Existence and uniqueness of rectilinear slit maps
Carl
H.
FitzGerald;
Frederick
Weening
485-513
Abstract: We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' ${\theta}$ module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.
The Calabi invariant and the Euler class
Takashi
Tsuboi
515-524
Abstract: We show the following relationship between the Euler class for the group of the orientation preserving diffeomorphisms of the circle and the Calabi invariant for the group of area preserving diffeomorphisms of the disk which are the identity along the boundary. A diffeomorphism of the circle admits an extension which is an area preserving diffeomorphism of the disk. For a homomorphism $\psi$ from the fundamental group $\langle a_{1}, \cdots , a_{2g} ; [a_{1},a_{2}]\cdots [a_{2g-1},a_{2g}]\rangle$ of a closed surface to the group of the diffeomorphisms of the circle, by taking the extensions $\widetilde {\psi (a}_{i})$ for the generators $a_{i}$, one obtains the product $[\widetilde {\psi (a}_{1}),\widetilde {\psi (a}_{2})]\cdots [\widetilde {\psi (a}_{2g-1}),\widetilde {\psi (a}_{2g})]$ of their commutators, and this is an area preserving diffeomorphism of the disk which is the identity along the boundary. Then the Calabi invariant of this area preserving diffeomorphism is a non-zero multiple of the Euler class of the associated circle bundle evaluated on the fundamental cycle of the surface.
Quantization of presymplectic manifolds and circle actions
Ana
Cannas
da Silva;
Yael
Karshon;
Susan
Tolman
525-552
Abstract: We prove several versions of ``quantization commutes with reduction'' for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin$^c$ structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
Factorization in generalized power series
Alessandro
Berarducci
553-577
Abstract: The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group $\mathbf{G}$ is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring $\mathbf{R}((\mathbf{G}^{\leq 0}))$ consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): $\sum _n t^{-1/n}+1$. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway's series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If $\mathbf{G}= (\mathbf{R}, +, 0, \leq)$ we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either $\omega$ or of the form $\omega^{\omega^\alpha}$ and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case $\mathbf{G}=\mathbf{R}$. In the final part of the paper we study the irreducibility of series with finite support.
Resolutions of monomial ideals and cohomology over exterior algebras
Annetta
Aramova;
Luchezar
L.
Avramov;
Jürgen
Herzog
579-594
Abstract: This paper studies the homology of finite modules over the exterior algebra $E$ of a vector space $V$. To such a module $M$ we associate an algebraic set $V_E(M)\subseteq V$, consisting of those $v\in V$ that have a non-minimal annihilator in $M$. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a `depth formula'. Explicit results are obtained for $M=E/J$, when $J$ is generated by products of elements of a basis $e_1,\dots,e_n$ of $V$. A (infinite) minimal free resolution of $E/J$ is constructed from a (finite) minimal resolution of $S/I$, where $I$ is the squarefree monomial ideal generated by `the same' products of the variables in the polynomial ring $S=K[x_1,\dots,x_n]$. It is proved that $V_E(E/J)$ is the union of the coordinate subspaces of $V$, spanned by subsets of $\{\,e_1,\dots,e_n\,\}$ determined by the Betti numbers of $S/I$ over $S$.
Automorphism scheme of a finite field extension
Pedro
J. Sancho
de Salas
595-608
Abstract: Let $k\to K$ be a finite field extension and let us consider the automorphism scheme $Aut_kK$. We prove that $Aut_kK$ is a complete $k$-group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions $K_1, K_2$ of $k$, not being separable of degree 2 or 6, the following equivalence: \begin{equation*}K_1\simeq K_2 \Leftrightarrow Aut_kK_1\simeq Aut_kK_2.\end{equation*}
Rarified sums of the Thue-Morse sequence
Michael
Drmota;
Mariusz
Skalba
609-642
Abstract: Let $q$ be an odd number and $S_{q,0}(n)$ the difference between the number of $k<n$, $k\equiv 0\bmod\,q$, with an even binary digit sum and the corresponding number of $k<n$, $k\equiv 0\bmod\,q$, with an odd binary digit sum. A remarkable theorem of Newman says that $S_{3,0}(n)>0$ for all $n$. In this paper it is proved that the same assertion holds if $q$ is divisible by 3 or $q=4^N+1$. On the other hand, it is shown that the number of primes $q\le x$ with this property is $o(x/\log x)$. Finally, analoga for ``higher parities'' are provided.
A hereditarily indecomposable tree-like continuum without the fixed point property
Piotr
Minc
643-654
Abstract: A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.
Closed incompressible surfaces in knot complements
Elizabeth
Finkelstein;
Yoav
Moriah
655-677
Abstract: In this paper we show that given a knot or link $K$ in a $2n$-plat projection with $n\ge 3$ and $m\ge 5$, where $m$ is the length of the plat, if the twist coefficients $a_{i,j}$ all satisfy $|a_{i,j}|>1$ then $S^3-N(K)$ has at least $2n-4$ nonisotopic essential meridional planar surfaces. In particular if $K$ is a knot then $S^3-N(K)$ contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in $\mathbb{Z}\oplus\mathbb{Z}$.
Two special cases of Ganea's conjecture
Jeffrey
A.
Strom
679-688
Abstract: Ganea conjectured that for any finite CW complex $X$ and any $k>0$, $\operatorname{cat}(X\times S^k) =\operatorname{cat}(X) + 1$. In this paper we prove two special cases of this conjecture. The main result is the following. Let $X$ be a $(p-1)$-connected $n$-dimensional CW complex (not necessarily finite). We show that if $\operatorname{cat}(X) = \left\lfloor {n \over p} \right\rfloor + 1$ and $n\not\equiv -1 \operatorname{mod} p$(which implies $p>1$), then $\operatorname{cat}(X\times S^k) =\operatorname{cat}(X) +1$. This is proved by showing that $\operatorname{wcat}(X\times S^k) =\operatorname{wcat}(X) + 1$ in a much larger range, and then showing that under the conditions imposed, $\operatorname{cat}(X)=\operatorname{wcat}(X)$. The second special case is an extension of Singhof's earlier result for manifolds.
Products and duality in Waldhausen categories
Michael
S.
Weiss;
Bruce
Williams
689-709
Abstract: The natural transformation $\Xi$ from $\mathbf{L}$-theory to the Tate cohomology of $\mathbb{Z}/2$ acting on $\mathbf{K}$-theory commutes with external products. Corollary: The Tate cohomology of $\mathbb{Z}/2$ acting on the $\mathbf{K}$-theory of any ring with involution is a generalized Eilenberg-Mac Lane spectrum, and it is 4-periodic.
The $\mathcal U$-Lagrangian of a convex function
Claude
Lemaréchal;
François
Oustry;
Claudia
Sagastizábal
711-729
Abstract: At a given point ${\overline{p}}$, a convex function $f$ is differentiable in a certain subspace $\mathcal{U}$ (the subspace along which $\partial f({\overline{p}})$ has 0-breadth). This property opens the way to defining a suitably restricted second derivative of $f$ at ${\overline{p}}$. We do this via an intermediate function, convex on $\mathcal{U}$. We call this function the $\mathcal{U}$-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.
Remarks on a Nonlinear Parabolic Equation
Matania
Ben-Artzi;
Jonathan
Goodman;
Arnon
Levy
731-751
Abstract: The equation $u_{t} =\Delta u +\mu |\nabla u |$, $\mu \in \mathbb{R}$, is studied in $\mathbb{R}^{n}$ and in the periodic case. It is shown that the equation is well-posed in $L^{1}$ and possesses regularizing properties. For nonnegative initial data and $\mu <0$ the solution decays in $L^{1}(\mathbb{R}^{n})$ as $t\to \infty$. In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.
Operating points in infinite nonlinear networks approximated by finite networks
Bruce
D.
Calvert;
Armen
H.
Zemanian
753-780
Abstract: Given a nonlinear infinite resistive network, an operating point can be determined by approximating the network by finite networks obtained by shorting together various infinite sets of nodes, and then taking a limit of the nodal potential functions of the finite networks. Initially, by taking a completion of the node set of the infinite network under a metric given by the resistances, limit points are obtained that represent generalized ends, which we call ``terminals,'' of the infinite network. These terminals can be shorted together to obtain a generalized kind of node, a special case of a 1-node. An operating point will involve Kirchhoff's current law holding at 1-nodes, and so the flow of current into these terminals is studied. We give existence and bounds for an operating point that also has a nodal potential function, which is continuous at the 1-nodes. The existence is derived from the said approximations.
$L^p$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients
Marco
Bramanti;
Luca
Brandolini
781-822
Abstract: Let $X_1,X_2,\ldots,X_q$ be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain $\Omega\subset\mathbb{R}^n$ ($n>q$). We consider the differential operator \begin{equation*}\mathcal{L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where the coefficients $a_{ij}(x)$ are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: \begin{equation*}\mu|\xi|^2\leq\sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq\mu^{-1}|\xi|^2 \end{equation*} for a.e. $x\in\Omega$, every $\xi\in\mathbb{R}^q$, some constant $\mu$. Moreover, we assume that the coefficients $a_{ij}$ belong to the space VMO (``Vanishing Mean Oscillation''), defined with respect to the subelliptic metric induced by the vector fields $X_1,X_2,\ldots,X_q$. We prove the following local $\mathcal{L}^p$-estimate: \begin{equation*}\left\|X_iX_jf\right\|_{\mathcal{L}^p(\Omega')}\leq c\left\{\left\|\mathcal{L}f\right\|_{\mathcal{L}^p(\Omega)}+\left\|f\right \|_{\mathcal{L}^p(\Omega)}\right\} \end{equation*} for every $\Omega'\subset\subset\Omega$, $1<p<\infty$. We also prove the local Hölder continuity for solutions to $\mathcal{L}f=g$ for any $g\in\mathcal{L}^p$ with $p$ large enough. Finally, we prove $\mathcal{L}^p$-estimates for higher order derivatives of $f$, whenever $g$ and the coefficients $a_{ij}$ are more regular.
The set of idempotents in the weakly almost periodic compactification of the integers is not closed
B.
Bordbar;
J.
Pym
823-842
Abstract: This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of $(\mathbb{N}, +)$ and $(\mathbb{Z} ,+)$ are closed.
Rates of mixing for potentials of summable variation
Mark
Pollicott
843-853
Abstract: It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials.
Banach spaces with the Daugavet property
Vladimir
M.
Kadets;
Roman
V.
Shvidkoy;
Gleb
G.
Sirotkin;
Dirk
Werner
855-873
Abstract: A Banach space $X$ is said to have the Daugavet property if every operator $T:\allowbreak X\to X$ of rank $1$ satisfies $\|\operatorname{Id}+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell _{1}$. However, $X$ need not contain a copy of $L_{1}$. We also study pairs of spaces $X\subset Y$ and operators $T:\allowbreak X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J:\allowbreak X\to Y$ is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with $\|\operatorname{Id}+T\|=1+\|T\|$ is as small as possible and give characterisations in terms of a smoothness condition.
Spin $\mathrm{L}$-functions on $GSp_8$ and $Gsp_{10}$
Daniel
Bump;
David
Ginzburg
875-899
Abstract: The ``spin'' L-function of an automorphic representation of $GSp_{2n}$ is an Euler product of degree $2^{n}$ associated with the spin representation of the L-group $\mathrm{GSpin}(2n+1)$. If $n=4$ or $5$, and the automorphic representation is generic in the sense of having a Whittaker model, the analytic properties of these L-functions are studied by the Rankin-Selberg method.
On the module structure of free Lie algebras
R.
M.
Bryant;
Ralph
Stöhr
901-934
Abstract: We study the free Lie algebra $L$ over a field of non-zero characteristic $p$ as a module for the cyclic group of order $p$ acting on $L$ by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of $L$ as a direct sum of indecomposable modules.
A quantum octonion algebra
Georgia
Benkart;
José
M.
Pérez-Izquierdo
935-968
Abstract: Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_{q}$(D$_{4}$) of D$_{4}$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_{q}$(D$_{4}$). The product in the quantum octonions is a $U_{q}$(D$_{4}$)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation theory'' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.