On invariants of graphs with applications to knot theory
Kunio
Murasugi
1-49
Abstract: To each weighted graph $\Gamma$, two invariants, a polynomial ${P_\Gamma }(x,y,z)$ and the signature $\sigma (\Gamma)$, are defined. The various partial degress of $ {P_\Gamma }(x,y,z)$ and $\sigma (\Gamma)$ are expressed in terms of maximal spanning graphs of $\Gamma$. Furthermore, one unexpected property of Tutte's dichromate is proved. These results are applied to knots or links in ${S^3}$.
Propagation of $L\sp q\sb k$-smoothness for solutions of the Euler equation
Gustavo
Ponce
51-61
Abstract: The motion of an ideal incompressible fluid is described by a system of partial differential equations known as the Euler equation. Considering the initial value problem for this equation, we prove that in a classical solution the $ L_k^q$-regularity of the data propagates along the fluid lines. Our method consists of combining properties of the $\varepsilon$-approximate solution with ${L^q}$-energy estimates and simple results of classical singular integral operators. In particular, for the two-dimensional case we present an elementary proof.
Mel\cprime nikov transforms, Bernoulli bundles, and almost periodic perturbations
Kenneth R.
Meyer;
George R.
Sell
63-105
Abstract: In this paper we study nonlinear time-varying perturbations of an autonomous vector field in the plane ${R^2}$ . We assume that the unperturbed equation, i.e. the given vector field has a homoclinic orbit and we present a generalization of the Melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory. The key to our generalization is the concept of the Melnikov transform, which is a linear transformation on the space of perturbation functions. The appropriate dynamical setting for studying these perturbation is the concept of a skew product flow. The concept of transversality we require is best understood in this context. Under conditions whereby the perturbed equation admits a transversal homoclinic trajectory, we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow. We show the dynamics near this trajectory can have the exotic behavior of the Bernoulli shift. The exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle, which we call, the Bernoulli bundle. We allow all perturbations which are bounded and uniformly continuous in time. Thus our theory includes the classical periodic perturbations studied by Melnikov, quasi periodic and almost periodic perturbations, as well as toroidal perturbations which are close to quasi periodic perturbations.
Global solvability of the derivative nonlinear Schr\"odinger equation
Jyh-Hao
Lee
107-118
Abstract: The derivative nonlinear Schrödinger equation $($DNLS$)$ \begin{displaymath}\begin{array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q... ...sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \end{array} \end{displaymath} was first derived by plasma physicists [9,10]. This equation was used to interpret the propagation of circular polarized nonlinear Alfvén waves in plasma. Kaup and Newell obtained the soliton solutions of DNLS in 1978 [5]. The author obtained the local solvability of DNLS in his dissertation [6]. In this paper we obtain global existence (in time $ t$) of Schwartz class solutions of DNLS if the ${L^2}$-norm of the generic initial data $ q(x,0)$ is bounded.
Summability of Hermite expansions. I
S.
Thangavelu
119-142
Abstract: We study the summability of one-dimensional Hermite expansions. We prove that the critical index for the Riesz summability is $ 1/6$. We also prove analogues of the Fejér-Lebesgue theorem and Riemann's localisation principle.
Summability of Hermite expansions. II
S.
Thangavelu
143-170
Abstract: We study the summability of $n$-dimensional Hermite expansions where $n > 1$. We prove that the critical index for the Riesz summability is $ (n - 1)/2$. We also prove analogues of the Fejér-Lebesgue theorem and Riemann's localisation principle when the index $ \alpha$ of the Riesz means is $> (3n - 2)/6$ .
Equivalent conditions to the spectral decomposition property for closed operators
I.
Erdélyi;
Sheng Wang
Wang
171-186
Abstract: The spectral decomposition property has been instrumental in developing a local spectral theory for closed operators acting on a complex Banach space. This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property.
On the Cauchy problem and initial traces for a degenerate parabolic equation
E.
DiBenedetto;
M. A.
Herrero
187-224
Abstract: We consider the Cauchy problem (f) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \operatorname{div}(\ver... ...= {u_0}(x),} & {x \in {{\mathbf{R}}^N},} \end{array} } \right.$ and discuss existence of solutions in some strip ${S_T} \equiv {{\mathbf{R}}^N} \times (0,T)$, $0 < T \leq \infty$, in terms of the behavior of $x \to {u_0}(x)$ as $\vert x\vert \to \infty $. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of $ L_{{\text{loc}}}^1({{\mathbf{R}}^N})$.
Similarity, quasisimilarity, and operator factorizations
Raúl E.
Curto;
Lawrence A.
Fialkow
225-254
Abstract: We introduce and illustrate an operator factorization technique to study similarity and quasisimilarity of Hilbert space operators. The technique allows one to generate, in a systematic way, families of "test" operators, and to check for similarity and quasisimilarity with a given model. In the case of the unilateral shift ${U_ + }$, we obtain a one-parameter family of nonhyponormal, noncontractive, shift-like operators in the similarity orbit of ${U_ + }$. We also obtain new characterizations of quasisimilarity and similarity in terms of invariant operator ranges, and conditions for spectral and essential spectral inclusions.
Superprocesses and their linear additive functionals
E. B.
Dynkin
255-282
Abstract: Let $X = ({X_t},P)$ be a measure-valued stochastic process. Linear functionals of $X$ are the elements of the minimal closed subspace $ L$ of ${L^2}(P)$ which contains all $ {X_t}(B)$ with $\smallint {{X_t}{{(B)}^2}\;dP\; < \infty }$. Various classes of $L$-valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case--"weighted occupation times"--such equations have been established earlier by Iscoe. We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.
Index formulas for elliptic boundary value problems in plane domains with corners
Gregory
Eskin
283-348
Abstract: We derive the conditions for the operator corresponding to a general elliptic boundary value problem in a plane domain with corners to be Fredholm and give an explicit formula for the index of this operator.
Variational problems on contact Riemannian manifolds
Shukichi
Tanno
349-379
Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable ${\text{CR}}$ manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.
Singular relaxation moduli and smoothing in three-dimensional viscoelasticity
Wolfgang
Desch;
Ronald
Grimmer
381-404
Abstract: We develop a semigroup setting for linear viscoelasticity in three-dimensional space with tensor-valued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.
On $H\sb *(\Omega\sp {n+2}S\sp {n+1};{\bf F}\sb 2)$
Thomas J.
Hunter
405-420
Abstract: In this paper, we study ${H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ . Here $ \Omega X$ denotes the space of pointed maps $ {S^1} \to X$, and $ {H_\ast}$ represents homology modulo $2$. We show that the Eilenberg-Moore spectral sequence $\operatorname{Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast\ast}({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}$ collapses, and we identify the kernel of the Whitehead product map ${\Omega ^{n + 1}}{p_\ast}:{H_\ast}{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast}{\Omega ^{n + 1}}{S^n}$ . These observations yield two different descriptions of ${H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ up to extension.
Limitation topologies on function spaces
Philip L.
Bowers
421-431
Abstract: Four competing definitions for limitation topologies on the set of continuous functions $C(X,Y)$ are compared.