Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary pathalgebra, over some field k. Denote by H-reg the category offinite-dimensional regular H-modules, that is, the categoryof modules M with for all integers m, where _H denotes the Auslander–Reiten translation.Call a filtration of a regular H-module M a regular filtration if all subquotients M_i/M_i+1are regular. Call a regular filtration (*) a regular compositionseries if it is strictly decreasing and has no proper refinement.A regular component C in the Auslander–Reiten quiver (H) of H-mod is called filtration closed if, for each M addC, the additive closure of C, and each regular filtration (*)of M, all the subquotients M_i/M_i+1 are also in add C. We showthat most wild hereditary algebras have filtration-closed Auslander–Reitencomponents. Moreover, we deduce from this that there are alsoalmost serial components, that is regular components C, suchthat any indecomposable XC has a unique regular compositionseries. This composition series coincides with the Auslander–Reitenfiltration of X, given by the maximal chain of irreducible monosending at X. 1991 Mathematics Subject Classification: 16G70,16G20, 16G60, 16E30.     

Arens Regularity of the Algebra of Operators on a Banach Space

A short proof is given that if E is a super-reflexive Banachspace, then B(E), the Banach algebra of operators on E withcomposition product, is Arens regular. Some remarks are madeon necessary conditions on E for B(E) to be Arens regular. 2000Mathematics Subject Classification 47L10 (primary), 46H05 (secondary).     

On the Multiplicities of Graph Eigenvalues

Star complements and associated quadratic functions are usedto obtain a sharp upper bound for the order of a graph withan eigenspace of prescribed codimension. It is shown that forregular graphs the bound can be reduced by 1, and that thisreduced bound is attained by a regular graph G if and only ifG is an extremal strongly regular graph. 2000 Mathematics SubjectClassification 05C50.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

Inscribing Smooth Knots with Regular Polygons

A regular n-gon inscribing a knot is a sequence of n pointson a knot, such that the distances between adjacent points areall the same. It is shown that any smooth knot is inscribedby a regular n-gon for any n. 2000 Mathematics Subject Classification57M25 (primary).     

Exact Rates of Convergence to a Stable Law

Let F be a non-lattice distribution function which lies in thedomain of attraction of a non-normal stable distribution. Exactuniform convergence rates are obtained for the convergence ofthe normalised partial sums of random variables with distributionF. Second order regular variation conditions are assumed relatedto the domain of attraction conditions.     

A Variational Method For The Existence of Bounded Solutions of a Sublinear Forced Oscillator

We prove that, for every bounded and measurable forcing p(t),the differential equation has bounded solutions with arbitrarily large amplitude. In generalit is not possible to say that all solutions are bounded, asshown by an example due to Littlewood. The proof is based on a variational method which can be seenas a dual version of Nehari's method for boundary value problemson compact intervals. 2000 Mathematics Subject Classification34C11, 34B14, 49J35.     

Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.     

Cone-Decompositions and Degenerate Critical Points

Let M be a compact, smooth, 2-connected, 2m-dimensional manifoldwith M simply connected. If M has the homotopytype of an m-dimensionalCW-complex, then it supports a smooth, self-indexed function,maximal, constant and regular on M with at most cat(M) + 2 criticalpoints, all of which are of a certain ‘reasonable’type. To such a critical point there corresponds, homotopically,the attachment of a cone. Conversely, to a cone attachment wemay associate, under certain dimensionality and connectivityconditions, a ‘reasonable’ critical point. 1991Mathematics Subject Classification: primary 55M30, 57R70; secondary58E05, 55P50, 55P62.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convexn-dimensional real space. We prove that every continuous functionfrom T into the closed unit ball of X can be expressed as anaverage of eight continuous functions from T into the sphereof X if and only if dim (T) n–1, where dim(T) denotesthe covering dimension of T. The proof we give can be used toprove the same fact, without hypotheses on T, when X is infinite-dimensional,although in this case it has been proved recently that a betterresult can be obtained.     

Study at high frequencies of a stratified waveguide

A waveguide in integrated optics is defined by its refractiveindex. The guide is assumed to be invariant in the propagationdirection while in the transverse direction it is supposed tobe a compact perturbation of an unbounded stratified medium.We are interested in the high frequency modes guided by thisdevice. We consider the problem under the assumptions of weak guidance,so that it reduces to a two-dimensional eigenvalue problem fora scalar field. While a general study has been done in a previouspaper (Bonnet-BenDhia et al., IMAJAM 60, 1998), our goal hereis to present an asymptotic study at high frequencies, whichillustrates the dispersive character of the stratified guide.We will give the limit as the frequency tends to of the guidedmodes and characterize this limit as the solution of an eigenproblem.The technical difficulty lies in the stratified character ofthe unbounded reference medium.     

Projective Prime Ideals and Localisation in PI-Rings

The results here generalise [2, Proposition 4.3] and [9, Theorem5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotentprime ideal of R such that P_R is projective. Then P is leftlocalisable and R_P is a prime principal left and right idealring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotentmaximal ideal of R such that M_R is projective. Then M has theleft AR-property and M contains a right regular element of R.     

Convolution complementarity problems with application to impact problems

^** Email: dstewart{at}math.uiowa.edu. Part of this work was carried out while visiting CMAF at the University of Lisbon and while visiting the University of Lyons 1. Convolution complementarity problems (CCPs) have the followingform: given a matrix-valued function k and a vector-valued functionq, find a vector-valued function u satisfying 0 u(t) (k*u)(t)+ q(t) 0 for all t. In this paper CCPs are applied to a mechanicalimpact problem, but they can also be applied to other dynamicproblems with hard constraints. CCPs are shown to have solutionsprovided q(0) 0 and q is sufficiently regular, k has locallybounded variation and k(0⁺) is a P-matrix. Uniqueness also holdsprovided, in addition, k(0⁺) is symmetric positive definite.This theory shows that the impact problem studied here has aunique solution, and that energy is conserved. Numerical methodshave been devised and implemented for the impact problem, andthe results are presented.     

Volumes of Projections of unit Cubes

Let Z and be the images of a regular unit n-cube in Eⁿ under orthogonal projection on toorthogonal complementary subspaces of dimensions d and n —d respectively. It is shown that the d-volume of Z and the (n— d)-volume of are equal. This generalizes to a connexion between the volumes of stronglyassociated zonotopes.     

Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

On the Magnitude of the Sum of Bases of the Natural Numbers

We construct two bases of the natural numbers B₁ and B₂, eachof order two, such that (B₁ + B₂ (n) <n^{+c/(log n)}. For alower estimate, it is proved that if B₂ and are two bases, eachof order two, then (B₁+B₂)(n) > n. Generalisations to sumsof bases of order h > 2 are also given.     

On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.     

Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.