The classification of the simple modular Lie algebras: VI. Solving the final case
H.
Strade
2553-2628
Abstract: We investigate the structure of simple Lie algebras $L$ over an algebraically closed field of characteristic $p>7$. Let $T$ denote a torus in the $p$-envelope of $L$ in $\operatorname{Der}L$ of maximal dimension. We classify all $L$ for which every 1-section with respect to every such torus $T$ is solvable. This settles the remaining case of the classification of these algebras.
The Regular Complex in the $BP\rangle 1 \langle$-Adams Spectral Sequence
Jesús
González
2629-2664
Abstract: We give a complete description of the quotient complex ${\cal C}$ obtained by dividing out the ${\mathbb F}_{p}$ Eilenberg-Mac Lane wedge summands in the first term of the $BP\langle 1\rangle$-Adams spectral sequence for the sphere spectrum $S^0$. We also give a detailed computation of the cohomology groups $H^{s,t}({\cal C})$ and obtain as a consequence a vanishing line of slope $(p^{2}-p-1)^{-1}$ in their usual $(t-s, s)$ representation. These calculations are interpreted as giving general simple conditions to lift homotopy classes through a $BP\langle 1 \rangle$ resolution of $S^0$.
Generators and relations of direct products of semigroups
E.
F.
Robertson;
N.
Ruskuc;
J.
Wiegold
2665-2685
Abstract: The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup $S$ of order 11 such that $S\times T$ is finitely generated but not finitely presented for every finitely generated infinite semigroup $T$. By way of contrast we show that, if $S$ and $T$ belong to a wide class of semigroups, then $S\times T$ is finitely presented if and only if both $S$ and $T$ are finitely presented, exactly as in the case of groups and monoids.
A weak-type inequality for differentially subordinate harmonic functions
Changsun
Choi
2687-2696
Abstract: Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder $\mu(|v|\ge 1)\le 2\|u\|_1$ for two harmonic functions $u$ and $v$. That is, we prove the sharp weak-type inequality $\mu(|v|\ge 1)\le K\|u\|_1$ under the assumptions that $|v(\xi)|\le |u(\xi)|$, $|\nabla v|\le|\nabla u|$ and the extra assumption that $\nabla u\cdot\nabla v=0$. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant $K$ is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.
Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems
Abbas
Bahri;
Iskander
A.
Taimanov
2697-2717
Abstract: A Lagrangian system describing a motion of a charged particle on a Riemannian manifold is studied. For this flow an analog of a Ricci curvature is introduced, and for Ricci positively curved flows the existence of periodic orbits is proved.
The $\Pi_3$-theory of the computably enumerable Turing degrees is undecidable
Steffen
Lempp;
André
Nies;
Theodore
A.
Slaman
2719-2736
Abstract: We show the undecidability of the $\Pi _{3}$-theory of the partial order of computably enumerable Turing degrees.
Picard groups and infinite matrix rings
Gene
Abrams;
Jeremy
Haefner
2737-2752
Abstract: We describe a connection between the Picard group of a ring with local units $T$ and the Picard group of the unital overring $End(_TT)$. Using this connection, we show that the three groups $Pic(R)$, $Pic(FM(R))$, and $Pic(RFM(R))$ are isomorphic for any unital ring $R$. Furthermore, each element of $Pic(RFM(R))$ arises from an automorphism of $RFM(R)$, which yields an isomorphsm between $Pic(RFM(R))$ and $Out(RFM(R))$. As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group $Out_R(RFM(R))$ is abelian for any commutative unital ring $R$.
Scattering theory for twisted automorphic functions
Ralph
Phillips
2753-2778
Abstract: The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $\rho$ and satisfying $u(\gamma z)=\rho(\gamma)u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim(\rho)>1$ this is no longer the case.
On the measure theoretic structure of compact groups
S.
Grekas;
S.
Mercourakis
2779-2796
Abstract: If $G$ is a compact group with $w(G)=a\geq \omega$, we show the following results: (i) There exist direct products $\displaystyle{\prod _{\xi<a}G_{\xi}, \prod _{\xi<a}H_{\xi}}$ of compact metric groups and continuous open surjections $\displaystyle{\prod _{\xi<a}G_{\xi} \stackrel{p}{\rightarrow }G \stackrel{q}{\rightarrow }\prod _{\xi<a}H_{\xi}}$ with respect to Haar measure; and (ii) the Haar measure on $G$ is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups. Applications of the above results in measure theory are given.
A Hilbert-Nagata theorem in noncommutative invariant theory
Mátyás
Domokos;
Vesselin
Drensky
2797-2811
Abstract: Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.
The Castelnuovo regularity of the Rees algebra and the associated graded ring
Ngô
Viêt
Trung
2813-2832
Abstract: It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of $d$-sequences.
On the averages of Darboux functions
Aleksander
Maliszewski
2833-2846
Abstract: Let $\mathbf{A}$ be the family of functions which can be written as the average of two comparable Darboux functions. In 1974 A. M. Bruckner, J. G. Ceder, and T. L. Pearson characterized the family $\mathbf{A} $ and showed that if $\alpha \ge 2$, then $\mathbf{A} \cap {\mathbf B}_\alpha$ is the family of the averages of comparable Darboux functions in Baire class $\alpha$. They also asked whether the latter result holds true also for $\alpha =1$. The main goal of this paper is to answer this question in the negative and to characterize the family of the averages of comparable Darboux Baire one functions.
Kruzkov's estimates for scalar conservation laws revisited
F.
Bouchut;
B.
Perthame
2847-2870
Abstract: We give a synthetic statement of Kruzkov-type estimates for multi-dimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in $h^{1/4}$ known for finite volume methods on unstructured grids. Les estimations de Kruzkov pour les lois de conservation scalaires revisitées Résumé Nous donnons un énoncé synthétique des estimations de type de Kruzkov pour les lois de conservation scalaires multidimensionnelles. Nous l'appliquons pour obtenir d'estimations nombreuses pour problèmes différents d'approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en $h^{1/4}$ connue pour les méthodes de volumes finis sur des maillages non structurés.
A Note on the Monomial Conjecture
S.
P.
Dutta
2871-2878
Abstract: Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed.
Bounds for multiplicities
Jürgen
Herzog;
Hema
Srinivasan
2879-2902
Abstract: Let $R=K[x_1,x_2,\ldots, x_n]$ and $S=R/I$ be a homogeneous $K$-algebra. We establish bounds for the multiplicity of certain homogeneous $K$-algebras $S$ in terms of the shifts in a free resolution of $S$ over $R$. Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasi-pure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals $I$. As a consequence, we get a bound for the regularity of $S$. Further, when $S$ is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a $p$-linear resolution.
Projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits
T.
Nakano
2903-2924
Abstract: The birational geometry of projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the $\mathbf{SL}(2)$-birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.
A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II
Y.
S.
Choi;
P.
J.
McKenna
2925-2937
Abstract: Let $\Omega \subset \mathbf{R}^N$ with $N \geq 2$. We consider the equations \begin{displaymath}\begin{array}{rcl} \displaystyle \sum _{i=1}^{N} u^{a_i} \frac{\partial^2 u}{\partial x_i^2} +p(\mathbf{x})& = & 0, u|_{\partial\Omega} & = & 0, \end{array} \end{displaymath} with $a_1 \geq a_2 \geq .... \geq a_N \geq 0$ and $a_1>a_N$. We show that if $\Omega$ is a convex bounded region in $\mathbf{R}^N$, there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in $\mathbf{R}^2$ are also given.
A probabilistic approach to some of Euler's number theoretic identities
Don
Rawlings
2939-2951
Abstract: Probabilistic proofs and interpretations are given for the $q$-binomial theorem, $q$-binomial series, two of Euler's fundamental partition identities, and for $q$-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.
Sum theorems for monotone operators and convex functions
S.
Simons
2953-2972
Abstract: In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.
Pairs of monotone operators
S.
Simons
2973-2980
Abstract: This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.