Nonsmooth sequential analysis in Asplund spaces
Boris
S.
Mordukhovich;
Yongheng
Shao
1235-1280
Abstract: We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.
Duality and Polynomial Testing of Tree Homomorphisms
P.
Hell;
J.
Nesetril;
X.
Zhu
1281-1297
Abstract: Let $H$ be a fixed digraph. We consider the $H$-colouring problem, i.e., the problem of deciding which digraphs $G$ admit a homomorphism to $H$. We are interested in a characterization in terms of the absence in $G$ of certain tree-like obstructions. Specifically, we say that $H$ has tree duality if, for all digraphs $G$, $G$ is not homomorphic to $H$ if and only if there is an oriented tree which is homomorphic to $G$ but not to $H$. We prove that if $H$ has tree duality then the $H$-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the $\underline X$-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when $H$ itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads $H$ for which the $H$-colouring problem is $NP$-complete. We contrast these with several families of oriented triads $H$ which have tree duality, or bounded treewidth duality, and hence polynomial $H$-colouring problems. If $P \neq NP$, then no oriented triad $H$ with an $NP$-complete $H$-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad $H$. We prove that none of the oriented triads $H$ with $NP$-complete $H$-colouring problems given in the companion paper has tree duality.
Shellable Nonpure Complexes and Posets. I
Anders
Björner;
Michelle
L.
Wachs
1299-1327
Abstract: The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed $f$-vectors and $h$-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the $k$-equal partition lattice (the intersection lattice of the $k$-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the $k$-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
On universal relations in gauge theory
Selman
Akbulut
1329-1355
Abstract: In this paper we study the algebraic topology of gauge group, and as a corollary we deduce some universal relations among Donaldson polynomials of smooth 4-manifolds.
The space of $\omega$-limit sets of a continuous map of the interval
Alexander
Blokh;
A.
M.
Bruckner;
P.
D.
Humke;
J.
Smítal
1357-1372
Abstract: We first give a geometric characterization of $\omega$-limit sets. We then use this characterization to prove that the family of $\omega$-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.
Twisted Groups and Locally Toroidal Regular Polytopes
Peter
McMullen;
Egon
Schulte
1373-1410
Abstract: In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks $5$ and $6$. For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.
Some remarks on a probability limit theorem for continued fractions
Jorge
D.
Samur
1411-1428
Abstract: It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.
Analysis of the Wu metric. I: The case of convex Thullen domains
C.
K.
Cheung;
Kang-Tae
Kim
1429-1457
Abstract: We present an explicit description of the Wu metric on the convex Thullen domains which turns out to be the first natural example of a purely Hermitian, non-Kählerian invariant metric. Also, we show that the Wu metric on these Thullen domains is in fact real analytic everywhere except along a lower dimensional subvariety, and is $C^{1}$ smooth overall. Finally, we show that the holomorphic curvature of the Wu metric on these Thullen domains is strictly negative where the Wu metric is real analytic, and is strictly negative everywhere in the sense of current.
Comparative asymptotics for perturbed orthogonal polynomials
Franz
Peherstorfer;
Robert
Steinbauer
1459-1486
Abstract: Let $\{\Phi_n\}_{n\in\mathbb N_0}$ and $\{\widetilde\Phi_n\}_{n\in\mathbb N_0}$ be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials $\widetilde\Phi_n$ behave asymptotically like those of $\Phi_n$. We give, under weak assumptions on the system $\{\Phi_n\}_{n\in\mathbb N_0}$ and the perturbations, comparative asymptotics as for $\widetilde\Phi_n^*(z)/ \Phi_n^*(z)$ etc., $\Phi_n^*(z):= z^n\bar\Phi_n(\frac 1z)$, on the open unit disk and on the circumference mainly off the support of the measure $\sigma$ with respect to which the $\Phi_n$'s are orthonormal. In particular these results apply if the comparative system $\{\Phi_n\} _{n\in\mathbb N_0}$ has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.
A Periodic Point Free Homeomorphism of a Tree-Like Continuum
Piotr
Minc
1487-1519
Abstract: An example of a homeomorphism without periodic points is constructed on a tree-like continuum.
Whitehead test modules
Jan
Trlifaj
1521-1554
Abstract: A (right $R$-) module $N$ is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module $M$, $Ext_R(M,N)=0$ implies $M$ is projective. Dually, i-test modules are defined. For example, $\mathbb{Z}$ is a p-test abelian group iff each Whitehead group is free. Our first main result says that if $R$ is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring $R$ , there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring $R$ is said to be fully saturated ($\kappa$-saturated) provided that all non-projective ($\le\kappa$-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, $GT(1,n,p,S,T)$. The four parameters involved here are skew-fields $S$ and $T$, and natural numbers $n,p$. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of $\ast$-modules.
Further Nice Equations for Nice Groups
Shreeram
S.
Abhyankar
1555-1577
Abstract: Nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic $p$ with P$\Omega ^{-}(2m,q)$ and $\Omega ^{-}(2m,q)$ as Galois groups where $m>3$ is any integer and $q>1$ is any power of $p>2$.
A Groenewold-Van Hove Theorem for $S^2$
Mark
J.
Gotay;
Hendrik
Grundling;
C.
A.
Hurst
1579-1597
Abstract: We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold $S^2$ which is irreducible on the su(2) subalgebra generated by the components $\{S_1,S_2,S_3\}$ of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra $\cal P$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\cal P$ containing $\{1,S_1,S_2,S_3\}$ that can be so quantized is just that generated by $\{1,S_1,S_2,S_3\}$.
Transfer operators acting on Zygmund functions
Viviane
Baladi;
Yunping
Jiang;
Oscar
E.
Lanford III
1599-1615
Abstract: We obtain a formula for the essential spectral radius $\rho _{\text{ess}}$ of transfer-type operators associated with families of $C^{1+\delta }$ diffeomorphisms of the line and Zygmund, or Hölder, weights acting on Banach spaces of Zygmund (respectively Hölder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive.
Epigraphical and Uniform Convergence of Convex Functions
Jonathan
M.
Borwein;
Jon
D.
Vanderwerff
1617-1631
Abstract: We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of $\ell _{1}$.
Totally real submanifolds in $S^6(1)$ satisfying Chen's equality
Franki
Dillen;
Luc
Vrancken
1633-1646
Abstract: In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb{C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb{C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen's equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen's equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$.
Cohomological dimension and metrizable spaces. II
Jerzy
Dydak
1647-1661
Abstract: The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A,B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{{\mathbf R} }(A\cup B)\le \dim _{{\mathbf R} }A+\dim _{{\mathbf R} }B+1 \end{equation*} for any ring ${\mathbf R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then a. Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. b. There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. c. There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z'$ with $K_{i}\in AE(Z')$ for all $i\ge 1$ embeds in $Z$ as a closed subset.