Composition Preserves Rigidity

Say that f in H^p (1 p ) of the unit disk is rigid if it isdetermined in H^p by its argument modulo 2 on the unit circle.We show that if f is rigid and u is an inner function, thenthe composition f o u is rigid. The proof uses the disintegrationof Lebesgue measure with respect to u to compute the adjointof the operator of composition with u. This result is a generalization of the work of Younis, who provedthe special case p = 1 using operator theoretic methods.     

On the Left and Right Cohomological Dimension of Monoids

In this paper we show that, given 0m, n, any monoid can be embeddedinto a monoid with left cohomological dimension m and rightcohomological dimension n. The embedding can be chosen to preservevarious monoid-theoretical properties. 1991 Mathematics SubjectClassification 20M50.     

Differentiability Properties of an Abstract Autonomous Composition Operator

The autonomous composition operator is the nonlinear map whichtakes a pair of functions into its composite function. The compositionoperator often appears in problems of nonlinear analysis andto analyse such problems it is often important to know whetherthe composition operator is continuous or differentiable. Afairly large number of papers in the literature have been devotedto the study of composition operators. For fullscale references,we refer the reader to the extensive monographs of Appell andZabrejko [1] and Runst and Sickel [8]. To exemplify a typicalsituation, we consider the semilinear Dirichlet boundary valueproblem where denotes a sufficiently regular bounded open subset ofR^N, and h₀ a map of R to R, and where u is the unknown of theproblem. We assume that we know that a certain function u₀ belongingto a certain function space X solves (1.1). Then if we wishto know whether by perturbing h₀ in a certain function space,say Y, the solutions u depend on h continuously, with differentiability,with analyticity or bifurcate, we could set G[h, u] u+h u,recast problem (1.1) into the abstract form G[h, u] (1.2) and study the solution set of equation (1.2) around the pair(h₀, u₀) by means of the implicit function theorem or by localbifurcation theorems in a Banach space setting.     

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT are Not Equivalent

A finite rewriting system is presented that does not satisfythe homotopical finiteness condition FDT, although it satisfiesthe homological finiteness condition FHT. This system is obtainedfrom a group G and a finitely generated subgroup H of G througha monoid extension that is almost an HNN extension. The FHTproperty of the extension is closely related to the FP₂ propertyfor the subgroup H, while the FDT property of the extensionis related to the finite presentability of H. The example systemseparating the FDT property from the FHT property is then obtainedby applying this construction to an example group.     

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topologysuch that every group morphism from the free group onto a finitediscrete group is continuous. It was shoen by M.Hall Jr thatevery finitely generated subgroup of the free group is closedfor this topology. We conjecture that if H₁, H₂,...,H_n are finitelygenerated subgroups of the free group, then the product H₁ H₂...H_n is closed. We discuss some consequences of this conjecture.First, it would give a nice and simple algorithm to computethe closure of a given rational subset of the free group. Next,it implies a similar conjecture for the free monoid, which inturn is equivalent to a deep conjecture on finite semigroupsfor the solution of which J. Rhodes has offered $100. We hopethat our new conjecture will shed some light on Rhodes' conjecture.     

The Monoid of Families of Quiver Representations

A monoid structure on families of representations of a quiveris introduced by taking extensions of representations in families,that is, subvarieties of the varieties of representations. Thestudy of this monoid leads to interesting interactions betweenrepresentation theory, algebraic geometry and quantum grouptheory. For example, it produces a wealth of interesting examplesof families of quiver representations, which can be analysedby representation-theoretic and geometric methods. Conversely,results from representation theory, in particular A. Schofield'swork on general properties of quiver representations, allowus to relate the monoid to certain degenerate forms of quantizedenveloping algebras. 2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.     

Abel Equations: Composition Conjectures and the Model Problem

We consider the Abel equation where p and q are polynomials, and a is a fixed constant. Wedenote the solution of (1) by y(x, c), where y(a, c) = c. Standardexistence theorems ensure that y(x, c) is well-defined and analyticin both its arguments, for c sufficiently small. If y(b, c)= c, then we call y(x, c) a periodic solution. Likewise, ify(b, c) c for all c close to 0, then we say that the systemhas a centre between a and b. The numbers a and b are not important;by a simple transformation, we can always choose a = 0 and b= 1, and we shall usually do so from now on. Abel equations arise in several circumstances, but perhaps themain reason for their recent study is connected to the familyof systems where M and N are homogeneous polynomials of the same degreen. A transformation due to Cerkas allows us to bring these systemsto the form (1), where p and q are now trigonometric polynomials.It is not hard to show that, setting a = 0 and b = 2, the definitionsof periodic solution and centre for (1) coincide with theirusual definitions in the planar system (2). There are also transformationsto Abel-type equations for more general systems; see [8, 10]. This trigonometric Abel equation has been used in a large numberof works in order to estimate the number of limit cycles orobtain centre conditions, as well as in more general investigationsrelating the derivatives of the return map with iterated integrals.However, studying system (2) in whatever form is by no meanseasy, and a natural question is to ask whether we can stillcapture the essence of this problem if we take p and q to bepolynomials, in the hope that the calculations will become easier.The recent series of investigations by Briskin, Françoiseand Yomdin [3, 4, 5] seems to indicate that this could be thecase. Our interest here is to see what conditions the existence ofa centre in (1) imposes on the defining equations. For easeof reference, we shall always denote the antiderivative of thepolynomials p and q as P and Q; that is, 2000 Mathematics SubjectClassification 34C25, 34C99.     

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

Copolymer Composition in N-Monomer Systems

A matrix formulation of the differential equations describingcopolymer composition in n-monomer systems in steady state approximationenables properties to be derived from those of positive matrices.We discuss the role of reactivity ratios, existence of solutionsand the conditions for an azeotrope. Some computational aspectsof the solution of the non-linear equations for transient behaviourare examined. Illustrative applications to ternary systems aremade.     

Virtual Algebraic Lie Theory: Tilting Modules and Ringel Duals for Blob Algebras

In this paper we describe a tensor space representation of theblob algebra over a ring allowing base change to every interesting(that is, non-semisimple) specialisation. We show that in quasihereditaryspecialisations this passes to a full tilting module. We considerthe nature of the resultant Ringel dual algebra. 2000 MathematicsSubject Classification 20C08.     

Jacobiennes de Courbes Algebriques de Genre 2 et 3 de Grand Rang Sur Q

Infinite families of curves are constructed of genus 2 and 3over Q whose jacobians have high rank over Q. More precisely,if E is an elliptic curve with rank at least r over Q, an infinitefamily of curves are constructed of genus 2 whose jacobianshave rank at least r+4 over Q, and, under certain conditions,an infinite family of curves are constructed of genus 3 whosejacobians have rank at least 2r over Q. On specialisation, afamily of curves are obtained of genus 2 whose jacobians haverank at least 27 and a family of curves are obtained of genus3 whose jacobians have rank at least 26; one of these has rankat least 42.     

Cyclic Artinian Modules without a Composition Series

Let R be a ring (always understood to be associative with aunit element 1). It is well known that an R-module is Noetherianif and only if all its submodules are finitely generated andthat it has a finite composition series if and only if it isNoetherian and Artinian. This raises the question whether everyfinitely generated Artinian module is Noetherian; here it isenough to consider cyclic Artinian modules, by an inductionon the length. This question has been answered (negatively)by Brian Hartley [5], who gives a construction of an Artinianuniserial module of uncountable composition-length over thegroup algebra of a free group of countable rank. If we are justinterested in finding cyclic modules that are Artinian but notNoetherian, there is a very simple construction based on thefact that over a free algebra every countably generated Artinianmodule can be embedded in a cyclic module which is again Artinian.This is described in 2 below.     

Connectedness and Stability of Julia Sets of the Composition of Polynomials of the Form z2+cn

For a sequence (c_n) of complex numbers, the quadratic polynomialsf_cn(z) := z² + c_n and the sequence (F_n) of iterates F_n := f_cn...f_c1areconsidered. The Fatou set F_(cn) is by definition the set ofall z C^ such that (F_n) is normal in some neighbourhood ofz, while the complement of F_(cn) is called the Julia set J_(cn).The aim of this article is to study the connectedness and stabilityof the Julia set J_(cn) provided that the sequence (c_n) is bounded.     

Algebras and Modules in Monoidal Model Categories

In recent years the theory of structured ring spectra (formerlyknown as A- and E-ring spectra) has been simplified by the discoveryof categories of spectra with strictly associative and commutativesmash products. Now a ring spectrum can simply be defined asa monoid with respect to the smash product in one of these newcategories of spectra. In this paper we provide a general methodfor constructing model category structures for categories ofring, algebra, and module spectra. This provides the necessaryinput for obtaining model categories of symmetric ring spectra,functors with smash product, Gamma-rings, and diagram ring spectra.Algebraic examples to which our methods apply include the stablemodule category over the group algebra of a finite group andunbounded chain complexes over a differential graded algebra.1991 Mathematics Subject Classification: primary 55U35; secondary18D10.     

Composition Operators with Closed Image on Spaces of Real Analytic Functions

We characterize composition operators on spaces of real analyticfunctions which at the same time have closed image and are openonto their images. Under some mild assumptions, we also characterizecomposition operators with closed range and composition operatorsopen onto their images. 2000 Mathematics Subject Classification46E10, 47B33, 32C07 (primary), 26E05, 32E30, 32D15 (secondary).     

On the inverse braid and reflection monoids of type <Emphasis Type="Italic">B</Emphasis>

There are well-known relations between braid and symmetric groups as well as Artin-Brieskorn braid groups and Coxeter groups: the latter are the factor-groups of the Artin-Brieskorn braid groups. The inverse braid monoid is related to the inverse symmetric monoid in the same way. We show that similar relations exist between the inverse braid monoid of type B and the inverse reflection monoid of type B. This gives a presentation of the latter monoid.     

Complete congruence lattices of two related modular lattices

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable \({\omega}\)-categorical structures. Finally, we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable \({\omega}\)-categorical structures.     

Vines and partial transformations

We introduce the partial vine monoid PVn. This monoid is related to the partial transformation semigroup PTn in the same way as the braid group Bn is related to the symmetric group Sn, and contains both the vine monoid [T.G. Lavers, The theory of vines, Comm. Algebra 25 (4) (1997) 1257-1284] and the inverse braid monoid [D. Easdown, T.G. Lavers, The inverse braid monoid, Adv. Math. 186 (2) (2004) 438-455]. We give a presentation for PVn in terms of generators and relations, as well as a faithful representation in a monoid of endomorphisms of a free group. We also derive a new presentation for PTn.     

Symmetrization Inequalities for Composition Operators of Caratheodory Type

Let F:(0, ) x [0, ) R be a function of Carathéodorytype. We establish the inequality where u^* denotes the Schwarz symmetrization of u, under hypotheseson F that seem quite natural when this inequality is used toobtain existence results in the context of elliptic partialdifferential equations. We also treat the case where R^N is replacedby a set of finite measure. The identity is also discussed under the assumption that G: [0,) R is a Borel function. 2000 Mathematics Subject Classification26D20, 42C20, 46E30.     

Structure on the set of closure operations of a commutative ring

We investigate the algebraic structure on the set of closure operations of a ring. We show the set of closure operations is not a monoid under composition for a discrete valuation ring. Even the set of semiprime operations over a DVR is not a monoid; however, it is the union of two monoids, one being the left but not right act of the other. We also determine all semiprime operations over the ring $K?t^2,t^3?$.