Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

LAX CONSTRAINTS IN SEMISIMPLE LIE GROUPS

Instead of studying Lax equations as such, a solution Z of aLax equation is assumed to be given. Then Z is regarded as defininga constraint on a non-autonomous linear differential equationassociated with the Lax equation. In generic cases, quadratureand sometimes algebraic formulae in terms of Z are then provedfor solution x of the linear differential equation, and examplesare given where these formulae lead to new results in higher-ordervariational problems for curves in general semisimple Lie groupsG, extending results previously obtained by different methodsfor the case where G has dimension 3. The new construction isexplored in detail for G = SU(m).     

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.     

Finite Subsequences of Shift Register Sequences

In a stream cipher a cryptogram is produced from a binary datastream by modulo-2-adding it to a keystream sequence. The securityof the system relies on the inability of an interceptor to determinethis keystream sequence. One obvious requirement for such asystem is that there should be sufficiently many possibilitiesfor the keystream sequence that the interceptor cannot possiblytry them all. In this paper we consider the likelihood of an interceptor beingable to decipher the cryptogram correctly even though he maybe trying the wrong keystream sequence. This possibility arisesbecause the length of any particular message is likely to beconsiderably shorter than the period of the keystream sequence,and thus only a comparatively small section of the keystreamsequence is used. Hence, if the interceptor tries a sequencewhich intersects (i.e. agrees) with the keystream sequence inthe appropriate positions, he will deduce the message correctly. A number of the standard methods for generating keystream sequencesuse shift registers as ‘building blocks’. So welook in considerable detail at the number of intersections (ofvarious lengths) for sequences generated by two different shiftregisters. We also show that if a keystream sequence has linearequivalence n, then the local linear equivalence of any subsequenceof length at least 2n is n. This means that if the message haslength at least 2n and the keystream sequence has linear equivalencen, then there is no other sequence of linear equivalence lessthan n+1 which can be used to decipher correctly.     

Approximately Local Derivations

Certain linear operators from a Banach algebra A into a BanachA-bimodule X, which are called approximately local derivations,are studied. It is shown that when A is a C^*-algebra, a Banachalgebra generated by idempotents, a semisimple annihilator Banachalgebra, or the group algebra of a SIN or a totally disconnectedgroup, bounded approximately local derivations from A into Xare derivations. This, in particular, extends a result of B.E. Johnson that ‘local derivations on C^*-algebras arederivations’ and provides an alternative proof of it.     

Reduced-order filtering with energy-to-peak performance for discrete-time Markovian jumping systems

In this paper, the l₂–l (energy-to-peak) performanceof the discrete-time Markovian jump linear system is investigated.The jump parameters are modelled by a discrete-time Markov process.Furthermore, we study the l₂–l reduced-order filteringproblem for the Markovian jump linear system. A reduced-orderfilter with the same randomly jumping parameters is proposedwhich can make the error systems with Markovian jump parametersstochastically stable with a prescribed l₂–lperformance.Sufficient conditions in terms of linear matrix inequalities(LMIs) and a coupling non-convex rank constraint are derivedfor the existence of a solution to the reduced-order filteringproblems. A numerical example is given to illustrate the designprocedures.     

Convex Bodies in Exceptional Relative Positions

Two convex bodies K and K' in Euclidean space Eⁿ can be saidto be in exceptional relative position if they have a commonboundary point at which the linear hulls of their normal coneshave a non-trivial intersection. It is proved that the set ofrigid motions g for which K and gK' are in exceptional relativeposition is of Haar measure zero. A similar result holds trueif ‘exceptional relative position’ is defined viacommon supporting hyperplanes. Both results were conjecturedby S. Glasauer; they have applications in integral geometry.     

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to beunitary if u is invertible in A and satisfies ||u|| = ||u^–1||= 1. The norm-unital Banach algebra A is called unitary if theconvex hull of the set of its unitary elements is norm-densein the closed unit ball of A. If X is a complex Hilbert space,then the algebra BL(X) of all bounded linear operators on Xis unitary by the Russo–Dye theorem. The question of whetherthis property characterizes complex Hilbert spaces among complexBanach spaces seems to be open. Some partial affirmative answersto this question are proved here. In particular, a complex Banachspace X is a Hilbert space if (and only if) BL(X) is unitaryand, for Y equal to X, X* or X** there exists a biholomorphicautomorphism of the open unit ball of Y that cannot be extendedto a surjective linear isometry on Y. 2000 Mathematics SubjectClassification 46B04, 46B10, 46B20.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

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.     

Discriminant of a Germ {Phi}: (C2, 0)->(C2, 0) and Seifert Fibred Manifolds

Let f and g be two analytic function germs without common branches.We define the Jacobian quotients of (g, f), which are ‘firstorder invariants’ of the discriminant curve of (g, f),and we prove that they only depend on the topological type of(g, f). We compute them with the help of the topology of (g,f). If g is a linear form transverse to f, the Jacobian quotientsare exactly the polar quotients of f and we affirm the resultsof D. T. Lê, F. Michel and C. Weber.     

A safeguarded dual weighted residual method

Andreas Veeser ^{The dual weighted residual (DWR) method yields reliable a posteriorierror bounds for linear output functionals provided that theerror incurred by the numerical approximation of the dual solutionis negligible. In that case, its performance is generally superiorthan that of global ‘energy norm’ error estimatorswhich are ‘unconditionally’ reliable. We presenta simple numerical example for which neglecting the approximationerror leads to severe underestimation of the functional error,thus showing that the DWR method may be unreliable. We proposea remedy that preserves the original performance, namely a DWRmethod safeguarded by additional asymptotically higher ordera posteriori terms. In particular, the enhanced estimator isunconditionally reliable and asymptotically coincides with theoriginal DWR method. These properties are illustrated via theaforementioned example.}     

A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

Derangements and Eigenvalue-Free Elements in Finite Classical Groups

Permutations that have no fixed points have been known for avery long time as ‘derangements’. Under that headingRouse-Ball [10, p. 46] puts the matter in the following charmingway: ‘Suppose you have written a letter to each of n differentfriends, and addressed the n corresponding envelopes. In howmany ways can you make the regrettable mistake of putting everyletter into a wrong envelope?’ He traces this problemback to 1713 and since then it has occurred in one form or anotherin many elementary texts on probability theory (see for example[12, p. 57] and references cited by Rouse-Ball). Let p_n be theprobability that all the letters are put into the wrong envelope.It is well known that p_ne^–1 as n, and that, moreover,convergence is very fast. In fact, , so that . A derangement can be thought of as a fixed-point-free elementof the symmetric group Sym(n). In this paper we turn our attentionto eigenvalue-free elements of finite linear groups. Since aneigenvector of a linear transformation X in a vector space Vcorresponds to a fixed point of X in the projective space whosepoints are the 1-dimensional subspaces of V, eigenvalue-freeinvertible matrices correspond to derangements in projectivegroups.     

Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.     

A One-step Method of Order 10 for y' = f(x, y)

In some situations, especially if one demands the solution ofthe differential equation with a great precision, it is preferableto use high-order methods. The methods considered here are similarto Runge—Kutta methods, but for the second-order equationy'= f(x, y). As for Runge—Kutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a non—linear system of 440 algebraicequations to obtain a tenth—order method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors.     

Asymptotic Error Expansions for Numerical Solutions of Integral Equations

A general theorem dealing with asymptotic error expansions fornumerical solutions of linear operator equations is proved.This is applied to the Nystr?m, collocation, and Galerkin methodsfor second kind, Fredholm integral equations. For example, weshow that when piecewise polynomials of degree m–1 areused, the iterated Galerkin solution admits an error expansionin even powers of the step-size h, beginning with a term inh^2m.     

Linear Systems and Quotients of Projective Space

A complete characterization of the categorical quotients of(P¹)ⁿ by the diagonal action of SL(2, C) with respect to anypolarization is given by M. Polito, in ‘SL(2, C)-quotientsde (P¹)ⁿ’, C. R. Acad. Sci. Paris Sér. I 321 (1995)1577–1582. In this paper, these categorical quotientsare obtained by certain linear systems on P^n–3, dependingon the given polarization. 2000 Mathematics Subject Classification14L24, 14L30     

Systems of Inequalities and Numerical Semigroups

A one-to-one correspondence is described between the set S(m)of numerical semigroups with multiplicity m and the set of non-negativeinteger solutions of a system of linear Diophantine inequalities.This correspondence infers in S(m) a semigroup structure andthe resulting semigroup is isomorphic to a subsemigroup of N^m–1.Finally, this result is particularized to the symmetric case.     

A Characterization of Strongly Continuous Groups of Linear Operators on a Hilbert Space

It is proved that the infinitesimal generator A of a stronglycontinuous semigroup of linear operators on a Hilbert spacealso generates a strongly continuous group if and only if theresolvent of –A, ( + A)^–1, is also a bounded functionon some right-hand-side half plane of complex numbers, and convergesstrongly to zero as the real part of tends to infinity. Anapplication to a partial differential equation is given. 1991Mathematics Subject Classification 47D03.