Lower central series and free resolutions of hyperplane arrangements
Henry
K.
Schenck;
Alexander
I.
Suciu
3409-3433
Abstract: If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk)$is the cohomology ring of $M$ over a field $\Bbbk$ of characteristic $0$, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim \operatorname{Tor}^A_i(\Bbbk,\Bbbk)_i$, of the linear strand in a minimal free resolution of $\Bbbk$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers,
On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups
M.
E.
Harris;
M.
Linckelmann
3435-3453
Abstract: If $G$ is a finite $p$-solvable group for some prime $p$, $A$ a solvable subgroup of the automorphism group of $G$ of order prime to $\vert G\vert$such that $A$ stabilises a $p$-block $b$ of $G$ and acts trivially on a defect group $P$ of $b$, then there is a Morita equivalence between the block $b$ and its Watanabe correspondent $w(b)$ of $C_{G}(A)$, given by a bimodule $M$ with vertex $\Delta P$ and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).
Braid pictures for Artin groups
Daniel
Allcock
3455-3474
Abstract: We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams $A_n$, $B_n=C_n$ and $D_n$ and the affine diagrams $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ as subgroups of the braid groups of various simple orbifolds. The cases $D_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except $\tilde{A}_n$ the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type $D_n$.
Weyl--Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
Steve
Clark;
Fritz
Gesztesy
3475-3534
Abstract: We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\mathbb{R}$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.
On the finite-dimensional dynamical systems with limited competition
Xing
Liang;
Jifa
Jiang
3535-3554
Abstract: The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.
On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals
László
Hatvani
3555-3571
Abstract: Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovski{\u{\i}}\kern.15em functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either ${x(t)}$, the norm of the instantaneous value of the solutions or $\Vert x_t\Vert _2$, the $L_2$-norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.
On the profile of the changing sign mountain pass solutions for an elliptic problem
E.
N.
Dancer;
Shusen
Yan
3573-3600
Abstract: We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.
Diffusive logistic equation with constant yield harvesting, I: Steady States
Shobha
Oruganti;
Junping
Shi;
Ratnasingham
Shivaji
3601-3619
Abstract: We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.
Global existence and nonexistence for nonlinear wave equations with damping and source terms
Mohammad
A.
Rammaha;
Theresa
A.
Strei
3621-3637
Abstract: We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term $\left\vert u\right\vert^{m-1}u_t$ and a source term of the form $\left\vert u\right\vert^{p-1}u$, with $m,\,p>1$. We show that whenever $m\geq p$, then local weak solutions are global. On the other hand, we prove that whenever $p>m$ and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.
Attractors for graph critical rational maps
Alexander
Blokh;
Michal
Misiurewicz
3639-3661
Abstract: We call a rational map $f$ graph critical if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$.
Isomorphisms of function modules, and generalized approximation in modulus
David
Blecher;
Krzysztof
Jarosz
3663-3701
Abstract: For a function algebra $A$ we investigate relations between the following three topics: isomorphisms of singly generated $A$-modules, Morita equivalence bimodules, and ``real harmonic functions'' with respect to $A$. We also consider certain groups which are naturally associated with a uniform algebra $A$. We illustrate the notions considered with several examples.
Compactness properties for families of quasistationary solutions of some evolution equations
Giuseppe
Savaré
3703-3722
Abstract: The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions $\{\theta_n\}, \{{\raise.3ex\hbox{$\chi$}}_n\}$ (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum $\theta_n + {\raise.3ex\hbox{$\chi$}}_n$ converges in some $L^p$-space as $n\uparrow+\infty$ and that the time integrals of a suitable ``space'' functional evaluated on $\theta_n, {\raise.3ex\hbox{$\chi$}}_n$ are uniformly bounded with respect to $n$. Can we deduce that $\theta_n$ and ${\raise.3ex\hbox{$\chi$}}_n$ converge separately? LUCKHAUS (1990) gave a positive answer to this question in the framework of the two-phase Stefan problem with Gibbs-Thompson law for the melting temperature. PLOTNIKOV (1993) proposed an abstract result employing the original idea of Luckhaus and arguments of compactness and reflexivity type. We present a general setting for this and other related problems, providing necessary and sufficient conditions for their solvability: these conditions rely on general topological and coercivity properties of the functionals and the norms involved, and do not require reflexivity.
A note on Meyers' Theorem in $W^{k,1}$
Irene
Fonseca;
Giovanni
Leoni;
Jan
Malý;
Roberto
Paroni
3723-3741
Abstract: Lower semicontinuity properties of multiple integrals \begin{displaymath}u\in W^{k,1}(\Omega;\mathbb{R}^{d})\mapsto\int_{\Omega}f(x,u(x), \cdots,\nabla^{k}u(x))\,dx\end{displaymath} are studied when $f$ may grow linearly with respect to the highest-order derivative, $\nabla^{k}u,$ and admissible $W^{k,1}(\Omega;\mathbb{R}^{d})$ sequences converge strongly in $W^{k-1,1}(\Omega;\mathbb{R}^{d}).$ It is shown that under certain continuity assumptions on $f,$ convexity, $1$-quasiconvexity or $k$-polyconvexity of \begin{displaymath}\xi\mapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\xi)\end{displaymath} ensures lower semicontinuity. The case where $f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\cdot)$ is $k$-quasiconvex remains open except in some very particular cases, such as when $f(x,u(x),\cdots,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$
Homogeneous weak solenoids
Robbert
Fokkink;
Lex
Oversteegen
3743-3755
Abstract: A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.
Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group
David
Feldman;
Alexander
Wilce
3757-3769
Abstract: Applied to a continuous surjection $\pi : E \rightarrow B$ of completely regular Hausdorff spaces $E$ and $B$, the Stone-Cech compactification functor $\beta$ yields a surjection $\beta \pi: \beta E \rightarrow \beta B$. For an $n$-fold covering map $\pi$, we show that the fibres of $\beta \pi$, while never containing more than $n$ points, may degenerate to sets of cardinality properly dividing $n$. In the special case of the universal bundle $\pi:EG \rightarrow BG$ of a $p$-group $G$, we show more precisely that every possible type of $G$-orbit occurs among the fibres of $\beta \pi$. To prove this, we use a weak form of the so-called generalized Sullivan conjecture.
Submersions, fibrations and bundles
Gaël
Meigniez
3771-3787
Abstract: When does a submersion have the homotopy lifting property? When is it a locally trivial fibre bundle? We establish characterizations in terms of consistency in the topology of the neighbouring fibres.
The Chromatic Ext Groups $\Ext_{\Gamma(m+1)}^{0}(BP_{*},M_2^{1})$
Ippei
Ichigi;
Hirofumi
Nakai;
Douglas
C.
Ravenel
3789-3813
Abstract: We compute a certain Ext group related to the chromatic spectral sequence for $T (m)$, the spectrum whose $BP$-homology is $BP_{*}[t_{1},\cdots ,t_{m}]$ for each $m\ge 3$. The answer we get displays a kind of periodicity not seen in the corresponding computation for the sphere spectrum.
Euler characters and submanifolds of constant positive curvature
John
Douglas
Moore
3815-3834
Abstract: This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if $M^n$ is an $n$-dimensional compact connected Riemannian submanifold of constant positive curvature in ${\mathbb E}^{2n-1}$, then $M^n$ must be simply connected. It also gives a conformal version of this theorem.