On the local regularity of solutions in linear viscoelasticity of several space dimensions
Jong Uhn
Kim
359-398
Abstract: In this paper we discuss the local regularity of solutions of a nonlocal system of equations which describe the motion of a viscoelastic medium in several space dimensions. Our main tool is the microlocal analysis combined with MacCamy's trick and the argument of the classical energy method.
On the theory of internal waves of permanent form in fluids of great depth
C. J.
Amick
399-419
First- and second-order necessary conditions for control problems with constraints
Zsolt
Páles;
Vera
Zeidan
421-453
Abstract: Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form $G(x,u) \in \Gamma$, where $ \Gamma$ is a closed convex set of a Banach space with nonempty interior. The inequality constraints $ g(s,x,u) \leqslant 0$ depend on a parameter $s$ belonging to a compact metric space $S$. The equality constraints are split into two sets of equations $K(x,u) = 0$ and $ H(x,u) = 0$, where the first equation is an abstract control equation, and $ H$ is assumed to have a full rank property in $u$. The objective function is ${\max _{t \in T}}f(t,x,u)$ where $ T$ is a compact metric space, $f$ is upper semicontinuous in $ t$ and Lipschitz in $ (x,u)$. The results are in terms of a function $\sigma$ that disappears when the parameter spaces $ T$ and $S$ are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.
Porous sets and null sets for elliptic harmonic measures
Jang-Mei
Wu
455-473
Abstract: We give a genuinely $ n$-dimensional construction of uniformly elliptic operators $L$ in $ \mathbb{R}_ + ^n$ (of divergence form, and of nondivergence form), which have positive $L$-harmonic measures on a class of porous sets on $\partial \mathbb{R}_ + ^n$ with zero surface measure. The porosity condition given is sharp. The earlier methods were all two dimensional.
An elliptic regularity coefficient estimate for a problem arising from a frequency domain treatment of waves
Xiaobing
Feng;
Dongwoo
Sheen
475-487
Abstract: We consider a sequence of noncoercive elliptic problems, which are the wave equation in the frequency domain, in a rectangular or cubic domain with an absorbing boundary condition. The elliptic regularity coefficient depends on the frequency, and it has a singularity for both zero and infinite frequency. In this paper we derive an elliptic regularity estimate as the frequency tends to zero and infinity.
$L\sp p$-boundedness of pseudo-differential operators of class $S\sb {0,0}$
I. L.
Hwang;
R. B.
Lee
489-510
Abstract: We study the $ {L^p}$-boundedness of pseudo-differential operators with the support of their symbols being contained in $E \times {{\mathbf{R}}^n}$, where $ E$ is a compact subset of $ {{\mathbf{R}}^n}$, and their symbols have derivatives with respect to $ x$ only up to order $ k$, in the Hölder continuous sense, where $k > n/2$ (the case $1 < p \leqslant 2$) and $k > n/p$ (the case $2 < p < \infty$). We also give a new proof of the ${L^p}$-boundedness, $ 1 < p < \infty$, of pseudo-differential operators of class $ S_{0,0}^m$, where $m = m(p) = - n\vert 1/p - 1/2\vert$, and $ a \in S_{0,0}^m$ satisfies $\vert\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )\vert \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}$ for $(x,\xi ) \in {{\mathbf{R}}^n} \times {{\mathbf{R}}^n},\vert\alpha \vert \leqslant k$ and $1 < p \leqslant 2$) and $2 < p < \infty $).
Generalization of the Whitney-Mahowald theorem
Bang He
Li
511-521
Abstract: The Whitney-Mahowald theorem gave normal Euler number $(\bmod \,4)$ for embeddings of a closed $ 2n$-manifold in Euclidean $ 4n$-space. We generalize this theorem to embeddings of closed $2n$-manifolds in an oriented $ 4n$-manifold with an approach in the framework of unoriented bordism groups of maps.
Global oscillatory waves for second order quasilinear wave equations
Paul
Godin
523-547
Abstract: In this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of Cauchy problems for a class of scalar second order quasilinear wave equations, when the space dimension is odd and at least equal to $3$. If time is bounded, corresponding results for quasilinear first order systems were obtained by Guès; to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion. To obtain global results, suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time.
Algebras associated to the Young-Fibonacci lattice
Soichi
Okada
549-568
Abstract: The algebra ${\mathcal{F}_n}$ generated by ${E_1},\; \ldots \;,\;{E_{n - 1}}$ subject to the defining relations $ E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}}... ...\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(\vert i - j\vert \geqslant 2)$ is shown to be a semisimple algebra of dimension $n!$ if the parameters ${x_1},\; \ldots \;,\;{x_{n - 1}},\;{y_1},\; \ldots \;,\;{y_{n - 2}}$ are generic. We also prove that the Bratteli diagram of the tower ${({\mathcal{F}_n})_{n \geqslant 0}}$ of these algebras is the Hasse diagram of the Young-Fibonacci lattice, which is an interesting example, as well as Young's lattice, of a differential poset introduced by $\operatorname{R}$. Stanley. A Young-Fibonacci analogue of the ring of symmetric functions is given and studied.
Modular Schur functions
Grant
Walker
569-604
Abstract: A new family of symmetric functions is considered. These functions are analogous to the classical Schur functions, but depend on an integer modulus $p \geqslant 2$, as well as on a partition $ \lambda$. In the case where $p$ is prime, certain of these functions are shown to be irreducible characters of the general linear group $ GL(n,K)$ in the natural characteristic $p$ of the field $K$. This dualises a wellknown criterion of G. D. James for such characters to be given by classical Schur functions.
Upper bound for distortion of capacity under conformal mapping
Robert E.
Thurman
605-616
Abstract: For a finitely-connected domain $\Omega$ containing $\infty$, with boundary $\Gamma$, the logarithmic capacity $d(\Gamma )$ is invariant under normalized conformal maps of $\Omega$. But the capacity of a subset $A \subset \Gamma$ will likely be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of the capacity of such a set is the so-called "Robin capacity" of the set $ A$. We present here the sharp upper bound for the distortion, in terms of conformal invariants of $\Omega$: the harmonic measures of the boundary components of $\Omega$ and the periods of their harmonic conjugates (the Riemann matrix), and the capacity of $ \Gamma$. In particular, the upper bound depends only on knowing which components of $\Gamma$ contain parts of $A$, not on the specific distribution of $ A$. An extremal configuration is described explicitly for a special case.
The nilpotency class of finite groups of exponent $p$
Michael
Vaughan-Lee
617-640
Abstract: We investigate the properties of Lie algebras of characteristic $ p$ which satisfy the Engel identity $x{y^n} = 0$ for some $n < p$. We establish a criterion which (when satisfied) implies that if $a$ and $b$ are elements of an Engel-$n$ Lie algebra $L$ then $ a{b^{n - 2}}$ generates a nilpotent ideal of $L$. We show that this criterion is satisfied for $n = 6,\,p = 7$, and we deduce that if $ G$ is a finite $ m$-generator group of exponent $7$ then $G$ is nilpotent of class at most $51{m^8}$.
Geometric invariants for Seifert fibred $3$-manifolds
Ming Qing
Ouyang
641-659
Abstract: In this paper, we obtain a formula for the $\eta$-invariant of the signature operator for some circle bundles over Riemannian $2$-orbifolds. We then apply it to Seifert fibred $3$-manifolds endowed with one of the six Seifert geometries. By using a relation between the Chern-Simons invariant and the $\eta$-invariant, we also derive some elementary formulae for the Chern-Simons invariant of these manifolds. As applications, we show that some families of these manifolds cannot be conformally immersed into the Euclidean space ${{\mathbf{E}}^4}$.
BMO on strongly pseudoconvex domains: Hankel operators, duality and $\overline\partial$-estimates
Huiping
Li;
Daniel H.
Luecking
661-691
Abstract: We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman space of a strongly pseudoconvex domain and show that it is equivalent to $BMO$ plus analytic. (Here we mean the Bergman metric $BMO$ of Berger, Coburn and Zhu.) In the course of the proof we obtain new $\overline \partial $-estimates that may be of independent interest. Some applications include a decomposition of $BMO$ similar to the classical ${L^\infty } + \widetilde{{L^\infty }}$, and two characterizations of the dual of $VMO$ (which is also a predual of $ BMO$). In addition, we obtain some partial results on the boundedness of Hankel operators in ${L^1}$ norm.
The Mackey obstruction and the coadjoint orbits
Zongyi
Li
693-705
Abstract: This paper studies the Mackey obstruction representation theory at the coadjoint orbit level. It shows how to get rid of such obstructions and to get orbits of the "little groups". Such little group data is essential for inductive construction of coadjoint orbits of general Lie groups.
On square-preserving isometries of convolution algebras
Sadahiro
Saeki
707-718
Abstract: Let $S$ and $S'$ be two semigroups, each contained in a locally compact group. Under certain conditions on $S$ and $S'$, we shall characterize those isometric additive surjections
Nonproduct type analytic TUHF algebras
Belisario A.
Ventura
719-738
Abstract: We construct examples of nonproduct type real valued cocycles on a UHF groupoid, and show that the analytic triangular algebras associated to those cocycles, can only correspond to nonproduct type cocycles.