Stability of travelling wave solutions of diffusive predator-prey systems
R.
Gardner;
C. K. R. T.
Jones
465-524
Abstract: The stability of travelling wave solutions of singularly perturbed, diffusive predator-prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at $\lambda = 0$ is simple. The proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues.
The structure of open continuous mappings having two valences
A. K.
Lyzzaik;
Kenneth
Stephenson
525-566
Abstract: The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either $p$ or $q$ times for some positive integers $ p$, $q$. Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.
A formal Mellin transform in the arithmetic of function fields
David
Goss
567-582
Abstract: The Mellin transform is a fundamental tool of classical arithmetic. We would also like such a tool in the arithmetic of function fields based on Drinfeld modules, although a construction has not yet been found. One formal approach to finding Mellin transforms in classical theory is through $ p$-adic measures. It turns out that this approach also works for function fields. Thus this paper is devoted to exploring what can be learned this way. We will establish some very enticing connections with gamma functions and the Kummer-Vandiver conjecture for function fields.
Linear topological classifications of certain function spaces
Vesko M.
Valov
583-600
Abstract: Some linear classification results for the spaces ${C_P}(X)$ and $C_P^{\ast}\,(X)$ are proved.
On the range of the Radon $d$-plane transform and its dual
Fulton B.
Gonzalez
601-619
Abstract: We present direct, group-theoretic proofs of the range theorem for the Radon $d$-plane transform $f \to \hat f$ on $ \mathcal{S}({\mathbb{R}^n})$. (The original proof, by Richter, involves extensive use of local coordinate calculations on $ G(d,n)$, the Grassmann manifold of affine $d$-planes in $ {\mathbb{R}^n}$.) We show that moment conditions are not sufficient to describe this range when $d < n - 1$, in contrast to the compactly supported case. Finally, we show that the dual $ d$-plane transform maps $ \mathcal{E}(G(d,n))$ surjectively onto $ \mathcal{E}({\mathbb{R}^n})$.
Mod\`ele minimal \'equivariant et formalit\'e
Thierry
Lambre
621-639
Abstract: We study the rational equivariant homotopy type of a topological space $ X$ equipped with an action of the group of integers modulo $n$. For $n= {p^k}$ ($p$ prime, $k$ a positive integer), we build an algebraic model which gives the rational equivariant homotopy type of $ X$. The homotopical fixed-point set appears in the construction of a model of the fixed-points set. In general, this model is different from $ {\text{G}}$. Triantafillou's model $ [{\text{T1}}]$. For $ n= p$ ($p$ prime), we then give a notion of equivariant formality. We prove that this notion is equivalent to the formalizability of the inclusion of fixed-points set $i:{X^{{\mathbb{Z}_p}}} \to X$. Examples and counterexamples of $ {\mathbb{Z}_p}$-formal spaces are given.
$k$-cobordism for links in $S\sp 3$
Tim D.
Cochran
641-654
Abstract: We give an explicit finite set of (based) links which generates, under connected sum, the $k$-cobordism classes of links. We show that the union of these generating sets, $2 \leq k < \infty$, is not a generating set for $\omega$-cobordism classes or even $ \infty$-cobordism classes. For $2$-component links in ${S^3}$ we define $(2,k)$-corbordism and show that the concordance invariants ${\beta ^i},i \in {\mathbb{Z}^+}$, previously defined by the author, are invariants under $ (2,i + 1)$-cobordism. Moreover we show that the $(2,k)$-cobordism classes of links (with linking number 0) is a free abelian group of rank $k - 1$, detected precisely by ${\beta ^1} \times \cdots \times {\beta^{k - 1}}$. We write down a basis. The union of these bases $(2 \leq k < \infty)$ is not a generating set for $ (2,\infty)$ or $(2,\omega)$-cobordism classes. However, we can show that $\prod _{i = 1}^\infty {\beta ^i}(\;)$ is an isomorphism from the group of $(2,\infty)$-cobordism classes to the subgroup $\mathcal{R} \subset \prod _{i = 1}^\infty \mathbb{Z}$ of linearly recurrent sequences, so a basis exists by work of T. Jin.
Isotopy invariants of graphs
D.
Jonish;
K. C.
Millett
655-702
Abstract: The development of oriented and semioriented algebraic invariants associated to a class of embeddings of regular four valent graphs is given. These generalize the analogous invariants for classical knots and links, can be determined from them by means of a weighted averaging process, and define them by means of a new state model. This development includes the elucidation of the elementary spatial equivalences (generalizations of the classical Reidemeister moves), and the extension of fundamental concepts in classical knot theory, such as the linking number, to this class spatial graphs.
Derived functors of unitary highest weight modules at reduction points
Pierluigi Möseneder
Frajria
703-738
Abstract: The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the ${\mathbf{K}}$-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest ${\mathbf{K}}$-type are also easily computed.
Reflected Brownian motion in a cone with radially homogeneous reflection field
Y.
Kwon;
R. J.
Williams
739-780
Abstract: This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (i) The state space is a cone in $d$-dimensions $ (d \geq 3)$, and the process behaves in the interior of the cone like ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone. (iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure). The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter $\alpha$ which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether $\alpha > 0$. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if $ \alpha < 2$, and that starting away from the vertex, this process does not reach the vertex if $ \alpha \leq 0$ and does reach the vertex almost surely if $0 < \alpha < 2$. If $\alpha \geq 2$, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases. The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.
Terms in the Selberg trace formula for ${\rm SL}(3,{\scr Z})\backslash{\rm SL}(3,{\scr R})/{\rm SO}(3,{\scr R})$ associated to Eisenstein series coming from a minimal parabolic subgroup
D. I.
Wallace
781-793
Abstract: In this paper we compute the contribution to the trace formula for $SL(3,\mathcal{Z})$ of the integrals associated to inner products of Eisenstein series. We show these reduce to corresponding integrals for a lower rank trace formula plus a few residual terms.
Convex optimization and the epi-distance topology
Gerald
Beer;
Roberto
Lucchetti
795-813
Abstract: Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $X$, equipped with the completely metrizable topology $\tau$ of uniform convergence of distance functions on bounded sets. A function $f$ in $ \Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $\tau$-approximating functions to the minimum of $ f$. Moreover, we show that most functions in $\langle \Gamma (X),{\tau _{aw}}\rangle$ are well-posed, and that this fails if $ \Gamma (X)$ is topologized by the weaker topology of Mosco convergence, whenever $ X$ is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.
Stable rank and approximation theorems in $H\sp \infty$
Leonardo A.
Laroco
815-832
Abstract: It is conjectured that for ${H^\infty }$ the Bass stable rank $($bsr$)$ is $1$ and the topological stable rank $($tsr$)$ is $2$. ${\text{bsr}}({H^\infty })= 1$ if and only if for every $({f_1},{f_2})\; \in {H^\infty } \times {H^\infty }$ which is a corona pair (i.e., there exist $ {g_{1}}$, ${g_2} \in {H^\infty }$ such that ${f_1}{g_1} + {f_2}{g_2}= 1$) there exists a $g \in {H^\infty }$ such that ${f_1} + {f_2}g \in {({H^\infty })^{ - 1}}$, the invertibles in $ {H^\infty }$; however, it suffices to consider corona pairs $({f_1},{f_2})$ where ${f_1}$ is a Blaschke product. It is also shown that there exists a $ g \in {H^\infty }$ such that $ {f_1} + {f_2}g \in \exp ({H^\infty })$ if and only if $ \log {f_1}$ can be boundedly, analytically defined on $\{ {z \in \mathbb{D}:\vert\,{f_2}(z)\vert < \delta } \}$, for some $\delta > 0$. ${\text{tsr}}({H^\infty })= 2$ if and only if the corona pairs are uniformly dense in ${H^\infty } \times {H^\infty }$; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For $b$ an inner function, $K= {H^2} \ominus b\,{H^2}$ is an ${H^\infty }$-module via the compressed Toeplitz operators $ {C_f}= {P_K}{T_f}{\vert _K}$, for $ f \in {H^\infty }$, where $ {T_f}$ is the Toeplitz operator ${T_f}g= f\,g$, for $g \in {H^2}$. Some stable rank questions can be recast as lifting questions: for $({f_i})_1^n \subset{H^\infty }$, there exist $ ({g_i})_1^n$, $({h_i})_1^n \subset{H^\infty }$ such that $ \sum\nolimits_{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1}$ if and only if the compressed Toeplitz operators $ ({C_{{f_i}}})_1^n$ may be lifted to Toeplitz operators $({T_{{F_i}}})_1^n$ which generate $B({H^2})$ as an ideal.
A multidimensional Wiener-Wintner theorem and spectrum estimation
John J.
Benedetto
833-852
Abstract: Sufficient conditions are given for a bounded positive measure $ \mu$ on ${\mathbb{R}^d}$ to be the power spectrum of a function $ \varphi$. Applications to spectrum estimation are made for the cases in which a signal $\varphi$ is known or its autocorrelation $ {P_\phi }$ is known. In the first case, it is shown that $\displaystyle \int {\vert\hat f(\gamma)\vert^2}d{\mu _\phi}(\gamma)= \mathop {\... ...}\,\frac{1}{\vert B(R )\vert}\,\int_{B(R)} \vert f \ast \varphi (t)\vert^2\;dt,$ where ${\hat P}_{\varphi }= {\mu _\varphi }$, $ B(R)$ is the $ d$-dimensional ball of radius $R$, and $f$ ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is $\displaystyle \forall f \in {L^1}({\mathbb{R}^d})\, \cap \,{L^p}({\mathbb{R}^d}... ...ght)\;\left(\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right),$ where $1 \leq p < 2d/(d + 1)$ and the power spectrum ${\mu _{d - 1}}$ is the compactly supported restriction of surface measure to the unit sphere $\sum\nolimits_{d - 1} { \subseteq } \;{{\hat{\mathbb{R}}}^d}$.
Asymptotic integrations of nonoscillatory second order differential equations
Shao Zhu
Chen
853-865
Abstract: The linear differential equation (1) $(r(t)x^{\prime})^{\prime} + (f(t) + q(t))x= 0$ is viewed as a perturbation of the equation (2) $ (r(t)y^{\prime})^{\prime} + (f(t)y = 0$, where $r > 0$, $f$ and $q$ are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and $ {y_1}$, ${y_2}$ are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution ${x_1}$ of (1). Under some mild assumptions, we characterize that (1) has a principal solution $ {x_1}$ satisfying ${x_1}= {y_1}(1 + o(1))$. Sufficient (sometimes necessary) conditions under which the nonprincipal solution ${x_2}$ of (1) behaves, in three different degrees, like ${y_2}$ as $ t \to \infty$ are also established.
On subordinated holomorphic semigroups
Alfred S.
Carasso;
Tosio
Kato
867-878
Abstract: If $[{e^{ - tA}}]$ is a uniformly bounded $ {C_0}$ semigroup on a complex Banach space $X$, then $- {A^\alpha },$, $0 < \alpha < 1$, generates a holomorphic semigroup on $X$, and $ [{e^{ - t{A^\alpha }}}]$ is subordinated to $ [{e^{ - tA}}]$ through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property.
Radon-Nikod\'ym properties associated with subsets of countable discrete abelian groups
Patrick N.
Dowling
879-890
Abstract: With any subset of a countable discrete abelian we associate with it three Banach space properties. These properties are Radon-Nikodym type properties. The relationship between these properties is investigated. The results are applied to give new characterizations of Riesz subsets and Rosenthal subsets of countable discrete abelian groups.
Shadows of convex bodies
Keith
Ball
891-901
Abstract: It is proved that if $C$ is a convex body in ${\mathbb{R}^n}$ then $C$ has an affine image $\tilde C$ (of nonzero volume) so that if $ P$ is any $ 1$-codimensional orthogonal projection, $\displaystyle \vert P\tilde C\vert \geq \,\vert\tilde C{\vert^{(n - 1)\,/\,n}}.$ It is also shown that there is a pathological body, $ K$, all of whose orthogonal projections have volume about $\sqrt n$ times as large as $\vert K{\vert^{(n - 1)\,/\,n}}$.
Area integral estimates for the biharmonic operator in Lipschitz domains
Jill
Pipher;
Gregory
Verchota
903-917
Abstract: Let $D \subseteq {{\mathbf{R}}^n}$ be a Lipschitz domain and let $u$ be a function biharmonic in $D$, i.e., $\Delta \Delta u= 0$ in $D$. We prove that the nontangential maximal function and the square function of the gradient of $ u$ have equivalent ${L^p}(d\mu)$ norms, where $d\mu \in {A^\infty }\,(d\sigma)$ and $ d\sigma$ is surface measure on $\partial D$.
Erratum to: ``Tiled orders of finite global dimension'' [Trans. Amer. Math. Soc. {\bf 322} (1990), no. 1, 329--342; MR0968884 (91b:16016)]
Hisaaki
Fujita
919-920