Quadratic forms with values in invertible modules

LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the type 1 subgroup).     

Piecewise Linear Bases and Besov Spaces on Fractal Sets

For a class of closed sets F R ⁿ admitting a regular sequence of triangulations or generalized triangulations, the analogues on F of the Faber—Schauder and Franklin bases are discussed. The characterizations of the Besov spaces on F in the terms of coefficients of functions with respect to these bases are proved. As a consequence, analogous characterizations of the Besov spaces on some fractal domains (including the Sierpinski gasket and the von Koch curve) by coefficients of functions with respect to the wavelet bases constructed in [26] are obtained.     

Fourier transforms and generalized Lipschitz classes in uniform metric

For functions f∈L¹(R)∩C(R) with Fourier transforms in L¹(R) we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes Hω,m and hω,m in terms of behavior of .     

On the $$A_{\infty }$$ conditions for general bases

We discuss several characterizations of the \(A_\infty \) class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual \(A_\infty \) weights.     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.     

Dimension in diffeology

We define the dimension function for diffeological spaces, a simple but new invariant. We show then how it can be applied to prove that, for two different integers m and n the quotient spaces R^m/O(m) and Rⁿ/O(n) are not diffeomorphic, and not diffeomorphic to the half-line [0, ∞[ R.     

A characterization of the generalized quadrangle Q(4,q), q ODD

In [3], [ 4 ] we introduced the concept of (0,2)-set in generalized quadrangles, in order to obtain characterizations for P(S,()) and T ₂ ^* (O). Using these sets we are now able to formulate a characterization for Q(4,q), q odd, by assuming local conditions in an antiregular point x of a generalized quadrangle of order s.     

On the Small Ball Inequality in all dimensions

Let hR denote an L^∞ normalized Haar function adapted to a dyadic rectangle R⊂d[0,1]. We show that for choices of coefficients α(R), we have the following lower bound on the L^∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:

     

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.     

Higher cohomology of parabolic actions on certain homogeneous spaces

We show that for a parabolic R ^d -action on PSL(2,R) ^d /Γ, the cohomologies in degrees 1 through d ? 1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers, we have established these results for certain Anosov systems. This work extends the methods of those papers to systems that are not Anosov. The main new idea is defining special elements of representation spaces that allow us to modify the arguments from the previous papers. We discuss how to generalize this strategy to R ^d -systems coming from a product of Lie groups, as in the systems analyzed here.     

Conic formulations of graph homomorphisms

Given graphs X and Y, we define two conic feasibility programs which we show have a solution over the completely positive cone if and only if there exists a homomorphism from X to Y. By varying the cone, we obtain similar characterizations of quantum/entanglement-assisted homomorphisms and three previously studied relaxations of these relations. Motivated by this, we investigate the properties of these “conic homomorphisms” for general (suitable) cones. We also consider two generalized versions of the Lovász theta function, and how they interact with these conic homomorphisms. We prove analogs of several results on classical graph homomorphisms as well as some monotonicity theorems. We also show that one of the generalized theta functions is multiplicative on lexicographic and disjunctive graph products.     

Euler Characteristic and Fixed-Point Theorems

We propose a geometric definition of the Euler characteristic (M) for the class of compact epi-Lipschitzian sets MRⁿ and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.     

A calculus of Fourier integral operators with inhomogeneous phase functions on R<Superscript> <Emphasis Type="Italic">d</Emphasis> </Superscript>

We construct a calculus for generalized SG Fourier integral operators, extending known results to a broader class of symbols of SG type. In particular, we do not require that the phase functions are homogeneous. An essential ingredient in the proofs is a general criterion for asymptotic expansions within the Weyl-Hörmander calculus. We also prove the L²(R^d)-boundedness of the generalized SG Fourier integral operators having regular phase functions and amplitudes uniformly bounded on R^2d.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

Trace operators,Feynman distributions,and multiparameter white noise

Within the framework of white noise analysis on the probability space = ^* R ^d R ^M , the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case:d>1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tensor product of Hilbert spaces from functional analysis.     

The Structure of Frobenius Algebras and Separable Algebras

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R¹²R²³=R²³R¹³=R¹³R¹², called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang–Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) – the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B A is Frobenius if and only if A has a B-coring structure (A, , ) such that the comultiplication : A A_B A is an A-bimodule map.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld     

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.