rank ordered groups. This is used to study the conditions under which $\lq G$ is a dimension group. Finally we discuss the relation between $G$ and the ${K_0}$-group of the crossed product ${C^{\ast}}$-algebra associated to the system $ (X,T)$.
On the linear representation of braid groups
D. D.
Long
535-560
Abstract: We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.
The connection matrix theory for Morse decompositions
Robert D.
Franzosa
561-592
Abstract: The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of Morse decompositions is established, and examples illustrating applications of the connection matrix are provided.
Characterization of nonlinear semigroups associated with semilinear evolution equations
Shinnosuke
Oharu;
Tadayasu
Takahashi
593-619
Abstract: Nonlinear continuous perturbations of linear dissipative operators are considered from the point of view of the nonlinear semigroup theory. A general class of nonlinear perturbations of linear contraction semigroups in a Banach space $ X$ is introduced by means of a lower semicontinuous convex functional $ [{\text{unk}}]:X \to [0,\infty ]$ and two notions of semilinear infinitesimal generators of the associated nonlinear semigroups are formulated. Four types of necessary and sufficient conditions are given for a semilinear operator $A + B$ of the class to be the infinitesimal generator of a nonlinear semigroup $\{ S(t):t \geqslant 0\}$ on the domain $C$ of $B$ such that for $x \in C$ the $C$-valued function $ S( \cdot )x$ on $[0,\infty )$ provides a unique mild solution of the semilinear evolution equation $[{\text{unk]}}(u( \cdot ))$. It turns out that various types of characterizations of nonlinear semigroups associated with semilinear evolution equations are obtained and, in particular, a semilinear version of the Hille-Yosida theorem is established in a considerably general form.
Equivariant Morse theory for starshaped Hamiltonian systems
Claude
Viterbo
621-655
Abstract: Let $\Sigma$ be a starshaped hypersurface in $ {R^{2n}}$; the problem of finding closed characteristics of $ \Sigma$ can be classically reduced to a variational problem. This leads to studying an ${S^1}$-equivariant functional on a Hilbert space. The equivariant Morse theory of this functional, together with the assumption that $\Sigma$ only has finitely many geometrically distinct characteristics, leads to a remarkable formula relating the average indices of the characteristics. Using this formula one can prove, at least for $n$ even, that genetically there are infinitely many characteristics (cf. [E1] for the convex case).
On the monoid of tame extensions
Cornelius
Greither;
D. K.
Harrison
657-682
Abstract: This paper deals with not necessarily maximal orders in abelian extensions of number fields. We restrict our attention to orders invariant under the Galois group $G$. Based on recent work of Childs and Hurley [CH], we introduce a notion of tameness for such orders (actually this is done in a slightly more general setting). The maximal order is tame in this sense if and only if the field extension is tamely ramified.
Stability of viscous scalar shock fronts in several dimensions
Jonathan
Goodman
683-695
Abstract: We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation.
Invariants of graphs in three-space
Louis H.
Kauffman
697-710
Abstract: By associating a collection of knots and links to a graph in three-dimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigid-vertex isotopy. Rigid-vertex graphs are a category mixing topological flexibility with mechanical rigidity. Both categories constitute steps toward models for chemical and biological networks. We discuss chirality in both rigid and topological contexts.
Nonlinear second order elliptic partial differential equations at resonance
R.
Iannacci;
M. N.
Nkashama;
J. R.
Ward
711-726
Abstract: In this paper we study the solvability of boundary value problems for semilinear second order elliptic partial differential equations of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy the Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded and some crossing of eigenvalues is allowed. Selfadjoint and nonselfadjoint resonance problems are considered.
Infix congruences on a free monoid
C. M.
Reis
727-737
Abstract: A congruence $ \rho$ on a free monoid ${X^{\ast}}$ is said to be infix if each class $ C$ of $\rho$ satisfies $u \in C$ and $xuy \in C$ imply $xy = 1$. The main purpose of this paper is a characterization of commutative maximal infix congruences. These turn out to be congruences induced by homomorphisms $\tau$ from $ {X^{\ast}}$ to ${{\mathbf{N}}^0}$, the monoid of nonnegative integers under addition, with ${\tau ^{ - 1}}(0) = 1$.
Codimension two complete noncompact submanifolds with nonnegative curvature
Maria Helena
Noronha
739-748
Abstract: We study the topology of complete noncompact manifolds with non-negative sectional curvatures isometrically immersed in Euclidean spaces with codimension two. We investigate some conditions which imply that such a manifold is a topological product of a soul by a Euclidean space and this gives a complete topological description of this manifold.
Multiple solutions of perturbed superquadratic second order Hamiltonian systems
Yi Ming
Long
749-780
Abstract: In this paper we prove the existence of infinitely many distinct $ T$-periodic solutions for the perturbed second order Hamiltonian system $V:{{\mathbf{R}}^N} \to {\mathbf{R}}$ is continuously differentiable and superquadratic, and that $ f$ is square integrable and $T$-periodic. In the proof we use the minimax method of the calculus of variation combining with a priori estimates on minimax values of the corresponding functionals.
Approximating continuous functions by holomorphic and harmonic functions
Christopher J.
Bishop
781-811
Abstract: If $\Omega$ is a Widom domain in the plane (e.g., finitely connected) and $f$ is any bounded harmonic function on $ \Omega$ which is not holomorphic, then we prove the algebra ${H^\infty }(\Omega )[f]$ contains all the uniformly continuous functions on $\Omega$. The basic tools are the solution of the $\overline \partial$ equation with ${L^\infty }$ estimates and some estimates on the level sets of functions in BMOA.
Factorization of diffusions on fibre bundles
Ming
Liao
813-827
Abstract: Let $\pi :M \to N$ be a fibre bundle with a $G$-structure and a connection. A $ G$-invariant operator $ A$ on the standard fibre $ F$ is "shifted" to an operator ${A^{\ast}}$ on $M$ and a semielliptic operator $B$ on $N$ is "lifted" to an operator $\tilde B$ on $M$. Let ${X_t}$ be an $A$-diffusion on $F$, let ${Y_t}$ be a $B$-diffusion on $N$ which is independent of ${X_t}$ and let ${\Psi _t}$ be its horizontal lift in the associated principal bundle. Then ${Z_t} = {\Psi _t}({X_t})$ is a diffusion on $M$ with generator ${A^{\ast}} + \tilde B$. Conversely, such a factorization is possible only if the fibre bundle has a proper $ G$-structure. In the case of a Riemannian submersion, $X,\;Y$ and $Z$ can be taken to be Brownian motions and the existence of a $G$-structure then means that the fibres are totally geodesic.