Regular domains in homogeneous groups
Roberto
Monti;
Daniele
Morbidelli
2975-3011
Abstract: We study John, uniform and non-tangentially accessible domains in homogeneous groups of steps 2 and 3. We show that $C^{1,1}$ domains in groups of step 2 are non-tangentially accessible and we give an explicit condition which ensures the John property in groups of step 3.
Mixing times of the biased card shuffling and the asymmetric exclusion process
Itai
Benjamini;
Noam
Berger;
Christopher
Hoffman;
Elchanan
Mossel
3013-3029
Abstract: Consider the following method of card shuffling. Start with a deck of $N$cards numbered 1 through $N$. Fix a parameter $p$ between 0 and 1. In this model a ``shuffle'' consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability $p$. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all $p\ne 1/2$, the mixing time of this card shuffling is $O(N^2)$, as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.
Gauss-Manin connections for arrangements, III Formal connections
Daniel
C.
Cohen;
Peter
Orlik
3031-3050
Abstract: We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.
Descent representations and multivariate statistics
Ron
M.
Adin;
Francesco
Brenti;
Yuval
Roichman
3051-3082
Abstract: Combinatorial identities on Weyl groups of types $A$ and $B$ are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type $A$ we give a new construction of the Solomon descent representations. An extension of the descent basis to type $B$, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.
A converse to Dye's theorem
Greg
Hjorth
3083-3103
Abstract: Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of $\mathbb{F} _2$ on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the $\leq_B$ ordering.
A computer-assisted proof of Saari's conjecture for the planar three-body problem
Richard
Moeckel
3105-3117
Abstract: The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.
A countable Teichmüller modular group
Katsuhiko
Matsuzaki
3119-3131
Abstract: We construct an example of a Riemann surface of infinite topological type for which the Teichmüller modular group consists of only a countable number of elements. We also consider distinguished properties which the Teichmüller space of this Riemann surface possesses.
Valence of complex-valued planar harmonic functions
Genevra
Neumann
3133-3167
Abstract: The valence of a function $f$ at a point $w$is the number of distinct, finite solutions to $f(z) = w$. Let $f$ be a complex-valued harmonic function in an open set $R \subseteq \mathbb{C}$. Let $S$ denote the critical set of $f$and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S) \cup C(f)$has empty interior. We also show that a component $R_0 \subseteq R \backslash f^{-1} (f(S) \cup C(f))$ is an $n_0$-fold covering of some component $\Omega_0 \subseteq \mathbb{C}\backslash (f(S) \cup C(f))$. If $\Omega_0$ is simply connected, then $f$ is univalent on $R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in $\mathbb{R} ^2$ are first stated in that form and then applied to the case of planar harmonic functions. If $f$ is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of $\mathbb{C}\backslash (f(S) \cup C(f))$sharing a common boundary arc in $f(S) \backslash C(f)$.
Entire solutions of certain partial differential equations and factorization of partial derivatives
Bao
Qin
Li
3169-3177
Abstract: We show that the problem of characterizing entire solutions of certain partial differential equations and the problem of characterizing common right factors of partial derivatives of meromorphic functions in $\mathbf{C}^{2}$ are closely related, and characterizations will be given using their relations.
Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations
Hisaaki
Endo;
Seiji
Nagami
3179-3199
Abstract: We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus $3, 4$ and $5$ which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.
Abelian categories, almost split sequences, and comodules
Mark
Kleiner;
Idun
Reiten
3201-3214
Abstract: The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length. (a) Each indecomposable injective has a simple subobject. (b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented. (c) The category has left almost split sequences.
Groups of units of integral group rings of Kleinian type
Antonio
Pita;
Ángel
del Río;
Manuel
Ruiz
3215-3237
Abstract: We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.
The smoothing property for a class of doubly nonlinear parabolic equations
Carsten
Ebmeyer;
José
Miguel
Urbano
3239-3253
Abstract: We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic $p$-Laplace equation. The result is obtained via regularization and a comparison theorem.
Representation dimension: An invariant under stable equivalence
Xiangqian
Guo
3255-3263
Abstract: In this paper, we prove that the representation dimension is an invariant under stable equivalence.
Spike-layered solutions for an elliptic system with Neumann boundary conditions
Miguel
Ramos;
Jianfu
Yang
3265-3284
Abstract: We prove the existence of nonconstant positive solutions for a system of the form $-\varepsilon^2\Delta u + u = g(v)$, $-\varepsilon^2\Delta v + v = f(u)$ in $\Omega$, with Neumann boundary conditions on $\partial \Omega$, where $\Omega$ is a smooth bounded domain and $f$, $g$are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of $\varepsilon$, the corresponding solutions $u_{\varepsilon}$ and $v_{\varepsilon}$ admit a unique maximum point which is located at the boundary of $\Omega$.
Seshadri constants at very general points
Michael
Nakamaye
3285-3297
Abstract: We study the local positivity of an ample line bundle at a very general point of a smooth projective variety. We obtain a slight improvement of the result of Ein, Küchle, and Lazarsfeld.
Generating the surface mapping class group by two elements
Mustafa
Korkmaz
3299-3310
Abstract: Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.
Thurston's weak metric on the Teichmüller space of the torus
Abdelhadi
Belkhirat;
Athanase
Papadopoulos;
Marc
Troyanov
3311-3324
Abstract: We define and study a natural weak metric on the Teichmüller space of the torus. A similar metric has been defined by W. Thurston on the Teichmüller space of higher genus surfaces and our definition is motivated by Thurston's definition. However, we shall see that in the case of the torus, this metric has a different behaviour than on higher genus surfaces.
Torsion subgroups of elliptic curves in short Weierstrass form
Michael
A.
Bennett;
Patrick
Ingram
3325-3337
Abstract: In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any $\varepsilon>0$, all but finitely many curves \begin{displaymath}E_{A,B} \; : \; y^2 = x^3 + A x + B, \end{displaymath} where $A$ and $B$ are integers satisfying $A>\vert B\vert^{1+\varepsilon}>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $\vert A\vert>\vert B\vert^{2+\varepsilon}>0$, then the $E_{A,B}$ now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.
Telescoping, rational-valued series, and zeta functions
J.
Marshall
Ash;
Stefan
Catoiu
3339-3358
Abstract: We give an effective procedure for determining whether or not a series $\sum_{n=M}^{N}r\left( n\right)$ telescopes when $r\left( n\right)$ is a rational function with complex coefficients. We give new examples of series $\left( \ast\right) \sum_{n=1}^{\infty}r\left( n\right)$, where $r\left( n\right)$ is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form $\left( \ast\right)$ are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set $\left\{ \zeta\left( s\right) :s=2,3,4,\dots\right\}$ is linearly independent, where $\zeta\left( s\right) =\sum n^{-s}$ is the Riemann zeta function. Some series of the form $\sum_{n}s\left( \sqrt[r]{n},\sqrt[r]{n+1} ,\cdots,\sqrt[r]{n+k}\right)$, where $s$ is a quotient of symmetric polynomials, are shown to be telescoping, as is $\sum1/(n!+\left( n-1\right) !)$. Quantum versions of these examples are also given.
On compact symplectic manifolds with Lie group symmetries
Daniel
Guan
3359-3373
Abstract: In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.
An approximate universal coefficient theorem
Huaxin
Lin
3375-3405
Abstract: An approximate Universal Coefficient Theorem (AUCT) for certain $C^*$-algebras is established. We present a proof that Kirchberg-Phillips's classification theorem for separable nuclear purely infinite simple $C^*$-algebras is valid for $C^*$-algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that $C^*$-algebras that are locally approximated by $C^*$-algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple $C^*$-algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional $C^*$-algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.