New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces
Athanassios
G.
Kartsatos
1663-1707
Abstract: Let $X$ be a real Banach space and $G$ a bounded, open and convex subset of $X.$ The solvability of the fixed point problem $(*)~Tx+Cx \owns x$ in $D(T)\cap \overline{G}$ is considered, where $T:X\supset D(T)\to 2^{X}$ is a possibly discontinuous $m$-dissipative operator and $C: \overline{G}\to X$ is completely continuous. It is assumed that $X$ is uniformly convex, $D(T)\cap G \not = \emptyset$ and $(T+C)(D(T)\cap \partial G)\subset \overline{G}.$ A result of Browder, concerning single-valued operators $T$ that are either uniformly continuous or continuous with $X^{*}$ uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let $\Gamma = \{\beta :\mathcal{R}_{+}\to \mathcal{R}_{+}~;~\beta (r)\to 0\text{ as }r\to \infty \}.$ The effect of a weak boundary condition of the type $\langle u+Cx,x\rangle \ge -\beta (\|x\|)\|x\|^{2}$ on the range of operators $T+C$ is studied for $m$-accretive and maximal monotone operators $T.$ Here, $\beta \in \Gamma ,~x\in D(T)$ with sufficiently large norm and $u\in Tx.$ Various new eigenvalue results are given involving the solvability of $Tx+ \lambda Cx\owns 0$ with respect to $(\lambda ,x)\in (0,\infty )\times D(T).$ Several results do not require the continuity of the operator $C.$ Four open problems are also given, the solution of which would improve upon certain results of the paper.
A class of counterexamples to the Gel'fand-Kirillov conjecture
Jacques
Alev;
Alfons
Ooms;
Michel
Van den Bergh
1709-1716
Abstract: Let $G$ be a connected non-special semisimple algebraic group and let $W$ be a finite dimensional $G$-representation such that $W$ has trivial generic stabilizer. Let $\mathfrak{g}=\text{Lie}(G)$. Then the semi-direct product $\mathfrak{g}\oplus W$ is a counter-example to the Gel´fand-Kirillov conjecture.
Simultaneous rational approximation to binomial functions
Michael
A.
Bennett
1717-1738
Abstract: We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel´dman and Rickert, proving, for example, that \begin{displaymath}\max \left\{ \left| \sqrt{2} - p_{1}/q \right| , \left| \sqrt{3} - p_{2}/q \right| \right\} > q^{-1.79155} \end{displaymath} for $q > q_{0}$ (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell's equations and norm form equations.
Orientation-reversing involutions on handlebodies
John
Kalliongis;
Darryl
McCullough
1739-1755
Abstract: The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the $I$-fibers of some $I$-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.
Groups Quasi-isometric to Complex Hyperbolic Space
Richard
Chow
1757-1769
Abstract: We show that any finitely generated group quasi-isometric to complex hyperbolic space is a finite extension of a properly discontinuous, cocompact subgroup of the isometry group.
The Bergman kernel function of some Reinhardt domains
Sheng
Gong;
Xuean
Zheng
1771-1803
Abstract: The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points $(z,\bar z)$. Let $D$ be the Reinhardt domain \begin{displaymath}D=\left\{ z\in\mathbf C^n\,|\,\|z\|_\alpha=\sum_{j=1}^n|z_j|^{2/\al_j}<1\right\} \end{displaymath} where $\al_j>0$, $j=1,2,\dots, n$; and let $K(z,\bar w)$ be the Bergman kernel function of $D$. Then there exist two positive constants $m$ and $M$ and a function $F$ such that \begin{displaymath}mF(z,\bar z) \le K(z,\bar z)\le MF(z,\bar z) \end{displaymath} holds for every $z\in D$. Here \begin{displaymath}F(z,\bar z)=(-r(z))^{-n-1} \prod_{j=1}^n (-r(z)+|z_j|^{2/\al_j})^{1-\al_j} \end{displaymath} and $r(z)=\|z\|_\alpha-1$ is the defining function for $D$. The constants $m$ and $M$ depend only on $\alpha=(\al_1,\dots, \al_n)$ and $n$, not on $z$.
Prox-regular functions in variational analysis
R.
A.
Poliquin;
R.
T.
Rockafellar
1805-1838
Abstract: The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-$\mathcal{C}^{2}$ functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.
Decompositions in Quantum Logic
John
Harding
1839-1862
Abstract: We present a method of constructing an orthomodular poset from a relation algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomodular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several known methods of producing orthomodular posets are shown to be special cases of this result. These include the construction of an orthomodular poset from a modular lattice and the construction of an orthomodular poset from the idempotents of a ring. Particular attention is paid to decompositions of groups and modules. We develop the notion of a norm on a group with operators and of a projection on such a normed group. We show that the projections of a normed group with operators form an orthomodular poset with a full set of states. If the group is abelian and complete under the metric induced by the norm, the projections form a $\sigma$-complete orthomodular poset with a full set of countably additive states. We also describe some properties special to those orthomodular posets constructed from relation algebras. These properties are used to give an example of an orthomodular poset which cannot be embedded into such a relational orthomodular poset, or into an orthomodular lattice. It had previously been an open question whether every orthomodular poset could be embedded into an orthomodular lattice.
Diagrams up to cohomology
W.
G.
Dwyer;
C.
W.
Wilkerson
1863-1883
Abstract: We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the $p$-completion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.
Noncomplete linear systems on abelian varieties
Christina
Birkenhake
1885-1908
Abstract: Let $X$ be a smooth projective variety. Every embedding $X\hookrightarrow \mathbb{P}_N$ is the linear projection of an embedding defined by a complete linear system. In this paper the geometry of such not necessarily complete embeddings is investigated in the special case of abelian varieites. To be more precise, the properties $N_p$ of complete embeddings are extended to arbitrary embeddings, and criteria for these properties to be satisfied are elaborated. These results are applied to abelian varieties. The main result is: Let $(X,L)$ be a general polarized abelian variety of type $(d_1,\dots,d_g)$ and $p\ge1$, $n\ge 2p+2$ such that $nd_g\ge 6$ is even, and $c\le n^{g-1}$. The general subvector space $V\subseteq H^0(L^n)$ of codimension $c$ satisfies the property $N_p$.
On homomorphisms from a fixed representation to a general representation of a quiver
William
Crawley-Boevey
1909-1919
Abstract: We study the dimension of the space of homomorphisms from a given representation $X$ of a quiver to a general representation of dimension vector $\beta$. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as $\beta$ increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring.
Hyperfinite transversal theory. II
Bosko
Zivaljevic
1921-1938
Abstract: We continue the investigation of validity of Hall's theorem in the case of the Loeb space $L({\mathcal{H}})$ of an internal, uniformly distributed, hyperfinite measure space ${\mathcal{H}}=(\Omega ,{\mathcal{A}},\mu )$ initiated in1992 by the author. Some new classes of graphs are introduced for which the measure theoretic version of Hall's theorem still holds.
On transversely flat conformal foliations with good measures
Taro
Asuke
1939-1958
Abstract: Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the $C^{1+{\operatorname{Lip}}}$ sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely $C^{\omega }$-foliations.
Linear additive functionals of superdiffusions and related nonlinear P.D.E.
E.
B.
Dynkin;
S.
E.
Kuznetsov
1959-1987
Abstract: Let $L$ be a second order elliptic differential operator in a bounded smooth domain $D$ in $\mathbb{R}^{d}$ and let $1<\alpha \le 2$. We get necessary and sufficient conditions on measures $\eta , \nu$ under which there exists a positive solution of the boundary value problem \begin{equation*}\begin{gathered} -Lv+v^{\alpha }=\eta \quad \text{ in } D, v=\nu \quad \text{ on } \partial D. \end{gathered}\tag{*} \end{equation*} The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called $(L,\alpha )$-superdiffusions). We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain $E$ in $\mathbb{R}^{d}$, we establish a 1-1 correspondence between a class of such functionals and a class of $L$-excessive functions $h$ (which we describe in terms of their Martin integral representation). The Laplace transform of $A$ satisfies an integral equation which can be considered as a substitute for (*).
Fuchsian Subgroups of Bianchi Groups
D.
G.
James;
C.
Maclachlan
1989-2002
Abstract: A maximal non-elementary Fuchsian subgroup of a Bianchi group $PSL(2,O_d)$ has an invariant circle or straight line under its linear fractional action on the complex plane, to which is associated a positive integer $D$, the discriminant, which, in turn, is an invariant of the wide commensurability class of the Fuchsian subgroup. In this paper, for all Bianchi groups, we classify the conjugacy classes of these maximal Fuchsian subgroups by determining the number with given discriminant.
Fractal Dimensions and Random Transformations
Yuri
Kifer
2003-2038
Abstract: I start with random base expansions of numbers from the interval $[0,1]$ and, more generally, vectors from $[0,1]^{d}$, which leads to random expanding transformations on the $d$-dimensional torus $\mathbb{T}^{d}$. As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets ``invariant'' with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley's book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.
Higher Lefschetz Traces and Spherical Euler Characteristics
Ross
Geoghegan;
Andrew
Nicas;
John
Oprea
2039-2062
Abstract: Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler.
Regularity of Lorentzian Busemann Functions
Gregory
J.
Galloway;
Arnaldo
Horta
2063-2084
Abstract: A general theory of regularity for Lorentzian Busemann functions in future timelike geodesically complete spacetimes is presented. This treatment simplifies and extends the local regularity developed by Eschenburg, Galloway and Newman to prove the Lorentzian splitting theorem. Criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of Bartnik.
$k$-spaces and Borel filters on the set of integers
Jean
Calbrix
2085-2090
Abstract: We say that a countable, Hausdorff, topological space with one and only one accumulation point is a point-space. For such a space, we give several properties which are equivalent to the property of being a k-space. We study some free filters on the set of integers and we determine if the associated point-spaces are k-spaces or not. We show that the filters of Lutzer-van Mill-Pol are k-filters. We deduce that, for each countable ordinal ${\alpha \geq 2}$, there exists a free filter of true additive class ${\alpha }$ (Baire's classification) and a free filter of true multiplicative class ${\alpha }$ for which the associated point-spaces are k-spaces but not ${\aleph _{0}}$, the existence being true in the additive case for ${\alpha =1}$. In particular, we answer negatively a question raised in J. Calbrix, C. R. Acad. Sci. Paris 305 (1987), 109--111.