Conormal and piecewise smooth solutions to quasilinear wave equations
Seong Joo
Kang
1-35
Abstract: In this paper, we show first that if a solution $u$ of the equation ${P_2}(t,x,u,Du,D)u = f(t,x,u,Du)$, where ${P_2}(t,x,u,Du,D)$ is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface $\Sigma$ of ${P_2}$ in the past and $\Sigma$ is smooth in the past, then $\Sigma$ is smooth and $u$ is conormal with respect to $ \Sigma$ for all time. Second, let ${\Sigma _0}$ and ${\Sigma _1}$ be characteristic hypersurfaces of ${P_2}$ which intersect transversally and let $ \Gamma = {\Sigma _0} \cap {\Sigma _1}$. If $ {\Sigma _0}$ and ${\Sigma _1}$ are smooth in the past and $ u$ is conormal with repect to $ \{ {\Sigma _0},{\Sigma _1}\}$ in the past, then $\Gamma$ is smooth, and $u$ is conormal with respect to $\{ {\Sigma _0},{\Sigma _1}\}$ locally in time outside of $\Gamma$, even though $ {\Sigma _0}$ and ${\Sigma _1}$ are no longer necessarily smooth across $\Gamma$. Finally, we show that if $u(0,x)$ and ${\partial _t}u(0,x)$ are in an appropriate Sobolev space and are piecewise smooth outside of $\Gamma$, then $u$ is piecewise smooth locally in time outside of ${\Sigma _0} \cup {\Sigma _1}$.
Further results on fixpoints and zeros of entire functions
Jian Hua
Zheng;
Chung-Chun
Yang
37-50
Abstract: In this paper, a quantitative estimation on the number of zeros of the function $f \circ g(z) - \alpha (z)$ is derived, where $f$ and $g$ are transcendental entire functions and $\alpha (z)$ a nonconstant polynomial. As an application of this and a further step towards an affirmative answer to a conjecture of Baker, a quantitative estimation on the number of period points of exact order $ n$ of ${f_n}$ ($n$th iterate of $f$) is obtained.
Bifurcation of minimal surfaces in Riemannian manifolds
Jürgen
Jost;
Xianqing
Li-Jost;
Xiao Wei
Peng
51-62
Abstract: We study the bifurcation of closed minimal surfaces in Riemannian manifolds through higher order variations of the area functional and relate it to elementary catastrophes.
Torsion classes and a universal constraint on Donaldson invariants for odd manifolds
Selman
Akbulut;
Tom
Mrowka;
Yongbin
Ruan
63-76
Abstract: This paper studies the topology of the gauge group and gives $ \bmod \,2$ universal relations along Donaldson polynomials of smooth $ 4$-manifolds, generalizing Y. Ruan's previous related result.
On the coefficient groups of equivariant $K$-theory
Yimin
Yang
77-98
Abstract: We calculated the coefficient groups of equivariant $K$-theory for any cyclic group, and we proved that, for any compact Lie group, the coefficient groups can only have $2$-torsion. If there is any $2$-torsion, it is detected by $ 2$-primary finite subgroups of $G$. The rationalization of the coefficient groups then can be easily calculated.
Homology operations on a new infinite loop space
Burt
Totaro
99-110
Abstract: Boyer et al. [1] defined a new infinite loop space structure on the space ${M_0} = {\prod _{n \geqslant 1}}K({\mathbf{Z}},2n)$ such that the total Chern class map $BU \to {M_0}$ is an infinite loop map. This is a sort of Riemann-Roch theorem without denominators: for example, it implies Fulton-MacPherson's theorem that the Chern classes of the direct image of a vector bundle $E$ under a given finite covering map are determined by the rank and Chern classes of $E$. We compute the Dyer-Lashof operations on the homology of ${M_0}$. They provide a new explanation for Kochman's calculation of the operations on the homology of $BU$, and they suggest a possible characterization of the infinite loop structure on ${M_0}$.
A new measure of growth for countable-dimensional algebras. I
John
Hannah;
K. C.
O’Meara
111-136
Abstract: A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values lie in the unit interval [0, 1]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite $ GK$dimension.
Circle actions on rational homology manifolds and deformations of rational homotopy types
Martin
Raussen
137-153
Abstract: The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold $X$ with submanifold $F$, I would like to construct nontrivial actions of cyclic groups on $X$ with $F$ as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group $T = {S^1}$ on $X$ with fixed point set $F$ was established in [Rau92]. In this paper, I assume that the rational homotopy types of $ F$ and $X$ are related by a deformation in the sense of [A1178] between their (Sullivan) models as graded differential algebras (cf. [Sul77, Hal83]). Under certain additional assumptions, it is then possible to construct a rational homotopy description of a $ T$-action on the complement $X\backslash F$ that fits together with a given $T$-bundle action on the normal bundle of $ F$ in $X$. In a subsequent paper [Rau94], I plan to show how to realize this $T$-action on an actual manifold $ Y$ rationally homotopy equivalent to $X$ with fixed point set $F$ and how to "propagate" all but finitely many of the restricted cyclic group actions to $ X$ itself.
Localization and genus in group theory
G.
Peschke
155-174
Abstract: We provide a unifying category theoretical framework to discuss various kinds of local global phenomena. Specializing to localization of groups at sets of primes $ P$, we identify a large class of groups for which localization supports a passage from local information to global information. Local global principles for groups in this class are established and used to calculate certain homomorphism sets as well as splittings of epimorphisms and monomorphisms from local data.
Free ideals of one-relator graded Lie algebras
John P.
Labute
175-188
Abstract: In this paper we show that a one-relator graded Lie algebra $\mathfrak{g} = L/(r)$, over a principal ideal domain $K$, has a homogeneous ideal $\mathfrak{h}$ with $ \mathfrak{g}/\mathfrak{h}$ a free $K$-module of finite rank if the relator $ r$ is not a proper multiple of another element in the free Lie algebra $ L$. As an application, we deduce that the center of a one-relator Lie algebra over $ K$ is trivial if the rank of $L$ is greater than two. As another application, we find a new class of one-relator pro-$ p$-groups which are of cohomological dimension $2$.
Residue classes of Lagrangian subbundles and Maslov classes
Haruo
Suzuki
189-202
Abstract: For Lagrangian subbundles with singularities in symplectic vector bundles, explicit formulas of relation between their residue classes and Maslov classes outside singularities are obtained. Then a Lagrangian subbundle with singularity is constructed where all possible Maslov classes are nonzero but residue classes vanish for dimension $> 2$. Moreover, a Lagrangian immersion with singularity is constructed, where the similar property for the associated Maslov classes and residue classes is shown.
On parametric evolution inclusions of the subdifferential type with applications to optimal control problems
Nikolaos S.
Papageorgiou
203-231
Abstract: In this paper we study parametric evolution inclusions of the subdifferential type and their applications to the sensitivity analysis of nonlinear, infinite dimensional optimal control problems. The parameter appears in all the data of the problem, including the subdifferential operator. First we establish several continuity results for the solution multifunction of the subdifferential inclusion. Then we study how these results can be used to examine the sensitivity properties (variational stability) of certain broad classes of nonlinear infinite dimensional optimal control problems. Some examples are worked out in detail, illustrating the applicability of our work. These include obstacle problems (with time varying obstacles), optimal control of distributed parameter systems, and differential variational inequalities.
Generalizations of Browder's degree theory
Shou Chuan
Hu;
Nikolaos S.
Papageorgiou
233-259
Abstract: The starting point of this paper is the recent important work of F. E. Browder, who extended degree theory to operators of monotone type. The degree function of Browder is generalized to maps of the form $T + f + G$, where $T$ is maximal monotone, $f$ is of class ${(S)_ + }$ bounded, and $ G( \cdot )$ is an u.s.c. compact multifunction. It is also generalized to maps of the form $f + {N_G}$, with $f$ of class ${(S)_ + }$ and ${N_G}$ the Nemitsky operator of a multifunction $ G(x,r)$ satisfying various types of sign conditions. Some examples are also included to illustrate the abstract results.
Cohomologically symplectic spaces: toral actions and the Gottlieb group
Gregory
Lupton;
John
Oprea
261-288
Abstract: Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homotopical obstruction is described which detects when an action is Hamiltonian. This new entity, the ${\lambda _{\hat \alpha }}$-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree $ 2$ cohomology class which cups to a top class. Furthermore, new results in symplectic geometry also arise from this homotopical approach.
Singularities produced in conormal wave interactions
Linda M.
Holt
289-315
Abstract: Three problems on the interactions of conormal waves are considered. Two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal. In one case, two of the waves are tangential, and the other wave is transversal to the first two. The third result is a noninteraction theorem. It is shown that under certain conditions, no nonlinear spreading of the singularities will occur.
Polar $\sigma$-ideals of compact sets
Gabriel
Debs
317-338
Abstract: Let $E$ be a metric compact space. We consider the space $ \mathcal{K}(E)$ of all compact subsets of $E$ endowed with the topology of the Hausdorff metric and the space $ \mathcal{M}(E)$ of all positive measures on $E$ endowed with its natural ${w^{\ast}}$-topology. We study $\sigma $-ideals of $\mathcal{K}(E)$ of the form $I = {I_P} = \{ K \in \mathcal{K}(E):\mu (K) = 0,\;\forall \mu \in P\}$ where $P$ is a given family of positive measures on $E$. If $M$ is the maximal family such that $I = {I_M}$, then $M$ is a band. We prove that several descriptive properties of $I$: being Borel, and having a Borel basis, having a Borel polarity-basis, can be expressed by properties of the band $M$ or of the orthogonal band $M'$.
Curvature conditions on Riemannian manifolds with Brownian harmonicity properties
H. R.
Hughes
339-361
Abstract: The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius $ \varepsilon$ is considered and a general procedure is given for computing asymptotic expansions, as $ \varepsilon$ decreases to zero, for joint moments of the first exit time and place random variables. It is proven that asymptotic versions of exit time and place distribution properties that hold on harmonic spaces are equivalent to certain curvature conditions for harmonic spaces. In particular, it is proven that an asymptotic mean value condition involving first exit place is equivalent to certain levels of curvature conditions for harmonic spaces depending on the order of the asymptotics. Also, it is proven that an asymptotic uncorrelated condition for first exit time and place is equivalent to weaker curvature conditions at corresponding orders of asymptotics.