Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

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).     

An Application of the Fefferman-Stein Inequality to the Study of the Tent Spaces

The purpose of this note is to show how a result on tent spacesproved by Coifman, Meyer and Stein, namely the L^p-norm relationshipbetween the functionals A_q and C_q appearing in the definitionof such spaces, can be derived easily from the Fefferman-SteinL^p-inequality for the sharp maximal function.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

The determination of a coefficient in a parabolic equation from input sources

The authors consider the question of recovering the coefficientq from the equation u_t – u_xx + q(x)u = f_j(x) with homogeneousinitial and boundary conditions. The nonhomogeneous source terms form a basis for L²(0,1).It is proved that a unique determination is possible from datameasurements consisting of either the flux at one end of thebar or the net flux leaving the bar, taken at a single fixedtime for each input source. An algorithm that allows efficientnumerical reconstruction of q(x) from finite data is given.     

Some New and Old Asymptotic Representations of the Jost Solution and the Weyl m-Function For Schrodinger Operators on the Line

For the general one-dimensional Schrödinger operator –d²/dx²+q(x) with real q L₁(R), this paper presents a new series representationof the Jost solution which, in turn, implies a new asymptoticrepresentation of the Weyl m-function for locally summable q.This representation is then applied to smooth potentials q toobtain Weyl m-function power asymptotics. The condition q^(N) L₁(x₀, x₀ + ), for N N₀, allows one to derive the (N + 1)term for almost all x [x₀, x₀ + ), thereby refining a relevantresult by Danielyan, Levitan and Simon. 2000 Mathematics SubjectClassification 34E05, 34L40 (primary), 34B20, 34L25 (secondary).     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

Merit Factors of Character Polynomials

Let q be a prime and be a non-principal character modulo q.Let where 1 t q is the character polynomial associated to (cyclicallypermuted t places). The principal result is that for any non-principaland non-real character modulo q and 1 t q, where the implicit constant is independent of t and q. Here||·||₄ denotes the L₄ norm on the unit circle. It follows from this that all cyclically permuted characterpolynomials associated with non-principal and non-real charactershave merit factors that approach 3. This complements and completesresults of Golay, Høholdt and Jensen, and Turyn (andothers). These results show that the merit factors of cyclicallypermuted character polynomials associated with non-principalreal characters vary asymptotically between 3/2 and 6. The averages of the L₄ norms are also computed. Let q be a primenumber. Then where the summation is over all characters modulo q.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Classification of Quantum Groups SUq(n)

It is shown that the compact matrix quantum groups SU_q(2) arenon-isomorphic to each other for q[–1, 1]\{0}, and thatthe compact matrix quantum groups SU_q(n) are non-isomorphicto each other for q(0, 1]. Some invariants for compact quantumgroups are also discussed.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

Weighted estimates for the <Emphasis Type="Italic">k</Emphasis>-plane transform of radial functions on euclidean spaces

We discuss the L ^p − L ^q mapping property of k-plane transforms acting on radial functions in certain weighted L ^p spaces with power weight. We show that for all admissible power weights it is not always possible to get strong (p, q) boundedness of the k-plane transform. However, we prove the best possible estimates with respect to the Lorentz norms.     

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

A Lumped Mass Finite-element Method with Quadrature for a Non-linear Parabolic Problem

For a non-linear second-order parabolic equation in two spacedimensions we consider semidiscrete and totally discrete variantsof the lumped mass modification with quadrature of the standardGalerkin method using piecewise linear approximating functions.We demonstrate error estimates of optimal order in L₂ and ofalmost optimal order in L and discuss some positivity and monotonicityproperties of the discrete solution operator.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.