Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensionalrepresentation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situationthere exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that theimage of is isomorphicto the spherical subalgebra of a rational Cherednik algebrawhose multiplicity function is defined by the roots of the Bernstein–Satopolynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representationsof G occurring in andlowest-weight modules over the Lie algebra generated by f andthe "dual" differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space.If satisfies a Capelli-typecondition, some applications are given to holonomic and equivariantD-modules on V. These applications are related to results provedby Muro [34, 37, 38] or Nang [40, 42, 44] in special cases ofthe representation .     

Norm attaining polynomials

If L 0 is a continuous symmetric n-linear form on a Banachspace and is the associatedcontinuous n-homogeneous polynomial, the ratio always lies between 1 and nⁿ/n!. At one extreme,if L is defined on Hilbert space, then . If L attains norm on Hilbert space, then also attains norm; in this case, we give anexplicit construction to provide a unit vector x₀ with . At the other extreme, if and L attains norm, then attains norm. We prove that in general the converseis not true.     

Recurrence and asymptotics for orthonormal rational functions on an interval

^{Let µ be a positive bounded Borel measure on a subsetI of the real line and = {₁, ..., _n} a sequence of arbitrary ‘complex’poles outside I. Suppose {₁, ..., _n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of _n+1(x)/_n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for _n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]).}     

The integral Novikov conjectures for linear groups containing torsion elements

In this paper, we show that for any global field k, the generalizedintegral Novikov conjecture in both K- and L-theories holdsfor every finitely generated subgroup of GL(n, k). This impliesthat the conjecture holds for every finitely generated subgroupof , where is the algebraic closure of . We also show that for every linear algebraicgroup defined over k, every S-arithmetic subgroup satisfiesthis generalized integral Novikov conjecture. We note that theintegral Novikov conjecture implies the stable Borel conjecture,in particular, the stable Borel conjecture holds for all theabove torsion-free groups. Most of these subgroups are not discretesubgroups of Lie groups with finitely many connected components,and some of them are not finitely generated. When the fieldk is a function field such as , and the k-rank of is positive, many of these S-arithmeticsubgroups such as donot admit cofinite universal spaces for proper actions. Received February 15, 2007.     

Equations Defining Symmetric Varieties and Affine Grassmannians

Suppose be a simpleinvolution of a semisimple algebraic group and suppose H is the subgroup of G of pointsfixed by . If the restrictedroot system is of type or and G is simplyconnected, or if the restricted root system is of type and G is of adjoint type, then we describe astandard monomial theory and the equations for the coordinatering using the standardmonomial theory and the Plücker relations of an appropriate(maybe infinite-dimensional) Grassmann variety.     

On abelian, triangularizable, total reduction algebras

Suppose that is a complex Hilbert space and that () denotes the bounded linear operators on .In this paper we show that if () is an abelian algebrawith elements that admit representations under which they havea spanning set of eigenvectors, and if has the property that given any bounded representation: () of ona Hilbert space , every invariant subspace of () is topologically complemented by an invariantsubspace of (), then is similar to an abelian C^*-algebra. From thisit follows that any amenable Banach algebra of triangular operatorson a separable Hilbert space is similar to an abelian C^*-algebra.     

Operations on the A-theoretic nil-terms

For a space X, we define Frobenius and Verschiebung operationson the nil-terms inthe algebraic K-theory of spaces, in three different ways. Twoapplications are included. Firstly, we show that the homotopygroups of are eithertrivial or not finitely generated as abelian groups. Secondly,the Verschiebung operation defines a -module structure on the homotopy groups of , with the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms after p-completion foran odd prime p and their homotopy groups as -modules up to dimension 4p – 7. We obtainnon-trivial groups only in dimension 2p – 2, where itis finitely generated as a -module, and in dimension 2p – 1, where it is not finitelygenerated as a -module. Received April 12, 2007.     

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ⁿ⁺¹ and let W be a matrix weighton ⁿ, satisfying, for example, the matrix A_p condition. Definethe trace, or restriction, operator Tr by Tr (f)(x^')=f(x^', 0),where x^'n and f is a function on ⁿ⁺¹. If –1/p>n (1/p–1)₊+(β–n)/p,where β is the doubling exponent of W, then the trace operatoris bounded from into (matrix-weighted Besov spaces) if and only ifthe weights V and W uniformly satisfy an estimate controllingthe average of on anydyadic cube I ⁿ by the average of on Q(I)=Ix[0, (I)], for all . If V and W satisfy the converse inequality, then there existsa continuous linear map .If both inequalities hold, then Tr Ext is the identity on .     

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is two-fold. First we outline ageneral construction scheme for producing simply connected minimalsymplectic -manifolds with small Euler characteristics. Usingthis scheme, we illustrate how to obtain irreducible symplectic-manifolds homeomorphic but not diffeomorphic to for k = 1, ..., 4, or to for l = 1, ..., 6. Secondly, for each of thesehomeomorphism types, we show how to produce an infinite familyof pairwise nondiffeomorphic nonsymplectic 4-manifolds belongingto it. In particular, we prove that there are infinitely manyexotic irreducible nonsymplectic smooth structures on, and . Received April 17, 2007. Revised September 26, 2007.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.     

Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

We consider the initial boundary value problem for the focusingnonlinear Schrödinger equation in the quarter plane , inthe case of decaying initial data (for , as )and Dirichlet boundary data (for ) approaching a periodic (single-frequency) background as .We first provide admissibility conditions for the normal derivativeof the solution on the boundary, under the assumption that itbehaves asymptotically in a similar (single-frequency) manner.We then show that for the range , the long-time asymptotics of the solution inside the quarterplane exhibits decaying oscillations of Zakharov–Manakovtype.     

Zeta Functions for G2 and Their Zeros

The exceptional group G₂ has two maximal parabolic subgroups corresponding to theso-called long root and short root. In this paper, the secondnamed author introduces two zeta functions associated with and respectively, and the first named author proves that these zetassatisfy the Riemann hypothesis.     

One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras

The following results are proved for a non-compact, locallycompact group G: the dimension of every non-trivial right idealin L¹(G)** (equipped with the first Arens product) is at least, where (G) is the minimalnumber of compact sets required to cover G; there exist left ideals in L¹(G)** and in LUC(G)* with trivialintersections, and the linear span of right-cancellable elementsis weak*-dense in the annihilator of C₀(G) in LUC(G)* and inthe annihilator of (theL-functions that vanish at infinity) in L(G)*. The same resultsare proved for weighted algebras when the weight function isdiagonally bounded.     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Approximation of 1/x by exponential sums in [1, {infty})

^** Email: braess{at}num.rub.de^*** Email: wh{at}mis.mpg.de Approximations of 1/x by sums of exponentials are well studiedfor finite intervals. Here the error decreases like (exp(–ck))with the order k of the exponential sum. In this paper we investigateapproximations of 1/x in the interval [1, ). We prove estimatesof the error by and confirm this asymptotic estimate by numerical results. Numericalresults lead to the conjecture that the constant in the exponentequals .     

Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum     

Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Spaces , , of ultradecreasing ultradifferentiable (or forshort, ultra-) functions, depending on a weight e^(x), are introducedin the context of quantum statistics. The corresponding coefficientspaces in the Fock basis are identified, and it is shown thatthe Hermite expansion is a tame isomorphism between these spaces.These results are used to link decay properties of density matricesto corresponding properties of the Wigner distribution.