Maxwell's equations in a periodic structure
Xinfu
Chen;
Avner
Friedman
465-507
Abstract: Consider a diffraction of a beam of particles in ${\mathbb{R}^3}$ when the dielectric coefficient is a constant $ {\varepsilon _1}$ above a surface $S$ and a constant $ {\varepsilon _2}$ below a surface $S$, and the magnetic permeability is constant throughout $ {\mathbb{R}^3}$. $ S$ is assumed to be periodic in the ${x_1}$ direction and of the form $ {x_1} = {f_1}(s),\,{x_3} = {f_3}(s),\,{x_2}$ arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in $ {\mathbb{R}^3}$ having the form of refracted waves for $ {x_3} \ll 1$ and of transmitted waves for $ - {x_3} \gg 1$ if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the $ \varepsilon$'s, except for a discrete number, there exists a unique solution to the Maxwell equations.
Cavitational flows and global injectivity of conformal maps
Massimo
Lanza de Cristoforis
509-527
Abstract: This paper treats some new mathematical aspects of the two-dimensional cavitational problem of the flow of a perfect fluid past an obstacle. Natural regularity conditions of very general type are found to ensure the global injectivity of the complex-potential and the presence of at most one zero of its derivative on the boundary of the flow. This derivative is the complex velocity. Previous authors have hypothesized the properties obtained here. The same regularity conditions are then shown to be satisfied by the classical solutions found via Villat's integral equation. A simple counterexample in $ \S4$ shows that the global injectivity of a holomorphic map defined on an unbounded Jordan domain cannot be deduced solely from its injectivity on the boundary. This simple fact raises new questions on the relation between cavitational flows and Villat's integral equation, which are discussed in $\S3$.
On the homology of ${\rm SU}(n)$ instantons
Charles P.
Boyer;
Benjamin M.
Mann;
Daniel
Waggoner
529-561
Abstract: In this paper we study the homology of the moduli spaces of instantons associated to principal $ {\mathbf{SU}}(n)$ bundles over the four-sphere. This is accomplished by exploiting an "iterated loop space" structure implicit in the disjoint union of all moduli spaces associated to a fixed $ {\mathbf{SU}}(n)$ with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on $ B{\mathbf{SU}}(n)$.
Complex representations of matrix semigroups
Jan
Okniński;
Mohan S.
Putcha
563-581
Abstract: Let $M$ be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units $G$. The multiplicative semigroup ${\mathcal{M}_n}(F)$, where $F$ is a finite field, is a particular example. Using Harish-Chandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of $M$ is completely reducible. Using this we characterize the representations of $ G$ extending to irreducible representations of $M$ as being those induced from the irreducible representations of certain parabolic subgroups of $ G$. We go on to show that if $F$ is any field and $S$ any multiplicative subsemigroup of ${\mathcal{M}_n}(F)$, then the semigroup algebra of $S$ over any field of characteristic zero has nilpotent Jacobson radical. If $S = {\mathcal{M}_n}(F)$, then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of $ {\mathcal{M}_n}(F)$ over a field of characteristic zero is regular if and only if $\operatorname{ch} (F) = p > 0$ and $F$ is algebraic over its prime field.
An $L\sp 2$-cohomology construction of unitary highest weight modules for ${\rm U}(p,q)$
Lisa A.
Mantini
583-603
Abstract: In this paper a geometric construction is given of all unitary highest weight modules of $ G = \operatorname{U} (p,q)$. The construction is based on the unitary model of the $ k$th tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over $G/K$, and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.
Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface
Carsten
Thomassen
605-635
Abstract: We describe all regular tilings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface $S$, all (but finitely many) vertex-transitive graphs which can be drawn on $S$ but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each $g \geqslant 3$, there are only finitely many vertex-transitive graphs of genus $g$. In fact, they all have order $< {10^{10}}g$. The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz' theorem that, for each $g \geqslant 2$, there are only finitely many groups that act on the surface of genus $g$. We also derive a nonorientable version of Hurwitz' theorem.
Weak type $(1,1)$ estimates for heat kernel maximal functions on Lie groups
Michael
Cowling;
Garth
Gaudry;
Saverio
Giulini;
Giancarlo
Mauceri
637-649
Abstract: For a Lie group $ G$ with left-invariant Haar measure and associated Lebesgue spaces $ {L^p}(G)$, we consider the heat kernels $ {\{ {p_t}\} _{t > 0}}$ arising from a right-invariant Laplacian $ \Delta$ on $G$: that is, $u(t, \cdot ) = {p_t}{\ast}f$ solves the heat equation $(\partial /\partial t - \Delta )u = 0$ with initial condition $ u(0, \cdot ) = f( \cdot )$. We establish weak-type $(1,1)$ estimates for the maximal operator $\mathcal{M}(\mathcal{M}\;f = {\sup _{t > 0}}\vert{p_t}{\ast}f\vert)$ and for related Hardy-Littlewood maximal operators in a variety of contexts, namely for groups of polynomial growth and for a number of classes of Iwasawa $AN$ groups. We also study the "local" maximal operator $ {\mathcal{M}_0}({\mathcal{M}_0}f = {\sup _{0 < t < 1}}\vert{p_t}{\ast}f\vert)$ and related Hardy-Littlewood operators for all Lie groups.
Characteristic numbers for unoriented ${\bf Z}$-homology manifolds
Sandro
Buoncristiano;
Derek
Hacon
651-663
Abstract: It is shown that the analogue of Thom's theorem on Stiefel-Whitney numbers holds for $ {\mathbf{Z}}$-homology manifolds
Concentrated cyclic actions of high periodicity
Daniel
Berend;
Gabriel
Katz
665-689
Abstract: The class of concentrated periodic diffeomorphisms $ g:M \to M$ is introduced. A diffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small (with respect to the period of $g$ and the dimension of $M$) arc on the circle. In many ways, the cyclic action generated by such a $g$ behaves on the one hand as a circle action and on the other hand as a generic prime power order cyclic action. For example, as for circle actions, $\operatorname{Sign} (g,M) = \operatorname{Sign} ({M^g})$, provided that the left-hand side is an integer; as for prime power order actions, $g$ cannot have a single fixed point if $ M$ is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of ${M^g}$ in $M$ to $\operatorname{Sign} (g,M)$ via the normal $ g$-representations, is established.
A cubic counterpart of Jacobi's identity and the AGM
J. M.
Borwein;
P. B.
Borwein
691-701
Abstract: We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $\displaystyle {a_{n + 1}}: = \frac{{{a_n} + 2{b_n}}} {3}\quad {\text{and}}\quad... ...}: = \sqrt[3]{{{b_n}\left( {\frac{{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right).}}$ The limit of this iteration is identified in terms of the hypergeometric function $ {}_2{F_1}(1/3,2/3;1; \cdot )$, which supports a particularly simple cubic transformation.
Distal functions and unique ergodicity
Ebrahim
Salehi
703-713
Abstract: A. Knapp [5] has shown that the set, $D(S)$, of all distal functions on a group $ S$ is a norm closed subalgebra of ${l^\infty }(S)$ that contains the constants and is closed under the complex conjugation and left translation by elements of $S$. Also it is proved that [7] for any $k \in \mathbb{N}$ and any $\lambda \in \mathbb{R}$ the function $f:\mathbb{Z} \to \mathbb{C}$ defined by $f(n) = {e^{i\lambda {n^k}}}$ is distal on $ \mathbb{Z}$. Now let ${\mathbf{W}}$ be the norm closure of the algebra generated by the set of functions $\displaystyle \{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb{N},\;\lambda \in \mathbb{R}\} ,$ which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on $\mathbb{Z}$. In this paper, we will show that any element of $ {\mathbf{W}}$ is uniquely ergodic (Theorem 2.13) and that the set ${\mathbf{W}}$ does not exhaust all the distal functions on $ \mathbb{Z}$ (Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6]. The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.
Roots of unity and the Adams-Novikov spectral sequence for formal $A$-modules
Keith
Johnson
715-726
Abstract: The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal $A$-modules is studied in the special case in which $ A$ is the ring of integers in the field obtained by adjoining $p$th roots of unity to $ {\widehat{\mathbb{Q}}_p}$, the $p$-adic numbers. Information about these cohomology groups is used to give new proofs of results about the ${E_2}$ term of the Adams spectral sequence based on $ 2$-local complex $ K$-theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence.
Infinitely many co-existing sinks from degenerate homoclinic tangencies
Gregory J.
Davis
727-748
Abstract: The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families $\{ {F_t}\}$ of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, ${P_t};\;F_t^n({P_t}) = {P_t}$, and $\vert\det DF_t^n({P_t})\vert < 1$. We also require the stable and unstable manifolds of $ {P_t}$ to form homoclinic tangencies as the parameter $t$ varies through ${t_0}$. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values $t$ near ${t_0}$. We show that there are parameter values $ t$ near ${t_0}$ at which ${F_t}$ has infinitely many co-existing periodic sinks.
Generalizations of Picard's theorem for Riemann surfaces
Pentti
Järvi
749-763
Abstract: Let $D$ be a plane domain, $E \subset D$ a compact set of capacity zero, and $ f$ a holomorphic mapping of $D\backslash E$ into a hyperbolic Riemann surface $W$. Then there is a Riemann surface $ W'$ containing $ W$ such that $ f$ extends to a holomorphic mapping of $D$ into $W'$. The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of $ W$ be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.
Strong shape for topological spaces
Jerzy
Dydak;
Sławomir
Nowak
765-796
Abstract: Strong shape equivalences for topological spaces are introduced in a way which generalizes easily to inverse systems of topological spaces. Each space is then mapped via a strong shape equivalence into a fibrant inverse system of ANRs. This leads naturally to defining the strong shape category SSh for topological spaces. Other descriptions of SSh are also provided.
Conformal automorphisms and conformally flat manifolds
William M.
Goldman;
Yoshinobu
Kamishima
797-810
Abstract: A geometric structure on a smooth $n$-manifold $M$ is a maximal collection of distinguished charts modeled on a $1$-connected $n$-dimensional homogeneous space $ X$ of a Lie group $ G$ where coordinate changes are restrictions of transformations from $ G$. There exists a developing map $dev:wm \to X$ which is always locally a diffeomorphism. It is in general far from globally being a diffeomorphism. We study the rigid property of developing maps of $(G,X)$-manifolds. As an application we shall classify closed conformally flat manifolds $M$ when the universal covering space $ \tilde M$ supports a one parameter group of conformal transformations.
A countably compact topological group $H$ such that $H\times H$ is not countably compact
Klaas Pieter
Hart;
Jan
van Mill
811-821
Abstract: Using $ {\mathbf{M}}{{\mathbf{A}}_{{\text{countable}}}}$ we construct a topological group with the properties mentioned in the title.
M\"obius invariant Hilbert spaces of holomorphic functions in the unit ball of ${\bf C}\sp n$
Ke He
Zhu
823-842
Abstract: We prove that there exists a unique Hilbert space of holomorphic functions in the open unit ball of ${\mathbb{C}^n}$ whose (semi-) inner product is invariant under Möbius transformations.
Generating modules efficiently over noncommutative Noetherian rings
S. C.
Coutinho
843-856
Abstract: The Forster-Swan Theorem gives an upper bound on the number of generators of a module over a commutative ring in terms of local data. Stafford showed that this theorem could be generalized to arbitrary right and left noetherian rings. In this paper a similar result is proved for right noetherian rings with finite Krull dimension. A new dimension function--the basic dimension--is the main tool used in the proof of this result.
A nonstandard resonance problem for ordinary differential equations
Shair
Ahmad
857-875
Abstract: Necessary and sufficient conditions are established for the existence of bounded solutions for a class of second order differential equations.
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
J.
García Azorero;
I.
Peral Alonso
877-895
Abstract: We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain $ \Omega \subset {{\mathbf{R}}^N}$ $\displaystyle - \operatorname{div} (\vert\nabla u{\vert^{p - 2}}\nabla u) = \ve... ...{\vert^{{p^{\ast}} - 2}}u + \lambda \vert u{\vert^{q - 2}}u,\qquad \lambda > 0,$ where ${p^{\ast}}$ is the critical Sobolev exponent, and $ u{\vert _{\delta \Omega }} \equiv 0$. By using critical point methods we obtain the existence of solutions in the following cases: If $ p < q < {p^{\ast}}$, there exists $ {\lambda _0} > 0$ such that for all $ \lambda > {\lambda _0}$ there exists a nontrivial solution. If $ \max (p,{p^{\ast}} - p/(p - 1)) < q < {p^{\ast}}$, there exists nontrivial solution for all $ \lambda > 0$. If $ 1 < q < p$ there exists ${\lambda _1}$ such that, for $0 < \lambda < {\lambda _1}$, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.
On the spectral character of Toeplitz operators on multiply connected domains
Kevin F.
Clancey
897-910
Abstract: An explicit resolvent formula is given for selfadjoint Toeplitz operators acting on the least harmonic majorant Hardy spaces of a multiply connected planar domain. This formula is obtained by using theta functions associated with the double of the domain. Several consequences concerning the spectral resolutions of selfadjoint Toeplitz operators are deduced.
Locally flat $2$-knots in $S\sp 2\times S\sp 2$ with the same fundamental group
Yoshihisa
Sato
911-920
Abstract: We consider a locally flat $2$-sphere in $ {S^2} \times {S^2}$ representing a primitive homology class $\xi$, which is referred to as a $ 2$-knot in ${S^2} \times {S^2}$ representing $\xi$. Then for any given primitive class $\xi$, there exists a $2$-knot in $ {S^2} \times {S^2}$ representing $\xi$ with simply-connected complement. In this paper, we consider the classification of $ 2$-knots in ${S^2} \times {S^2}$ whose complements have a fixed fundamental group. We show that if the complement of a $2$-knot $S$ in $ {S^2} \times {S^2}$ is simply connected, then the ambient isotopy type of $ S$ is determined. In the case of nontrivial ${\pi _1}$, however, we show that the ambient isotopy type of a $2$-knot in $ {S^2} \times {S^2}$ with nontrivial ${\pi _1}$ is not always determined by ${\pi _1}$.