Embedding l{infty} into the Space of Bounded Operators on Certain Banach Spaces

Sufficient conditions are given on a Banach space X which ensurethat embeds in L(X), the space of all bounded linear operatorson X. A basic sequence e_n is said to be quasisubsymmetric iffor any two increasing sequences (k_n) and (l_n) of positive integerswith (k_n) (l_n) for all n, () dominates (). If a Banach space X has a seminormalized quasisubsymmetric basis then l embedsin l(X). 2000 Mathematics Subject Classification 46B28 (primary),46B03 (secondary).     

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.     

Mahler Measure of Alexander Polynomials

Let l be an oriented link of d components in a homology 3-sphere.For any nonnegative integer q, let l(q) be the link of d–1components obtained from l by performing 1/q surgery on itsdth component l_d. The Mahler measure of the multivariable Alexanderpolynomial _l(q) converges to the Mahler measure of _l as q goesto infinity, provided that l_d has nonzero linking number withsome other component. If l_d has zero linking number with eachof the other components, then the Mahler measure of _l(q) hasa well defined but different limiting behavior. Examples aregiven of links l such that the Mahler measure of _l is small.Possible connections with hyperbolic volume are discussed.     

Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

It is proved that the dual of a Banach space with the Mazurintersection property is almost weak* Asplund. Analogously,the predual of a dual space with the weak* Mazur intersectionproperty is almost Asplund. Through the use of these arguments,it is found that, in particular, almost all (in the Baire sense)equivalent norms on l₁() and l() are Fréchet differentiableon a dense G subset. Necessary conditions for Mazur intersectionproperties in terms of convex sets satisfying a Krein–Milmantype condition are also discussed. It is also shown that, ifa Banach space has the Mazur intersection property, then everysubspace of countable codimension can be equivalently renormedto satisfy this property.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

Drinfeld modules and torsion in the Chow groups of certain threefolds

Let E B be an elliptic surface defined over the algebraic closureof a finite field of characteristic greater than 5. Let W bea resolution of singularities of E x _B E. We show that the l-adicAbel–Jacobi map from the l-power-torsion in the secondChow group of W to H³(W, _l(2)) _l/_l is an isomorphism for almostall primes l. A main tool in the proof is the assertion thatcertain CM-cycles in fibres of W B are torsion, which is provenusing results from the theory of Drinfeld modular curves.     

An Augmented Galerkin Method for First Kind Fredholm Equations

We describe an augmented Galerkin technique for the numericalsolution of first kind Fredholm equations, which is simple touse and which has the considerable advantage of providing acheaply computed numerical criterion for the existence of asolution of the equations under study. The method has guaranteedstability, and leads to a standard linear programming problem(when posed in the l₁ or l norms). It is much faster than themethod recommended in a recent review by Lewis (1975); numericalcomparisons indicate that it achieves comparable accuracy. Theexistence criterion also appears effective in practice.     

Error Bounds for the Truncation of Infinite Linear Differential Systems

An infinite system of linear differential equations of the form = Ax + f, x(0) = y is considered, wherex, y and f are infinite column vectors in E, and A is a constantinfinite matrix defining a bounded operator on E, where E isl₁, (c₀), or l. Explicit error bounds are obtained for the approximationof the solution of the infinite system by the solutions of thefinite truncated systems.     

Homology of GLn over algebraically closed fields

In this paper we define higher pre-Bloch groups _n(F) of a fieldF. When the base field is algebraically closed, we study itsconnection to the homology of the general linear groups withcoefficients in /l , where l is a positive integer. As a resultof our investigation we give a necessary and sufficient conditionfor the natural map H_n(GL_n–1(F), /l ) H_n(GL_n(F), /l )to be bijective. We prove that this map is bijective for n4.We also demonstrate that a certain property of _n() is equivalentto the validity of the Friedlander–Milnor isomorphismconjecture for (n+1)th homology of GL_n().     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

Symmetric Polynomials on Rearrangement-Invariant Function Spaces

The exact representation of symmetric polynomials on Banachspaces with symmetric basis and also on separable rearrangement-invariantfunction spaces over [0, 1] and [0, ) is given. As a consequenceof this representation it is obtained that, among these spaces,l_2n, L_2n[0, 1], L_2n[0, ) and L_2n[0, )L_2m[0, ) where n, m areboth integers are the only spaces that admit separating polynomials.     

Linear Automorphisms that are Symplectomorphisms

Let K be the field of real or complex numbers. Let (X K²ⁿ,) be a symplectic vector space and take 0 < k < n,N =. Let L₁,...,L_N X be 2k-dimensionallinear subspaces which are in a sufficiently general position.It is shown that if F : X X is a linear automorphism whichpreserves the form ^k on all subspaces L₁,...,L_N, then F is an_k-symplectomorphism (that is, F^* = _k, where ). In particular, if K = R and k is odd then F mustbe a symplectomorphism. The unitary version of this theoremis proved as well. It is also observed that the set A_l,2r ofall l-dimensional linear subspaces on which the form has rank 2r is linear in the Grassmannian G(l,2n), that is, there isa linear subspace L such that A_l,2r = L G(l, 2n). In particular,the set A_l,2r can be computed effectively. Finally, the notionof symplectic volume is introduced and it is proved that itis another strong invariant.     

Generating tuples of free products

Grushko's theorem [Mat. Sb. 8 (1940) 169–182] says thatany generating tuple (g₁, ..., g_m) of a free product H*K isNielsen-equivalent to a tuple (h₁, ..., h_l, k_l+1, ..., k_m) withh_i H and k_i K for all i. The h_i and k_i are clearly not unique.In this paper we address the extent of this non-uniqueness.     

(k, l)-Algebraic Stability of Runge-Kutta Methods

For ordinary differential equations satisfying a one-sided Lipschitzcondition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimatesof the form ||y_n||²k(l)||y_n–1||² are desirable. Burrage(1986) has investigated the behaviour of the error-boundingfunction k for positive l for the family of s-stage Gauss methodsof order 2s, and has shown that k(l)=exp 2l+O(l³) (l0) for s3.In this paper, we extend the analysis of k to any irreduciblealgebraically stable Runge-Kutta method, and obtain resultsabout the maximum order of k as an approximation to exp 2l.As a particular example, we investigate the function k for allalgebraically stable methods of order 2s–1.     

(k, l)-Algebraic Stability of Gauss Methods

It is well known that the s-stage Gauss Runge-Kutta methodsof order 2s are algebraically stable, or equivalently (1, 0)-algebraicallystable. In this paper, we show that there exists some l_s >0 such that the Gauss methods are (k, l) algebraically stablefor l [0, l_s) with k(l)=e^2l+O(l^p+1, where p=2s if s=1 or s=2,and p=2 if s>3.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Binomial Sums and Functions of Exponential Type

Let (a_n)_n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(b_n) and (c_n) to that of (a_n). For example, given p0, if b_n= O(n^p and for all > 0,then a_n=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if b_n,c_n=O(rßⁿ), then a_n=O(ⁿ),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e^(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH