Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

On Principal Blocks of p-Constrained Groups

A theorem of K. W. Roggenkamp and L. L. Scott shows that fora finite group G with a normal p-subgroup containing its owncentralizer, any two group bases of the integral group ringZG are conjugate in the units of Z_pG. Though the theorem presentsitself in the work of others and appears to be needed, thereis no published account. There seems to be a flaw in the proof,because a ‘theorem’ appearing in the survey [K.W. Roggenkamp, ‘The isomorphism problem for integral grouprings of finite groups’, Progress in Mathematics 95 (Birkhäuser,Basel, 1991), pp. 193--220], where the main ingredients of aproof are given, is false. In this paper, it is shown how toclose this gap, at least if one is only interested in the conclusionmentioned above. Therefore, some consequences of the resultsof A. Weiss on permutation modules are stated. The basic stepsof which any proof should consist are discussed in some detail.In doing so, a complete, yet short, proof of the theorem isgiven in the case that G has a normal Sylow p-subgroup. 2000Mathematical Subject Classification: primary 16U60; secondary20C05.     

Boundary value problems for systems of linear evolution equations

It is shown that the new method for solving initial-boundaryvalue problems for scalar evolution equations recently introducedby one of the authors can also be applied to systems of evolutionequations. The novel step needed in this case is the constructionof a scalar Lax pair by using a suitable parametrization ofthe dispersion relation as well as certain linear transformations.The simultaneous spectral analysis of the Lax pair yields thesolution of a given initial-boundary value problem in termsof an integral in the complex spectral plane which involvesan appropriate x-transform of the initial conditions and anappropriate t-transform of the boundary conditions. These transformsare neither the x-Fourier transform nor the t-Laplace transform,rather they are new transforms custom made for the given systemof partial differential equations (PDEs) and the given domain.This method is illustrated by solving on the half-line the linearizedequations governing infinitesimal deformations in a heat conductingbar. Received 8 January 2003. Revised 18 July 2003. ^* Email: p.a.treharne{at}damtp.cam.ac.uk Email: t.fokas{at}damtp.cam.ac.uk     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

On Holomorphic Principal Bundles Over a Compact Riemann Surface Admitting a Flat Connection, II

Holomorphic principal bundles over a compact Riemann surfaceX that admits a flat connection are considered. A holomorphicG-bundle over X, where G is a connected semisimple linear algebraicgroup over C, admits a flat connection if and only if the adjointvector bundle admits one. More generally, for a complex reductivegroup G, the necessary and sufficient condition on a G-bundleto admit a flat connection is described. This simplifies thecriterion obtained by the authors and given in Math. Ann. 322(2002) 333–346. 2000 Mathematics Subject Classification53C05, 32L05.     

Some Applications of the Lyapunov Matrix Equation

The matrix equation A'P+PA = –Q originates in the stabilityanalysis of the system of linear differential equations x =Ax by Lyapunov's direct method. Many other aspects of the stabilityof such systems and of related problems in matrix theory canalso be examined by this matrix equation. Some of these arediscussed in this paper and new applications to the stabilityof second order damped dynamic systems and to stable quasi-Jacobimatrices are given.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

A Class of Functional Equations on a Locally Compact Group

Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation, E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications.     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

High-Order Embedded Runge-Kutta-Nystrom Formulae

New efficient embedded Runge-Kutta-Nystrom processes of orders8(6) and 12(10) are presented for the numerical solution ofthe special second-order differential equation y'(x) = f[x,y(x)]. Test results indicate their improved efficiency relativeto other RKN formulae in current use.     

Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation , where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant. Received April 29, 1997; in final form September 22, 1997     

Perfect and Acyclic Subgroups of Finitely Presentable Groups

Acyclic groups of low dimension are considered. To indicatethe results simply, let G' be the nontrivial perfect commutatorsubgroup of a finitely presentable group G. Then def(G)1. Whendef(G)=1, G' is acyclic provided that it has no integral homologyin dimensions above 2 (a sufficient condition for this is thatG' be finitely generated); moreover, G/G' is then Z or Z². Naturalexamples are the groups of knots and links with Alexander polynomial1. A further construction is given, based on knots in S²x S¹.In these geometric examples, G' cannot be finitely generated;in general, it cannot be finitely presentable. When G is a 3-manifoldgroup it fails to be acyclic; on the other hand, if G' is finitelygenerated it has finite index in the group of a Q-homology 3-sphere.     

On the behavior of the oscillatory solutions of first or second order delay differential equations

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x(t)=O(v(t)) as t→∞, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented.     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.     

Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularityX is facilitated by the fact that the group Aut X of automorphismsof X has a maximal reductive algebraic subgroup G which containsevery reductive algebraic subgroup of Aut X up to conjugation.If X is not weighted homogeneous then this maximal group G isfinite (Scheja, Wiebe). It has been determined for cusp singularitiesby Wall. On the other hand, if X is weighted homogeneous butnot a cyclic quotient singularity then the connected componentG₁ of the unit coincides with the C* defining the weighted homogeneousstructure (Scheja, Wiebe, Wahl). Thus the main interest liesin the finite group G/G₁. Not much is known about G/G₁. Ganterhas given a bound on its order valid for Gorenstein singularitieswhich are not log-canonical. Aumann-Körber has determinedG/G₁ for all quotient singularities. We propose to study G/G₁ through the action of G on the minimalgood resolution of X. If X is weightedhomogeneous but not a cyclic quotient singularity, let E₀ bethe central curve of the exceptional divisor of . We show that the natural homomorphism GAut E₀ haskernel C* and finite image. In particular, this re-proves therest of Scheja, Wiebe and Wahl mentioned above. Moreover, itallows us to view G/G₁ as a subgroup of Aut E₀. For simple ellipticsingularities it equals (Z_bxZ_b)Aut₀ E₀ where –b is theself-intersection number of E₀, Z_bxZ_b is the group of b-torsionpoints of the elliptic curve E₀ acting by translations, andAut₀ E₀ is the group of automorphisms fixing the zero elementof E₀. If E₀ is rational then G/G₁ is the group of automorphismsof E₀ which permute the intersection points with the branchesof the exceptional divisor while preserving the Seifert invariantsof these branches. When there are exactly three branches weconclude that G/G₁ is isomorphic to the group of automorphismsof the weighted resolution graph. This applies to all non-cyclicquotient singularities as well as to triangle singularities.We also investigate whether the maximal reductive automorphismgroup is a direct product GG₁xG/G₁. This is the case, for instance,if the central curve E₀ is rational of even self-intersectionnumber or if X is Gorenstein such that its nowhere-zero 2-form has degree ±1. In the latter case there is a ‘natural’section G/G₁G of GG/G₁ given by the group of automorphisms inG which fix . For a simple elliptic singularity one has GG₁xG/G₁if and only if –E₀ · E₀ = 1.     

Subgroups of Word Hyperbolic Groups in Dimension 2

If G is a word hyperbolic group of cohomological dimension 2,then every subgroup of G of type FP₂ is also word hyperbolic.Isoperimetric inequalities are defined for groups of type FP₂and it is shown that the linear isoperimetric inequality inthis generalized context is equivalent to word hyperbolicity.A sufficient condition for hyperbolicity of a general graphis given along with an application to ‘relative hyperbolicity’.Finitely presented subgroups of Lyndon's small cancellationgroups of hyperbolic type are word hyperbolic. Finitely presentedsubgroups of hyperbolic 1-relator groups are hyperbolic. Finitelypresented subgroups of free Burnside groups are finite in thestable range.     

The gradient iteration for approximation in reproducing kernel Hilbert spaces

Consider the bounded linear operator, L: F Z, where Z R^N andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, L_i:F R. By the Riesz representation theorem, there exist functionsk(x_i, ^·) F : L_if = f, k(x_i, ^·)_F. The functions,k(x_i, ^·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.     

A Nonoscillation Theorem for Second-Order Nonlinear Differential Equations with Decaying Coefficients

The purpose of this paper is to give sufficient conditions forall nontrivial solutions of the nonlinear differential equationx ' a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfiesthe sign condition xg(x) > 0 if x 0, but is not assumedto be monotone increasing. This differential equation includesthe generalized Emden–Fowler equation as a special case.Our main result extends some nonoscillation theorems for thegeneralized Emden–Fowler equation. Proof is given by meansof some Liapunov functions and phase-plane analysis.