Some combinatorial aspects of reduced words in finite Coxeter groups
John
R.
Stembridge
1285-1332
Abstract: We analyze the structure of reduced expressions in the Coxeter groups $A_n$, $B_n$ and $D_n$. Several special classes of elements are singled out for their connections with symmetric functions or the theory of $P$-partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.
Disjoint paths, planarizing cycles, and spanning walks
Xingxing
Yu
1333-1358
Abstract: We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.
Randomness and semigenericity
John
T.
Baldwin;
Saharon
Shelah
1359-1376
Abstract: Let $L$ contain only the equality symbol and let $L^+$ be an arbitrary finite symmetric relational language containing $L$. Suppose probabilities are defined on finite $L^+$ structures with `edge probability' $n^{-\alpha }$. By $T^{\alpha }$, the almost sure theory of random $L^+$-structures we mean the collection of $L^+$-sentences which have limit probability 1. $T_{\alpha }$ denotes the theory of the generic structures for ${\mathbf {K}} _{\alpha }$ (the collection of finite graphs $G$ with $\delta _{\alpha }(G) =|G| - \alpha \cdot |\text { edges of$G$}|$ hereditarily nonnegative). Theorem.. $T^{\alpha }$, the almost sure theory of random $L^+$-structures, is the same as the theory $T_{\alpha }$ of the ${\mathbf {K}} _{\alpha }$-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.
There are no piecewise linear maps of type $2^{\infty}$
Víctor
Jiménez
López;
L'ubomír
Snoha
1377-1387
Abstract: The aim of this paper is to show that there are no piecewise linear maps of type $2^{\infty }$. For this purpose we use the fact that any piecewise monotone map of type $2^{\infty }$ has an infinite $\omega$-limit set which is a subset of a doubling period solenoid. Then we prove that piecewise linear maps cannot have any doubling period solenoids.
Group actions on arrangements of linear subspaces and applications to configuration spaces
Sheila
Sundaram;
Volkmar
Welker
1389-1420
Abstract: For an arrangement of linear subspaces in ${\mathbb R} ^n$ that is invariant under a finite subgroup of the general linear group $Gl_n({\mathbb R} )$ we develop a formula for the $G$-module structure of the cohomology of the complement ${\mathcal M} _{\mathcal A}$. Our formula specializes to the well known Goresky-MacPherson theorem in case $G = 1$, but for $G \neq 1$ the formula shows that the $G$-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space ${\mathcal M} _{\mathcal A} /G$. Our motivating examples are arrangements in ${\mathbb C} ^n$ that are invariant under the action of $S_n$ by permuting coordinates. A particular case is the ``$k$-equal'' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases ${\mathcal M} _{\mathcal A}$ and ${\mathcal M} _{\mathcal A} /S_n$ are spaces of ordered and unordered point configurations in ${\mathbb C} ^n$ many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in ${\mathbb R} ^d$ and provide explicit results for the ``$k$-equal'' and the ``$k$-divisible'' cases.
Convex integral functionals
Nikolaos
S.
Papageorgiou
1421-1436
Abstract: We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their $\varepsilon$-subdifferential $(\varepsilon \ge 0)$ and their $\varepsilon$-directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.
$p$-adic Power Series which Commute under Composition
Hua-Chieh
Li
1437-1446
Abstract: When two noninvertible series commute to each other, they have same set of roots of iterates. Most of the results of this paper will be concerned with the problem of which series commute with a given noninvertible series. Our main theorem is a generalization of Lubin's result about isogenies of formal groups.
Integer translation of meromorphic functions
Jeong
H.
Kim;
Lee
A.
Rubel
1447-1462
Abstract: Let $G$ be a given open set in the complex plane. We prove that there is an entire function such that its integer translations forms a normal family in a neighborhood of $z$ exactly for $z$ in $G$ if and only if $G$ is periodic with period 1, i.e., $z\pm 1\in G$ for all $z\in G$.
Essential laminations in $I$-bundles
Mark
Brittenham
1463-1485
Abstract: We show that, with a few familiar exceptions, every essential lamination in an interval-bundle over a closed surface can be isotoped to lie everywhere transverse to the $I$-fibers of the bundle.
On The Homotopy Type of $BG$ for Certain Finite 2-Groups $G$
Carlos
Broto;
Ran
Levi
1487-1502
Abstract: We consider the homotopy type of classifying spaces $BG$, where $G$ is a finite $p$-group, and we study the question whether or not the mod $p$ cohomology of $BG$, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of $BG$. This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of $BG$.
The Group of Galois Extensions Over Orders in $KC_{p^2}$
Robert
Underwood
1503-1514
Abstract: In this paper we characterize all Galois extensions over $H$ where $H$ is an arbitrary $R$-Hopf order in $KC_{p^{2}}$. We conclude that the abelian group of $H$-Galois extensions is isomorphic to a certain quotient of units groups in $R\times R$. This result generalizes the classification of $H$-Galois extensions, where $H\subset KC_{p}$, due to Roberts, and also to Hurley and Greither.
A four-dimensional deformation of a numerical Godeaux surface
Caryn
Werner
1515-1525
Abstract: A numerical Godeaux surface is a surface of general type with invariants $p_g =q =0$ and $K^2 =1$. In this paper the moduli space of a numerical Godeaux surface with order two torsion is computed to be eight-dimensional; whether or not the moduli space of such a surface is irreducible is still unknown. The surface in this paper is constructed as one member of a four parameter family of double planes. There is a natural involution on the surface, inherited from the double plane construction, which acts on the moduli space. We show that the invariant subspace is four-dimensional and coincides with the family of double planes.
Intersection Lawson homology
Pawel
Gajer
1527-1550
Abstract: The aim of this paper is to construct and describe basic properties of a theory that unifies Lawson homology and intersection homology. It is shown that this theory has a localization sequence, is functorial, satisfies a property analogous to the Lawson Suspension Theorem, and is equipped with an operation analogous to the Friedlander-Mazur $\mathbf s$-operation.
Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces
Indranil
Biswas;
Pablo
Arés-Gastesi;
Suresh
Govindarajan
1551-1560
Abstract: In this paper we study the relation between parabolic Higgs vector bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\mathbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of $k=2$ is the usual Teichmüller space of the punctured surface.
On Poincaré Type Inequalities
Roger
Chen;
Peter
Li
1561-1585
Abstract: Using estimates of the heat kernel we prove a Poincaré inequality for star-shape domains on a complete manifold. The method also gives a lower bound for the gap of the first two Neumann eigenvalues of a Schrödinger operator.
On the cohomology of split extensions of finite groups
Stephen
F.
Siegel
1587-1609
Abstract: Let $G=H\rtimes Q$ be a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentials $d_2$ in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a field $k$. We generalize this to give an explicit description of all the $d_r$ ($r\geq 2$) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in the $kG$-endomorphism algebra of the minimal $kH$-projective resolution induced from $H$ to $G$. This twisting cochain not only determines the differentials, but also allows one to construct an explicit $kG$-projective resolution of $k$.
Cohomological construction of quantized universal enveloping algebras
Joseph
Donin;
Steven
Shnider
1611-1632
Abstract: Given an associative algebra $A$ and the category $\mathcal C$ of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\mathcal C$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_{A}$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $\Phi \in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\mathcal R\in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode) $S$ of $A$ satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $\Phi$, $\mathcal R$ and $S$. Drinfeld defined such a structure on $A=U(\mathcal G)[[h]]$ for any semisimple Lie algebra $\mathcal {G}$ with the usual comultiplication and antipode but nontrivial $\mathcal R$ and $\Phi$, and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), $U_{h}(\mathcal G)$. In the paper we give a direct cohomological construction of the $F$ which reduces $\Phi$ to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that $F$ can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of $U(\mathcal G )[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements $F$, $R$ and $\Phi$ can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to $\mathcal G$ with $\Phi$ depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.
Topological conjugacy of linear endomorphisms of the 2-torus
Roy
Adler;
Charles
Tresser;
Patrick
A.
Worfolk
1633-1652
Abstract: We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincaré.
De Rham cohomology of logarithmic forms on arrangements of hyperplanes
Jonathan
Wiens;
Sergey
Yuzvinsky
1653-1662
Abstract: The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic.
One and two dimensional Cantor-Lebesgue type theorems
J.
Marshall
Ash;
Gang
Wang
1663-1674
Abstract: Let $\varphi (n)$ be any function which grows more slowly than exponentially in $n,$ i.e., $\mathop {limsup}\limits _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1.$ There is a double trigonometric series whose coefficients grow like $\varphi (n),$ and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like $\varphi (n),$ and which has the everywhere convergent partial sum subsequence $S_{2^j}.$ For any $p>1,$ there is a one dimensional trigonometric series whose coefficients grow like $\varphi (n^{\frac {p-1}p}),$ and which has the everywhere convergent partial sum subsequence $S_{[j^p]}.$ All these examples exhibit, in a sense, the worst possible behavior. If $m_j$ is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence $S_{m_j}.$
Primitive higher order embeddings of abelian surfaces
Th.
Bauer;
T.
Szemberg
1675-1683
Abstract: In recent years several concepts of higher order embeddings have been studied: $k$-spannedness, $k$-very ampleness and $k$-jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.
Nonexistence of global solutions of a nonlinear hyperbolic system
Keng
Deng
1685-1696
Abstract: Consider the initial value problem \begin{equation*}\begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0, [2\jot ] u(x,0)=f(x),&v(x,0)=h(x),&{}&{} [2\jot ] u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), &{}&{} \end {array} \end{equation*} with $1\le n\le 3$ and $p,q>0$. We show that there exists a bound $B(n) (\le \infty )$ such that if $1<pq<B(n)$ all nontrivial solutions with compact support blow up in finite time.