Invariance results for delay and Volterra equations in fractional order Sobolev spaces
F.
Kappel;
K.
Kunisch
1-51
Abstract: Invariance of the trajectories of infinite delay- and Volterra-type equations in fractional order Sobolev spaces are derived under minimal assumptions on the problem data. Properties of fractional order Sobolev spaces defined over intervals are summarized.
Oscillatory integrals and Fourier transforms of surface carried measures
Michael
Cowling;
Giancarlo
Mauceri
53-68
Abstract: We suppose that $ S$ is a smooth hypersurface in $ {{\mathbf{R}}^{n + 1}}$ with Gaussian curvature $\kappa$ and surface measure $dS$, $w$ is a compactly supported cut-off function, and we let $ {\mu _\alpha }$ be the surface measure with $d{\mu _\alpha } = w{\kappa ^\alpha }\,dS$. In this paper we consider the case where $S$ is the graph of a suitably convex function, homogeneous of degree $d$, and estimate the Fourier transform $ {\hat \mu _\alpha }$. We also show that if $S$ is convex, with no tangent lines of infinite order, then $ {\hat \mu _\alpha }(\xi )$ decays as $ \vert\xi {\vert^{ - n / 2}}$ provided $\alpha \geqslant [(n + 3)/2]$. The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.
Application of group cohomology to space constructions
Paul
Igodt;
Kyung Bai
Lee
69-82
Abstract: From a short exact sequence of crossed modules $1 \to K \to H \to \bar H \to 1$ and a $ 2$-cocycle $(\phi ,\,h) \in {Z^2}(G;\,H)$, a $ 4$-term cohomology exact sequence $H_{ab}^1(G;\,Z) \to H_{(\bar \phi ,\,\bar h)}^1(G;\,\bar H,\bar Z)\mathop \to ... ...(G;\,K):{\psi _{{\text{out}}}} = {\phi _{{\text{out}}}}\} \to H_{ab}^2(G;\,Z)}$ is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and $Z$ is the center of the crossed module $ H$. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.
On strongly summable ultrafilters and union ultrafilters
Andreas
Blass;
Neil
Hindman
83-97
Abstract: We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of $ P$-points and therefore cannot be established in ZFC.
Holomorphic kernels and commuting operators
Ameer
Athavale
101-110
Abstract: Necessary and sufficient conditions in terms of operator polynomials are obtained for an $m$-tuple $T = ({T_1}, \ldots ,{T_m})$ of commuting bounded linear operators on a separable Hilbert space $ \mathcal{H}$ to extend to an $\dot m$-tuple $S = ({S_1}, \ldots ,{S_m})$ of operators on some Hilbert space $ \mathcal{K}$, where each $ {S_i}$ is realized as a $ {\ast}$-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for $ T$ to have a commuting normal extension.
Taming wild extensions with Hopf algebras
Lindsay N.
Childs
111-140
Abstract: Let $K \subset L$ be a Galois extension of number fields with abelian Galois group $G$ and rings of integers $R \subset S$, and let $\mathcal{A}$ be the order of $S$ in $KG$. If $ \mathcal{A}$ is a Hopf $ R$-algebra with operations induced from $KG$, then $S$ is locally isomorphic to $\mathcal{A}$ as $ \mathcal{A}$-module. Criteria are found for $ \mathcal{A}$ to be a Hopf algebra when $ K = {\mathbf{Q}}$ or when $ L/K$ is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over $R$ in $L$ which are tame or Galois $ H$-extensions, $ H$ a Hopf order in $ KG$, using a generalization of the discriminant.
The Szeg\H o kernel as a singular integral kernel on a family of weakly pseudoconvex domains
Katharine Perkins
Diaz
141-170
Abstract: The Szegö kernels on the weakly pseudoconvex domains $\{ \operatorname{Im} {z_2} > \vert{z_1}{\vert^{2k}}\}$, $k \in {Z^ + }$, have been computed by Greiner and Stein. After constructing a global, nonisotropic pseudometric suitable for Calderón-Zygmund singular integral theory on the boundaries of the domains, we study principal value operators associated to these Szegö kernels. We prove that the principal value operators are bounded on ${L^p}$ for $1 < p < \infty$, and that they preserve certain nonisotropic Lipschitz classes. We then derive a Plemelj formula that relates the principal value operators to the Szegö projections. From this formula we deduce that the Szegö projections are also bounded on ${L^p}$, for $ 1 < p < \infty$, and that they preserve the same nonisotropic Lipschitz classes.
A strong generalization of Helgason's theorem
Kenneth D.
Johnson
171-192
Abstract: Let $G$ be a simple Lie group with $ KAN$ an Iwasawa decomposition of $G$, and let $M$ be the centralizer of $A$ in $K$. Suppose ${K_1}$ is a fixed, closed, normal, analytic subgroup of $K$, and set $ {\mathbf{P}}({K_1})$ equal to the set of all parabolic subgroups $P$ of $G$ which contain $MAN$ such that $ {K_1}P = G$ and ${K_1} \cap P$ is normal in the reductive part of $ P$. Suppose $\pi :G \to GL(V)$ is an irreducible representation of $G$. Then, if ${\mathbf{P}}({K_1}) \ne \emptyset$, we obtain necessary and sufficient conditions for ${V^{{K_1}}}$, the space of ${K_1}$-fixed vectors, to be $\ne (0)$. Moreover, reciprocity formulas are obtained which determine $ \dim {V^{{K_1}}}$.
The Morava $K$-theories of some classifying spaces
Nicholas J.
Kuhn
193-205
Abstract: Let $P$ be a finite abelian $ p$-group with classifying space $BP$. We compute, in representation theoretic terms, the Morava $K$-theories of the stable wedge summands of $ BP$. In particular, we obtain a simple, and purely group theoretic, description of the rank of $ K{(s)^{\ast}}(BG)$ for any finite group $G$ with an abelian $p$-Sylow subgroup. A minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup ${M_n}({\mathbf{Z}} / p)$.
Equivariant geometry and Kervaire spheres
Allen
Back;
Wu-Yi
Hsiang
207-227
Abstract: The intrinsic geometry of metrics on the Kervaire sphere which are invariant under a large transformation group (cohomogeneity one) is studied. Invariant theory is used to describe the behavior of these metrics near the singular orbits. Nice expressions for the Ricci and sectional curvatures are obtained. The nonexistence of invariant metrics of positive sectional curvature is proven, and Cheeger's construction of metrics of positive Ricci curvature is discussed.
Universal Loeb-measurability of sets and of the standard part map with applications
D.
Landers;
L.
Rogge
229-243
Abstract: It is shown in this paper that for $K$-saturated models many important external sets of nonstandard analysis--such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points--are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction. Moreover, the standard part map can be used as a measure preserving transformation for all $ \tau$-smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of $\tau$-smooth Baire-measures to $ \tau$-smooth Borel-measures, the decomposition theorems for $\tau$-smooth Baire-measures and $ \tau$-smooth Borel-measures and Kakutani's theorem for product measures.
Contributions to the theory of set valued functions and set valued measures
Nikolaos S.
Papageorgiou
245-265
Abstract: Measurable multifunctions and multimeasures with values in a Banach space are studied. We start by proving a variation of the known Dunford theorem for weak compactness in $ {L^1}(X)$. With a similar technique we prove that the range of certain vector valued integrals that appear in applications is $ w$-compact and convex. Also we obtain Dunford-Pettis type theorems for sequences of integrably bounded multifunctions. Some pointwise $w$-compactness theorems are also obtained for certain families of measurable multifunctions. Then we prove a representation theorem for additive, set valued operators defined on ${L^1}(X)$. Finally, in the last section, a detailed study of transition multimeasures is conducted and several representation theorems are proved.
Riemann problems for nonstrictly hyperbolic $2\times 2$ systems of conservation laws
David G.
Schaeffer;
Michael
Shearer
267-306
Abstract: The Riemann problem is solved for $2 \times 2$ systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.
Visibility and rank one in homogeneous spaces of $K\leq 0$
María J.
Druetta
307-321
Abstract: In this paper we study relationships between the visibility axiom and rank one in homogeneous spaces of nonpositive curvature. We obtain a complete classification (in terms of rank) of simply connected homogeneous spaces of nonpositive curvature and dimension $\leqslant 4$. We provide examples, in every $\dim \geqslant 4$, of simply connected, irreducible homogeneous spaces $ (K \leqslant 0)$ which are neither visibility manifolds nor symmetric spaces.
Ideals of holomorphic functions with $C\sp \infty$ boundary values on a pseudoconvex domain
Edward
Bierstone;
Pierre D.
Milman
323-342
Abstract: We give natural sufficient conditions for the solution of several problems concerning division in the space ${\mathcal{A}^\infty }(\Omega )$ of holomorphic functions with $ {\mathcal{C}^\infty }$ boundary values on a pseudoconvex domain $ \Omega$ with smooth boundary. The sufficient conditions come from upper semicontinuity with respect to the analytic Zariski topology of a local invariant of coherent analytic sheaves (the "invariant diagram of initial exponents"), and apply to division in the space of ${\mathcal{C}^\infty }$ Whitney functions on an arbitrary closed set. Our theorem on division in $ {\mathcal{A}^\infty }(\Omega )$ follows using Kohn's theorem on global regularity in the $\bar \partial $-Neumann problem.
Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations
Gary M.
Lieberman
343-353
Abstract: We consider solutions (and subsolutions and supersolutions) of the boundary value problem \begin{displaymath}\begin{array}{*{20}{c}} {{a^{ij}}(x,\,u,\,Du){D_{ij}}u + a(x,... ...(x)u = g(x)\quad {\text{on}}\;\partial \Omega } \end{array} \end{displaymath} for a Lipschitz domain $\Omega$, a positive-definite matrix-valued function $ [{a^{ij}}]$, and a vector field $\beta$ which points uniformly into $ \Omega$. Without making any continuity assumptions on the known functions, we prove Harnack and Hölder estimates for $u$ near $ \partial \Omega$. In addition we bound the $ {L^\infty }$ norm of $ u$ near $\partial \Omega$ in terms of an appropriate $ {L^p}$ norm and the known functions. Our approach is based on that for the corresponding interior estimates of Trudinger.
All infinite groups are Galois groups over any field
Manfred
Dugas;
Rüdiger
Göbel
355-384
Abstract: Let $G$ be an arbitrary monoid with $ 1$ and right cancellation, and $K$ be a given field. We will construct extension fields $F \supseteq K$ with endomorphism monoid End $F$ isomorphic to $G$ modulo Frobenius homomorphisms. If $ G$ is a group, then Aut $ F = G$. Let ${F^G}$ denote the fixed elements of $ F$ under the action of $ G$. In the case that $ G$ is an infinite group, also ${F^G} = K$ and $G$ is the Galois group of $F$ over $K$. If $G$ is an arbitrary group, and $G = 1$, respectively, this answers an open problem (R. Baer 1967, E. Fried, C. U. Jensen, J. Thompson) and if $G$ is infinite, the result is an infinite analogue of the still unsolved Hilbert-Noether conjecture inverting Galois theory. Observe that our extensions $K \subset F$ are not algebraic. We also suggest to consider the case $K = {\mathbf{C}}$ and $ G = \{ 1\}$.
Existence domains of holomorphic functions of restricted growth
M.
Jarnicki;
P.
Pflug
385-404
Abstract: The paper presents a large class of domains $G$ of holomorphy in $ {{\mathbf{C}}^n}$ such that, for any $N > 0$, there exists a nonextendable holomorphic function $f$ on $G$ with $ \vert f\vert\delta _G^N$ bounded where ${\delta _G}(z): = \min ({(1 + \vert z{\vert^2})^{ - 1 / 2}},\,\operatorname{dist} (z,\,\partial G))$. Any fat Reinhardt domain of holomorphy belongs to this class. On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.
Traveling waves in combustion processes with complex chemical networks
Steffen
Heinze
405-416
Abstract: The existence of traveling waves for laminar flames with complex chemistry is proved. The crucial assumptions are that all reactions have to be exothermic and that no cycles occur in the graph of the reaction network. The method is to solve the equations first in a bounded interval by a degree argument and then taking the infinite domain limit.
Classifying spaces for foliations with isolated singularities
Peter
Greenberg
417-429
Abstract: Let ${\Gamma ^a} \subset \Gamma $ be transitive pseudogroups on $ {{\mathbf{R}}^n}$, such that, for any element $ g:\,U \to V$ of $ \Gamma$, there is a locally finite subset $S \subset U$, such that $g{\vert _{U - S}}$ is an element of ${\Gamma ^a}$. We construct $B\Gamma$, up to weak homotopy type, from $B{\Gamma ^a}$ and the classifying spaces of certain groups of germs. As an application, the classifying space of the pseudogroup of orientation-preserving, piecewise linear homeomorphisms between open subsets of ${\mathbf{R}}$ is weakly homotopy equivalent to $ B{\mathbf{R}}{\ast}B{\mathbf{R}}$.