Homotopy coherent category theory
Jean-Marc
Cordier;
Timothy
Porter
1-54
Abstract: This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.
Expansive Subdynamics
Mike
Boyle;
Douglas
Lind
55-102
Abstract: This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let $\alpha$ be a continuous action of ${\mathbb Z}^d$ on an infinite compact metric space. For each subspace $V$ of ${\mathbb R}^d$ we introduce a notion of expansiveness for $\alpha$ along $V$, and show that there are nonexpansive subspaces in every dimension $\le d-1$. For each $k\le d$ the set ${\mathbb E} _k(\alpha )$ of expansive $k$-dimensional subspaces is open in the Grassmann manifold of all $k$-dimensional subspaces of ${\mathbb R}^d$. Various dynamical properties of $\alpha$ are constant, or vary nicely, within a connected component of ${\mathbb E} _k(\alpha )$, but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,'' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For $d=2$ we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an ${\mathbb E} _1(\alpha )$. The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.
On the strong equality between supercompactness and strong compactness
Arthur
W.
Apter;
Saharon
Shelah
103-128
Abstract: We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models ``\kappa$ is $\lambda$ supercompact'', then $V[G] \models ``\kappa$ is $\lambda$ supercompact'' and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models ``\kappa$ is $\lambda$ strongly compact'' iff $V[{G}] \models ``\kappa$ is $\lambda$ supercompact'', except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.
Integration of Correspondences on Loeb Spaces
Yeneng
Sun
129-153
Abstract: We study the Bochner and Gel$^{\prime }$fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework.
On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures
Alessandra
Lunardi
155-169
Abstract: We consider a class of elliptic and parabolic differential operators with unbounded coefficients in $\mathbb R^n$, and we study the properties of the realization of such operators in suitable weighted $L^2$ spaces.
Bifurcation problems for the $p$-Laplacian in $R^n$
Pavel
Drábek;
Yin
Xi
Huang
171-188
Abstract: In this paper we consider the bifurcation problem \begin{equation*}-\text {div } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u+f({\lambda } , x, u), \end{equation*} in ${R^N}$ with $p>1$. We show that a continuum of positive solutions bifurcates out from the principal eigenvalue ${\lambda } _{1}$ of the problem \begin{equation*}-\text {div } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u. \end{equation*} Here both functions $f$ and $g$ may change sign.
$\beta\mathbf{nbc}$-bases for cohomology of local systems on hyperplane complements
Michael
Falk;
Hiroaki
Terao
189-202
Abstract: We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes ${\mathcal A}$. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement ${\mathcal A}$. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of ${\mathcal A}$. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.
Shadowing orbits of ordinary differential equations on invariant submanifolds
Brian
A.
Coomes
203-216
Abstract: A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function $g$ mapping phase space into $\mathbb {R}^{m}$ with the property that for a particular regular value $\boldsymbol c$ of $g$ the submanifold $g^{-1}(\boldsymbol c)$ is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to $g^{-1}(\boldsymbol c)$ is uniformly close to a true solution lying in $g^{-1}(\boldsymbol c)$. Applications of this theorem to computer generated approximate orbits are discussed.
Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator
Chunlan
Jiang;
Shunhua
Sun;
Zongyao
Wang
217-233
Abstract: It is shown that given an essentially normal operator $T$ with connected spectrum, there exists a compact operator $K$ such that $T+K$ is strongly irreducible.
Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces
Eleonor
Harboure;
Oscar
Salinas;
Beatriz
Viviani
235-255
Abstract: Necessary and sufficient conditions are given for the fractional integral operator $I_\alpha$ to be bounded from weighted strong and weak $% L^p$ spaces within the range $p\geq n/\alpha$ into suitable weighted $BMO$ and Lipschitz spaces. We also characterize the weights for which $% I_\alpha$ can be extended to a bounded operator from weighted $BMO$ into a weighted Lipschitz space of order $\alpha$. Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of $I_\alpha$ between weighted Lipschitz spaces.
Multidimensional stability of planar travelling waves
Todd
Kapitula
257-269
Abstract: The multidimensional stability of planar travelling waves for systems of reaction-diffusion equations is considered in the case that the diffusion matrix is the identity. It is shown that if the wave is exponentially orbitally stable in one space dimension, then it is stable for $x\in % \mathbf {R}^n,\,n\ge 2$. Furthermore, it is shown that the perturbation of the wave decays like $t^{-(n-1)/4}$ as $t\to \infty$. The result is proved via an application of linear semigroup theory.
Every semigroup is isomorphic to a transitive semigroup of binary relations
Ralph
McKenzie;
Boris
M.
Schein
271-285
Abstract: Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).
Sums of Three or More Primes
J.
B.
Friedlander;
D.
A.
Goldston
287-310
Abstract: It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error $\sum _{p \le x} \log p - x$ in the Prime Number Theorem, such bounds being within a factor of $(\log x)^{2}$ of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis'' is replaced by ``Generalized Riemann Hypothesis'', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of $k$ primes for $k \ge 4$, and, in a mean square sense, for $k \ge 3$. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``Quasi-Riemann Hypothesis''. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.
Automorphism Groups and Invariant Subspace Lattices
Paul
S.
Muhly;
Baruch
Solel
311-330
Abstract: Let $(B,\mathbf {R},\alpha )$ be a $C^{*}$- dynamical system and let $% A=B^\alpha ([0,\infty ))$ be the analytic subalgebra of $B$. We extend the work of Loebl and the first author that relates the invariant subspace structure of $\pi (A),$ for a $C^{*}$-representation $\pi$ on a Hilbert space $\mathcal {H}_\pi$, to the possibility of implementing $\alpha$ on $% \mathcal {H}_\pi .$ We show that if $\pi$ is irreducible and if lat $\pi (A)$ is trivial, then $\pi (A)$ is ultraweakly dense in $\mathcal {L(H}_\pi ).$ We show, too, that if $A$ satisfies what we call the strong Dirichlet condition, then the ultraweak closure of $\pi (A)$ is a nest algebra for each irreducible representation $\pi .$ Our methods give a new proof of a ``density'' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid $C^{*}$-algebras.
On the Complete Integrability of some Lax Equations on a Periodic Lattice
Luen-Chau
Li
331-372
Abstract: We consider some Lax equations on a periodic lattice with $N=2$ sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.
Decomposition of Birational Toric Maps in Blow-Ups and Blow-Downs
Jaroslaw
Wlodarczyk
373-411
Abstract: We prove that a toric birational map between two complete smooth toric varieties of the same dimension can be decomposed in a sequence of equivariant blow-ups and blow-downs along smooth centers.
Linear isometries between subspaces of continuous functions
Jesús
Araujo;
Juan
J.
Font
413-428
Abstract: We say that a linear subspace $A$ of $C_0 (X)$ is strongly separating if given any pair of distinct points $x_1, x_2$ of the locally compact space $X$, then there exists $f \in A$ such that $\left | f(x_1 ) \right | \neq \left | f(x_2 ) \right |$. In this paper we prove that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain singular subspaces of the Shilov boundaries of $B$ and $A$, sending the Choquet boundary of $B$ onto the Choquet boundary of $A$. We also provide an example which shows that the above result is no longer true if we do not assume $A$ to be strongly separating. Furthermore we obtain the following multiplicative representation of $T$: $(Tf)(y)=a(y)f(h(y))$ for all $y \in \partial B$ and all $f \in A$, where $a$ is a unimodular scalar-valued continuous function on $\partial B$. These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.