POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

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.     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

On Littlewood's Constants

In two papers, Littlewood studied seemingly unrelated constants:(i) the best such that for any polynomial f, of degree n, theareal integral of its spherical derivative is at most ·n,and (ii) the extremal growth rate rß of the lengthof Green's equipotentials for simply connected domains. Thesetwo constants are shown to coincide, thus greatly improvingknown estimates on . 2000 Mathematics Subject Classification30C50 (primary), 30C85, 30D35 (secondary).     

Analytical Functional Models and Local Spectral Theory

In 1959 E. Bishop used a Banach-space version of the analyticduality principle established by e Silva, Köthe, Grothendieckand others to study connections between spectral decompositionproperties of a Banach-space operator and its adjoint. Accordingto Bishop a continuous linear operator T L(X) on a Banach spaceX satisfies property (rß) if the multiplication operator is injective with closed range for each open set U in the complex plane. In the present articlethe analytic duality principle in its original locally convexform is used to develop a complete duality theory for property(rß). At the same time it is shown that, up to similarity,property (rß) characterizes those operators occurringas restrictions of operators decomposable in the sense of C.Foias, and that its dual property, formulated as a spectraldecomposition property for the spectral subspaces of the givenoperator, characterizes those operators occurring as quotientsof decomposable operators. It is proved that, unlike the situationfor commuting subnormal operators, each finite commuting systemof operators with property (rß) can be extended toa finite commuting system of decomposable operators. Meanwhilethe results of this paper have been used to prove the existenceof invariant subspaces for subdecomposable operators with sufficientlyrich spectrum. 1991 Mathematics Subject Classification: 47A11,47B40.     

On the Location of Zeros of Certain Classes of Polynomials with Applications to Numerical Analysis

A polynomial is said to be of type (p₁, p₂, p₃) relative tothe unit circle if it has p₁ zeros interior to, p₂ on, and p₃exterior to the unit circle. Stability criteria frequently arisewhere a polynomial or a family of polynomials must be shownto be of type (p₁, p₂, 0) or of type (p₁, 0, 0). Here we reconsiderthe practical problem of showing that a polynomial is of oneor other of these types, and we show that the testing of a polynomialof degree n may always be reduced to the testing of one of degreen–1. The simplicity of the method is illustrated by itsapplication to several well known difference schemes for partialdifferential equations.     

On the computation of the generalized inverse of a polynomial matrix

The main purpose of this paper is to present a quicker and less memory-expensive algorithm for the generalized inversionof polynomial matrices than those presented earlier (Karampetakis,1997a Computation of the generalized inverse of a polynomialmatrix and applications. Linear Algebr. Appl. 252, 35–60 and Karampetakis, 1997b Generalized inverses of two variable polynomial matrices and applications. Circuit Syst. & Signal Process. 16, 439–453). Received 24 January, 1999. ⁺ karampetakis@ccf.auth.gr     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

W. Kotarski Institute of Informatics, Silesian University, Bedzinska 60, 41-200 Sosnowiec, Poland Email: bahaa_gm{at}hotmail.com Email: kotarski{at}gate.math.us.edu.pl Received on March 14, 2006; Accepted on December 20, 2006 A distributed control problem for n x n parabolic coupled systemsinvolving operators with infinite order is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem, the necessary and sufficientconditions of optimality are derived for the Dirichlet problem.Yet, the problem considered here is more general than the problemsin El-Saify & Bahaa (2002, Optimal control for n x n hyperbolicsystems involving operators of infinite order. Math. Slovaca,52, 409–424), El-Zahaby (2002, Optimal control of systemsgoverned by infinite order operators. Proceeding (Abstracts)of the International Conference of Mathematics (Trends and Developments)of the Egyptian Mathematical Society, Cairo, Egypt, 28–31December 2002. J. Egypt. Math. Soc. (submitted)), Gali &El-Saify (1983, Control of system governed by infinite orderequation of hyperbolic type. Proceeding of the InternationalConference on Functional-Differential Systems and Related Topics,vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributedcontrol of a system governed by Dirichlet and Neumann problemsfor elliptic equations of infinite order. Proceeding of theInternational Conference on Functional-Differential Systemsand Related Topics, vol. III. Poland, pp. 83–87) and Kotarskiet al. (200b, Optimal control problem for a hyperbolic systemwith mixed control-state constraints involving operator of infiniteorder. Int. J. Pure Appl. Math., 1, 241–254).     

On the Group of Symplectic Matrices Over a Free Associative Algebra

Symplectic groups are well known as the groups of isometriesof a vector space with a non-singular bilinear alternating form.These notions can be extended by replacing the vector spaceby a module over a ring R, but if R is non-commutative, it willalso have to have an involution. We shall here be concernedwith symplectic groups over free associative algebras (witha suitably defined involution). It is known that the generallinear group GL_n over the free algebra is generated by the setof all elementary and diagonal matrices (see [1, Proposition2.8.2, p. 124]). Our object here is to prove that the symplecticgroup over the free algebra is generated by the set of all elementarysymplectic matrices. For the lowest order this result was obtainedin [4]; the general case is rather more involved. It makes useof the notion of transduction (see [1, 2.4, p. 105]). When thereis only a single variable over a field, the free algebra reducesto the polynomial ring and the weak algorithm becomes the familiardivision algorithm. In that case the result has been provedin [3, Anhang 5].     

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.     

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.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

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.     

Mappings Preserving Submodules of Hilbert C*-Modules

A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||^1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by A_X the algebra of all mappings on X of the form (1.1) where m is an integer and a_iA, b_iB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology.     

Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures

We investigate Riemann–Liouville processes R_H, with H> 0, and fractional Brownian motions B_H, for 0 < H <1, and study their small deviation properties in the spacesL_q([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of Riemann–Liouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary).     

Deformation Theory and The Computation of Zeta Functions

We present a new approach to the problem of computing the zetafunction of a hypersurface over a finite field. For a hypersurfacedefined by a polynomial of degree d in n variables over thefield of q elements, one desires an algorithm whose runningtime is a polynomial function of dⁿ log(q). (Here we assumed 2, for otherwise the problem is easy.) The case n = 1 isrelated to univariate polynomial factorisation and is comparativelystraightforward. When n = 2 one is counting points on curves,and the method of Schoof and Pila yields a complexity of , where the function C_d depends exponentiallyon d. For arbitrary n, the theorem of the author and Wan givesa complexity which is a polynomial function of (pdⁿ log(q))ⁿ,where p is the characteristic of the field. A complexity estimateof this form can also be achieved for smooth hypersurfaces usingthe method of Kedlaya, although this has only been worked outin full for curves. The new approach we present should yielda complexity which is a small polynomial function of pdⁿ log(q).In this paper, we work this out in full for Artin–Schreierhypersurfaces defined by equations of the form Z^p – Z= f, where the polynomial f has a diagonal leading form. Themethod utilises a relative p-adic cohomology theory for familiesof hypersurfaces, due in essence to Dwork. As a corollary ofour main theorem, we obtain the following curious result. Letf be a multivariate polynomial with integer coefficients whoseleading form is diagonal. There exists an explicit deterministicalgorithm which takes as input a prime p, outputs the numberof solutions to the congruence equation f = 0 op, and runs in bit operations, for any >0. This improves upon the elementary estimate of bit operations, where n is the number of variables,which can be achieved using Berlekamp's root counting algorithm.2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.     

On Second Order Bifurcations of Limit Cycles

The paper derives a formula for the second variation of thedisplacement function for polynomial perturbations of Hamiltoniansystems with elliptic or hyperelliptic Hamiltonians H(x, y)=y²–U(x)in terms of the coefficients of the perturbation. As an application,the conjecture stated by C. Chicone that a specific cubic systemappearing in a deformation of singularity with two zero eigenvalueshas at most two limit cycles is proved.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

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.