Functions for parametrization of solutions of an equation in a free monoid
Gennady
S.
Makanin;
Tatiana
A.
Makanina
1-54
Abstract: In this paper we introduce recursive functions \begin{align*}&{}^{\mathbf{Fi}}(x_1,x_2)^{\lambda _1,\dotsc,\lambda _s}\qquad(s\ge 0), &{}^{\mathbf{Th}}(x_1,x_2,x_3)_i^{\lambda _1,\dotsc,\lambda _{2s}} \qquad(i=1,2,3;s\ge 0), &{}^{\mathbf{Ro}}(x_1,x_2,x_3)_i^{\mu _1,\dotsc,\mu _s}\qquad(i=1,2,3;s\ge 0) \end{align*} of the word variables $x_1,x_2,x_3$, natural number variables $\lambda _k$ and variables $\mu _k$ whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation \begin{displaymath}x_1x_2x_3x_4=\zeta(x_1,x_2,x_3)x_5,\end{displaymath} where $\zeta(x_1,x_2,x_3)$ is an arbitrary word in the alphabet $x_1,x_2,x_3$, in a free monoid.
The metric projection onto the soul
Luis
Guijarro;
Gerard
Walschap
55-69
Abstract: We study geometric properties of the metric projection $\pi :M\to S$ of an open manifold $M$ with nonnegative sectional curvature onto a soul $S$. $\pi$ is shown to be $C^{\infty }$ up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of $S$ also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of $M$, and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of $M$.
Even valuations on convex bodies
Daniel
A.
Klain
71-93
Abstract: The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.
Tight closure, plus closure and Frobenius closure in cubical cones
Moira
A.
McDermott
95-114
Abstract: We consider tight closure, plus closure and Frobenius closure in the rings $R = K[[x,y,z]]/(x^{3} + y^{3} +z^{3})$, where $K$ is a field of characteristic $p$ and $p \neq 3$. We use a $\mathbb{Z}_3$-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring $K[[x,y]]$. We show that Frobenius closure is the same as tight closure in certain classes of ideals when $p \equiv 2 \text{mod} 3$. Since $I^{F} \subseteq IR^{+} \cap R \subseteq I^{*}$, we conclude that $IR^{+} \cap R = I^{*}$ for these ideals. Using injective modules over the ring $R^{\infty }$, the union of all ${p^{e}}$th roots of elements of $R$, we reduce the question of whether $I^{F} = I^{*}$ for $\mathbb{Z}_3$-graded ideals to the case of $\mathbb{Z}_3$-graded irreducible modules. We classify the irreducible $m$-primary $\mathbb{Z}_3$-graded ideals. We then show that $I^{F} = I^{*}$ for most irreducible $m$-primary $\mathbb{Z}_3$-graded ideals in $K[[x,y,z]]/(x^3+y^3+z^3)$, where $K$ is a field of characteristic $p$ and $p \equiv 2 \text{mod} 3$. Hence $I^{*} = IR^{+} \cap R$ for these ideals.
The spectrum of infinite regular line graphs
Tomoyuki
Shirai
115-132
Abstract: Let $G$ be an infinite $d$-regular graph and $L(G)$ its line graph. We consider discrete Laplacians on $G$ and $L(G)$, and show the exact relation between the spectrum of $-\Delta _G$ and that of $-\Delta _{L(G)}$. Our method is also applicable to $(d_1,d_2)$-semiregular graphs, subdivision graphs and para-line graphs.
Cantor sets and numbers with restricted partial quotients
S.
Astels
133-170
Abstract: For $j=1,\dots,k$ let $C_j$ be a Cantor set constructed from the interval $I_j$, and let $\epsilon _j=\pm 1$. We derive conditions under which \begin{equation*}\epsilon _1 C_1+\dots+\epsilon _k C_k = \epsilon _1 I_1+\dots+\epsilon _k I_k \quad\text{and}\quad C_1^{\epsilon _1}\dotsb C_k^{\epsilon _k}= I_1^{\epsilon _1}\dotsb I_k^{\epsilon _k}.\end{equation*} When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets $F(B_j)$, where $B_j$ is a set of positive integers and $F(B_j)$ is the set of real numbers $x$ such that all partial quotients of $x$, except possibly the first, are members of $B_j$.
Almost normal surfaces in 3-manifolds
Michelle
Stocking
171-207
Abstract: J. H. Rubinstein introduced the theory of almost normal surfaces to solve several homeomorphism problems for 3-manifolds. A. Thompson simplified Rubinstein's algorithm for recognizing the 3-sphere by using almost normal surface theory and thin position. This paper discusses higher genus analogues to A. Thompson's work.
A multiplication in cyclic homology
Kiyoshi
Igusa
209-242
Abstract: We define a multiplication on the cyclic homology of a commutative, cocommutative bialgebra $H$ with ``superproduct.'' In the case when $H$ is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator. (The Loday-Quillen multiplication is trivial in that case.)
Absolutely continuous spectrum of perturbed Stark operators
Alexander
Kiselev
243-256
Abstract: We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate $(1+x) ^{-\frac{1}{3}-\epsilon}$ or if it is continuously differentiable with derivative from the Hölder space $C_{\alpha}(R),$ with any $\alpha>0.$
Orbit equivalence of global attractors of semilinear parabolic differential equations
Bernold
Fiedler;
Carlos
Rocha
257-284
Abstract: We consider global attractors ${\cal A}_f$ of dissipative parabolic equations \begin{equation*}u_t=u_{xx}+f(x,u,u_x) \end{equation*} on the unit interval $0\leq x\leq 1$ with Neumann boundary conditions. A permutation $\pi _f$ is defined by the two orderings of the set of (hyperbolic) equilibrium solutions $u_t\equiv 0$ according to their respective values at the two boundary points $x=0$ and $x=1.$ We prove that two global attractors, ${\cal A}_f$ and ${\cal A}_g$, are globally $C^0$ orbit equivalent, if their equilibrium permutations $\pi _f$ and $\pi _g$ coincide. In other words, some discrete information on the ordinary differential equation boundary value problem $u_t\equiv 0$ characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.
Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations
José
M.
Arrieta;
Alexandre
N.
Carvalho
285-310
Abstract: We prove a local existence and uniqueness theorem for abstract parabolic problems of the type $\dot x=Ax+f(t,x)$ when the nonlinearity $f$ satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.
Fuchsian Groups, Quasiconformal Groups, and Conical Limit Sets
Peter
W.
Jones;
Lesley
A.
Ward
311-362
Abstract: We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.
Hankel Operators on Bounded Analytic Functions
James
Dudziak;
T.
W.
Gamelin;
Pamela
Gorkin
363-377
Abstract: For $U$ a domain in the complex plane and $g$ a bounded measurable function on $U$, the generalized Hankel operator $S_g$ on $H^\infty(U)$ is the operator of multiplication by $g$ followed by projection into $L^\infty/H^\infty$. Under certain conditions on $U$ we show that either $S_g$ is compact or there is an embedded $\ell^\infty$ on which $S_g$ is bicontinuous. We characterize those $g$'s for which $S_g$ is compact in the case that $U$ is a Behrens roadrunner domain.
Operator ideal norms on $L^p$
L.
Rodríguez-Piazza;
M.
C.
Romero-Moreno
379-395
Abstract: Let $p$ be a real number such that $p \in (1,+\infty )$ and its conjugate exponent $q\not =4,6,8\ldots$. We prove that for an operator $T$ defined on $L^{p}(\lambda )$ with values in a Banach space, the image of the unit ball determines whether $T$ belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of $p$. Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.
Vector lattices of weakly compact operators on Banach lattices
Z.
L.
Chen;
A.
W.
Wickstead
397-412
Abstract: A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.
A local Peter-Weyl theorem
Leonard
Gross
413-427
Abstract: An $Ad\, K$ invariant inner product on the Lie algebra of a compact connected Lie group $K$ extends to a Hermitian inner product on the Lie algebra of the complexified Lie group $K_{c}$. The Laplace-Beltrami operator, $\Delta$, on $K_{c}$ induced by the Hermitian inner product determines, for each number $a>0$, a Green's function $r_{a}$ by means of the identity $(a^{2} -\Delta /4 )^{-1} = r_{a} *$. The Hilbert space of holomorphic functions on $K_{c}$ which are square integrable with respect to $r_{a} (x)dx$ is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of $K$ whose Casimir operator is appropriately related to $a$.
On the existence of a unipotent support for the irreducible characters of a finite group of Lie type
Meinolf
Geck;
Gunter
Malle
429-456
Abstract: In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig in the case where the characteristic of the field over which the group is defined is large enough. The first named author extended this to the case where the characteristic is good. It is the purpose of this paper to remove this condition as well, so that the existence of unipotent supports is established in complete generality.
Finite groups of matrices over group rings
Gerald
Cliff;
Alfred
Weiss
457-475
Abstract: We investigate certain finite subgroups $\Gamma$ of $GL_{n}(\mathbf{Z}\Pi )$, where $\Pi$ is a finite nilpotent group. Such a group $\Gamma$ gives rise to a $\mathbf{Z}[\Gamma \times \Pi]$-module; we study the characters of these modules to limit the structure of $\Gamma$. We also exhibit some exotic subgroups $\Gamma$.
Polynomial Retracts and the Jacobian Conjecture
Vladimir
Shpilrain;
Jie-Tai
Yu
477-484
Abstract: Let $K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a retraction, or projection) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R.$ The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.